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content/article/alkhatib03_activ_struc_vibrat_contr.md
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@@ -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:

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content/article/bibel92_guidel_h.md
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{{< 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>

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content/article/bryson93_contr_spacec_aircr.md
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## 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")

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content/article/butler11_posit_contr_lithog_equip.md
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## 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>

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@@ -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_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_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>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
</div>

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content/article/collette11_review_activ_vibrat_isolat_strat.md
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@@ -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>

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## 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>

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content/article/collette15_sensor_fusion_method_high_perfor.md
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@@ -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>

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<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>

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content/article/devasia07_survey_contr_issues_nanop.md
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## 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>

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content/article/fleming10_nanop_system_with_force_feedb.md
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@@ -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>

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## 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>

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@@ -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>

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## 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>

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## 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>
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@@ -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>
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{{< 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>

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## 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>
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## 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>

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@@ -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>

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## 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>
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## 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>
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@@ -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
@@ -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}
@@ -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_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_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>

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@@ -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>:
Definition of quantities on Figure <a href="#org84f1a50">2</a>
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span>
Definition of quantities on <a href="#orgffe7e8f">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
@@ -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_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_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>

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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>
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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 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 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
<a id="figure--fig:preumont02-force-acc-fb-light"></a>

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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.

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## 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>
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## 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>
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## 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>
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@@ -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}
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<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>
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</div>

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@@ -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>

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@@ -35,7 +35,7 @@ Combines:
- the FxLMS-based adaptive inverse control =&gt; suppress transmission of periodic vibrations
- direct feedback of integrated forces =&gt; 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**

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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>

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## 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>
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## 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>