Update Content - 2025-07-18

config.toml

@@ -26,11 +26,11 @@ changefreq = "weekly"
 priority = 0.5
 filename = "sitemap.xml"
 
-[[menu.main]]
+# [[menu.main]]
-name = "Home"
+# name = "Home"
-weight = 10
+# weight = 10
-identifier = "home"
+# identifier = "home"
-url = "/"
+# url = "/"
 
 [[menu.main]]
 name = "Zettels"
@@ -121,9 +121,9 @@ enable = false
 hint = 30
 warn = 180
 
-[params.utterances]       # https://utteranc.es/
+# [params.utterances]       # https://utteranc.es/
-repo = "tdehaeze/brain-dump-comments"
+# repo = "tdehaeze/brain-dump-comments"
-theme = "boxy-light"
+# theme = "boxy-light"
 
 [params.valine]
 enable = false

content/article/alkhatib03_activ_struc_vibrat_contr.md

@@ -75,7 +75,7 @@ The major restriction to the application of feedforward adaptive filtering is th
 
 <a id="table--table:comparison-constrol-strat"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--table:comparison-constrol-strat">Table 1</a></span>:
+  <span class="table-number"><a href="#table--table:comparison-constrol-strat">Table 1</a>:</span>
   Comparison of control strategies
 </div>
 
@@ -123,7 +123,7 @@ Uncertainty can be divided into four types:
 -   neglected nonlinearities
 
 The \\(\mathcal{H}\_\infty\\)  controller is developed to address uncertainty by systematic means.
-A general block diagram of the control system is shown figure [1](#figure--fig:alkhatib03-hinf-control).
+A general block diagram of the control system is shown [Figure 1](#figure--fig:alkhatib03-hinf-control).
 
 A **frequency shaped filter** \\(W(s)\\) coupled to selected inputs and outputs of the plant is included.
 The outputs of this frequency shaped filter define the error ouputs used to evaluate the system performance and generate the **cost** that will be used in the design process.
@@ -204,7 +204,7 @@ Two different methods
 
 {{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="<span class=\"figure-number\">Figure 2: </span>1 DoF control of a spring-mass-damping system" >}}
 
-Consider the control system figure [2](#figure--fig:alkhatib03-1dof-control), the equation of motion of the system is:
+Consider the control system [Figure 2](#figure--fig:alkhatib03-1dof-control), the equation of motion of the system is:
 \\[ m\ddot{x} + c\dot{x} + kx = f\_a + f \\]
 
 The controller force can be expressed as: \\(f\_a = -g\_a \ddot{x} + g\_v \dot{x} + g\_d x\\). The equation of motion becomes:

content/article/bibel92_guidel_h.md

@@ -23,7 +23,7 @@ Year
 
 {{< figure src="/ox-hugo/bibel92_control_diag.png" caption="<span class=\"figure-number\">Figure 1: </span>Control System Diagram" >}}
 
-From the figure [1](#figure--fig:bibel92-control-diag), we have:
+From the [Figure 1](#figure--fig:bibel92-control-diag), we have:
 
 \begin{align\*}
 y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\
@@ -78,7 +78,7 @@ Usually, reference signals and disturbances occur at low frequencies, while nois
 
 {{< figure src="/ox-hugo/bibel92_general_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>\\(\mathcal{H}\_\infty\\) control framework" >}}
 
-New design framework (figure [2](#figure--fig:bibel92-general-plant)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
+New design framework ([Figure 2](#figure--fig:bibel92-general-plant)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
 
 -   \\(w\\): exogenous inputs
 -   \\(z\\): regulated performance output
@@ -104,7 +104,7 @@ The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\
 
 ## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
 
-Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#figure--fig:bibel92-hinf-weights)).
+Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs ([Figure 3](#figure--fig:bibel92-hinf-weights)).
 
 <a id="figure--fig:bibel92-hinf-weights"></a>
 
@@ -148,13 +148,13 @@ When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the
 
 ## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
 
-Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#figure--fig:bibel92-unmodeled-dynamics)).
+Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output ([Figure 4](#figure--fig:bibel92-unmodeled-dynamics)).
 
 <a id="figure--fig:bibel92-unmodeled-dynamics"></a>
 
 {{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="<span class=\"figure-number\">Figure 4: </span>Unmodeled dynamics model" >}}
 
-The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#figure--fig:bibel92-weight-dynamics).
+The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown [Figure 5](#figure--fig:bibel92-weight-dynamics).
 
 <a id="figure--fig:bibel92-weight-dynamics"></a>
 

content/article/bryson93_contr_spacec_aircr.md

@@ -40,7 +40,7 @@ Year
 
 ## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac.md" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
 
-> Figure <fig:bryson93_hac_lac> shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
+> [Figure 1](#figure--fig:bryson93-hac-lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
 > LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
 > HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
 

content/article/butler11_posit_contr_lithog_equip.md

@@ -20,5 +20,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” <i>Ieee Control Systems</i> 31 (5): 28–47. doi:<a href="https://doi.org/10.1109/mcs.2011.941882">10.1109/mcs.2011.941882</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” <i>IEEE Control Systems</i> 31 (5): 28–47. doi:<a href="https://doi.org/10.1109/mcs.2011.941882">10.1109/mcs.2011.941882</a>.</div>
 </div>

content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md

@@ -103,6 +103,6 @@ where
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 Icra. Millennium Conference. Ieee International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00ch37065)</i>, nil. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)</i>. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
 </div>

content/article/collette11_review_activ_vibrat_isolat_strat.md

@@ -51,7 +51,7 @@ The general expression of the force delivered by the actuator is \\(f = g\_a \dd
 
 <a id="table--table:active-isolation"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--table:active-isolation">Table 1</a></span>:
+  <span class="table-number"><a href="#table--table:active-isolation">Table 1</a>:</span>
   Active isolation techniques
 </div>
 

content/article/collette14_vibrat.md

@@ -28,7 +28,7 @@ Year
 
 ## Different types of sensors {#different-types-of-sensors}
 
-In this paper, three types of sensors are used. Their advantages and disadvantages are summarized table [1](#table--tab:sensors).
+In this paper, three types of sensors are used. Their advantages and disadvantages are summarized [Table 1](#table--tab:sensors).
 
 > Several types of sensors can be used for the feedback control of vibration isolation systems:
 >
@@ -38,7 +38,7 @@ In this paper, three types of sensors are used. Their advantages and disadvantag
 
 <a id="table--tab:sensors"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:sensors">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:sensors">Table 1</a>:</span>
   Types of sensors
 </div>
 
@@ -51,11 +51,11 @@ In this paper, three types of sensors are used. Their advantages and disadvantag
 
 ## Inertial Control and sensor fusion configurations {#inertial-control-and-sensor-fusion-configurations}
 
-For a simple 1DoF model, two fusion-sensor configuration are studied. The results are summarized Table [2](#table--tab:fusion-trade-off).
+For a simple 1DoF model, two fusion-sensor configuration are studied. The results are summarized [Table 2](#table--tab:fusion-trade-off).
 
 <a id="table--tab:fusion-trade-off"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:fusion-trade-off">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:fusion-trade-off">Table 2</a>:</span>
   Sensor fusion configurations
 </div>
 
@@ -103,5 +103,5 @@ Three types of sensors have been considered for the high frequency part of the f
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In <i>International Conference on Noise and Vibration Engineering (Isma2014)</i>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In <i>International Conference on Noise and Vibration Engineering (ISMA2014)</i>.</div>
 </div>

content/article/collette15_sensor_fusion_method_high_perfor.md

@@ -28,5 +28,5 @@ The stability margins of the controller can be significantly increased with no o
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” <i>Journal of Sound and Vibration</i> 342 (nil): 1–21. doi:<a href="https://doi.org/10.1016/j.jsv.2015.01.006">10.1016/j.jsv.2015.01.006</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” <i>Journal of Sound and Vibration</i> 342: 1–21. doi:<a href="https://doi.org/10.1016/j.jsv.2015.01.006">10.1016/j.jsv.2015.01.006</a>.</div>
 </div>

content/article/csencsics20_explor_paret_front_actuat_techn.md

@@ -0,0 +1,123 @@
++++
+title = "Exploring the pareto fronts of actuation technologies for high performance mechatronic systems"
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Csencsics and Schitter 2020</a>)
+
+Author(s)
+: Csencsics, E., &amp; Schitter, G.
+
+Year
+: 2020
+
+
+## Abstract {#abstract}
+
+> This paper proposes a novel method for estimating the limitations of individual actuation technologies for a desired system class based on analytically obtained relations, which can be used to systematically trade off desired range and speed specifications in the design phase.
+> The method is presented along the example of **fast steering mirrors** with the tradeoff limit curves estimated for the established **piezoelectric**, **lorentz force** and **hybrid reluctance** actuation technologies.
+
+<a id="figure--fig:csencsics20-fsm-schematic"></a>
+
+{{< figure src="/ox-hugo/csencsics20_fsm_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Fast Steering Mirror system. The main components are: mirror, actuators, position sensors and suspension system." >}}
+
+
+## Fast Steering Mirrors {#fast-steering-mirrors}
+
+
+### Application area and performance specification {#application-area-and-performance-specification}
+
+<a id="table--tab:fsm-requirements"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:fsm-requirements">Table 1</a>:</span>
+  FSM performance requirements for two application
+</div>
+
+| Application       | Pointing        | Scanning |
+|-------------------|-----------------|----------|
+| System Range      | large           | large    |
+| System Dimensions | arbitrary       | compact  |
+| Main objective    | dist. rejection | tracking |
+| Bandwidth         | high            | high     |
+| Motion amplitude  | small           | large    |
+| Mover inertia     | arbitrary       | small    |
+| Precision         | high            | high     |
+
+
+### Safe operating area {#safe-operating-area}
+
+The concept of the Safe Operating Area (SOA) relates the frequency of a sinusoidal reference to the maximum admissible scan amplitude that still stays within the limits of the system.
+
+From figure [2](#figure--fig:csencsics20-soa) we can already see that piezo are typically used for system with high bandwidth and small range.
+
+<a id="figure--fig:csencsics20-soa"></a>
+
+{{< figure src="/ox-hugo/csencsics20_soa.png" caption="<span class=\"figure-number\">Figure 1: </span>Measured safe operating area of closed-loop FSM systems with sinusoidal reference signals. Piezo actuated in blue, lorentz force actuated in red and hybrid reluctance actuated in green." >}}
+
+
+## Limitations of actuator technology {#limitations-of-actuator-technology}
+
+
+### Piezo actuation {#piezo-actuation}
+
+Piezo actuated FMS are in general **high stiffness** system, for which the **bandwidth limitation** for feedback control is typically given by the **first mechanical resonance**.
+
+<a id="figure--fig:csencsics20-typical-piezo-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_piezo_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Piezo actuated FSM cross section" >}}
+
+The angular range of the FSM is:
+
+\begin{equation}
+\phi = \frac{L/1000}{2 d}
+\end{equation}
+
+with \\(L\\) the length of the stack, and d the distance between the stacks and the center of rotation (the factor 1000 is linked to the fact that typical piezo stack have a store equal to 0.1% of their length).
+
+The first resonance frequency is:
+
+\begin{equation}
+f\_{PZA} = \frac{1}{2\pi L}\sqrt{\frac{3E}{\rho\_\text{piezo}}}
+\end{equation}
+
+with \\(E\\) the elastic modulus and \\(\rho\_\text{piezo}\\) the density of the piezo material.
+
+As the resonance limits the achievable bandwidth, we therefore have that \\(f\_{\text{max,PZA}} \propto 1/\phi\\).
+
+
+### Lorentz force actuation {#lorentz-force-actuation}
+
+Lorentz force actuated FSM are in general **low stiffness** systems, which typically have a control bandwidth beyond the suspension mode that is usually limited by the **internal modes of the moving part**.
+
+The mover's mass is dominating the dynamics of low stiffness systems beyond the suspension mode.
+
+<a id="figure--fig:csencsics20-typical-lorentz-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_lorentz_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Lorentz force actuator designs." >}}
+
+\begin{equation}
+f\_\text{max,LFA} = \frac{1}{2\pi} k\_\text{LFA} \sqrt{\frac{1}{\phi J\_\text{init} + \Delta\_J + 2 d \phi^2}}
+\end{equation}
+
+
+### Hybrid reluctance force actuation {#hybrid-reluctance-force-actuation}
+
+<a id="figure--fig:csencsics20-typical-hybrid-reluctance-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_hybrid_reluctance_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Hybrid reluctance actuator designs" >}}
+
+
+## Pareto front estimates for FSM systems {#pareto-front-estimates-for-fsm-systems}
+
+<a id="figure--fig:csencsics20-pareto-estimate"></a>
+
+{{< figure src="/ox-hugo/csencsics20_pareto_estimate.png" caption="<span class=\"figure-number\">Figure 1: </span>Two dimensional performance space for FSM systems showing the tradeoff between range and bandwidth. Commercially available (symbols) as well as academically reported systems (dots) actuated by piezo (blue), Lorentz force (red) and reluctance actuators (green) are depicted." >}}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Csencsics, Ernst, and Georg Schitter. 2020. “Exploring the Pareto Fronts of Actuation Technologies for High Performance Mechatronic Systems.” <i>IEEE/ASME Transactions on Mechatronics</i>. IEEE.</div>
+</div>

content/article/dasgupta00_stewar_platf_manip.md

@@ -18,7 +18,7 @@ Year
 
 <a id="table--tab:parallel-vs-serial-manipulators"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:parallel-vs-serial-manipulators">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:parallel-vs-serial-manipulators">Table 1</a>:</span>
   Parallel VS serial manipulators
 </div>
 

content/article/devasia07_survey_contr_issues_nanop.md

@@ -30,5 +30,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” <i>Ieee Transactions on Control Systems Technology</i> 15 (5). IEEE: 802–23.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” <i>IEEE Transactions on Control Systems Technology</i> 15 (5). IEEE: 802–23.</div>
 </div>

content/article/fleming10_nanop_system_with_force_feedb.md

@@ -124,5 +124,5 @@ The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
 </div>

content/article/fleming12_estim.md

@@ -21,5 +21,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In <i>2012 Ieee International Conference on Robotics and Automation</i>, nil. doi:<a href="https://doi.org/10.1109/icra.2012.6224850">10.1109/icra.2012.6224850</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In <i>2012 IEEE International Conference on Robotics and Automation</i>. doi:<a href="https://doi.org/10.1109/icra.2012.6224850">10.1109/icra.2012.6224850</a>.</div>
 </div>

content/article/fleming13_review_nanom_resol_posit_sensor.md

@@ -147,7 +147,7 @@ The empirical rule states that there is a \\(99.7\\%\\) probability that a sampl
 This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
 
 Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
-There is usually a trade-off between bandwidth and resolution (figure [3](#figure--fig:tradeoff-res-bandwidth)).
+There is usually a trade-off between bandwidth and resolution ([Figure 3](#figure--fig:tradeoff-res-bandwidth)).
 
 <a id="figure--fig:tradeoff-res-bandwidth"></a>
 
@@ -166,7 +166,7 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
 
 <a id="table--tab:summary-position-sensors"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a>:</span>
   Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
 </div>
 
@@ -185,5 +185,5 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190 (nil): 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190: 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
 </div>

content/article/fleming15_low_order_dampin_track_contr.md

@@ -21,5 +21,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” <i>Frontiers in Mechanical Engineering</i> 1 (nil): nil. doi:<a href="https://doi.org/10.3389/fmech.2015.00014">10.3389/fmech.2015.00014</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” <i>Frontiers in Mechanical Engineering</i> 1. doi:<a href="https://doi.org/10.3389/fmech.2015.00014">10.3389/fmech.2015.00014</a>.</div>
 </div>

content/article/gao15_measur_techn_precis_posit.md

@@ -20,5 +20,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” <i>Cirp Annals</i> 64 (2): 773–96. doi:<a href="https://doi.org/10.1016/j.cirp.2015.05.009">10.1016/j.cirp.2015.05.009</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” <i>CIRP Annals</i> 64 (2): 773–96. doi:<a href="https://doi.org/10.1016/j.cirp.2015.05.009">10.1016/j.cirp.2015.05.009</a>.</div>
 </div>

content/article/garg07_implem_chall_multiv_contr.md

@@ -38,5 +38,5 @@ The control rate should be weighted appropriately in order to not saturate the s
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In <i>Aiaa Guidance, Navigation and Control Conference and Exhibit</i>, nil. doi:<a href="https://doi.org/10.2514/6.2007-6334">10.2514/6.2007-6334</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In <i>AIAA Guidance, Navigation and Control Conference and Exhibit</i>. doi:<a href="https://doi.org/10.2514/6.2007-6334">10.2514/6.2007-6334</a>.</div>
 </div>

content/article/geraldes23_fly_scan_orien_motion_analy.md

@@ -0,0 +1,84 @@
++++
+title = "Fly-scan-oriented motion analyses and upgraded beamline integration architecture for the high-dynamic double-crystal monochromator at sirius/lnls"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Geraldes et al. 2023</a>)
+
+Author(s)
+: Geraldes, R. R., Luiz, S. A. L., Neto, J. L. d. B., Telles Ren\\'e Silva Soares, Reis, R. D. d., Calligaris, G. A., Witvoet, G., …
+
+Year
+: 2023
+
+
+## Effect of different d spacing {#effect-of-different-d-spacing}
+
+> Thus, if different d-spacings are found in the two crystals, an ideal energy matching for maximum flux would be related to slightly different \\(\theta\_B\\) in the crystals, such that the monochromatic beam would no longer be exactly parallel to the incoming beam, and **the magnitude of the deviation would be variable over the operational energy range**.
+
+
+## Effect of pitch error on source motion {#effect-of-pitch-error-on-source-motion}
+
+> Then, considering that variations of the virtual source are often proportionally related to shifts of the beam at the sample through the beamline optics, **a common requirement is having them small compared with the source size**.
+> With **X-ray source sizes of about 5 um** and **L commonly of the order of 30m** for modern beamlines, a typical budget of 10% pushes **pitch errors to the range of 10 nrad** only.
+
+
+## Correct pitch errors with gap adjustments {#correct-pitch-errors-with-gap-adjustments}
+
+> It can be seen that displacements in the virtual source related to pitch errors may be at least partly compensated by energy-dependent beam offset corrections via gap adjustments.
+
+
+## Allow some flux loss in order to have a more stable beam {#allow-some-flux-loss-in-order-to-have-a-more-stable-beam}
+
+> The angular boundaries for pitch around an ideal energy tuning, which might be already out or perfect parallelism due to d-spacing variations, can be derived as a fraction of the angular bandwidth of the Darwin width of the crystals.
+> This can be used, for example, to **evaluate acceptable flux losses in trying to keep the incoming and outgoing beam parallel despite thermal effects**.
+
+The pitch bandwidth for typical Si111 and Si311 can vary from 100urad at low energy to &lt;1urad at high energy.
+
+
+## Analytical effect of miss-cut on the change of beam height {#analytical-effect-of-miss-cut-on-the-change-of-beam-height}
+
+> This indicates that in reality the **gap motion range may need to be larger by a few percent than nominally expected**, that sensitivities at low angles may vary by more than one order of magnitude, that **calibrations for fixed exit may require more than the simpler trigonometric relation** of (2), and that the required velocities and accelerations related to the fly scan are in practice different from nominal ones.
+
+
+### Estimate the effect of the miss-cut on the beam error for our values of angles and miss-cut {#estimate-the-effect-of-the-miss-cut-on-the-beam-error-for-our-values-of-angles-and-miss-cut}
+
+
+## High dynamic range: low energy and high energy issues {#high-dynamic-range-low-energy-and-high-energy-issues}
+
+> Hence, **differences of three to four orders of magnitude occur for the gap velocity for a given energy variation rate** within the operational range of the HD-DCM.
+>
+> For a control-based instrument like the HD-DCM, these aspects place demanding specifications on metrology and acquisition hardware, since very high resolution and low noise are required for the lower angular (higher energy) range, whereas high rates are necessary at the opposite limit.
+>
+> For example, while the angular resolution in the Bragg angle quadrature encoder is 50nrad for high angular resolution and small control errors, for an energy scan of 1keV/s, the crystal angular speed requirements would be around 0.1deg/s at the high energy range and as much as 40deg/s at the low energy limit.
+> In the latter case, the counting rates would have to be higher than the current electronics capacity of 10 MHz.
+>
+> Similarly for the gap, with a resolution of 0.1 nm from the quadrature laser interferometers for the nanometre-level control performance, an equivalent energy rate scan speed with Si(111) crystals without a miscut would translate to about 0.8 mm/s and 20 mm/s at the high and low energy limits, respectively.
+> In the latter case, counting rates would need to reach 200 MHz.
+
+
+## Bragg control has a bandwidth of 20Hz {#bragg-control-has-a-bandwidth-of-20hz}
+
+
+## Crystal control has a bandwidth between 150Hz and 250Hz {#crystal-control-has-a-bandwidth-between-150hz-and-250hz}
+
+
+## They are using the Bragg angle reference signal to measure the wanted crystal distance {#they-are-using-the-bragg-angle-reference-signal-to-measure-the-wanted-crystal-distance}
+
+They are not using the encoder signal as we are doing.
+
+
+## Modes of operation {#modes-of-operation}
+
+1.  Standalone (similar as what we are using).
+2.  Follower: follows an encoder signal from the ID
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geraldes, Renan Ramalho, Sergio Augusto Lordano Luiz, João Leandro de Brito Neto, Telles René Silva Soares, Ricardo Donizeth dos Reis, Guilherme A. Calligaris, Gert Witvoet, and J. P. M. B. Vermeulen. 2023. “Fly-Scan-Oriented Motion Analyses and Upgraded Beamline Integration Architecture for the High-Dynamic Double-Crystal Monochromator at Sirius/Lnls.” <i>Journal of Synchrotron Radiation</i> 30 (1): 90–110. doi:<a href="https://doi.org/10.1107/s1600577522010724">10.1107/s1600577522010724</a>.</div>
+</div>

content/article/hauge04_sensor_contr_space_based_six.md

@@ -28,11 +28,11 @@ Year
 
 {{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Hexapod for active vibration isolation" >}}
 
-**Stewart platform** (Figure [1](#figure--fig:hauge04-stewart-platform)):
+**Stewart platform** ([Figure 1](#figure--fig:hauge04-stewart-platform)):
 
 -   Low corner frequency
 -   Large actuator stroke (\\(\pm5mm\\))
--   Sensors in each strut (Figure [2](#figure--fig:hauge05-struts)):
+-   Sensors in each strut ([Figure 2](#figure--fig:hauge05-struts)):
     -   three-axis load cell
     -   base and payload geophone in parallel with the struts
     -   LVDT
@@ -87,7 +87,7 @@ With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and
 
 <a id="table--tab:hauge05-comp-load-cell-geophone"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a>:</span>
   Typical characteristics of sensors used for isolation in hexapod systems
 </div>
 

content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md

@@ -20,5 +20,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 10 (2): 135–45. doi:<a href="https://doi.org/10.1109/tmech.2005.844702">10.1109/tmech.2005.844702</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 10 (2): 135–45. doi:<a href="https://doi.org/10.1109/tmech.2005.844702">10.1109/tmech.2005.844702</a>.</div>
 </div>

content/article/ito16_compar_class_high_precis_actuat.md

@@ -20,7 +20,7 @@ Year
 ## Classification of high-precision actuators {#classification-of-high-precision-actuators}
 
 <div class="table-caption">
-  <span class="table-number">Table 1</span>:
+  <span class="table-number">Table 1:</span>
   Zero/Low and High stiffness actuators
 </div>
 
@@ -70,5 +70,5 @@ In contrast, the frequency band between the first and the other resonances of Lo
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>Ieee/Asme Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>IEEE/ASME Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
 </div>

content/article/legnani12_new_isotr_decoup_paral_manip.md

@@ -34,5 +34,5 @@ Example of generated isotropic manipulator (not decoupled).
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58 (nil): 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58: 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
 </div>

content/article/li01_simul_vibrat_isolat_point_contr.md

@@ -22,5 +22,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In <i>Smart Structures and Materials 2001: Smart Structures and Integrated Systems</i>, nil. doi:<a href="https://doi.org/10.1117/12.436521">10.1117/12.436521</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In <i>Smart Structures and Materials 2001: Smart Structures and Integrated Systems</i>. doi:<a href="https://doi.org/10.1117/12.436521">10.1117/12.436521</a>.</div>
 </div>

content/article/mcinroy00_desig_contr_flexur_joint_hexap.md

@@ -21,5 +21,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” <i>Ieee Transactions on Robotics and Automation</i> 16 (4): 372–81. doi:<a href="https://doi.org/10.1109/70.864229">10.1109/70.864229</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” <i>IEEE Transactions on Robotics and Automation</i> 16 (4): 372–81. doi:<a href="https://doi.org/10.1109/70.864229">10.1109/70.864229</a>.</div>
 </div>

content/article/mcinroy02_model_desig_flexur_joint_stewar.md

@@ -40,11 +40,11 @@ This short paper is very similar to (<a href="#citeproc_bib_item_1">McInroy 1999
 
 {{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="<span class=\"figure-number\">Figure 1: </span>The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
 
-The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#figure--fig:mcinroy02-leg-model)).
+The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass ([Figure 1](#figure--fig:mcinroy02-leg-model)).
 
 Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
 
-The model of the strut are shown in Figure [1](#figure--fig:mcinroy02-leg-model) with:
+The model of the strut are shown in [Figure 1](#figure--fig:mcinroy02-leg-model) with:
 
 -   \\(m\_{s\_i}\\) moving strut mass
 -   \\(k\_i\\) spring constant
@@ -115,7 +115,7 @@ where:
 -   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
 -   \\(g\\) is the gravity vector
 
-By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
+By combining \eqref{eq:strut\_dynamics\_vec}, \eqref{eq:payload\_dynamics} and \eqref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
 
 \begin{equation}
   {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
@@ -136,12 +136,12 @@ This section establishes design guidelines for the spherical flexure joint to gu
 
 {{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="<span class=\"figure-number\">Figure 2: </span>A simplified dynamic model of a strut and its joint" >}}
 
-Figure [2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
+[Figure 2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
 The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
 Damping terms are neglected.
 \\(k\_r\\) denotes the rotational stiffness of the spherical joint.
 
-From Figure [2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
+From [Figure 2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
 
 \begin{equation}
   f\_p = \begin{bmatrix}
@@ -200,7 +200,7 @@ In order to get dominance at low frequencies, the hexapod must be designed so th
 
 This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
 
-By satisfying <eq:cond_stiff>, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
+By satisfying \eqref{eq:cond\_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
 \\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
 At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
 \\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
@@ -225,14 +225,14 @@ In this case, it is reasonable to use:
 
 <div class="important">
 
-By designing the flexure jointed hexapod and its controller so that both <eq:cond_stiff> and <eq:cond_bandwidth> are met, the dynamics of the hexapod can be greatly reduced in complexity.
+By designing the flexure jointed hexapod and its controller so that both \eqref{eq:cond\_stiff} and \eqref{eq:cond\_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
 
 </div>
 
 
 ## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
 
-Equation <eq:eom_fjh> is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
+Equation \eqref{eq:eom\_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
 
 This section will derive this implicit relationship.
 Let denote:
@@ -269,6 +269,6 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>Ieee/Asme Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
 </div>

content/article/mcinroy99_dynam.md

@@ -42,7 +42,7 @@ The actuators for FJHs can be divided into two categories:
 
 {{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="<span class=\"figure-number\">Figure 1: </span>A general Stewart Platform" >}}
 
-Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#figure--fig:mcinroy99-strut-model).
+Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in [Figure 2](#figure--fig:mcinroy99-strut-model).
 
 In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
 
@@ -52,8 +52,8 @@ In order to provide low frequency passive vibration isolation, the hard actuator
 
 <a id="table--tab:mcinroy99-strut-model"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span>
-  Definition of quantities on Figure <a href="#org84f1a50">2</a>
+  Definition of quantities on <a href="#org1f8da5d">2</a>
 </div>
 
 | **Symbol**                   | **Meaning**                                |
@@ -74,7 +74,7 @@ It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction
 This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used.
 This allows to reduce considerably the complexity of the model.
 
-From Figure [2](#figure--fig:mcinroy99-strut-model) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
+From [Figure 2](#figure--fig:mcinroy99-strut-model) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
 
 \begin{equation}
   m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
@@ -142,7 +142,7 @@ where:
 -   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
 -   \\(g\\) is the gravity vector
 
-By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
+By combining \eqref{eq:strut\_dynamics\_vec}, \eqref{eq:payload\_dynamics} and \eqref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
 
 \begin{aligned}
   & {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\
@@ -165,6 +165,6 @@ In the next section, a connection between the two will be found to complete the
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>Ieee/Asme Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
 </div>

content/article/oomen18_advan_motion_contr_precis_mechat.md

@@ -21,12 +21,12 @@ Year
 
 Control of positioning systems is traditionally simplified by an excellent mechanical design.
 In particular, the mechanical design is such that the system is stiff and highly reproducible.
-In conjunction with moderate performance requirements, the control bandwidth is well-below the resonance frequency of the flexible mechanics as is shown in Figure [1](#figure--fig:oomen18-next-gen-loop-gain) (a).
+In conjunction with moderate performance requirements, the control bandwidth is well-below the resonance frequency of the flexible mechanics as is shown in [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (a).
 As a result, the system can often be completely **decoupled** in the frequency range relevant for control.
 Consequently, the control design is divided into well-manageable SISO control loops.
 
 Although motion control design is well developed, presently available techniques mainly apply to positioning systems that behave as a rigid body in the relevant frequency range.
-On one hand, increasing performance requirements hamper the validity of this assumption, since the bandwidth has to increase, leading to flexible dynamics in the cross-over region, see Figure [1](#figure--fig:oomen18-next-gen-loop-gain) (b).
+On one hand, increasing performance requirements hamper the validity of this assumption, since the bandwidth has to increase, leading to flexible dynamics in the cross-over region, see [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (b).
 
 <a id="figure--fig:oomen18-next-gen-loop-gain"></a>
 
@@ -55,7 +55,7 @@ In this case, matrices \\(T\_u\\) and \\(T\_y\\) can be selected such that:
 G = T\_y G\_m T\_u = \frac{1}{s^2} I\_{n\_{RB}} + G\_{\text{flex}}
 \end{equation}
 
-A tradition motion control architecture is shown in Figure [2](#figure--fig:oomen18-control-architecture).
+A tradition motion control architecture is shown in [Figure 2](#figure--fig:oomen18-control-architecture).
 
 <a id="figure--fig:oomen18-control-architecture"></a>
 
@@ -119,7 +119,7 @@ This leads to several challenges for motion control design:
 
 A generalized plant framework allows for a systematic way to address the future challenges in advanced motion control.
 
-The generalized plant is depicted in Figure [3](#figure--fig:oomen18-generalized-plant):
+The generalized plant is depicted in [Figure 3](#figure--fig:oomen18-generalized-plant):
 
 -   \\(z\\) are the performance variables
 -   \\(y\\) and \\(u\\) are the measured variables and measured variables, respectively
@@ -180,8 +180,6 @@ This motivates a robust control design, where the **model quality is explicitly
 
 ## Feedforward and learning {#feedforward-and-learning}
 
-## References
-
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” <i>Ieej Journal of Industry Applications</i> 7 (2): 127–40. doi:<a href="https://doi.org/10.1541/ieejjia.7.127">10.1541/ieejjia.7.127</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” <i>IEEJ Journal of Industry Applications</i> 7 (2): 127–40. doi:<a href="https://doi.org/10.1541/ieejjia.7.127">10.1541/ieejjia.7.127</a>.</div>
 </div>

content/article/preumont02_force_feedb_versus_accel_feedb.md

@@ -26,8 +26,8 @@ The force applied to a **rigid body** is proportional to its acceleration, thus
 Thus force feedback and acceleration feedback are equivalent for solid bodies.
 When there is a flexible payload, the two sensing options are not longer equivalent.
 
--   For light payload (Figure [1](#figure--fig:preumont02-force-acc-fb-light)), the acceleration feedback gives larger damping on the higher mode.
+-   For light payload ([Figure 1](#figure--fig:preumont02-force-acc-fb-light)), the acceleration feedback gives larger damping on the higher mode.
--   For heavy payload (Figure [2](#figure--fig:preumont02-force-acc-fb-heavy)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
+-   For heavy payload ([Figure 2](#figure--fig:preumont02-force-acc-fb-heavy)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
 
 <a id="figure--fig:preumont02-force-acc-fb-light"></a>
 

content/article/preumont07_six_axis_singl_stage_activ.md

@@ -18,15 +18,15 @@ Year
 
 Summary:
 
--   **Cubic** Stewart platform (Figure [3](#figure--fig:preumont07-stewart-platform))
+-   **Cubic** Stewart platform ([Figure 3](#figure--fig:preumont07-stewart-platform))
     -   Provides uniform control capability
     -   Uniform stiffness in all directions
     -   minimizes the cross-coupling among actuators and sensors of different legs
--   Flexible joints (Figure [2](#figure--fig:preumont07-flexible-joints))
+-   Flexible joints ([Figure 2](#figure--fig:preumont07-flexible-joints))
 -   Piezoelectric force sensors
 -   Voice coil actuators
 -   Decentralized feedback control approach for vibration isolation
--   Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#figure--fig:preumont07-iff-effect-stiffness))
+-   Effect of parasitic stiffness of the flexible joints on the IFF performance ([Figure 1](#figure--fig:preumont07-iff-effect-stiffness))
 -   The Stewart platform has 6 suspension modes at different frequencies.
     Thus the gain of the IFF controller cannot be optimal for all the modes.
     It is better if all the modes of the platform are near to each other.

content/article/saxena12_advan_inter_model_contr_techn.md

@@ -88,5 +88,5 @@ The interesting feature regarding IMC is that the design scheme is identical to
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” <i>Iete Technical Review</i> 29 (6): 461. doi:<a href="https://doi.org/10.4103/0256-4602.105001">10.4103/0256-4602.105001</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” <i>IETE Technical Review</i> 29 (6): 461. doi:<a href="https://doi.org/10.4103/0256-4602.105001">10.4103/0256-4602.105001</a>.</div>
 </div>

content/article/schroeck01_compen_desig_linear_time_invar.md

@@ -21,5 +21,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” <i>Ieee/Asme Transactions on Mechatronics</i> 6 (1): 50–57. doi:<a href="https://doi.org/10.1109/3516.914391">10.1109/3516.914391</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” <i>IEEE/ASME Transactions on Mechatronics</i> 6 (1): 50–57. doi:<a href="https://doi.org/10.1109/3516.914391">10.1109/3516.914391</a>.</div>
 </div>

content/article/sebastian12_nanop_with_multip_sensor.md

@@ -20,5 +20,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” <i>Ieee Transactions on Control Systems Technology</i> 20 (2): 382–94. doi:<a href="https://doi.org/10.1109/tcst.2011.2177982">10.1109/tcst.2011.2177982</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” <i>IEEE Transactions on Control Systems Technology</i> 20 (2): 382–94. doi:<a href="https://doi.org/10.1109/tcst.2011.2177982">10.1109/tcst.2011.2177982</a>.</div>
 </div>

content/article/souleille18_concep_activ_mount_space_applic.md

@@ -23,7 +23,7 @@ This article discusses the use of Integral Force Feedback with amplified piezoel
 
 ## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
 
-Figure [1](#figure--fig:souleille18-model-piezo) shows a picture of the amplified piezoelectric stack.
+[Figure 1](#figure--fig:souleille18-model-piezo) shows a picture of the amplified piezoelectric stack.
 The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
 
 <a id="figure--fig:souleille18-model-piezo"></a>
@@ -31,7 +31,7 @@ The piezoelectric actuator is divided into two parts: one is used as an actuator
 {{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="<span class=\"figure-number\">Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
 
 <div class="table-caption">
-  <span class="table-number">Table 1</span>:
+  <span class="table-number">Table 1:</span>
   Parameters used for the model of the APA 100M
 </div>
 
@@ -61,7 +61,7 @@ and the control force is given by:
   f = F\_s G(s) = F\_s \frac{g}{s}
 \end{equation}
 
-The effect of the controller are shown in Figure [2](#figure--fig:souleille18-tf-iff-result):
+The effect of the controller are shown in [Figure 2](#figure--fig:souleille18-tf-iff-result):
 
 -   the resonance peak is almost critically damped
 -   the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
@@ -79,14 +79,14 @@ The effect of the controller are shown in Figure [2](#figure--fig:souleille18-tf
 
 ## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators}
 
-A heavy payload is mounted on a set of three isolators (Figure [4](#figure--fig:souleille18-setup-flexible-payload)).
+A heavy payload is mounted on a set of three isolators ([Figure 4](#figure--fig:souleille18-setup-flexible-payload)).
 The payload consists of two  masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
 
 <a id="figure--fig:souleille18-setup-flexible-payload"></a>
 
 {{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="<span class=\"figure-number\">Figure 4: </span>Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
 
-As shown in Figure [5](#figure--fig:souleille18-result-damping-transmissibility), both the suspension modes and the flexible modes of the payload can be critically damped.
+As shown in [Figure 5](#figure--fig:souleille18-result-damping-transmissibility), both the suspension modes and the flexible modes of the payload can be critically damped.
 
 <a id="figure--fig:souleille18-result-damping-transmissibility"></a>
 
@@ -96,5 +96,5 @@ As shown in Figure [5](#figure--fig:souleille18-result-damping-transmissibility)
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>Ceas Space Journal</i> 10 (2). Springer: 157–65.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer: 157–65.</div>
 </div>

content/article/spanos95_soft_activ_vibrat_isolat.md

@@ -16,7 +16,7 @@ Author(s)
 Year
 : 1995
 
-**Stewart Platform** (Figure [1](#figure--fig:spanos95-stewart-platform)):
+**Stewart Platform** ([Figure 1](#figure--fig:spanos95-stewart-platform)):
 
 -   Voice Coil
 -   Flexible joints (cross-blades)
@@ -52,7 +52,7 @@ The controller used consisted of:
 -   first order lag filter to provide adequate phase margin at the low frequency crossover
 -   a first order high pass filter to attenuate the excess gain resulting from the low frequency zero
 
-The results in terms of transmissibility are shown in Figure [3](#figure--fig:spanos95-results).
+The results in terms of transmissibility are shown in [Figure 3](#figure--fig:spanos95-results).
 
 <a id="figure--fig:spanos95-results"></a>
 
@@ -62,5 +62,5 @@ The results in terms of transmissibility are shown in Figure [3](#figure--fig:sp
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In <i>Proceedings of 1995 American Control Conference - Acc’95</i>, nil. doi:<a href="https://doi.org/10.1109/acc.1995.529280">10.1109/acc.1995.529280</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In <i>Proceedings of 1995 American Control Conference - ACC’95</i>. doi:<a href="https://doi.org/10.1109/acc.1995.529280">10.1109/acc.1995.529280</a>.</div>
 </div>

content/article/wang16_inves_activ_vibrat_isolat_stewar.md

@@ -35,7 +35,7 @@ Combines:
 -   the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations
 -   direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations
 
-Force Feedback (Figure [2](#figure--fig:wang16-force-feedback)).
+Force Feedback ([Figure 2](#figure--fig:wang16-force-feedback)).
 
 -   the force sensor is mounted **between the base and the strut**
 

content/article/yang19_dynam_model_decoup_contr_flexib.md

@@ -25,10 +25,10 @@ Year
 The joint stiffness impose a limitation on the control performance using force sensors as it adds a zero at low frequency in the dynamics.
 Thus, this stiffness is taken into account in the dynamics and compensated for.
 
-**Stewart platform** (Figure [1](#figure--fig:yang19-stewart-platform)):
+**Stewart platform** ([Figure 1](#figure--fig:yang19-stewart-platform)):
 
 -   piezoelectric actuators
--   flexible joints (Figure [2](#figure--fig:yang19-flexible-joints))
+-   flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints))
 -   force sensors (used for vibration isolation)
 -   displacement sensors (used to decouple the dynamics)
 -   cubic (even though not said explicitly)
@@ -41,11 +41,11 @@ Thus, this stiffness is taken into account in the dynamics and compensated for.
 
 {{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="<span class=\"figure-number\">Figure 2: </span>Flexible Joints" >}}
 
-The stiffness of the flexible joints (Figure [2](#figure--fig:yang19-flexible-joints)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
+The stiffness of the flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints)) are computed with an FEM model and shown in [Table 1](#table--tab:yang19-stiffness-flexible-joints).
 
 <a id="table--tab:yang19-stiffness-flexible-joints"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:yang19-stiffness-flexible-joints">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:yang19-stiffness-flexible-joints">Table 1</a>:</span>
   Stiffness of flexible joints obtained by FEM
 </div>
 
@@ -105,7 +105,7 @@ In order to apply this control strategy:
 -   The jacobian has to be computed
 -   No information about modal matrix is needed
 
-The block diagram of the control strategy is represented in Figure [3](#figure--fig:yang19-control-arch).
+The block diagram of the control strategy is represented in [Figure 3](#figure--fig:yang19-control-arch).
 
 <a id="figure--fig:yang19-control-arch"></a>
 
@@ -121,7 +121,7 @@ Substituting \\(H(s)\\) in the equation of motion gives that:
 
 **Experimental Validation**:
 An external Shaker is used to excite the base and accelerometers are located on the base and mobile platforms to measure their motion.
-The results are shown in Figure [4](#figure--fig:yang19-results).
+The results are shown in [Figure 4](#figure--fig:yang19-results).
 In theory, the vibration performance can be improved, however in practice, increasing the gain causes saturation of the piezoelectric actuators and then the instability occurs.
 
 <a id="figure--fig:yang19-results"></a>

content/article/yong12_invit_review_artic.md

@@ -0,0 +1,22 @@
++++
+title = "Invited review article: high-speed flexure-guided nanopositioning: mechanical design and control issues"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Yong et al. 2012</a>)
+
+Author(s)
+: Yong, Y. K., Moheimani, S. O. R., Kenton, B. J., &amp; Leang, K. K.
+
+Year
+: 2012
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yong, Y. K., S. O. R. Moheimani, B. J. Kenton, and K. K. Leang. 2012. “Invited Review Article: High-Speed Flexure-Guided Nanopositioning: Mechanical Design and Control Issues.” <i>Review of Scientific Instruments</i> 83 (12): 121101. doi:<a href="https://doi.org/10.1063/1.4765048">10.1063/1.4765048</a>.</div>
+</div>

content/article/yun20_inves_two_stage_vibrat_suppr.md

@@ -21,5 +21,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” <i>Aerospace Science and Technology</i> 96 (nil): 105543. doi:<a href="https://doi.org/10.1016/j.ast.2019.105543">10.1016/j.ast.2019.105543</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” <i>Aerospace Science and Technology</i> 96: 105543. doi:<a href="https://doi.org/10.1016/j.ast.2019.105543">10.1016/j.ast.2019.105543</a>.</div>
 </div>

content/article/zhang11_six_dof.md

@@ -33,5 +33,5 @@ Year
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six Dof Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In <i>2011 6th Ieee Conference on Industrial Electronics and Applications</i>, nil. doi:<a href="https://doi.org/10.1109/iciea.2011.5975679">10.1109/iciea.2011.5975679</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In <i>2011 6th IEEE Conference on Industrial Electronics and Applications</i>. doi:<a href="https://doi.org/10.1109/iciea.2011.5975679">10.1109/iciea.2011.5975679</a>.</div>
 </div>

content/book/du19_multi_actuat_system_contr.md

@@ -83,18 +83,18 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
 
 #### Secondary Actuators {#secondary-actuators}
 
-We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#figure--fig:pzt-actuator)) and the microactuator.
+We here consider two types of secondary actuators: the PZT milliactuator ([Figure 1](#figure--fig:pzt-actuator)) and the microactuator.
 
 <a id="figure--fig:pzt-actuator"></a>
 
 {{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class=\"figure-number\">Figure 1: </span>A PZT-actuator suspension" >}}
 
 There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
-There characteristics are shown on table [1](#table--tab:microactuator).
+There characteristics are shown on [Table 1](#table--tab:microactuator).
 
 <a id="table--tab:microactuator"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:microactuator">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:microactuator">Table 1</a>:</span>
   Performance comparison of microactuators
 </div>
 
@@ -107,7 +107,7 @@ There characteristics are shown on table [1](#table--tab:microactuator).
 
 ### Single-Stage Actuation Systems {#single-stage-actuation-systems}
 
-A typical closed-loop control system is shown on figure [2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
+A typical closed-loop control system is shown on [Figure 2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
 
 <a id="figure--fig:single-stage-control"></a>
 
@@ -145,7 +145,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
 
 ### Control Schemes {#control-schemes}
 
-A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#figure--fig:decoupled-control).
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in [Figure 4](#figure--fig:decoupled-control).
 
 -   \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
 -   \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@@ -175,7 +175,7 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
 And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
 Thus, the dual-stage system control design can be decoupled into two independent controller designs.
 
-Another type of control scheme is the **parallel structure** as shown in figure [5](#figure--fig:parallel-control-structure).
+Another type of control scheme is the **parallel structure** as shown in [Figure 5](#figure--fig:parallel-control-structure).
 The open-loop transfer function from \\(pes\\) to \\(y\\) is
 \\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
 
@@ -192,7 +192,7 @@ Because of the limited displacement range of the secondary actuator, the control
 ### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
 
 \\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
-The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
+The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in [Figure 6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
 
 For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
 
@@ -206,7 +206,7 @@ is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\
 
 {{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
 
-Equation [1](#org563f2ec) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+Equation [ 1](#orgcf76ccd) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
 One form of \\(W(s)\\) is taken as
 
 \begin{equation}
@@ -219,13 +219,13 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
 
 The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
 But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
-As seen in figure [7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
+As seen in [Figure 7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
 
 <a id="figure--fig:dual-stage-loop-gain"></a>
 
 {{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class=\"figure-number\">Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
 
-The sensitivity functions are shown in figure [8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
+The sensitivity functions are shown in [Figure 8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
 This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
 
 <a id="figure--fig:dual-stage-sensitivity"></a>
@@ -261,7 +261,7 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
 
 ### Control Strategy and Controller Design {#control-strategy-and-controller-design}
 
-Figure [9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
+[Figure 9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
 
 The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
 The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
@@ -296,13 +296,13 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
 The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
 
 The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
-The open-loop frequency responses of the three stages are shown on figure [10](#figure--fig:open-loop-three-stage).
+The open-loop frequency responses of the three stages are shown on [Figure 10](#figure--fig:open-loop-three-stage).
 
 <a id="figure--fig:open-loop-three-stage"></a>
 
 {{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class=\"figure-number\">Figure 10: </span>Frequency response of the open-loop transfer function" >}}
 
-The obtained sensitivity function is shown on figure [11](#figure--fig:sensitivity-three-stage).
+The obtained sensitivity function is shown on [Figure 11](#figure--fig:sensitivity-three-stage).
 
 <a id="figure--fig:sensitivity-three-stage"></a>
 
@@ -319,7 +319,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
 Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
 Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
 
-For the three-stage control architecture as shown on figure [9](#figure--fig:three-stage-control), the position error is
+For the three-stage control architecture as shown on [Figure 9](#figure--fig:three-stage-control), the position error is
 \\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
 The control signals and positions of the actuators are given by
 
@@ -335,11 +335,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
 
 ### Different Configurations of the Control System {#different-configurations-of-the-control-system}
 
-A decoupled control structure can be used for the three-stage actuation system (see figure [12](#figure--fig:three-stage-decoupled)).
+A decoupled control structure can be used for the three-stage actuation system (see [Figure 12](#figure--fig:three-stage-decoupled)).
 
 The overall sensitivity function is
 \\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
-with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org9bf2b8d) and
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [ 1](#org40d0f02) and
 \\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
 
 Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -360,7 +360,7 @@ The control signals and the positions of the three actuators are
   u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
 \end{align\*}
 
-The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#figure--fig:three-stage-decoupled-loop-gain)).
+The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see [Figure 13](#figure--fig:three-stage-decoupled-loop-gain)).
 
 <a id="figure--fig:three-stage-decoupled-loop-gain"></a>
 

content/book/fleming14_desig_model_contr_nanop_system.md

@@ -732,7 +732,7 @@ Year
 
 {{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
 
-Consider the electrical circuit shown in Figure [1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
+Consider the electrical circuit shown in [Figure 1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
 The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
 The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
 
@@ -768,11 +768,11 @@ If the inductance \\(L\\) is neglected, the transfer function from source voltag
 This is thus highly dependent of the load.
 
 The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
-A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see Table [1](#table--tab:piezo-limitation-Rs) for values).
+A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see [Table 1](#table--tab:piezo-limitation-Rs) for values).
 
 <a id="table--tab:piezo-limitation-Rs"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
   Bandwidth limitation due to \(R_s\)
 </div>
 
@@ -784,11 +784,11 @@ A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-of
 
 The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
 Ideally, low inductance cables should be used.
-It is however usually quite high compare to \\(\omega\_c\\) as shown in Table [2](#table--tab:piezo-limitation-L).
+It is however usually quite high compare to \\(\omega\_c\\) as shown in [Table 2](#table--tab:piezo-limitation-L).
 
 <a id="table--tab:piezo-limitation-L"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
   Bandwidth limitation due to \(R_s\)
 </div>
 
@@ -827,7 +827,7 @@ For sinusoidal signals, the maximum positive and negative current is equal to:
 
 <a id="table--tab:piezo-required-current"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
   Minimum current requirements for a 10V sinusoid
 </div>
 

content/book/hatch00_vibrat_matlab_ansys.md

@@ -23,14 +23,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
 
 ## Introduction {#introduction}
 
-<span class="org-target" id="org-target--sec:introduction"></span>
+<span class="org-target" id="org-target--sec-introduction"></span>
 
 The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
 
 
 ### Modal Analysis {#modal-analysis}
 
-The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
+The diagram in [Figure 1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
 
 <div class="important">
 
@@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
 
 <div class="important">
 
-Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
+[Figure 2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
 
 -   start with the finite element model
 -   compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -78,11 +78,11 @@ Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the s
 
 ### Notations {#notations}
 
-Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
+[Figure 3](#figure--fig:hatch00-n-dof-zeros), [Table 2](#table--tab:notations-eigen-vectors-values) and [Table 3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
 
 <a id="table--tab:notations-modes-nodes"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a>:</span>
   Notation for the modes and nodes
 </div>
 
@@ -97,7 +97,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
 
 <a id="table--tab:notations-eigen-vectors-values"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a>:</span>
   Notation for the dofs, eigenvectors and eigenvalues
 </div>
 
@@ -112,7 +112,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
 
 <a id="table--tab:notations-stiffness-mass"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a>:</span>
   Notation for the mass and stiffness matrices
 </div>
 
@@ -127,12 +127,12 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
 
 ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
 
-<span class="org-target" id="org-target--sec:zeros_siso_systems"></span>
+<span class="org-target" id="org-target--sec-zeros-siso-systems"></span>
 The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
 
 We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
 
-Figure [3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
+[Figure 3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
 The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
 
 <a id="figure--fig:hatch00-n-dof-zeros"></a>
@@ -150,12 +150,12 @@ The resonances of the "overhanging appendages" of the constrained system create
 
 ## State Space Analysis {#state-space-analysis}
 
-<span class="org-target" id="org-target--sec:state_space_analysis"></span>
+<span class="org-target" id="org-target--sec-state-space-analysis"></span>
 
 
 ## Modal Analysis {#modal-analysis}
 
-<span class="org-target" id="org-target--sec:modal_analysis"></span>
+<span class="org-target" id="org-target--sec-modal-analysis"></span>
 
 Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
 
@@ -193,7 +193,7 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
 
 #### Equation of Motion {#equation-of-motion}
 
-Let's consider the model shown in Figure [4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
+Let's consider the model shown in [Figure 4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
 
 <a id="figure--fig:hatch00-undamped-tdof-model"></a>
 
@@ -252,7 +252,7 @@ where:
 
 #### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
 
-Re-injecting normal modes <eq:principal_mode> into the equation of motion <eq:tdof_eom> gives the eigenvalue problem:
+Re-injecting normal modes \eqref{eq:principal\_mode} into the equation of motion \eqref{eq:tdof\_eom} gives the eigenvalue problem:
 
 \begin{equation}
   (\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
@@ -293,7 +293,7 @@ One then find:
   \end{bmatrix}
 \end{equation}
 
-Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig:hatch00-tdof-mode-1), [6](#figure--fig:hatch00-tdof-mode-2) and [7](#figure--fig:hatch00-tdof-mode-3).
+Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:hatch00-tdof-mode-1), [Figure 6](#figure--fig:hatch00-tdof-mode-2) and [Figure 7](#figure--fig:hatch00-tdof-mode-3).
 
 <a id="figure--fig:hatch00-tdof-mode-1"></a>
 
@@ -310,7 +310,7 @@ Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig
 
 #### Modal Matrix {#modal-matrix}
 
-The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. <eq:modal_matrix>
+The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal\_matrix}
 
 \begin{equation}
   \bm{z}\_m = \begin{bmatrix}
@@ -341,7 +341,7 @@ There are many options for change of basis, but we will show that **when eigenve
 The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
 The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
 
-This procedure is schematically shown in Figure [8](#figure--fig:hatch00-schematic-modal-solution).
+This procedure is schematically shown in [Figure 8](#figure--fig:hatch00-schematic-modal-solution).
 
 <a id="figure--fig:hatch00-schematic-modal-solution"></a>
 
@@ -687,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
 
 ## Frequency Response: Modal Form {#frequency-response-modal-form}
 
-<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
+<span class="org-target" id="org-target--sec-frequency-response-modal-form"></span>
 
 The procedure to obtain the frequency response from a modal form is as follow:
 
@@ -695,7 +695,7 @@ The procedure to obtain the frequency response from a modal form is as follow:
 -   use Laplace transform to obtain the transfer functions in principal coordinates
 -   back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
 
-This will be applied to the model shown in Figure [9](#figure--fig:hatch00-tdof-model).
+This will be applied to the model shown in [Figure 9](#figure--fig:hatch00-tdof-model).
 
 <a id="figure--fig:hatch00-tdof-model"></a>
 
@@ -873,11 +873,11 @@ If modes have some damping:
   \frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
 \end{equation}
 
-Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
+Equations \eqref{eq:general\_add\_tf} and \eqref{eq:general\_add\_tf\_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
 
 </div>
 
-Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
+[Figure 10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
 
 <a id="figure--fig:hatch00-z11-tf-example"></a>
 
@@ -888,12 +888,12 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
 
 ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
 
-<span class="org-target" id="org-target--sec:siso_state_space"></span>
+<span class="org-target" id="org-target--sec-siso-state-space"></span>
 
 
 ### Introduction {#introduction}
 
-In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
+In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in [Figure 11](#figure--fig:hatch00-cantilever-beam).
 A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
 The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
 
@@ -976,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
 
 ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
 
-<span class="org-target" id="org-target--sec:ground_acceleration"></span>
+<span class="org-target" id="org-target--sec-ground-acceleration"></span>
 
 
 ### Model Description {#model-description}
@@ -990,9 +990,9 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
 
 ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
 
-<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
+<span class="org-target" id="org-target--sec-siso-disk-drive"></span>
 
-In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-model)).
+In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator ([Figure 12](#figure--fig:hatch00-disk-drive-siso-model)).
 
 
 ### Actuator Description {#actuator-description}
@@ -1001,14 +1001,14 @@ In this section we wish to extract a SISO state space model from a Finite Elemen
 
 {{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
 
-The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
+The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident ([Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
 
 <a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
 
 {{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
 
 For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
-Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
+[Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
 The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
 The arrows at the nodes indicate the direction of forces.
 
@@ -1074,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
 
 ## Balanced Reduction {#balanced-reduction}
 
-<span class="org-target" id="org-target--sec:balanced_reduction"></span>
+<span class="org-target" id="org-target--sec-balanced-reduction"></span>
 
 In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
 
@@ -1189,9 +1189,9 @@ The **states to be kept are the states with the largest diagonal terms**.
 
 ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
 
-<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
+<span class="org-target" id="org-target--sec-mimo-disk-drive"></span>
 
-In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
+In this section, a MIMO two-stage actuator model is derived from a finite element model ([Figure 14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
 
 
 ### Actuator Description {#actuator-description}
@@ -1217,7 +1217,7 @@ Since the same forces are being applied to both piezo elements, they represent t
 
 ### Ansys Model Description {#ansys-model-description}
 
-In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
+In [Figure 15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
 
 <a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
 
@@ -1440,7 +1440,7 @@ State Space Model
 
 ### Simple mode truncation {#simple-mode-truncation}
 
-Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
+Let's plot the frequency of the modes ([Figure 18](#figure--fig:hatch00-cant-beam-modes-freq)).
 
 <a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
 
@@ -2123,6 +2123,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using MATLAB and ANSYS</i>. CRC Press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
 </div>

content/book/morrison16_groun_shiel.md

@@ -148,7 +148,7 @@ In a few elements, the atomic structure is such that atoms align to generate a n
 The flow of electrons is another way to generate a magnetic field.
 
 The letter \\(H\\) is reserved for the magnetic field generated by a current.
-Figure [6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
+[Figure 6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
 
 <a id="figure--fig:morrison16-H-field"></a>
 
@@ -167,7 +167,7 @@ Ampere's law states that the integral of the \\(H\\) field intensity in a closed
 \boxed{\oint H dl = I}
 \end{equation}
 
-The simplest path to use for this integration is the one of the concentric circles in Figure [6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
+The simplest path to use for this integration is the one of the concentric circles in [Figure 6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
 Solving for \\(H\\), we obtain
 
 \begin{equation}
@@ -179,10 +179,10 @@ And we see that \\(H\\) has units of amperes per meter.
 
 ### The solenoid {#the-solenoid}
 
-The magnetic field of a solenoid is shown in Figure [7](#figure--fig:morrison16-solenoid).
+The magnetic field of a solenoid is shown in [Figure 7](#figure--fig:morrison16-solenoid).
 The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
 
-Using Ampere's law <eq:ampere_law>:
+Using Ampere's law \eqref{eq:ampere\_law:}
 
 \begin{equation}
 \oint H dl \approx n I l
@@ -196,7 +196,7 @@ Using Ampere's law <eq:ampere_law>:
 ### Faraday's law and the induction field {#faraday-s-law-and-the-induction-field}
 
 When a conducting coil is moved through a magnetic field, a voltage appears at the open ends of the coil.
-This is illustrated in Figure [8](#figure--fig:morrison16-voltage-moving-coil).
+This is illustrated in [Figure 8](#figure--fig:morrison16-voltage-moving-coil).
 The voltage depends on the number of turns in the coil and the rate at which the flux is changing.
 
 <a id="figure--fig:morrison16-voltage-moving-coil"></a>
@@ -236,7 +236,7 @@ The unit of inductance if the henry.
 
 </div>
 
-For the  coil in Figure [7](#figure--fig:morrison16-solenoid):
+For the  coil in [Figure 7](#figure--fig:morrison16-solenoid):
 
 \begin{equation} \label{eq:inductance\_coil}
 V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
@@ -244,12 +244,12 @@ V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
 
 where \\(k\\) relates to the geometry of the coil.
 
-Equation <eq:inductance_coil> states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
+Equation \eqref{eq:inductance\_coil} states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
 
 
 ### The energy stored in an inductance {#the-energy-stored-in-an-inductance}
 
-One way to calculate the work stored in a magnetic field is to use Eq. <eq:inductance_coil>.
+One way to calculate the work stored in a magnetic field is to use Eq. \eqref{eq:inductance\_coil}.
 The voltage \\(V\\) applied to a coil results in a linearly increasing current.
 At any time \\(t\\), the power \\(P\\) supplied is equal to \\(VI\\).
 Power is the rate of change of energy or \\(P = d\bm{E}/dt\\) where \\(\bm{E}\\) is the stored energy in the inductance.
@@ -476,21 +476,21 @@ For example, signals that overload an input stage can produce noise that may loo
 
 ### The basic shield enclosure {#the-basic-shield-enclosure}
 
-Consider the simple amplifier circuit shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
+Consider the simple amplifier circuit shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
 
 -   \\(V\_1\\) the input lead
 -   \\(V\_2\\) the output lead
 -   \\(V\_3\\) the conducting enclosure which is floating and taken as the reference conductor
 -   \\(V\_4\\) a signal common or reference conductor
 
-Every conductor pair has a mutual capacitance, which are shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
+Every conductor pair has a mutual capacitance, which are shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
-The equivalent circuit is shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
+The equivalent circuit is shown in [Figure 9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
 
 <a id="figure--fig:morrison16-parasitic-capacitance-amp"></a>
 
 {{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="<span class=\"figure-number\">Figure 9: </span>Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
 
-It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#figure--fig:morrison16-grounding-shield-amp)).
+It is common practice in analog design to connect the enclosure to circuit common ([Figure 10](#figure--fig:morrison16-grounding-shield-amp)).
 When this connection is made, the feedback is removed and the enclosure no longer couples signals into the feedback structure.
 The conductive enclosure is called a **shield**.
 Connecting the signal common to the conductive enclosure is called "**grounding the shield**".
@@ -502,13 +502,13 @@ This "grounding" usually removed "hum" from the circuit.
 
 Most practical circuits provide connections to external points.
 To see the effect of making a _single_ external connection, open the conductive enclosure and connect the input circuit common to an external ground.
-Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
+[Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
 A problem can be caused by an incorrect location of the connection between the cable shield and the enclosure.
-In Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
+In [Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
 This conductor being the common ground that might have a resistance \\(R\\) or \\(1\\,\Omega\\), this current induced voltage that it added to the transmitted signal.
 Our goal in this chapter is to find ways of keeping interference currents from flowing in any input signal conductor.
 To remove this coupling, the shield connection to circuit common must be made at the point, where the circuit common connects to the external ground.
-This connection is shown in Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
+This connection is shown in [Figure 11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
 This connection keeps the circulation of interference current on the outside of the shield.
 
 There is only one point of zero signal potential external to the enclosure and that is where the signal common connects to an external hardware ground.
@@ -655,7 +655,7 @@ If the resistors are replaced by capacitors, the gain is the ratio of reactances
 This feedback circuit is called a **charge converter**.
 The charge on the input capacitor is transferred to the feedback capacitor.
 If the feedback capacitor is smaller than the transducer capacitance by a factor of 100, then the voltage across the feedback capacitor will be 100 times greater than the open-circuit transducer voltage.
-This feedback arrangement is shown in Figure [17](#figure--fig:morrison16-charge-amplifier).
+This feedback arrangement is shown in [Figure 17](#figure--fig:morrison16-charge-amplifier).
 The open-circuit input signal voltage is \\(Q/C\_T\\).
 The output voltage is \\(Q/C\_{FB}\\).
 The voltage gain is therefore \\(C\_T/C\_{FB}\\).

content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md

@@ -67,7 +67,7 @@ There are two radically different approached to disturbance rejection: feedback
 
 {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
 
-The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
+The principle of feedback is represented on [Figure 1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
 The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
 
 In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@@ -94,18 +94,18 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
 {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>Principle of feedforward control" >}}
 
 The method relies on the availability of a **reference signal correlated to the primary disturbance**.
-The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#figure--fig:feedforward-adaptative).
+The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in [Figure 2](#figure--fig:feedforward-adaptative).
 
 The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
 
 There is no guarantee that the global response is reduced at other locations. This method is therefor considered as a local one.
 Because it is less sensitive to phase lag than feedback, it can be used at higher frequencies (\\(\omega\_c \approx \omega\_s/10\\)).
 
-The table [1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
+The [Table 1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
 
 <a id="table--tab:adv-dis-type-control"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:adv-dis-type-control">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:adv-dis-type-control">Table 1</a>:</span>
   Advantages and Disadvantages of some types of control
 </div>
 
@@ -129,7 +129,7 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
 
 {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>The various steps of the design" >}}
 
-The various steps of the design of a controlled structure are shown in figure [3](#figure--fig:design-steps).
+The various steps of the design of a controlled structure are shown in [Figure 3](#figure--fig:design-steps).
 
 The **starting point** is:
 
@@ -156,7 +156,7 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
 
 ### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
 
-From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
+From the block diagram of the control system ([Figure 4](#figure--fig:general-plant)):
 
 \begin{align\*}
 y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
@@ -186,7 +186,7 @@ Even more interesting for the design is the **Cumulative Mean Square** response
 It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
 \\(\sigma\_z(0)\\) is then the global RMS response.
 
-A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown [Figure 5](#figure--fig:cas-plot).
 It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
 
 The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
@@ -331,9 +331,9 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
 If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
 \\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
 
-One can verify that the Rayleigh damping <eq:rayleigh_damping> complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
+One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
 
-And we obtain decoupled modal equations <eq:modal_eom>.
+And we obtain decoupled modal equations \eqref{eq:modal\_eom}.
 
 <div class="cbox">
 
@@ -353,7 +353,7 @@ Typical values of the modal damping ratio are summarized on table <tab:damping_r
 
 <a id="table--tab:damping-ratio"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:damping-ratio">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:damping-ratio">Table 2</a>:</span>
   Typical Damping ratio
 </div>
 
@@ -366,15 +366,15 @@ Typical values of the modal damping ratio are summarized on table <tab:damping_r
 
 The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
 
-If one accepts the assumption of classical damping, the only difference between equation <eq:general_eom> and <eq:modal_eom> lies in the change of coordinates.
+If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general\_eom} and \eqref{eq:modal\_eom} lies in the change of coordinates.
 However, in physical coordinates, the number of degrees of freedom is usually very large.
-If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation <eq:modal_eom> can often be restricted to theses modes.
+If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal\_eom} can often be restricted to theses modes.
 Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
 
 
 #### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
 
-If we consider the steady-state response of equation <eq:general_eom> to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
+If we consider the steady-state response of equation \eqref{eq:general\_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
 \\[ X = G(\omega) F \\]
 
 Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@@ -398,7 +398,7 @@ With:
 
 {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
 
-If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
+If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on [Figure 6](#figure--fig:neglected-modes)).
 
 And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
 \\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@@ -422,7 +422,7 @@ A **collocated control system** is a control system where:
 
 <a id="table--tab:dual-actuator-sensor"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:dual-actuator-sensor">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:dual-actuator-sensor">Table 3</a>:</span>
   Examples of dual actuators and sensors
 </div>
 
@@ -436,7 +436,7 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
 If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
 \\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
 
-\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) ([Figure 7](#figure--fig:collocated-control-frf)).
 
 <a id="figure--fig:collocated-control-frf"></a>
 
@@ -451,7 +451,7 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
 
 </div>
 
-If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#figure--fig:collocated-zero).
+If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line [Figure 8](#figure--fig:collocated-zero).
 
 <a id="figure--fig:collocated-zero"></a>
 
@@ -467,7 +467,7 @@ The open-loop poles are independant of the actuator and sensor configuration whi
 
 </div>
 
-By looking at figure [7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
+By looking at [Figure 7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
 
 <a id="figure--fig:alternating-p-z"></a>
 
@@ -479,7 +479,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
   G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
 \end{equation}
 
-The corresponding Bode plot is represented in figure [9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
+The corresponding Bode plot is represented in [Figure 9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
 In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
 
 
@@ -501,7 +501,7 @@ Two broad categories of actuators can be distinguish:
 
 A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
 
-The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
+The system consists of (see [Figure 10](#figure--fig:voice-coil-schematic)):
 
 -   A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
 -   A coil which is free to move axially
@@ -543,7 +543,7 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
 
 #### Proof-Mass Actuator {#proof-mass-actuator}
 
-A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#figure--fig:proof-mass-actuator)).
+A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) ([Figure 11](#figure--fig:proof-mass-actuator)).
 
 <a id="figure--fig:proof-mass-actuator"></a>
 
@@ -574,7 +574,7 @@ with:
 
 </div>
 
-Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#figure--fig:proof-mass-tf)).
+Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** ([Figure 12](#figure--fig:proof-mass-tf)).
 
 <a id="figure--fig:proof-mass-tf"></a>
 
@@ -610,7 +610,7 @@ Designing geophones with very low corner frequency is in general difficult. Acti
 
 ### General Electromechanical Transducer {#general-electromechanical-transducer}
 
-The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
+The consitutive behavior of a wide class of electromechanical transducers can be modelled as in [Figure 14](#figure--fig:electro-mechanical-transducer).
 
 <a id="figure--fig:electro-mechanical-transducer"></a>
 
@@ -634,10 +634,10 @@ With:
 -   \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
 -   \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
 
-Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
+Equation \eqref{eq:gen\_trans\_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
 Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
 
-To do so, the bridge circuit as shown on figure [15](#figure--fig:bridge-circuit) can be used.
+To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used.
 
 We can show that
 
@@ -655,7 +655,7 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
 ### Smart Materials {#smart-materials}
 
 Smart materials have the ability to respond significantly to stimuli of different physical nature.
-Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
+[Figure 16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
 
 <a id="figure--fig:smart-materials"></a>
 
@@ -706,7 +706,7 @@ With:
 
 #### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
 
-The set of equations <eq:piezo_eq> can be written in a matrix form:
+The set of equations \eqref{eq:piezo\_eq} can be written in a matrix form:
 
 \begin{equation}
 \begin{bmatrix}D\\\S\end{bmatrix}
@@ -748,7 +748,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
 
 </div>
 
-If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
+If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo\_eq\_matrix\_bis} over the volume of the transducer:
 
 \begin{equation}
 \begin{bmatrix}Q\\\\Delta\end{bmatrix}
@@ -773,7 +773,7 @@ where
 
 {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}}
 
-Equation <eq:piezo_stack_eq> can be inverted to obtain
+Equation \eqref{eq:piezo\_stack\_eq} can be inverted to obtain
 
 \begin{equation}
 \begin{bmatrix}V\\\f\end{bmatrix}
@@ -789,7 +789,7 @@ Equation <eq:piezo_stack_eq> can be inverted to obtain
 
 #### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
 
-Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
+Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on [Figure 18](#figure--fig:piezo-discrete).
 
 The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
 
@@ -801,7 +801,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
 
 {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}}
 
-By integrating equation <eq:piezo_work> and using the constitutive equations <eq:piezo_stack_eq_inv>, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
+By integrating equation \eqref{eq:piezo\_work} and using the constitutive equations \eqref{eq:piezo\_stack\_eq\_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
 
 \begin{equation}
   W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@@ -831,7 +831,7 @@ The ratio between the remaining stored energy and the initial stored energy is
 
 #### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
 
-Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
+Consider the system of [Figure 19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
 The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
 
 <a id="figure--fig:piezo-stack-admittance"></a>
@@ -853,7 +853,7 @@ And one can see that
   \frac{z^2 - p^2}{z^2} = k^2
 \end{equation}
 
-Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
+Equation \eqref{eq:distance\_p\_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
 
 <a id="figure--fig:piezo-admittance-curve"></a>
 
@@ -1552,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
 
 When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
 
-The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in [Figure 21](#figure--fig:hac-lac-control).
 The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
 This approach has the following advantages:
 

content/book/schmidt20_desig_high_perfor_mechat_third_revis_edition.md

@@ -1,19 +1,19 @@
 +++
 title = "The design of high performance mechatronics - third revised edition"
-author = ["Thomas Dehaeze"]
+author = ["Dehaeze Thomas"]
 description = "Awesome book that gives great overview of high performance mechatronic systems"
 keywords = ["Metrology", "Mechatronics", "Control"]
 draft = false
 +++
 
 Tags
-: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
+: [Reference Books]({{< relref "reference_books.md" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
 
 Reference
-: ([Schmidt, Schitter, and Rankers 2020](#org4e5c703))
+: (<a href="#citeproc_bib_item_1">Schmidt, Schitter, and Rankers 2020</a>)
 
 Author(s)
-: Schmidt, R. M., Schitter, G., & Rankers, A.
+: Schmidt, R. M., Schitter, G., &amp; Rankers, A.
 
 Year
 : 2020
@@ -66,9 +66,9 @@ Year
 
 #### Electric Field {#electric-field}
 
-<a id="org16b370d"></a>
+<a id="figure--fig:schmidt20-electrical-field"></a>
 
-{{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="Figure 1: Charges have an electric field" >}}
+{{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="<span class=\"figure-number\">Figure 1: </span>Charges have an electric field" >}}
 
 
 ##### Potential Difference and Capacitance {#potential-difference-and-capacitance}
@@ -172,14 +172,13 @@ The term "rms" refers to Root Mean Square, named from the action of taking the r
 The RMS value is a well known term used to characterize the "useful" value of the energy supply with a signal by comparing it with an equivalent DC voltage that would cause the same power in a resistive load.
 
 <div class="exampl">
-  <div></div>
 
 For a sinusoidal signal \\(V(t) = V\_p \sin(\omega t)\\), the equivalent DC voltage becomes:
 
 \begin{equation}
 \begin{aligned}
-V\_{\text{rms}} &= \sqrt{\frac{1}{T} \int\_0^T \left< V\_p\sin(\omega t) \right>^2 dt} \\\\\\
+V\_{\text{rms}} &= \sqrt{\frac{1}{T} \int\_0^T \left< V\_p\sin(\omega t) \right>^2 dt} \\\\
-               &= \dots \\\\\\
+               &= \dots \\\\
                &= \frac{V\_p}{\sqrt{2}} \quad [V]
 \end{aligned}
 \end{equation}
@@ -213,7 +212,7 @@ Its inverse, the spatial period length called _wavelength_ \\(\lambda\\) [m] is
 The relation between the above defined terms is:
 
 \begin{align}
-  \lambda &= \frac{1}{\nu} = c\_p T = \frac{c\_p}{f} \quad [m] \\\\\\
+  \lambda &= \frac{1}{\nu} = c\_p T = \frac{c\_p}{f} \quad [m] \\\\
   v\_p     &= \frac{f}{\nu} \quad [m/s]
 \end{align}
 
@@ -221,11 +220,11 @@ The relation between the above defined terms is:
 ##### Mechanical Waves {#mechanical-waves}
 
 The propagation speed value of a mechanical wave is mostly determined by the density and elasticity of the medium.
-The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#orgaa33fb8).
+The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#figure--fig:schmidt20-mechanical-wave).
 
-<a id="orgaa33fb8"></a>
+<a id="figure--fig:schmidt20-mechanical-wave"></a>
 
-{{< figure src="/ox-hugo/schmidt20_mechanical_wave.svg" caption="Figure 2: Lumped element model of one wavelength of a mechanical wave." >}}
+{{< figure src="/ox-hugo/schmidt20_mechanical_wave.svg" caption="<span class=\"figure-number\">Figure 2: </span>Lumped element model of one wavelength of a mechanical wave." >}}
 
 To explain the principle of energy transfer, the longitudinal wave is taken as example.
 When a movement of mass \\(m\_1\\) is introduced in the propagation direction of the chain, this will first cause a compression of the elastic coupling \\(k\_1\\).
@@ -234,7 +233,6 @@ This process is repeated over the total chain until the original movement reache
 With this mechanism the kinetic energy from mass \\(m\_1\\) is converted into potential energy in \\(k\_1\\), which in turn is transferred into kinetic energy of \\(m\_2\\), and so on.
 
 <div class="important">
-  <div></div>
 
 This phenomenon of transfer of energy in an elastic body is important in mechatronic systems because driving forces are also transported through the body as a wave and as a consequence will experience a delay between the actuator and the sensor when they are located separately.
 
@@ -247,7 +245,6 @@ The propagation speed \\(c\_p\\) is determined by the density \\(\rho\_m\\) and
 \end{equation}
 
 <div class="exampl">
-  <div></div>
 
 The propagation speed in stainless steel varies between 3500 m/s for transversal waves and 5500 m/s for longitudinal waves.
 With for instance half a meter of steel this gives a delay of about 0.1ms, resulting in a phase delay of 36 degrees at 1kHz, which can be significant from a control point of view.
@@ -384,7 +381,7 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
 
 <a id="table--tab:relation-slope-decade"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:relation-slope-decade">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:relation-slope-decade">Table 1</a>:</span>
   Relation between the order of the slope of a bode plot and the magnitude ration in dB, amplitude ratio and power ration, per <b>decade</b> (\(f_1 = 10 f_2\))
 </div>
 
@@ -398,7 +395,7 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
 
 <a id="table--tab:relation-slope-octave"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:relation-slope-octave">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:relation-slope-octave">Table 2</a>:</span>
   Relation between the order of the slope of a bode plot and the magnitude ration in dB, amplitude ratio and power ration, per <b>octave</b> (\(f_1 = 2 f_2\))
 </div>
 
@@ -589,7 +586,8 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
 
 ### Summary on Dynamics {#summary-on-dynamics}
 
-<summary>
+<div class="sum">
+
 In this chapter some important lessons have been learned, which are summarised as follows:
 
 -   Stiffness, whether it is created mechanically or by means of a control system, is determinative for precision
@@ -604,7 +602,8 @@ In this chapter some important lessons have been learned, which are summarised a
 -   Modal analysis is a powerful and widely applied tool to investigate the dynamics of a mechanical structure.
 
 Finally it can be concluded, that these insights help in designing actively controlled dynamic motion systems with optimally located actuators and sensors, which reduce the sensitivity for modal dynamic problems.
-</summary>
+
+</div>
 
 
 ## Motion Control {#motion-control}
@@ -612,15 +611,15 @@ Finally it can be concluded, that these insights help in designing actively cont
 
 ### A Walk around the Control Loop {#a-walk-around-the-control-loop}
 
-Figure [3](#org6432052) shows a basic control loop of a positioning system.
+Figure [3](#figure--fig:schmidt20-walk-control-loop) shows a basic control loop of a positioning system.
 First, the A/D and D/A converters are used to translate analog signals into time-discrete digital signals and vice versa.
 Secondly, the impact locations of several disturbances are shown, which play a large role in determining what reqwuirements the controller needs to fulfil.
 The core of the control system is the _plant_, which is the physical system that needs to be controlled.
 
 <a id="table--tab:walk-control-loop"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:walk-control-loop">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:walk-control-loop">Table 3</a>:</span>
-  Symbols used in Figure <a href="#org6432052">10</a>
+  Symbols used in Figure <a href="#org4b1b612">3</a>
 </div>
 
 | Symbol     | Meaning                              | Unit |
@@ -634,16 +633,16 @@ The core of the control system is the _plant_, which is the physical system that
 | \\(y\\)    | Measured output motion               | [m]  |
 | \\(y\_m\\) | Measurement value                    | [m]  |
 
-<a id="org6432052"></a>
+<a id="figure--fig:schmidt20-walk-control-loop"></a>
 
-{{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="Figure 3: Block diagram of a motion control system, including feedforward and feedback control." >}}
+{{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="<span class=\"figure-number\">Figure 3: </span>Block diagram of a motion control system, including feedforward and feedback control." >}}
 
-The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [4](#org21aa9c3)).
+The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [4](#figure--fig:schmidt20-energy-actuator-system)).
 They interact in both directions in such a way that each element not only determines the input of the next element, but also influences the previous element by its dynamic load.
 
-<a id="org21aa9c3"></a>
+<a id="figure--fig:schmidt20-energy-actuator-system"></a>
 
-{{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="Figure 4: The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
+{{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="<span class=\"figure-number\">Figure 4: </span>The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
 
 
 #### Poles and Zeros in Motion Control {#poles-and-zeros-in-motion-control}
@@ -672,7 +671,7 @@ Fortunately the effect is mostly so small that it can be neglected.
 
 #### Overview Feedforward Control {#overview-feedforward-control}
 
-Figure [5](#org4b3a329) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
+Figure [5](#figure--fig:schmidt20-feedforward-control-diagram) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
 
 The reference signal \\(r\\) [m] is applied to the controller, which as a reference transfer function \\(C\_{ff}(s)\\) in [N/m].
 The output \\(u\\) in [N] of the controller is connected to the input of the motion system, which has a transfer function \\(G(s)\\) in [m/N] giving the output \\(x\\) in [m].
@@ -684,12 +683,11 @@ If one would like to achieve perfect control, which means that there is no diffe
 G\_{t,ff}(s) = \frac{x}{r} = C\_{ff}(s)G(s) = 1 \quad \Longrightarrow \quad C\_{ff}(s) = G^{-1}(s)
 \end{equation}
 
-<a id="org4b3a329"></a>
+<a id="figure--fig:schmidt20-feedforward-control-diagram"></a>
 
-{{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="Figure 5: Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
+{{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="<span class=\"figure-number\">Figure 5: </span>Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
 
 <div class="important">
-  <div></div>
 
 Feedforward control is a very useful and preferred first step in the control of a complex dynamic motion system as it provides the following advantages:
 
@@ -702,7 +700,6 @@ Feedforward control is a very useful and preferred first step in the control of
 </div>
 
 <div class="important">
-  <div></div>
 
 The drawbacks and limitations of feedforward control are:
 
@@ -718,21 +715,20 @@ The drawbacks and limitations of feedforward control are:
 
 In feedback control the actuator status of the motion system is monitored by a sensor and the controller generates a control action based on the difference between the desired motion (reference signal) and the actuator system status (sensor signal).
 
-The block diagram of Figure [6](#org3bccc77) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
+The block diagram of Figure [6](#figure--fig:schmidt20-feedback-control-diagram) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
 The output \\(x\\) in [m] is the total motion of the plant on all its parts and details, while \\(y\\) is the measured motion with a measured value \\(y\_m\\) measured on a selected location in the plant.
 This measured is compared with \\(r\_f\\), which is the reference \\(r\\) after filtering.
 The result of this comparison is used as input for the feedback controller.
 
 <div class="note">
-  <div></div>
 
 The transfer function of any input to any output in a closed-loop feedback controlled dynamic system is equal to the forward path from the input to the output divided by one plus the transfer function of the total feedback path.
 
 </div>
 
-<a id="org3bccc77"></a>
+<a id="figure--fig:schmidt20-feedback-control-diagram"></a>
 
-{{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="Figure 6: Block diagram of a SISO feedback controlled motion system." >}}
+{{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram of a SISO feedback controlled motion system." >}}
 
 In control design, one has the freedom to choose \\(F(s)\\) and particularly \\(C\_{fb}(s)\\) such that the total transfer function fulfills the desired specifications.
 Feedback control allows to directly place the system poles at values that are more useful for the operation of the motion system that their natural locations.
@@ -743,7 +739,6 @@ It is mainly used to present unwanted signals from entering the system.
 This can be signals that drive the system into its "incapability" region where the system can no longer perform as required due to limitations in the hardware.
 
 <div class="important">
-  <div></div>
 
 Feedback is an addition to feedforward control with the following benefits:
 
@@ -754,7 +749,6 @@ Feedback is an addition to feedforward control with the following benefits:
 </div>
 
 <div class="important">
-  <div></div>
 
 Also, some pitfalls have to be dealt with:
 
@@ -769,9 +763,9 @@ Also, some pitfalls have to be dealt with:
 
 #### Summary {#summary}
 
-<a id="table--tab:feedback-feedforward-summary"></a>
+<a id="table--tab:feedback-feedforward-sum"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:feedback-feedforward-summary">Table 4</a></span>:
+  <span class="table-number"><a href="#table--tab:feedback-feedforward-sum">Table 4</a>:</span>
   Summary of Feedback and Feedforward control
 </div>
 
@@ -795,11 +789,11 @@ Also, some pitfalls have to be dealt with:
 #### Model-Based Feedforward Control {#model-based-feedforward-control}
 
 In the following an example of a model-based feedforward controller is introduced.
-The measured frequency-response of the scanning unit taken as as an example is shown in Figure [7](#org77061d7).
+The measured frequency-response of the scanning unit taken as as an example is shown in Figure [7](#figure--fig:schmidt20-bode-plot-scanning).
 
-<a id="org77061d7"></a>
+<a id="figure--fig:schmidt20-bode-plot-scanning"></a>
 
-{{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="Figure 7: Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
+{{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="<span class=\"figure-number\">Figure 7: </span>Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
 
 A mathematical model of a seconder-order mass-spring system with a force input is fitted to this measured response:
 
@@ -815,7 +809,7 @@ This means that the transfer function of the feedforward controller is:
 C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{\omega\_0^2}
 \end{equation}
 
-However, such controller needs to be modified in such a way that it becomes realizable.
+However, such controller needs to be modified in such a way that it becomes _realizable_.
 In this case, it is decided to create a resulting overall transfer function of the controller and the plant that acts like a well damped mass-spring system with the same natural frequency as the plant and an additional reduction of the excitation of high frequency eigen-modes.
 In order to realize this controller, first two poles have to be added:
 
@@ -835,17 +829,17 @@ C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0)(s^2
 Then this controller is connected in series with the scanning unit, the anti-resonance of the controller and the resonance of the piezo-scanner cancel each other out:
 
 \begin{align}
-G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\\\
+G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\
-            &= \frac{C\_f}{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\\\
+            &= \frac{C\_f}{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\
             &= \frac{C\_f}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}}
 \end{align}
 
-The bode plot of the resulting dynamics is shown in Figure [8](#org1f513e4).
+The bode plot of the resulting dynamics is shown in Figure [8](#figure--fig:schmidt20-bode-plot-feedfoward-example).
 The controlled system has low-pass characteristics, rolling of at the scanner's natural frequency.
 
-<a id="org1f513e4"></a>
+<a id="figure--fig:schmidt20-bode-plot-feedfoward-example"></a>
 
-{{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="Figure 8: Bode plot of the feedforward-controlled scanning unit" >}}
+{{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="<span class=\"figure-number\">Figure 8: </span>Bode plot of the feedforward-controlled scanning unit" >}}
 
 
 #### Input-Shaping {#input-shaping}
@@ -861,16 +855,16 @@ The oscillation caused by each individual step are 180 degrees out of phase and
 This method is clearly very different form pole-zero cancellation.
 In the frequency domain, these sampled adaptations to the input create a frequency spectrum with a multiple of notch filters at the harmonic of the frequency where these adaptations are applied.
 
-Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [9](#orgddd25e4).
+Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [9](#figure--fig:schmidt20-input-shaping-example).
 
-<a id="orgddd25e4"></a>
+<a id="figure--fig:schmidt20-input-shaping-example"></a>
 
-{{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="Figure 9: Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
+{{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="<span class=\"figure-number\">Figure 9: </span>Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
 
 
 #### Adaptive Feedforward Control {#adaptive-feedforward-control}
 
-Both examples of feedforward control, the model-based pole-zero cancellation and the input-shaping, only work reliably as long as the dynamic properties of the total plant are known and remain constant.
+Both examples of feedforward control, the model-based pole-zero cancellation and the input-shaping, only work reliably as long as the dynamic properties of the total plant are known and _remain constant_.
 There is always some deviation between the parameters in the model and the reality.
 This deviation can be partly solved by _adaptive feedforward control_, adapting the feedforward signal by measuring the real behavior of the system.
 This method requires a sensor to obtain information about the response of the system and for that reason it is often applied in combination with feedback.
@@ -891,13 +885,13 @@ The limitations of the actuators and electronics in a controlled motion system a
 Of at least the levels of Jerk and preferable also Snap should be limited.
 The standard method to cope with these limitations involves shaping the input of a mechatronic motion system by means of _trajectory profile generation_ or _path-planning_.
 
-Figure [10](#orge30b109) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
+Figure [10](#figure--fig:schmidt20-trajectory-profile) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
 A third order trajectory would show a square profile for the jerk indicating an infinite Snap and the round of the acceleration would be gone.
 A second order trajectory would show a square acceleration profile with infinite Jerk and sharp edges on the velocity.
 
-<a id="orge30b109"></a>
+<a id="figure--fig:schmidt20-trajectory-profile"></a>
 
-{{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="Figure 10: Figure caption" >}}
+{{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="<span class=\"figure-number\">Figure 10: </span>Figure caption" >}}
 
 
 ### Feedback Control {#feedback-control}
@@ -915,29 +909,29 @@ Feedback control is more complex and critical to design than feedforward control
 In general, a feedback controlled motion system is to perform a certain predetermined motion task defined by the reference input \\(r\\), while reducing the effects of other inputs like external vibrations and noise from the electronics.
 All these input signals, whether desired of undesired, are treated by the feedback loop as disturbances and it is the sensitivity of the desired output signal to all input signals that determine the performance of the feedback controller.
 
-<a id="org39f635c"></a>
+<a id="figure--fig:schmidt20-feedback-full-simplified"></a>
 
-{{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="Figure 11: Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
+{{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="<span class=\"figure-number\">Figure 11: </span>Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
 
 Several standard sensitivity functions have been defined to quantify the performance of feedback controlled dynamic systems.
-There are derived from a simplified version of the generic feedback loop as shown in Figure [11](#org39f635c).
+There are derived from a simplified version of the generic feedback loop as shown in Figure [11](#figure--fig:schmidt20-feedback-full-simplified).
 The first simplification is made by approximating the measurement system to have a unity-gain transfer function.
 For further simplification the sensor disturbance in the measurement system is included in the output disturbance \\(n\\), thereby defining the output of the system \\(y\\) as the measured output.
 With this simplified model, the transfer functions of the different inputs of the system to three relevant output variables in the loop are written down in a set of equations.
-Six different transfer functions are obtained and summarized in equation \eqref{eq:gang_of_six}.
+Six different transfer functions are obtained and summarized in equation <eq:gang_of_six>.
 
 \begin{equation} \label{eq:gang\_of\_six}
 \begin{aligned}
- \frac{x}{r} &= \frac{y}{r} = \frac{GCF}{1 + GC} \\\\\\
+ \frac{x}{r} &= \frac{y}{r} = \frac{GCF}{1 + GC} \\\\
--\frac{x}{n} &= -\frac{u}{d} = \frac{GC}{1 + GC} \\\\\\
+-\frac{x}{n} &= -\frac{u}{d} = \frac{GC}{1 + GC} \\\\
- \frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\\\
+ \frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\
- \frac{u}{r} &= \frac{CF}{1 + GC} \\\\\\
+ \frac{u}{r} &= \frac{CF}{1 + GC} \\\\
- \frac{u}{n} &= \frac{C}{1 + GC} \\\\\\
+ \frac{u}{n} &= \frac{C}{1 + GC} \\\\
  \frac{y}{n} &= \frac{1}{1 + GC}
 \end{aligned}
 \end{equation}
 
-In case no input filter is applied \\(F\\) is equal to one and the set of six equations is reduced to a set of four equations as shown in equation \eqref{eq:gang_of_four}.
+In case no input filter is applied \\(F\\) is equal to one and the set of six equations is reduced to a set of four equations as shown in equation <eq:gang_of_four>.
 This short set of equations also corresponds to the situation without a reference signal.
 
 The most important transfer function is named the _Sensitivity Function_ (no unit):
@@ -968,9 +962,9 @@ For that reason the most relevant motion system performance criteria are the Sen
 
 \begin{equation} \label{eq:gang\_of\_four}
 \boxed{\begin{aligned}
-\frac{x}{r} &= \frac{y}{r} = -\frac{x}{n} = -\frac{u}{d} = \frac{GC}{1 + GC} \\\\\\
+\frac{x}{r} &= \frac{y}{r} = -\frac{x}{n} = -\frac{u}{d} = \frac{GC}{1 + GC} \\\\
-\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\\\
+\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\
-\frac{u}{r} &= \frac{u}{n} = \frac{C}{1 + GC} \\\\\\
+\frac{u}{r} &= \frac{u}{n} = \frac{C}{1 + GC} \\\\
 \frac{y}{n} &= \frac{1}{1 + GC}
 \end{aligned}}
 \end{equation}
@@ -1001,17 +995,17 @@ To achieve sufficient robustness against instability in closed-loop feedback con
 
 The condition for robustness of closed-loop stability is that the total phase-lag of the **total feedback-loop**, consisting of the feedback controller in series with the mechatronic system, must be less than 180 degrees in the frequency region of the _unity-gain cross-over frequency_.
 
-The Nyquist plot of the feedback loop, like the example shown in Figure [12](#org6e48553), is most appropriate to analyze the robustness on stability of a feedback system.
+The Nyquist plot of the feedback loop, like the example shown in Figure [12](#figure--fig:schmidt20-nyquist-plot-stable), is most appropriate to analyze the robustness on stability of a feedback system.
 It is an analysis tool that shows the frequency response of the **feedback-loop** combining magnitude and phase in one plot.
 In this figure, two graphs are shown, designed for a different purpose.
 The first graph from the left shows margin circles related to the capability of the closed-loop feedback controlled system to follow a reference according to the complementary sensitivity.
 The second graph shows a margin circle related to the capability of the closed-loop feedback controlled system to suppress disturbances according to the sensitivity function.
 
-<a id="org6e48553"></a>
+<a id="figure--fig:schmidt20-nyquist-plot-stable"></a>
 
-{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="Figure 12: Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
+{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="<span class=\"figure-number\">Figure 12: </span>Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
 
-Three values are shown in Figure [12](#org6e48553) related to the robustness of the closed-loop feedback system:
+Three values are shown in Figure [12](#figure--fig:schmidt20-nyquist-plot-stable) related to the robustness of the closed-loop feedback system:
 
 -   **The gain margin** determines by which factor the feedback loop gain additionally can increase before the closed-loop goes unstable.
 -   **The phase margin** determines how much additional phase-lab at the unity-gain cross-over frequency is acceptable before the closed-loop system becomes unstable.
@@ -1024,14 +1018,14 @@ Higher margins corresponds to a higher level of damping.
 The Nyquist plot has one significant disadvantage as it does not show directly the frequency along the plot.
 For that reason many designers prefer to use the Bode plot.
 
-Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [13](#orgc932364).
+Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [13](#figure--fig:schmidt20-phase-gain-margin-bode).
 
 In many not too complicated cases, these two margins are sufficient to tune a feedback motion controller.
 In more complicated control systems, it remains useful to also use the Nyquist plot as it also gives the Modulus margin.
 
-<a id="orgc932364"></a>
+<a id="figure--fig:schmidt20-phase-gain-margin-bode"></a>
 
-{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="Figure 13: The gain and phase margin in the Bode plot" >}}
+{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="<span class=\"figure-number\">Figure 13: </span>The gain and phase margin in the Bode plot" >}}
 
 
 ### PID Feedback Control {#pid-feedback-control}
@@ -1152,11 +1146,11 @@ However, analogue controllers have three important disadvantages:
 
 The digital implementation of filters overcome these problems as well as allows more complex algorithm such as adaptive control, real-time optimization, nonlinear control and learning control methods.
 
-In Figure [14](#org4b95e2a) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
+In Figure [14](#figure--fig:schmidt20-digital-implementation) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
 
-<a id="org4b95e2a"></a>
+<a id="figure--fig:schmidt20-digital-implementation"></a>
 
-{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="Figure 14: Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
+{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="<span class=\"figure-number\">Figure 14: </span>Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
 
 Anti-aliasing filter is needed at the input of the ADC to limit the frequency range at the input to less than half the sampling frequency, according to the Nyquist-Shannon sampling theorem.
 
@@ -1176,7 +1170,7 @@ Fixed point arithmetic has been favored in the past, because of the less complex
 A main drawback is, that the developer must pay attention to truncation, overflow, underflow and round-off errors that occur during mathematical operations.
 Fixed points numbers are equally spaced over the whole range, separated by the gap which is denoted by the least significant bit.
 The two's complement is the most used format for representing positive and negative numbers.
-For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [15](#orgf73916a)).
+For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [15](#figure--fig:schmidt20-digital-number-representation)).
 \\(m\\) denotes the number of integer bits and \\(n\\) denotes the number of fractional bits.
 \\(m+n+1=N\\) bits are necessary to store a signed \\(Q\_{m,n}\\) number.
 If the binary representation is given, the decimal value can be calculated to:
@@ -1185,11 +1179,11 @@ If the binary representation is given, the decimal value can be calculated to:
 x = \frac{1}{2^n} \left( -2^{N-1 }b\_{N-1} + \sum\_{i=0}^{N-2} 2^i b\_i \right)
 \end{equation}
 
-where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [15](#orgf73916a).
+where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [15](#figure--fig:schmidt20-digital-number-representation).
 
-<a id="orgf73916a"></a>
+<a id="figure--fig:schmidt20-digital-number-representation"></a>
 
-{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="Figure 15: Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
+{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="<span class=\"figure-number\">Figure 15: </span>Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
 
 Floating point arithmetic has a higher dynamic range than fixed point arithmetic, given by the largest and smallest number that can be represented, has a higher precision due to the smaller gaps between adjacent numbers, less quantization noise, and it is easier to handle in terms of programming.
 A floating point number is represented by a multiplication of a _mantissa_ \\(M\\) with a _base_ \\(b\\) to the power of the _exponent_ \\(q\\):
@@ -1209,41 +1203,41 @@ x = -1^i M 2^{E-127}
 The term \\(E\\) in the exponent is stored as a positive number ranging from \\(0 \le E < 256\\) with 8 bits.
 An offset of \\(-127\\) is added in order to allow very small to very large numbers.
 The decimal value is normalized, meaning that only one nonzero digit is noted at the left of the decimal point.
-The storage register is divided into three groups, as shown in Figure [15](#orgf73916a).
+The storage register is divided into three groups, as shown in Figure [15](#figure--fig:schmidt20-digital-number-representation).
 1 bit represents the sign, the exponent term \\(E\\) is represented by 8 bits, and the mantissa is stored in 23 bits.
 
 
 #### Digital Filter Theory {#digital-filter-theory}
 
-<a id="org3a2480f"></a>
+<a id="figure--fig:schmidt20-s-z-planes"></a>
 
-{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="Figure 16: Corresponding points and area in s and z planes" >}}
+{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="<span class=\"figure-number\">Figure 16: </span>Corresponding points and area in s and z planes" >}}
 
 
 #### Finite Impulse Response (FIR) Filter {#finite-impulse-response--fir--filter}
 
-<a id="org4983bdd"></a>
+<a id="figure--fig:schmidt20-transversal-filter-structure"></a>
 
-{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="Figure 17: Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
+{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="<span class=\"figure-number\">Figure 17: </span>Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
 
-<a id="org936becf"></a>
+<a id="figure--fig:schmidt20-optimized-fir-filter-structure"></a>
 
-{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="Figure 18: Optimized FIR filter structure with symmetric filter coefficients" >}}
+{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="<span class=\"figure-number\">Figure 18: </span>Optimized FIR filter structure with symmetric filter coefficients" >}}
 
-<a id="org8ea00c7"></a>
+<a id="figure--fig:schmidt20-dir-filter-cascaded-sos"></a>
 
-{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="Figure 19: Higher-order FIR filter realization with cascade SOS filter structures" >}}
+{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="<span class=\"figure-number\">Figure 19: </span>Higher-order FIR filter realization with cascade SOS filter structures" >}}
 
 
 #### Infinite Impulse Response (IIR) Filter {#infinite-impulse-response--iir--filter}
 
-<a id="org696a8aa"></a>
+<a id="figure--fig:schmidt20-irr-structure"></a>
 
-{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="Figure 20: (a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
+{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="<span class=\"figure-number\">Figure 20: </span>(a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
 
-<a id="orge99cdac"></a>
+<a id="figure--fig:schmidt20-irr-sos-structure"></a>
 
-{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="Figure 21: IIR SOS structure in DF-2 realization" >}}
+{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="<span class=\"figure-number\">Figure 21: </span>IIR SOS structure in DF-2 realization" >}}
 
 
 #### Converting Continuous to Discrete-Time Filters {#converting-continuous-to-discrete-time-filters}
@@ -1275,7 +1269,8 @@ The storage register is divided into three groups, as shown in Figure [15](#orgf
 
 ### Conclusion on Motion Control {#conclusion-on-motion-control}
 
-<summary>
+<div class="sum">
+
 Motion control is essential for Precision Mechatronic Systems and consists of two complementary elements:
 
 -   **Extremely accurate Feedforward Control** is required when the motion system must execute a user defined motion to within maximum user defined position error limits.
@@ -1283,7 +1278,8 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
 -   **High Performance Feedback Control** is required when the motion system must be able to follow an unknown motion of a target, stabilize an otherwise unstable system and reduce the impact of disturbing forces and vibrations, such that the position error remains below a maximum user defined level.
     Due to the fact that a feedback controller can become unstable, sufficient robustness must be guaranteed.
     These is a conflicting relation between stability and performance.
-</summary>
+
+</div>
 
 
 ## Electromechanic Actuators {#electromechanic-actuators}
@@ -2237,4 +2233,6 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
 
 ## Bibliography {#bibliography}
 
-<a id="org4e5c703"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
+</div>

content/book/skogestad07_multiv_feedb_contr.md

@@ -56,7 +56,7 @@ draft = false
 
 ## Introduction {#introduction}
 
-<span class="org-target" id="org-target--sec:introduction"></span>
+<span class="org-target" id="org-target--sec-introduction"></span>
 
 
 ### The Process of Control System Design {#the-process-of-control-system-design}
@@ -183,11 +183,11 @@ In order to obtain a linear model from the "first-principle", the following appr
 
 ### Notation {#notation}
 
-Notations used throughout this note are summarized in tables [1](#table--tab:notation-conventional), [2](#table--tab:notation-general) and [3](#table--tab:notation-tf).
+Notations used throughout this note are summarized in [Table 1](#table--tab:notation-conventional), [Table 2](#table--tab:notation-general) and [Table 3](#table--tab:notation-tf).
 
 <a id="table--tab:notation-conventional"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notation-conventional">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:notation-conventional">Table 1</a>:</span>
   Notations for the conventional control configuration
 </div>
 
@@ -204,7 +204,7 @@ Notations used throughout this note are summarized in tables [1](#table--tab:not
 
 <a id="table--tab:notation-general"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notation-general">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:notation-general">Table 2</a>:</span>
   Notations for the general configuration
 </div>
 
@@ -218,7 +218,7 @@ Notations used throughout this note are summarized in tables [1](#table--tab:not
 
 <a id="table--tab:notation-tf"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:notation-tf">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:notation-tf">Table 3</a>:</span>
   Notations for transfer functions
 </div>
 
@@ -231,7 +231,7 @@ Notations used throughout this note are summarized in tables [1](#table--tab:not
 
 ## Classical Feedback Control {#classical-feedback-control}
 
-<span class="org-target" id="org-target--sec:classical_feedback"></span>
+<span class="org-target" id="org-target--sec-classical-feedback"></span>
 
 
 ### Frequency Response {#frequency-response}
@@ -272,7 +272,7 @@ We note \\(N(\w\_0) = \left( \frac{d\ln{|G(j\w)|}}{d\ln{\w}} \right)\_{\w=\w\_0}
 
 #### One Degree-of-Freedom Controller {#one-degree-of-freedom-controller}
 
-The simple one degree-of-freedom controller negative feedback structure is represented in Fig.&nbsp;[1](#figure--fig:classical-feedback-alt).
+The simple one degree-of-freedom controller negative feedback structure is represented in [Figure 1](#figure--fig:classical-feedback-alt).
 
 The input to the controller \\(K(s)\\) is \\(r-y\_m\\) where \\(y\_m = y+n\\) is the measured output and \\(n\\) is the measurement noise.
 Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
@@ -592,13 +592,13 @@ For reference tracking, we typically want the controller to look like \\(\frac{1
 
 We cannot achieve both of these simultaneously with a single feedback controller.
 
-The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig.&nbsp;[2](#figure--fig:classical-feedback-2dof-alt)), rather than operating on their difference \\(r - y\_m\\).
+The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller ([Figure 2](#figure--fig:classical-feedback-2dof-alt)), rather than operating on their difference \\(r - y\_m\\).
 
 <a id="figure--fig:classical-feedback-2dof-alt"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="<span class=\"figure-number\">Figure 2: </span>2 degrees-of-freedom control architecture" >}}
 
-The controller can be slit into two separate blocks (Fig.&nbsp;[3](#figure--fig:classical-feedback-sep)):
+The controller can be slit into two separate blocks ([Figure 3](#figure--fig:classical-feedback-sep)):
 
 -   the **feedback controller** \\(K\_y\\) that is used to **reduce the effect of uncertainty** (disturbances and model errors)
 -   the **prefilter** \\(K\_r\\) that **shapes the commands** \\(r\\) to improve tracking performance
@@ -672,7 +672,7 @@ Which can be expressed as an \\(\mathcal{H}\_\infty\\):
   W\_P(s) = \frac{s/M + \w\_B^\*}{s + \w\_B^\* A}
 \end{equation\*}
 
-With (see Fig.&nbsp;[4](#figure--fig:performance-weigth)):
+With (see [Figure 4](#figure--fig:performance-weigth)):
 
 -   \\(M\\): maximum magnitude of \\(\abs{S}\\)
 -   \\(\w\_B\\): crossover frequency
@@ -714,7 +714,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
 
 ## Introduction to Multivariable Control {#introduction-to-multivariable-control}
 
-<span class="org-target" id="org-target--sec:multivariable_control"></span>
+<span class="org-target" id="org-target--sec-multivariable-control"></span>
 
 
 ### Introduction {#introduction}
@@ -750,7 +750,7 @@ The main rule for evaluating transfer functions is the **MIMO Rule**: Start from
 
 #### Negative Feedback Control Systems {#negative-feedback-control-systems}
 
-For negative feedback system (Fig.&nbsp;[5](#figure--fig:classical-feedback-bis)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
+For negative feedback system ([Figure 5](#figure--fig:classical-feedback-bis)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant:
 
 -   \\(L = G K\\)
 -   \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
@@ -1109,7 +1109,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
 
 ### General Control Problem Formulation {#general-control-problem-formulation}
 
-The general control problem formulation is represented in Fig.&nbsp;[6](#figure--fig:general-control-names) (introduced in (<a href="#citeproc_bib_item_1">Doyle 1983</a>)).
+The general control problem formulation is represented in [Figure 6](#figure--fig:general-control-names) (introduced in (<a href="#citeproc_bib_item_1">Doyle 1983</a>)).
 
 <a id="figure--fig:general-control-names"></a>
 
@@ -1141,7 +1141,7 @@ Then we have to break all the "loops" entering and exiting the controller \\(K\\
 
 #### Controller Design: Including Weights in \\(P\\) {#controller-design-including-weights-in-p}
 
-In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) (Fig.&nbsp;[7](#figure--fig:general-plant-weights)).
+In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) ([Figure 7](#figure--fig:general-plant-weights)).
 We consider:
 
 -   The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
@@ -1199,7 +1199,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
 
 #### A General Control Configuration Including Model Uncertainty {#a-general-control-configuration-including-model-uncertainty}
 
-The general control configuration may be extended to include model uncertainty as shown in Fig.&nbsp;[8](#figure--fig:general-config-model-uncertainty).
+The general control configuration may be extended to include model uncertainty as shown in [Figure 8](#figure--fig:general-config-model-uncertainty).
 
 <a id="figure--fig:general-config-model-uncertainty"></a>
 
@@ -1228,7 +1228,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
 
 ## Elements of Linear System Theory {#elements-of-linear-system-theory}
 
-<span class="org-target" id="org-target--sec:linear_sys_theory"></span>
+<span class="org-target" id="org-target--sec-linear-sys-theory"></span>
 
 
 ### System Descriptions {#system-descriptions}
@@ -1595,14 +1595,14 @@ RHP-zeros therefore imply high gain instability.
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="<span class=\"figure-number\">Figure 9: </span>Block diagram used to check internal stability" >}}
 
-Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in Fig.&nbsp;[9](#figure--fig:block-diagram-for-stability) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
+Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in [Figure 9](#figure--fig:block-diagram-for-stability) is **internally stable** if and only if all four closed-loop transfer matrices are stable.
 
 \begin{align\*}
  &(I+KG)^{-1} & -K&(I+GK)^{-1} \\\\
 G&(I+KG)^{-1} &   &(I+GK)^{-1}
 \end{align\*}
 
-Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in Fig.&nbsp;[9](#figure--fig:block-diagram-for-stability) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
+Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in [Figure 9](#figure--fig:block-diagram-for-stability) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable.
 
 
 ### Stabilizing Controllers {#stabilizing-controllers}
@@ -1761,7 +1761,7 @@ It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \
 
 ## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems}
 
-<span class="org-target" id="org-target--sec:perf_limit_siso"></span>
+<span class="org-target" id="org-target--sec-perf-limit-siso"></span>
 
 
 ### Input-Output Controllability {#input-output-controllability}
@@ -2234,7 +2234,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="<span class=\"figure-number\">Figure 10: </span>Feedback control system" >}}
 
-Consider the control system in Fig.&nbsp;[10](#figure--fig:classical-feedback-meas).
+Consider the control system in [Figure 10](#figure--fig:classical-feedback-meas).
 Here \\(G\_m(s)\\) denotes the measurement transfer function and we assume \\(G\_m(0) = 1\\) (perfect steady-state measurement).
 
 <div class="important">
@@ -2285,7 +2285,7 @@ The rules may be used to **determine whether or not a given plant is controllabl
 
 ## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems}
 
-<span class="org-target" id="org-target--sec:perf_limit_mimo"></span>
+<span class="org-target" id="org-target--sec-perf-limit-mimo"></span>
 
 
 ### Introduction {#introduction}
@@ -2654,7 +2654,7 @@ The issues are the same for SISO and MIMO systems, however, with MIMO systems th
 
 In practice, the difference between the true perturbed plant \\(G^\prime\\) and the plant model \\(G\\) is caused by a number of different sources.
 We here focus on input and output uncertainty.
-In multiplicative form, the input and output uncertainties are given by (see Fig.&nbsp;[12](#figure--fig:input-output-uncertainty)):
+In multiplicative form, the input and output uncertainties are given by (see [Figure 12](#figure--fig:input-output-uncertainty)):
 
 \begin{equation\*}
   G^\prime = (I + E\_O) G (I + E\_I)
@@ -2801,7 +2801,7 @@ However, the situation is usually the opposite with model uncertainty because fo
 
 ## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
 
-<span class="org-target" id="org-target--sec:uncertainty_robustness_siso"></span>
+<span class="org-target" id="org-target--sec-uncertainty-robustness-siso"></span>
 
 
 ### Introduction to Robustness {#introduction-to-robustness}
@@ -2873,7 +2873,7 @@ In most cases, we prefer to lump the uncertainty into a **multiplicative uncerta
   G\_p(s) = G(s)(1 + w\_I(s)\Delta\_I(s)); \quad \abs{\Delta\_I(j\w)} \le 1 \\, \forall\w
 \end{equation\*}
 
-which may be represented by the diagram in Fig.&nbsp;[13](#figure--fig:input-uncertainty-set).
+which may be represented by the diagram in [Figure 13](#figure--fig:input-uncertainty-set).
 
 </div>
 
@@ -2940,7 +2940,7 @@ This is of course conservative as it introduces possible plants that are not pre
 
 #### Uncertain Regions {#uncertain-regions}
 
-To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in Fig.&nbsp;[14](#figure--fig:uncertainty-region) the Nyquist plots generated by the following set of plants
+To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in [Figure 14](#figure--fig:uncertainty-region) the Nyquist plots generated by the following set of plants
 
 \begin{equation\*}
    G\_p(s) = \frac{k}{\tau s + 1} e^{-\theta s}, \quad 2 \le k, \theta, \tau \le 3
@@ -2968,7 +2968,7 @@ The disc-shaped regions may be generated by **additive** complex norm-bounded pe
   \end{aligned}
 \end{equation}
 
-At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig.&nbsp;[15](#figure--fig:uncertainty-disc-generated).
+At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in [Figure 15](#figure--fig:uncertainty-disc-generated).
 
 </div>
 
@@ -3044,7 +3044,7 @@ To simplify subsequent controller design, we select a delay-free nominal model
 \end{equation\*}
 
 To obtain \\(l\_I(\w)\\), we consider three values (2, 2.5 and 3) for each of the three parameters (\\(k, \theta, \tau\\)).
-The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) (Fig.&nbsp;[16](#figure--fig:uncertainty-weight)).
+The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) ([Figure 16](#figure--fig:uncertainty-weight)).
 To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_I(j\w)}\\) lies above all the dotted lines.
 
 </div>
@@ -3092,7 +3092,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
 
 ##### Neglected delay {#neglected-delay}
 
-Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency (Fig.&nbsp;[17](#figure--fig:neglected-time-delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
+Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency ([Figure 17](#figure--fig:neglected-time-delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
 
 \begin{equation\*}
    l\_I(\w) = \begin{cases} \abs{1 - e^{-j\w\theta\_{\text{max}}}} & \w < \pi/\theta\_{\text{max}} \\\ 2 & \w \ge \pi/\theta\_{\text{max}} \end{cases}
@@ -3105,7 +3105,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
 
 ##### Neglected lag {#neglected-lag}
 
-Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) (Fig.&nbsp;[18](#figure--fig:neglected-first-order-lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
+Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) ([Figure 18](#figure--fig:neglected-first-order-lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
 
 \begin{equation\*}
    w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1}
@@ -3131,7 +3131,7 @@ There is an exact expression, its first order approximation is
    w\_I(s) = \frac{(1+\frac{r\_k}{2})\theta\_{\text{max}} s + r\_k}{\frac{\theta\_{\text{max}}}{2} s + 1}
 \end{equation\*}
 
-However, as shown in Fig.&nbsp;[19](#figure--fig:lag-delay-uncertainty), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor:
+However, as shown in [Figure 19](#figure--fig:lag-delay-uncertainty), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor:
 
 \begin{equation\*}
    w\_I^\prime(s) = w\_I \cdot \frac{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.838 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1}{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.685 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1}
@@ -3167,7 +3167,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
 
 #### RS with Multiplicative Uncertainty {#rs-with-multiplicative-uncertainty}
 
-We want to determine the stability of the uncertain feedback system in Fig.&nbsp;[20](#figure--fig:feedback-multiplicative-uncertainty) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\).
+We want to determine the stability of the uncertain feedback system in [Figure 20](#figure--fig:feedback-multiplicative-uncertainty) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\).
 The loop transfer function becomes
 
 \begin{equation\*}
@@ -3189,7 +3189,7 @@ We use the Nyquist stability condition to test for robust stability of the close
 
 ##### Graphical derivation of RS-condition {#graphical-derivation-of-rs-condition}
 
-Consider the Nyquist plot of \\(L\_p\\) as shown in Fig.&nbsp;[21](#figure--fig:nyquist-uncertainty). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc.
+Consider the Nyquist plot of \\(L\_p\\) as shown in [Figure 21](#figure--fig:nyquist-uncertainty). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc.
 Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
 
 \begin{align\*}
@@ -3236,7 +3236,7 @@ And we obtain the same condition as before.
 
 #### RS with Inverse Multiplicative Uncertainty {#rs-with-inverse-multiplicative-uncertainty}
 
-We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig.&nbsp;[22](#figure--fig:inverse-uncertainty-set)) in which
+We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty ([Figure 22](#figure--fig:inverse-uncertainty-set)) in which
 
 \begin{equation\*}
    G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1}
@@ -3290,7 +3290,7 @@ The condition for **nominal performance** when considering performance in terms
 </div>
 
 Now \\(\abs{1 + L}\\) represents at each frequency the distance of \\(L(j\omega)\\) from the point \\(-1\\) in the Nyquist plot, so \\(L(j\omega)\\) must be at least a distance of \\(\abs{w\_P(j\omega)}\\) from \\(-1\\).
-This is illustrated graphically in Fig.&nbsp;[23](#figure--fig:nyquist-performance-condition).
+This is illustrated graphically in [Figure 23](#figure--fig:nyquist-performance-condition).
 
 <a id="figure--fig:nyquist-performance-condition"></a>
 
@@ -3312,7 +3312,7 @@ For robust performance, we require the performance condition to be satisfied for
 
 </div>
 
-Let's consider the case of multiplicative uncertainty as shown on Fig.&nbsp;[24](#figure--fig:input-uncertainty-set-feedback-weight-bis).
+Let's consider the case of multiplicative uncertainty as shown on [Figure 24](#figure--fig:input-uncertainty-set-feedback-weight-bis).
 The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
 
 \begin{equation\*}
@@ -3326,7 +3326,7 @@ The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \D
 
 ##### Graphical derivation of RP-condition {#graphical-derivation-of-rp-condition}
 
-As illustrated on Fig.&nbsp;[23](#figure--fig:nyquist-performance-condition), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\).
+As illustrated on [Figure 23](#figure--fig:nyquist-performance-condition), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\).
 Since \\(L\_p\\) at each frequency stays within a disk of radius \\(|w\_I(j\omega) L(j\omega)|\\) centered on \\(L(j\omega)\\), the condition for RP becomes:
 
 \begin{align\*}
@@ -3524,7 +3524,7 @@ In the transfer function form:
 
 with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
 
-This is illustrated in the block diagram of Fig.&nbsp;[25](#figure--fig:uncertainty-state-a-matrix), which is in the form of an inverse additive perturbation.
+This is illustrated in the block diagram of [Figure 25](#figure--fig:uncertainty-state-a-matrix), which is in the form of an inverse additive perturbation.
 
 <a id="figure--fig:uncertainty-state-a-matrix"></a>
 
@@ -3544,7 +3544,7 @@ We also derived a condition for robust performance with multiplicative uncertain
 
 ## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
 
-<span class="org-target" id="org-target--sec:robust_perf_mimo"></span>
+<span class="org-target" id="org-target--sec-robust-perf-mimo"></span>
 
 
 ### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
@@ -3562,13 +3562,13 @@ The starting point for our robustness analysis is a system representation in whi
 
 where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g. input uncertainty \\(\Delta\_I\\) or parametric uncertainty \\(\delta\_i\\).
 
-If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig.&nbsp;[26](#figure--fig:general-control-delta). This form is useful for controller synthesis.
+If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in [Figure 26](#figure--fig:general-control-delta). This form is useful for controller synthesis.
 
 <a id="figure--fig:general-control-delta"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="<span class=\"figure-number\">Figure 26: </span>General control configuration used for controller synthesis" >}}
 
-If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig.&nbsp;[27](#figure--fig:general-control-Ndelta).
+If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in [Figure 27](#figure--fig:general-control-Ndelta).
 
 <a id="figure--fig:general-control-Ndelta"></a>
 
@@ -3588,7 +3588,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
     &\triangleq N\_{22} + N\_{21} \Delta (I - N\_{11} \Delta)^{-1} N\_{12}
 \end{align\*}
 
-To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig.&nbsp;[28](#figure--fig:general-control-Mdelta-bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
+To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in [Figure 28](#figure--fig:general-control-Mdelta-bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
 
 <a id="figure--fig:general-control-Mdelta-bis"></a>
 
@@ -3627,7 +3627,7 @@ However, the inclusion of parametric uncertainty may be more significant for MIM
 Unstructured perturbations are often used to get a simple uncertainty model.
 We here define unstructured uncertainty as the use of a "full" complex perturbation matrix \\(\Delta\\), usually with dimensions compatible with those of the plant, where at each frequency any \\(\Delta(j\w)\\) satisfying \\(\maxsv(\Delta(j\w)) < 1\\) is allowed.
 
-Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nbsp;[4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
+Three common forms of **feedforward unstructured uncertainty** are shown [Table 4](#table--fig:feedforward-uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
 
 <div class="important">
 
@@ -3643,15 +3643,15 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb
 
 <a id="table--fig:feedforward-uncertainty"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--fig:feedforward-uncertainty">Table 4</a></span>:
+  <span class="table-number"><a href="#table--fig:feedforward-uncertainty">Table 4</a>:</span>
   Common feedforward unstructured uncertainty
 </div>
 
 | ![](/ox-hugo/skogestad07_additive_uncertainty.png)                                              | ![](/ox-hugo/skogestad07_input_uncertainty.png)                                                          | ![](/ox-hugo/skogestad07_output_uncertainty.png)                                                           |
 |-------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------|
-| <span class="org-target" id="org-target--fig:additive_uncertainty"></span> Additive uncertainty | <span class="org-target" id="org-target--fig:input_uncertainty"></span> Multiplicative input uncertainty | <span class="org-target" id="org-target--fig:output_uncertainty"></span> Multiplicative output uncertainty |
+| <span class="org-target" id="org-target--fig-additive-uncertainty"></span> Additive uncertainty | <span class="org-target" id="org-target--fig-input-uncertainty"></span> Multiplicative input uncertainty | <span class="org-target" id="org-target--fig-output-uncertainty"></span> Multiplicative output uncertainty |
 
-In Fig.&nbsp;[5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
+In [Table 5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
 
 <div class="important">
 
@@ -3667,13 +3667,13 @@ In Fig.&nbsp;[5](#table--fig:feedback-uncertainty), three **feedback or inverse
 
 <a id="table--fig:feedback-uncertainty"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--fig:feedback-uncertainty">Table 5</a></span>:
+  <span class="table-number"><a href="#table--fig:feedback-uncertainty">Table 5</a>:</span>
   Common feedback unstructured uncertainty
 </div>
 
 | ![](/ox-hugo/skogestad07_inv_additive_uncertainty.png)                                                      | ![](/ox-hugo/skogestad07_inv_input_uncertainty.png)                                                                  | ![](/ox-hugo/skogestad07_inv_output_uncertainty.png)                                                                   |
 |-------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------|
-| <span class="org-target" id="org-target--fig:inv_additive_uncertainty"></span> Inverse additive uncertainty | <span class="org-target" id="org-target--fig:inv_input_uncertainty"></span> Inverse multiplicative input uncertainty | <span class="org-target" id="org-target--fig:inv_output_uncertainty"></span> Inverse multiplicative output uncertainty |
+| <span class="org-target" id="org-target--fig-inv-additive-uncertainty"></span> Inverse additive uncertainty | <span class="org-target" id="org-target--fig-inv-input-uncertainty"></span> Inverse multiplicative input uncertainty | <span class="org-target" id="org-target--fig-inv-output-uncertainty"></span> Inverse multiplicative output uncertainty |
 
 
 ##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
@@ -3768,7 +3768,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
 
 ### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining-p-n-and-m}
 
-Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig.&nbsp;[29](#figure--fig:input-uncertainty-set-feedback-weight).
+Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown [Figure 29](#figure--fig:input-uncertainty-set-feedback-weight).
 \\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
 
 <a id="figure--fig:input-uncertainty-set-feedback-weight"></a>
@@ -3906,7 +3906,7 @@ Then the \\(M\Delta\text{-system}\\) is stable for all perturbations \\(\Delta\\
 
 #### Application of the Unstructured RS-condition {#application-of-the-unstructured-rs-condition}
 
-We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in Figs&nbsp;[4](#table--fig:feedforward-uncertainty) and&nbsp;[5](#table--fig:feedback-uncertainty) with
+We will now present necessary and sufficient conditions for robust stability for each of the six single unstructured perturbations in [Table 4](#table--fig:feedforward-uncertainty) and [Table 5](#table--fig:feedback-uncertainty) with
 
 \begin{equation\*}
    E = W\_2 \Delta W\_1, \quad \hnorm{\Delta} \le 1
@@ -3951,7 +3951,7 @@ In order to get tighter condition we must use a tighter uncertainty description
 Robust stability bound in terms of the \\(\hinf\\) norm (\\(\text{RS}\Leftrightarrow\hnorm{M}<1\\)) are in general only tight when there is a single full perturbation block.
 An "exception" to this is when the uncertainty blocks enter or exit from the same location in the block diagram, because they can then be stacked on top of each other or side-by-side, in an overall \\(\Delta\\) which is then full matrix.
 
-One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in Fig.&nbsp;[30](#figure--fig:coprime-uncertainty), for which the set of plants is
+One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in [Figure 30](#figure--fig:coprime-uncertainty), for which the set of plants is
 
 \begin{equation\*}
    G\_p = (M\_l + \Delta\_M)^{-1}(Nl + \Delta\_N), \quad \hnorm{[\Delta\_N, \ \Delta\_N]} \le \epsilon
@@ -4007,7 +4007,7 @@ To this effect, introduce the block-diagonal scaling matrix
 
 where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same dimension as the \\(i\\)'th perturbation block \\(\Delta\_i\\).
 
-Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig.&nbsp;[31](#figure--fig:block-diagonal-scalings).
+Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in [Figure 31](#figure--fig:block-diagonal-scalings).
 This clearly has no effect on stability.
 
 <a id="figure--fig:block-diagonal-scalings"></a>
@@ -4302,7 +4302,7 @@ Note that \\(\mu\\) underestimate how bad or good the actual worst case performa
 
 ### Application: RP with Input Uncertainty {#application-rp-with-input-uncertainty}
 
-We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity (Fig.&nbsp;[29](#figure--fig:input-uncertainty-set-feedback-weight)).
+We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity ([Figure 29](#figure--fig:input-uncertainty-set-feedback-weight)).
 
 The performance requirement is then
 
@@ -4416,7 +4416,7 @@ with the decoupling controller we have:
    \overline{\sigma}(N\_{22}) = \overline{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right|
 \end{equation\*}
 
-and we see from Fig.&nbsp;[32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
+and we see from [Figure 32](#figure--fig:mu-plots-distillation) that the NP-condition is satisfied.
 
 <a id="figure--fig:mu-plots-distillation"></a>
 
@@ -4431,7 +4431,7 @@ In this case \\(w\_I T\_I = w\_I T\\) is a scalar times the identity matrix:
    \mu\_{\Delta\_I}(w\_I T\_I) = |w\_I t| = \left|0.2 \frac{5s + 1}{(0.5s + 1)(1.43s + 1)}\right|
 \end{equation\*}
 
-and we see from Fig.&nbsp;[32](#figure--fig:mu-plots-distillation) that RS is satisfied.
+and we see from [Figure 32](#figure--fig:mu-plots-distillation) that RS is satisfied.
 
 The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may increase the uncertainty by a factor of \\(1/0.53 = 1.89\\) before the worst case uncertainty yields instability.
 
@@ -4439,7 +4439,7 @@ The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may in
 ##### RP {#rp}
 
 Although the system has good robustness margins and excellent nominal performance, the robust performance is poor.
-This is shown in Fig.&nbsp;[32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
+This is shown in [Figure 32](#figure--fig:mu-plots-distillation) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\).
 The peak value is close to 6, meaning that even with 6 times less uncertainty, the weighted sensitivity will be about 6 times larger than what we require.
 
 
@@ -4447,7 +4447,7 @@ The peak value is close to 6, meaning that even with 6 times less uncertainty, t
 
 We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller.
 We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity.
-With \\(N\\) given by <eq:n_delta_structure_clasic>, we have:
+With \\(N\\) given by \eqref{eq:n\_delta\_structure\_clasic}, we have:
 
 \begin{equation\*}
    \overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\overline{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\overline{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k})
@@ -4576,7 +4576,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
 #### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example-mu-text-synthesis-with-dk-iteration}
 
 For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
-The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig.&nbsp;[33](#figure--fig:weights-distillation).
+The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in [Figure 33](#figure--fig:weights-distillation).
 
 <a id="figure--fig:weights-distillation"></a>
 
@@ -4592,8 +4592,8 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
 
 -   Iteration No. 1.
     Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights).
-    Step 2: the upper \\(\mu\text{-bound}\\) is shown in Fig.&nbsp;[34](#figure--fig:dk-iter-mu).
+    Step 2: the upper \\(\mu\text{-bound}\\) is shown in [Figure 34](#figure--fig:dk-iter-mu).
-    Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in Fig.&nbsp;[35](#figure--fig:dk-iter-d-scale)
+    Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in [Figure 35](#figure--fig:dk-iter-d-scale)
 -   Iteration No. 2.
     Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller.
     Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\).
@@ -4609,7 +4609,7 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
 
 {{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="<span class=\"figure-number\">Figure 35: </span>Change in D-scale \\(d\_1\\) during DK-iteration" >}}
 
-The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in Fig.&nbsp;[36](#figure--fig:mu-plot-optimal-k3).
+The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in [Figure 36](#figure--fig:mu-plot-optimal-k3).
 The objectives of RS and NP are easily satisfied.
 The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\overline{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
 
@@ -4617,7 +4617,7 @@ The peak value of \\(\mu\\) is just slightly over 1, so the performance specific
 
 {{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="<span class=\"figure-number\">Figure 36: </span>\\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}}
 
-To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig.&nbsp;[37](#figure--fig:perturb-s-k3).
+To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in [Figure 37](#figure--fig:perturb-s-k3).
 
 <a id="figure--fig:perturb-s-k3"></a>
 
@@ -4686,7 +4686,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
 
 ## Controller Design {#controller-design}
 
-<span class="org-target" id="org-target--sec:controller_design"></span>
+<span class="org-target" id="org-target--sec-controller-design"></span>
 
 
 ### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
@@ -4696,7 +4696,7 @@ By multivariable transfer function shaping, therefore, we mean the shaping of th
 
 The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values.
 
-Consider the one degree-of-freedom system as shown in Fig.&nbsp;[38](#figure--fig:classical-feedback-small).
+Consider the one degree-of-freedom system as shown in [Figure 38](#figure--fig:classical-feedback-small).
 We have the following important relationships:
 
 \begin{align}
@@ -4750,7 +4750,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
 
 </div>
 
-Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in Fig.&nbsp;[39](#figure--fig:design-trade-off-mimo-gk).
+Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in [Figure 39](#figure--fig:design-trade-off-mimo-gk).
 
 <a id="figure--fig:design-trade-off-mimo-gk"></a>
 
@@ -4810,7 +4810,7 @@ The optimal state estimate is given by a **Kalman filter**.
 
 The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\).
 
-We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig.&nbsp;[40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
+We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in [Figure 40](#figure--fig:lqg-separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
 
 <a id="figure--fig:lqg-separation"></a>
 
@@ -4842,7 +4842,7 @@ and \\(X\\) is the unique positive-semi definite solution of the algebraic Ricca
 
 <div class="important">
 
-The **Kalman filter** has the structure of an ordinary state-estimator, as shown on Fig.&nbsp;[41](#figure--fig:lqg-kalman-filter), with:
+The **Kalman filter** has the structure of an ordinary state-estimator, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), with:
 
 \begin{equation} \label{eq:kalman\_filter\_structure}
   \dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x})
@@ -4866,7 +4866,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
 
 {{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="<span class=\"figure-number\">Figure 41: </span>The LQG controller and noisy plant" >}}
 
-The structure of the LQG controller is illustrated in Fig.&nbsp;[41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
+The structure of the LQG controller is illustrated in [Figure 41](#figure--fig:lqg-kalman-filter), its transfer function from \\(y\\) to \\(u\\) is given by
 
 \begin{align\*}
   L\_{\text{LQG}}(s) &= \left[ \begin{array}{c|c}
@@ -4881,7 +4881,7 @@ The structure of the LQG controller is illustrated in Fig.&nbsp;[41](#figure--fi
 
 It has the same degree (number of poles) as the plant.<br />
 
-For the LQG-controller, as shown on Fig.&nbsp;[41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig.&nbsp;[42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
+For the LQG-controller, as shown on [Figure 41](#figure--fig:lqg-kalman-filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in [Figure 42](#figure--fig:lqg-integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
 
 <a id="figure--fig:lqg-integral"></a>
 
@@ -4896,16 +4896,16 @@ Their main limitation is that they can only be applied to minimum phase plants.
 
 ### \\(\htwo\\) and \\(\hinf\\) Control {#htwo-and-hinf-control}
 
-<span class="org-target" id="org-target--sec:htwo_and_hinf"></span>
+<span class="org-target" id="org-target--sec-htwo-and-hinf"></span>
 
 
 #### General Control Problem Formulation {#general-control-problem-formulation}
 
-<span class="org-target" id="org-target--sec:htwo_inf_assumptions"></span>
+<span class="org-target" id="org-target--sec-htwo-inf-assumptions"></span>
 There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
 It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
 
-Such a general formulation is afforded by the general configuration shown in Fig.&nbsp;[43](#figure--fig:general-control).
+Such a general formulation is afforded by the general configuration shown in [Figure 43](#figure--fig:general-control).
 
 <a id="figure--fig:general-control"></a>
 
@@ -5085,7 +5085,7 @@ Then the LQG cost function is
 
 #### \\(\hinf\\) Optimal Control {#hinf-optimal-control}
 
-With reference to the general control configuration on Fig.&nbsp;[43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
+With reference to the general control configuration on [Figure 43](#figure--fig:general-control), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize
 
 \begin{equation\*}
    \hnorm{F\_l(P, K)} = \max\_{\omega} \maxsv\big(F\_l(P, K)(j\omega)\big)
@@ -5196,7 +5196,7 @@ In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can b
 This can be useful for **systems with channels of quite different bandwidths**.
 In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.<br />
 
-To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in Fig.&nbsp;[44](#figure--fig:mixed-sensitivity-dist-rejection).
+To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection).
 We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required.
 The elements of the generalized plant are
 
@@ -5217,10 +5217,10 @@ The elements of the generalized plant are
 
 {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="<span class=\"figure-number\">Figure 44: </span>\\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
 
-Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of Fig.&nbsp;[45](#figure--fig:mixed-sensitivity-ref-tracking).
+Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of [Figure 45](#figure--fig:mixed-sensitivity-ref-tracking).
 Here we consider a tracking problem.
 The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
-As the regulation problem of Fig.&nbsp;[44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
+As the regulation problem of [Figure 44](#figure--fig:mixed-sensitivity-dist-rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
 
 <a id="figure--fig:mixed-sensitivity-ref-tracking"></a>
 
@@ -5235,7 +5235,7 @@ Another useful mixed sensitivity optimization problem, is to find a stabilizing
 The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation.
 It is also important for robust stability with respect to multiplicative perturbations at the plant output.
 
-The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in Fig.&nbsp;[46](#figure--fig:mixed-sensitivity-s-t).
+The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in [Figure 46](#figure--fig:mixed-sensitivity-s-t).
 
 The elements of the generalized plant are
 
@@ -5277,7 +5277,7 @@ The focus of attention has moved to the size of signals and away from the size a
 </div>
 
 Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals.
-Weights are also used if a perturbation is used to model uncertainty, as in Fig.&nbsp;[47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
+Weights are also used if a perturbation is used to model uncertainty, as in [Figure 47](#figure--fig:input-uncertainty-hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
 
 <a id="figure--fig:input-uncertainty-hinf"></a>
 
@@ -5288,9 +5288,9 @@ As we have seen, the weights \\(Q\\) and \\(R\\) are constant, but LQG can be ge
 
 When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm.
 In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm.
-However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in Fig.&nbsp;[47](#figure--fig:input-uncertainty-hinf).<br />
+However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in [Figure 47](#figure--fig:input-uncertainty-hinf).<br />
 
-A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of Fig.&nbsp;[48](#figure--fig:hinf-signal-based).
+A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of [Figure 48](#figure--fig:hinf-signal-based).
 \\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed.
 The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals.
 The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\).
@@ -5315,7 +5315,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
 
 {{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="<span class=\"figure-number\">Figure 48: </span>A signal-based \\(\hinf\\) control problem" >}}
 
-Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in Fig.&nbsp;[49](#figure--fig:hinf-signal-based-uncertainty).
+Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in [Figure 49](#figure--fig:hinf-signal-based-uncertainty).
 The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\).
 We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**.
 This problem is a non-standard \\(\hinf\\) optimization.
@@ -5378,7 +5378,7 @@ The objective of robust stabilization is to stabilize not only the nominal model
 
 where \\(\epsilon > 0\\) is then the **stability margin**.<br />
 
-For the perturbed feedback system of Fig.&nbsp;[50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
+For the perturbed feedback system of [Figure 50](#figure--fig:coprime-uncertainty-bis), the stability property is robust if and only if the nominal feedback system is stable and
 
 \begin{equation\*}
   \gamma \triangleq \hnorm{\begin{bmatrix}
@@ -5439,13 +5439,13 @@ for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
   L &= (1-\gamma^2) I + XZ
 \end{align}
 
-The Matlab function `coprimeunc` can be used to generate the controller in <eq:control_coprime_factor>.
+The Matlab function `coprimeunc` can be used to generate the controller in \eqref{eq:control\_coprime\_factor}.
-It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from <eq:gamma_min_coprime> we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
+It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \eqref{eq:gamma\_min\_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
 
 
 #### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic-hinf-loop-shaping-design-procedure}
 
-<span class="org-target" id="org-target--sec:hinf_loop_shaping_procedure"></span>
+<span class="org-target" id="org-target--sec-hinf-loop-shaping-procedure"></span>
 Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements.
 
 To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant.
@@ -5456,7 +5456,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
   G\_s = W\_2 G W\_1
 \end{equation}
 
-as shown in Fig.&nbsp;[51](#figure--fig:shaped-plant).
+as shown in [Figure 51](#figure--fig:shaped-plant).
 
 <a id="figure--fig:shaped-plant"></a>
 
@@ -5491,7 +5491,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
     -   A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements
 7.  **Analyze the design** and if not all the specification are met, make further modifications to the weights
 8.  **Implement the controller**.
-    The configuration shown in Fig.&nbsp;[52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in Fig.&nbsp;[38](#figure--fig:classical-feedback-small).
+    The configuration shown in [Figure 52](#figure--fig:shapping-practical-implementation) has been found useful when compared with the conventional setup in [Figure 38](#figure--fig:classical-feedback-small).
     This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
     The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
 
@@ -5518,17 +5518,17 @@ Many control design problems possess two degrees-of-freedom:
 Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output.
 But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.<br />
 
-A general two degrees-of-freedom feedback control scheme is depicted in Fig.&nbsp;[53](#figure--fig:classical-feedback-2dof-simple).
+A general two degrees-of-freedom feedback control scheme is depicted in [Figure 53](#figure--fig:classical-feedback-2dof-simple).
 The commands and feedbacks enter the controller separately and are independently processed.
 
 <a id="figure--fig:classical-feedback-2dof-simple"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="<span class=\"figure-number\">Figure 53: </span>General two degrees-of-freedom feedback control scheme" >}}
 
-The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section&nbsp;is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
+The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
 However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.<br />
 
-The design problem is illustrated in Fig.&nbsp;[54](#figure--fig:coprime-uncertainty-hinf).
+The design problem is illustrated in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
 The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
 A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
 
@@ -5536,7 +5536,7 @@ A prefilter is introduced to force the response of the closed-loop system to fol
 
 {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="<span class=\"figure-number\">Figure 54: </span>Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
 
-The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in Fig.&nbsp;[54](#figure--fig:coprime-uncertainty-hinf).
+The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in [Figure 54](#figure--fig:coprime-uncertainty-hinf).
 This problem is easily cast into the general configuration.
 
 The control signal to the shaped plant \\(u\_s\\) is given by:
@@ -5559,14 +5559,14 @@ The purpose of the prefilter is to ensure that:
 
 The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller are:
 
-1.  Design a one degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller (section&nbsp;) but without a post-compensator \\(W\_2\\)
+1.  Design a one degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller (section ) but without a post-compensator \\(W\_2\\)
 2.  Select a desired closed-loop transfer function \\(T\_{\text{ref}}\\) between the commands and controller outputs
 3.  Set the scalar parameter \\(\rho\\) to a small value greater than \\(1\\); something in the range \\(1\\) to \\(3\\) will usually suffice
 4.  For the shaped \\(G\_s = G W\_1\\), the desired response \\(T\_{\text{ref}}\\), and the scalar parameter \\(\rho\\), solve the standard \\(\mathcal{H}\_\infty\\) optimization problem to a specified tolerance to get \\(K = [K\_1,\ K\_2]\\)
 5.  Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state.
 6.  Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\)
 
-The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig.&nbsp;[55](#figure--fig:hinf-synthesis-2dof).
+The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in [Figure 55](#figure--fig:hinf-synthesis-2dof).
 
 <a id="figure--fig:hinf-synthesis-2dof"></a>
 
@@ -5650,7 +5650,7 @@ When implemented in Hanus form, the expression for \\(u\\) becomes
 
 where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation.
 
-The situation is illustrated in Fig.&nbsp;[56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
+The situation is illustrated in [Figure 56](#figure--fig:weight-anti-windup), where the actuators are each modeled by a unit gain and a saturation.
 
 <a id="figure--fig:weight-anti-windup"></a>
 
@@ -5708,12 +5708,12 @@ Moreover, one should be careful about combining controller synthesis and analysi
 
 ## Controller Structure Design {#controller-structure-design}
 
-<span class="org-target" id="org-target--sec:controller_structure_design"></span>
+<span class="org-target" id="org-target--sec-controller-structure-design"></span>
 
 
 ### Introduction {#introduction}
 
-In previous sections, we considered the general problem formulation in Fig.&nbsp;[57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
+In previous sections, we considered the general problem formulation in [Figure 57](#figure--fig:general-control-names-bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
 
 <a id="figure--fig:general-control-names-bis"></a>
 
@@ -5748,31 +5748,31 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
 -   **Optimization layer**: computes the desired reference commands \\(r\\)
 -   **Control layer**: implements these commands to achieve \\(y \approx r\\)
 
-Additional layers are possible, as is illustrated in Fig.&nbsp;[58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
+Additional layers are possible, as is illustrated in [Figure 58](#figure--fig:control-system-hierarchy) which shows a typical control hierarchy for a chemical plant.
 
 <a id="figure--fig:control-system-hierarchy"></a>
 
 {{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="<span class=\"figure-number\">Figure 58: </span>Typical control system hierarchy in a chemical plant" >}}
 
-In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see Fig.&nbsp;[6](#org-target--fig:optimize_control_b)).
+In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see [ 6](#org-target--fig-optimize-control-b)).
 There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.<br />
 
 The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information.
 The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.<br />
 
-From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see Fig.&nbsp;[6](#org-target--fig:optimize_control_c)).
+From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see [ 6](#org-target--fig-optimize-control-c)).
 All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant.
 However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power.
 
 <a id="table--fig:optimize-control"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--fig:optimize-control">Table 6</a></span>:
+  <span class="table-number"><a href="#table--fig:optimize-control">Table 6</a>:</span>
   Alternative structures for optimization and control
 </div>
 
 | ![](/ox-hugo/skogestad07_optimize_control_a.png)                                                | ![](/ox-hugo/skogestad07_optimize_control_b.png)                                                                                | ![](/ox-hugo/skogestad07_optimize_control_c.png)                                                             |
 |-------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------|
-| <span class="org-target" id="org-target--fig:optimize_control_a"></span> Open loop optimization | <span class="org-target" id="org-target--fig:optimize_control_b"></span> Closed-loop implementation with separate control layer | <span class="org-target" id="org-target--fig:optimize_control_c"></span> Integrated optimization and control |
+| <span class="org-target" id="org-target--fig-optimize-control-a"></span> Open loop optimization | <span class="org-target" id="org-target--fig-optimize-control-b"></span> Closed-loop implementation with separate control layer | <span class="org-target" id="org-target--fig-optimize-control-c"></span> Integrated optimization and control |
 
 
 ### Selection of Controlled Outputs {#selection-of-controlled-outputs}
@@ -5885,7 +5885,7 @@ Thus, the selection of controlled and measured outputs are two separate issues.
 
 ### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements}
 
-We are here concerned with the variable sets \\(u\\) and \\(v\\) in Fig.&nbsp;[57](#figure--fig:general-control-names-bis).
+We are here concerned with the variable sets \\(u\\) and \\(v\\) in [Figure 57](#figure--fig:general-control-names-bis).
 Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to:
 
 -   Stabilize the plant, or more generally change its dynamics
@@ -5977,19 +5977,19 @@ Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degre
 A cascade control structure results when either of the following two situations arise:
 
 -   The reference \\(r\_i\\) is an output from another controller.
-    This is the **conventional cascade control** (Fig.&nbsp;[7](#org-target--fig:cascade_extra_meas))
+    This is the **conventional cascade control** ([ 7](#org-target--fig-cascade-extra-meas))
 -   The "measurement" \\(y\_i\\) is an output from another controller.
-    This is referred to as **input resetting** (Fig.&nbsp;[7](#org-target--fig:cascade_extra_input))
+    This is referred to as **input resetting** ([ 7](#org-target--fig-cascade-extra-input))
 
 <a id="table--fig:cascade-implementation"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--fig:cascade-implementation">Table 7</a></span>:
+  <span class="table-number"><a href="#table--fig:cascade-implementation">Table 7</a>:</span>
   Cascade Implementations
 </div>
 
 | ![](/ox-hugo/skogestad07_cascade_extra_meas.png)                                                       | ![](/ox-hugo/skogestad07_cascade_extra_input.png)                                                 |
 |--------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
-| <span class="org-target" id="org-target--fig:cascade_extra_meas"></span> Extra measurements \\(y\_2\\) | <span class="org-target" id="org-target--fig:cascade_extra_input"></span> Extra inputs \\(u\_2\\) |
+| <span class="org-target" id="org-target--fig-cascade-extra-meas"></span> Extra measurements \\(y\_2\\) | <span class="org-target" id="org-target--fig-cascade-extra-input"></span> Extra inputs \\(u\_2\\) |
 
 
 #### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
@@ -6013,7 +6013,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c
 
 ##### Cascade implementation {#cascade-implementation}
 
-To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in Fig.&nbsp;[7](#org-target--fig:cascade_extra_meas):
+To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in [ 7](#org-target--fig-cascade-extra-meas):
 
 \begin{align\*}
   r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\
@@ -6023,12 +6023,12 @@ To obtain an implementation with two SISO controllers, we may cascade the contro
 Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\).
 Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**.
 
-In the general case (Fig.&nbsp;[7](#org-target--fig:cascade_extra_meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
+In the general case ([ 7](#org-target--fig-cascade-extra-meas)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_.
-However, it is common to encounter the situation in Fig.&nbsp;[59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of Fig.&nbsp;[7](#org-target--fig:cascade_extra_meas).
+However, it is common to encounter the situation in [Figure 59](#figure--fig:cascade-control) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of [ 7](#org-target--fig-cascade-extra-meas).
 
 <div class="important">
 
-With reference to the special (but common) case of cascade control shown in Fig.&nbsp;[59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
+With reference to the special (but common) case of cascade control shown in [Figure 59](#figure--fig:cascade-control), the use of **extra measurements** is useful under the following circumstances:
 
 -   The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
     If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
@@ -6065,7 +6065,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t
 
 ##### Cascade implementation {#cascade-implementation}
 
-To obtain an implementation with two SISO controllers we may cascade the controllers as shown in Fig.&nbsp;[7](#org-target--fig:cascade_extra_input).
+To obtain an implementation with two SISO controllers we may cascade the controllers as shown in [ 7](#org-target--fig-cascade-extra-input).
 We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**.
 The fast control loop is then
 
@@ -6086,7 +6086,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
 
 <div class="exampl">
 
-Consider the system in Fig.&nbsp;[60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
+Consider the system in [Figure 60](#figure--fig:cascade-control-two-layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
 Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
 Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
 The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
@@ -6173,7 +6173,7 @@ Four applications of partial control are:
     The outputs \\(y\_1\\) have an associated control objective but are not measured.
     Instead, we aim at indirectly controlling \\(y\_1\\) by controlling the secondary measured variables \\(y\_2\\).
 
-The table&nbsp;[8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
+The table [Table 8](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
 In all cases, there is a control objective associated with \\(y\_1\\) and a feedback involving measurement and control of \\(y\_2\\) and we want:
 
 -   The effect of disturbances on \\(y\_1\\) to be small (when \\(y\_2\\) is controlled)
@@ -6181,7 +6181,7 @@ In all cases, there is a control objective associated with \\(y\_1\\) and a feed
 
 <a id="table--tab:partial-control"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:partial-control">Table 8</a></span>:
+  <span class="table-number"><a href="#table--tab:partial-control">Table 8</a>:</span>
   Applications of partial control
 </div>
 
@@ -6201,7 +6201,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w
   \end{aligned}
 \end{equation}
 
-Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) (Fig.&nbsp;[61](#figure--fig:partial-control)).
+Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) ([Figure 61](#figure--fig:partial-control)).
 We get:
 
 \begin{equation} \label{eq:partial\_control}
@@ -6270,10 +6270,10 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
 
 ##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems}
 
-Consider the conventional cascade control system in Fig.&nbsp;[7](#org-target--fig:cascade_extra_meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
+Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
 The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
 
-From <eq:partial_control>, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
+From \eqref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
 These arguments particularly apply at high frequencies.
 More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\).
 
@@ -6289,7 +6289,7 @@ A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all
 
 </div>
 
-To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in <eq:partial_control_partitioning> and <eq:partial_control>, and compute \\(P\_d\\) using <eq:tight_control_y2>.
+To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \eqref{eq:partial\_control\_partitioning} and \eqref{eq:partial\_control}, and compute \\(P\_d\\) using \eqref{eq:tight\_control\_y2}.
 
 
 #### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control}
@@ -6338,7 +6338,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
 
 ### Decentralized Feedback Control {#decentralized-feedback-control}
 
-In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig.&nbsp;[62](#figure--fig:decentralized-diagonal-control)).
+In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller ([Figure 62](#figure--fig:decentralized-diagonal-control)).
 
 <a id="figure--fig:decentralized-diagonal-control"></a>
 
@@ -6459,7 +6459,7 @@ We then derive **necessary conditions for stability** which may be used to elimi
 
 For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time.
 Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable).
-Using the **spectral radius condition** on the factorized \\(S\\) in <eq:S_factorization>, we have that the overall system is stable (\\(S\\) is stable) if
+Using the **spectral radius condition** on the factorized \\(S\\) in \eqref{eq:S\_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
 
 \begin{equation}
   \rho(E\tilde{T}(j\omega)) < 1, \forall\omega
@@ -6684,7 +6684,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
       |1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
     \end{equation}
 
-    To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by <eq:decent_contr_one_loop> is achievable.
+    To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \eqref{eq:decent\_contr\_one\_loop} is achievable.
     Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
     Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)
 6.  Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
@@ -6700,7 +6700,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
 If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4.
 If one still cannot find any pairing which are controllable, then one should consider multivariable control.
 
-7.  If the chosen pairing is controllable, then <eq:decent_contr_one_loop> tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
+7.  If the chosen pairing is controllable, then \eqref{eq:decent\_contr\_one\_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
     This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)
 
 
@@ -6720,7 +6720,7 @@ Thus sequential design may involve many iterations.
 
 #### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
 
-A number of **conditions for the stability**, e.g. <eq:decent_contr_cond_stability> and <eq:decent_contr_necessary_cond_stability>, and **performance**, e.g. <eq:decent_contr_cond_perf_dist> and <eq:decent_contr_cond_perf_ref>, of decentralized control systems have been derived.
+A number of **conditions for the stability**, e.g. \eqref{eq:decent\_contr\_cond\_stability} and \eqref{eq:decent\_contr\_necessary\_cond\_stability}, and **performance**, e.g. \eqref{eq:decent\_contr\_cond\_perf\_dist} and \eqref{eq:decent\_contr\_cond\_perf\_ref}, of decentralized control systems have been derived.
 
 The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.
 
@@ -6729,7 +6729,7 @@ The conditions are also useful in an **input-output controllability analysis** f
 
 ## Model Reduction {#model-reduction}
 
-<span class="org-target" id="org-target--sec:model_reduction"></span>
+<span class="org-target" id="org-target--sec-model-reduction"></span>
 
 
 ### Introduction {#introduction}

content/book/taghirad13_paral.md

@@ -24,7 +24,7 @@ PDF version
 
 ## Introduction {#introduction}
 
-<span class="org-target" id="org-target--sec:introduction"></span>
+<span class="org-target" id="org-target--sec-introduction"></span>
 
 This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
 A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
@@ -49,14 +49,14 @@ The control of parallel robots is elaborated in the last two chapters, in which
 
 ## Motion Representation {#motion-representation}
 
-<span class="org-target" id="org-target--sec:motion_representation"></span>
+<span class="org-target" id="org-target--sec-motion-representation"></span>
 
 
 ### Spatial Motion Representation {#spatial-motion-representation}
 
 Six independent parameters are sufficient to fully describe the spatial location of a rigid body.
 
-Consider a rigid body in a spatial motion as represented in Figure [1](#figure--fig:rigid-body-motion).
+Consider a rigid body in a spatial motion as represented in [Figure 1](#figure--fig:rigid-body-motion).
 Let us define:
 
 -   A **fixed reference coordinate system** \\((x, y, z)\\) denoted by frame \\(\\{\bm{A}\\}\\) whose origin is located at point \\(O\_A\\)
@@ -89,7 +89,7 @@ It can be **represented in several different ways**: the rotation matrix, the sc
 ##### Rotation Matrix {#rotation-matrix}
 
 We consider a rigid body that has been exposed to a pure rotation.
-Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#figure--fig:rotation-matrix)).
+Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) ([Figure 2](#figure--fig:rotation-matrix)).
 
 A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
 
@@ -264,7 +264,7 @@ If the pose of a rigid body \\(\\{{}^A\bm{R}\_B, {}^A\bm{P}\_{O\_B}\\}\\) is giv
 
 ### Homogeneous Transformations {#homogeneous-transformations}
 
-To describe general transformations, we introduce the \\(4\times1\\) **homogeneous coordinates**, and Eq. <eq:chasles_therorem> is generalized to
+To describe general transformations, we introduce the \\(4\times1\\) **homogeneous coordinates**, and Eq. \eqref{eq:chasles\_therorem} is generalized to
 
 \begin{equation} \label{eq:homogeneous\_transformation}
   \boxed{{}^A\bm{P} = {}^A\bm{T}\_B {}^B\bm{P}}
@@ -363,7 +363,7 @@ There exist transformations to from screw displacement notation to the transform
 
 ##### Consecutive transformations {#consecutive-transformations}
 
-Let us consider the motion of a rigid body described at three locations (Figure [5](#figure--fig:consecutive-transformations)).
+Let us consider the motion of a rigid body described at three locations ([Figure 5](#figure--fig:consecutive-transformations)).
 Frame \\(\\{\bm{A}\\}\\) represents the initial location, frame \\(\\{\bm{B}\\}\\) is an intermediate location, and frame \\(\\{\bm{C}\\}\\) represents the rigid body at its final location.
 
 <a id="figure--fig:consecutive-transformations"></a>
@@ -429,7 +429,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
 
 ## Kinematics {#kinematics}
 
-<span class="org-target" id="org-target--sec:kinematics"></span>
+<span class="org-target" id="org-target--sec-kinematics"></span>
 
 
 ### Introduction {#introduction}
@@ -536,7 +536,7 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
 
 {{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="<span class=\"figure-number\">Figure 6: </span>Geometry of a Stewart-Gough platform" >}}
 
-The geometry of the manipulator is shown Figure [6](#figure--fig:stewart-schematic).
+The geometry of the manipulator is shown [Figure 6](#figure--fig:stewart-schematic).
 
 
 #### Inverse Kinematics {#inverse-kinematics}
@@ -583,7 +583,7 @@ The complexity of the problem depends widely on the manipulator architecture and
 
 ## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
 
-<span class="org-target" id="org-target--sec:jacobian"></span>
+<span class="org-target" id="org-target--sec-jacobian"></span>
 
 
 ### Introduction {#introduction}
@@ -620,7 +620,7 @@ The direction of \\(\bm{\Omega}\\) indicates the instantaneous axis of rotation
 
 </div>
 
-The angular velocity vector is related to the screw formalism by equation <eq:angular_velocity_vector>.
+The angular velocity vector is related to the screw formalism by equation \eqref{eq:angular\_velocity\_vector}.
 
 \begin{equation} \label{eq:angular\_velocity\_vector}
   \boxed{\bm{\Omega} \triangleq \dot{\theta} \hat{\bm{s}}}
@@ -676,7 +676,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
   \end{bmatrix}}
 \end{equation}
 
-Now consider the general motion of a rigid body shown in Figure [7](#figure--fig:general-motion), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
+Now consider the general motion of a rigid body shown in [Figure 7](#figure--fig:general-motion), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
 
 <a id="figure--fig:general-motion"></a>
 
@@ -695,7 +695,7 @@ The time derivative of the rotation matrix \\({}^A\dot{\bm{R}}\_B\\) is:
   \boxed{{}^A\dot{\bm{R}}\_B = {}^A\bm{\Omega}^\times \ {}^A\bm{R}\_B}
 \end{equation}
 
-And we finally obtain equation <eq:absolute_velocity_formula>.
+And we finally obtain equation \eqref{eq:absolute\_velocity\_formula}.
 
 <div class="important">
 
@@ -755,7 +755,7 @@ The **general Jacobian matrix** is defined as:
   \dot{\bm{q}} = \bm{J} \dot{\bm{\mathcal{X}}}
 \end{equation}
 
-From equation <eq:jacobians>, we have:
+From equation \eqref{eq:jacobians}, we have:
 
 \begin{equation}
   \bm{J} = {\bm{J}\_q}^{-1} \bm{J}\_x
@@ -842,7 +842,7 @@ Moreover, we have:
 -   \\({}^A\dot{\bm{R}}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{R}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{b}\_i\\) in which \\({}^A\bm{\omega}\\) denotes the angular velocity of the moving platform expressed in the fixed frame \\(\\{\bm{A}\\}\\).
 -   \\(l\_i {}^A\dot{\hat{\bm{s}}}\_i = l\_i \left( {}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i \right)\\) in which \\({}^A\bm{\omega}\_i\\) is the angular velocity of limb \\(i\\) express in fixed frame \\(\\{\bm{A}\\}\\).
 
-Then, the velocity loop closure <eq:loop_closure_limb_diff> simplifies to
+Then, the velocity loop closure \eqref{eq:loop\_closure\_limb\_diff} simplifies to
 \\[ {}^A\bm{v}\_p + {}^A\bm{\omega} \times {}^A\bm{b}\_i = \dot{l}\_i {}^A\hat{\bm{s}}\_i + l\_i ({}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i) \\]
 
 By dot multiply both side of the equation by \\(\hat{\bm{s}}\_i\\):
@@ -880,9 +880,9 @@ We then omit the superscript \\(A\\) and we can rearrange the 6 equations into a
 #### Singularity Analysis {#singularity-analysis}
 
 It is of primary importance to avoid singularities in a given workspace.
-To study the singularity configurations of the Stewart-Gough platform, we consider the Jacobian matrix determined with the equation <eq:jacobian_formula_stewart>.<br />
+To study the singularity configurations of the Stewart-Gough platform, we consider the Jacobian matrix determined with the equation \eqref{eq:jacobian\_formula\_stewart}.<br />
 
-From equation <eq:jacobians>, it is clear that for the Stewart-Gough platform, \\(\bm{J}\_q = \bm{I}\\) and \\(\bm{J}\_x = \bm{J}\\).
+From equation \eqref{eq:jacobians}, it is clear that for the Stewart-Gough platform, \\(\bm{J}\_q = \bm{I}\\) and \\(\bm{J}\_x = \bm{J}\\).
 Hence the manipulator has **no inverse kinematic singularities** within the manipulator workspace, but **may possess forward kinematic singularity** when \\(\bm{J}\\) becomes rank deficient. This may occur when
 \\[ \det \bm{J} = 0 \\]
 
@@ -935,7 +935,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
 
 #### Static Forces of the Stewart-Gough Platform {#static-forces-of-the-stewart-gough-platform}
 
-As shown in Figure [8](#figure--fig:stewart-static-forces), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
+As shown in [Figure 8](#figure--fig:stewart-static-forces), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
 
 <a id="figure--fig:stewart-static-forces"></a>
 
@@ -1001,7 +1001,7 @@ The relation between the applied actuator force \\(\tau\_i\\) and the correspond
 
 in which \\(k\_i\\) denotes the **stiffness constant of the actuator**.<br />
 
-Re-writing the equation <eq:stiffness_actuator> for all limbs in a matrix form result in
+Re-writing the equation \eqref{eq:stiffness\_actuator} for all limbs in a matrix form result in
 
 \begin{equation} \label{eq:stiffness\_matrix\_relation}
   \boxed{\bm{\tau} = \mathcal{K} \cdot \Delta \bm{q}}
@@ -1010,7 +1010,7 @@ Re-writing the equation <eq:stiffness_actuator> for all limbs in a matrix form r
 in which \\(\bm{\tau}\\) is the vector of actuator forces, and \\(\Delta \bm{q}\\) corresponds to the actuator deflections.
 \\(\mathcal{K} = \text{diag}\left[ k\_1 \ k\_2 \dots k\_m \right]\\) is an \\(m \times m\\) diagonal matrix composed of the actuator stiffness constants.<br />
 
-Writing the Jacobian relation given in equation <eq:jacobian_disp> for infinitesimal deflection read
+Writing the Jacobian relation given in equation \eqref{eq:jacobian\_disp} for infinitesimal deflection read
 
 \begin{equation} \label{eq:jacobian\_disp\_inf}
   \Delta \bm{q} = \bm{J} \cdot \Delta \bm{\mathcal{X}}
@@ -1018,19 +1018,19 @@ Writing the Jacobian relation given in equation <eq:jacobian_disp> for infinites
 
 in which \\(\Delta \bm{\mathcal{X}} = [\Delta x\ \Delta y\ \Delta z\ \Delta\theta x\ \Delta\theta y\ \Delta\theta z]\\) is the infinitesimal linear and angular deflection of the moving platform.
 
-Furthermore, rewriting the Jacobian as the projection of actuator forces to the moving platform <eq:jacobian_forces> gives
+Furthermore, rewriting the Jacobian as the projection of actuator forces to the moving platform \eqref{eq:jacobian\_forces} gives
 
 \begin{equation} \label{eq:jacobian\_force\_inf}
   \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
 \end{equation}
 
-Hence, by substituting <eq:stiffness_matrix_relation> and <eq:jacobian_disp_inf> in <eq:jacobian_force_inf>, we obtain:
+Hence, by substituting \eqref{eq:stiffness\_matrix\_relation} and \eqref{eq:jacobian\_disp\_inf} in \eqref{eq:jacobian\_force\_inf}, we obtain:
 
 \begin{equation} \label{eq:stiffness\_jacobian}
   \boxed{\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}\_{\bm{K}} \cdot \Delta \bm{\mathcal{X}}}
 \end{equation}
 
-Equation <eq:stiffness_jacobian> implies that the moving platform output wrench is related to its deflection by the **stiffness matrix** \\(K\\).
+Equation \eqref{eq:stiffness\_jacobian} implies that the moving platform output wrench is related to its deflection by the **stiffness matrix** \\(K\\).
 
 <div class="important">
 
@@ -1093,7 +1093,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
 
 #### Stiffness Analysis of the Stewart-Gough Platform {#stiffness-analysis-of-the-stewart-gough-platform}
 
-In this section, we restrict our analysis to a 3-6 structure (Figure [9](#figure--fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
+In this section, we restrict our analysis to a 3-6 structure ([Figure 9](#figure--fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
 
 <a id="figure--fig:stewart36"></a>
 
@@ -1125,7 +1125,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
 
 ## Dynamics {#dynamics}
 
-<span class="org-target" id="org-target--sec:dynamics"></span>
+<span class="org-target" id="org-target--sec-dynamics"></span>
 
 
 ### Introduction {#introduction}
@@ -1208,7 +1208,7 @@ where \\(\\{\theta, \hat{\bm{s}}\\}\\) are the screw parameters representing the
 
 </div>
 
-As shown by <eq:angular_acceleration>, the angular acceleration of the rigid body is also along the screw axis \\(\hat{\bm{s}}\\) with a magnitude equal to \\(\ddot{\theta}\\).
+As shown by \eqref{eq:angular\_acceleration}, the angular acceleration of the rigid body is also along the screw axis \\(\hat{\bm{s}}\\) with a magnitude equal to \\(\ddot{\theta}\\).
 
 
 ##### Linear Acceleration of a Point {#linear-acceleration-of-a-point}
@@ -1222,7 +1222,7 @@ Linear acceleration of a point \\(P\\) can be easily determined by time derivati
 Note that this is correct only if the derivative is taken with respect to a **fixed** frame.<br />
 
 Now consider the general motion of a rigid body, in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and the problem is to find the absolute acceleration of point \\(P\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
-The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [7](#figure--fig:general-motion)).
+The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see [Figure 7](#figure--fig:general-motion)).
 To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
 
 \begin{equation}
@@ -1305,7 +1305,7 @@ in which
 
 ##### Principal Axes {#principal-axes}
 
-As seen in equation <eq:moment_inertia>, the inertia matrix elements are a function of mass distribution of the rigid body with respect to the frame \\(\\{\bm{A}\\}\\).
+As seen in equation \eqref{eq:moment\_inertia}, the inertia matrix elements are a function of mass distribution of the rigid body with respect to the frame \\(\\{\bm{A}\\}\\).
 Hence, it is possible to find **orientations of frame** \\(\\{\bm{A}\\}\\) in which the product of inertia terms vanish and inertia matrix becomes **diagonal**:
 
 \begin{equation} \label{eq:inertia\_matrix\_diagonal}
@@ -1347,7 +1347,7 @@ On the other hand, if the reference frame \\(\\{B\\}\\) has **pure rotation** wi
 
 ##### Linear Momentum {#linear-momentum}
 
-Linear momentum of a material body, shown in Figure [11](#figure--fig:angular-momentum-rigid-body), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
+Linear momentum of a material body, shown in [Figure 11](#figure--fig:angular-momentum-rigid-body), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
 
 \begin{equation}
   {}^A\bm{G} = \int\_V \frac{d\bm{p}}{dt} \rho dV
@@ -1376,7 +1376,7 @@ This highlights the important of the center of mass in dynamic formulation of ri
 
 ##### Angular Momentum {#angular-momentum}
 
-Consider the solid body represented in Figure [11](#figure--fig:angular-momentum-rigid-body).
+Consider the solid body represented in [Figure 11](#figure--fig:angular-momentum-rigid-body).
 Angular momentum of the differential masses \\(\rho dV\\) about a reference point \\(A\\), expressed in the reference frame \\(\\{\bm{A}\\}\\) is defined as
 \\[ {}^A\bm{H} = \int\_V \left(\bm{p} \times \frac{d\bm{p}}{dt} \right) \rho dV \\]
 in which \\(d\bm{p}/dt\\) denotes the velocity of differential mass with respect to the reference frame \\(\\{\bm{A}\\}\\).
@@ -1393,10 +1393,10 @@ Therefore, angular momentum of the rigid body about point \\(A\\) is reduced to
 in which
 \\[ {}^C\bm{H} = \int\_V \bm{r} \times (\bm{\Omega} \times \bm{r}) \rho dV = {}^C\bm{I} \cdot \bm{\Omega} \\]
 
-Equation <eq:angular_momentum> reveals that angular momentum of a rigid body about a point \\(A\\) can be written as \\(\bm{p}\_c \times \bm{G}\_c\\), which is the contribution of linear momentum of the rigid body about point \\(A\\), and \\({}^C\bm{H}\\) which is the angular momentum of the rigid body about the center of mass.
+Equation \eqref{eq:angular\_momentum} reveals that angular momentum of a rigid body about a point \\(A\\) can be written as \\(\bm{p}\_c \times \bm{G}\_c\\), which is the contribution of linear momentum of the rigid body about point \\(A\\), and \\({}^C\bm{H}\\) which is the angular momentum of the rigid body about the center of mass.
 
 This also highlights the important of the center of mass in the dynamic analysis of rigid bodies.
-If the center of mass is taken as the reference point, the relation describing angular momentum <eq:angular_momentum> is very analogous to that of linear momentum <eq:linear_momentum>.
+If the center of mass is taken as the reference point, the relation describing angular momentum \eqref{eq:angular\_momentum} is very analogous to that of linear momentum \eqref{eq:linear\_momentum}.
 
 
 ##### Kinetic Energy {#kinetic-energy}
@@ -1504,7 +1504,7 @@ With \\(\bm{v}\_{b\_{i}}\\) an **intermediate variable** corresponding to the ve
   \bm{v}\_{b\_{i}} = \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i}
 \end{equation}
 
-As illustrated in Figure [12](#figure--fig:free-body-diagram-stewart), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
+As illustrated in [Figure 12](#figure--fig:free-body-diagram-stewart), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
 The position vector of these two center of masses can be determined by the following equations:
 
 \begin{align}
@@ -1539,7 +1539,7 @@ We assume that each limb consists of two parts, the cylinder and the piston, whe
 We also assume that the centers of masses of the cylinder and the piston are located at a distance of \\(c\_{i\_1}\\) and \\(c\_{i\_2}\\) above their foot points, and their masses are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
 Moreover, consider that the pistons are symmetric about their axes, and their centers of masses lie at their midlengths.
 
-The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#figure--fig:free-body-diagram-stewart).
+The free-body diagrams of the limbs and the moving platforms is given in [Figure 12](#figure--fig:free-body-diagram-stewart).
 The reaction forces at fixed points \\(A\_i\\) are denoted by \\(\bm{f}\_{a\_i}\\), the internal force at moving points \\(B\_i\\) are dentoed by \\(\bm{f}\_{b\_i}\\), and the internal forces and moments between cylinders and pistons are denoted by \\(\bm{f}\_{c\_i}\\) and \\(\bm{M\_{c\_i}}\\) respectively.
 
 Assume that the only existing external disturbance wrench is applied on the moving platform and is denoted by \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\).
@@ -1566,7 +1566,7 @@ in which \\(m\_{c\_e}\\) is defined as
 ##### Dynamic Formulation of the Moving Platform {#dynamic-formulation-of-the-moving-platform}
 
 Assume that the **moving platform center of mass is located at the center point** \\(P\\) and it has a mass \\(m\\) and moment of inertia \\({}^A\bm{I}\_{P}\\).
-Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in Figure [12](#figure--fig:free-body-diagram-stewart).
+Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in [Figure 12](#figure--fig:free-body-diagram-stewart).
 
 The Newton-Euler formulation of the moving platform is as follows:
 
@@ -1608,7 +1608,7 @@ in which \\(\bm{\mathcal{X}} = [\bm{x}\_P, \bm{\theta}]^T\\) is the motion varia
 
 #### Closed-Form Dynamics {#closed-form-dynamics}
 
-While dynamic formulation in the form of Equation <eq:dynamic_formulation_implicit> can be used to simulate inverse dynamics of the Stewart-Gough platform, its implicit nature makes it unpleasant for the dynamic analysis and control.
+While dynamic formulation in the form of Equation \eqref{eq:dynamic\_formulation\_implicit} can be used to simulate inverse dynamics of the Stewart-Gough platform, its implicit nature makes it unpleasant for the dynamic analysis and control.
 
 
 ##### Closed-Form Dynamics of the Limbs {#closed-form-dynamics-of-the-limbs}
@@ -1666,7 +1666,7 @@ It is preferable to use the **screw coordinates** for representing the angular m
   \ddot{\bm{\mathcal{X}}} = \begin{bmatrix}\bm{a}\_p \\\ \dot{\bm{\omega}}\end{bmatrix}}
 \end{equation}
 
-Equations <eq:dyn_form_implicit_trans> and <eq:dyn_form_implicit_rot> can be simply converted into a closed form of Equation <eq:close_form_dynamics_platform> with the following terms:
+Equations \eqref{eq:dyn\_form\_implicit\_trans} and \eqref{eq:dyn\_form\_implicit\_rot} can be simply converted into a closed form of Equation \eqref{eq:close\_form\_dynamics\_platform} with the following terms:
 
 \begin{equation} \label{eq:close\_form\_dynamics\_stewart\_terms}
   \begin{aligned}
@@ -1723,20 +1723,20 @@ in which
 
 ##### Forward Dynamics Simulations {#forward-dynamics-simulations}
 
-As shown in Figure [13](#figure--fig:stewart-forward-dynamics), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
+As shown in [Figure 13](#figure--fig:stewart-forward-dynamics), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
 
 <a id="figure--fig:stewart-forward-dynamics"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="<span class=\"figure-number\">Figure 13: </span>Flowchart of forward dynamics implementation sequence" >}}
 
-The closed-form dynamic formulation of the Stewart-Gough platform corresponds to the set of equations given in <eq:closed_form_dynamic_stewart_wanted>, whose terms are given in <eq:close_form_dynamics_stewart_terms>.
+The closed-form dynamic formulation of the Stewart-Gough platform corresponds to the set of equations given in \eqref{eq:closed\_form\_dynamic\_stewart\_wanted}, whose terms are given in \eqref{eq:close\_form\_dynamics\_stewart\_terms}.
 
 
 ##### Inverse Dynamics Simulation {#inverse-dynamics-simulation}
 
 In inverse dynamics simulations, it is assumed that the **trajectory of the manipulator is given**, and the **actuator forces required to generate such trajectories are to be determined**.
 
-As illustrated in Figure [14](#figure--fig:stewart-inverse-dynamics), inverse dynamic formulation is implemented in the following sequence.
+As illustrated in [Figure 14](#figure--fig:stewart-inverse-dynamics), inverse dynamic formulation is implemented in the following sequence.
 The first step is trajectory generation for the manipulator moving platform.
 Many different algorithms are developed for a smooth trajectory generation.
 For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \\(\dot{\bm{\mathcal{X}}}\_{d}(t)\\), \\(\ddot{\bm{\mathcal{X}}}\_{d}(t)\\) are known.
@@ -1744,11 +1744,11 @@ For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \
 The next step is to solve the inverse kinematics of the manipulator and to find the limbs' linear and angular positions, velocity and acceleration as a function of the manipulator trajectory.
 The manipulator Jacobian matrix \\(\bm{J}\\) is also calculated in this step.
 
-Next, the dynamic matrices given in the closed-form formulations of the limbs and the moving platform are calculated using equations <eq:closed_form_intermediate_parameters> and <eq:close_form_dynamics_stewart_terms>, respectively.<br />
+Next, the dynamic matrices given in the closed-form formulations of the limbs and the moving platform are calculated using equations \eqref{eq:closed\_form\_intermediate\_parameters} and \eqref{eq:close\_form\_dynamics\_stewart\_terms}, respectively.<br />
 
-To combine the corresponding matrices, an to generate the whole manipulator dynamics, it is necessary to find intermediate Jacobian matrices \\(\bm{J}\_i\\), given in <eq:jacobian_intermediate>, and then compute compatible matrices for the limbs given in <eq:closed_form_stewart_manipulator>.
+To combine the corresponding matrices, an to generate the whole manipulator dynamics, it is necessary to find intermediate Jacobian matrices \\(\bm{J}\_i\\), given in \eqref{eq:jacobian\_intermediate}, and then compute compatible matrices for the limbs given in \eqref{eq:closed\_form\_stewart\_manipulator}.
 Now that all the terms required to **computed to actuator forces required to generate such a trajectory** is computed, let us define \\(\bm{\mathcal{F}}\\) as the resulting Cartesian wrench applied to the moving platform.
-This wrench can be calculated from the summation of all inertial and external forces **excluding the actuator torques** \\(\bm{\tau}\\) in the closed-form dynamic formulation <eq:closed_form_dynamic_stewart_wanted>.
+This wrench can be calculated from the summation of all inertial and external forces **excluding the actuator torques** \\(\bm{\tau}\\) in the closed-form dynamic formulation \eqref{eq:closed\_form\_dynamic\_stewart\_wanted}.
 
 By this definition, \\(\bm{\mathcal{F}}\\) can be viewed as the projector of the actuator forces acting on the manipulator, mapped to the Cartesian space.
 Since there is no redundancy in actuation in the Stewart-Gough manipulator, the Jacobian matrix \\(\bm{J}\\), squared and actuator forces can be uniquely determined from this wrench, by \\(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\\), provided \\(\bm{J}\\) is non-singular.
@@ -1783,7 +1783,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
 
 ## Motion Control {#motion-control}
 
-<span class="org-target" id="org-target--sec:motion_control"></span>
+<span class="org-target" id="org-target--sec-motion-control"></span>
 
 
 ### Introduction {#introduction}
@@ -1804,7 +1804,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
 
 ### Controller Topology {#controller-topology}
 
-<span class="org-target" id="org-target--sec:control_topology"></span>
+<span class="org-target" id="org-target--sec-control-topology"></span>
 
 <div class="important">
 
@@ -1847,7 +1847,7 @@ In general, the desired motion of the moving platform may be represented by the
 To perform such motion in closed loop, it is necessary to **measure the output motion** \\(\bm{\mathcal{X}}\\) of the manipulator by an instrumentation system.
 Such instrumentation usually consists of two subsystems: the first subsystem may use accurate accelerometers, or global positioning systems to calculate the position of a point on the moving platform; and a second subsystem may use inertial or laser gyros to determine orientation of the moving platform.<br />
 
-Figure [15](#figure--fig:general-topology-motion-feedback) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
+[Figure 15](#figure--fig:general-topology-motion-feedback) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
 In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
 The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.<br />
 
@@ -1859,7 +1859,7 @@ However, it is usually much **easier to measure the active joint variable** rath
 The relation between the **joint variable** \\(\bm{q}\\) and **motion variable** of the moving platform \\(\bm{\mathcal{X}}\\) is dealt with the **forward and inverse kinematics**.
 The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) and \\(\dot{\bm{\mathcal{X}}}\\) is studied through the **Jacobian analysis**.<br />
 
-It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [16](#figure--fig:general-topology-motion-feedback-bis) to implement such a controller.
+It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis) to implement such a controller.
 
 <a id="figure--fig:general-topology-motion-feedback-bis"></a>
 
@@ -1870,9 +1870,9 @@ As described earlier, this is a **complex task** for parallel manipulators.
 It is even more complex when a solution has to be found in real time.<br />
 
 However, as shown herein before, the inverse kinematic analysis of parallel manipulators is much easier to carry out.
-To overcome the implementation problem of the control topology in Figure [16](#figure--fig:general-topology-motion-feedback-bis), another control topology is usually implemented for parallel manipulators.
+To overcome the implementation problem of the control topology in [Figure 16](#figure--fig:general-topology-motion-feedback-bis), another control topology is usually implemented for parallel manipulators.
 
-In this topology, depicted in Figure [17](#figure--fig:general-topology-motion-feedback-ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
+In this topology, depicted in [Figure 17](#figure--fig:general-topology-motion-feedback-ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
 Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
 
 <a id="figure--fig:general-topology-motion-feedback-ter"></a>
@@ -1881,12 +1881,12 @@ Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_
 
 Therefore, the **structure and characteristics** of the controller in this topology is totally **different** from that given in the first two topologies.
 
-The **input and output** of the controller depicted in Figure [17](#figure--fig:general-topology-motion-feedback-ter) are **both in the joint space**.
+The **input and output** of the controller depicted in [Figure 17](#figure--fig:general-topology-motion-feedback-ter) are **both in the joint space**.
 However, this is not the case in the previous topologies where the input to the controller is the motion error in task space, while its output is in the joint space.
 
-For the topology in Figure [17](#figure--fig:general-topology-motion-feedback-ter), **independent controllers** for each joint may be suitable.<br />
+For the topology in [Figure 17](#figure--fig:general-topology-motion-feedback-ter), **independent controllers** for each joint may be suitable.<br />
 
-To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#figure--fig:general-topology-motion-feedback-quater).
+To generate a **direct input to output relation in the task space**, consider the topology depicted in [Figure 18](#figure--fig:general-topology-motion-feedback-quater).
 A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
 
 <a id="figure--fig:general-topology-motion-feedback-quater"></a>
@@ -1899,12 +1899,12 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
 
 ### Motion Control in Task Space {#motion-control-in-task-space}
 
-<span class="org-target" id="org-target--sec:control_task_space"></span>
+<span class="org-target" id="org-target--sec-control-task-space"></span>
 
 
 #### Decentralized PD Control {#decentralized-pd-control}
 
-In the control structure in Figure [19](#figure--fig:decentralized-pd-control-task-space), a number of linear PD controllers are used in a feedback structure on each error component.
+In the control structure in [Figure 19](#figure--fig:decentralized-pd-control-task-space), a number of linear PD controllers are used in a feedback structure on each error component.
 The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
 The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
 
@@ -1927,7 +1927,7 @@ The controller gains are generally tuned experimentally based on physical realiz
 
 #### Feed Forward Control {#feed-forward-control}
 
-A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#figure--fig:feedforward-control-task-space).
+A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in [Figure 20](#figure--fig:feedforward-control-task-space).
 This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
 \\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
 
@@ -1986,7 +1986,7 @@ By this means, **nonlinear and coupling behavior of the robotic manipulator is s
 
 </div>
 
-General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#figure--fig:inverse-dynamics-control-task-space).
+General structure of IDC applied to a parallel manipulator is depicted in [Figure 21](#figure--fig:inverse-dynamics-control-task-space).
 A corrective wrench \\(\bm{\mathcal{F}}\_{fl}\\) is added in a **feedback structure** to the closed-loop system, which is calculated from the Coriolis and centrifugal matrix and gravity vector of the manipulator dynamic formulation.
 
 Furthermore, mass matrix is added in the forward path in addition to the desired trajectory acceleration \\(\ddot{\bm{\mathcal{X}}}\_d\\).
@@ -2043,9 +2043,9 @@ These are the reasons why, in practice, IDC control is extended to different for
 
 To develop the simplest possible implementable IDC, let us recall dynamic formulation complexities:
 
--   the manipulator mass matrix \\(\bm{M}(\bm{\mathcal{X}})\\) is derived from kinetic energy of the manipulator (Eq. <eq:kinetic_energy>)
+-   the manipulator mass matrix \\(\bm{M}(\bm{\mathcal{X}})\\) is derived from kinetic energy of the manipulator (Eq. \eqref{eq:kinetic\_energy})
--   the gravity vector \\(\bm{G}(\bm{\mathcal{X}})\\) is derived from potential energy (Eq. <eq:gravity_vectory>)
+-   the gravity vector \\(\bm{G}(\bm{\mathcal{X}})\\) is derived from potential energy (Eq. \eqref{eq:gravity\_vectory})
--   the Coriolis and centrifugal matrix \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) is derived from Eq. <eq:gravity_vectory>
+-   the Coriolis and centrifugal matrix \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) is derived from Eq. \eqref{eq:gravity\_vectory}
 
 The computation of the Coriolis and centrifugal matrix is more intensive than that of the mass matrix.
 Gravity vector is more easily computable.
@@ -2053,7 +2053,7 @@ Gravity vector is more easily computable.
 However, it is shown that certain properties hold for mass matrix, gravity vector and Coriolis and centrifugal matrix, which might be directly used in the control techniques developed for parallel manipulators.
 One of the most important properties of dynamic matrices is the skew-symmetric property of the matrix \\(\dot{\bm{M}} - 2 \bm{C}\\) .<br />
 
-Consider dynamic formulation of parallel robot given in Eq. <eq:closed_form_dynamic_formulation>, in which the skew-symmetric property of dynamic matrices is satisfied.
+Consider dynamic formulation of parallel robot given in Eq. \eqref{eq:closed\_form\_dynamic\_formulation}, in which the skew-symmetric property of dynamic matrices is satisfied.
 The simplest form of IDC control effort \\(\bm{\mathcal{F}}\\) consists of:
 \\[ \bm{\mathcal{F}} = \bm{\mathcal{F}}\_{pd} + \bm{\mathcal{F}}\_{fl} \\]
 in which the first term \\(\bm{\mathcal{F}}\_{pd}\\) is generated by the simplified PD form on the motion error:
@@ -2091,7 +2091,7 @@ A global understanding of the trade-offs involved in each method is needed to em
 
 Various sources of uncertainties such as unmodelled dynamics, unknown parameters, calibration error, unknown disturbance wrenches, and varying payloads may exist, and are not seen in dynamic model of the manipulator.
 
-To consider these modeling uncertainty in the closed-loop performance of the manipulator, recall the general closed-form dynamic formulation of the manipulator given in Eq. <eq:closed_form_dynamic_formulation>, and modify the inverse dynamics control input \\(\bm{\mathcal{F}}\\) as
+To consider these modeling uncertainty in the closed-loop performance of the manipulator, recall the general closed-form dynamic formulation of the manipulator given in Eq. \eqref{eq:closed\_form\_dynamic\_formulation}, and modify the inverse dynamics control input \\(\bm{\mathcal{F}}\\) as
 
 \begin{align\*}
   \bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a}\_r + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}})\\\\
@@ -2133,7 +2133,7 @@ If this measurement is available without any doubt, such topologies are among th
 However, as explained in Section , in many practical situations measurement of the motion variable \\(\bm{\mathcal{X}}\\) is difficult or expensive, and usually just the active joint variables \\(\bm{q}\\) are measured.
 In such cases, the controllers developed in the joint space may be recommended for practical implementation.<br />
 
-To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#figure--fig:general-topology-motion-feedback-bis).
+To generate a direct input to output relation in the joint space, consider the topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis).
 In this topology, the controller input is the joint variable error vector \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), and the controller output is directly the actuator force vector \\(\bm{\tau}\\), and hence there exists a **one-to-one correspondence between the controller input to its output**.<br />
 
 The general form of dynamic formulation of parallel robot is usually given in the task space.
@@ -2183,7 +2183,7 @@ with:
 
 </div>
 
-Equation <eq:dynamics_joint_space> represents the closed form dynamic formulation of a general parallel robot in the joint space.<br />
+Equation \eqref{eq:dynamics\_joint\_space} represents the closed form dynamic formulation of a general parallel robot in the joint space.<br />
 
 Note that the dynamic matrices are **not** explicitly represented in terms of the joint variable vector \\(\bm{q}\\).
 In fact, to fully derive these matrices, the Jacobian matrices must be computed and are generally derived as a function of the motion variables \\(\bm{\mathcal{X}}\\).
@@ -2191,7 +2191,7 @@ Furthermore, the main dynamic matrices are all functions of the motion variable
 Hence, in practice, to find the dynamic matrices represented in the joint space, **forward kinematics** should be solved to find the motion variable \\(\bm{\mathcal{X}}\\) for any given joint motion vector \\(\bm{q}\\).<br />
 
 Since in parallel robots the forward kinematic analysis is computationally intensive, there exist inherent difficulties in finding the dynamic matrices in the joint space as an explicit function of \\(\bm{q}\\).
-In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#figure--fig:general-topology-motion-feedback-bis), and implement control law design in the task space.<br />
+In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [Figure 16](#figure--fig:general-topology-motion-feedback-bis), and implement control law design in the task space.<br />
 
 However, one implementable alternative to calculate the dynamic matrices represented in the joint space is to use the **desired motion trajectory** \\(\bm{\mathcal{X}}\_d\\) instead of the true value of motion vector \\(\bm{\mathcal{X}}\\) in the calculations.
 This approximation significantly reduces the computational cost, with the penalty of having mismatch between the estimated values of these matrices to their true values.
@@ -2200,7 +2200,7 @@ This approximation significantly reduces the computational cost, with the penalt
 #### Decentralized PD Control {#decentralized-pd-control}
 
 The first control strategy introduced in the joint space consists of the simplest form of feedback control in such manipulators.
-In this control structure, depicted in Figure [24](#figure--fig:decentralized-pd-control-joint-space), a number of PD controllers are used in a feedback structure on each error component.
+In this control structure, depicted in [Figure 24](#figure--fig:decentralized-pd-control-joint-space), a number of PD controllers are used in a feedback structure on each error component.
 
 The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.<br />
 
@@ -2224,7 +2224,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
 #### Feedforward Control {#feedforward-control}
 
 The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
-To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [25](#figure--fig:feedforward-pd-control-joint-space).
+To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in [Figure 25](#figure--fig:feedforward-pd-control-joint-space).
 
 <a id="figure--fig:feedforward-pd-control-joint-space"></a>
 
@@ -2266,7 +2266,7 @@ By this means, the **nonlinear and coupling characteristics** of robotic manipul
 
 </div>
 
-The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in Figure [26](#figure--fig:inverse-dynamics-control-joint-space).
+The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in [Figure 26](#figure--fig:inverse-dynamics-control-joint-space).
 
 A corrective torque \\(\bm{\tau}\_{fl}\\) is added in a **feedback** structure to the closed-loop system, which is calculated from the Coriolis and Centrifugal matrix, and the gravity vector of the manipulator dynamic formulation in the joint space.
 Furthermore, the mass matrix is acting in the **forward path**, in addition to the desired trajectory acceleration \\(\ddot{\bm{q}}\_q\\).
@@ -2547,7 +2547,7 @@ Hence, it is recommended to design and implement controllers in the task space,
 
 ## Force Control {#force-control}
 
-<span class="org-target" id="org-target--sec:force:control"></span>
+<span class="org-target" id="org-target--sec-force-control"></span>
 
 
 ### Introduction {#introduction}
@@ -2596,7 +2596,7 @@ However, note that the motion control of the robot when the robot is in interact
 
 To follow **two objectives** with different properties in one control system, usually a **hierarchy** of two feedback loops is used in practice.
 This kind of control topology is called **cascade control**, which is used when there are **several measurements and one prime control variable**.
-Cascade control is implemented by **nesting** the control loops, as shown in Figure [27](#figure--fig:cascade-control).
+Cascade control is implemented by **nesting** the control loops, as shown in [Figure 27](#figure--fig:cascade-control).
 The output control loop is called the **primary loop**, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system.
 
 </div>
@@ -2628,9 +2628,9 @@ Consider the force control schemes, in which **force tracking is the prime objec
 In such a case, it is advised that the outer loop of cascade control structure is constructed by wrench feedback, while the inner loop is based on position feedback.
 Since different types of measurement units may be used in parallel robots, different control topologies may be constructed to implement such a cascade structure.<br />
 
-Consider first the cascade control topology shown in Figure [28](#figure--fig:taghira13-cascade-force-outer-loop) in which the measured variables are both in the **task space**.
+Consider first the cascade control topology shown in [Figure 28](#figure--fig:taghira13-cascade-force-outer-loop) in which the measured variables are both in the **task space**.
 The inner loop is constructed by position feedback while the outer loop is based on force feedback.
-As seen in Figure [28](#figure--fig:taghira13-cascade-force-outer-loop), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
+As seen in [Figure 28](#figure--fig:taghira13-cascade-force-outer-loop), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
 
 The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.<br />
 
@@ -2639,7 +2639,7 @@ The output of motion controller is also designed in the task space, and to conve
 {{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="<span class=\"figure-number\">Figure 28: </span>Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
 
 Other alternatives for force control topology may be suggested based on the variations of position and force measurements.
-If the force is measured in the joint space, the topology suggested in Figure [29](#figure--fig:taghira13-cascade-force-outer-loop-tau) can be used.
+If the force is measured in the joint space, the topology suggested in [Figure 29](#figure--fig:taghira13-cascade-force-outer-loop-tau) can be used.
 In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).<br />
 
 <a id="figure--fig:taghira13-cascade-force-outer-loop-tau"></a>
@@ -2647,7 +2647,7 @@ In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped i
 {{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="<span class=\"figure-number\">Figure 29: </span>Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
 
 Consider the case where the force and motion variables are both measured in the **joint space**.
-Figure [30](#figure--fig:taghira13-cascade-force-outer-loop-tau-q) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
+[Figure 30](#figure--fig:taghira13-cascade-force-outer-loop-tau-q) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
 In this topology, it is advised that the force controller is designed in the **task** space, and the Jacobian transpose mapping is used to project the measured actuator force vector into its corresponding wrench in the task space.
 However, as the inner loop is constructed in the joint space, the desired motion variable \\(\bm{\mathcal{X}}\_d\\) is mapped into joint space using **inverse kinematic** solution.
 
@@ -2665,7 +2665,7 @@ In such a case, force tracking is not the primary objective, and it is advised t
 
 Since different type of measurement units may be used in parallel robots, different control topologies may be constructed to implement such cascade controllers.<br />
 
-Figure [31](#figure--fig:taghira13-cascade-force-inner-loop-F) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
+[Figure 31](#figure--fig:taghira13-cascade-force-inner-loop-F) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
 The inner loop is loop is constructed by force feedback while the outer loop is based on position feedback.
 By this means, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed through the primary controller.
 However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
@@ -2676,14 +2676,14 @@ This configuration may be seen as if the **outer loop generates a desired force
 {{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="<span class=\"figure-number\">Figure 31: </span>Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
 
 Other alternatives for control topology may be suggested based on the variations of position and force measurements.
-If the force is measured in the joint space, control topology shown in Figure [32](#figure--fig:taghira13-cascade-force-inner-loop-tau) can be used.
+If the force is measured in the joint space, control topology shown in [Figure 32](#figure--fig:taghira13-cascade-force-inner-loop-tau) can be used.
 In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.<br />
 
 <a id="figure--fig:taghira13-cascade-force-inner-loop-tau"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="<span class=\"figure-number\">Figure 32: </span>Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
 
-If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [33](#figure--fig:taghira13-cascade-force-inner-loop-tau-q) is suggested.
+If the force and motion variables are both measured in the **joint** space, the control topology shown in [Figure 33](#figure--fig:taghira13-cascade-force-inner-loop-tau-q) is suggested.
 The inner loop is based on the measured actuator force vector in the joint space \\(\bm{\tau}\\), and the outer loop is based on the measured actuated joint position vector \\(\bm{q}\\).
 In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
 Thus, independent controllers for each joint may be suitable for this topology.
@@ -2766,7 +2766,7 @@ Nevertheless, note that Laplace transform is only applicable for **linear time i
 
 <div class="exampl">
 
-Consider an RLC circuit depicted in Figure [35](#figure--fig:taghirad13-impedance-control-rlc).
+Consider an RLC circuit depicted in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc).
 The differential equation relating voltage \\(v\\) to the current \\(i\\) is given by
 \\[ v = L\frac{di}{dt} + Ri + \int\_0^t \frac{1}{C} i(\tau)d\tau \\]
 in which \\(L\\) denote the inductance, \\(R\\) the resistance and \\(C\\) the capacitance.
@@ -2781,7 +2781,7 @@ The impedance of the system may be found from the Laplace transform of the above
 
 <div class="exampl">
 
-Consider the mass-spring-damper system depicted in Figure [35](#figure--fig:taghirad13-impedance-control-rlc).
+Consider the mass-spring-damper system depicted in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc).
 The governing dynamic formulation for this system is given by
 \\[ m \ddot{x} + c \dot{x} + k x = f \\]
 in which \\(m\\) denote the body mass, \\(c\\) the damper viscous coefficient and \\(k\\) the spring stiffness.
@@ -2811,7 +2811,7 @@ An impedance \\(\bm{Z}(s)\\) is called
 
 </div>
 
-Hence, for the mechanical system represented in Figure [35](#figure--fig:taghirad13-impedance-control-rlc):
+Hence, for the mechanical system represented in [Figure 35](#figure--fig:taghirad13-impedance-control-rlc):
 
 -   mass represents inductive impedance
 -   viscous friction represents resistive impedance
@@ -2844,9 +2844,9 @@ In the impedance control scheme, **regulation of the motion-force dynamic relati
 Therefore, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed by a primary controller.
 However, when there is an interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure may be designed to control the force-motion dynamic relation.<br />
 
-As a possible impedance control scheme, consider the closed-loop system depicted in Figure [36](#figure--fig:taghira13-impedance-control), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
+As a possible impedance control scheme, consider the closed-loop system depicted in [Figure 36](#figure--fig:taghira13-impedance-control), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
 This structure is advised when a desired impedance relation between the force and motion variables is required that consists of desired inductive, resistive, and capacitive impedances.
-As shown in Figure [36](#figure--fig:taghira13-impedance-control), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
+As shown in [Figure 36](#figure--fig:taghira13-impedance-control), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
 
 Moreover, direct force-tracking objective is not assigned in this control scheme, and therefore the desired force trajectory \\(\bm{\mathcal{F}}\_d\\) is absent in this scheme.
 However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
@@ -2859,9 +2859,9 @@ By this means, no prescribed force trajectory is tracked, while the **motion con
 The required wrench \\(\bm{\mathcal{F}}\\) in the impedance control scheme, is based on inverse dynamics control and consists of three main parts.
 In the inner loop, the force control scheme is based on a feedback linearization part in addition to a mass matrix adjustment, while in the outer loop usually a linear motion controller is considered based on the desired impedance requirements.
 
-Although many different impedance structures may be considered as the basis of the control law, in Figure [36](#figure--fig:taghira13-impedance-control), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
+Although many different impedance structures may be considered as the basis of the control law, in [Figure 36](#figure--fig:taghira13-impedance-control), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
 
-According to Figure [36](#figure--fig:taghira13-impedance-control), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
+According to [Figure 36](#figure--fig:taghira13-impedance-control), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
 \\[ \bm{\mathcal{F}} = \hat{\bm{M}} \bm{M}\_d^{-1} \bm{e}\_F + \bm{\mathcal{F}}\_{fl} \\]
 with:
 

content/inproceedings/abramovitch23_tutor_real_time_comput_issues_contr_system.md

@@ -0,0 +1,22 @@
++++
+title = "A tutorial on real-time computing issues for control systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Abramovitch et al. 2023</a>)
+
+Author(s)
+: Abramovitch, D. Y., Andersson, S., Leang, K. K., Nagel, W., &amp; Ruben, S.
+
+Year
+: 2023
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Abramovitch, Daniel Y., Sean Andersson, Kam K. Leang, William Nagel, and Shalom Ruben. 2023. “A Tutorial on Real-Time Computing Issues for Control Systems.” In <i>2023 American Control Conference (ACC)</i>, 3751–68. doi:<a href="https://doi.org/10.23919/acc55779.2023.10156102">10.23919/acc55779.2023.10156102</a>.</div>
+</div>

content/phdthesis/jabben07_mechat.md

@@ -43,9 +43,9 @@ This approach allows frequency dependent error budgeting, which is why it is ref
 #### Ground vibrations {#ground-vibrations}
 
 
-#### Electronic Noise {#electronic-noise}
+#### [Electronic Noise]({{< relref "electronic_noise.md" >}}) {#electronic-noise--electronic-noise-dot-md}
 
-**Thermal Noise** (or Johson noise).
+**Thermal Noise** (or Johnson noise).
 This noise can be modeled as a voltage source in series with the system impedance.
 The noise source has a PSD given by:
 \\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
@@ -65,7 +65,7 @@ with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{
 
 <div class="exampl">
 
-An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
+A current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
 
 </div>
 
@@ -159,7 +159,7 @@ Three factors influence the performance:
 The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
 
 The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
-In Figure [1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
+In [Figure 1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
 By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
 
 <a id="figure--fig:jabben07-general-plant"></a>
@@ -169,11 +169,11 @@ By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the
 
 #### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
 
-Since disturbances are generally not white, the system of Figure [1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
+Since disturbances are generally not white, the system of [Figure 1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
 
 A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
 
-This is illustrated in Figure [2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
+This is illustrated in [Figure 2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
 
 The generalized plant framework also allows to include **weighting filters for the performance channels**.
 This is useful for three reasons:
@@ -207,7 +207,7 @@ So, to obtain feasible controllers, the performance channel is a combination of
 By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
 \\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
 
-By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
+By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in [Figure 3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
 
 <a id="figure--fig:jabben07-pareto-curve-H2"></a>
 

content/phdthesis/li01_simul_fault_vibrat_isolat_point.md

@@ -24,13 +24,13 @@ Year
 
 ### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
 
-A general flexible jointed hexapod is shown in Figure [1](#figure--fig:li01-flexure-hexapod-model).
+A general flexible jointed hexapod is shown in [Figure 1](#figure--fig:li01-flexure-hexapod-model).
 
 <a id="figure--fig:li01-flexure-hexapod-model"></a>
 
 {{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
 
-Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#figure--fig:li01-quet-dirty-box).
+Flexure jointed hexapods have been developed to meet two needs illustrated in [Figure 2](#figure--fig:li01-quet-dirty-box).
 
 <a id="figure--fig:li01-quet-dirty-box"></a>
 
@@ -43,7 +43,7 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
 Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
 Moreover, base and/or payload vibrations become significant contributors to the motion.
 
-The University of Wyoming hexapods (example in Figure [3](#figure--fig:li01-stewart-platform)) are:
+The University of Wyoming hexapods (example in [Figure 3](#figure--fig:li01-stewart-platform)) are:
 
 -   Cubic (mutually orthogonal)
 -   Flexure Jointed
@@ -87,7 +87,7 @@ J = \begin{bmatrix}
 \end{bmatrix}
 \end{equation}
 
-where (see Figure [1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
+where (see [Figure 1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
 To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
 Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
 
@@ -131,7 +131,7 @@ Define a new input and a new output:
 u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
 \end{equation}
 
-Equation <eq:hexapod_eq_motion> can be rewritten as:
+Equation \eqref{eq:hexapod\_eq\_motion} can be rewritten as:
 
 \begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
 \begin{split}
@@ -140,7 +140,7 @@ Equation <eq:hexapod_eq_motion> can be rewritten as:
 \end{split}
 \end{equation}
 
-If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#figure--fig:li01-decoupling-conf)).
+If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled ([Figure 4](#figure--fig:li01-decoupling-conf)).
 
 <a id="figure--fig:li01-decoupling-conf"></a>
 
@@ -152,7 +152,7 @@ Alternatively, a new set of inputs and outputs can be defined:
 u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
 \end{equation}
 
-And another decoupled plant is found (Figure [5](#figure--fig:li01-decoupling-conf-bis)):
+And another decoupled plant is found ([Figure 5](#figure--fig:li01-decoupling-conf-bis)):
 
 \begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
 \begin{split}
@@ -200,13 +200,13 @@ The control bandwidth is divided as follows:
 
 ### Vibration Isolation {#vibration-isolation}
 
-The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#figure--fig:li01-vibration-isolation-control).
+The system is decoupled into six independent SISO subsystems using the architecture shown in [Figure 6](#figure--fig:li01-vibration-isolation-control).
 
 <a id="figure--fig:li01-vibration-isolation-control"></a>
 
 {{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="<span class=\"figure-number\">Figure 6: </span>Vibration isolation control strategy" >}}
 
-One of the subsystem plant transfer function is shown in Figure [6](#figure--fig:li01-vibration-isolation-control)
+One of the subsystem plant transfer function is shown in [Figure 6](#figure--fig:li01-vibration-isolation-control)
 
 <a id="figure--fig:li01-vibration-isolation-control"></a>
 
@@ -243,7 +243,7 @@ The reason is not explained.
 
 ### Pointing Control Techniques {#pointing-control-techniques}
 
-A block diagram of the pointing control system is shown in Figure [8](#figure--fig:li01-pointing-control).
+A block diagram of the pointing control system is shown in [Figure 8](#figure--fig:li01-pointing-control).
 
 <a id="figure--fig:li01-pointing-control"></a>
 
@@ -252,7 +252,7 @@ A block diagram of the pointing control system is shown in Figure [8](#figure--f
 The plant is decoupled into two independent SISO subsystems.
 The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
 
-Figure [9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
+[Figure 9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
 
 <a id="figure--fig:li01-transfer-function-angle"></a>
 
@@ -268,7 +268,7 @@ A typical compensator consists of the following elements:
 
 The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
 
-A feedforward control is added as shown in Figure [10](#figure--fig:li01-feedforward-control).
+A feedforward control is added as shown in [Figure 10](#figure--fig:li01-feedforward-control).
 \\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
 Ideally, the feedforward compensator is an invert of the plant dynamics.
 
@@ -284,7 +284,7 @@ The simultaneous vibration isolation and pointing control is approached in two w
 1.  **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
 2.  **Closing the pointing loop first**: Reverse order.
 
-Figure [11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
+[Figure 11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
 
 <a id="figure--fig:li01-parallel-control"></a>
 
@@ -302,16 +302,16 @@ However, the interaction between loops may affect the transfer functions of the
 The dynamic interaction effect:
 
 -   Only happens in the unity bandwidth of the loop transmission of the first closed loop.
--   Affect the closed loop transmission of the loop first closed (see Figures [12](#figure--fig:li01-closed-loop-pointing) and [13](#figure--fig:li01-closed-loop-vibration))
+-   Affect the closed loop transmission of the loop first closed (see [Figure 12](#figure--fig:li01-closed-loop-pointing) and [Figure 13](#figure--fig:li01-closed-loop-vibration))
 
-As shown in Figure [12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
+As shown in [Figure 12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
 The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
 
 <a id="figure--fig:li01-closed-loop-pointing"></a>
 
 {{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="<span class=\"figure-number\">Figure 12: </span>Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
 
-The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#figure--fig:li01-closed-loop-vibration)).
+The same happens when first closing the vibration isolation loop and after the pointing loop ([Figure 13](#figure--fig:li01-closed-loop-vibration)).
 The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
 
 <a id="figure--fig:li01-closed-loop-vibration"></a>
@@ -328,7 +328,7 @@ Thus, it is recommended to design and implement the isolation control system fir
 
 ### Experimental results {#experimental-results}
 
-Two hexapods are stacked (Figure [14](#figure--fig:li01-test-bench)):
+Two hexapods are stacked ([Figure 14](#figure--fig:li01-test-bench)):
 
 -   the bottom hexapod is used to generate disturbances matching candidate applications
 -   the top hexapod provide simultaneous vibration isolation and pointing control
@@ -338,7 +338,7 @@ Two hexapods are stacked (Figure [14](#figure--fig:li01-test-bench)):
 {{< figure src="/ox-hugo/li01_test_bench.png" caption="<span class=\"figure-number\">Figure 14: </span>Stacked Hexapods" >}}
 
 First, the vibration isolation and pointing controls were implemented separately.
-Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#figure--fig:li01-vibration-isolation-control-results).
+Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in [Figure 15](#figure--fig:li01-vibration-isolation-control-results).
 
 <a id="figure--fig:li01-vibration-isolation-control-results"></a>
 
@@ -349,7 +349,7 @@ The simultaneous control is of dual use:
 -   it provide simultaneous pointing and isolation control
 -   it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
 
-The results of simultaneous control is shown in Figure [16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
+The results of simultaneous control is shown in [Figure 16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
 
 <a id="figure--fig:li01-simultaneous-control-results"></a>
 

content/phdthesis/monkhorst04_dynam_error_budget.md

@@ -106,7 +106,7 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
 
 In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
 
-This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#figure--fig:monkhorst04-weighting-filter)).
+This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise ([Figure 1](#figure--fig:monkhorst04-weighting-filter)).
 
 <a id="figure--fig:monkhorst04-weighting-filter"></a>
 
@@ -119,7 +119,7 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
 Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
 However, this can be avoided if the modeling of the disturbances is directly done in terms of weighting filters.
 
-Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#figure--fig:monkhorst04-general-weighted-plant)).
+Output weighting filters can also be used to scale different outputs relative to each other ([Figure 2](#figure--fig:monkhorst04-general-weighted-plant)).
 
 <a id="figure--fig:monkhorst04-general-weighted-plant"></a>
 
@@ -128,7 +128,7 @@ Output weighting filters can also be used to scale different outputs relative to
 
 #### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
 
-In this research, the outputs of the closed loop system (Figure [3](#figure--fig:monkhorst04-closed-loop-H2)) are:
+In this research, the outputs of the closed loop system ([Figure 3](#figure--fig:monkhorst04-closed-loop-H2)) are:
 
 -   the performance (error) signal \\(e\\)
 -   the controller output \\(u\\)

content/phdthesis/rankers98_machin.md

@@ -170,7 +170,7 @@ The basic questions that are addressed in this thesis are:
 
 ### Basic Control Aspects {#basic-control-aspects}
 
-A block diagram representation of a typical servo-system is shown in Figure [1](#figure--fig:rankers98-basic-el-mech-servo).
+A block diagram representation of a typical servo-system is shown in [Figure 1](#figure--fig:rankers98-basic-el-mech-servo).
 The main task of the system is to achieve a desired positional relation between two or more components of the system.
 Therefore, a sensor measures the position which is then compared to the desired value, and the resulting error is used to generate correcting forces.
 In most systems, the "actual output" (e.g. position of end-effector) cannot be measured directly, and the feedback will therefore be based on a "measured output" (e.g. encoder signal at the motor).
@@ -187,7 +187,7 @@ The correction force \\(F\\) is defined by:
 F = k\_p \epsilon + k\_d \dot{\epsilon} + k\_i \int \epsilon dt
 \end{equation}
 
-It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#figure--fig:rankers98-basic-elastic-struct)).
+It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points ([Figure 2](#figure--fig:rankers98-basic-elastic-struct)).
 If \\(c\\) and \\(d\\) represent the constant mechanical stiffness and damping between points \\(A\\) and \\(B\\), and a reference position profile \\(h(t)\\) is applied at \\(A\\), then an opposing force \\(F\\) is generated as soon as the position \\(x\\) and speed \\(\dot{x}\\) of point \\(B\\) does not correspond to \\(h(t)\\) and \\(\dot{h}(t)\\).
 
 <a id="figure--fig:rankers98-basic-elastic-struct"></a>
@@ -206,7 +206,7 @@ These properties are very essential since they introduce the issue of **servo st
 
 An important aspect of a feedback controller is the fact that control forces can only result from an error signal.
 Thus any desired set-point profile first leads to a position error before the corresponding driving forces are generated.
-Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in Figure [3](#figure--fig:rankers98-feedforward-example).
+Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in [Figure 3](#figure--fig:rankers98-feedforward-example).
 
 <a id="figure--fig:rankers98-feedforward-example"></a>
 
@@ -253,7 +253,7 @@ Basically, machine dynamics can have two deterioration effects in mechanical ser
 
 #### Actuator Flexibility {#actuator-flexibility}
 
-The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#figure--fig:rankers98-actuator-flexibility)) due to **compliance between the motor and the load**.
+The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body ([Figure 4](#figure--fig:rankers98-actuator-flexibility)) due to **compliance between the motor and the load**.
 
 <a id="figure--fig:rankers98-actuator-flexibility"></a>
 
@@ -265,7 +265,7 @@ The basic characteristics of what is called "actuator flexibility" is the fact t
 The second category of dynamic phenomena results from the **limited stiffness of the guiding system** in combination with the fact the the device is driven in such a way that it has to rely on the guiding system to suppress motion in an undesired direction (in case of a linear direct drive system this occurs if the driving force is not applied at the center of gravity).
 
 In general, a rigid actuator possesses six degrees of freedom, five of which need to be suppressed by the guiding system in order to leave one mobile degree of freedom.
-In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#figure--fig:rankers98-guiding-flexibility-planar)).
+In the present discussion, a planar actuator with three degrees of freedom will be considered ([Figure 5](#figure--fig:rankers98-guiding-flexibility-planar)).
 
 <a id="figure--fig:rankers98-guiding-flexibility-planar"></a>
 
@@ -287,7 +287,7 @@ The last category of dynamic phenomena results from the **limited mass and stiff
 In contrast to many textbooks on mechanics and machine dynamics, it is good practice always to look at the combination of driving force on the moving part, and **reaction force** on the stationary part, of a positioning device.
 When doing so, one has to consider what the effect of the reaction force on the systems performance will be.
 In the discussion of the previous two dynamic phenomena, the stationary part of the machine was assumed to be infinitely stiff and heavy, and therefore the effect of the reaction force was negligible.
-However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#figure--fig:rankers98-limited-m-k-stationary-machine-part)).
+However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces ([Figure 6](#figure--fig:rankers98-limited-m-k-stationary-machine-part)).
 
 <a id="figure--fig:rankers98-limited-m-k-stationary-machine-part"></a>
 
@@ -302,7 +302,7 @@ The effect of frame vibrations is even worse where the quality of positioning of
 
 To understand and describe the behaviour of a mechanical system in a quantitative way, one usually sets up a model of the system.
 The mathematical description of such a model with a finite number of DoF consists of a set of ordinary differential equations.
-Although in the case of simple systems, such as illustrated in Figure [7](#figure--fig:rankers98-1dof-system) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
+Although in the case of simple systems, such as illustrated in [Figure 7](#figure--fig:rankers98-1dof-system) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
 
 <a id="figure--fig:rankers98-1dof-system"></a>
 
@@ -335,17 +335,17 @@ These eigenvectors have the following orthogonality properties, or can always be
 \phi\_i^T M \phi\_j = 0 \quad (i \neq j)
 \end{equation}
 
-For \\(i=j\\) the result of the multiplication according to equation <eq:eigenvector_orthogonality_mass> yields a non-zero result, which is normally indicated as **modal mass** \\(\mathit{m}\_i\\):
+For \\(i=j\\) the result of the multiplication according to equation \eqref{eq:eigenvector\_orthogonality\_mass} yields a non-zero result, which is normally indicated as **modal mass** \\(\mathit{m}\_i\\):
 
 \begin{equation} \label{eq:modal\_mass}
 \phi\_i^T M \phi\_i = \mathit{m}\_i
 \end{equation}
 
-Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation <eq:modal_mass>:
+Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation \eqref{eq:modal\_mass:}
 
--   \\(|\phi\_i| = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation <eq:modal_mass>.
+-   \\(|\phi\_i| = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation \eqref{eq:modal\_mass}.
--   \\(\max(\phi\_i) = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation <eq:modal_mass>.
+-   \\(\max(\phi\_i) = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation \eqref{eq:modal\_mass}.
--   \\(m\_i = 1\\). The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation <eq:modal_mass>.
+-   \\(m\_i = 1\\). The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation \eqref{eq:modal\_mass}.
 
 The orthogonality properties also apply to the stiffness matrix \\(K\\):
 
@@ -450,7 +450,7 @@ In such an analysis one is typically interested in the transfer function between
 Applying the principle of modal decomposition, any transfer function can be derived by first calculating the behavior of the individual modes, and then **summing all modal contributions**.
 
 The contribution of one single mode \\(i\\) to the transfer function \\(x\_l/f\_k\\) can be derived by first considering the response of the modal DoF \\(q\_i\\) to a force vector \\(f\\) with only one non-zero component \\(f\_k\\).
-In that case, equation <eq:eoq_modal_i> is reduced to:
+In that case, equation \eqref{eq:eoq\_modal\_i} is reduced to:
 
 \begin{equation}
 m\_i \ddot{q}\_i(t) + k\_i q\_i(t) = \phi\_{ik} f\_k(t)
@@ -481,17 +481,17 @@ The overall transfer function can be found by summation of the individual modal
 
 ### Graphical Representation {#graphical-representation}
 
-Due to the equivalence with the differential equations of a single mass spring system, equation <eq:eoq_modal_i> is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts.
+Due to the equivalence with the differential equations of a single mass spring system, equation \eqref{eq:eoq\_modal\_i} is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts.
 However, this representation implies an important loss of information because it neglects all information about the mode-shape vector.
 
-Consider the system in Figure [8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation.
+Consider the system in [Figure 8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation.
 In this representation, the physical DoF are located at fixed positions and the mode shapes displacement is indicated by the length of an arrow.
 
 <a id="figure--fig:rankers98-mode-trad-representation"></a>
 
 {{< figure src="/ox-hugo/rankers98_mode_trad_representation.png" caption="<span class=\"figure-number\">Figure 8: </span>System and traditional graphical representation of modes" >}}
 
-Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** (Figure [9](#figure--fig:rankers98-mode-new-representation)).
+Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** ([Figure 9](#figure--fig:rankers98-mode-new-representation)).
 
 <div class="important">
 
@@ -503,20 +503,20 @@ System with no, very little, or proportional damping exhibit real mode shape vec
 Consequently, the respective DoF can only be in phase or in opposite phase.
 All DoF on the same side of the rotation point have identical phases, whereas DoF on opposite sides have opposite phases.
 
-The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#figure--fig:rankers98-mode-new-representation)).
+The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point ([Figure 9](#figure--fig:rankers98-mode-new-representation)).
 
 <a id="figure--fig:rankers98-mode-new-representation"></a>
 
 {{< figure src="/ox-hugo/rankers98_mode_new_representation.png" caption="<span class=\"figure-number\">Figure 9: </span>System and new graphical representation of mode-shape" >}}
 
-In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in Figure [10](#figure--fig:rankers98-mode-2-lumped-masses) (a).
+In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in [Figure 10](#figure--fig:rankers98-mode-2-lumped-masses) (a).
 The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is:
 
 \begin{equation}
 J\_i = \sum\_{k=1}^n m\_k \phi\_{ik}^2
 \end{equation}
 
-This result is identical to the modal mass \\(m\_i\\) found with Equation <eq:modal_mass>, because the mass matrix \\(M\\) is a diagonal matrix of physical masses \\(m\_k\\), and consequently the expression for the modal mass \\(m\_i\\) yields:
+This result is identical to the modal mass \\(m\_i\\) found with Equation \eqref{eq:modal\_mass}, because the mass matrix \\(M\\) is a diagonal matrix of physical masses \\(m\_k\\), and consequently the expression for the modal mass \\(m\_i\\) yields:
 
 \begin{equation}
 m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2
@@ -524,7 +524,7 @@ m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2
 
 As a result of this, the **modal mass** \\(m\_i\\) could be interpreted as the resulting mass moment of inertia of the modal lever, or alternatively as a **mass located at a distance "1" from the pivot point**.
 
-The transition from physical masses to modal masses is illustrated in Figure [10](#figure--fig:rankers98-mode-2-lumped-masses) for the mode 2 of the example system.
+The transition from physical masses to modal masses is illustrated in [Figure 10](#figure--fig:rankers98-mode-2-lumped-masses) for the mode 2 of the example system.
 The modal stiffness \\(k\_2\\) is simply calculated via the relation between natural frequency, mass and stiffness:
 
 \begin{equation}
@@ -537,7 +537,7 @@ k\_i = \omega\_i^2 m\_i
 
 Let's now consider the effect of excitation forces that act on the physical DoF.
 The scalar product \\(\phi\_{ik}f\_k\\) of each force component with the corresponding element of the mode shape vector can be seen as the moment that acts on the modal level, or as an equivalent force that acts at the location of \\(q\_i\\) on the lever.
-Based on the graphical representation in Figure [11](#figure--fig:rankers98-lever-representation-with-force), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
+Based on the graphical representation in [Figure 11](#figure--fig:rankers98-lever-representation-with-force), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
 
 \begin{equation}
 \boxed{\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}}
@@ -556,7 +556,7 @@ This linear combination of physical DoF, which will be called "User DoF" can be
 x\_u = b\_1 x\_1 + \dots + b\_n x\_n = b^T x
 \end{equation}
 
-User DoF can be indicated on the modal lever, as illustrated in Figure [12](#figure--fig:rankers98-representation-user-dof) for a user DoF \\(x\_u = x\_3 - x\_2\\).
+User DoF can be indicated on the modal lever, as illustrated in [Figure 12](#figure--fig:rankers98-representation-user-dof) for a user DoF \\(x\_u = x\_3 - x\_2\\).
 The location of this user DoF \\(x\_u\\) with respect to the pivot point of modal lever \\(i\\) is defined by \\(\phi\_{iu}\\):
 
 \begin{equation}
@@ -575,7 +575,7 @@ Even though the dimension mode vector can be very large, only **three user DoF**
 
 <div class="exampl">
 
-To illustrate this, a servo controlled positioning device is shown in Figure [13](#figure--fig:rankers98-servo-system).
+To illustrate this, a servo controlled positioning device is shown in [Figure 13](#figure--fig:rankers98-servo-system).
 The task of the device is to position the payload with respect to a tool that is mounted to the machine frame.
 The actual accuracy of the machine is determined by the relative motion of these two components (actual output).
 However, direct measurement of the distance between the tool and the payload is not possible and therefore the control action is based on the measured distance between a sensor and the slide on which the payload is mounted (measured output).
@@ -605,7 +605,7 @@ These effective modal parameters can be used very effectively in understanding t
 
 <div class="exampl">
 
-The eigenvalue analysis of the two mass spring system in Figure [14](#figure--fig:rankers98-example-2dof) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#figure--fig:rankers98-example-2dof-modal).
+The eigenvalue analysis of the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) leads to the modal results summarized in [Table 1](#table--tab:2dof-example-modal-params) and which are graphically represented in [Figure 15](#figure--fig:rankers98-example-2dof-modal).
 
 <a id="figure--fig:rankers98-example-2dof"></a>
 
@@ -622,7 +622,7 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
 
 <a id="table--tab:2dof-example-modal-params"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:2dof-example-modal-params">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:2dof-example-modal-params">Table 1</a>:</span>
   Modal results for the two mass spring system
 </div>
 
@@ -637,12 +637,12 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
 
 {{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="<span class=\"figure-number\">Figure 15: </span>Graphical representation of modes and modal parameters of the two mass spring system" >}}
 
-From these results, the effective modal parameters for each mode, and for each individual DoF can be defined using equations <eq:m_modal_eff> and <eq:k_modal_eff>.
+From these results, the effective modal parameters for each mode, and for each individual DoF can be defined using equations \eqref{eq:m\_modal\_eff} and \eqref{eq:k\_modal\_eff}.
-The results are summarized in Table [2](#table--tab:2dof-example-modal-params-eff).
+The results are summarized in [Table 2](#table--tab:2dof-example-modal-params-eff).
 
 <a id="table--tab:2dof-example-modal-params-eff"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:2dof-example-modal-params-eff">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:2dof-example-modal-params-eff">Table 2</a>:</span>
   Effective modal parameters for the two mass spring system
 </div>
 
@@ -653,8 +653,8 @@ The results are summarized in Table [2](#table--tab:2dof-example-modal-params-ef
 | Effective stiff - DoF 1 | \\(k\_{\text{eff},11} = 1.02 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},21} = 1.02 \cdot 10^9 N/m\\) |
 | Effective stiff - DoF 2 | \\(k\_{\text{eff},12} = 0.84 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},22} = 1.25 \cdot 10^7 N/m\\) |
 
-The effective modal parameters can then be used in the graphical representation of Figure [16](#figure--fig:rankers98-example-2dof-effective-modal).
+The effective modal parameters can then be used in the graphical representation of [Figure 16](#figure--fig:rankers98-example-2dof-effective-modal).
-Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system (Figure [17](#figure--fig:rankers98-2dof-example-frf)).
+Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system ([Figure 17](#figure--fig:rankers98-2dof-example-frf)).
 
 <a id="figure--fig:rankers98-example-2dof-effective-modal"></a>
 
@@ -662,7 +662,7 @@ Based on this representation, it is now very easy to construct the individual mo
 
 One can observe that the low frequency part of each modal contribution corresponds to the inverse of the calculated effective modal mass stiffness at DoF \\(x\_1\\) whereas the high frequency contribution is defined by the effective modal mass.
 
-In the final Bode diagram (Figure [17](#figure--fig:rankers98-2dof-example-frf), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
+In the final Bode diagram ([Figure 17](#figure--fig:rankers98-2dof-example-frf), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
 
 <a id="figure--fig:rankers98-2dof-example-frf"></a>
 
@@ -690,7 +690,7 @@ The technique furthermore gives an indication of the amount of frequency shift t
 
 <div class="exampl">
 
-Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#figure--fig:rankers98-example-3dof-sensitivity) by attaching a linear spring \\(k\\) between two of the three represented DoF.
+Assuming that one is asked to increase the natural frequency of the mode corresponding to [Figure 18](#figure--fig:rankers98-example-3dof-sensitivity) by attaching a linear spring \\(k\\) between two of the three represented DoF.
 As the relative motion between \\(x\_A\\) and \\(x\_B\\) is the largest of all possible combinations, this is the choice that will maximize the natural frequency of the mode.
 
 <a id="figure--fig:rankers98-example-3dof-sensitivity"></a>
@@ -713,26 +713,26 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
 
 <div class="exampl">
 
-Let's use the two mass spring system in Figure [14](#figure--fig:rankers98-example-2dof) as an example.
+Let's use the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) as an example.
 
-In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see Table [2](#table--tab:2dof-example-modal-params-eff)).
+In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see [Table 2](#table--tab:2dof-example-modal-params-eff)).
-Then using equation <eq:sensitivity_add_m>, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\).
+Then using equation \eqref{eq:sensitivity\_add\_m}, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\).
 
 To estimate the influence of extra stiffness between the two DoF, one needs to calculate the effective modal stiffness that corresponds to the relative motion between \\(x\_2\\) and \\(x\_1\\).
-This can be graphically done as shown in Figure [19](#figure--fig:rankers98-example-sensitivity-2dof):
+This can be graphically done as shown in [Figure 19](#figure--fig:rankers98-example-sensitivity-2dof):
 
 \begin{align}
 k\_{\text{eff},1,(2-1)} &= 0.46 \cdot 10^7 / 0.07^2 = 93.9 \cdot 10^7   N/m \\\\
 k\_{\text{eff},2,(2-1)} &= 1.23 \cdot 10^7 / 1.1^2 = 1.0 \cdot 10^7   N/m
 \end{align}
 
-And using equation <eq:sensitivity_add_m>, the effect of additional stiffness on the frequency of the two modes can be computed.
+And using equation \eqref{eq:sensitivity\_add\_m}, the effect of additional stiffness on the frequency of the two modes can be computed.
 
-The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-results).
+The results are summarized in [Table 3](#table--tab:example-sensitivity-2dof-results).
 
 <a id="table--tab:example-sensitivity-2dof-results"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:example-sensitivity-2dof-results">Table 3</a></span>:
+  <span class="table-number"><a href="#table--tab:example-sensitivity-2dof-results">Table 3</a>:</span>
   Sensitivity analysis results
 </div>
 
@@ -752,7 +752,7 @@ The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-res
 ### Modal Superposition {#modal-superposition}
 
 Previously, the lever representation was used only to represent the individual mode shapes.
-In the mechanism shown in Figure [20](#figure--fig:rankers98-addition-of-motion), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
+In the mechanism shown in [Figure 20](#figure--fig:rankers98-addition-of-motion), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
 
 <a id="figure--fig:rankers98-addition-of-motion"></a>
 
@@ -764,7 +764,7 @@ This approach can be applied to the concept of modal superposition, which expres
 x\_k(t) = \sum\_{i=1}^n \phi\_{ik} q\_i(t) = \sum\_{i=1}^n x\_{ki}(t)
 \end{equation}
 
-Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to Figure [21](#figure--fig:rankers98-conversion-modal-to-physical), which is a visualization of the transformation between the modal and the physical domains.
+Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to [Figure 21](#figure--fig:rankers98-conversion-modal-to-physical), which is a visualization of the transformation between the modal and the physical domains.
 
 <a id="figure--fig:rankers98-conversion-modal-to-physical"></a>
 
@@ -777,7 +777,7 @@ The "rigid body modes" usually refer to the lower natural frequencies of a machi
 This is misleading at it suggests that the structure exhibits no internal deformation.
 A better term for such a mode would be **suspension mode**.
 
-To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#figure--fig:rankers98-suspension-mode-machine)).
+To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered ([Figure 22](#figure--fig:rankers98-suspension-mode-machine)).
 
 <a id="figure--fig:rankers98-suspension-mode-machine"></a>
 
@@ -828,7 +828,7 @@ The interaction between the desired (rigid body) motion and the dynamics of one
 
 ### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
 
-Consider the position control loop of Figure [23](#figure--fig:rankers98-mechanical-servo-system).
+Consider the position control loop of [Figure 23](#figure--fig:rankers98-mechanical-servo-system).
 
 <a id="figure--fig:rankers98-mechanical-servo-system"></a>
 
@@ -840,7 +840,7 @@ In the ideal situation the mechanical system behaves as one rigid body with mass
 \frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2}
 \end{equation}
 
-The corresponding Bode and Nyquist plots and shown in Figure [24](#figure--fig:rankers98-ideal-bode-nyquist).
+The corresponding Bode and Nyquist plots and shown in [Figure 24](#figure--fig:rankers98-ideal-bode-nyquist).
 
 <a id="figure--fig:rankers98-ideal-bode-nyquist"></a>
 
@@ -860,15 +860,15 @@ Let's introduce a variable \\(\alpha\\), which **relates the high-frequency cont
 \alpha = \frac{\frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i}}{\frac{1}{m}}
 \end{equation}
 
-which simplifies equation <eq:effect_one_mode> to:
+which simplifies equation \eqref{eq:effect\_one\_mode} to:
 
 \begin{equation} \label{eq:effect\_one\_mode\_simplified}
 \boxed{\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}}
 \end{equation}
 
-Equation <eq:effect_one_mode_simplified> will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability.
+Equation \eqref{eq:effect\_one\_mode\_simplified} will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability.
 
-Three different types of intersection pattern can be found in the amplitude plot as shown in Figure [25](#figure--fig:rankers98-frf-effect-alpha).
+Three different types of intersection pattern can be found in the amplitude plot as shown in [Figure 25](#figure--fig:rankers98-frf-effect-alpha).
 Depending on the absolute value of \\(\alpha\\) one can observe:
 
 -   \\(|\alpha| < 1\\): two intersections
@@ -881,7 +881,7 @@ The interaction between the rigid body motion and the additional mode will not o
 
 {{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="<span class=\"figure-number\">Figure 25: </span>Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
 
-The general shape of the overall FRF can be constructed for all cases (Figure [26](#figure--fig:rankers98-final-frf-alpha)).
+The general shape of the overall FRF can be constructed for all cases ([Figure 26](#figure--fig:rankers98-final-frf-alpha)).
 Interesting points are the interaction of the two parts at the frequency that corresponds to an intersection in the amplitude plot.
 At this frequency the magnitudes are equal, so it depends on the phase of the two contributions whether they cancel each other, thus leading to a zero, or just add up.
 
@@ -891,7 +891,7 @@ At this frequency the magnitudes are equal, so it depends on the phase of the tw
 
 <div class="important">
 
-When analyzing the plots of Figure [26](#figure--fig:rankers98-final-frf-alpha), four different types of FRF can be found:
+When analyzing the plots of [Figure 26](#figure--fig:rankers98-final-frf-alpha), four different types of FRF can be found:
 
 -   -2 slope / zero / pole / -2 slope (\\(\alpha > 0\\))
 -   -2 slope / pole / zero / -2 slope (\\(-1 < \alpha < 0\\))
@@ -900,7 +900,7 @@ When analyzing the plots of Figure [26](#figure--fig:rankers98-final-frf-alpha),
 
 </div>
 
-All cases are shown in Figure [27](#figure--fig:rankers98-interaction-shapes).
+All cases are shown in [Figure 27](#figure--fig:rankers98-interaction-shapes).
 
 <a id="figure--fig:rankers98-interaction-shapes"></a>
 
@@ -921,13 +921,13 @@ f\_{lp} &= 4 \cdot f\_b
 \end{align\*}
 
 with \\(f\_b\\) the bandwidth frequency.
-The asymptotic amplitude plot is shown in Figure [28](#figure--fig:rankers98-pid-amplitude).
+The asymptotic amplitude plot is shown in [Figure 28](#figure--fig:rankers98-pid-amplitude).
 
 <a id="figure--fig:rankers98-pid-amplitude"></a>
 
 {{< figure src="/ox-hugo/rankers98_pid_amplitude.png" caption="<span class=\"figure-number\">Figure 28: </span>Typical crossover frequencies of a PID controller with 2nd order low pass filtering" >}}
 
-With these settings, the open loop response of the position loop (controller + mechanics) looks like Figure [29](#figure--fig:rankers98-ideal-frf-pid).
+With these settings, the open loop response of the position loop (controller + mechanics) looks like [Figure 29](#figure--fig:rankers98-ideal-frf-pid).
 
 <a id="figure--fig:rankers98-ideal-frf-pid"></a>
 
@@ -937,10 +937,10 @@ With these settings, the open loop response of the position loop (controller + m
 
 Conclusions are:
 
--   A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency (Figure [30](#figure--fig:rankers98-zero-pole-low-freq)) and high frequency (Figure [32](#figure--fig:rankers98-zero-pole-high-freq)) regions.
+-   A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency ([Figure 30](#figure--fig:rankers98-zero-pole-low-freq)) and high frequency ([Figure 32](#figure--fig:rankers98-zero-pole-high-freq)) regions.
-    In the medium frequency region (Figure [31](#figure--fig:rankers98-zero-pole-medium-freq)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
+    In the medium frequency region ([Figure 31](#figure--fig:rankers98-zero-pole-medium-freq)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
 -   A "-2 slope / pole / zero / -2 slope" combination has the reverse effect.
-    It is potentially destabilizing in the medium frequency range (Figure [34](#figure--fig:rankers98-pole-zero-medium-freq)) and is harmless in the low (Figure [33](#figure--fig:rankers98-pole-zero-low-freq)) and high frequency (Figure [35](#figure--fig:rankers98-pole-zero-high-freq)) ranges.
+    It is potentially destabilizing in the medium frequency range ([Figure 34](#figure--fig:rankers98-pole-zero-medium-freq)) and is harmless in the low ([Figure 33](#figure--fig:rankers98-pole-zero-low-freq)) and high frequency ([Figure 35](#figure--fig:rankers98-pole-zero-high-freq)) ranges.
 -   The "-2 slope / pole / -4 slope" behavior always has a devastating effect on the stability of the loop if located in the low of medium frequency ranges.
 
 These conclusions may differ for different mass ratio \\(\alpha\\).
@@ -977,13 +977,13 @@ These conclusions may differ for different mass ratio \\(\alpha\\).
 
 #### Actuator Flexibility {#actuator-flexibility}
 
-Figure [36](#figure--fig:rankers98-2dof-actuator-flexibility) shows the schematic representation of a system with a certain compliance between the motor and the load.
+[Figure 36](#figure--fig:rankers98-2dof-actuator-flexibility) shows the schematic representation of a system with a certain compliance between the motor and the load.
 
 <a id="figure--fig:rankers98-2dof-actuator-flexibility"></a>
 
 {{< figure src="/ox-hugo/rankers98_2dof_actuator_flexibility.png" caption="<span class=\"figure-number\">Figure 36: </span>Servo system with actuator flexibility - Schematic representation" >}}
 
-The corresponding modes are shown in Figure [37](#figure--fig:rankers98-2dof-modes-act-flex).
+The corresponding modes are shown in [Figure 37](#figure--fig:rankers98-2dof-modes-act-flex).
 
 <a id="figure--fig:rankers98-2dof-modes-act-flex"></a>
 
@@ -998,7 +998,7 @@ The following transfer function must be considered:
 \end{align}
 
 with \\(\alpha = m\_2/m\_1\\) (mass ratio) relates the "mass" of the additional modal contribution to the mass of the rigid body motion.
-The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" (Figure [38](#figure--fig:rankers98-2dof-act-flex-frf)).
+The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" ([Figure 38](#figure--fig:rankers98-2dof-act-flex-frf)).
 
 <a id="figure--fig:rankers98-2dof-act-flex-frf"></a>
 
@@ -1029,7 +1029,7 @@ Now we are interested by the following transfer function:
 \frac{x\_2}{F\_{\text{servo}}} = = \frac{1}{m\_1 + m\_2} \left( \frac{1}{s^2} - \frac{1}{s^2 + \omega\_2^2} \right)
 \end{equation}
 
-The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" (Figure [39](#figure--fig:rankers98-2dof-act-flex-meas-load-frf)).
+The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" ([Figure 39](#figure--fig:rankers98-2dof-act-flex-meas-load-frf)).
 
 <a id="figure--fig:rankers98-2dof-act-flex-meas-load-frf"></a>
 
@@ -1046,7 +1046,7 @@ Guideline in presence of actuator flexibility with measurement at the load posit
 
 #### Guiding System Flexibility {#guiding-system-flexibility}
 
-Here, the influence of a limited guiding stiffness (Figure [40](#figure--fig:rankers98-2dof-guiding-flex)) on the FRF of such an actuator system will be analyzed.
+Here, the influence of a limited guiding stiffness ([Figure 40](#figure--fig:rankers98-2dof-guiding-flex)) on the FRF of such an actuator system will be analyzed.
 
 The servo force \\(F\_{\text{servo}}\\) acts at a certain distance \\(a\_F\\) with respect to the center of gravity, and the servo position is measured at a distance \\(a\_s\\) with respect to the center of gravity.
 Due to the symmetry of the system, the Y motion is decoupled from the X and \\(\phi\\) motions and can therefore be omitted in this analysis.
@@ -1055,7 +1055,7 @@ Due to the symmetry of the system, the Y motion is decoupled from the X and \\(\
 
 {{< figure src="/ox-hugo/rankers98_2dof_guiding_flex.png" caption="<span class=\"figure-number\">Figure 40: </span>2DoF rigid body model of actuator with flexibility of the guiding system" >}}
 
-Considering the two relevant modes (Figures [41](#figure--fig:rankers98-2dof-guiding-flex-x-mode) and [42](#figure--fig:rankers98-2dof-guiding-flex-rock-mode)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
+Considering the two relevant modes ([Figure 41](#figure--fig:rankers98-2dof-guiding-flex-x-mode) and [Figure 42](#figure--fig:rankers98-2dof-guiding-flex-rock-mode)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
 
 \begin{equation}
 \frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{a\_s a\_F}{Js^2 + 2cb^2}
@@ -1103,7 +1103,7 @@ As this point, the resonance will not be present in the FRF.
 
 #### Limited Mass and Stiffness of Stationary Machine Part {#limited-mass-and-stiffness-of-stationary-machine-part}
 
-Figure [43](#figure--fig:rankers98-frame-dynamics-2dof) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
+[Figure 43](#figure--fig:rankers98-frame-dynamics-2dof) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
 
 <a id="figure--fig:rankers98-frame-dynamics-2dof"></a>
 
@@ -1138,7 +1138,7 @@ Guidelines regarding frame motion:
 
 <div class="sum">
 
-The amount of contribution of a certain mode (Figure [44](#figure--fig:rankers98-mode-representation-guideline)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
+The amount of contribution of a certain mode ([Figure 44](#figure--fig:rankers98-mode-representation-guideline)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
 
 <a id="figure--fig:rankers98-mode-representation-guideline"></a>
 
@@ -1167,7 +1167,7 @@ If such a modification is not required and the modes are not excited by some oth
 
 ### Steps in a Modelling Activity {#steps-in-a-modelling-activity}
 
-One can distinguish at least four steps in any modelling activity (Figure [45](#figure--fig:rankers98-steps-modelling)).
+One can distinguish at least four steps in any modelling activity ([Figure 45](#figure--fig:rankers98-steps-modelling)).
 
 <a id="figure--fig:rankers98-steps-modelling"></a>
 
@@ -1176,10 +1176,10 @@ One can distinguish at least four steps in any modelling activity (Figure [45](#
 1.  The first step consists of a translation of the real structure or initial design drawing of a structure in a **physical model**.
     Such a physical model is a simplification of the reality, but contains all relations that are considered to be important to describe the investigated phenomenon.
     This step requires experience and engineering judgment in order to determine which simplifications are valid.
-    See for example Figure [46](#figure--fig:rankers98-illustration-first-two-steps).
+    See for example [Figure 46](#figure--fig:rankers98-illustration-first-two-steps).
 2.  Once a physical model has been derived, the second step consists of translating this physical  model into a **mathematical model**.
     The real world is now represented by a set of differential equations.
-    This step is fairly straightforward, because it is based and existing approaches and rules (Example in Figure [46](#figure--fig:rankers98-illustration-first-two-steps)).
+    This step is fairly straightforward, because it is based and existing approaches and rules (Example in [Figure 46](#figure--fig:rankers98-illustration-first-two-steps)).
 3.  The third step consists of actuator **simulation run**, the outcome of which is the value of some quantity (for instance stress of some part, resonance frequency, FRF, etc.).
 4.  The final step is the **interpretation** of results.
     Here, the calculated results and previously defined specifications are compared.
@@ -1233,7 +1233,7 @@ Therefore, computer simulations should be regarded as a means to guide the desig
 
 <div class="exampl">
 
-This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece (Figure [47](#figure--fig:rankers98-pattern-generator)).
+This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece ([Figure 47](#figure--fig:rankers98-pattern-generator)).
 
 <a id="figure--fig:rankers98-pattern-generator"></a>
 
@@ -1251,7 +1251,7 @@ Based on the required throughput of the machine, an acceleration level of \\(1m/
 <div class="important">
 
 One of the most crucial step in the modelling process is the **definition of proper criteria on the basis of which the simulation results can be judged**.
-In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** (Figure [48](#figure--fig:rankers98-system-performance-spec)).
+In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** ([Figure 48](#figure--fig:rankers98-system-performance-spec)).
 
 </div>
 
@@ -1292,14 +1292,14 @@ In this stage, the designer only has a rough idea about the outlines of the mach
 
 <div class="exampl">
 
-One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions (Figure [49](#figure--fig:rankers98-pattern-generator-concept)).
+One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions ([Figure 49](#figure--fig:rankers98-pattern-generator-concept)).
 In the X direction, two driving forces are applied to the slides, whereas the position is measured by two linear encoders mounted between the slide and the granite frame.
 
 <a id="figure--fig:rankers98-pattern-generator-concept"></a>
 
 {{< figure src="/ox-hugo/rankers98_pattern_generator_concept.png" caption="<span class=\"figure-number\">Figure 49: </span>One of the possible concepts of the pattern generator" >}}
 
-In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility (Figure [50](#figure--fig:rankers98-concept-1dof-evaluation)).
+In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility ([Figure 50](#figure--fig:rankers98-concept-1dof-evaluation)).
 
 <a id="figure--fig:rankers98-concept-1dof-evaluation"></a>
 
@@ -1324,7 +1324,7 @@ Typically, such a model contains 5-10 rigid bodies connected by suitable connect
 
 <div class="exampl">
 
-Figure [51](#figure--fig:rankers98-pattern-generator-rigid-body) shows such a 3D model of a different concept for the pattern generator.
+[Figure 51](#figure--fig:rankers98-pattern-generator-rigid-body) shows such a 3D model of a different concept for the pattern generator.
 
 <a id="figure--fig:rankers98-pattern-generator-rigid-body"></a>
 
@@ -1373,7 +1373,7 @@ Due to the complexity of the structures it is normally not very practical to bui
 -   The resulting mass and stiffness matrices can easily have many thousands degrees of freedom, which puts high demands on the required computing capacity.
 
 A technique which overcomes these disadvantages is the co-called **sub-structuring technique**.
-In this approach, illustrated in Figure [52](#figure--fig:rankers98-substructuring-technique), the system is divided into substructures or components, which are analyzed separately.
+In this approach, illustrated in [Figure 52](#figure--fig:rankers98-substructuring-technique), the system is divided into substructures or components, which are analyzed separately.
 Then, the (reduced) models of the components are assembled to form the overall system.
 By doing so, the size of the final system model is significantly reduced.
 

content/phdthesis/verbaan15_robus.md

@@ -0,0 +1,443 @@
++++
+title = "Robust mass damper design for bandwidth increase of motion stages"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Verbaan 2015</a>)
+
+Author(s)
+: Verbaan, C.
+
+Year
+: 2015
+
+> This thesis addresses the challenge to increase the modal damping of the bandwidth limiting resonances of motions stages.
+> This modal damping increase is realized by adding passive elements, called robust tuned mass dampers, at specific stage locations.
+>
+> [...]
+>
+> The damper parameters that have to be determined are mass, stiffness, and damping.
+> The optimal parameters are obtained by executing optimization algorithm.
+>
+> The first motion stage design is optimized based on an open-loop criterion for modal damping increase between 1 and 4kHz.
+> Experimental validation shows that a suppression factor of over 24dB is obtained.
+
+
+## Robust Mass Damper and broad banded damping {#robust-mass-damper-and-broad-banded-damping}
+
+> In high tech motion systems, the finite stiffness of mechanical components results in natural frequencies which limit the bandwidth of the control system.
+> This is usually counteracted by increasing the controller complexity by adding notch filters.
+> The height of the non-rigid body modes in the frequency response function and the amount of damping significantly affect the achievable bandwidth.
+> This chapter described a method to add damping to the flexible behavior of a motion stage, by using robust mass dampers which are mass-spring-damper systems with an **over-critical** damping value.
+> This high damping results in robust dynamic behavior with respect to stiffness and damping variations for both the motion stage and the damper mechanisms.
+> The main result is a significant increase in modal damping over a broad band of resonance frequencies.
+
+
+### Tuned mass damper {#tuned-mass-damper}
+
+The effectiveness of the TMD is related to the mass ratio between \\(m\\) and \\(M\\).
+To obtain a substantial suppression factor in combination with a relatively small increase in mass, the mass ratio is usually determined to be approximately 5 to 10% of the main structural mass.
+The undamped natural frequency of the TMD has to be tuned close to the targeted natural frequency of the main structure.
+
+A drawback of the TMD is the relatively **large sensitivity of the suppression factor for variations in stiffness and damping values**.
+This sensitivity also holds for natural frequency variations of the main structure.
+
+<a id="figure--fig:verbaan15-tmd-principle"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_principle.png" caption="<span class=\"figure-number\">Figure 1: </span>TMD principle" >}}
+
+
+### Damper design and validation {#damper-design-and-validation}
+
+This damper is designed and tested to prove that it is possible to create dampers with over-critical damping values and with natural frequencies that are high enough to be useful.
+The spring and damper are assumed to behave linearly.
+In addition, the vibration amplitudes of high-tech positioning tables are small, which allows for assuming linear system theory.
+These small vibration amplitudes lead to small damper strokes.
+Therefore **flexures** can be used to provide for the guidance of the moving mass.
+The dimensions of the flexures determine the spring stiffness and therefore the natural frequency of the TMD.
+An additional advantage of flexures is the lack of hysteresis, which **enables the damper to work even if the damper strokes are very small**.
+
+The dampers are intended to act purely in z-direction.
+The natural frequency in this direction is determined at 1250Hz and the natural frequency in the other directions should be as high as possible.
+
+<a id="figure--fig:verbaan15-tmd-modes"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_modes.png" caption="<span class=\"figure-number\">Figure 2: </span>Natural frequency of the TMD. First natural frequency at 1250Hz and the second at 8100Hz." >}}
+
+The second challenge is to create a damping mechanism with a high damping coefficient in a relatively small volume.
+The damper is designed to be **passive**.
+This guarantees stability of the damper system itself and preserves from increasing complexity.
+As damping concept, a **viscous fuild damper** is chosen due to the following properties:
+
+-   the linear time independent behavior
+-   the ability to create an extremely large damping constant in a small volume
+-   separation of stiffness and damping
+-   the supreme damping properties of fuilds with respect to other damping materials
+
+The guild applied is Rocol Kilopoise 0868 and is chosen based on the extremely high viscosity of 220 Pas.
+
+In order to measure the damping the measurement bench shown in [3](#figure--fig:verbaan15-tmd-mech-system) is used.
+The measured FRF are shown in [4](#figure--fig:verbaan15-obtained-damping-bench).
+The measurement clearly shows that the damper mechanism is over-critically damped.
+
+<a id="figure--fig:verbaan15-tmd-mech-system"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_mech_system.png" caption="<span class=\"figure-number\">Figure 3: </span>Damper test setup to measure the damping characteristics" >}}
+
+<a id="figure--fig:verbaan15-obtained-damping-bench"></a>
+
+{{< figure src="/ox-hugo/verbaan15_obtained_damping_bench.png" caption="<span class=\"figure-number\">Figure 4: </span>Obtained damping results" >}}
+
+
+## Linear viscoelastic characterisation of an ultra-high viscosity fluid {#linear-viscoelastic-characterisation-of-an-ultra-high-viscosity-fluid}
+
+> This chapter presents the use of a state of the art damper for high precision motion stages as a sliding plate rheometer for measuring linear viscoelastic properties in the frequency range of 10Hz to 10kHz.
+> This design is flexure based to minimize parasitic nonlinear forces.
+> Design and the damping mechanism are elaborated and a model is presented that describes the dynamic behavior.
+
+The damper shown in [5](#figure--fig:verbaan15-damper-parts) can be used as a sliding plate rheometer to measure the linear viscoelastic properties of ultra-high viscosity fluids in the frequency range 10Hz to 10kHz.
+
+<a id="figure--fig:verbaan15-damper-parts"></a>
+
+{{< figure src="/ox-hugo/verbaan15_damper_parts.png" caption="<span class=\"figure-number\">Figure 5: </span>Damper parts" >}}
+
+The full damper assembly consists of a mass, mounted on two springs and a damper in parallel configuration.
+The mass can make small strokes in the x-direction and is fixed in all other directions.
+The spring is a double leaf spring guide.
+The space between the lead springs is used to accommodate for the damping mechanism.
+
+<a id="figure--fig:verbaan15-tmd-slot-fin-parts"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_slot_fin_parts.png" caption="<span class=\"figure-number\">Figure 6: </span>Exploded view of the damper parts" >}}
+
+A high-viscosity fluid is applied to create a velocity dependent force.
+For this purpose, the sliding plate principle is used which induces a **shear flow**: the fluid is placed between two slot plates and a fin is positioned between these two plates ([7](#figure--fig:verbaan15single-double-fin)).
+A **flexible encapsulation** is used to hold the fluid between find and slot part.
+
+To study different damping values with the same fluid, two damper designs with different geometries are used (see [7](#figure--fig:verbaan15single-double-fin)).
+
+<a id="figure--fig:verbaan15single-double-fin"></a>
+
+{{< figure src="/ox-hugo/verbaan15single_double_fin.png" caption="<span class=\"figure-number\">Figure 7: </span>Cross-sectional views of the two different damping mechanims. The single fin (left) and double fin (right)." >}}
+
+To excite the damper mass, a voice coil is mounted to the hardware.
+The damper position is measured with a laser vibrometer.
+
+A sliding plate damper for high frequencies introduces side effects:
+
+1.  geometry related effects
+2.  frequency dependent effects
+
+A first geometrical effect is due to the **finite length of the plates**.
+The ratio length/gap here is more than 100 which makes this effect negligible.
+A second geometrical effect is due to the difficulty to get the **plates parallel to each other**, especially with the normal forces acting on the moving fin, induced by the flow.
+This design counteracts this problem in two-ways: the damper part is **symmetric**, which means that the fin normal forces cancel each other.
+In addition, the double leaf spring mechanism has a **very high lateral stiffness**, which minimizes lateral displacements.
+A third geometrical effect is pumping of the fluid, which appears in the case of closed ends and introduces a flow opposite to the fin velocity, and therefore introduces a parasitic damping force.
+This problem is avoided by letting the gaps' ends open.
+The **fin is shorted than the slot** to maintain the same damping area over the damper stroke.
+
+These effects all arise at low frequencies, at which the flow can be assumed homogeneous.
+The ratio between inertial and viscous effects determines up to which frequency the flow can be assumed homogeneous:
+
+\begin{equation}
+t\_c = \frac{10 \rho h^2}{\eta}
+\end{equation}
+
+in which \\(\rho\\) describes the fluid density in \\(kg/m^3\\), \\(\eta\\) the dynamic viscosity in \\(Pa s\\) and \\(h\\) the gap width in \\(m\\).
+Dimensions are provided in [1](#table--tab:single-fin-parameters).
+This estimate results in a frequency above 100kHz.
+It shows that high fluid viscosities and small gap widths enable high frequencies without losing homogeneous flow conditions.
+
+<a id="table--tab:single-fin-parameters"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:single-fin-parameters">Table 1</a>:</span>
+  Parameters for the single fin design
+</div>
+
+| Dimension      | Value [mm] |
+|----------------|------------|
+| Length \\(l\\) | 16         |
+| Width \\(w\\)  | 8.5        |
+| Gap \\(h\\)    | 0.12       |
+
+**Conclusion**:
+A design of a sliding plate damper that can be used to characterize fluid behavior of high viscosity fluids in the frequency range between 10Hz and 10kHz.
+The drawbacks of standard sliding plate devices are taken care off by the mechanical design.
+The flexure mechanism very precisely determines the position of the fin with respect to the slot part.
+A three mode Maxwell model can accurately describe the behavior.
+
+
+## Damping optimization of a complex motion stage {#damping-optimization-of-a-complex-motion-stage}
+
+
+### Stage and damper dynamic models {#stage-and-damper-dynamic-models}
+
+This chapter presents the results of a robust mass damper implementation on a complex motion stage with realistic natural frequencies to increase the modal damping of flexible modes.
+A design approach is presented which results in parameter values for the dampers to improve the modal damping over a specified frequency range.
+
+[8](#figure--fig:verbaan15-stage-undamped) shows a collocated FRF of the stage's corner.
+The goal is to increase the modal damping of modes 7, 9, 10/11 and 13.
+
+<a id="figure--fig:verbaan15-stage-undamped"></a>
+
+{{< figure src="/ox-hugo/verbaan15_stage_undamped.png" caption="<span class=\"figure-number\">Figure 8: </span>FRF at the stage corner in the z-direction, undamped" >}}
+
+The transfer function \\(T\_i(s)\\) is defined as the contribution of the a single mode \\(i\\) in an input/output transfer function:
+
+\begin{equation}
+T\_i(s) = \frac{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}{s^2 + 2 \xi \omega\_i s + \omega\_i^2} = \frac{1}{m\_i s^2 + c\_i s + k\_i}
+\end{equation}
+
+With \\(\phi\_i^{\text{act}}\\) and \\(\phi\_i^{\text{sen}}\\) the modal factors of the actuator and sensor.
+
+From this equation, it appears that the modal mass of a mode in a certain transfer function equals:
+
+\begin{equation}
+m\_i = \frac{1}{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}
+\end{equation}
+
+This equation shows that a certain mode's modal mass depends on the locations of the actuator and sensor.
+Since a TMD can be seen as a local control loop, the actuator and sensor location are equal.
+This results in the following equation for the apparent modal mass for mode \\(i\\) at the TMD location:
+
+\begin{equation}
+m\_i = \frac{}{(\phi\_i^{\text{TMD}})^2}
+\end{equation}
+
+It is known from literature that the efficiency of a TMD depends on the **mass ratio** of the TMD and the mode that has to be damped.
+It follows that the efficiency of a TMD to damp a certain resonance depends on the position of the damper on the stage in a quadratic sense.
+The TMD has to be located at the maximum displacement of the mode(s) to be damped.
+
+The damper configuration consists of an inertial mass \\(m\\), a transnational flexible guide designed as a double leaf spring mechanism with total stiffness \\(c\\) and a part that creates the damping force with damping constant \\(d\\) (model shown in [9](#figure--fig:verbaan15-maxwell-fluid-model)).
+The velocity dependent damper force is the result of two parameters:
+
+-   the fluid's mechanical properties
+-   the damper geometry
+
+The fluid model is presented in [10](#figure--fig:verbaan15-fluid-lve-model).
+This figure shows the viscous and elastic properties of the fluid as a function of the frequency.
+The damper principle is chosen to be a parallel plate damper based on the shear principle with the viscous fluid in between the two parallel plates.
+In case of a velocity difference between these plates, a velocity gradient is created in the fluid causing a specific force per unit of area, which, multiplied by the effective area submerged in the fluid, leads to a damping force.
+
+The damping can be expressed with a geometrical damping factor (GDF) in meters:
+
+\begin{equation}
+\text{GDF} = \frac{A}{h} = \frac{2 n l w}{h}
+\end{equation}
+
+with \\(A\\) the total area of the damper fins, \\(n\\) is the number of fins, \\(l\\) is the fin length, \\(w\\) is the fin width and \\(h\\) is the effective gap width in which the fluid is applied.
+
+This GDF, combined with the fluid properties in Pas and Pa, lead to a spring stiffness in N/m and a damping constant in N/(m/s).
+
+In general, larger suppression factors can be obtained with larger TMD masses.
+In the example, the modal mass is 3.5kg and the damper mass is 110g (useful inertial mass of 65g).
+
+<a id="figure--fig:verbaan15-maxwell-fluid-model"></a>
+
+{{< figure src="/ox-hugo/verbaan15_maxwell_fluid_model.png" caption="<span class=\"figure-number\">Figure 9: </span>Damper model with multi-mode Maxwell fluid model included" >}}
+
+<a id="figure--fig:verbaan15-fluid-lve-model"></a>
+
+{{< figure src="/ox-hugo/verbaan15_fluid_lve_model.png" caption="<span class=\"figure-number\">Figure 10: </span>Storage and loss modulus of the 3 Maxwell mode LVE fluid model" >}}
+
+
+### TMD and RMD optimisation {#tmd-and-rmd-optimisation}
+
+An algorithm is used to optimize the damping and is used in two cases:
+
+-   a small banded optimisation which includes a single resonance.
+    This results in a **tuned mass damper** optimal design
+-   a broad banded optimization which includes a range of resonances.
+    This results in a **robust mass damper** optimal design
+
+The algorithm is first used to calculate the optimal parameters to suppress a **single** resonance frequency.
+The result is shown in [11](#figure--fig:verbaan15-tmd-optimization) and shows **Tuned Mass Damper** behavior.
+
+For this single frequency, stiffness and damping values can be calculated by hand.
+
+<a id="figure--fig:verbaan15-tmd-optimization"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_optimization.png" caption="<span class=\"figure-number\">Figure 11: </span>Result of the optimization procedure. The cost function is specified between 1kHz and 2kHz. This implies that the first mode is suppressed by the damper." >}}
+
+To obtain broad banded damping, the cost function is redefined between 1 and 4kHz.
+[12](#figure--fig:verbaan15-broadbanded-damping-results) presents the resulting bode diagram.
+
+<a id="figure--fig:verbaan15-broadbanded-damping-results"></a>
+
+{{< figure src="/ox-hugo/verbaan15_broadbanded_damping_results.png" caption="<span class=\"figure-number\">Figure 12: </span>Result of the optimization procedure with the cost function specified between 1 and 4kHz. The result is a range of resonances that are suppressed by the dampers." >}}
+
+Results of optimizations for increasing damper mass, in the range from 10 to 250g per damper are shown in [13](#figure--fig:verbaan15-results-fct-mass).
+
+<a id="figure--fig:verbaan15-results-fct-mass"></a>
+
+{{< figure src="/ox-hugo/verbaan15_results_fct_mass.png" caption="<span class=\"figure-number\">Figure 13: </span>Optimal damper parameters as a function of the damper mass. The upper graph shows the suppression factor in dB, the second graph shows the natural frequency of the damper in Hz and the lower graph shows the geometrical damping factor in m." >}}
+
+
+### Damper Design and Validation {#damper-design-and-validation}
+
+A damper mechanism is design which contains the following properties:
+
+-   a moving mass \\(m\_d = 65\\,g\\)
+-   a mounting mass \\(m\_m = 45\\,g\\)
+-   a natural frequency \\(\omega\_0 = 1270\\,Hz\\)
+-   other natural frequencies as high as possible
+-   a geometrical damping factor of 14.3m
+-   an encapsulation to contain the fluid
+
+[14](#figure--fig:verbaan15-RMD-mechanical-parts) shows an exploded view of the RMD design.
+The mechanism part is monolithically designed and consists of:
+
+1.  a mounting side
+2.  leaf spring pair
+3.  the damper side
+
+The fluid is surrounded by a flexible encapsulation, which prevents it from running out.
+
+<a id="figure--fig:verbaan15-RMD-mechanical-parts"></a>
+
+{{< figure src="/ox-hugo/verbaan15_RMD_mechanical_parts.png" caption="<span class=\"figure-number\">Figure 14: </span>Exploded view of the robust mass damper design with different parts indicated" >}}
+
+<a id="figure--fig:verbaan15-RMD-design-modes"></a>
+
+{{< figure src="/ox-hugo/verbaan15_RMD_design_modes.png" caption="<span class=\"figure-number\">Figure 15: </span>Four lowest natural frequencies and corresponding mode shapes of the RMD while mounted to a stage corner" >}}
+
+<a id="figure--fig:verbaan15-tmd-side-front-views"></a>
+
+{{< figure src="/ox-hugo/verbaan15_tmd_side_front_views.png" caption="<span class=\"figure-number\">Figure 16: </span>A side view and a front view of the fin and slot parts" >}}
+
+| Dimension   | Value | Unit |
+|-------------|-------|------|
+| Length fin  | 17    | mm   |
+| Height fins | 4     | mm   |
+| Gap width   | 50    | um   |
+| GDF         | 14    | m    |
+
+<a id="figure--fig:verbaan15-damped-undamped-frf"></a>
+
+{{< figure src="/ox-hugo/verbaan15_damped_undamped_frf.png" caption="<span class=\"figure-number\">Figure 17: </span>Measured undamped and damped FRF" >}}
+
+
+### Conclusion {#conclusion}
+
+This chapter shows an approach to add damping to a range of resonances of a motion stage by adding robust mass dampers.
+Analysis is performed to calculate the damping increase beforehand, and experiments are conducted to validate the behavior of both the damper and the stage with dampers added.
+
+The broadbanded solution shows a resonance suppression of at least 24.3dB between 1kHz and 4kHz.
+The overall mass increase is less than 2%.
+
+The robustness, as one of the most important properties of the RMD, is proven: the suppression factor is well predictable despite different errors and estimations:
+
+-   stage model errors (the natural frequencies resulting from the FEM are an overestimation of the real frequencies)
+-   fluid model errors
+-   a simplified 1DoF model is applied as a damper model
+-   production tolerances for the dampers
+
+Tuned mass dampers are well known in literature.
+The equations are proven to calculate the optimal suppression factor, natural frequency and damping ratio.
+In these equations, the damper behavior is assumed to be purely viscous.
+We shows that larger suppression factors are possible by using visco-elastic fluids as damping medium.
+Although this effect is relatively small for single resonance suppression, it is larger for broadbanded suppression.
+The damper benefits from the frequency dependent stiffness of the fluid.
+
+
+## Conclusion {#conclusion}
+
+In this thesis, the opportunities to increase the performance of high-tech motion systems are investigated by increasing the modal damping of non-rigid body resonances by introducing robust mass dampers (RMD), which provides damping over a broad frequency band.
+A combination of techniques is applied to improve the performance of motion stages in a systematical way, including mechanical design, dynamic modeling, material characterization and optimization procedures.
+Theoretical improvement factors are calculated and experimental validation is provided to support the theory.
+The main conclusions of the previous chapters are summarized and listed by subject.
+
+
+### Robust Mass Dampers {#robust-mass-dampers}
+
+Robust mass dampers have proven to be able to provide **broad banded damping**.
+In addition, **robust behavior** is proven in case of parameter variations of both the motion stage and/or the parameters of the RMDs.
+This property explicitly underlines the suitability of RMDs to improve the behavior of motion stages that are operated in closed-loop conditions: parameter sensitive designs will result in a performance decrease and might eventually lead to destabilization of the closed-loop system.
+
+The RMDs in this thesis are **passive and stand-alone devices**.
+Advantages of these types of devices are
+
+1.  the stabilizing behavior due to the principle of energy dissipation.
+2.  The stand-alone property implies that no connection between any structural part and the motion stage is created, and no signal or power cables are needed which prevents the introduction of disturbance forces.
+3.  The damper design by application of LVE behavior enables larger suppression factors than purely viscous fluid behavior.
+
+At least in case of motion stages with a relatively large length-height ratio it appears that an overall mass contribution by the RMDs of 2 % of the stage mass is sufficient to improve the stage performance significantly.
+This is proven by experiments.
+
+
+### Influence on stage dynamics {#influence-on-stage-dynamics}
+
+The relatively high modal damping of the RMDs prevents for visible effects in the rigid body mass line of the frequency response functions.
+In other directions, the natural frequencies of the RMDs can be designed above 6 kHz for dampers of 65 g.
+This is usually high enough to prevent for detrimental properties in the direction of motion
+
+
+### RMD locations {#rmd-locations}
+
+The **location of an RMD on the mechanical stage is a significant factor in the performance increase factor**.
+The effectiveness of the RMD to improve the modal damping factor scales quadratically with the stage displacement at the damper location.
+Therefore, if the limiting natural frequencies are determined, **the locations with large displacements for the corresponding mode shapes have to be found**.
+In case of more than one resonance this might be a weighted criterion for the different modes.
+This approach is applicable for both open- loop and closed-loop performance criteria.
+
+
+### The fluid model {#the-fluid-model}
+
+A **linear visco-elastic fluid model** is derived from measurements and applied in the optimization formulations.
+The results show that the model quality is good enough to predict the system’s damped behavior quite accurately.
+
+
+### Open-loop modal damping improvement {#open-loop-modal-damping-improvement}
+
+The principle of **broad banded damping** is well applicable for practical cases: the intended damping range was 1-4 kHz.
+In addition, a damping increase is visible up to 6 kHz.
+This frequency range abundantly covers the range in which performance limiting flexibilities usually arise in motion stage designs.
+An optimization criterion in terms of resonance suppression is applied and works well: this criterion inherently only optimizes the visible resonances at the actuator and sensor location.
+The choice which resonances should be suppressed, therefore, is specified in the cost function by the frequency response function.
+Robustness of the solution and broad banded effect in practical cases is proven by the experimental validation.
+The calculated suppression factor compares well to the measured ones.
+The suppression factor amounts approximately 24 dB between 1 and 4 kHz, which indicates a modal damping increase factor of 16.
+
+
+### Closed-loop performance increase {#closed-loop-performance-increase}
+
+The principle of closed-loop performance increase is formulated in an optimization formulation which accurately estimates the bandwidth improvement factor.
+The optimization formulation is non-convex, however, a hybrid optimization procedure is able to solve this specific problem in a limited amount of time.
+In addition to the improvements in the intended control loops, other control loops often benefit from the damping increase.
+
+
+### Advantages in analysis {#advantages-in-analysis}
+
+A more general observation regarding the analyses method is presented.
+The approach with separate RMDs is an efficient approach which contains two large advantages: It enables to continue with the current applied mechanical design approach for high natural frequencies and increase the modal damping afterwards.
+This enables to still apply the materials with high specific stiffness and low damping.
+
+In the analysis phase the advantages are enormous:
+
+1.  Undamped natural frequencies and mode shapes can be calculated and are valid for the low damped stage’s mechanical design.
+    These algorithms are very efficient and large models can be solved.
+2.  State space models can be created which contain the complexity of the FEM model and can be validated by calculating the responses by means of superposition of the undamped modes in the FEM software.
+3.  RMDs can be added at specific locations.
+    This results in non-proportional damping and complex mode shapes, which are correctly calculated by the state space model.
+4.  This enables to apply optimization algorithms and compare different RMDs very quickly.
+
+The complete model including dampers can be solved in FEM, however, this approach contains serious drawbacks:
+
+1.  The mode shapes change from real normal modes to complex modes due to the damping at specific locations.
+    This implies that complex solvers have to be applied.
+    These solvers are much more time consuming than the solvers for real natural modes.
+2.  The frequency response functions can be calculated using fully harmonic solvers.
+    This results in the most accurate solution because the model is not truncated as in case of a state space model with a limited number of modes.
+    However, this algorithm solves the complete model for every frequency point in the frequency response function and, therefore, this approach is extremely time-consuming.
+3.  Therefore, in this approach the ability to implement different RMD parameters and execute optimization algorithms practically vanishes due to the limitations listed above.
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Verbaan, C.A.M. 2015. “Robust mass damper design for bandwidth increase of motion stages.” Mechanical Engineering; Technische Universiteit Eindhoven.</div>
+</div>

content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md

@@ -7,7 +7,7 @@ ref_year = 2004
 +++
 
 Tags
-: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Eddy Current Damping]({{< relref "eddy_current_damping.md" >}})
 
 Reference
 : (<a href="#citeproc_bib_item_1">Zuo 2004</a>)
@@ -28,21 +28,47 @@ Year
 > They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
 > Note that reaction force actuators can also work with soft mounts or hard mounts.
 
+
+## Passive Vibration Isolation {#passive-vibration-isolation}
+
+
+### The Role of damping and its practical constructions {#the-role-of-damping-and-its-practical-constructions}
+
+
+#### Viscous damping {#viscous-damping}
+
+
+#### Eddy-current damper {#eddy-current-damper}
+
+<a id="figure--fig:zuo04-eddy-current-magnets"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class=\"figure-number\">Figure 1: </span>(left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
+
+<a id="figure--fig:zuo04-eddy-current-setup"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class=\"figure-number\">Figure 2: </span>Single DoF system damped by eddy current damper" >}}
+
+
+## Elements and configurations for active vibration systems {#elements-and-configurations-for-active-vibration-systems}
+
+
+### System architectures {#system-architectures}
+
 <a id="figure--fig:zuo04-piezo-spring-series"></a>
 
-{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 1: </span>PZT actuator and spring in series" >}}
+{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 3: </span>PZT actuator and spring in series" >}}
 
 <a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
 
-{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 2: </span>Voice coil actuator and spring in parallel" >}}
+{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 4: </span>Voice coil actuator and spring in parallel" >}}
 
 <a id="figure--fig:zuo04-piezo-plant"></a>
 
-{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 3: </span>Transmission from PZT voltage to geophone output" >}}
+{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 5: </span>Transmission from PZT voltage to geophone output" >}}
 
 <a id="figure--fig:zuo04-voice-coil-plant"></a>
 
-{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Transmission from voice coil voltage to geophone output" >}}
+{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 6: </span>Transmission from voice coil voltage to geophone output" >}}
 
 
 ## Bibliography {#bibliography}

content/posts/test.md

@@ -17,7 +17,7 @@ This is a test for a blog post.
 
 You can make words **bold**, _italic_, <span class="underline">underlined</span>, `verbatim` and `code`, and, if you must, ~~strike-through~~.
 
-Here is some inline code Matlab code: `[K,CL,gamma] = mixsyn(G,W1,[],W3);`.
+Here is some inline code Matlab code: <span class="inline-src language-matlab" data-lang="matlab">`[K,CL,gamma] = mixsyn(G,W1,[],W3);`</span>.
 
 
 ### Links to Footnotes {#links-to-footnotes}
@@ -87,13 +87,13 @@ Here is some inline mathematics: \\(z = 2\\).
 Unumbered equation:
 \\[ F(x) = \int\_0^x f(t) dt \\]
 
-Using the `equation` environment in Eq. <eq:numbered>.
+Using the `equation` environment in Eq. \eqref{eq:numbered}.
 
-\begin{equation}
+\begin{equation} \label{eq:numbered}
-  F(s) = \int\_0^\infty f(t) e^{-st} dt \label{eq:numbered}
+  F(s) = \int\_0^\infty f(t) e^{-st} dt
 \end{equation}
 
-Using the `align` environment Equations <eq:align_1> and <eq:align_2>.
+Using the `align` environment Equations \eqref{eq:align\_1} and \eqref{eq:align\_2}.
 
 \begin{align}
   \mathcal{F}(a) &= \frac{1}{2\pi i}\oint\_\gamma \frac{f(z)}{z - a}\\,dz \label{eq:align\_1} \\\\
@@ -105,7 +105,7 @@ Using the `align` environment Equations <eq:align_1> and <eq:align_2>.
 
 Below is a verse.
 
-<p class="verse">
+<div class="verse">
 
 Great clouds overhead<br />
 Tiny black birds rise and fall<br />
@@ -113,7 +113,7 @@ Snow covers Emacs<br />
 <br />
 &nbsp;&nbsp;&nbsp;---AlexSchroeder<br />
 
-</p>
+</div>
 
 Below is a quote.
 

content/posts/test2.md

@@ -2,7 +2,7 @@
 title = "Second Blog Post"
 author = ["Dehaeze Thomas"]
 date = 2021-05-01T00:00:00+02:00
-lastmod = 2022-03-15T16:31:43+01:00
+lastmod = 2024-12-17T16:38:28+01:00
 tags = ["hugo", "org"]
 categories = ["emacs"]
 draft = false

content/zettels/.org.md

@@ -0,0 +1,14 @@
++++
+title = "Electrical Impedance"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/acquisition_systems.md

@@ -11,11 +11,14 @@ Tags
 
 ## Manufacturers {#manufacturers}
 
+<https://dewesoft.com/daq/list-of-data-acquisition-companies>
+
 | Manufacturers                                                                                      | Country  |
 |----------------------------------------------------------------------------------------------------|----------|
 | [Dewesoft](https://dewesoft.com/)                                                                  | Slovenia |
 | [Oros](https://www.oros.com/)                                                                      | France   |
 | [National Instruments](https://www.ni.com/fr-fr/shop/pc-based-measurement-and-control-system.html) | USA      |
+| [Gantner](https://www.gantner-instruments.com/products/daq-systems/)                               | Austria  |
 
 
 ## Bibliography {#bibliography}

content/zettels/actuator_fusion.md

@@ -7,14 +7,15 @@ draft = false
 Tags
 : [Complementary Filters]({{< relref "complementary_filters.md" >}})
 
-(<a href="#citeproc_bib_item_2">Beijen et al. 2019</a>)
+Mention in the literature:
 
-(<a href="#citeproc_bib_item_1">Beijen 2018</a>) (section 6.3.1)
+-   (<a href="#citeproc_bib_item_2">Beijen et al. 2019</a>)
+-   (<a href="#citeproc_bib_item_1">Beijen 2018</a>) (section 6.3.1)
 
 
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” <i>Mechatronics</i> 61: 106–16. doi:<a href="https://doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005">https://doi.org/10.1016/j.mechatronics.2019.06.005</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” <i>Mechatronics</i> 61: 106–16. doi:<a href="https://doi.org/10.1016/j.mechatronics.2019.06.005">10.1016/j.mechatronics.2019.06.005</a>.</div>
 </div>

content/zettels/actuators.md

@@ -24,7 +24,7 @@ For short stroke and very high dynamic applications, mainly two types of actuato
 
 Rotational drives can be combined with ball-screw mechanisms for long (infinite) axial motion:
 
--   Brush-less DC Motor. See (<a href="#citeproc_bib_item_2">Yedamale 2003</a>) and this [working principle](https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html).
+-   Brush-less DC Motor. See (<a href="#citeproc_bib_item_3">Yedamale 2003</a>) and this [working principle](https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html).
 -   [Stepper Motor]({{< relref "stepper_motor.md" >}})
 
 
@@ -33,11 +33,13 @@ Rotational drives can be combined with ball-screw mechanisms for long (infinite)
 For vibration isolation:
 
 -   In (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}}))
+-   (<a href="#citeproc_bib_item_2">Murugesan 1981</a>) On overview of electric motors for space applications
 
 
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>Ieee/Asme Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>IEEE/ASME Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (Bldc) Motor Fundamentals.” <i>Microchip Technology Inc</i> 20: 3–15.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Murugesan, S. 1981. “An Overview of Electric Motors for Space Applications.” <i>IEEE Transactions on Industrial Electronics and Control Instrumentation</i> IECI-28 (4): 260–65. doi:<a href="https://doi.org/10.1109/TIECI.1981.351050">10.1109/TIECI.1981.351050</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” <i>Microchip Technology Inc</i> 20: 3–15.</div>
 </div>

content/zettels/air_bearing.md

@@ -0,0 +1,72 @@
++++
+title = "Air Bearing"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Advantages of air bearing {#advantages-of-air-bearing}
+
+Advantages of air bearings compared to roller bearings:
+
+-   **low friction**: because air bearing have almost zero static friction, this enables infinite resolution of motion that is highly repeatable.
+    Friction in air bearing is a function of air shear, which is itself a function of velocity.
+-   **zero wear**: non-contact motion means virtually zero wear owing to friction, resulting in consistent machine and minimal particulate generation
+-   **straighter motion**: rolling element bearings are directly influence by surface finishing and irregularities on the guide surface. The air bearing's fluid film layer averages these errors resulting in straighter motion
+-   **silent and smooth operation**: recirculating rollers or balls create noise and vibration as hard elements are loaded, unloaded and change directions in return tubes. Air bearings have no dynamic components resulting in virtually silent operation
+-   **higher damping**: being fluid film bearings, air bearings have a squeeze film damping effect resulting in higher dynamic stiffness and stability
+-   **no lubrication**:
+
+
+## Air bearing stiffness {#air-bearing-stiffness}
+
+Observing [1](#figure--fig:air-bearing-stiffness-gap), we see that air bearings do not have a linear stiffness curve but rather an exponential one, producing higher and higher stiffness values as the film becomes thinner and the loading becomes higher.
+
+<a id="figure--fig:air-bearing-stiffness-gap"></a>
+
+{{< figure src="/ox-hugo/air_bearing_stiffness_gap.png" caption="<span class=\"figure-number\">Figure 1: </span>Lift/load curve of a typical air bearing. The slope of the curve is representative of the bearing stiffness. A vertical line represent infinite stiffness and an horizontal line would represent zero stiffness" >}}
+
+Because air is a compressible fluid, it possesses its own spring rate, or stiffness.
+Higher pressures effectively act as a preload on the "air spring", and if we thing of the air column as a spring of arbitrary height, compressing or shortening the spring will increase its stiffness as the air attempts to "push back".
+Stiffness in an air bearing system is a product of pressure in the air gap, thickness of the air gap and the projected surface area of the bearing.
+
+
+## Orifice and porous technology {#orifice-and-porous-technology}
+
+Air bearings generally fall into one of two categories: orifice or porous media bearings.
+
+In orifice compensation bearings, the precisely sized orifices are strategically placed on the bearing, and are often combined with groove to distribute the pressurized air as evenly as possible across the bearing face.
+However, should the bearing face become scratched across a groove or near an orifice, the volume or air which escapes via the scratch in the surface may be more than the orifice can supply, causing a bearing crash.
+
+Porous media air bearings control the airflow across the entire bearing surface through millions of sub-micron holes in the porous material.
+Due to the porous nature, even if some of the holes become clogged or damaged, the air will continue to be supplied to the majority of the bearing face.
+
+
+## Air Bearing Components {#air-bearing-components}
+
+| Manufacturer                                                                                                                                 | Country |
+|----------------------------------------------------------------------------------------------------------------------------------------------|---------|
+| [New way](https://www.newwayairbearings.com/catalog/components/) ([IBSPE](https://www.ibspe.com/air-bearings/flat-air-bearings) distributor) | USA     |
+| [Positechnics](http://positechnics.fr/index.adml?r=176)                                                                                      |         |
+| [Huber](https://www.xhuber.com/en/products/4-accessories/41-mechanics/airpads/)                                                              |         |
+| [Specialty Components](https://www.specialtycomponents.com/)                                                                                 | USA     |
+
+
+## Linear Air Bearing Stages {#linear-air-bearing-stages}
+
+-   <https://microplan-group.com/en/>
+-   <https://www.aerotech.com/motion-and-positioning/stages-actuators-products/?pagenum=1&CATEGORY=Linear+Motion&AXIS+CONFIGURATION=Single+Axis&AXIS+ORIENTATION=Horizontal&BEARING+TYPE=Air+Bearing>
+-   <https://www.pi-usa.us/en/products/air-bearings-ultra-high-precision-stages/a-10x-piglide-rb-linear-air-bearing-module-900716>
+-   <https://www.ibspe.com/air-bearings/air-slides>
+
+
+## Spindle Air Bearing {#spindle-air-bearing}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/analog_to_digital_converters.md

@@ -33,7 +33,7 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati
 Interestingly, the noise amplitude is uniformly distributed.
 
 The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
-Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#figure--fig:probability-density-function-adc)).
+Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [Figure 1](#figure--fig:probability-density-function-adc)).
 
 <a id="figure--fig:probability-density-function-adc"></a>
 
@@ -84,12 +84,83 @@ The quantization is:
 
 {{< youtube b9lxtOJj3yU >}}
 
+Also see (<a href="#citeproc_bib_item_4">Kester 2005</a>).
+
+
+## Link between required dynamic range and effective number of bits {#link-between-required-dynamic-range-and-effective-number-of-bits}
+
+<a id="figure--fig:dynamic-range-enob"></a>
+
+{{< figure src="/ox-hugo/dynamic_range_enob.png" caption="<span class=\"figure-number\">Figure 2: </span>Relation between Dynamic range and required number of bits (effective)" >}}
+
 
 ## Oversampling {#oversampling}
 
+(<a href="#citeproc_bib_item_5">Lab 2013</a>)
+
+To have additional \\(w\\) bits of resolution, the oversampling frequency \\(f\_{os}\\) should be:
+
+\begin{equation}
+f\_{os} = 4^w \cdot f\_s
+\end{equation}
+
+(<a href="#citeproc_bib_item_3">Hauser 1991</a>)
+
+
+### When Oversampling and Averaging Will Work {#when-oversampling-and-averaging-will-work}
+
+> Key points to consider are:
+>
+> -   The noise must approximate **white noise** with uniform power spectral density over the frequency band of interest.
+> -   The **noise amplitude must be sufficient** to cause the input signal to change randomly from sample to sample by amounts comparable to at least the distance between two adjacent codes (i.e., 1 LSB).
+> -   The input signal can be represented as a random variable that has equal probability of existing at any value between two adjacent ADC codes.
+
+
+## Sigma Delta ADC {#sigma-delta-adc}
+
+(<a href="#citeproc_bib_item_7">Pisani 2018</a>)
+
+From (<a href="#citeproc_bib_item_8">Schmidt, Schitter, and Rankers 2020</a>):
+
+> The low cost and excellent linearity properties of the Sigma-Delta ADC have replaced other ADC types in many measurement and registration systems, especially where storage of data is more important than real-time measurement.
+> This has typically been the case in audio recording and reproduction.
+> The reason why this principle is less applied with real-time measurements is the time delay between the bitstream representing the actual value and the availability of the corresponding value after the decimation filter.
+> The resulting **latency** amounts with a low cost sigma-delta ADC approximately **twenty times the sampling period of the decimated digital output**.
+
+<div class="exampl">
+
+A 50kHz decimated sampling frequency has a sample period of 20us, resulting in a total latency of more than 400us.
+This would cause almost 180 degrees phase delay for a 1kHz signal frequency, which is not acceptable with high bandwidth motion control systems.
+This phenomenon clearly illustrates the necessity to distinguish sample frequency from speed.
+
+</div>
+
+Therefore, even though there are sigma-delta ADC with high precision and sampling rate, they add large latency (i.e. time delay) that are very problematic for feedback systems.
+
+> The SAR-ADC (Successive approximation ADCs) is still the mostly applied type for data-acquisition and feedback systems because of its single sample latency.
+
+<https://www.crystalinstruments.com/antialiasing-filter-and-phase-match>
+
+
+## Anti-Aliasing Filters {#anti-aliasing-filters}
+
+(<a href="#citeproc_bib_item_6">Microchip 1999</a>)
+
+
+## State of the art ADC {#state-of-the-art-adc}
+
+(<a href="#citeproc_bib_item_2">Beev 2018</a>)
+
 
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” <i>Analog Applications</i> 7.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Beev, Nikolai. 2018. “Analog-to-Digital Conversion beyond 20 Bits.” In <i>2018 IEEE International Instrumentation and Measurement Technology Conference (I2MTC)</i>, nil. doi:<a href="https://doi.org/10.1109/i2mtc.2018.8409543">10.1109/i2mtc.2018.8409543</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Hauser, Max. 1991. “Principles of Oversampling a/D Conversion.” <i>Journal of Audio Engineering Society</i>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_4"></a>Kester, Walt. 2005. “Taking the Mystery out of the Infamous Formula, $snr = 6.02 N + 1.76 Db$, and Why You Should Care.”</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_5"></a>Lab, Silicon. 2013. “Improving the ADC Resolution by Oversampling and Averaging.” Silicon Laboratories.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_6"></a>Microchip. 1999. “Anti-Aliasing, Analog Filters for Data Acquisition Systems.”</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_7"></a>Pisani, Brian. 2018. “Accounting for Delay from Multiple Sources in Delta-Sigma ADCs.”</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_8"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
 </div>

content/zettels/brushless_dc_motor.md

@@ -0,0 +1,23 @@
++++
+title = "Brushless DC Motor"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Manufacturers {#manufacturers}
+
+| Manufacturers                                                                              | Country |
+|--------------------------------------------------------------------------------------------|---------|
+| [Faulhaber](https://www.faulhaber.com/en/products/brushless-dc-motors/)                    |         |
+| [Maxon](https://www.maxongroup.com/maxon/view/content/Overview-brushless-DC-motors)        |         |
+| [OrientalMotors](https://www.orientalmotor.com/brushless-dc-motors-gear-motors/index.html) |         |
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/cables.md

@@ -32,6 +32,12 @@ Tags
 -   [WireViz](https://github.com/formatc1702/WireViz) is a nice software to easily document cables and wiring harnesses
 
 
+## Cable Chains {#cable-chains}
+
+-   <https://www.gore.com/products/gore-high-flex-cables-assemblies-lithography>
+-   <https://www.igus.eu/info/energychains>
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

content/zettels/capacitive_sensors.md

@@ -27,10 +27,20 @@ Tags
 | [Fogale](http://www.fogale.fr/brochures.html)                                                          | USA         |
 | [Queensgate](https://www.nanopositioning.com/product-category/nanopositioning/nanopositioning-sensors) | UK          |
 | [Capacitec](https://www.capacitec.com/Displacement-Sensing-Systems)                                    | USA         |
-| [MTIinstruments](https://www.mtiinstruments.com/products/non-contact-measurement/capacitance-sensors/) | USA         |
+| [MTIinstruments](https://vitrek.com/mti-instruments/non-contact-measurement/)                          | USA         |
 | [Althen](https://www.althensensors.com/sensors/linear-position-sensors/capacitive-position-sensors/)   | Netherlands |
 
 
+## Comparison {#comparison}
+
+|                      | RMS noise           | Noise Density (%/Hz^.5) | Linearity |
+|----------------------|---------------------|-------------------------|-----------|
+| Fogale MC900         | < 0.005% (10kHz) | 0.000050                | <0.2%  |
+| Micro-Epsilon DL6230 | 0.005% (5kHz)       | 0.000070                | 0.05%     |
+| Lion CPL290          | 0.003% (15kHz)      | 0.000025                | <0.2%  |
+| PI                   | 0.002% (3kHz)       | 0.000036                | <0.1   |
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

content/zettels/charge_amplifiers.md

@@ -18,7 +18,7 @@ This can be typically used to interface with piezoelectric sensors.
 
 ## Basic Circuit {#basic-circuit}
 
-Two basic circuits of charge amplifiers are shown in Figure [1](#figure--fig:charge-amplifier-circuit) (taken from (<a href="#citeproc_bib_item_1">Fleming 2010</a>)) and Figure [2](#figure--fig:charge-amplifier-circuit-bis) (taken from (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2014</a>))
+Two basic circuits of charge amplifiers are shown in [Figure 1](#figure--fig:charge-amplifier-circuit) (taken from (<a href="#citeproc_bib_item_1">Fleming 2010</a>)) and [Figure 2](#figure--fig:charge-amplifier-circuit-bis) (taken from (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2014</a>))
 
 <a id="figure--fig:charge-amplifier-circuit"></a>
 
@@ -30,7 +30,7 @@ Two basic circuits of charge amplifiers are shown in Figure [1](#figure--fig:cha
 
 The input impedance of the charge amplifier is very small (unlike when using a voltage amplifier).
 
-The gain of the charge amplified (Figure [1](#figure--fig:charge-amplifier-circuit)) is equal to:
+The gain of the charge amplified ([Figure 1](#figure--fig:charge-amplifier-circuit)) is equal to:
 \\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\]
 
 
@@ -45,12 +45,12 @@ The gain of the charge amplified (Figure [1](#figure--fig:charge-amplifier-circu
 | [DJB](https://www.djbinstruments.com/products/instrumentation/view/9-Channel-Charge-Voltage-Amplifier-IEPE-Signal-Conditioning-Rack-Mounted) | UK      |
 | [MTI Instruments](https://www.mtiinstruments.com/products/turbine-balancing-vibration-analysis/charge-amplifiers/ca1800/)                    | USA     |
 | [Sinocera](http://www.china-yec.net/instruments/signal-conditioner/multi-channels-charge-amplifier.html)                                     | China   |
-| [L-Card](https://en.lcard.ru/products/accesories/le-41)                                                                                      | Rusia   |
+| [Physimetron](http://www.physimetron.de/produkte_en.html)                                                                                    | Germany |
 
 
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. <i>The Design of High Performance Mechatronics - 2nd Revised Edition</i>. Ios Press.</div>
 </div>

content/zettels/collocated_control.md

@@ -19,7 +19,7 @@ According to (<a href="#citeproc_bib_item_1">Preumont 2018</a>):
 
 ## Nearly Collocated Actuator Sensor Pair {#nearly-collocated-actuator-sensor-pair}
 
-From Figure [1](#figure--fig:preumont18-nearly-collocated-schematic), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
+From [Figure 1](#figure--fig:preumont18-nearly-collocated-schematic), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
 
 <a id="figure--fig:preumont18-nearly-collocated-schematic"></a>
 

content/zettels/communication_protocol.md

@@ -0,0 +1,52 @@
++++
+title = "Communication Protocol"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Value Communication {#value-communication}
+
+Typically used for encoders.
+
+|  |   |
+|--|---|
+|  |   |
+
+
+### Quadrature (A-Quad-B) {#quadrature--a-quad-b}
+
+
+### Step-Dir {#step-dir}
+
+
+### SSI and BISS {#ssi-and-biss}
+
+
+### EnDAT {#endat}
+
+
+### HSSL {#hssl}
+
+
+### SPI {#spi}
+
+
+### RS232 and RS422 {#rs232-and-rs422}
+
+ASCII
+
+
+### RS485 {#rs485}
+
+
+### EtherCAT {#ethercat}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/complementary_filters.md

@@ -13,8 +13,14 @@ Tags
 The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis (<a href="#citeproc_bib_item_1">Dehaeze, Vermat, and Collette 2019</a>).
 
 
+## First Order complementary filters {#first-order-complementary-filters}
+
+
+## Second Order complementary filters {#second-order-complementary-filters}
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dehaeze, Thomas, Mohit Vermat, and Christophe Collette. 2019. “Complementary Filters Shaping Using $h_\Infty$ Synthesis.” In <i>7th International Conference on Control, Mechatronics and Automation (Iccma)</i>, 459–64. doi:<a href="https://doi.org/10.1109/ICCMA46720.2019.8988642">10.1109/ICCMA46720.2019.8988642</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dehaeze, Thomas, Mohit Vermat, and Christophe Collette. 2019. “Complementary Filters Shaping Using $H_\Infty$ Synthesis.” In <i>7th International Conference on Control, Mechatronics and Automation (ICCMA)</i>, 459–64. doi:<a href="https://doi.org/10.1109/ICCMA46720.2019.8988642">10.1109/ICCMA46720.2019.8988642</a>.</div>
 </div>

content/zettels/connectors.md

@@ -20,7 +20,7 @@ Tags
 
 ## BNC {#bnc}
 
-BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in Figure [1](#figure--fig:bnc-50-75-ohms).
+BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in [Figure 1](#figure--fig:bnc-50-75-ohms).
 
 <a id="figure--fig:bnc-50-75-ohms"></a>
 

content/zettels/cubic_architecture.md

@@ -22,5 +22,5 @@ Cubic Stewart Platforms can be decoupled provided that (from (<a href="#citeproc
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 Icra. Millennium Conference. Ieee International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00ch37065)</i>, nil. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)</i>. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
 </div>

content/zettels/digital_filters.md

@@ -10,6 +10,80 @@ Tags
 A nice open access book on digital filter is accessible here: <https://ccrma.stanford.edu/~jos/filters/filters.html>
 
 
+## Analog to Digital Filter {#analog-to-digital-filter}
+
+In order to convert an analog filter (Laplace domain) to a digital filter (z-domain), the `c2d` command can be used ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html)).
+
+<div class="exampl">
+
+Let's define a simple first order low pass filter in the Laplace domain:
+
+```matlab
+s = tf('s');
+G = 1/(1 + s/(2*pi*10));
+```
+
+To obtain the equivalent digital filter:
+
+```matlab
+Ts = 1e-3; % Sampling Time [s]
+Gz = c2d(G, Ts, 'tustin');
+```
+
+</div>
+
+There are several methods to go from the analog to the digital domain, `Tustin` is the one I use the most as it ensures the stability of the digital filter provided that the analog filter is stable.
+
+
+## Bilinear transform {#bilinear-transform}
+
+The bilinear transform also known as the Tustin's method (see the [wikipedia page](https://en.wikipedia.org/wiki/Bilinear_transform)) is used to convert a continuous-time system representations to discrete-time.
+
+It uses the fact that \\(z = e^{sT} \approx \frac{1 + sT/2}{1-sT/2}\\).
+
+To go from the Laplace domain to the z-domain, we just have to use the following approximation:
+
+\begin{equation}
+\boxed{s \approx \frac{2}{T\_s} \frac{z - 1} {z + 1} = \frac{2}{T\_s}\frac{1 - z^{-1}}{1 + z^{-1}}}
+\end{equation}
+
+
+## Standard Digital Filters {#standard-digital-filters}
+
+
+### First order low pass filter {#first-order-low-pass-filter}
+
+\begin{equation}
+G(s) = \frac{1}{1 + s/\omega\_0}
+\end{equation}
+
+Using the bilinear transform, we obtain:
+
+\begin{equation}
+G(z) = \frac{a(1 + z^{-1})}{1 + b z^{-1}}
+\end{equation}
+
+with:
+
+\begin{align}
+a &= \frac{2}{T\_s\omega\_0} + 1\\\\
+b &= \frac{2}{T\_s\omega\_0} - 1
+\end{align}
+
+If we want to compute how the filter output \\(y[n]\\) depends on previous output \\(y[n-1]\\), previous input \\(x[n-1]\\) and current input \\(x[n]\\) we can write:
+
+\begin{equation}
+y[n] = G(z) x[n]
+\end{equation}
+
+By developing the relation and using the fact that \\(z^{-1} x[n] = x[n-1]\\), we obtain:
+
+\begin{align}
+y[n] &= a (x[n] + x[n-1]) + b y[n-1] \\\\
+     &= \left( \frac{2}{T\_s \omega\_0} + 1 \right) (x[n] + x[n-1]) + \left(\frac{2}{T\_s\omega\_0} - 1\right) y[n-1]
+\end{align}
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

content/zettels/digital_signal_processing.md

@@ -1,8 +1,22 @@
 +++
 title = "Digital Signal Processing"
-author = ["Thomas Dehaeze"]
+author = ["Dehaeze Thomas"]
 draft = false
 +++
 
 Tags
 :
+
+
+## References {#references}
+
+Books:
+
+-   (<a href="#citeproc_bib_item_1">Lyons 2011</a>)
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lyons, Richard. 2011. <i>Understanding Digital Signal Processing</i>. Upper Saddle River, NJ: Prentice Hall.</div>
+</div>

content/zettels/discrete_transfer_functions.md

@@ -8,6 +8,278 @@ Tags
 : [Digital Filters]({{< relref "digital_filters.md" >}})
 
 
+## Continuous to discrete transfer function {#continuous-to-discrete-transfer-function}
+
+In order to convert an analog filter (Laplace domain) to a digital filter (z-domain), the `c2d` command can be used ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html)).
+
+<div class="exampl">
+
+Let's define a simple first order low pass filter in the Laplace domain:
+
+```matlab
+s = tf('s');
+G = 1/(1 + s/(2*pi*10));
+```
+
+To obtain the equivalent digital filter:
+
+```matlab
+Ts = 1e-3; % Sampling Time [s]
+Gz = c2d(G, Ts, 'tustin');
+```
+
+</div>
+
+There are several methods to go from the analog to the digital domain, `Tustin` is the one I use the most as it ensures the stability of the digital filter provided that the analog filter is stable.
+
+
+## Obtaining analytical formula of filter {#obtaining-analytical-formula-of-filter}
+
+
+### Procedure {#procedure}
+
+The Matlab [Symbolic Toolbox](https://fr.mathworks.com/help/symbolic/) can be used to obtain analytical formula for discrete transfer functions.
+
+Let's consider a notch filter:
+
+\begin{equation}
+  G(s) = \frac{s^2 + 2 g\_c \xi \omega\_n s + \omega\_n^2}{s^2 + 2 \xi \omega\_n s + \omega\_n^2}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): frequency of the notch
+-   \\(g\_c\\): gain at the notch frequency
+-   \\(\xi\\): damping ratio (notch width)
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms gc wn xi Ts s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Notch Filter - Symbolic representation
+Ga = (s^2 + 2*xi*gc*s*wn + wn^2)/(s^2 + 2*xi*s*wn + wn^2);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+org_babel_eoe
+```
+
+```text
+den = (Ts^2*wn^2 - 4*Ts*wn*xi + 4) + (2*Ts^2*wn^2 - 8) * z + (Ts^2*wn^2 + 4*Ts*wn*xi + 4) * z^2
+```
+
+
+### Second Order Low Pass Filter {#second-order-low-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_n} + \frac{s^2}{\omega\_n^2}}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): Cut off frequency
+-   \\(\xi\\): damping ratio
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms wn xi Ts s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = 1/(1 + 2*xi*s/wn + s^2/wn^2);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(Ts^2*wn^2 + 4*Ts*wn*xi + 4)
+```
+
+```text
+num = (Ts^2*wn^2) + (2*Ts^2*wn^2) * z^-1 + (Ts^2*wn^2) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*wn^2 - 8) * z^-1 + (Ts^2*wn^2 - 4*Ts*wn*xi + 4) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
+### Second Order Low Pass Filter {#second-order-low-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{g}{ms^2 + cs + k}
+\end{equation}
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms Ts g m c k s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = g/(m*s^2 + c*s + k)
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(4*m + 2*Ts*c + Ts^2*k)
+```
+
+```text
+num = (Ts^2*g) + (2*Ts^2*g) * z^-1 + (Ts^2*g) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*k - 8*m) * z^-1 + (4*m - 2*Ts*c + Ts^2*k) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
+### Second Order High Pass Filter {#second-order-high-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_n} + \frac{s^2}{\omega\_n^2}}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): Cut off frequency
+-   \\(\xi\\): damping ratio
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms wn xi Ts s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = (s^2/wn^2)/(1 + 2*xi*s/wn + s^2/wn^2);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(Ts^2*wn^2 + 4*Ts*wn*xi + 4)
+```
+
+```text
+num = (4) + (-8) * z^-1 + (4) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*wn^2 - 8) * z^-1 + (Ts^2*wn^2 - 4*Ts*wn*xi + 4) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
+## Variable Discrete Filter {#variable-discrete-filter}
+
+Once the analytical formula of a discrete transfer function is obtained, it is possible to vary some parameters in real time.
+
+This is easily done in Simulink (see [Figure 1](#figure--fig:variable-controller-simulink)) where a `Discrete Varying Transfer Function` block is used.
+The coefficients are simply computed with a Matlab function.
+
+<a id="figure--fig:variable-controller-simulink"></a>
+
+{{< figure src="/ox-hugo/variable_controller_simulink.png" caption="<span class=\"figure-number\">Figure 1: </span>Variable Discrete Filter in Simulink" >}}
+
+
 ## Typical Transfer functions {#typical-transfer-functions}
 
 

content/zettels/eddy_current_damping.md

@@ -0,0 +1,109 @@
++++
+title = "Eddy Current Damping"
+draft = false
++++
+
+Tags
+: [Passive Damping]({{< relref "passive_damping.md" >}})
+
+<https://courses.lumenlearning.com/suny-physics/chapter/23-4-eddy-currents-and-magnetic-damping/>
+
+
+## Vacuum compatible magnets {#vacuum-compatible-magnets}
+
+<https://www.mceproducts.com/articles/magnets-in-vacuum-applications>
+
+
+## Estimate the damping {#estimate-the-damping}
+
+
+### Formulas {#formulas}
+
+From (<a href="#citeproc_bib_item_1">Zuo 2004</a>):
+The empirical formula for damping coefficient (Ns/m) of an eddy current damper is:
+
+\begin{equation} \label{eq:damping\_formula}
+C = C\_0 B^2 t A \sigma
+\end{equation}
+
+with:
+
+-   \\(B\\) is the magnetic flux density in [T] or in [Vs/m2]
+-   \\(t\\) is the thickness of the conductor plate in [m]
+-   \\(A\\) is the area of the conductor intersected by the magnetic field in [m2]
+-   \\(\sigma\\) is the electrical conductivity of the conductor material [S/m]
+-   \\(C\_0\\) is a dimensionless coefficient to account for the shapes and sizes of the conductor and magnetic field
+
+\\(C\_0 = 1\\) corresponds to a conductor with conductivity \\(\sigma\\) inside a uniform magnetic field and conductivity infinite outside this field.
+A typical value of \\(C\_0\\) is about 0.25-0.4 for a conductor plate with area 2 to 5 times that of the magnetic field.
+
+From <eq:damping_formula>, we see that the damping coefficient is proportional to:
+
+-   the square of the magnetic flux density \\(B\\). Therefore it is very important to have large magnetic field strengh
+-   the thickness \\(t\\) of the conductor. However due to **skin depth effect**,  the benefit of increasing the thickness is limited.
+    The apparent conductivity \\(\sigma\_e\\) is:
+
+    \begin{equation}
+      \sigma\_e = \frac{2\delta\_s}{t}(1 - e^{-\frac{t}{2\delta\_s}})\sigma
+    \end{equation}
+
+    where \\(\delta\_s\\) is the skin depth in [m] of the conductor with permeability \\(\mu\\) in [H/m] at frequency \\(f\\) in [Hz]:
+
+    \begin{equation}
+      \delta\_s = \sqrt{\frac{2}{2 \pi f \cdot \mu \cdot \sigma}}
+    \end{equation}
+
+An eddy current damper is developed in (<a href="#citeproc_bib_item_1">Zuo 2004</a>).
+The magnets have alternating poles to optimize the eddy current damping (stronger varying magnetic field).
+See Figures [1](#figure--fig:zuo04-eddy-current-magnets) and [2](#figure--fig:zuo04-eddy-current-setup).
+
+<a id="figure--fig:zuo04-eddy-current-magnets"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class=\"figure-number\">Figure 1: </span>(left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
+
+<a id="figure--fig:zuo04-eddy-current-setup"></a>
+
+{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class=\"figure-number\">Figure 1: </span>Single DoF system damped by eddy current damper" >}}
+
+
+### Numerical Simulation {#numerical-simulation}
+
+It is possible to estimate that with FEM simulation: <https://www.youtube.com/watch?v=_1pgyj4lD7Q>
+
+An approximation is done bellow.
+
+```matlab
+B = 1.0; % Magnetic Flux Density [T]
+t = 5e-3; % Thickness [m]
+A = 50e-3*50e-3; % Area [m2]
+sigma = 6e7; % Copper conductivity [S/m]
+C0 = 0.5; % [-]
+```
+
+```matlab
+C = C0*B^2*t*A*sigma; % Damping in [N/(m/s)]
+```
+
+```text
+C = 375 [N/(m/s)]
+```
+
+```matlab
+m = 10; % [kg]
+k = m*(2*pi*10)^2; % [N/m]
+```
+
+```matlab
+xi = 1/2*C/sqrt(k*m);
+```
+
+```text
+xi = 0.298
+```
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.</div>
+</div>

content/zettels/electrical_impedance.md

@@ -0,0 +1,24 @@
++++
+title = "Electrical Impedance"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Impedance Analyzer {#impedance-analyzer}
+
+-   <https://www.zhinst.com/europe/en/instruments/product-finder/type/impedance_analyzers>
+
+
+## LCR Meter {#lcr-meter}
+
+-   <https://www.thinksrs.com/products/sr715720.html>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/electromagnetism.md

@@ -0,0 +1,49 @@
++++
+title = "Electromagnetism"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Maxwell equations for magnetics {#maxwell-equations-for-magnetics}
+
+
+### Gauss law {#gauss-law}
+
+"Magnetic fieldlines are closed loop."
+
+\begin{equation}
+\oiint\_S (\bm{B} \cdot \hat{\bm{n}}) dS = 0
+\end{equation}
+
+
+### Faraday's law {#faraday-s-law}
+
+A changing magnetic field causes an electric field over a wire
+
+\begin{equation}
+\oint\_L \bm{E} \cdot d\bm{l} = -\frac{d}{dt} \iint\_S(\bm{B} \cdot \bm{n}) dS
+\end{equation}
+
+The line-integral of the electrical field over a closed loop L equals the change of the field through the open surface S bounded by the loop L.
+This is a voltage source (EMF), where the current is driven in the direction of the electric field.
+
+
+### Ampère's law {#ampère-s-law}
+
+"Current through a wire gives a magnetic field".
+
+\begin{equation}
+\oint\_L \bm{B} \cdot dl = \mu\_0 I
+\end{equation}
+
+The line integral of the magnetic field over a closed loop L is proportional to the current through the surface S enclosed by the loop L.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/electronic_active_filters.md

@@ -7,14 +7,8 @@ draft = false
 Tags
 : [Operational Amplifiers]({{< relref "operational_amplifiers.md" >}})
 
-TODOS:
 
--   [X] Electronics circuits containing input voltage, output voltage, Op-amp, RLC components
+## Second Order Low Pass Filter {#second-order-low-pass-filter}
--   [ ] Bode plots of the filters
--   [ ] Inputs and output impedance
-
-
-## Low Pass Filter {#low-pass-filter}
 
 \begin{equation}
   \frac{V\_o}{V\_i}(s) = \frac{1}{R^2 C\_1 C\_2 s^2 + 2 R C\_2 s + 1}
@@ -29,14 +23,16 @@ With:
 -   \\(\omega\_0 = \frac{1}{R\sqrt{C\_1 C\_2}}\\)
 -   \\(\xi = \frac{C\_2}{C\_1}\\)
 
-<a id="figure--fig:elec-active-second-order-low-pass-filter"></a>
+The input impedance is \\(V\_i/i\_i\\).
 
-{{< figure src="/ox-hugo/elec_active_second_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>Second Order Low Pass Filter" >}}
+<a id="figure--fig:analog-act-filt-second-order-lpf"></a>
+
+{{< figure src="/ox-hugo/analog_act_filt_second_order_lpf.png" caption="<span class=\"figure-number\">Figure 1: </span>Second Order Low Pass Filter" >}}
 
 
-## High Pass Filter {#high-pass-filter}
+## Second Order High Pass Filter {#second-order-high-pass-filter}
 
-Same as [1](#figure--fig:elec-active-second-order-low-pass-filter) but by exchanging R1 with C1 and R2 with C2
+Same as [Figure 1](#figure--fig:analog-act-filt-second-order-lpf) but by exchanging R1 with C1 and R2 with C2
 
 \begin{equation}
   \frac{V\_o}{V\_i}(s) = \frac{R^2 C\_1 C\_2 s^2}{R^2 C\_1 C\_2 s^2 + 2 R C\_2 s + 1}
@@ -48,6 +44,11 @@ With:
 -   \\(\xi = \frac{C\_2}{C\_1}\\)
 
 
+## PID controller {#pid-controller}
+
+See [The design of high performance mechatronics - third revised edition]({{< relref "schmidt20_desig_high_perfor_mechat_third_revis_edition.md" >}}) (Chapter 6.2.6).
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

content/zettels/electronic_noise.md

@@ -0,0 +1,77 @@
++++
+title = "Electronic Noise"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Electronics]({{< relref "electronics.md" >}}), [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio.md" >}})
+
+
+## Thermal (Johnson) Noise {#thermal--johnson--noise}
+
+Thermal noise is generated by the thermal agitation of the electrons inside the electrical conductor.
+Its Power Spectral Density is equal to:
+
+\begin{equation}
+S\_T \approx 4 k T \text{Re}(Z(f)) \quad [V^2/Hz]
+\end{equation}
+
+with:
+with \\(k = 1.38 \cdot 10^{-23} \\,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
+
+This noise can be modeled as a voltage source in series with the system impedance.
+
+| Resistance      | PSD \\([V^2 / Hz]\\)     | ASD \\([V/\sqrt{Hz}]\\)  | RMS (1kHz) | RMS (10kHz) |
+|-----------------|--------------------------|--------------------------|------------|-------------|
+| \\(1 \Omega\\)  | \\(1.6 \cdot 10^{-20}\\) | \\(1.2 \cdot 10^{-10}\\) | 4nV        | 130nV       |
+| \\(1 k\Omega\\) | \\(1.6 \cdot 10^{-17}\\) | \\(4 \cdot 10^{-9}\\)    | 130nV      | 4uV         |
+| \\(1 M\Omega\\) | \\(1.6 \cdot 10^{-14}\\) | \\(1.2 \cdot 10^{-7}\\)  | 4uV        | 130uV       |
+
+
+## Shot Noise {#shot-noise}
+
+Seen with junctions in a transistor.
+It has a white spectral density:
+
+\begin{equation}
+S\_S = 2 q\_e i\_{dc} \ [A^2/Hz]
+\end{equation}
+
+with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
+
+<div class="exampl">
+
+A current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
+
+</div>
+
+
+## Excess Noise (or \\(1/f\\) noise) {#excess-noise--or-1-f-noise}
+
+It results from fluctuating conductivity due to imperfect contact between two materials.
+The PSD of excess noise increases when the frequency decreases:
+\\[ S\_E = \frac{K\_f}{f^\alpha}\ [V^2/Hz] \\]
+where \\(K\_f\\) is dependent on the average voltage drop over the resistor and the index \\(\alpha\\) is usually between 0.8 and 1.4, and often set to unity for approximate calculation.
+
+
+## Noise of Amplifiers {#noise-of-amplifiers}
+
+The noise of amplifiers can be modelled as shown in Figure [1](#figure--fig:electronic-amplifier-noise).
+
+<a id="figure--fig:electronic-amplifier-noise"></a>
+
+{{< figure src="/ox-hugo/electronic_amplifier_noise.png" caption="<span class=\"figure-number\">Figure 1: </span>Amplifier noise model" >}}
+
+The identification of this noise is a two steps process:
+
+1.  The amplifier input is short-circuited such that only \\(V^2(f)\\) has an impact on the output.
+    The output noise is measured and \\(V^2\\) in \\([V^2/Hz]\\) is identified
+2.  The amplifier input is open-circuited such that only \\(I^2(f)\\) has an impact on the output.
+    The output noise is measured and \\(I^2(f)\\) in \\([A^2/Hz]\\) is identified.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/encoders.md

@@ -13,12 +13,30 @@ There are two main types of encoders: optical encoders, and magnetic encoders.
 
 ## Manufacturers {#manufacturers}
 
-| Manufacturers                                                                   | Country |
+| Manufacturers                                                            | Country     |
-|---------------------------------------------------------------------------------|---------|
+|--------------------------------------------------------------------------|-------------|
-| [Heidenhain](https://www.heidenhain.com/en_US/products/linear-encoders/)        | Germany |
+| [Heidenhain](https://www.heidenhain.com/en_US/products/linear-encoders/) | Germany     |
-| [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA     |
+| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440)       | UK          |
-| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440)              | UK      |
+| [Celera Motion](https://www.celeramotion.com/microe/)                    | USA         |
-| [Celera Motion](https://www.celeramotion.com/microe/)                           | USA     |
+| [Magnescale](https://www.magnescale.com/en/)                             | Japanese    |
+| [Posic](https://www.posic.com/EN/)                                       | Switzerland |
+| [RLS](https://www.rls.si/eng/products/rotary-magnetic-encoders)          | Slovenia    |
+| [AMO](https://www.amo-gmbh.com/en/)                                      | Australia   |
+| [NumerikJena](https://www.numerikjena.de/en/)                            | Germany     |
+| [RSF Elektronik](https://www.rsf.at/en/)                                 | Austria     |
+| [Flux](https://flux.gmbh/products/gmi-rotary-encoder/)                   | Austria     |
+| [Lika](https://www.lika.it/eng/)                                         | Italy       |
+
+
+## Incremental vs Absolute {#incremental-vs-absolute}
+
+Incremental:
+
+-   Less delay
+
+Absolute:
+
+-   No problem of "missed" steps
 
 
 ## Bibliography {#bibliography}

content/zettels/ethercat.md

@@ -0,0 +1,39 @@
++++
+title = "EtherCAT"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Manufacturers {#manufacturers}
+
+General purpose / PLC:
+
+| Manufacturer                                                                                 |
+|----------------------------------------------------------------------------------------------|
+| [Bechoff](https://www.beckhoff.com/fr-fr/products/i-o/ethercat-terminals/)                   |
+| [Wago](https://www.wago.com/global/i-o-systems/fieldbus-coupler-ethercat/p/750-354)          |
+| [Rexroth](https://apps.boschrexroth.com/microsites/ctrlx-automation/en/portfolio/ctrlx-i-o/) |
+
+Acquisition systems:
+
+| Manufacturer                                                               |
+|----------------------------------------------------------------------------|
+| [National Instrument](https://www.ni.com/fr-fr/support/model.ni-9145.html) |
+| [Dewesoft](https://dewesoft.com/products/daq-systems)                      |
+
+
+## Cycle Time {#cycle-time}
+
+See (<a href="#citeproc_bib_item_1">Robert et al. 2012</a>).
+There is a nice [online calculator](https://developer.acontis.com/ethercat-cycle-time-calculator.html).
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Robert, Jérémy, Jean-Philippe Georges, Eric Rondeau, and Thierry Divoux. 2012. “Minimum Cycle Time Analysis of Ethernet-Based Real-Time protocols.” <i>International Journal of Computers, Communications and Control</i> 7 (4). Agora University of Oradea: 743–57. <a href="https://hal.archives-ouvertes.fr/hal-00714560">https://hal.archives-ouvertes.fr/hal-00714560</a>.</div>
+</div>

content/zettels/extremum_seeking_control.md

@@ -0,0 +1,194 @@
++++
+title = "Extremum Seeking Control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Nonlinear Control]({{< relref "nonlinear_control.md" >}})
+
+The idea behind "Extremum Seeking Control" is to use a controlled signal \\(u\\) to explore the relation \\(y(u)\\), estimate its gradient and find its minimum or maximum.
+This relation should be convex or concave for the architecture to work properly.
+
+See (<a href="#citeproc_bib_item_1">Tan et al. 2010</a>) for a good overview.
+
+
+## Control Architecture {#control-architecture}
+
+There are many extremum seeking control architectures.
+One of the simplest one is shown in Figure [1](#figure--fig:extremum-seeking-control-architecture).
+
+```latex
+\begin{tikzpicture}[
+  triangle/.style = {regular polygon, regular polygon sides=3},
+  right/.style = {shape border rotate=-90},
+  left/.style = {shape border rotate=90},
+  top/.style = {shape border rotate=0}
+  bottom/.style = {shape border rotate=180}
+  ]
+  % Extremum Seeking
+  \node[draw, minimum width=10cm,minimum height=4.2cm, dashed, label=Extremum Seeking Control] (extremum) at (0, 0) {};
+  \begin{scope}[shift={(-2.3, 0.7)}]
+    \node[draw, label=Adapt] (Adapt) at (0, 0) {$\displaystyle\frac{1}{A_p}\frac{K_I}{s}$};
+    \node[draw, label=LPF] (LPF) at (2, 0) {$\displaystyle\frac{1}{s/\omega_{L} + 1}$};
+    \node[addb={\times}{}{}{}{}] (multiply) at (4, 0) {};
+    \node[draw, label=HPF] (HPF) at (6, 0) {$\displaystyle\frac{s/\omega_{H}}{s/\omega_{H} + 1}$};
+    \node[addb={+}{}{}{}{}] (add) at (-2, 0) {};
+    \node[draw, triangle, left, inner sep=0pt] (gain_a) at (1, -2) {$A_p$};
+
+    \node[] (sinus) at (4, -2) {$\sin(\omega_p t)$};
+
+    \draw[<-] (add) -- node[above]{$\overline{u}$} (Adapt);
+    \draw[<-] (Adapt) -- node[above left]{} (LPF);
+    \draw[<-] (LPF) -- node[above left]{} (multiply);
+    \draw[<-] (multiply) -- node[above left]{} (HPF);
+    \draw[<-] (multiply) -- node[above right]{} (sinus);
+    \draw[<-] (gain_a) -- node[above left]{} (sinus);
+    \draw[<-] (add) -- node[above right]{$du$} (-2, -2) -- (gain_a);
+  \end{scope}
+
+  % Système
+  \node[draw, fill=black!20!white, minimum width=10cm,minimum height=3cm, label=System] (system) at (0, 4.5) {};
+  \begin{scope}[shift={(-2.5, 0.6)}]
+    % AC function of C_phi
+    \draw[domain=-1.5:1.5, shift={(2.5, 3)}] plot (2*\x, 0.8*\x*\x);
+    % Axes of plot
+    \draw[->] (-1, 2.8) -- (6, 2.8) node[above right]{$u$};
+    \draw[->] (-0.9, 2.7) -- (-0.9, 5) node[below left]{$y$};
+  \end{scope}
+
+  % Connections
+  \draw[<-] (system.west) node[above left](command){$u$} -- ++(-0.7, 0) |- (add);
+  \draw[->] (system.east) node[above right](measure){$y$} -- ++(0.7, 0) |- (HPF);
+\end{tikzpicture}
+```
+
+<a id="figure--fig:extremum-seeking-control-architecture"></a>
+
+{{< figure src="/ox-hugo/extremum_seeking_control_architecture.png" caption="<span class=\"figure-number\">Figure 1: </span>Extremum seeking control algorithm" >}}
+
+Its working principle is schematically shown in Figures [2](#figure--fig:extremum-seeking-control-non-minimum) and [3](#figure--fig:extremum-seeking-control-minimum).
+
+```latex
+\begin{tikzpicture}
+  % AC function of phi0
+  \draw[domain=-2:2] plot (\x, \x*\x);
+  % Axes of plot
+  \draw[->, >=latex] (-2.5 ,-0.5) -- (2.5, -0.5) node[below]{$u$};
+  \draw[->, >=latex] (-2.4 ,-0.6) -- (-2.4, 4.5) node[left]{$y$};
+
+  % Perturbation Signal
+  \draw[domain=-pi/2:pi/2, shift={(1, -1.5)}, rotate=90, samples=200] plot (\x,{0.5*sin(4*deg(\x))});
+  % Legend of perturbation Signal
+  \node[align=center] (pert_signal) at (-2, -1.8) {$\overline{u}$\\[-0.4em]+\\[-0.4em]$A_p\sin(\omega_p t)$};
+  \draw[<-, >=latex, dashed] ($(pert_signal.east)+(1.0, 0)$) -- ++(-1.5, 0);
+
+  % Dashed lines to show limits of the signals
+  \draw[dashed] (1.0, -3.2) -- ($(1.0, 1.0*1.0)$) -- ($(4.0, 1.0*1.0)$);
+  \draw[dashed] (0.5, -3.2) -- ($(0.5, 0.5*0.5)$) -- ($(8.0, 0.5*0.5)$);
+  \draw[dashed] (1.5, -3.2) -- ($(1.5, 1.5*1.5)$) -- ($(8.0, 1.5*1.5)$);
+
+  \begin{scope}[shift={(-0.5, 0)}]
+    % Image of the perturbation signal on AC
+    \draw[domain=-pi/2:pi/2, shift={(6, 0)}, samples=200] plot (\x,{(1+0.5*sin(4*deg(\x)))^2});
+    \draw[->, >=latex] (4, 1) -- (8, 1) node[below]{$t$};
+    \draw[->, >=latex] (4.1, 0.9) -- (4.1, 1.5*1.5+0.4) node[left]{$y$};
+
+    % Sinus omega_p
+    \draw[domain=-pi/2:pi/2, shift={(6, 4)}, samples=200] plot (\x,{0.5*sin(4*deg(\x))});
+    \draw[->, >=latex] (4, 4) -- (8, 4) node[below]{$t$};
+    \draw[->, >=latex] (4.1, 3.75) -- (4.1, 4.5) node[left]{$\sin(\omega_p t)$};
+
+    % Product of the sinus and the AC Command
+    \draw[domain=-pi/2:pi/2, shift={(6, -2.5)}, samples=200] plot (\x,{0.5*sin(4*deg(\x))*(1+0.5*sin(4*deg(\x)))^2});
+    \draw[->, >=latex] (4,-2.5) -- (8, -2.5) node[below]{$t$};
+    \filldraw[fill=gray!20] (4,-2.5) -- plot [domain=-pi/2:pi/2, shift={(6, -2.5)}, samples=200] (\x,{0.5*sin(4*deg(\x))*(1+0.5*sin(4*deg(\x)))^2}) -- cycle;
+    \draw[->, >=latex] (4.1,-2.6) -- (4.1, -1.0) node[left]{$ $};
+
+    % Multiply and equal signs
+    \node[addb={\times}{}{}{}{}] at (6, 3) {};
+    \node[addb={=}{}{}{}{}] at (6, -0.6) {};
+  \end{scope}
+\end{tikzpicture}
+```
+
+<a id="figure--fig:extremum-seeking-control-non-minimum"></a>
+
+{{< figure src="/ox-hugo/extremum_seeking_control_non_minimum.png" caption="<span class=\"figure-number\">Figure 2: </span>\\(\bar{u}\\) is not at the minimum of the \\(y(u)\\) relation. In that case the integral of the product between the sinusoidal excitation and the measured \\(y\\) is the image of the local gradient of the \\(y(u)\\) relationship." >}}
+
+```latex
+\begin{tikzpicture}
+  % AC function of phi0
+  \draw[domain=-2:2] plot (\x, \x*\x);
+  % Axes of plot
+  \draw[->, >=latex] (-2.5 ,-0.5) -- (2.5, -0.5) node[below]{$u$};
+  \draw[->, >=latex] (-2.4 ,-0.6) -- (-2.4, 4.5) node[left]{$y$};
+
+  % Perturbation Signal
+  \draw[domain=-pi/2:pi/2, shift={(0, -1.5)}, rotate=90, samples=200] plot (\x,{0.5*sin(4*deg(\x))});
+  % Legend of perturbation Signal
+  \node[] (pert_signal) at (-2, -1.0) {};
+
+  % Dashed lines to show limits of the signals
+  \draw[dashed] (0.0, -3.2) -- ($(0, 0)$) -- ($(4.0, 0)$);
+  \draw[dashed] (-0.5, -3.2) -- ($(-0.5, 0.5*0.5)$) -- ($(8.0, 0.5*0.5)$);
+  \draw[dashed] (0.5, -3.2) -- ($(0.5, 0.5*0.5)$);
+
+  \begin{scope}[shift={(-0.5, 0)}]
+    % Image of the perturbation signal on AC
+    \draw[domain=-pi/2:pi/2, shift={(6, 0)}, samples=200] plot (\x,{(0.5*sin(4*deg(\x)))^2});
+    \draw[->, >=latex] (4, 0) -- (8,  0) node[below]{$t$};
+    \draw[->, >=latex] (4.1, -0.1) -- (4.1, 1.0) node[left]{$y$};
+
+    % Sinus omega_p
+    \draw[domain=-pi/2:pi/2, shift={(6, 2.5)}, samples=200] plot (\x,{0.5*sin(4*deg(\x))});
+    \draw[->, >=latex] (4, 2.5) -- (8, 2.5) node[below]{$t$};
+    \draw[->, >=latex] (4.1, 2.25) -- (4.1, 3.0) node[above]{$\sin(\omega_p t)$};
+
+    % Product of the sinus and the AC Command
+    \draw[domain=-pi/2:pi/2, shift={(6, -2.5)}, samples=200] plot (\x,{0.7*(sin(4*deg(\x)))^3});
+    \draw[->, >=latex] (4,-2.5) -- (8, -2.5) node[below]{$t$};
+    \filldraw[fill=gray!20] (4,-2.5) -- plot [domain=-pi/2:pi/2, shift={(6, -2.5)}, samples=200] (\x,{0.7*(sin(4*deg(\x))^3}) -- cycle;
+    \draw[->, >=latex] (4.1,-2.6) -- (4.1, -1.0) node[left]{$ $};
+
+    % Multiply and equal signs
+    \node[addb={\times}{}{}{}{}] at (6, 1.2) {};
+    \node[addb={=}{}{}{}{}] at (6, -1.0) {};
+  \end{scope}
+\end{tikzpicture}
+```
+
+<a id="figure--fig:extremum-seeking-control-minimum"></a>
+
+{{< figure src="/ox-hugo/extremum_seeking_control_minimum.png" caption="<span class=\"figure-number\">Figure 3: </span>\\(\bar{u}\\) is at the minimum of the \\(y(u)\\) relation. In that case the integral of the product between the sinusoidal excitation and the measured \\(y\\) is null." >}}
+
+
+## Tuning of the Extremum Seeking Control {#tuning-of-the-extremum-seeking-control}
+
+The \\(y(u)\\) relationship should be static compared to \\(\omega\_p\\) the frequency of the sinusoidal excitation.
+
+When this architecture is to be applied the following two signals have to be properly chosen:
+
+-   what is the controlled signal \\(u\\)
+-   what is the measured signal to be minimized \\(y\\)
+
+Then, the following parameters should be tuned:
+
+-   \\(\omega\_p\\): the frequency of the perturbation, that should be small compared to the system dynamics
+-   \\(A\_p\\): amplitude of the sinusoidal perturbation, that should be small compared to the allowed deviation from the minimum
+-   \\(\omega\_H\\) and \\(\omega\_L\\): cut-off frequency of the high pass and low pass filters. As \\(\omega\_p\\) should be in the pass band of both filters, \\(\omega\_H\\) and \\(\omega\_L\\) should be chosen such that:
+    \\[ \omega\_H \ll \omega\_p \quad \text{and} \quad \omega\_L \gg \omega\_p \\]
+-   \\(K\_I\\): gain for the integrator that should be tuned such that the control loop is stable and converges to the minimum as fast as wanted.
+
+There are three time scales present in this control algorithm:
+
+-   Fast time scale that corresponds to the system variations (\\(y(u)\\) relationship)
+-   Medium time scale that corresponds to the perturbations on \\(u\\) (frequency \\(\omega\_p\\))
+-   Slow time scale that corresponds to the variations of \\(\bar{u}\\)
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tan, Y, WH Moase, C Manzie, D Nešić, and IMY Mareels. 2010. “Extremum Seeking from 1922 to 2010.” In <i>Control Conference (CCC), 2010 29th Chinese</i>, 14–26. IEEE.</div>
+</div>

content/zettels/feedback_control.md

@@ -7,7 +7,5 @@ draft = false
 Tags
 :
 
-## References
-
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
 </div>

content/zettels/feedforward_control.md

@@ -7,7 +7,227 @@ draft = false
 Tags
 :
 
-## References
+Depending on the physical system to be controlled, several feedforward controllers can be used:
+
+-   [Rigid body feedforward](#org-target--sec-rigid-body-feedforward)
+-   [Fourth order feedforward](#org-target--sec-fourth-order-feedforward)
+-   [Model based feedforward](#org-target--sec-model-based-feedforward)
+
+
+## Rigid Body Feedforward {#rigid-body-feedforward}
+
+<span class="org-target" id="org-target--sec-rigid-body-feedforward"></span>
+
+Second order trajectory planning: the acceleration and velocity can be bound to wanted values.
+
+Such trajectory is shown in [Figure 1](#figure--fig:feedforward-second-order-trajectory).
+
+<a id="figure--fig:feedforward-second-order-trajectory"></a>
+
+{{< figure src="/ox-hugo/feedforward_second_order_trajectory.png" caption="<span class=\"figure-number\">Figure 1: </span>Second order trajectory" >}}
+
+Here, it is supposed that the driven system is a simple mass \\(m\\) with a damper \\(c\\).
+In that case, the feedforward force should be:
+
+\begin{equation}
+F\_{ff} = m a + c v
+\end{equation}
+
+
+## Fourth Order Feedforward {#fourth-order-feedforward}
+
+<span class="org-target" id="org-target--sec-fourth-order-feedforward"></span>
+
+The main advantage of "fourth order feedforward" is that it takes into account the flexibility in the system (one resonance between the actuation point and the measurement point, see [Figure 2](#figure--fig:feedforward-double-mass-system)).
+This can lead to better results than second order trajectory planning as demonstrated [here](https://www.20sim.com/control-engineering/snap-feedforward/).
+
+<a id="figure--fig:feedforward-double-mass-system"></a>
+
+{{< figure src="/ox-hugo/feedforward_double_mass_system.png" caption="<span class=\"figure-number\">Figure 2: </span>Double mass system" >}}
+
+The equations of motion are:
+
+\begin{align}
+m\_1 \ddot{x}\_1 &= -c\_1 \dot{x}\_1 - k(x\_1 - x\_2) - c (\dot{x}\_1 - \dot{x}\_2) + F \\\\
+m\_2 \ddot{x}\_2 &= k(x\_1 - x\_2) + c (\dot{x}\_1 - \dot{x}\_2)
+\end{align}
+
+From the equation of motion, two transfer functions are computed:
+
+\begin{align}
+\frac{x\_2}{F}(s) &= \frac{c s + k}{(m\_1 s^2 + c\_1 s)(m\_2 s^2 + c s + k) + m\_2 s^2 (cs + k)} \\\\
+\frac{x\_1}{F}(s) &= \frac{m\_2 s^2 + c s + k}{(m\_1 s^2 + c\_1 s)(m\_2 s^2 + c s + k) + m\_2 s^2 (cs + k)}
+\end{align}
+
+Depending on whether \\(x\_1\\) or \\(x\_2\\) is to be positioned, two feedforward controllers can be used.
+
+If \\(x\_2\\) is to be positioned, the ideal feedforward force \\(F\_{f2}\\) is:
+
+\begin{equation}
+F\_{f2} = \frac{q\_1 s^4 + q\_2 s^3 + q\_3 s^2 + q\_4 s}{k\_{12} s + c} \cdot x\_2
+\end{equation}
+
+with:
+
+\begin{align}
+q\_1 &= m\_1 m\_2 \\\\
+q\_2 &= (m\_1 + m\_2) k\_{12} + m\_1 k\_2 + m\_2 k\_1 \\\\
+q\_3 &= (m\_1 + m\_2)c + k\_1 k\_2 + (k\_1 + k\_2) k\_{12} \\\\
+q\_4 &= (k\_1 + k\_2) c
+\end{align}
+
+This means that if a fourth-order trajectory for \\(x\_2\\) is used, the feedforward architecture shown in [Figure 3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used:
+
+\begin{equation}
+F\_{f2} = \frac{1}{k\_12 s + c} (q\_1 d + q\_2 j + q\_3 q + q\_4 v)
+\end{equation}
+
+<a id="figure--fig:feedforward-fourth-order-feedforward-architecture"></a>
+
+{{< figure src="/ox-hugo/feedforward_fourth_order_feedforward_architecture.png" caption="<span class=\"figure-number\">Figure 3: </span>Fourth order feedforward implementation" >}}
+
+Similarly, if \\(x\_1\\) is to be positioned, the perfect feedforward force \\(F\_{f1}\\) is:
+
+\begin{equation}
+F\_{f1} = \frac{1}{m\_2 s^2 + c s + k} \cdot (q\_1 s + q\_2 j + q\_3 a + q\_4 v)
+\end{equation}
+
+with:
+
+\begin{align}
+q\_1 &= m\_1 m\_2 \\\\
+q\_2 &= (m\_1 + m\_2) c + m\_2 c\_1 \\\\
+q\_3 &= (m\_1 + m\_2) k + c\_1 c \\\\
+q\_4 &= c\_1 k
+\end{align}
+
+and \\(s\\) the snap, \\(j\\) the jerk, \\(a\\) the acceleration and \\(v\\) the velocity.
+
+The same architecture shown in [Figure 3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used.
+
+In order to implement a fourth order trajectory, look at [this](https://www.mathworks.com/matlabcentral/fileexchange/16352-advanced-setpoints-for-motion-systems) nice implementation in Simulink of fourth-order trajectory planning (see also (<a href="#citeproc_bib_item_1">Lambrechts, Boerlage, and Steinbuch 2004</a>)).
+
+
+## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant}
+
+<span class="org-target" id="org-target--sec-model-based-feedforward"></span>
+
+See (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
+
+Suppose we have a second order plant (could typically be a piezoelectric stage):
+\\[ G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\xi \omega\_0 s + \omega\_0^2} \\]
+
+<a id="figure--fig:feedforward-second-order-plant"></a>
+
+{{< figure src="/ox-hugo/feedforward_second_order_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Bode plot of a second order system with fitted model" >}}
+
+The idea is to design a feedforward controller that corresponds to the plant inverse:
+\\[ C\_{ff}(s) = \frac{s^2 + 2\xi \omega\_0 s + \omega\_0^2}{C\_f \omega\_0^2} \\]
+
+This controller has a pair of zeros, corresponding to an anti-resonance at the eigenfrequency of the first eigenmode of the system, with equal damping.
+The controller needs to be modified in such a way that it becomes realisable.
+In this case it is decided to create a resulting overall transfer function of the controller and the plant that acts like a well damped mass-spring system with the same natural frequency as the plant and an additional reduction of the excitation of higher frequency eigenmodes.
+In order to realise this controller first two poles have to be added, placed at the same frequency as the resonance but with a higher damping ratio.
+Typically a damping ratio between aperiodic and critical (\\(0.7 < \xi < 1\\)) is applied to avoid oscillations.
+For \\(\xi = 1\\) this results in the following transfer function:
+\\[ C\_{ff}(s) = \frac{s^2 + 2\xi \omega\_0 s + \omega\_0^2}{s^2 + 2 \cdot 1 \cdot \omega\_0 s + \omega\_0^2}\\]
+
+<a id="figure--fig:feedforward-compensated-system"></a>
+
+{{< figure src="/ox-hugo/feedforward_compensated_system.png" caption="<span class=\"figure-number\">Figure 5: </span>Bode plot of the feedforward controlled system" >}}
+
+
+## Advanced Feedforward (from MIMO training) {#advanced-feedforward--from-mimo-training}
+
+A typical control configuration for motion systems consists of:
+
+-   A setpoint generator (SPG)
+-   A feedback controller (\\(K\_{fb}\\))
+-   A feedforward controller (\\(K\_{ff}\\))
+
+{{< figure src="/ox-hugo/feedforward_schematic.png" >}}
+
+The closed-loop error (no disturbances) is:
+\\[ e(s) = (1 + G(s)K\_{fb})^{-1} (1 - G(s)K\_{ff}(s)) r(s) \\]
+It therefore depends on:
+
+1.  the setpoint \\(r\\)
+2.  the feedforward controller
+3.  the feedback controller
+
+**Setpoint generation**:
+
+-   It can be 2nd order, 3rd order or 4th order
+-   For 4th order, derivative of jerk is generated over time, and then integrated 4 times to give: jerk, acceleration, velocity and position.
+
+**2nd order setpoint generation**:
+If we compute the fourier transform of the generated acceleration, we get the following signal (-20db/dec).
+
+{{< figure src="/ox-hugo/feedforward_2nd_order_fourier.png" >}}
+
+Notches are at \\(f\_1\\), \\(2f\_1\\), \\(3f\_1\\), ... with \\(f\_1 = \frac{a\_{\text{max}}}{v\_{\text{max}}}\\).
+It is therefore possible to choose the velocity and acceleration such that \\(f\_1\\) (or one of its integral multiple) matches the resonance frequency of the system.
+Therefore, the acceleration time constant can be chosen at the inverse of the plant resonance.
+
+**3rd order setpoint generation**:
+There is a drawback of having an extra time of \\(\frac{a\_{max}}{J\_{max}}\\) seconds.
+However, we get an additional -20db/dec at high frequency, and additional notches at \\(f\_2 = \frac{j\_{max}}{a\_{max}}\\).
+This new notch has larger "damping" and can be used to be more robust against resonances of the plant.
+
+**Feedforward control**:
+Plant inversion: if \\(K\_{ff} = G^{-1}(s) \Longrightarrow e(s) = 0\\)
+Challenges:
+
+-   Model required
+-   High order
+-   Delay/non-minimum phase?
+
+**Rigid body dynamics**:
+\\(G(s) = \frac{1}{ms^2}\\)
+In that case, \\(G^{-1}(s) = ms^2\\), and with 2nd order setpoint, a feedforward controller \\(K\_{ff}(s) = m\\) gives good performances.
+
+{{< figure src="/ox-hugo/feedforward_schematic_rigid_body.png" >}}
+
+**Discrete time implementation**.
+The DAC can usually be modelled by a "Zero Order Hold" (ZOH) and the ADC with a "sampler".
+This adds **1.5 samples of delay**: \\(0.5z^{-1} + 0.5z^{-2}\\).
+
+It seems the ZOH can be modelled by an "half-sample delay".
+There is an additional one sample delay.
+
+{{< figure src="/ox-hugo/feedforward_schematic_zoh_sampler.png" >}}
+
+Therefore, it is very important to match the delay of the plant:
+
+> In high-performance control system, it can be useful to consider propagation delay when designing the feedforward and feedback controllers.
+> Feedforward control directly uses the reference trajectory and does not depend on any measurement data.
+> However, the feedback controller uses (delayed) measured position data.
+> Due to propagation delay in the control system (caused by the controller, actuator or sensor), it can take multiple cycles for the effect of feedforward control to be observed in the measured position.
+> In the meantime, the feedback control is already seeing a tracking error and is compensating for it.
+> Essentially, the result of the feedforward action arrives too late, resulting in possible overcompensation by the feedback control.
+> When the propagation delay in the control system is known, it can be compensated for by applying this same delay to the demand position in the tracking error calculation.
+
+{{< figure src="/ox-hugo/feedforward_schematic_delay.png" >}}
+
+**Feedforward for flexible dynamics**:
+4th order dynamics:
+
+{{< figure src="/ox-hugo/feedforward_4th_order.png" >}}
+
+Dynamics from \\(F\\) to \\(x\_2\\) is:
+\\[ G(s) = \frac{x\_2}{F} = \frac{cs + k}{m\_1m\_2 s^2(s^2 + 2\xi\omega\_0 s + \omega\_0^2)} \\]
+We take the inverse for the feedforward controller:
+\\[ K\_{ff}(s) = G^{-1}(s) = \frac{m\_1m\_2 s^2(s^2 + 2\xi\omega\_0 s + \omega\_0^2)}{cs + k} \\]
+If we neglect damped: \\(\xi = 0\\), and we get:
+\\[ K\_{ff}(s) = \underbrace{\frac{m\_1m\_2}{k} s^4}\_{\text{snap FF}} + \underbrace{(m\_1 + m\_2) s^2}\_{\text{acc FF}} \\]
+This can be solved by using **snap feedforward**
+
+{{< figure src="/ox-hugo/feedforward_schematic_snap.png" >}}
+
+
+## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lambrechts, P., M. Boerlage, and M. Steinbuch. 2004. “Trajectory Planning and Feedforward Design for High Performance Motion Systems.” In <i>Proceedings of the 2004 American Control Conference</i>. doi:<a href="https://doi.org/10.23919/acc.2004.1384042">10.23919/acc.2004.1384042</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
 </div>

content/zettels/finite_element_model.md

@@ -14,7 +14,7 @@ Some resources:
 
 -   (<a href="#citeproc_bib_item_1">Hatch 2000</a>) ([Notes]({{< relref "hatch00_vibrat_matlab_ansys.md" >}}))
 -   (<a href="#citeproc_bib_item_2">Khot and Yelve 2011</a>)
--   (<a href="#citeproc_bib_item_3">Kovarac et al. 2015</a>)
+-   (NO_ITEM_DATA:kosarac15_creat_siso_ansys)
 
 The idea is to extract reduced state space model from Ansys into Matlab.
 
@@ -22,7 +22,7 @@ The idea is to extract reduced state space model from Ansys into Matlab.
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using MATLAB and ANSYS</i>. CRC Press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Khot, SM, and Nitesh P Yelve. 2011. “Modeling and Response Analysis of Dynamic Systems by Using Ansys and Matlab.” <i>Journal of Vibration and Control</i> 17 (6). SAGE Publications Sage UK: London, England: 953–58.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Kovarac, A, M Zeljkovic, C Mladjenovic, and A Zivkovic. 2015. “Create Siso State Space Model of Main Spindle from Ansys Model.” In <i>12th International Scientific Conference, Novi Sad, Serbia</i>, 37–41.</div>
+  <div class="csl-entry">NO_ITEM_DATA:kosarac15_creat_siso_ansys</div>
 </div>

content/zettels/flexible_joints.md

@@ -24,7 +24,7 @@ Presentations:
 -   (<a href="#citeproc_bib_item_4">Henein 2010</a>)
 
 
-## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms}
+## Flexure Joints for Stewart Platforms {#flexure-joints-for-stewart-platforms}
 
 From (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>):
 
@@ -32,19 +32,31 @@ From (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>):
 > A flexure joint bends material to achieve motion, rather than sliding of rolling across two surfaces.
 > This does eliminate friction and backlash, but adds spring dynamics and limits the workspace.
 
+<https://www.youtube.com/watch?v=tenxq7N5q3k>
+
 
 ## Materials {#materials}
 
+Typical materials used for flexible joints are:
+
+-   Steel
+-   Aluminum
+-   Titanium
+
+
+## Manufacturers {#manufacturers}
+
+<https://www.flexpivots.com/>
+Prototyping kits: <https://www.motusmechanical.com/>
+
 
 ## Bibliography {#bibliography}
 
-## References
-
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 Icra. Millennium Conference. Ieee International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00ch37065)</i>, nil. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)</i>. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Cosandier, Florent. 2017. <i>Flexure Mechanism Design</i>. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Henein, Simon. 2003. <i>Conception Des Guidages Flexibles</i>. Lausanne, Suisse: Presses polytechniques et universitaires romandes.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_4"></a>———. 2010. “Flexures: Simply Subtle.” In <i>Diamond Light Source Proceedings, Medsi 2010</i>. Cambridge University Press.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_4"></a>———. 2010. “Flexures: Simply Subtle.” In <i>Diamond Light Source Proceedings, MEDSI 2010</i>. Cambridge University Press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_5"></a>Lobontiu, Nicolae. 2002. <i>Compliant Mechanisms: Design of Flexure Hinges</i>. CRC press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_6"></a>Smith, Stuart T. 2000. <i>Flexures: Elements of Elastic Mechanisms</i>. Crc Press.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_7"></a>———. 2005. <i>Foundations of Ultra-Precision Mechanism Design</i>. Vol. 2. CRC Press.</div>

content/zettels/force_sensors.md

@@ -18,7 +18,7 @@ There are two main technique for force sensors:
 
 The choice between the two is usually based on whether the measurement is static (strain gauge) or dynamics (piezoelectric).
 
-Main differences between the two are shown in Figure [1](#figure--fig:force-sensor-piezo-vs-strain-gauge).
+Main differences between the two are shown in [Figure 1](#figure--fig:force-sensor-piezo-vs-strain-gauge).
 
 <a id="figure--fig:force-sensor-piezo-vs-strain-gauge"></a>
 
@@ -79,5 +79,5 @@ However, if a charge conditioner is used, the signal will be doubled.
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
 </div>

content/zettels/fractional_order_transfer_functions.md

@@ -37,7 +37,7 @@ G =
 Continuous-time transfer function.
 ```
 
-Few examples of different slopes are shown in Figure [1](#figure--fig:approximate-deriv-int).
+Few examples of different slopes are shown in [Figure 1](#figure--fig:approximate-deriv-int).
 
 <a id="figure--fig:approximate-deriv-int"></a>
 

content/zettels/granite.md

@@ -11,10 +11,11 @@ Tags
 
 ## Manufacturers {#manufacturers}
 
-| Manufacturers                                    | Country |
+| Manufacturers                                    | Country     |
-|--------------------------------------------------|---------|
+|--------------------------------------------------|-------------|
-| [Microplan](https://www.microplan-group.com/fr/) | France  |
+| [Microplan](https://www.microplan-group.com/fr/) | France      |
-| [Zali](http://zali-precision.it/en/products/)    | Italy   |
+| [Zali](http://zali-precision.it/en/products/)    | Italy       |
+| [Mytri](https://www.mytri.nl/en)                 | Netherlands |
 
 
 ## Bibliography {#bibliography}

content/zettels/gravity_compensation.md

@@ -0,0 +1,50 @@
++++
+title = "Gravity Compensation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Reviews {#reviews}
+
+<https://www.zaber.com/articles/magnetic-counterbalances-for-vertical-applications>
+
+
+## Counterweight {#counterweight}
+
+<&yoshioka17_newly_devel_zero_gravit_vertic>
+
+
+## Magnetic {#magnetic}
+
+-   <&hol06_desig_magnet_gravit_compen_system>
+-   <https://linmot.com/products/magspring/>
+-   <&westhoff24_desig_elect_linear_drive_with>
+
+
+## Simple Spring {#simple-spring}
+
+
+## Constant force spring {#constant-force-spring}
+
+<https://www.inexal.be/fr/ressorts-force-constante>
+<https://www.leespring.com/constant-force-springs>
+<https://aimcoil.com/constant-force-springs/>
+
+
+## Variable Gravity Compensation {#variable-gravity-compensation}
+
+As the mass / position of the load may change during operation, a variable gravity compensation mechanism is very useful.
+
+
+## Commercial products {#commercial-products}
+
+<https://linmot.com/blog/floating-weights-with-magspring/>
+
+
+## Bibliography {#bibliography}
+
+<./biblio/references.bib>

content/zettels/hac_hac.md

@@ -11,7 +11,7 @@ High-Authority Control/Low-Authority Control
 
 From (<a href="#citeproc_bib_item_2">Preumont 2018</a>):
 
-> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#figure--fig:hac-lac-control-architecture). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:
+> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in [Figure 1](#figure--fig:hac-lac-control-architecture). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:
 >
 > -   The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth
 > -   The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
@@ -32,5 +32,5 @@ Nice papers:
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Aubrun, J.N. 1980. “Theory of the Control of Structures by Low-Authority Controllers.” <i>Journal of Guidance and Control</i> 3 (5): 444–51. doi:<a href="https://doi.org/10.2514/3.56019">10.2514/3.56019</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In <i>Proceedings of the 28th Ieee Conference on Decision and Control</i>, nil. doi:<a href="https://doi.org/10.1109/cdc.1989.70563">10.1109/cdc.1989.70563</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In <i>Proceedings of the 28th IEEE Conference on Decision and Control</i>. doi:<a href="https://doi.org/10.1109/cdc.1989.70563">10.1109/cdc.1989.70563</a>.</div>
 </div>

content/zettels/heat_transfer.md

@@ -0,0 +1,215 @@
++++
+title = "Heat Transfer"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Electrical Analogy - Lumped Mass Modeling {#electrical-analogy-lumped-mass-modeling}
+
+The difference in temperature \\(\Delta T\\) is driving potential energy flow \\(Q\\) (in watts):
+
+\begin{equation}
+  \Delta T = R\_{th} \cdot Q
+\end{equation}
+
+\\(R\_{th}\\) is the analogy of a "thermal resistance", and is expressed in K/W.
+
+
+## Conduction (diffusion) {#conduction--diffusion}
+
+The _conduction_ corresponds to the heat transfer \\(Q\\) (in watt) through molecular agitation within a material.
+
+\begin{equation}
+R\_{th} = \frac{d}{\lambda A}
+\end{equation}
+
+with:
+
+-   \\(\lambda\\) the thermal conductivity in \\([W/m \cdot K]\\)
+-   \\(A\\) the surface area in \\([m^2]\\)
+-   \\(d\\) the length of the barrier in \\([m]\\)
+
+
+## Convection {#convection}
+
+The convection corresponds to the heat transfer \\(Q\\) through flow of a fluid.
+It can be either _natural_ or _forced_.
+
+\begin{equation}
+R\_{th} = \frac{1}{h A}
+\end{equation}
+
+with:
+
+-   \\(h\\) the convection heat transfer coefficient in \\([W/m^2 \cdot K]\\).
+    \\(h \approx 10.5 - v + 10\sqrt{v}\\) with \\(v\\) the velocity of the object through the fluid in \\([m/s]\\)
+    Typically:
+    -   \\(h = 5 - 10\ W/m^2/K\\) for free convection with air
+    -   \\(h = 500 - 5000\ W/m^2/K\\) for forced water cooling in a tube of 5mm diameter
+-   \\(A\\) the surface area in \\([m^2]\\)
+
+Note that clean-room air flow should be considered as forced convection, and \\(h \approx 10 W/m^2/K\\).
+
+
+## Radiation {#radiation}
+
+_Radiation_ corresponds to the heat transfer \\(Q\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings.
+
+In the general case, we have:
+\\[ Q = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
+with:
+
+-   \\(\epsilon\\) the emissivity which corresponds to the ability of a surface to emit energy through radiation relative to a black body surface at equal temperature.
+    It is between 0 (no emissivity) and 1 (maximum emissivity)
+-   \\(\sigma\\) the Stefan-Boltzmann constant: \\(\sigma = 5.67 \cdot 10^{-8} \\, \frac{W}{m^2 K^4}\\)
+-   \\(T\_r\\) the temperature of the emitter in \\([K]\\)
+-   \\(T\_s\\) the temperature of the surrounding in \\([K]\\)
+
+In order to use the lumped mass approximation, the equations can be linearized to obtain:
+
+\begin{equation}
+R\_{th} = \frac{1}{h\_{rad} A}
+\end{equation}
+
+with:
+
+-   \\(h\_{rad}\\) the effective heat transfer coefficient for radiation in \\(W/m^2 \cdot K\\)
+-   \\(A\\) the surface in \\([m^2]\\)
+
+
+### Practical Cases {#practical-cases}
+
+Two parallel plates:
+
+\begin{equation}
+h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + 1/\epsilon\_2 - 1} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+Two concentric cylinders:
+
+\begin{equation}
+h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + r\_1/r\_2 (1/\epsilon\_2 - 1)} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+A small object enclosed in a large volume:
+
+\begin{equation}
+h\_{rad} = \epsilon\_1 \sigma (T\_1^2 + T\_2^2)(T\_1 + T\_2)
+\end{equation}
+
+
+### Emissivity {#emissivity}
+
+The emissivity of materials highly depend on the surface finish (the more polished, the lower the emissivity).
+Some examples are given in <tab:emissivity_examples>.
+
+Gold coating gives also a very low emissivity and is typically used in cryogenic applications.
+
+<a id="table--tab:emissivity-examples"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:emissivity-examples">Table 1</a>:</span>
+  Some examples of emissivity (specified at 25 degrees)
+</div>
+
+| Substance                  | Emissivity |
+|----------------------------|------------|
+| Silver (polished)          | 0.005      |
+| Silver (oxidized)          | 0.04       |
+| Stainless Steel (polished) | 0.02       |
+| Aluminium (polished)       | 0.02       |
+| Aluminium (oxidized)       | 0.2        |
+| Aluminium (anodized)       | 0.9        |
+| Copper (polished)          | 0.03       |
+| Copper (oxidized)          | 0.87       |
+
+<div class="exampl">
+
+Let's take a polished aluminum plate (20 by 20 cm) at 125K (temperature of zero thermal expansion coefficient of silicon) surrounded by elements are 25 degrees (300 K):
+\\[ P = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) = 0.36\\, J \\]
+
+</div>
+
+
+## Heat {#heat}
+
+The _heat_ \\(Q\\) (in Joules) corresponds to the energy necessary to change the temperature of the mass with a certain material specific heat capacity:
+\\[ Q = m \cdot c \cdot \Delta T \\]
+with:
+
+-   \\(m\\) the mass in \\([kg]\\)
+-   \\(c\\) the specific heat capacity in \\([J/kg \cdot K]\\)
+-   \\(\Delta T\\) the temperature different \\([K]\\)
+
+<div class="exampl">
+
+Let's compute the heat (i.e. energy) necessary to increase a 1kg granite by 1 degree.
+The specific heat capacity of granite is \\(c = 790\\,[J/kg\cdot K]\\).
+The required heat is then:
+\\[ Q = m\cdot c \cdot \Delta T = 790 \\,J \\]
+
+</div>
+
+<a id="table--tab:specific-heat-capacity"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:specific-heat-capacity">Table 2</a>:</span>
+  Some examples of specific heat capacity
+</div>
+
+| Substance           | Specific heat capacity [J/kg.K] |
+|---------------------|---------------------------------|
+| Air                 | 1012                            |
+| Aluminium           | 897                             |
+| Copper              | 385                             |
+| Granite             | 790                             |
+| Steel               | 466                             |
+| Water at 25 degrees | 4182                            |
+
+
+## Heat Transport (i.e. Water cooling) {#heat-transport--i-dot-e-dot-water-cooling}
+
+<a id="figure--fig:heat-transfer-fluid"></a>
+
+{{< figure src="/ox-hugo/heat_transfer_fluid.png" caption="<span class=\"figure-number\">Figure 1: </span>Heat transfered to the fluid" >}}
+
+\begin{equation}
+Q\_{in} = h \cdot A \cdot (T\_{wall} - T\_{mean})
+\end{equation}
+
+<a id="figure--fig:heat-transport"></a>
+
+{{< figure src="/ox-hugo/heat_transport.png" caption="<span class=\"figure-number\">Figure 2: </span>Heat Transport in the fluid" >}}
+
+\begin{equation}
+Q\_{out} = \phi \rho c\_p (T\_{mean,in} - T\_{mean,out})
+\end{equation}
+
+with:
+
+-   \\(Q\_{out}\\) the transported heat in W
+-   \\(\phi\\) the flow in \\(m^3/s\\)
+-   \\(\rho\\) the fluid density in \\(kg/m^3\\)
+-   \\(c\_p\\) the specific heat capacity of the fluid in \\(J/(kg \cdot K)\\)
+-   \\(T\_{mean}\\) the mean incoming and outgoing fluid temperature
+
+Because of energy balance, we have in the stationary condition: \\(Q\_{in} = Q\_{out}\\)
+
+
+## Heat flow {#heat-flow}
+
+The heat flow \\(P\\) (in watt) is the derivative of the heat:
+\\[ P = \cdot{Q} = \frac{dQ}{dt} = \frac{dT}{R\_T} = C\_T \cdot dT \\]
+with:
+
+-   \\(Q\\) the heat in [W]
+-   \\(R\_T\\) the thermal resistance in \\([K/W]\\)
+-   \\(C\_T\\) the thermal conductance in \\([W/K]\\)
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/heaters.md

@@ -0,0 +1,48 @@
++++
+title = "Heaters"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Commercial products {#commercial-products}
+
+-   <https://www.tcdirect.fr/product-2-300-1/Cartouche-chauffante>
+-   <https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=305>
+
+
+### Cryogenic temperature (~120K, UHV compatible) {#cryogenic-temperature--120k-uhv-compatible}
+
+
+### 20degC temperature {#20degc-temperature}
+
+<https://fr.rs-online.com/web/p/elements-chauffants/7983769>
+
+
+### 20degC temperature (UHV) {#20degc-temperature--uhv}
+
+From <https://confluence.esrf.fr/display/~moyne/GRATING+MIRRORS+COOLING+SYSTEM+DESIGN>:
+
+-   <https://www.watlow.com/Products/Heaters/Specialty-Heaters/ULTRAMIC-Ceramic-Heaters>
+
+From (<a href="#citeproc_bib_item_1">Neto et al. 2022</a>)
+
+> UHV-compatible Kapton heaters from Taiwan KLC (part number TSC013D003GR36Z01), with nominal resistances of 36 Ω and 14.4 Ω for 4 W and 10 W power at 12 V, respectively
+> Although having an easy integration and proven vacuum compatibility, along with low cost, the flexible nature of the Kapton heaters made the clamping to the components a potential source of failure.
+
+<!--quoteend-->
+
+> Therefore, a new heating element is under development for higher reliability.
+> As depicted in Fig. 2, it consists of an **SMD nickel thin film and alumina power resistor from Susumu**, soldered over a small aluminium metalcore PCB (Printed Circuit Board) using a lead free (SAC305) solder paste.
+> The board is then encapsulated inside a small aluminium case using the Stycast 2850FT epoxy resin along with CAT11 catalyser.
+> The aluminum PCB and housing serve as efficient heat condutors to the part of interest
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Neto, Joao Brito, Renan Geraldes, Francesco Lena, Marcelo Moraes, Antonio Piccino Neto, Marlon Saveri Silva, and Lucas Volpe. 2022. “Temperature Control for Beamline Precision Systems of Sirius/Lnls.” <i>Proceedings of the 18th International Conference on Accelerator and Large Experimental Physics Control Systems</i> ICALEPCS2021 (nil): China. doi:<a href="https://doi.org/10.18429/JACOW-ICALEPCS2021-WEPV001">10.18429/JACOW-ICALEPCS2021-WEPV001</a>.</div>
+</div>

content/zettels/inertial_sensors.md

@@ -36,7 +36,7 @@ Wireless Accelerometers
 
 -   <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
 
-Several commercial accelerometers are compared in Table [2](#figure--fig:characteristics-accelerometers) (see (<a href="#citeproc_bib_item_1">Collette et al. 2011</a>)).
+Several commercial accelerometers are compared in Table [Figure 2](#figure--fig:characteristics-accelerometers) (see (<a href="#citeproc_bib_item_1">Collette et al. 2011</a>)).
 
 <a id="figure--fig:characteristics-accelerometers"></a>
 
@@ -67,5 +67,5 @@ Several commercial accelerometers are compared in Table [2](#figure--fig:charact
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C, K Artoos, M Guinchard, S Janssens, P Carmona Fernandez, and C Hauviller. 2011. “Review of Sensors for Low Frequency Seismic Vibration Measurement.” CERN.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” <i>Bulletin of the Seismological Society of America</i> 102 (4): 1289–1300. doi:<a href="https://doi.org/10.1785/0120110223">10.1785/0120110223</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In <i>International Conference on Noise and Vibration Engineering (Isma)</i>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In <i>International Conference on Noise and Vibration Engineering (ISMA)</i>.</div>
 </div>

content/zettels/integral_force_feedback.md

@@ -16,6 +16,6 @@ This can be done with a [Voice Coil Actuator]({{< relref "voice_coil_actuators.m
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Jansen, Bas, Hans Butler, and Ruben Di Filippo. 2019. “Active Damping of Dynamical Structures Using Piezo Self Sensing.” <i>Ifac-Papersonline</i> 52 (15). Elsevier: 543–48.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Jansen, Bas, Hans Butler, and Ruben Di Filippo. 2019. “Active Damping of Dynamical Structures Using Piezo Self Sensing.” <i>IFAC-PapersOnLine</i> 52 (15). Elsevier: 543–48.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Verma, Mohit, Vicente Lafarga, and Christophe Collette. 2020. “Perfect Collocation Using Self-Sensing Electromagnetic Actuator: Application to Vibration Control of Flexible Structures.” <i>Sensors and Actuators a: Physical</i> 313: 112210. doi:<a href="https://doi.org/10.1016/j.sna.2020.112210">10.1016/j.sna.2020.112210</a>.</div>
 </div>

content/zettels/interferometers.md

@@ -21,6 +21,8 @@ Tags
 | [Sios](https://sios-de.com/products/length-measurement/laser-interferometer/)                                | Germany     |
 | [Keysight](https://www.keysight.com/en/pc-1000000393%3Aepsg%3Apgr/laser-heads?nid=-536900395.0&cc=FR&lc=fre) | USA         |
 | [Optics11](https://optics11.com/)                                                                            | Netherlands |
+| [Prodrive](https://prodrive-technologies.com/motion/products/interferometer/)                                | Netherlands |
+| [Agito](https://agito-akribis.com/voice-coil-motors/)                                                        |             |
 
 
 ## Reviews {#reviews}
@@ -30,11 +32,11 @@ Tags
 
 ## Effect of Refractive Index - Environmental Units {#effect-of-refractive-index-environmental-units}
 
-The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in Table [1](#table--tab:index-air) (Taken from (<a href="#citeproc_bib_item_5">Thurner et al. 2015</a>)).
+The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in [Table 1](#table--tab:index-air) (Taken from (<a href="#citeproc_bib_item_5">Thurner et al. 2015</a>)).
 
 <a id="table--tab:index-air"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:index-air">Table 1</a></span>:
+  <span class="table-number"><a href="#table--tab:index-air">Table 1</a>:</span>
   Dependence of Refractive Index \(n\) of Air from Temperature \(T\), pressure \(p\), Humidity \(h\), and CO2 content \(x_c\). Taken around \(T = 20^oC\), \(p=101kPa\), \(h = 50\%\), \(x_c = 400 ppm\) and \(\lambda = 1530nm\)
 </div>
 
@@ -48,11 +50,11 @@ The measured distance is proportional to the refractive index of the air that de
 
 In order to limit the measurement uncertainty due to variation of air parameters, an Environmental Unit can be used that typically measures the temperature, pressure and humidity and compensation for the variation of refractive index in real time.
 
-Typical characteristics of commercial environmental units are shown in Table [2](#table--tab:environmental-units).
+Typical characteristics of commercial environmental units are shown in [Table 2](#table--tab:environmental-units).
 
 <a id="table--tab:environmental-units"></a>
 <div class="table-caption">
-  <span class="table-number"><a href="#table--tab:environmental-units">Table 2</a></span>:
+  <span class="table-number"><a href="#table--tab:environmental-units">Table 2</a>:</span>
   Characteristics of Environmental Units
 </div>
 
@@ -65,7 +67,7 @@ Typical characteristics of commercial environmental units are shown in Table [2]
 
 ## Interferometer Precision {#interferometer-precision}
 
-Figure [1](#figure--fig:position-sensor-interferometer-precision) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from (<a href="#citeproc_bib_item_3">Jang and Kim 2017</a>)).
+[Figure 1](#figure--fig:position-sensor-interferometer-precision) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from (<a href="#citeproc_bib_item_3">Jang and Kim 2017</a>)).
 
 <a id="figure--fig:position-sensor-interferometer-precision"></a>
 
@@ -84,7 +86,7 @@ It includes:
     -   Pressure: \\(K\_P \approx 0.27 ppm hPa^{-1}\\)
     -   Humidity: \\(K\_{HR} \approx 0.01 ppm \\% RH^{-1}\\)
     -   These errors can partially be compensated using an environmental unit.
--   Air turbulence (Figure [2](#figure--fig:interferometers-air-turbulence))
+-   Air turbulence ([Figure 2](#figure--fig:interferometers-air-turbulence))
 -   Non linearity
 
 <a id="figure--fig:interferometers-air-turbulence"></a>
@@ -98,6 +100,6 @@ It includes:
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bobroff, N. 1993. “Recent Advances in Displacement Measuring Interferometry.” <i>Measurement Science and Technology</i> 4 (9): 907–26. doi:<a href="https://doi.org/10.1088/0957-0233/4/9/001">10.1088/0957-0233/4/9/001</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” doi:<a href="https://doi.org/10.13140/rg.2.2.21044.35205">10.13140/rg.2.2.21044.35205</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” <i>International Journal of Precision Engineering and Manufacturing</i> 18 (12): 1881–90. doi:<a href="https://doi.org/10.1007/s12541-017-0217-y">10.1007/s12541-017-0217-y</a>.</div>
-  <div class="csl-entry"><a id="citeproc_bib_item_4"></a>Loughridge, Russell, and Daniel Y. Abramovitch. 2013. “A Tutorial on Laser Interferometry for Precision Measurements.” In <i>2013 American Control Conference</i>, nil. doi:<a href="https://doi.org/10.1109/acc.2013.6580402">10.1109/acc.2013.6580402</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_4"></a>Loughridge, Russell, and Daniel Y. Abramovitch. 2013. “A Tutorial on Laser Interferometry for Precision Measurements.” In <i>2013 American Control Conference</i>. doi:<a href="https://doi.org/10.1109/acc.2013.6580402">10.1109/acc.2013.6580402</a>.</div>
   <div class="csl-entry"><a id="citeproc_bib_item_5"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” <i>Applied Optics</i> 54 (10). Optical Society of America: 3051–63.</div>
 </div>

content/zettels/interpolation.md

@@ -0,0 +1,36 @@
++++
+title = "Interpolation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Band limited interpolation {#band-limited-interpolation}
+
+<https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula>
+
+```matlab
+rng default
+```
+
+```matlab
+t = 1:10; % Time Vector [s]
+x = randn(size(t))'; % Sampled data [V]
+
+ts = linspace(-5,15,600); % New time vector [s]
+[Ts,T] = ndgrid(ts,t);
+y = sinc(Ts - T)*x;
+```
+
+<a id="figure--fig:interpolation-perfect-example"></a>
+
+{{< figure src="/ox-hugo/interpolation_perfect_example.png" caption="<span class=\"figure-number\">Figure 1: </span>Sampled and interpolated signals" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>