Update many posts

config.toml

@@ -31,11 +31,7 @@ name = "Home"
 weight = 10
 identifier = "home"
 url = "/"
-# [[menu.main]]
+
-# name = "Blog"
-# weight = 20
-# identifier = "posts"
-# url = "/posts/"
 [[menu.main]]
 name = "Zettels"
 weight = 30
@@ -43,21 +39,14 @@ identifier = "zettels"
 url = "/zettels/"
 
 [[menu.main]]
-name = "Books"
+name = "Bibliography"
-weight = 40
-identifier = "book"
-url = "/book/"
-
-[[menu.main]]
-name = "Papers"
 weight = 50
-identifier = "paper"
+identifier = "bibliography"
-url = "/paper/"
+url = "/bibliography/"
-
 
 [[menu.main]]
 name = "Search"
-weight = 60
+weight = 70
 identifier = "search"
 url = "/search/"
 

content/article/alkhatib03_activ_struc_vibrat_contr.md

@@ -2,8 +2,6 @@
 title = "Active structural vibration control: a review"
 author = ["Thomas Dehaeze"]
 draft = false
-tags = ["tag1", "tag2"]
-categories = ["cat1", "cat2"]
 +++
 
 Tags
@@ -11,7 +9,7 @@ Tags
 
 
 Reference
-: <sup id="279b5558de3a8131b329a9ba1a99e4f8"><a href="#alkhatib03_activ_struc_vibrat_contr" title="Rabih Alkhatib \&amp; Golnaraghi, Active Structural Vibration Control: a Review, {The Shock and Vibration Digest}, v(5), 367-383 (2003).">(Rabih Alkhatib \& Golnaraghi, 2003)</a></sup>
+: <sup id="279b5558de3a8131b329a9ba1a99e4f8"><a class="reference-link" href="#alkhatib03_activ_struc_vibrat_contr" title="Rabih Alkhatib \&amp; Golnaraghi, Active Structural Vibration Control: a Review, {The Shock and Vibration Digest}, v(5), 367-383 (2003).">(Rabih Alkhatib \& Golnaraghi, 2003)</a></sup>
 
 Author(s)
 : Alkhatib, R., & Golnaraghi, M. F.
@@ -125,12 +123,12 @@ 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](#org95c575a).
+A general block diagram of the control system is shown figure [1](#orgb7a9ee5).
 
 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.
 
-<a id="org95c575a"></a>
+<a id="orgb7a9ee5"></a>
 
 {{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="Figure 1: Block diagram for robust control" >}}
 
@@ -202,11 +200,11 @@ Two different methods
 
 ## Active Control Effects on the System {#active-control-effects-on-the-system}
 
-<a id="org7c357dd"></a>
+<a id="org352d1a3"></a>
 
 {{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="Figure 2: 1 DoF control of a spring-mass-damping system" >}}
 
-Consider the control system figure [2](#org7c357dd), the equation of motion of the system is:
+Consider the control system figure [2](#org352d1a3), 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:
@@ -227,4 +225,4 @@ The problem of optimizing the locations of the actuators can be more significant
 If the actuator is placed at the wrong location, the system will require a greater force control. In that case, the system is said to have a **low degree of controllability**.
 
 # Bibliography
-<a id="alkhatib03_activ_struc_vibrat_contr"></a>Alkhatib, R., & Golnaraghi, M. F., *Active structural vibration control: a review*, The Shock and Vibration Digest, *35(5)*, 367–383 (2003).  http://dx.doi.org/10.1177/05831024030355002 [↩](#279b5558de3a8131b329a9ba1a99e4f8)
+<a class="bibtex-entry" id="alkhatib03_activ_struc_vibrat_contr">Alkhatib, R., & Golnaraghi, M. F., *Active structural vibration control: a review*, The Shock and Vibration Digest, *35(5)*, 367–383 (2003).  http://dx.doi.org/10.1177/05831024030355002</a> [↩](#279b5558de3a8131b329a9ba1a99e4f8)

content/article/bibel92_guidel_h.md

@@ -8,7 +8,7 @@ Tags
 : [H Infinity Control]({{< relref "h_infinity_control" >}})
 
 Reference
-: <sup id="5b41da575e27e6e86f1a1410a0170836"><a href="#bibel92_guidel_h" title="Bibel \&amp; Malyevac, Guidelines for the selection of weighting functions for  H-infinity control, NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA, (1992).">(Bibel \& Malyevac, 1992)</a></sup>
+: <sup id="5b41da575e27e6e86f1a1410a0170836"><a class="reference-link" href="#bibel92_guidel_h" title="Bibel \&amp; Malyevac, Guidelines for the selection of weighting functions for  H-infinity control, NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA, (1992).">(Bibel \& Malyevac, 1992)</a></sup>
 
 Author(s)
 : Bibel, J. E., & Malyevac, D. S.
@@ -19,11 +19,11 @@ Year
 
 ## Properties of feedback control {#properties-of-feedback-control}
 
-<a id="org82bead2"></a>
+<a id="org5999225"></a>
 
 {{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
 
-From the figure [1](#org82bead2), we have:
+From the figure [1](#org5999225), we have:
 
 \begin{align\*}
 y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
@@ -77,11 +77,11 @@ Usually, reference signals and disturbances occur at low frequencies, while nois
 
 </div>
 
-<a id="org71ea720"></a>
+<a id="org4e0009c"></a>
 
 {{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
 
-New design framework (figure [2](#org71ea720)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
+New design framework (figure [2](#org4e0009c)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
 
 -   \\(w\\): exogenous inputs
 -   \\(z\\): regulated performance output
@@ -108,9 +108,9 @@ 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](#org549c59f)).
+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](#orgdd8fae0)).
 
-<a id="org549c59f"></a>
+<a id="orgdd8fae0"></a>
 
 {{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
 
@@ -154,15 +154,15 @@ 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](#org379d5b1)).
+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](#org0d13a20)).
 
-<a id="org379d5b1"></a>
+<a id="org0d13a20"></a>
 
 {{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: 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](#orgcc65489).
+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](#org45b0983).
 
-<a id="orgcc65489"></a>
+<a id="org45b0983"></a>
 
 {{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
 
@@ -182,4 +182,4 @@ Typically actuator input weights are constant over frequency and set at the inve
 **The order of the weights should be kept reasonably low** to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.
 
 # Bibliography
-<a id="bibel92_guidel_h"></a>Bibel, J. E., & Malyevac, D. S., *Guidelines for the selection of weighting functions for h-infinity control* (1992). [↩](#5b41da575e27e6e86f1a1410a0170836)
+<a class="bibtex-entry" id="bibel92_guidel_h">Bibel, J. E., & Malyevac, D. S., *Guidelines for the selection of weighting functions for h-infinity control* (1992).</a> [↩](#5b41da575e27e6e86f1a1410a0170836)

content/article/bryson93_contr_spacec_aircr.md

@@ -8,7 +8,7 @@ Tags
 : [HAC-HAC]({{< relref "hac_hac" >}})
 
 Reference
-: <sup id="4970865a21830fff7b1daeec187bfa68"><a href="#bryson93_contr_spacec_aircr" title="Bryson, Control of Spacecraft and Aircraft, Princeton university press Princeton, New Jersey (1993).">(Bryson, 1993)</a></sup>
+: <sup id="4970865a21830fff7b1daeec187bfa68"><a class="reference-link" href="#bryson93_contr_spacec_aircr" title="Bryson, Control of Spacecraft and Aircraft, Princeton university press Princeton, New Jersey (1993).">(Bryson, 1993)</a></sup>
 
 Author(s)
 : Bryson, A. E.
@@ -41,11 +41,11 @@ Year
 > 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")
 
-<a id="orgf5c85db"></a>
+<a id="orgdde3e8f"></a>
 
 {{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="Figure 1: HAC-LAC control concept" >}}
 
 > LAC/HAC is usually insensitive to small deviation of the plant dynamics away from the design values, that is, it is **robust** to plant parameter changes.
 
 # Bibliography
-<a id="bryson93_contr_spacec_aircr"></a>Bryson, A. E., *Control of spacecraft and aircraft* (1993), : Princeton university press Princeton, New Jersey. [↩](#4970865a21830fff7b1daeec187bfa68)
+<a class="bibtex-entry" id="bryson93_contr_spacec_aircr">Bryson, A. E., *Control of spacecraft and aircraft* (1993), : Princeton university press Princeton, New Jersey.</a> [↩](#4970865a21830fff7b1daeec187bfa68)

content/article/butler11_posit_contr_lithog_equip.md

@@ -0,0 +1,20 @@
++++
+title = "Position control in lithographic equipment"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Multivariable Control]({{< relref "multivariable_control" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
+
+Reference
+: <sup id="6a014e3a2ee3e41d20bd0644654c56f0"><a class="reference-link" href="#butler11_posit_contr_lithog_equip" title="Hans Butler, Position Control in Lithographic Equipment, {IEEE Control Systems}, v(5), 28-47 (2011).">(Hans Butler, 2011)</a></sup>
+
+Author(s)
+: Butler, H.
+
+Year
+: 2011
+
+# Bibliography
+<a class="bibtex-entry" id="butler11_posit_contr_lithog_equip">Butler, H., *Position control in lithographic equipment*, IEEE Control Systems, *31(5)*, 28–47 (2011).  http://dx.doi.org/10.1109/mcs.2011.941882</a> [↩](#6a014e3a2ee3e41d20bd0644654c56f0)

content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md

@@ -0,0 +1,108 @@
++++
+title = "Identification and decoupling control of flexure jointed hexapods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: <sup id="ba05ff213f8e5963d91559d95becfbdb"><a class="reference-link" href="#chen00_ident_decoup_contr_flexur_joint_hexap" title="Yixin Chen \&amp; McInroy, Identification and Decoupling Control of Flexure Jointed  Hexapods, nil, in in: {Proceedings 2000 ICRA. Millennium Conference. IEEE
+                      International Conference on Robotics and Automation. Symposia
+                      Proceedings (Cat. No.00CH37065)}, edited by (2000)">(Yixin Chen \& McInroy, 2000)</a></sup>
+
+Author(s)
+: Chen, Y., & McInroy, J.
+
+Year
+: 2000
+
+
+## Abstract {#abstract}
+
+> By exploiting properties of the joint space mass-inertia matrix of flexure jointed hexapods, a new **decoupling method** is proposed.
+> The new decoupling method, through a **static** input-output mapping, transforms the highly coupled 6 inputs 6 outputs dynamics into 6 independent single-input single-output channels.
+> Prior decoupling control algorithms imposed severe constraints on the allowable geometry, workspace and payload.
+> This paper derives a new algorithm which removes these constraints, thus greatly expanding the applications.
+> Based on the new decoupling algorithm, an **identification algorithm** is introduced to identify the **joint space mass-inertia matrix** using payload acceleration and base forces.
+> This algorithm can be used for precision payload calibration, thus improving performance and removing the labor required to design the control for different payloads.
+> The new decoupling algorithm is experimentally compared to earlier techniques.
+> These experimental results indicate that the new approach is practical, and improves performance.
+
+
+## Introduction {#introduction}
+
+Typical decoupling algorithm impose two constraints:
+
+-   the payload mass/inertia matrix is diagonal
+-   the geometry of the platform and attachment of the payload must be carefully chosen
+
+This limits the applications significantly.
+
+The algorithm derived herein removes these constraints, thus greatly expanding the potential applications.
+
+
+## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
+
+The derivation of the dynamic model is done in <sup id="5da427f78c552aa92cd64c2a6df961f1"><a class="reference-link" href="#mcinroy99_dynam" title="McInroy, Dynamic modeling of flexure jointed hexapods for control  purposes, nil, in in: {Proceedings of the 1999 IEEE International Conference on
+                  Control Applications (Cat. No.99CH36328)}, edited by (1999)">(McInroy, 1999)</a></sup> ([Notes]({{< relref "mcinroy99_dynam" >}})).
+
+<a id="org81e0a96"></a>
+
+{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="Figure 1: A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
+
+In the joint space, the dynamics of a flexure jointed hexapod are written as:
+
+\begin{equation}
+  \vec{f}\_b = \vec{f}\_m - \bm{K}(\vec{l} - \vec{l}\_r) - \bm{B} \dot{\vec{l}}
+\end{equation}
+
+\begin{aligned}
+  & \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\\\
+  & {}^U\_B\bm{R} \bm{J}^T \bm{B} \dot{\vec{l}} + {}^U\_B\bm{R}\bm{J}^T \bm{K}(\vec{l} - \vec{l}\_r) = \\\\\\
+  & {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\\\
+  & \left( {}^U\_B\bm{R} \bm{J}^T \bm{M}\_s + {}^U\_P\bm{R} {}^P\bm{M}\_x {}^U\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1} \right) \ddot{\vec{q}}\_s
+\end{aligned}
+
+where:
+
+-   \\(\bm{J}\\) is the \\(6 \times 6\\) hexapod Jacobian relating payload Cartesian movements, expressed in {P}, to strut length changes in the joint space
+-   \\({}^B\_U\bm{R}\\) is the \\(6 \times 6\\) rotation matrix from the base frame {B} to the universal inertial frame of reference {U} (it consists of two identical \\(3 \times 3\\) rotation matrices forming a block diagonal \\(6 \times 6\\) matrix)
+-   \\(\bm{J}\_c\\) and \\(\bm{J}\_B\\) are \\(6 \times 6\\) Jacobian matrices capturing base motion
+-   \\({}^P\bm{M}\_x\\) is the \\(6 \times 6\\) mass-inertia matrix of the payload found with respect to the payload frame {P}
+-   \\(\bm{M}\_s\\) is a diagonal \\(6 \times 6\\) matrix containing the moving mass of each strut
+-   \\(\bm{B}\\) and \\(\bm{K}\\) are \\(6 \times 6\\) diagonal matrices containing the damping of stiffness, respectively, of each strut
+-   \\(\vec{l}\\) is the \\(6 \times 1\\) vector of strut lengths, and \\(\vec{l}\_r\\) is the constant vector of relaxed strut length
+-   \\(\vec{f}\_b\\) is the vector of forces exerted at the bottom of the strut
+-   \\(\vec{f}\_m\\) is the vector of strut motor forces
+-   \\(\ddot{\vec{q}}\_s\\) is a \\(6 \times 1\\) vector of base accelerations along each strut plus some Coriolis terms
+-   \\(\vec{\mathcal{F}}\_e\\) is a vector of payload exogenous generalized forces
+-   \\(\vec{\mathcal{C}}\\) is a vector containing all the Coriolis and centripetal terms except the Coriolis terms in \\(\ddot{\vec{q}}\_s\\)
+-   \\(\vec{\mathcal{G}}\\) is a vector containing all gravity terms
+
+\begin{aligned}
+  \bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\\\
+  & \bm{M}\_q \ddot{\vec{q}}\_s + \bm{B} \dot{\vec{q}}\_s + \bm{J}^{-T} {}^U\_B\bm{R}^T \vec{\mathcal{F}}\_e
+\end{aligned}
+
+where
+
+-   \\(\bm{M}\_p = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} + \bm{M}\_s\\)
+-   \\(\bm{M}\_q = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} - \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1}\\)
+
+\\(\bm{M}\_p\\) and \\(\bm{M}\_q\\) are joint space mass-inertia matrices.
+
+
+## Decoupling the Dynamics of Flexure Jointed Hexapods {#decoupling-the-dynamics-of-flexure-jointed-hexapods}
+
+
+## Identification of Joint Space Mass-Inertia Matrix {#identification-of-joint-space-mass-inertia-matrix}
+
+
+## Experimental Results {#experimental-results}
+
+# Bibliography
+<a class="bibtex-entry" id="chen00_ident_decoup_contr_flexur_joint_hexap">Chen, Y., & McInroy, J., *Identification and decoupling control of flexure jointed hexapods*, In , Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065) (pp. ) (2000). : .</a> [↩](#ba05ff213f8e5963d91559d95becfbdb)
+
+<a class="bibtex-entry" id="mcinroy99_dynam">McInroy, J., *Dynamic modeling of flexure jointed hexapods for control purposes*, In , Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328) (pp. ) (1999). : .</a> [↩](#5da427f78c552aa92cd64c2a6df961f1)

content/article/claeyssen07_amplif_piezoel_actuat.md

@@ -0,0 +1,21 @@
++++
+title = "Amplified piezoelectric actuators: static & dynamic applications"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="5decd2b31c4a9842b80c58b56f96590a"><a class="reference-link" href="#claeyssen07_amplif_piezoel_actuat" title="Frank Claeyssen, Le Letty, Barillot, \&amp; Sosnicki, Amplified Piezoelectric Actuators: Static \&amp; Dynamic  Applications, {Ferroelectrics}, v(1), 3-14 (2007).">(Frank Claeyssen {\it et al.}, 2007)</a></sup>
+
+Author(s)
+: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
+
+Year
+: 2007
+
+# Bibliography
+<a class="bibtex-entry" id="claeyssen07_amplif_piezoel_actuat">Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O., *Amplified piezoelectric actuators: static \& dynamic applications*, Ferroelectrics, *351(1)*, 3–14 (2007).  http://dx.doi.org/10.1080/00150190701351865</a> [↩](#5decd2b31c4a9842b80c58b56f96590a)

content/article/collette11_review_activ_vibrat_isolat_strat.md

@@ -8,7 +8,7 @@ Tags
 : [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="2d69d483f210ca387ca8061596ec27ea"><a href="#collette11_review_activ_vibrat_isolat_strat" title="Christophe Collette, Stef Janssens \&amp; Kurt Artoos, Review of Active Vibration Isolation Strategies, {Recent Patents on Mechanical Engineeringe}, v(3), 212-219 (2011).">(Christophe Collette {\it et al.}, 2011)</a></sup>
+: <sup id="2d69d483f210ca387ca8061596ec27ea"><a class="reference-link" href="#collette11_review_activ_vibrat_isolat_strat" title="Christophe Collette, Stef Janssens \&amp; Kurt Artoos, Review of Active Vibration Isolation Strategies, {Recent Patents on Mechanical Engineeringe}, v(3), 212-219 (2011).">(Christophe Collette {\it et al.}, 2011)</a></sup>
 
 Author(s)
 : Collette, C., Janssens, S., & Artoos, K.
@@ -71,9 +71,9 @@ The general expression of the force delivered by the actuator is \\(f = g\_a \dd
 
 ## Conclusions {#conclusions}
 
-<a id="orgef29aaf"></a>
+<a id="org4270456"></a>
 
 {{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="Figure 1: Comparison of Active Vibration Isolation Strategies" >}}
 
 # Bibliography
-<a id="collette11_review_activ_vibrat_isolat_strat"></a>Collette, C., Janssens, S., & Artoos, K., *Review of active vibration isolation strategies*, Recent Patents on Mechanical Engineeringe, *4(3)*, 212–219 (2011).  http://dx.doi.org/10.2174/2212797611104030212 [↩](#2d69d483f210ca387ca8061596ec27ea)
+<a class="bibtex-entry" id="collette11_review_activ_vibrat_isolat_strat">Collette, C., Janssens, S., & Artoos, K., *Review of active vibration isolation strategies*, Recent Patents on Mechanical Engineeringe, *4(3)*, 212–219 (2011).  http://dx.doi.org/10.2174/2212797611104030212</a> [↩](#2d69d483f210ca387ca8061596ec27ea)

content/article/collette14_vibrat.md

@@ -8,7 +8,7 @@ Tags
 : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
 
 Reference
-: <sup id="1223611da2f9b157af97503a4fec7631"><a href="#collette14_vibrat" title="Collette \&amp; Matichard, Vibration control of flexible structures using fusion of  inertial sensors and hyper-stable actuator-sensor pairs, in in: {International Conference on Noise and Vibration Engineering
+: <sup id="1223611da2f9b157af97503a4fec7631"><a class="reference-link" href="#collette14_vibrat" title="Collette \&amp; Matichard, Vibration control of flexible structures using fusion of  inertial sensors and hyper-stable actuator-sensor pairs, in in: {International Conference on Noise and Vibration Engineering
                       (ISMA2014)}, edited by (2014)">(Collette \& Matichard, 2014)</a></sup>
 
 Author(s)
@@ -101,4 +101,4 @@ Three types of sensors have been considered for the high frequency part of the f
 -   The fusion with a **force sensor** can be used to increase the loop gain with little effect on the compliance and passive isolation, provided that the blend is possible and that no active damping of flexible modes is required.
 
 # Bibliography
-<a id="collette14_vibrat"></a>Collette, C., & Matichard, F., *Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs*, In , International Conference on Noise and Vibration Engineering (ISMA2014) (pp. ) (2014). : . [↩](#1223611da2f9b157af97503a4fec7631)
+<a class="bibtex-entry" id="collette14_vibrat">Collette, C., & Matichard, F., *Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs*, In , International Conference on Noise and Vibration Engineering (ISMA2014) (pp. ) (2014). : .</a> [↩](#1223611da2f9b157af97503a4fec7631)

content/article/collette15_sensor_fusion_method_high_perfor.md

@@ -8,7 +8,7 @@ Tags
 : [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="7772841a8f05142ec30f0f6daae20932"><a href="#collette15_sensor_fusion_method_high_perfor" title="Collette \&amp; Matichard, Sensor Fusion Methods for High Performance Active Vibration  Isolation Systems, {Journal of Sound and Vibration}, v(nil), 1-21 (2015).">(Collette \& Matichard, 2015)</a></sup>
+: <sup id="7772841a8f05142ec30f0f6daae20932"><a class="reference-link" href="#collette15_sensor_fusion_method_high_perfor" title="Collette \&amp; Matichard, Sensor Fusion Methods for High Performance Active Vibration  Isolation Systems, {Journal of Sound and Vibration}, v(nil), 1-21 (2015).">(Collette \& Matichard, 2015)</a></sup>
 
 Author(s)
 : Collette, C., & Matichard, F.
@@ -25,4 +25,4 @@ The stability margins of the controller can be significantly increased with no o
 -   there exists a bandwidth where we can superimpose the open loop transfer functions obtained with the two sensors.
 
 # Bibliography
-<a id="collette15_sensor_fusion_method_high_perfor"></a>Collette, C., & Matichard, F., *Sensor fusion methods for high performance active vibration isolation systems*, Journal of Sound and Vibration, *342(nil)*, 1–21 (2015).  http://dx.doi.org/10.1016/j.jsv.2015.01.006 [↩](#7772841a8f05142ec30f0f6daae20932)
+<a class="bibtex-entry" id="collette15_sensor_fusion_method_high_perfor">Collette, C., & Matichard, F., *Sensor fusion methods for high performance active vibration isolation systems*, Journal of Sound and Vibration, *342(nil)*, 1–21 (2015).  http://dx.doi.org/10.1016/j.jsv.2015.01.006</a> [↩](#7772841a8f05142ec30f0f6daae20932)

content/article/dasgupta00_stewar_platf_manip.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}})
 
 Reference
-: <sup id="ad17e03f0fbbcc1a070557d7b5a0e1e1"><a href="#dasgupta00_stewar_platf_manip" title="Bhaskar Dasgupta \&amp; Mruthyunjaya, The Stewart Platform Manipulator: a Review, {Mechanism and Machine Theory}, v(1), 15-40 (2000).">(Bhaskar Dasgupta \& Mruthyunjaya, 2000)</a></sup>
+: <sup id="ad17e03f0fbbcc1a070557d7b5a0e1e1"><a class="reference-link" href="#dasgupta00_stewar_platf_manip" title="Bhaskar Dasgupta \&amp; Mruthyunjaya, The Stewart Platform Manipulator: a Review, {Mechanism and Machine Theory}, v(1), 15-40 (2000).">(Bhaskar Dasgupta \& Mruthyunjaya, 2000)</a></sup>
 
 Author(s)
 : Dasgupta, B., & Mruthyunjaya, T.
@@ -34,4 +34,4 @@ Year
 The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platoform) connected through six extensible legs, each with sherical joints at both ends.
 
 # Bibliography
-<a id="dasgupta00_stewar_platf_manip"></a>Dasgupta, B., & Mruthyunjaya, T., *The stewart platform manipulator: a review*, Mechanism and Machine Theory, *35(1)*, 15–40 (2000).  http://dx.doi.org/10.1016/s0094-114x(99)00006-3 [↩](#ad17e03f0fbbcc1a070557d7b5a0e1e1)
+<a class="bibtex-entry" id="dasgupta00_stewar_platf_manip">Dasgupta, B., & Mruthyunjaya, T., *The stewart platform manipulator: a review*, Mechanism and Machine Theory, *35(1)*, 15–40 (2000).  http://dx.doi.org/10.1016/s0094-114x(99)00006-3</a> [↩](#ad17e03f0fbbcc1a070557d7b5a0e1e1)

content/article/devasia07_survey_contr_issues_nanop.md

@@ -9,7 +9,7 @@ Tags
 
 
 Reference
-: <sup id="8ce53b8a612ce8ae3eb616cd1ed05630"><a href="#devasia07_survey_contr_issues_nanop" title="Devasia, Eleftheriou, Moheimani \&amp; SO Reza, A Survey of Control Issues in Nanopositioning, {IEEE Transactions on Control Systems Technology}, v(5), 802--823 (2007).">(Devasia {\it et al.}, 2007)</a></sup>
+: <sup id="8ce53b8a612ce8ae3eb616cd1ed05630"><a class="reference-link" href="#devasia07_survey_contr_issues_nanop" title="Devasia, Eleftheriou, Moheimani \&amp; SO Reza, A Survey of Control Issues in Nanopositioning, {IEEE Transactions on Control Systems Technology}, v(5), 802--823 (2007).">(Devasia {\it et al.}, 2007)</a></sup>
 
 Author(s)
 : Devasia, S., Eleftheriou, E., & Moheimani, S. R.
@@ -22,9 +22,9 @@ Year
 -   Interesting analysis about Bandwidth-Precision-Range tradeoffs
 -   Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
 
-<a id="orga0f4b4e"></a>
+<a id="org103305d"></a>
 
 {{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="Figure 1: Tradeoffs between bandwidth, precision and range" >}}
 
 # Bibliography
-<a id="devasia07_survey_contr_issues_nanop"></a>Devasia, S., Eleftheriou, E., & Moheimani, S. R., *A survey of control issues in nanopositioning*, IEEE Transactions on Control Systems Technology, *15(5)*, 802–823 (2007).  [↩](#8ce53b8a612ce8ae3eb616cd1ed05630)
+<a class="bibtex-entry" id="devasia07_survey_contr_issues_nanop">Devasia, S., Eleftheriou, E., & Moheimani, S. R., *A survey of control issues in nanopositioning*, IEEE Transactions on Control Systems Technology, *15(5)*, 802–823 (2007). </a> [↩](#8ce53b8a612ce8ae3eb616cd1ed05630)

content/article/fleming10_nanop_system_with_force_feedb.md

@@ -8,7 +8,7 @@ Tags
 : [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
 
 Reference
-: <sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup>
+: <sup id="c823f68dd2a72b9667a61b3c046b4731"><a class="reference-link" href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup>
 
 Author(s)
 : Fleming, A.
@@ -30,7 +30,7 @@ Summary:
 
 ## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
 
-<a id="orgf7e4ab9"></a>
+<a id="org3f4c96b"></a>
 
 {{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
 
@@ -113,7 +113,7 @@ The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\
 The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
 
 # Bibliography
-<a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)
+<a class="bibtex-entry" id="fleming10_nanop_system_with_force_feedb">Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422</a> [↩](#c823f68dd2a72b9667a61b3c046b4731)
 
 
 ## Backlinks {#backlinks}

content/article/fleming12_estim.md

@@ -0,0 +1,22 @@
++++
+title = "Estimating the resolution of nanopositioning systems from frequency domain data"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="a1cc9b70316a7dda2f652efd146caf84"><a class="reference-link" href="#fleming12_estim" title="Andrew Fleming, Estimating the resolution of nanopositioning systems from  frequency domain data, nil, in in: {2012 IEEE International Conference on Robotics and
+                      Automation}, edited by (2012)">(Andrew Fleming, 2012)</a></sup>
+
+Author(s)
+: Fleming, A. J.
+
+Year
+: 2012
+
+# Bibliography
+<a class="bibtex-entry" id="fleming12_estim">Fleming, A. J., *Estimating the resolution of nanopositioning systems from frequency domain data*, In , 2012 IEEE International Conference on Robotics and Automation (pp. ) (2012). : .</a> [↩](#a1cc9b70316a7dda2f652efd146caf84)

content/article/fleming13_review_nanom_resol_posit_sensor.md

@@ -8,7 +8,7 @@ Tags
 : [Position Sensors]({{< relref "position_sensors" >}})
 
 Reference
-: <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors:  Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup>
+: <sup id="3fb5b61524290e36d639a4fac65703d0"><a class="reference-link" href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors:  Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup>
 
 Author(s)
 : Fleming, A. J.
@@ -33,7 +33,7 @@ Usually quoted as a percentage of the fill-scale range (FSR):
 
 With \\(e\_m(v)\\) is the mapping error.
 
-<a id="org18802f9"></a>
+<a id="org6e00657"></a>
 
 {{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
 
@@ -42,7 +42,7 @@ With \\(e\_m(v)\\) is the mapping error.
 
 If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
 
-<a id="org65fb6f9"></a>
+<a id="org076fb4b"></a>
 
 {{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
 
@@ -147,9 +147,9 @@ The empirical rule states that there is a \\(99.7\%\\) probability that a sample
 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](#org954f29f)).
+There is usually a trade-off between bandwidth and resolution (figure [3](#org92eeb72)).
 
-<a id="org954f29f"></a>
+<a id="org92eeb72"></a>
 
 {{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
 
@@ -182,4 +182,9 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
 | Encoder        | Meters                           |         | 6 nm       | >100kHz  | 5 ppm FSR |
 
 # Bibliography
-<a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013).  http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)
+<a class="bibtex-entry" id="fleming13_review_nanom_resol_posit_sensor">Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013).  http://dx.doi.org/10.1016/j.sna.2012.10.016</a> [↩](#3fb5b61524290e36d639a4fac65703d0)
+
+
+## Backlinks {#backlinks}
+
+-   [Position Sensors]({{< relref "position_sensors" >}})

content/article/furqan17_studies_stewar_platf_manip.md

@@ -0,0 +1,22 @@
++++
+title = "Studies on stewart platform manipulator: a review"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}})
+
+Reference
+: <sup id="cc10fe9545c7c381cc2b610e8f91a071"><a class="reference-link" href="#furqan17_studies_stewar_platf_manip" title="Mohd Furqan, Mohd Suhaib \&amp; Nazeer Ahmad, Studies on Stewart Platform Manipulator: a Review, {Journal of Mechanical Science and Technology}, v(9), 4459-4470 (2017).">(Mohd Furqan {\it et al.}, 2017)</a></sup>
+
+Author(s)
+: Furqan, M., Suhaib, M., & Ahmad, N.
+
+Year
+: 2017
+
+Lots of references.
+
+# Bibliography
+<a class="bibtex-entry" id="furqan17_studies_stewar_platf_manip">Furqan, M., Suhaib, M., & Ahmad, N., *Studies on stewart platform manipulator: a review*, Journal of Mechanical Science and Technology, *31(9)*, 4459–4470 (2017).  http://dx.doi.org/10.1007/s12206-017-0846-1</a> [↩](#cc10fe9545c7c381cc2b610e8f91a071)

content/article/furutani04_nanom_cuttin_machin_using_stewar.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
 
 Reference
-: <sup id="bedab298599c84f60236313ebaad2714"><a href="#furutani04_nanom_cuttin_machin_using_stewar" title="Katsushi Furutani, Michio Suzuki \&amp; Ryusei Kudoh, Nanometre-Cutting Machine Using a Stewart-Platform Parallel  Mechanism, {Measurement Science and Technology}, v(2), 467-474 (2004).">(Katsushi Furutani {\it et al.}, 2004)</a></sup>
+: <sup id="bedab298599c84f60236313ebaad2714"><a class="reference-link" href="#furutani04_nanom_cuttin_machin_using_stewar" title="Katsushi Furutani, Michio Suzuki \&amp; Ryusei Kudoh, Nanometre-Cutting Machine Using a Stewart-Platform Parallel  Mechanism, {Measurement Science and Technology}, v(2), 467-474 (2004).">(Katsushi Furutani {\it et al.}, 2004)</a></sup>
 
 Author(s)
 : Furutani, K., Suzuki, M., & Kudoh, R.
@@ -35,4 +35,4 @@ To minimize the errors, a calibration is done between the required leg length an
 Then, it is fitted with 4th order polynomial and included in the control architecture.
 
 # Bibliography
-<a id="furutani04_nanom_cuttin_machin_using_stewar"></a>Furutani, K., Suzuki, M., & Kudoh, R., *Nanometre-cutting machine using a stewart-platform parallel mechanism*, Measurement Science and Technology, *15(2)*, 467–474 (2004).  http://dx.doi.org/10.1088/0957-0233/15/2/022 [↩](#bedab298599c84f60236313ebaad2714)
+<a class="bibtex-entry" id="furutani04_nanom_cuttin_machin_using_stewar">Furutani, K., Suzuki, M., & Kudoh, R., *Nanometre-cutting machine using a stewart-platform parallel mechanism*, Measurement Science and Technology, *15(2)*, 467–474 (2004).  http://dx.doi.org/10.1088/0957-0233/15/2/022</a> [↩](#bedab298599c84f60236313ebaad2714)

content/article/gao15_measur_techn_precis_posit.md

@@ -0,0 +1,20 @@
++++
+title = "Measurement technologies for precision positioning"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Position Sensors]({{< relref "position_sensors" >}})
+
+Reference
+: <sup id="b820b918ced36901ea0ad4bf653202c6"><a class="reference-link" href="#gao15_measur_techn_precis_posit" title="Gao, Kim, Bosse, Haitjema, , Chen, Lu, Knapp, Weckenmann, , Estler \&amp; Kunzmann, Measurement Technologies for Precision Positioning, {CIRP Annals}, v(2), 773-796 (2015).">(Gao {\it et al.}, 2015)</a></sup>
+
+Author(s)
+: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
+
+Year
+: 2015
+
+# Bibliography
+<a class="bibtex-entry" id="gao15_measur_techn_precis_posit">Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …, *Measurement technologies for precision positioning*, CIRP Annals, *64(2)*, 773–796 (2015).  http://dx.doi.org/10.1016/j.cirp.2015.05.009</a> [↩](#b820b918ced36901ea0ad4bf653202c6)

content/article/garg07_implem_chall_multiv_contr.md

@@ -8,7 +8,7 @@ Tags
 : [Multivariable Control]({{< relref "multivariable_control" >}})
 
 Reference
-: <sup id="07f63c751c1d9fcfe628178688f7ec24"><a href="#garg07_implem_chall_multiv_contr" title="Sanjay Garg, Implementation Challenges for Multivariable Control: What  you did not learn in school!, nil, in in: {AIAA Guidance, Navigation and Control Conference and
+: <sup id="07f63c751c1d9fcfe628178688f7ec24"><a class="reference-link" href="#garg07_implem_chall_multiv_contr" title="Sanjay Garg, Implementation Challenges for Multivariable Control: What  you did not learn in school!, nil, in in: {AIAA Guidance, Navigation and Control Conference and
                       Exhibit}, edited by (2007)">(Sanjay Garg, 2007)</a></sup>
 
 Author(s)
@@ -36,4 +36,4 @@ The control rate should be weighted appropriately in order to not saturate the s
 -   importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles
 
 # Bibliography
-<a id="garg07_implem_chall_multiv_contr"></a>Garg, S., *Implementation challenges for multivariable control: what you did not learn in school!*, In , AIAA Guidance, Navigation and Control Conference and Exhibit (pp. ) (2007). : . [↩](#07f63c751c1d9fcfe628178688f7ec24)
+<a class="bibtex-entry" id="garg07_implem_chall_multiv_contr">Garg, S., *Implementation challenges for multivariable control: what you did not learn in school!*, In , AIAA Guidance, Navigation and Control Conference and Exhibit (pp. ) (2007). : .</a> [↩](#07f63c751c1d9fcfe628178688f7ec24)

content/article/geng95_intel_contr_system_multip_degree.md

@@ -16,9 +16,9 @@ Author(s)
 Year
 : 1995
 
-<a id="org1384437"></a>
+<a id="org9d6018b"></a>
 
 {{< figure src="/ox-hugo/geng95_control_structure.png" caption="Figure 1: Local force feedback and adaptive acceleration feedback for active isolation" >}}
 
 # Bibliography
-<a class="bibtex-entry" id="geng95_intel_contr_system_multip_degree"></a>Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A., *An intelligent control system for multiple degree-of-freedom vibration isolation*, Journal of Intelligent Material Systems and Structures, *6(6)*, 787–800 (1995).  http://dx.doi.org/10.1177/1045389x9500600607 [↩](#76af0f5c88615842fa91864c8618fb58)
+<a class="bibtex-entry" id="geng95_intel_contr_system_multip_degree">Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A., *An intelligent control system for multiple degree-of-freedom vibration isolation*, Journal of Intelligent Material Systems and Structures, *6(6)*, 787–800 (1995).  http://dx.doi.org/10.1177/1045389x9500600607</a> [↩](#76af0f5c88615842fa91864c8618fb58)

content/article/hanieh03_activ_stewar.md

@@ -0,0 +1,20 @@
++++
+title = "Active isolation and damping of vibrations via stewart platform"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}})
+
+Reference
+: <sup id="10e535e895bdcd6b921bff33ef68cd81"><a class="reference-link" href="#hanieh03_activ_stewar" title="Hanieh, Active isolation and damping of vibrations via Stewart  platform (2003).">(Hanieh, 2003)</a></sup>
+
+Author(s)
+: Hanieh, A. A.
+
+Year
+: 2003
+
+# Bibliography
+<a class="bibtex-entry" id="hanieh03_activ_stewar">Hanieh, A. A., *Active isolation and damping of vibrations via stewart platform* (2003). Universit{\'e} Libre de Bruxelles, Brussels, Belgium.</a> [↩](#10e535e895bdcd6b921bff33ef68cd81)

content/article/hauge04_sensor_contr_space_based_six.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
 
 Reference
-: <sup id="f9698a1741fe7492aa9b7b42c7724670"><a href="#hauge04_sensor_contr_space_based_six" title="Hauge \&amp; Campbell, Sensors and Control of a Space-Based Six-Axis Vibration  Isolation System, {Journal of Sound and Vibration}, v(3-5), 913-931 (2004).">(Hauge \& Campbell, 2004)</a></sup>
+: <sup id="f9698a1741fe7492aa9b7b42c7724670"><a class="reference-link" href="#hauge04_sensor_contr_space_based_six" title="Hauge \&amp; Campbell, Sensors and Control of a Space-Based Six-Axis Vibration  Isolation System, {Journal of Sound and Vibration}, v(3-5), 913-931 (2004).">(Hauge \& Campbell, 2004)</a></sup>
 
 Author(s)
 : Hauge, G., & Campbell, M.
@@ -24,20 +24,20 @@ Year
 -   Vibration isolation using a Stewart platform
 -   Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
 
-<a id="org666133a"></a>
+<a id="org342e642"></a>
 
 {{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
 
-**Stewart platform** (Figure [1](#org666133a)):
+**Stewart platform** (Figure [1](#org342e642)):
 
 -   Low corner frequency
 -   Large actuator stroke (\\(\pm5mm\\))
--   Sensors in each strut (Figure [2](#org4d96564)):
+-   Sensors in each strut (Figure [2](#orge1d3dcd)):
     -   three-axis load cell
     -   base and payload geophone in parallel with the struts
     -   LVDT
 
-<a id="org4d96564"></a>
+<a id="orge1d3dcd"></a>
 
 {{< figure src="/ox-hugo/hauge05_struts.png" caption="Figure 2: Strut" >}}
 
@@ -64,7 +64,7 @@ With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and
 
 -   single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems
 
-<a id="org74432f8"></a>
+<a id="org5bf1a1a"></a>
 
 {{< figure src="/ox-hugo/hauge04_strut_model.png" caption="Figure 3: Strut model" >}}
 
@@ -136,9 +136,9 @@ And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain
 -   The performance requirements are met
 -   Good robustness
 
-<a id="orgca6905f"></a>
+<a id="org52ac01d"></a>
 
 {{< figure src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="Figure 4: Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)" >}}
 
 # Bibliography
-<a id="hauge04_sensor_contr_space_based_six"></a>Hauge, G., & Campbell, M., *Sensors and control of a space-based six-axis vibration isolation system*, Journal of Sound and Vibration, *269(3-5)*, 913–931 (2004).  http://dx.doi.org/10.1016/s0022-460x(03)00206-2 [↩](#f9698a1741fe7492aa9b7b42c7724670)
+<a class="bibtex-entry" id="hauge04_sensor_contr_space_based_six">Hauge, G., & Campbell, M., *Sensors and control of a space-based six-axis vibration isolation system*, Journal of Sound and Vibration, *269(3-5)*, 913–931 (2004).  http://dx.doi.org/10.1016/s0022-460x(03)00206-2</a> [↩](#f9698a1741fe7492aa9b7b42c7724670)

content/article/holler12_instr_x_ray_nano_imagin.md

@@ -8,7 +8,7 @@ Tags
 : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
 
 Reference
-: <sup id="66ab0e7602a1dedda963d7da60533b0d"><a href="#holler12_instr_x_ray_nano_imagin" title="Holler, Raabe, Diaz, Guizar-Sicairos, , Quitmann, Menzel \&amp; Bunk, An Instrument for 3d X-Ray Nano-Imaging, {Review of Scientific Instruments}, v(7), 073703 (2012).">(Holler {\it et al.}, 2012)</a></sup>
+: <sup id="66ab0e7602a1dedda963d7da60533b0d"><a class="reference-link" href="#holler12_instr_x_ray_nano_imagin" title="Holler, Raabe, Diaz, Guizar-Sicairos, , Quitmann, Menzel \&amp; Bunk, An Instrument for 3d X-Ray Nano-Imaging, {Review of Scientific Instruments}, v(7), 073703 (2012).">(Holler {\it et al.}, 2012)</a></sup>
 
 Author(s)
 : Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O.
@@ -19,7 +19,7 @@ Year
 Instrument similar to the NASS.
 Obtain position stability of 10nm (standard deviation).
 
-<a id="orgba4a339"></a>
+<a id="org16e51fb"></a>
 
 {{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
 
@@ -39,4 +39,4 @@ Obtain position stability of 10nm (standard deviation).
 -   **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts.
 
 # Bibliography
-<a id="holler12_instr_x_ray_nano_imagin"></a>Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O., *An instrument for 3d x-ray nano-imaging*, Review of Scientific Instruments, *83(7)*, 073703 (2012).  http://dx.doi.org/10.1063/1.4737624 [↩](#66ab0e7602a1dedda963d7da60533b0d)
+<a class="bibtex-entry" id="holler12_instr_x_ray_nano_imagin">Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O., *An instrument for 3d x-ray nano-imaging*, Review of Scientific Instruments, *83(7)*, 073703 (2012).  http://dx.doi.org/10.1063/1.4737624</a> [↩](#66ab0e7602a1dedda963d7da60533b0d)

content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md

@@ -0,0 +1,20 @@
++++
+title = "Active damping based on decoupled collocated control"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping" >}})
+
+Reference
+: <sup id="cc7836a555fe4dbae791e17008c29bfd"><a class="reference-link" href="#holterman05_activ_dampin_based_decoup_colloc_contr" title="Holterman \&amp; deVries, Active Damping Based on Decoupled Collocated Control, {IEEE/ASME Transactions on Mechatronics}, v(2), 135-145 (2005).">(Holterman \& deVries, 2005)</a></sup>
+
+Author(s)
+: Holterman, J., & deVries, T.
+
+Year
+: 2005
+
+# Bibliography
+<a class="bibtex-entry" id="holterman05_activ_dampin_based_decoup_colloc_contr">Holterman, J., & deVries, T., *Active damping based on decoupled collocated control*, IEEE/ASME Transactions on Mechatronics, *10(2)*, 135–145 (2005).  http://dx.doi.org/10.1109/tmech.2005.844702</a> [↩](#cc7836a555fe4dbae791e17008c29bfd)

content/article/ito16_compar_class_high_precis_actuat.md

@@ -8,7 +8,7 @@ Tags
 : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Actuators]({{< relref "actuators" >}})
 
 Reference
-: <sup id="aad53368e29e8a519e2f63857044fa46"><a href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators  Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>
+: <sup id="aad53368e29e8a519e2f63857044fa46"><a class="reference-link" href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators  Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>
 
 Author(s)
 : Ito, S., & Schitter, G.
@@ -41,7 +41,7 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
 -   **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
 -   **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
 
-<a id="org6e18c94"></a>
+<a id="orgffb6d7f"></a>
 
 {{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
 
@@ -54,7 +54,7 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
 
 ## Controller Design {#controller-design}
 
-<a id="orgc911fbe"></a>
+<a id="org658b9e0"></a>
 
 {{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
 
@@ -67,7 +67,7 @@ In practice, this is difficult to achieve with piezoelectric actuators as their
 In contrast, the frequency band between the first and the other resonances of Lorentz actuators can be broad by design making them more suitable to construct a low-stiffness actuators.
 
 # Bibliography
-<a id="ito16_compar_class_high_precis_actuat"></a>Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016).  http://dx.doi.org/10.1109/tmech.2015.2478658 [↩](#aad53368e29e8a519e2f63857044fa46)
+<a class="bibtex-entry" id="ito16_compar_class_high_precis_actuat">Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016).  http://dx.doi.org/10.1109/tmech.2015.2478658</a> [↩](#aad53368e29e8a519e2f63857044fa46)
 
 
 ## Backlinks {#backlinks}

content/article/jiao18_dynam_model_exper_analy_stewar.md

@@ -0,0 +1,20 @@
++++
+title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: <sup id="ee917739f88877d6c2758c1c36565deb"><a class="reference-link" href="#jiao18_dynam_model_exper_analy_stewar" title="Jian Jiao, Ying Wu, Kaiping Yu \&amp; Rui Zhao, Dynamic Modeling and Experimental Analyses of Stewart  Platform With Flexible Hinges, {Journal of Vibration and Control}, v(1), 151-171 (2018).">(Jian Jiao {\it et al.}, 2018)</a></sup>
+
+Author(s)
+: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
+
+Year
+: 2018
+
+# Bibliography
+<a class="bibtex-entry" id="jiao18_dynam_model_exper_analy_stewar">Jiao, J., Wu, Y., Yu, K., & Zhao, R., *Dynamic modeling and experimental analyses of stewart platform with flexible hinges*, Journal of Vibration and Control, *25(1)*, 151–171 (2018).  http://dx.doi.org/10.1177/1077546318772474</a> [↩](#ee917739f88877d6c2758c1c36565deb)

content/article/legnani12_new_isotr_decoup_paral_manip.md

@@ -2,13 +2,14 @@
 title = "A new isotropic and decoupled 6-dof parallel manipulator"
 author = ["Thomas Dehaeze"]
 draft = false
+GHissueID = 1
 +++
 
 Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}})
 
 Reference
-: <sup id="17295cbc2858c65ecc60d51b450233e3"><a href="#legnani12_new_isotr_decoup_paral_manip" title="Legnani, Fassi, Giberti, Cinquemani, \&amp; Tosi, A New Isotropic and Decoupled 6-dof Parallel Manipulator, {Mechanism and Machine Theory}, v(nil), 64-81 (2012).">(Legnani {\it et al.}, 2012)</a></sup>
+: <sup id="17295cbc2858c65ecc60d51b450233e3"><a class="reference-link" href="#legnani12_new_isotr_decoup_paral_manip" title="Legnani, Fassi, Giberti, Cinquemani, \&amp; Tosi, A New Isotropic and Decoupled 6-dof Parallel Manipulator, {Mechanism and Machine Theory}, v(nil), 64-81 (2012).">(Legnani {\it et al.}, 2012)</a></sup>
 
 Author(s)
 : Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
@@ -22,13 +23,13 @@ Year
 
 Example of generated isotropic manipulator (not decoupled).
 
-<a id="orgd015b7e"></a>
+<a id="org9b13cfd"></a>
 
 {{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
 
-<a id="orgb3cab58"></a>
+<a id="org958618e"></a>
 
 {{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
 
 # Bibliography
-<a id="legnani12_new_isotr_decoup_paral_manip"></a>Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D., *A new isotropic and decoupled 6-dof parallel manipulator*, Mechanism and Machine Theory, *58(nil)*, 64–81 (2012).  http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.008 [↩](#17295cbc2858c65ecc60d51b450233e3)
+<a class="bibtex-entry" id="legnani12_new_isotr_decoup_paral_manip">Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D., *A new isotropic and decoupled 6-dof parallel manipulator*, Mechanism and Machine Theory, *58(nil)*, 64–81 (2012).  http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.008</a> [↩](#17295cbc2858c65ecc60d51b450233e3)

content/article/li01_simul_fault_vibrat_isolat_point.md

@@ -8,21 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Multivariable Control]({{< relref "multivariable_control" >}})
 
 Reference
-: <sup id="f885df380638b868e509fbbf75912d1e"><a href="#li01_simul_fault_vibrat_isolat_point" title="@phdthesis{li01_simul_fault_vibrat_isolat_point,
+: <sup id="f885df380638b868e509fbbf75912d1e"><a class="reference-link" href="#li01_simul_fault_vibrat_isolat_point" title="Li, Simultaneous, Fault-tolerant Vibration Isolation and  Pointing Control of Flexure Jointed Hexapods (2001).">(Li, 2001)</a></sup>
-      author          = {Li, Xiaochun},
-      school          = {University of Wyoming},
-      title           = {Simultaneous, Fault-tolerant Vibration Isolation and
-                      Pointing Control of Flexure Jointed Hexapods},
-      year            = 2001,
-      tags            = {parallel robot},
-    }">@phdthesis{li01_simul_fault_vibrat_isolat_point,
-      author          = {Li, Xiaochun},
-      school          = {University of Wyoming},
-      title           = {Simultaneous, Fault-tolerant Vibration Isolation and
-                      Pointing Control of Flexure Jointed Hexapods},
-      year            = 2001,
-      tags            = {parallel robot},
-    }</a></sup>
 
 Author(s)
 : Li, X.
@@ -38,7 +24,7 @@ Year
 -   Cubic (mutually orthogonal)
 -   Flexure Joints => eliminate friction and backlash but add complexity to the dynamics
 
-<a id="orgd72b050"></a>
+<a id="org24b3ba4"></a>
 
 {{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}}
 
@@ -52,18 +38,18 @@ Year
 The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload.
 
 **Decoupling**:
-If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org2d875d1) or in Figure [3](#org3e247bd), we obtain a decoupled plant provided that:
+If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org5d5e02c) or in Figure [3](#org0c14c06), we obtain a decoupled plant provided that:
 
 1.  the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\))
 2.  the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen
 
 > For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
 
-<a id="org2d875d1"></a>
+<a id="org5d5e02c"></a>
 
 {{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
 
-<a id="org3e247bd"></a>
+<a id="org0c14c06"></a>
 
 {{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
 
@@ -89,15 +75,15 @@ 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 [4](#org3c42849).
+The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#orgf519833).
 
-<a id="org3c42849"></a>
+<a id="orgf519833"></a>
 
 {{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 4: Figure caption" >}}
 
-One of the subsystem plant transfer function is shown in Figure [4](#org3c42849)
+One of the subsystem plant transfer function is shown in Figure [4](#orgf519833)
 
-<a id="orga10e0a5"></a>
+<a id="orgef0f6ef"></a>
 
 {{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 5: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
 
@@ -111,9 +97,9 @@ The unity control bandwidth of the isolation loop is designed to be from **5Hz t
 
 ### Pointing Control {#pointing-control}
 
-A block diagram of the pointing control system is shown in Figure [6](#org3c3e6ad).
+A block diagram of the pointing control system is shown in Figure [6](#org1c5bf82).
 
-<a id="org3c3e6ad"></a>
+<a id="org1c5bf82"></a>
 
 {{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}}
 
@@ -122,9 +108,9 @@ The compensators are design with inverse-dynamics methods.
 
 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 [7](#orgc8fa614).
+A feedforward control is added as shown in Figure [7](#org700dd8b).
 
-<a id="orgc8fa614"></a>
+<a id="org700dd8b"></a>
 
 {{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}}
 
@@ -136,17 +122,17 @@ The simultaneous vibration isolation and pointing control is approached in two w
 1.  design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators
 2.  the reverse design order
 
-Figure [8](#org987b709) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
+Figure [8](#orga79e625) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
 
-<a id="org987b709"></a>
+<a id="orga79e625"></a>
 
 {{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 8: A parallel scheme" >}}
 
 The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
 
-The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgb070c43).
+The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgbd95400).
 
-<a id="orgb070c43"></a>
+<a id="orgbd95400"></a>
 
 {{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 9: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}}
 
@@ -157,19 +143,19 @@ The effect of pointing control on the isolation plant has not much effect.
 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 [10](#org0d64bc7) and [11](#orgb43f022))
+-   affect the closed loop transmission of the loop first closed (see Figures [10](#org191e7e3) and [11](#org28140a0))
 
-As shown in Figure [10](#org0d64bc7), the peak resonance of the pointing loop increase after the isolation loop is closed.
+As shown in Figure [10](#org191e7e3), 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="org0d64bc7"></a>
+<a id="org191e7e3"></a>
 
 {{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 10: 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 [11](#orgb43f022)).
+The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org28140a0)).
 The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
 
-<a id="orgb43f022"></a>
+<a id="org28140a0"></a>
 
 {{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 11: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
 
@@ -179,18 +165,18 @@ The first peak resonance of the vibration isolation loop at 15Hz is increased wh
 
 ### Experimental results {#experimental-results}
 
-Two hexapods are stacked (Figure [12](#org12b1e53)):
+Two hexapods are stacked (Figure [12](#orgb11b2c6)):
 
 -   the bottom hexapod is used to generate disturbances matching candidate applications
 -   the top hexapod provide simultaneous vibration isolation and pointing control
 
-<a id="org12b1e53"></a>
+<a id="orgb11b2c6"></a>
 
 {{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 12: Stacked Hexapods" >}}
 
-Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org4b99c02).
+Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#orgdd443cd).
 
-<a id="org4b99c02"></a>
+<a id="orgdd443cd"></a>
 
 {{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
 
@@ -199,9 +185,9 @@ 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 [14](#orged11c63) where the bandwidth of the isolation control is expanded to very low frequency.
+The results of simultaneous control is shown in Figure [14](#org64f7223) where the bandwidth of the isolation control is expanded to very low frequency.
 
-<a id="orged11c63"></a>
+<a id="org64f7223"></a>
 
 {{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
 
@@ -230,4 +216,4 @@ Proposed future research areas include:
     -   **Geophones** to provide payload and base velocity information
 
 # Bibliography
-<a id="li01_simul_fault_vibrat_isolat_point"></a>Li, X., *Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods* (Doctoral dissertation) (2001). University of Wyoming, . [↩](#f885df380638b868e509fbbf75912d1e)
+<a class="bibtex-entry" id="li01_simul_fault_vibrat_isolat_point">Li, X., *Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods* (2001). University of Wyoming.</a> [↩](#f885df380638b868e509fbbf75912d1e)

content/article/li01_simul_vibrat_isolat_point_contr.md

@@ -0,0 +1,23 @@
++++
+title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: <sup id="e3df2691f750617c3995644d056d553a"><a class="reference-link" href="#li01_simul_vibrat_isolat_point_contr" title="Xiaochun Li, Jerry Hamann \&amp; John McInroy, Simultaneous Vibration Isolation and Pointing Control of  Flexure Jointed Hexapods, nil, in in: {Smart Structures and Materials 2001: Smart Structures and
+                      Integrated Systems}, edited by (2001)">(Xiaochun Li {\it et al.}, 2001)</a></sup>
+
+Author(s)
+: Li, X., Hamann, J. C., & McInroy, J. E.
+
+Year
+: 2001
+
+-   if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
+
+# Bibliography
+<a class="bibtex-entry" id="li01_simul_vibrat_isolat_point_contr">Li, X., Hamann, J. C., & McInroy, J. E., *Simultaneous vibration isolation and pointing control of flexure jointed hexapods*, In , Smart Structures and Materials 2001: Smart Structures and Integrated Systems (pp. ) (2001). : .</a> [↩](#e3df2691f750617c3995644d056d553a)

content/article/lin06_distur_atten_precis_hexap_point.md

@@ -0,0 +1,21 @@
++++
+title = "Disturbance attenuation in precise hexapod pointing using positive force feedback"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="3ab8ef7353729de315618a708ece8379"><a class="reference-link" href="#lin06_distur_atten_precis_hexap_point" title="Haomin Lin \&amp; John McInroy, Disturbance Attenuation in Precise Hexapod Pointing Using  Positive Force Feedback, {Control Engineering Practice}, v(11), 1377-1386 (2006).">(Haomin Lin \& John McInroy, 2006)</a></sup>
+
+Author(s)
+: Lin, H., & McInroy, J. E.
+
+Year
+: 2006
+
+# Bibliography
+<a class="bibtex-entry" id="lin06_distur_atten_precis_hexap_point">Lin, H., & McInroy, J. E., *Disturbance attenuation in precise hexapod pointing using positive force feedback*, Control Engineering Practice, *14(11)*, 1377–1386 (2006).  http://dx.doi.org/10.1016/j.conengprac.2005.10.002</a> [↩](#3ab8ef7353729de315618a708ece8379)

content/article/mcinroy00_desig_contr_flexur_joint_hexap.md

@@ -0,0 +1,21 @@
++++
+title = "Design and control of flexure jointed hexapods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="f6d310236552ee92579cf0673a2ca695"><a href="#mcinroy00_desig_contr_flexur_joint_hexap" title="McInroy \&amp; Hamann, Design and Control of Flexure Jointed Hexapods, {IEEE Transactions on Robotics and Automation}, v(4), 372-381 (2000).">(McInroy \& Hamann, 2000)</a></sup>
+
+Author(s)
+: McInroy, J., & Hamann, J.
+
+Year
+: 2000
+
+# Bibliography
+<a id="mcinroy00_desig_contr_flexur_joint_hexap"></a>McInroy, J., & Hamann, J., *Design and control of flexure jointed hexapods*, IEEE Transactions on Robotics and Automation, *16(4)*, 372–381 (2000).  http://dx.doi.org/10.1109/70.864229 [↩](#f6d310236552ee92579cf0673a2ca695)

content/article/mcinroy02_model_desig_flexur_joint_stewar.md

@@ -0,0 +1,228 @@
++++
+title = "Modeling and design of flexure jointed stewart platforms for control purposes"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="8bfe2d2dce902a584fa016e86a899044"><a class="reference-link" href="#mcinroy02_model_desig_flexur_joint_stewar" title="McInroy, Modeling and Design of Flexure Jointed Stewart Platforms  for Control Purposes, {IEEE/ASME Transactions on Mechatronics}, v(1), 95-99 (2002).">(McInroy, 2002)</a></sup>
+
+Author(s)
+: McInroy, J.
+
+Year
+: 2002
+
+This short paper is very similar to <sup id="5da427f78c552aa92cd64c2a6df961f1"><a class="reference-link" href="#mcinroy99_dynam" title="McInroy, Dynamic modeling of flexure jointed hexapods for control  purposes, nil, in in: {Proceedings of the 1999 IEEE International Conference on
+                  Control Applications (Cat. No.99CH36328)}, edited by (1999)">(McInroy, 1999)</a></sup>.
+
+> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
+
+
+## Introduction {#introduction}
+
+> When pursuing micro-meter/micro-radian scale motion, two new phenomena become important:
+>
+> 1.  joint friction and backlash can cause extremely nonlinear micro-dynamics
+> 2.  base and/or payload vibrations become significant contributor to the motion
+
+<!--quoteend-->
+
+> If the spherical flexure is not properly matched to the particular application, it is shown that the complexity of the dynamics can greatly increase, thus limiting the control performance.
+
+
+## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
+
+<a id="org1e5260a"></a>
+
+{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="Figure 1: 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](#org1e5260a)).
+
+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](#org1e5260a) with:
+
+-   \\(m\_{s\_i}\\) moving strut mass
+-   \\(k\_i\\) spring constant
+-   \\(b\_i\\) damping constant
+-   \\(f\_{m\_i}\\) force applied by the actuator
+-   \\(f\_{p\_i}\\) force exerted by the payload
+-   \\(p\_i\\) three dimensional position of the top
+-   \\(q\_i\\) three dimensional position of the bottom
+-   \\(l\_i\\) strut length
+-   \\(l\_{r\_i}\\) relaxed strut length
+
+In general, **the strut mass and spherical flexure stiffness will cause payload forces that are not perfectly aligned with the strut**.
+
+Applying Newton's second law and stacking the equations into a vector form gives:
+
+\begin{equation}
+  f\_p = 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 \label{eq:strut\_dynamics\_vec}
+\end{equation}
+
+where:
+
+-   \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
+-   \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
+-   \\(Ms = \text{diag}([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
+-   \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\) contains nonlinear Coriolis and centripetal accelerations
+
+
+### Payload Dynamics {#payload-dynamics}
+
+The payload is modeled as a rigid body:
+
+\begin{equation}
+\underbrace{\begin{bmatrix}
+m I\_3 & 0\_{3\times 3} \\\\\\
+0\_{3\times 3}   & {}^cI
+\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
+  0\_{3 \times 1} \\ \omega \times {}^cI\omega
+\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
+\end{equation}
+
+where:
+
+-   \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
+-   \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
+-   \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
+-   \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
+-   \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
+
+Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
+
+First, consider the **generalized force due to struts**.
+Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
+
+\begin{equation}
+  \mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
+\end{equation}
+
+where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
+
+The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
+
+\begin{equation}
+  \mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
+\end{equation}
+
+where:
+
+-   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
+-   \\(g\\) is the gravity vector
+
+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\\\\\\
+  & - 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)
+\end{aligned}
+
+Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
+In the next section, a connection between the two will be found to complete the formulation
+
+
+## Direction of Payload Force {#direction-of-payload-force}
+
+Many prior hexapod dynamic formulations assume that the strut exerts force only along its direction of motion.
+
+The flexure joints Hexapods transmit forces (or torques) proportional to the deflection of the joints.
+This section establishes design guidelines for the spherical flexure joint to guarantee that the dynamics remain tractable for control.
+
+<a id="orgbd4aaf0"></a>
+
+{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="Figure 2: A simplified dynamic model of a strut and its joint" >}}
+
+Figure [2](#orgbd4aaf0) 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](#orgbd4aaf0) (b), Newton's second law yields:
+
+\begin{equation}
+  f\_p = \begin{bmatrix}
+  -f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\\\
+  m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\\\
+  m\_s \Delta \ddot{z} + \frac{k\_r}{l^2} \Delta z
+\end{bmatrix}
+\end{equation}
+
+Note that the payload force is **not** in general aligned with the strut.
+The force is aligned perfectly with the strut only if \\(m\_s = 0\\) and \\(k\_r = 0\\) (i.e. the struts have negligible mass and the spherical joints have negligible rotational stiffness).
+
+To examine the passive behavior, let \\(f\_m = 0\\) and consider a sinusoidal motion:
+
+\begin{equation}
+  \begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} =
+  \begin{bmatrix} A\_x \cos \omega t \\ A\_y \cos \omega t \\  A\_z \cos \omega t \end{bmatrix}
+\end{equation}
+
+This yields:
+
+\begin{equation}
+  f\_p = \begin{bmatrix}
+    \Big( -m\_s \omega^2 + k \Big) A\_x \cos \omega t \\\\\\
+    \Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_y \cos \omega t \\\\\\
+    \Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_z \cos \omega t
+  \end{bmatrix}
+\end{equation}
+
+The direction of \\(f\_p\\) depends upon to motion specifications, leg inertia and control algorithm.
+
+The hypothesis that it is mostly along the strut direction can be tested by dividing the magnitude of the \\(x\\) component by the magnitude of the combined \\(y\\) and \\(z\\) components:
+
+\begin{equation}
+  x\_\text{gain} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|} \frac{|A\_x|}{\sqrt{A\_y^2 + A\_z^2}}
+\end{equation}
+
+Note that large \\(x\_\text{gain}\\) indicates \\(x\\) direction dominance.
+
+\\(x\_\text{gain}\\) is divided into two parts.
+The first part depends on the mechanical terms and the frequency of the movement:
+
+\begin{equation}
+  x\_{\text{gain}\_\omega} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|}
+\end{equation}
+
+> In order to get dominance at low frequencies, the hexapod must be designed so that:
+>
+> \begin{equation}
+>   \frac{k\_r}{l^2} \ll k \label{eq:cond\_stiff}
+> \end{equation}
+
+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 \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 \\]
+
+> To ensure that the control system acts only in the band of frequencies where dominance is retained, the control bandwidth can be selected so that:
+>
+> \begin{equation}
+>   \text{control bandwidth} \ll \sqrt{\frac{k\_r}{m\_s l^2}} \label{eq:cond\_bandwidth}
+> \end{equation}
+
+The control bandwidth can be increase for hexapods that are designed so that \\(x\_{\text{gain}\_\omega} \gg 1\\) for \\(\omega \ll \sqrt{k/m\_s}\\).
+This can be achieve, for instance, by adding damping.
+In this case, it is reasonable to use:
+
+\begin{equation}
+  \text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
+\end{equation}
+
+> 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.
+
+
+## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
+
+# Bibliography
+<a class="bibtex-entry" id="mcinroy02_model_desig_flexur_joint_stewar">McInroy, J., *Modeling and design of flexure jointed stewart platforms for control purposes*, IEEE/ASME Transactions on Mechatronics, *7(1)*, 95–99 (2002).  http://dx.doi.org/10.1109/3516.990892</a> [↩](#8bfe2d2dce902a584fa016e86a899044)
+
+<a class="bibtex-entry" id="mcinroy99_dynam">McInroy, J., *Dynamic modeling of flexure jointed hexapods for control purposes*, In , Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328) (pp. ) (1999). : .</a> [↩](#5da427f78c552aa92cd64c2a6df961f1)

content/article/oomen18_advan_motion_contr_precis_mechat.md

@@ -0,0 +1,24 @@
++++
+title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Motion Control]({{< relref "motion_control" >}})
+
+Reference
+: <sup id="73fd325bd20a6ef8972145e535f38198"><a class="reference-link" href="#oomen18_advan_motion_contr_precis_mechat" title="Tom Oomen, Advanced Motion Control for Precision Mechatronics:  Control, Identification, and Learning of Complex Systems, {IEEJ Journal of Industry Applications}, v(2), 127-140 (2018).">(Tom Oomen, 2018)</a></sup>
+
+Author(s)
+: Oomen, T.
+
+Year
+: 2018
+
+<a id="org5cf2052"></a>
+
+{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
+
+# Bibliography
+<a class="bibtex-entry" id="oomen18_advan_motion_contr_precis_mechat">Oomen, T., *Advanced motion control for precision mechatronics: control, identification, and learning of complex systems*, IEEJ Journal of Industry Applications, *7(2)*, 127–140 (2018).  http://dx.doi.org/10.1541/ieejjia.7.127</a> [↩](#73fd325bd20a6ef8972145e535f38198)

content/article/poel10_explor_activ_hard_mount_vibrat.md

@@ -0,0 +1,20 @@
++++
+title = "An exploration of active hard mount vibration isolation for precision equipment"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: <sup id="bcab548922e0e1ad6a2c310f63879596"><a class="reference-link" href="#poel10_explor_activ_hard_mount_vibrat" title="van der Poel, An Exploration of Active Hard Mount Vibration Isolation for  Precision Equipment (2010).">(van der Poel, 2010)</a></sup>
+
+Author(s)
+: van der Poel, G. W.
+
+Year
+: 2010
+
+# Bibliography
+<a class="bibtex-entry" id="poel10_explor_activ_hard_mount_vibrat">van der Poel, G. W., *An exploration of active hard mount vibration isolation for precision equipment* (2010). University of Twente.</a> [↩](#bcab548922e0e1ad6a2c310f63879596)

content/article/preumont02_force_feedb_versus_accel_feedb.md

@@ -8,7 +8,7 @@ Tags
 : [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="525e1e237b885f81fae3c25a3036ba6f"><a href="#preumont02_force_feedb_versus_accel_feedb" title="Preumont, Fran\ccois, Bossens, \&amp; Abu-Hanieh, Force Feedback Versus Acceleration Feedback in Active  Vibration Isolation, {Journal of Sound and Vibration}, v(4), 605-613 (2002).">(Preumont {\it et al.}, 2002)</a></sup>
+: <sup id="525e1e237b885f81fae3c25a3036ba6f"><a class="reference-link" href="#preumont02_force_feedb_versus_accel_feedb" title="Preumont, Fran\ccois, Bossens, \&amp; Abu-Hanieh, Force Feedback Versus Acceleration Feedback in Active  Vibration Isolation, {Journal of Sound and Vibration}, v(4), 605-613 (2002).">(Preumont {\it et al.}, 2002)</a></sup>
 
 Author(s)
 : Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
@@ -26,14 +26,14 @@ 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](#org7b4f6ee)), the acceleration feedback gives larger damping on the higher mode.
+-   For light payload (Figure [1](#org307b349)), the acceleration feedback gives larger damping on the higher mode.
--   For heavy payload (Figure [2](#org361b58f)), 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](#orgc0c4ad3)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
 
-<a id="org7b4f6ee"></a>
+<a id="org307b349"></a>
 
 {{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="Figure 1: Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
 
-<a id="org361b58f"></a>
+<a id="orgc0c4ad3"></a>
 
 {{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="Figure 2: Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
 
@@ -46,4 +46,4 @@ The same is true for the transfer function from the force actuator to the relati
 > According to physical interpretation of the zeros, they represent the resonances of the subsystem  constrained by the sensor and the actuator.
 
 # Bibliography
-<a id="preumont02_force_feedb_versus_accel_feedb"></a>Preumont, A., A. Fran\ccois, Bossens, F., & Abu-Hanieh, A., *Force feedback versus acceleration feedback in active vibration isolation*, Journal of Sound and Vibration, *257(4)*, 605–613 (2002).  http://dx.doi.org/10.1006/jsvi.2002.5047 [↩](#525e1e237b885f81fae3c25a3036ba6f)
+<a class="bibtex-entry" id="preumont02_force_feedb_versus_accel_feedb">Preumont, A., A. Fran\ccois, Bossens, F., & Abu-Hanieh, A., *Force feedback versus acceleration feedback in active vibration isolation*, Journal of Sound and Vibration, *257(4)*, 605–613 (2002).  http://dx.doi.org/10.1006/jsvi.2002.5047</a> [↩](#525e1e237b885f81fae3c25a3036ba6f)

content/article/preumont07_six_axis_singl_stage_activ.md

@@ -8,7 +8,7 @@ Tags
 : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
 
 Reference
-: <sup id="8096d5b2df73551d836ef96b7ca7efa4"><a href="#preumont07_six_axis_singl_stage_activ" title="Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, \&amp; Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on  Stewart Platform, {Journal of Sound and Vibration}, v(3-5), 644-661 (2007).">(Preumont {\it et al.}, 2007)</a></sup>
+: <sup id="8096d5b2df73551d836ef96b7ca7efa4"><a class="reference-link" href="#preumont07_six_axis_singl_stage_activ" title="Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, \&amp; Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on  Stewart Platform, {Journal of Sound and Vibration}, v(3-5), 644-661 (2007).">(Preumont {\it et al.}, 2007)</a></sup>
 
 Author(s)
 : Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@@ -18,32 +18,32 @@ Year
 
 Summary:
 
--   **Cubic** Stewart platform (Figure [3](#org32a4f7c))
+-   **Cubic** Stewart platform (Figure [3](#org2d41889))
     -   Provides uniform control capability
     -   Uniform stiffness in all directions
     -   minimizes the cross-coupling among actuators and sensors of different legs
--   Flexible joints (Figure [2](#orgf807976))
+-   Flexible joints (Figure [2](#orgf58a4b4))
 -   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](#org744bdc9))
+-   Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#org6835865))
 -   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.
 -   Discusses the design of the legs in order to maximize the natural frequency of the local modes.
 -   To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix
 
-<a id="org744bdc9"></a>
+<a id="org6835865"></a>
 
 {{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
 
-<a id="orgf807976"></a>
+<a id="orgf58a4b4"></a>
 
 {{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
 
-<a id="org32a4f7c"></a>
+<a id="org2d41889"></a>
 
 {{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
 
 # Bibliography
-<a id="preumont07_six_axis_singl_stage_activ"></a>Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …, *A six-axis single-stage active vibration isolator based on stewart platform*, Journal of Sound and Vibration, *300(3-5)*, 644–661 (2007).  http://dx.doi.org/10.1016/j.jsv.2006.07.050 [↩](#8096d5b2df73551d836ef96b7ca7efa4)
+<a class="bibtex-entry" id="preumont07_six_axis_singl_stage_activ">Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …, *A six-axis single-stage active vibration isolator based on stewart platform*, Journal of Sound and Vibration, *300(3-5)*, 644–661 (2007).  http://dx.doi.org/10.1016/j.jsv.2006.07.050</a> [↩](#8096d5b2df73551d836ef96b7ca7efa4)

content/article/saxena12_advan_inter_model_contr_techn.md

@@ -5,10 +5,10 @@ draft = false
 +++
 
 Tags
-: [Complementary Filters]({{< relref "complementary_filters" >}})
+: [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}})
 
 Reference
-: <sup id="14f767d8ba71d58fa8a3ec876628d750"><a href="#saxena12_advan_inter_model_contr_techn" title="Sahaj Saxena \&amp; YogeshV Hote, Advances in Internal Model Control Technique: a Review and  Future Prospects, {IETE Technical Review}, v(6), 461 (2012).">(Sahaj Saxena \& YogeshV Hote, 2012)</a></sup>
+: <sup id="14f767d8ba71d58fa8a3ec876628d750"><a class="reference-link" href="#saxena12_advan_inter_model_contr_techn" title="Sahaj Saxena \&amp; YogeshV Hote, Advances in Internal Model Control Technique: a Review and  Future Prospects, {IETE Technical Review}, v(6), 461 (2012).">(Sahaj Saxena \& YogeshV Hote, 2012)</a></sup>
 
 Author(s)
 : Saxena, S., & Hote, Y.
@@ -85,4 +85,4 @@ Issues:
 The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
 
 # Bibliography
-<a id="saxena12_advan_inter_model_contr_techn"></a>Saxena, S., & Hote, Y., *Advances in internal model control technique: a review and future prospects*, IETE Technical Review, *29(6)*, 461 (2012).  http://dx.doi.org/10.4103/0256-4602.105001 [↩](#14f767d8ba71d58fa8a3ec876628d750)
+<a class="bibtex-entry" id="saxena12_advan_inter_model_contr_techn">Saxena, S., & Hote, Y., *Advances in internal model control technique: a review and future prospects*, IETE Technical Review, *29(6)*, 461 (2012).  http://dx.doi.org/10.4103/0256-4602.105001</a> [↩](#14f767d8ba71d58fa8a3ec876628d750)

content/article/sayed01_survey_spect_factor_method.md

@@ -0,0 +1,21 @@
++++
+title = "A survey of spectral factorization methods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="e71cc5e3ec879813f2344a6dce1ac11f"><a href="#sayed01_survey_spect_factor_method" title="Sayed \&amp; Kailath, A Survey of Spectral Factorization Methods, {Numerical Linear Algebra with Applications}, v(6-7), 467-496 (2001).">(Sayed \& Kailath, 2001)</a></sup>
+
+Author(s)
+: Sayed, A. H., & Kailath, T.
+
+Year
+: 2001
+
+# Bibliography
+<a id="sayed01_survey_spect_factor_method"></a>Sayed, A. H., & Kailath, T., *A survey of spectral factorization methods*, Numerical Linear Algebra with Applications, *8(6-7)*, 467–496 (2001).  http://dx.doi.org/10.1002/nla.250 [↩](#e71cc5e3ec879813f2344a6dce1ac11f)

content/article/schellekens98_desig_precis.md

@@ -0,0 +1,20 @@
++++
+title = "Design for precision: current status and trends"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Precision Engineering]({{< relref "precision_engineering" >}})
+
+Reference
+: <sup id="89f7d8f4c31f79f83e3666017687f525"><a class="reference-link" href="#schellekens98_desig_precis" title="Schellekens, Rosielle, Vermeulen, , Vermeulen, Wetzels \&amp; Pril, Design for Precision: Current Status and Trends, {Cirp Annals}, v(2), 557-586 (1998).">(Schellekens {\it et al.}, 1998)</a></sup>
+
+Author(s)
+: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
+
+Year
+: 1998
+
+# Bibliography
+<a class="bibtex-entry" id="schellekens98_desig_precis">Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W., *Design for precision: current status and trends*, Cirp Annals, *(2)*, 557–586 (1998).  http://dx.doi.org/10.1016/s0007-8506(07)63243-0</a> [↩](#89f7d8f4c31f79f83e3666017687f525)

content/article/schroeck01_compen_desig_linear_time_invar.md

@@ -0,0 +1,21 @@
++++
+title = "On compensator design for linear time-invariant dual-input single-output systems"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="ee9f1b2ad5707e86bf7c26e8c325b324"><a class="reference-link" href="#schroeck01_compen_desig_linear_time_invar" title="Schroeck, Messner \&amp; McNab, On Compensator Design for Linear Time-Invariant Dual-Input  Single-Output Systems, {IEEE/ASME Transactions on Mechatronics}, v(1), 50-57 (2001).">(Schroeck {\it et al.}, 2001)</a></sup>
+
+Author(s)
+: Schroeck, S., Messner, W., & McNab, R.
+
+Year
+: 2001
+
+# Bibliography
+<a class="bibtex-entry" id="schroeck01_compen_desig_linear_time_invar">Schroeck, S., Messner, W., & McNab, R., *On compensator design for linear time-invariant dual-input single-output systems*, IEEE/ASME Transactions on Mechatronics, *6(1)*, 50–57 (2001).  http://dx.doi.org/10.1109/3516.914391</a> [↩](#ee9f1b2ad5707e86bf7c26e8c325b324)

content/article/sebastian12_nanop_with_multip_sensor.md

@@ -0,0 +1,20 @@
++++
+title = "Nanopositioning with multiple sensors: a case study in data storage"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion" >}})
+
+Reference
+: <sup id="eb5a15a8c900d93de0b9bab520e1b6da"><a class="reference-link" href="#sebastian12_nanop_with_multip_sensor" title="Abu Sebastian \&amp; Angeliki Pantazi, Nanopositioning With Multiple Sensors: a Case Study in Data  Storage, {IEEE Transactions on Control Systems Technology}, v(2), 382-394 (2012).">(Abu Sebastian \& Angeliki Pantazi, 2012)</a></sup>
+
+Author(s)
+: Sebastian, A., & Pantazi, A.
+
+Year
+: 2012
+
+# Bibliography
+<a class="bibtex-entry" id="sebastian12_nanop_with_multip_sensor">Sebastian, A., & Pantazi, A., *Nanopositioning with multiple sensors: a case study in data storage*, IEEE Transactions on Control Systems Technology, *20(2)*, 382–394 (2012).  http://dx.doi.org/10.1109/tcst.2011.2177982</a> [↩](#eb5a15a8c900d93de0b9bab520e1b6da)

content/article/souleille18_concep_activ_mount_space_applic.md

@@ -0,0 +1,96 @@
++++
+title = "A concept of active mount for space applications"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping" >}})
+
+Reference
+: <sup id="d5c1263eebe6caa1e91b078b620d72f1"><a class="reference-link" href="#souleille18_concep_activ_mount_space_applic" title="Souleille, Lampert, Lafarga, , Hellegouarch, Rondineau, Rodrigues, Gon\ccalo \&amp; Collette, A Concept of Active Mount for Space Applications, {CEAS Space Journal}, v(2), 157--165 (2018).">(Souleille {\it et al.}, 2018)</a></sup>
+
+Author(s)
+: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
+
+Year
+: 2018
+
+This article discusses the use of Integral Force Feedback with amplified piezoelectric stack actuators.
+
+> In the proposed configuration, it can also be noticed by the softening effect inherent to force control is limited by the metallic suspension.
+
+
+## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
+
+Figure [1](#orgec40a2d) 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="orgec40a2d"></a>
+
+{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: 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>:
+  Parameters used for the model of the APA 100M
+</div>
+
+|            | Value                 | Meaning                                                        |
+|------------|-----------------------|----------------------------------------------------------------|
+| \\(m\\)    | \\(1\,[kg]\\)         | Payload mass                                                   |
+| \\(k\_e\\) | \\(4.8\,[N/\mu m]\\)  | Stiffness used to adjust the pole of the isolator              |
+| \\(k\_1\\) | \\(0.96\,[N/\mu m]\\) | Stiffness of the metallic suspension when the stack is removed |
+| \\(k\_a\\) | \\(65\,[N/\mu m]\\)   | Stiffness of the actuator                                      |
+| \\(c\_1\\) | \\(10\,[N/(m/s)]\\)   | Added viscous damping                                          |
+
+The dynamic equation of the system is:
+
+\begin{equation}
+  m \ddot{x}\_1 = \left( k\_1 + \frac{k\_ek\_a}{k\_e + k\_a} \right) ( w - x\_1) + c\_1 (\dot{w} - \dot{x}\_1) + F + \left( \frac{k\_e}{k\_e + k\_a} \right)f
+\end{equation}
+
+The expression of the force measured by the force sensor is:
+
+\begin{equation}
+  F\_s = \left( -\frac{k\_e k\_a}{k\_e + k\_a} \right) x\_1 + \left( \frac{k\_e k\_a}{k\_e + k\_a} \right) w + \left( \frac{k\_e}{k\_e + k\_a} \right) f
+\end{equation}
+
+and the control force is given by:
+
+\begin{equation}
+  f = F\_s G(s) = F\_s \frac{g}{s}
+\end{equation}
+
+The effect of the controller are shown in Figure [2](#org656442f):
+
+-   the resonance peak is almost critically damped
+-   the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
+-   the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\)
+-   the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
+
+<a id="org656442f"></a>
+
+{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
+
+<a id="orgd1fa41a"></a>
+
+{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}}
+
+
+## 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](#org59a9fbf)).
+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="org59a9fbf"></a>
+
+{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: 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](#orgb30c1f0), both the suspension modes and the flexible modes of the payload can be critically damped.
+
+<a id="orgb30c1f0"></a>
+
+{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
+
+# Bibliography
+<a class="bibtex-entry" id="souleille18_concep_activ_mount_space_applic">Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gon\ccalo, & Collette, C., *A concept of active mount for space applications*, CEAS Space Journal, *10(2)*, 157–165 (2018). </a> [↩](#d5c1263eebe6caa1e91b078b620d72f1)

content/article/spanos95_soft_activ_vibrat_isolat.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="a48f6708d087625a42ca2375407a2bc4"><a href="#spanos95_soft_activ_vibrat_isolat" title="Spanos, Rahman \&amp; Blackwood, A Soft 6-axis Active Vibration Isolator, nil, in in: {Proceedings of 1995 American Control Conference - ACC'95}, edited by (1995)">(Spanos {\it et al.}, 1995)</a></sup>
+: <sup id="a48f6708d087625a42ca2375407a2bc4"><a class="reference-link" href="#spanos95_soft_activ_vibrat_isolat" title="Spanos, Rahman \&amp; Blackwood, A Soft 6-axis Active Vibration Isolator, nil, in in: {Proceedings of 1995 American Control Conference - ACC'95}, edited by (1995)">(Spanos {\it et al.}, 1995)</a></sup>
 
 Author(s)
 : Spanos, J., Rahman, Z., & Blackwood, G.
@@ -16,14 +16,14 @@ Author(s)
 Year
 : 1995
 
-**Stewart Platform** (Figure [1](#org4317d08)):
+**Stewart Platform** (Figure [1](#org2d8aec6)):
 
 -   Voice Coil
 -   Flexible joints (cross-blades)
 -   Force Sensors
 -   Cubic Configuration
 
-<a id="org4317d08"></a>
+<a id="org2d8aec6"></a>
 
 {{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
 
@@ -41,7 +41,7 @@ After redesign of the struts:
 -   low frequency zero at 2.6Hz but non-minimum phase (not explained).
     Small viscous damping material in the cross blade flexures made the zero minimum phase again.
 
-<a id="org67e505c"></a>
+<a id="org4f9f9d6"></a>
 
 {{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="Figure 2: Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
 
@@ -52,11 +52,11 @@ 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](#orgf128817).
+The results in terms of transmissibility are shown in Figure [3](#orgc669f80).
 
-<a id="orgf128817"></a>
+<a id="orgc669f80"></a>
 
 {{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
 
 # Bibliography
-<a id="spanos95_soft_activ_vibrat_isolat"></a>Spanos, J., Rahman, Z., & Blackwood, G., *A soft 6-axis active vibration isolator*, In , Proceedings of 1995 American Control Conference - ACC'95 (pp. ) (1995). : . [↩](#a48f6708d087625a42ca2375407a2bc4)
+<a class="bibtex-entry" id="spanos95_soft_activ_vibrat_isolat">Spanos, J., Rahman, Z., & Blackwood, G., *A soft 6-axis active vibration isolator*, In , Proceedings of 1995 American Control Conference - ACC'95 (pp. ) (1995). : .</a> [↩](#a48f6708d087625a42ca2375407a2bc4)

content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md

@@ -8,7 +8,7 @@ Tags
 : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
 
 Reference
-: <sup id="abb1be5f48179255f7d8c45b1784bcf8"><a href="#stankevic17_inter_charac_rotat_stages_x_ray_nanot" title="Tomas Stankevic, Christer Engblom, Florent Langlois, , Filipe Alves, Alain Lestrade, Nicolas Jobert, , Gilles Cauchon, Ulrich Vogt \&amp; Stefan Kubsky, Interferometric Characterization of Rotation Stages for  X-Ray Nanotomography, {Review of Scientific Instruments}, v(5), 053703 (2017).">(Tomas Stankevic {\it et al.}, 2017)</a></sup>
+: <sup id="abb1be5f48179255f7d8c45b1784bcf8"><a class="reference-link" href="#stankevic17_inter_charac_rotat_stages_x_ray_nanot" title="Tomas Stankevic, Christer Engblom, Florent Langlois, , Filipe Alves, Alain Lestrade, Nicolas Jobert, , Gilles Cauchon, Ulrich Vogt \&amp; Stefan Kubsky, Interferometric Characterization of Rotation Stages for  X-Ray Nanotomography, {Review of Scientific Instruments}, v(5), 053703 (2017).">(Tomas Stankevic {\it et al.}, 2017)</a></sup>
 
 Author(s)
 : Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
@@ -19,7 +19,7 @@ Year
 -   Similar Station than the NASS
 -   Similar Metrology with fiber based interferometers and cylindrical reference mirror
 
-<a id="orgc1f98d0"></a>
+<a id="orgc36dc2f"></a>
 
 {{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
 
@@ -30,4 +30,4 @@ Year
 -   Result: 40nm runout error
 
 # Bibliography
-<a id="stankevic17_inter_charac_rotat_stages_x_ray_nanot"></a>Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …, *Interferometric characterization of rotation stages for x-ray nanotomography*, Review of Scientific Instruments, *88(5)*, 053703 (2017).  http://dx.doi.org/10.1063/1.4983405 [↩](#abb1be5f48179255f7d8c45b1784bcf8)
+<a class="bibtex-entry" id="stankevic17_inter_charac_rotat_stages_x_ray_nanot">Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …, *Interferometric characterization of rotation stages for x-ray nanotomography*, Review of Scientific Instruments, *88(5)*, 053703 (2017).  http://dx.doi.org/10.1063/1.4983405</a> [↩](#abb1be5f48179255f7d8c45b1784bcf8)

content/article/tang18_decen_vibrat_contr_voice_coil.md

@@ -0,0 +1,21 @@
++++
+title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}})
+
+Reference
+: <sup id="85f81ff678aabc195636437548e4234a"><a class="reference-link" href="#tang18_decen_vibrat_contr_voice_coil" title="Jie Tang, Dengqing Cao \&amp; Tianhu Yu, Decentralized Vibration Control of a Voice Coil Motor-Based  Stewart Parallel Mechanism: Simulation and Experiments, {Proceedings of the Institution of Mechanical Engineers,
+                      Part C: Journal of Mechanical Engineering Science}, v(1), 132-145 (2018).">(Jie Tang {\it et al.}, 2018)</a></sup>
+
+Author(s)
+: Tang, J., Cao, D., & Yu, T.
+
+Year
+: 2018
+
+# Bibliography
+<a class="bibtex-entry" id="tang18_decen_vibrat_contr_voice_coil">Tang, J., Cao, D., & Yu, T., *Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments*, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, *233(1)*, 132–145 (2018).  http://dx.doi.org/10.1177/0954406218756941</a> [↩](#85f81ff678aabc195636437548e4234a)

content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md

@@ -8,7 +8,7 @@ Tags
 : [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="ef30bc07c91e9d46a42198757dc610de"><a href="#tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" title="Tjepkema, van Dijk \&amp; Soemers, Sensor Fusion for Active Vibration Isolation in Precision  Equipment, {Journal of Sound and Vibration}, v(4), 735-749 (2012).">(Tjepkema {\it et al.}, 2012)</a></sup>
+: <sup id="ef30bc07c91e9d46a42198757dc610de"><a class="reference-link" href="#tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" title="Tjepkema, van Dijk \&amp; Soemers, Sensor Fusion for Active Vibration Isolation in Precision  Equipment, {Journal of Sound and Vibration}, v(4), 735-749 (2012).">(Tjepkema {\it et al.}, 2012)</a></sup>
 
 Author(s)
 : Tjepkema, D., Dijk, J. v., & Soemers, H.
@@ -47,4 +47,4 @@ Heavier sensor => lower noise but it is harder to maintain collocation with the
 There is a compromise between sensor noise and the influence of the sensor size on the system's design and on the control bandwidth.
 
 # Bibliography
-<a id="tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip"></a>Tjepkema, D., Dijk, J. v., & Soemers, H., *Sensor fusion for active vibration isolation in precision equipment*, Journal of Sound and Vibration, *331(4)*, 735–749 (2012).  http://dx.doi.org/10.1016/j.jsv.2011.09.022 [↩](#ef30bc07c91e9d46a42198757dc610de)
+<a class="bibtex-entry" id="tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip">Tjepkema, D., Dijk, J. v., & Soemers, H., *Sensor fusion for active vibration isolation in precision equipment*, Journal of Sound and Vibration, *331(4)*, 735–749 (2012).  http://dx.doi.org/10.1016/j.jsv.2011.09.022</a> [↩](#ef30bc07c91e9d46a42198757dc610de)

content/article/wang12_autom_marker_full_field_hard.md

@@ -0,0 +1,29 @@
++++
+title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
+
+Reference
+: <sup id="1bccbe15e35ed02229afbc6528c5057e"><a class="reference-link" href="#wang12_autom_marker_full_field_hard" title="Jun Wang, Yu-chen Karen Chen, Qingxi Yuan, Andrei, Tkachuk, Can Erdonmez, Benjamin Hornberger, Michael \&amp; Feser, Automated Markerless Full Field Hard X-Ray Microscopic  Tomography At Sub-50 Nm 3-dimension Spatial Resolution, {Applied Physics Letters}, v(14), 143107 (2012).">(Jun Wang {\it et al.}, 2012)</a></sup>
+
+Author(s)
+: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
+
+Year
+: 2012
+
+**Introduction of Markers**:
+That limits the type of samples that is studied
+
+There is a need for markerless nano-tomography
+=> the key requirement is the precision and stability of the positioning stages.
+
+**Passive rotational run-out error system**:
+It uses calibrated metrology disc and capacitive sensors
+
+# Bibliography
+<a class="bibtex-entry" id="wang12_autom_marker_full_field_hard">Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M., *Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution*, Applied Physics Letters, *100(14)*, 143107 (2012).  http://dx.doi.org/10.1063/1.3701579</a> [↩](#1bccbe15e35ed02229afbc6528c5057e)

content/article/wang16_inves_activ_vibrat_isolat_stewar.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
 
 Reference
-: <sup id="db95fac7cd46c14e2b4f38e8ca4158fe"><a href="#wang16_inves_activ_vibrat_isolat_stewar" title="Wang, Xie, Chen, Zhang \&amp; Zhiyi, Investigation on Active Vibration Isolation of a Stewart  Platform With Piezoelectric Actuators, {Journal of Sound and Vibration}, v(), 1-19 (2016).">(Wang {\it et al.}, 2016)</a></sup>
+: <sup id="db95fac7cd46c14e2b4f38e8ca4158fe"><a class="reference-link" href="#wang16_inves_activ_vibrat_isolat_stewar" title="Wang, Xie, Chen, Zhang \&amp; Zhiyi, Investigation on Active Vibration Isolation of a Stewart  Platform With Piezoelectric Actuators, {Journal of Sound and Vibration}, v(), 1-19 (2016).">(Wang {\it et al.}, 2016)</a></sup>
 
 Author(s)
 : Wang, C., Xie, X., Chen, Y., & Zhang, Z.
@@ -25,7 +25,7 @@ Year
 The model is compared with a Finite Element model and is shown to give the same results.
 The proposed model is thus effective.
 
-<a id="orgbc70494"></a>
+<a id="orgd3fa417"></a>
 
 {{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
 
@@ -35,11 +35,11 @@ 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](#org4b1fbd9)).
+Force Feedback (Figure [2](#org55d173d)).
 
 -   the force sensor is mounted **between the base and the strut**
 
-<a id="org4b1fbd9"></a>
+<a id="org55d173d"></a>
 
 {{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
 
@@ -54,4 +54,4 @@ Sorts of HAC-LAC control:
 -   Effectiveness of control methods are shown
 
 # Bibliography
-<a id="wang16_inves_activ_vibrat_isolat_stewar"></a>Wang, C., Xie, X., Chen, Y., & Zhang, Z., *Investigation on active vibration isolation of a stewart platform with piezoelectric actuators*, Journal of Sound and Vibration, *383()*, 1–19 (2016).  http://dx.doi.org/10.1016/j.jsv.2016.07.021 [↩](#db95fac7cd46c14e2b4f38e8ca4158fe)
+<a class="bibtex-entry" id="wang16_inves_activ_vibrat_isolat_stewar">Wang, C., Xie, X., Chen, Y., & Zhang, Z., *Investigation on active vibration isolation of a stewart platform with piezoelectric actuators*, Journal of Sound and Vibration, *383()*, 1–19 (2016).  http://dx.doi.org/10.1016/j.jsv.2016.07.021</a> [↩](#db95fac7cd46c14e2b4f38e8ca4158fe)

content/article/yang19_dynam_model_decoup_contr_flexib.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
 
 Reference
-: <sup id="d39b6222c8dd2baf188d677733c2826c"><a href="#yang19_dynam_model_decoup_contr_flexib" title="Yang, Wu, Chen, Kang, ShengZheng \&amp; Cheng, Dynamic Modeling and Decoupled Control of a Flexible  Stewart Platform for Vibration Isolation, {Journal of Sound and Vibration}, v(), 398-412 (2019).">(Yang {\it et al.}, 2019)</a></sup>
+: <sup id="d39b6222c8dd2baf188d677733c2826c"><a class="reference-link" href="#yang19_dynam_model_decoup_contr_flexib" title="Yang, Wu, Chen, Kang, ShengZheng \&amp; Cheng, Dynamic Modeling and Decoupled Control of a Flexible  Stewart Platform for Vibration Isolation, {Journal of Sound and Vibration}, v(), 398-412 (2019).">(Yang {\it et al.}, 2019)</a></sup>
 
 Author(s)
 : Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
@@ -25,23 +25,23 @@ 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](#org936d8f9)):
+**Stewart platform** (Figure [1](#org96fb07f)):
 
 -   piezoelectric actuators
--   flexible joints (Figure [2](#orgd8c916a))
+-   flexible joints (Figure [2](#org62b30be))
 -   force sensors (used for vibration isolation)
 -   displacement sensors (used to decouple the dynamics)
 -   cubic (even though not said explicitly)
 
-<a id="org936d8f9"></a>
+<a id="org96fb07f"></a>
 
 {{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
 
-<a id="orgd8c916a"></a>
+<a id="org62b30be"></a>
 
 {{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
 
-The stiffness of the flexible joints (Figure [2](#orgd8c916a)) 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](#org62b30be)) 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">
@@ -105,9 +105,9 @@ 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](#orgeb7080e).
+The block diagram of the control strategy is represented in Figure [3](#org6a06ad2).
 
-<a id="orgeb7080e"></a>
+<a id="org6a06ad2"></a>
 
 {{< figure src="/ox-hugo/yang19_control_arch.png" caption="Figure 3: Control Architecture used" >}}
 
@@ -121,10 +121,10 @@ 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](#org48c287d).
+The results are shown in Figure [4](#orgb8bd696).
 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="org48c287d"></a>
+<a id="orgb8bd696"></a>
 
 {{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
 
@@ -134,4 +134,4 @@ In theory, the vibration performance can be improved, however in practice, incre
 > The proportional and integral gains in the sub-controller are used to separately regulate the vibration isolation bandwidth and active damping simultaneously for the six vibration modes.
 
 # Bibliography
-<a id="yang19_dynam_model_decoup_contr_flexib"></a>Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S., *Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation*, Journal of Sound and Vibration, *439()*, 398–412 (2019).  http://dx.doi.org/10.1016/j.jsv.2018.10.007 [↩](#d39b6222c8dd2baf188d677733c2826c)
+<a class="bibtex-entry" id="yang19_dynam_model_decoup_contr_flexib">Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S., *Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation*, Journal of Sound and Vibration, *439()*, 398–412 (2019).  http://dx.doi.org/10.1016/j.jsv.2018.10.007</a> [↩](#d39b6222c8dd2baf188d677733c2826c)

content/article/yun20_inves_two_stage_vibrat_suppr.md

@@ -0,0 +1,21 @@
++++
+title = "Investigation on two-stage vibration suppression and precision pointing for space optical payloads"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: <sup id="44caf201a37b1b3af63de65257785085"><a class="reference-link" href="#yun20_inves_two_stage_vibrat_suppr" title="Hai Yun, Lei Liu, Qing Li \&amp; Hongjie Yang, Investigation on Two-Stage Vibration Suppression and  Precision Pointing for Space Optical Payloads, {Aerospace Science and Technology}, v(nil), 105543 (2020).">(Hai Yun {\it et al.}, 2020)</a></sup>
+
+Author(s)
+: Yun, H., Liu, L., Li, Q., & Yang, H.
+
+Year
+: 2020
+
+# Bibliography
+<a class="bibtex-entry" id="yun20_inves_two_stage_vibrat_suppr">Yun, H., Liu, L., Li, Q., & Yang, H., *Investigation on two-stage vibration suppression and precision pointing for space optical payloads*, Aerospace Science and Technology, *96(nil)*, 105543 (2020).  http://dx.doi.org/10.1016/j.ast.2019.105543</a> [↩](#44caf201a37b1b3af63de65257785085)

content/article/zhang11_six_dof.md

@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
 
 Reference
-: <sup id="a457d4de462d2fe52a1bbb848182b554"><a href="#zhang11_six_dof" title="Zhen Zhang, J Liu, Jq Mao, Yx Guo \&amp; Yh Ma, Six DOF active vibration control using stewart platform  with non-cubic configuration, nil, in in: {2011 6th IEEE Conference on Industrial Electronics and
+: <sup id="a457d4de462d2fe52a1bbb848182b554"><a class="reference-link" href="#zhang11_six_dof" title="Zhen Zhang, J Liu, Jq Mao, Yx Guo \&amp; Yh Ma, Six DOF active vibration control using stewart platform  with non-cubic configuration, nil, in in: {2011 6th IEEE Conference on Industrial Electronics and
                       Applications}, edited by (2011)">(Zhen Zhang {\it et al.}, 2011)</a></sup>
 
 Author(s)
@@ -26,9 +26,9 @@ Year
 -   **Accelerometers** for active isolation
 -   Adaptive FIR filters for active isolation control
 
-<a id="orge1b0233"></a>
+<a id="org856ccde"></a>
 
 {{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
 
 # Bibliography
-<a id="zhang11_six_dof"></a>Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y., *Six dof active vibration control using stewart platform with non-cubic configuration*, In , 2011 6th IEEE Conference on Industrial Electronics and Applications (pp. ) (2011). : . [↩](#a457d4de462d2fe52a1bbb848182b554)
+<a class="bibtex-entry" id="zhang11_six_dof">Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y., *Six dof active vibration control using stewart platform with non-cubic configuration*, In , 2011 6th IEEE Conference on Industrial Electronics and Applications (pp. ) (2011). : .</a> [↩](#a457d4de462d2fe52a1bbb848182b554)

content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md

@@ -0,0 +1,46 @@
++++
+title = "Element and system design for active and passive vibration isolation"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: <sup id="e9037e3bf20089c45ab77215406558ca"><a class="reference-link" href="#zuo04_elemen_system_desig_activ_passiv_vibrat_isolat" title="Zuo, Element and System Design for Active and Passive Vibration  Isolation (2004).">(Zuo, 2004)</a></sup>
+
+Author(s)
+: Zuo, L.
+
+Year
+: 2004
+
+> Vibration isolation systems can have various system architectures.
+> When we configure an active isolation system, we can use compliant actuators (such as voice coils) or stiff actuators (such as PZT stacks).
+> We also need to consider how to **combine the active actuation with passive elements**: we can place the actuator in parallel or in series with the passive elements.
+> Most of the isolation systems fall into the category of soft active mounts, in which a compliant actuator is placed in parallel with a spring.
+> A second category is **hard active mounts**, in which the **payload mass is directly mounted to a stiff actuator**.
+> Soft active mounts generally have advantages for better passive performance; hard active mounts are favored for payload disturbance rejection, but combination with passive stages is required due to the lack of isolation performance out of the control bandwidth.
+> Beard, von Flotow and Schubert proposed another type of hard mount, wherein **a stiff PZT actuator is placed in series with a spring** stiffer than the top passive stage.
+> 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.
+
+<a id="org0286cf1"></a>
+
+{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
+
+<a id="org679f77c"></a>
+
+{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
+
+<a id="orged24ee6"></a>
+
+{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
+
+<a id="org9b75d10"></a>
+
+{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
+
+# Bibliography
+<a class="bibtex-entry" id="zuo04_elemen_system_desig_activ_passiv_vibrat_isolat">Zuo, L., *Element and system design for active and passive vibration isolation* (2004). Massachusetts Institute of Technology.</a> [↩](#e9037e3bf20089c45ab77215406558ca)

content/book/du19_multi_actuat_system_contr.md

@@ -68,9 +68,9 @@ 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](#org53c168b)) and the microactuator.
+We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#orgb678385)) and the microactuator.
 
-<a id="org53c168b"></a>
+<a id="orgb678385"></a>
 
 {{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
 
@@ -92,9 +92,9 @@ 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](#org5941a76), 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](#orgcf5d697), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
 
-<a id="org5941a76"></a>
+<a id="orgcf5d697"></a>
 
 {{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
 
@@ -104,7 +104,7 @@ A typical closed-loop control system is shown on figure [2](#org5941a76), where
 Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
 The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
 
-<a id="org8f04825"></a>
+<a id="orga011b51"></a>
 
 {{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
 
@@ -130,7 +130,7 @@ 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](#org03d53e8).
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org7def875).
 
 -   \\(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\\).
@@ -138,7 +138,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
 -   \\(n\\) is the measurement noise
 -   \\(d\_u\\) stands for external vibration
 
-<a id="org03d53e8"></a>
+<a id="org7def875"></a>
 
 {{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
 
@@ -160,14 +160,14 @@ 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](#org37116a9).
+Another type of control scheme is the **parallel structure** as shown in figure [5](#orgbb3c494).
 The open-loop transfer function from \\(pes\\) to \\(y\\) is
 \\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
 
 The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
 \\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
 
-<a id="org37116a9"></a>
+<a id="orgbb3c494"></a>
 
 {{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
 
@@ -177,7 +177,7 @@ 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](#org299b914) 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](#orge3f8703) 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
 
@@ -187,11 +187,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
 
 is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
 
-<a id="org299b914"></a>
+<a id="orge3f8703"></a>
 
 {{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
 
-Equation [1](#org361ec91) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+Equation [1](#orgc402b0c) 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}
@@ -204,16 +204,16 @@ 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](#org60ad057), 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](#org402df06), 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="org60ad057"></a>
+<a id="org402df06"></a>
 
 {{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: 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](#org1d6afb9), 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](#orge904ce1), 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="org1d6afb9"></a>
+<a id="orge904ce1"></a>
 
 {{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
 
@@ -246,13 +246,13 @@ 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](#org0e50764) shows the control structure for the three-stage actuation system.
+Figure [9](#org8c31dd5) 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.
 The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
 
-<a id="org0e50764"></a>
+<a id="org8c31dd5"></a>
 
 {{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
 
@@ -281,15 +281,15 @@ 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](#orgefe88f9).
+The open-loop frequency responses of the three stages are shown on figure [10](#orgd95bc97).
 
-<a id="orgefe88f9"></a>
+<a id="orgd95bc97"></a>
 
 {{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
 
-The obtained sensitivity function is shown on figure [11](#orgd0c25f8).
+The obtained sensitivity function is shown on figure [11](#org50990f8).
 
-<a id="orgd0c25f8"></a>
+<a id="org50990f8"></a>
 
 {{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
 
@@ -304,7 +304,7 @@ 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](#org0e50764), the position error is
+For the three-stage control architecture as shown on figure [9](#org8c31dd5), 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
 
@@ -320,11 +320,11 @@ 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](#org5bb499d)).
+A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org7ec3564)).
 
 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](#orga34ddfe) and
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org6bf8240) and
 \\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
 
 Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -333,7 +333,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
 The open-loop transfer function of the overall system is
 \\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
 
-<a id="org5bb499d"></a>
+<a id="org7ec3564"></a>
 
 {{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
 
@@ -345,9 +345,9 @@ 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](#org0a46272)).
+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](#org56aeb13)).
 
-<a id="org0a46272"></a>
+<a id="org56aeb13"></a>
 
 {{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
 

content/book/ewins00_modal.md

@@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
 
 The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
 
-<a id="orga6d0e6f"></a>
+<a id="org76193b4"></a>
 
 {{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
 
@@ -199,7 +199,7 @@ This process itself falls into two stages:
 Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis".
 
 We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system.
-The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation [eq:frf_modal](#eq:frf_modal) to the measured FRF and thereby finding the appropriate modal parameters.
+The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation \eqref{eq:frf_modal} to the measured FRF and thereby finding the appropriate modal parameters.
 
 A completely general curve-fitting approach is possible but generally inefficient.
 Mathematically, we can take an equation of the form
@@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
 This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
 At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
 
-<a id="orgdab4a6e"></a>
+<a id="org128748c"></a>
 
 {{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
 
@@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
 The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
 Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
 
-<a id="orgda09d3d"></a>
+<a id="org454ea68"></a>
 
 {{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
 
@@ -298,7 +298,7 @@ Three classes of system model will be described:
 The basic model for the SDOF system is shown in figure [fig:sdof_model](#fig:sdof_model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
 The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
 
-<a id="orgbe9f040"></a>
+<a id="org640feed"></a>
 
 {{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
 
@@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
 
 The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
 
-<a id="org1f289fa"></a>
+<a id="orga99ae3e"></a>
 
 {{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
 
@@ -418,7 +418,7 @@ The damping effect of such a component can conveniently be defined by the ratio
 
 | ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
 |-----------------------------------------------|----------------------------------------|------------------------------------------|
-| <a id="orgd01ea8f"></a> Material hysteresis   | <a id="org5fb7a29"></a> Dry friction   | <a id="org41cf290"></a> Viscous damper   |
+| <a id="org54caaf8"></a> Material hysteresis   | <a id="org0fc2b44"></a> Dry friction   | <a id="org0985c72"></a> Viscous damper   |
 | height=2cm                                    | height=2cm                             | height=2cm                               |
 
 Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
@@ -458,11 +458,11 @@ where \\(\eta\\) is the **structural damping loss factor** and replaces the crit
 
 #### Alternative Forms of FRF {#alternative-forms-of-frf}
 
-So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force [eq:receptance](#eq:receptance).
+So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force \eqref{eq:receptance}.
 This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\).
 
-We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function [eq:mobility](#eq:mobility).
+We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function \eqref{eq:mobility}.
-Similarly we could use the acceleration parameter so we could define a third FRF parameter [eq:inertance](#eq:inertance).
+Similarly we could use the acceleration parameter so we could define a third FRF parameter \eqref{eq:inertance}.
 
 <div class="cbox">
   <div></div>
@@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
 
 | ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
 |-------------------------------------------|-----------------------------------------|--------------------------------------------|
-| <a id="org1478c78"></a> Receptance FRF    | <a id="orgd7eb060"></a> Mobility FRF    | <a id="orgf95b5b3"></a> Accelerance FRF    |
+| <a id="orgea747d3"></a> Receptance FRF    | <a id="orgc5e3717"></a> Mobility FRF    | <a id="orgcf610b2"></a> Accelerance FRF    |
 | width=\linewidth                          | width=\linewidth                        | width=\linewidth                           |
 
 Each plot can be divided into three regimes:
@@ -560,13 +560,13 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
 
 | ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
 |------------------------------------------------|------------------------------------------------|
-| <a id="org35a1358"></a> Real part              | <a id="org69673f5"></a> Imaginary part         |
+| <a id="org695538e"></a> Real part              | <a id="org95c5960"></a> Imaginary part         |
 | width=\linewidth                               | width=\linewidth                               |
 
 
 ##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
 
-It can be seen from the expression of the inverse receptance [eq:dynamic_stiffness](#eq:dynamic_stiffness) that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
+It can be seen from the expression of the inverse receptance \eqref{eq:dynamic_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
 
 Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
 
@@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
 
 | ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
 |---------------------------------------------|-----------------------------------------------|
-| <a id="org8ba100f"></a> Mixed               | <a id="org9bed1e0"></a> Viscous               |
+| <a id="org9e0909f"></a> Mixed               | <a id="orge2690df"></a> Viscous               |
 | width=\linewidth                            | width=\linewidth                              |
 
 
@@ -595,7 +595,7 @@ The missing information (in this case, the frequency) must be added by identifyi
 
 | ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
 |------------------------------------------------------|---------------------------------------------------------|
-| <a id="org5cbe7df"></a> Viscous damping              | <a id="orgb5e61e4"></a> Structural damping              |
+| <a id="org86b8a60"></a> Viscous damping              | <a id="orgb0d3b09"></a> Structural damping              |
 | width=\linewidth                                     | width=\linewidth                                        |
 
 The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@@ -607,7 +607,7 @@ This makes the Nyquist plot very effective for modal testing applications.
 
 #### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties}
 
-For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form [eq:undamped_mdof](#eq:undamped_mdof).
+For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form \eqref{eq:undamped_mdof}.
 
 <div class="cbox">
   <div></div>
@@ -622,7 +622,7 @@ where \\([M]\\) and \\([K]\\) are \\(N\times N\\) mass and stiffness matrices, a
 
 We shall consider first the free vibration solution by taking \\(f(t) = 0\\).
 In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes.
-Substitution of this condition into [eq:undamped_mdof](#eq:undamped_mdof) leads to
+Substitution of this condition into \eqref{eq:undamped_mdof} leads to
 
 \begin{equation}
   \left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\}
@@ -632,7 +632,7 @@ for which the non trivial solutions are those which satisfy
 \\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\]
 from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**.
 
-Substituting any of these back into [eq:free_eom_mdof](#eq:free_eom_mdof) yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
+Substituting any of these back into \eqref{eq:free_eom_mdof} yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
 
 <div class="cbox">
   <div></div>
@@ -774,7 +774,7 @@ An alternative means of deriving the FRF parameters is used which makes use of t
 \\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\]
 Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain
 \\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\]
-which leads to [eq:receptance_modal](#eq:receptance_modal).
+which leads to \eqref{eq:receptance_modal}.
 
 <div class="cbox">
   <div></div>
@@ -783,7 +783,7 @@ which leads to [eq:receptance_modal](#eq:receptance_modal).
   [\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal}
 \end{equation}
 
-Equation [eq:receptance_modal](#eq:receptance_modal) permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
+Equation \eqref{eq:receptance_modal} permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
 
 \begin{subequations}
   \begin{align}
@@ -800,7 +800,7 @@ where \\({}\_rA\_{jk}\\) is called the **modal constant**.
 <div class="cbox">
   <div></div>
 
-It is clear from equation [eq:receptance_modal](#eq:receptance_modal) that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
+It is clear from equation \eqref{eq:receptance_modal} that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
 
 This principle of reciprocity applies to many structural characteristics.
 
@@ -938,7 +938,7 @@ From this full matrix equation, we have:
 
 Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**.
 
-The general behavior for this case is governed by equation [eq:force_response_eom](#eq:force_response_eom) with solution [eq:force_response_eom_solution](#eq:force_response_eom_solution).
+The general behavior for this case is governed by equation \eqref{eq:force_response_eom} with solution \eqref{eq:force_response_eom_solution}.
 However, a more explicit form of the solution is
 
 \begin{equation}
@@ -962,7 +962,7 @@ The properties of the normal modes of the undamped system are of interest becaus
 
 </div>
 
-We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in [eq:ods](#eq:ods) but one is zero.
+We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in \eqref{eq:ods} but one is zero.
 This can be attained if \\(\\{F\\}\\) is chosen such that
 \\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\]
 
@@ -1046,7 +1046,7 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
 When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
 \\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
 
-From equations [eq:viscous_damping_orthogonality](#eq:viscous_damping_orthogonality), we can obtain
+From equations \eqref{eq:viscous_damping_orthogonality}, we can obtain
 
 \begin{subequations}
   \begin{align}
@@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
 
 While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
 
-<a id="org71662dc"></a>
+<a id="org76fb154"></a>
 
 {{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
 
@@ -1118,7 +1118,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
 
 | ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png)    |
 |--------------------------------------------|--------------------------------------------|-----------------------------------------------|
-| <a id="orgdb142d3"></a> Almost-real mode   | <a id="org40e30d6"></a> Complex Mode       | <a id="org4c29a71"></a> Measure of complexity |
+| <a id="orgd9e3564"></a> Almost-real mode   | <a id="orgeedeefa"></a> Complex Mode       | <a id="org2d21384"></a> Measure of complexity |
 | width=\linewidth                           | width=\linewidth                           | width=\linewidth                              |
 
 
@@ -1235,7 +1235,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
 
 | ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
 |---------------------------------------------------|------------------------------------------------------|
-| <a id="org0d0c340"></a> Point FRF                 | <a id="org13ad8cd"></a> Transfer FRF                 |
+| <a id="org464f787"></a> Point FRF                 | <a id="orgd21bcd3"></a> Transfer FRF                 |
 | width=\linewidth                                  | width=\linewidth                                     |
 
 For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
@@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
 
 This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
 
-<a id="orgb88a5bd"></a>
+<a id="orgd6edca6"></a>
 
 {{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
 
@@ -1281,7 +1281,7 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
 
 | ![](/ox-hugo/ewins00_nyquist_point.png)  | ![](/ox-hugo/ewins00_nyquist_transfer.png)  |
 |------------------------------------------|---------------------------------------------|
-| <a id="org7bbfae7"></a> Point receptance | <a id="org0f31112"></a> Transfer receptance |
+| <a id="org5dbb609"></a> Point receptance | <a id="orgf225939"></a> Transfer receptance |
 | width=\linewidth                         | width=\linewidth                            |
 
 In the two figures [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
@@ -1296,7 +1296,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
 
 | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
 |-----------------------------------------------------|--------------------------------------------------------|
-| <a id="org460aa35"></a> Point receptance            | <a id="org01c8e2c"></a> Transfer receptance            |
+| <a id="orgae9806e"></a> Point receptance            | <a id="orgb532a2f"></a> Transfer receptance            |
 | width=\linewidth                                    | width=\linewidth                                       |
 
 
@@ -1343,7 +1343,7 @@ One these two series are available, the FRF can be defined at the same set of fr
 
 ##### Analysis via Fourier transform {#analysis-via-fourier-transform}
 
-For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from [eq:fourier_transform](#eq:fourier_transform).
+For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from \eqref{eq:fourier_transform}.
 
 \begin{equation}
   F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt
@@ -1450,7 +1450,7 @@ Examples of random signals, autocorrelation function and power spectral density
 
 | ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png)           |
 |---------------------------------------|--------------------------------------------------|------------------------------------------------|
-| <a id="org3a8665e"></a> Time history  | <a id="org5ade2bf"></a> Autocorrelation Function | <a id="org9f69a06"></a> Power Spectral Density |
+| <a id="org30bff26"></a> Time history  | <a id="org7e07ced"></a> Autocorrelation Function | <a id="orgcb31329"></a> Power Spectral Density |
 | width=\linewidth                      | width=\linewidth                                 | width=\linewidth                               |
 
 A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@@ -1493,10 +1493,10 @@ However, the same equation can be transform to the frequency domain
   \tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) }
 \end{equation}
 
-Although very convenient, equation [eq:psd_input_output](#eq:psd_input_output) does not provide a complete description of the random vibration conditions.
+Although very convenient, equation \eqref{eq:psd_input_output} does not provide a complete description of the random vibration conditions.
 Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula.
 
-A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained [eq:cross_relation_alternatives](#eq:cross_relation_alternatives).
+A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained \eqref{eq:cross_relation_alternatives}.
 
 <div class="cbox">
   <div></div>
@@ -1513,8 +1513,8 @@ A second equation is required and this may be obtain by a similar analysis, two
 
 ##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals}
 
-The pair of equations [eq:cross_relation_alternatives](#eq:cross_relation_alternatives) provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
+The pair of equations \eqref{eq:cross_relation_alternatives} provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
-Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities [eq:frf_estimates_spectral_densities](#eq:frf_estimates_spectral_densities).
+Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities \eqref{eq:frf_estimates_spectral_densities}.
 
 <div class="cbox">
   <div></div>
@@ -1547,11 +1547,11 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
 
 | ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png)     |
 |------------------------------------------|--------------------------------------------------|
-| <a id="org5183bee"></a> Basic SISO model | <a id="org7eda16f"></a> SISO model with feedback |
+| <a id="orgcf49de0"></a> Basic SISO model | <a id="orgad8dce0"></a> SISO model with feedback |
 | width=\linewidth                         | width=\linewidth                                 |
 
 In this configuration, it can be seen that there are two feedback mechanisms which apply.
-We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities [eq:H3](#eq:H3).
+We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities \eqref{eq:H3}.
 
 <div class="cbox">
   <div></div>
@@ -1580,7 +1580,7 @@ We obtain two alternative formulas:
 
 In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
 
-<a id="org1066c50"></a>
+<a id="org2388f52"></a>
 
 {{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
 
@@ -1694,7 +1694,7 @@ First, if we have a **modal incompleteness** (\\(m<N\\) modes included), then we
 
 However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices.
 
-In both reduced-model cases, it is not possible to use equation [eq:spatial_model_from_modal](#eq:spatial_model_from_modal) to re-construct the system mass and stiffness matrices.
+In both reduced-model cases, it is not possible to use equation \eqref{eq:spatial_model_from_modal} to re-construct the system mass and stiffness matrices.
 First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used.
 
 
@@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
 
 A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
 
-<a id="org257621c"></a>
+<a id="org1415164"></a>
 
 {{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
 
@@ -1909,7 +1909,7 @@ This can modify the response of the system in those directions.
 In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [fig:shaker_rod](#fig:shaker_rod).
 Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
 
-<a id="orga7d69b0"></a>
+<a id="orgbf524e6"></a>
 
 {{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
 
@@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
 
 | ![](/ox-hugo/ewins00_shaker_mount_1.png)    | ![](/ox-hugo/ewins00_shaker_mount_2.png)        | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
 |---------------------------------------------|-------------------------------------------------|------------------------------------------|
-| <a id="orge0af414"></a> Ideal Configuration | <a id="org21ea9d5"></a> Suspended Configuration | <a id="org272e483"></a> Unsatisfactory   |
+| <a id="orga9157bf"></a> Ideal Configuration | <a id="org4b90d28"></a> Suspended Configuration | <a id="org3061b55"></a> Unsatisfactory   |
 | width=\linewidth                            | width=\linewidth                                | width=\linewidth                         |
 
 
@@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
 When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
 A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
 
-<a id="orgfb698ca"></a>
+<a id="orgdb53d89"></a>
 
 {{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
 
@@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
 The force transducer is the simplest type of piezoelectric transducer.
 The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
 
-<a id="org6b146da"></a>
+<a id="org93aad2e"></a>
 
 {{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
 
@@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
 In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
 Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
 
-<a id="org29aa78f"></a>
+<a id="org84766b5"></a>
 
 {{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
 
@@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
 
 | ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png)     |
 |-----------------------------------------------------|------------------------------------------------------------|
-| <a id="org1a66b31"></a> Attachment methods          | <a id="org4e23863"></a> Frequency response characteristics |
+| <a id="org796e903"></a> Attachment methods          | <a id="org308a233"></a> Frequency response characteristics |
 | width=\linewidth                                    | width=\linewidth                                           |
 
 
@@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
 With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
 These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [fig:aliasing](#fig:aliasing).
 
-<a id="orgb59b07e"></a>
+<a id="orge489af5"></a>
 
 {{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
 
@@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
 
 | ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png)       |
 |--------------------------------------------------|-----------------------------------------------------|
-| <a id="org7e6ecc4"></a> True spectrum of signal  | <a id="org3c375b8"></a> Indicated spectrum from DFT |
+| <a id="org3c7851f"></a> True spectrum of signal  | <a id="orgd31d06c"></a> Indicated spectrum from DFT |
 | width=\linewidth                                 | width=\linewidth                                    |
 
 The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@@ -2165,7 +2165,7 @@ Leakage is a problem which is a direct **consequence of the need to take only a
 
 | ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png)  |
 |--------------------------------------|----------------------------------------|
-| <a id="org3c7efb4"></a> Ideal signal | <a id="orga61ea56"></a> Awkward signal |
+| <a id="org62f211a"></a> Ideal signal | <a id="orgd4e0fe1"></a> Awkward signal |
 | width=\linewidth                     | width=\linewidth                       |
 
 The problem is illustrated on figure [fig:leakage](#fig:leakage).
@@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
 
 The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
 
-<a id="org11d5f31"></a>
+<a id="orge28ad03"></a>
 
 {{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
 
@@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
 
 #### Improving Resolution {#improving-resolution}
 
-<a id="orgf44c4ab"></a>
+<a id="org81b4f25"></a>
 
 
 ##### Increasing transform size {#increasing-transform-size}
@@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
 
 | ![](/ox-hugo/ewins00_zoom_range.png)           | ![](/ox-hugo/ewins00_zoom_bandpass.png)  |
 |------------------------------------------------|------------------------------------------|
-| <a id="orgafd809b"></a> Spectrum of the signal | <a id="org7d44157"></a> Band-pass filter |
+| <a id="org28ed6ec"></a> Spectrum of the signal | <a id="org8a7e75c"></a> Band-pass filter |
 | width=\linewidth                               | width=\linewidth                         |
 
-<a id="org8e7e54a"></a>
+<a id="org60b3e9b"></a>
 
 {{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
 
@@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
 It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
 If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [fig:sweep_distortions](#fig:sweep_distortions).
 
-<a id="orgb233cbb"></a>
+<a id="orgeab1f57"></a>
 
 {{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
 
@@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
 
 This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [fig:coherence_resonance](#fig:coherence_resonance).
 
-<a id="org4d83015"></a>
+<a id="orgb72faa8"></a>
 
 {{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
 
@@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
 
 Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [fig:burst_excitation](#fig:burst_excitation)).
 
-<a id="org2152665"></a>
+<a id="org681a980"></a>
 
 {{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
 
@@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
 
 The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
 
-<a id="org56036c2"></a>
+<a id="org632f8cc"></a>
 
 {{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
 
@@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
 
 This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
 
-<a id="orga792905"></a>
+<a id="orgdecf769"></a>
 
 {{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
 
@@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
 On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
 In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [fig:double_hits](#fig:double_hits)).
 
-<a id="org43e765b"></a>
+<a id="orgea279f8"></a>
 
 {{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
 
@@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
 
 Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
 
-<a id="org1176320"></a>
+<a id="org3a6c052"></a>
 
 {{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
 
@@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
 
 This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
 
-<a id="org25ad6ad"></a>
+<a id="orgf6011aa"></a>
 
 {{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
 
@@ -2670,7 +2670,7 @@ There are two problems to be tackled:
 
 The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [fig:rotational_measurement](#fig:rotational_measurement).
 
-<a id="org92fa8a1"></a>
+<a id="org6c1a993"></a>
 
 {{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
 
@@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
 Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
 By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
 
-<a id="org97cd93f"></a>
+<a id="org19d9418"></a>
 
 {{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
 
@@ -2700,7 +2700,7 @@ Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this pr
 Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers.
 
 However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions.
-For example, the magnitude of the difference in equation [eq:rotational_diff](#eq:rotational_diff) is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
+For example, the magnitude of the difference in equation \eqref{eq:rotational_diff} is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
 
 
 ### Multi-point excitation methods {#multi-point-excitation-methods}
@@ -2739,7 +2739,7 @@ The two vectors are related by the system's FRF properties as:
   \\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1}
 \end{equation}
 
-However, it is not possible to derive the FRF matrix from the single equation [eq:mpss_equation](#eq:mpss_equation), because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
+However, it is not possible to derive the FRF matrix from the single equation \eqref{eq:mpss_equation}, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
 
 What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular.
 This can be assured if:
@@ -3031,7 +3031,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
 
 | ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
 |---------------------------------------------|---------------------------------------------|---------------------------------------------|
-| <a id="org3c82345"></a> FRF                 | <a id="orga931e29"></a> Singular Values     | <a id="orgf4a6ae7"></a> PRF                 |
+| <a id="org911bfc8"></a> FRF                 | <a id="org60f84fb"></a> Singular Values     | <a id="orgdf8522b"></a> PRF                 |
 | width=\linewidth                            | width=\linewidth                            | width=\linewidth                            |
 
 <a id="table--fig:PRF-measured"></a>
@@ -3042,7 +3042,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
 
 | ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
 |--------------------------------------------|--------------------------------------------|--------------------------------------------|
-| <a id="orge590e19"></a> FRF                | <a id="org176a55f"></a> Singular Values    | <a id="orge859b81"></a> PRF                |
+| <a id="org3d1c696"></a> FRF                | <a id="orgeb81dac"></a> Singular Values    | <a id="orgc25aeb3"></a> PRF                |
 | width=\linewidth                           | width=\linewidth                           | width=\linewidth                           |
 
 
@@ -3084,7 +3084,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
 -   the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
 -   the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
 
-<a id="org52ec446"></a>
+<a id="org5f7cb1f"></a>
 
 {{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
 
@@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
 It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
 Only real modal constants and thus real modes can be deduced by this method.
 
-<a id="org12489c1"></a>
+<a id="org0d4b46a"></a>
 
 {{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
 
@@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
 
 | ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png)                         |
 |----------------------------------------|--------------------------------------------------------------------|
-| <a id="orge6f933c"></a> Properties     | <a id="org852e910"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
+| <a id="org187efdc"></a> Properties     | <a id="org0e24b72"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
 | width=\linewidth                       | width=\linewidth                                                   |
 
 For any frequency \\(\omega\\), we have the following relationship:
@@ -3226,13 +3226,13 @@ For any frequency \\(\omega\\), we have the following relationship:
   \end{align}
 \end{subequations}
 
-From [eq:modal_circle_tan](#eq:modal_circle_tan), we obtain:
+From \eqref{eq:modal_circle_tan}, we obtain:
 
 \begin{equation}
   \omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right)
 \end{equation}
 
-If we differentiate [eq:modal_circle_omega](#eq:modal_circle_omega) with respect to \\(\theta\\), we obtain:
+If we differentiate \eqref{eq:modal_circle_omega} with respect to \\(\theta\\), we obtain:
 
 \begin{equation}
   \frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2}
@@ -3317,10 +3317,10 @@ The sequence is:
     Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle.
 3.  **Locate natural frequency, obtain damping estimate**.
     The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
-    At the same time, an estimate of the damping is derived using [eq:estimate_damping_sweep_rate](#eq:estimate_damping_sweep_rate).
+    At the same time, an estimate of the damping is derived using \eqref{eq:estimate_damping_sweep_rate}.
     A typical example is shown on figure [fig:circle_fit_natural_frequency](#fig:circle_fit_natural_frequency).
 4.  **Calculate multiple damping estimates, and scatter**.
-    A set of damping estimates using all possible combination of the selected data points are computed using [eq:estimate_damping](#eq:estimate_damping).
+    A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
     Then, we can choose the damping estimate to be the mean value.
     We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis:
     -   Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data.
@@ -3328,7 +3328,7 @@ The sequence is:
 5.  **Determine modal constant modulus and argument**.
     The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
 
-<a id="orgc0336ab"></a>
+<a id="org379e1a2"></a>
 
 {{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
 
@@ -3427,7 +3427,7 @@ we now have sufficient information to extract estimates for the four parameters
 <ol class="org-ol">
 <li value="3">Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\)</li>
 <li>Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\)</li>
-<li>Use these four quantities, and equation [eq:modal_parameters_formula](#eq:modal_parameters_formula), to determine the **four modal parameters** required for that mode</li>
+<li>Use these four quantities, and equation \eqref{eq:modal_parameters_formula}, to determine the **four modal parameters** required for that mode</li>
 </ol>
 
 This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region.
@@ -3453,7 +3453,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
 
 | ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
 |--------------------------------------------|-----------------------------------------|
-| <a id="org4fd3d88"></a> without residual   | <a id="orgdb69a63"></a> with residuals  |
+| <a id="orge96a388"></a> without residual   | <a id="org92a8b32"></a> with residuals  |
 | width=\linewidth                           | width=\linewidth                        |
 
 If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@@ -3484,7 +3484,7 @@ The three terms corresponds to:
 
 These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
 
-<a id="org85845f3"></a>
+<a id="org745f0a4"></a>
 
 {{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
 
@@ -3493,7 +3493,7 @@ From the sketch, it may be seen that within the frequency range of interest:
 -   the first term tends to approximate to a **mass-like behavior**
 -   the third term approximates to a **stiffness effect**
 
-Thus, we have a basis for the residual terms and shall rewrite equation [eq:sum_modes](#eq:sum_modes):
+Thus, we have a basis for the residual terms and shall rewrite equation \eqref{eq:sum_modes}:
 
 \begin{equation}
   H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R}
@@ -3554,7 +3554,7 @@ We can write the receptance in the frequency range of interest as:
 
 In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\).
 However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis.
-Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in [eq:second_term_refinement](#eq:second_term_refinement), we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
+Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in \eqref{eq:second_term_refinement}, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
 
 This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**.
 
@@ -3614,7 +3614,7 @@ If we further increase the generality by attaching a **weighting factor** \\(w\_
 
 is minimized.
 
-This is achieved by differentiating [eq:error_weighted](#eq:error_weighted) with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
+This is achieved by differentiating \eqref{eq:error_weighted} with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
 
 \begin{equation}
   \frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots
@@ -3663,7 +3663,7 @@ leading to the modified, but more convenient version actually used in the analys
 \end{equation}
 
 In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis.
-Equation [eq:rpf_error](#eq:rpf_error) can be rewritten as follows:
+Equation \eqref{eq:rpf_error} can be rewritten as follows:
 
 \begin{equation}
   \begin{aligned}
@@ -3717,7 +3717,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
 \end{equation}
 
 Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
-This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations [eq:frf_clasic](#eq:frf_clasic) and [eq:frf_rational](#eq:frf_rational):
+This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf_clasic} and \eqref{eq:frf_rational}:
 
 -   the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
 -   the numerator is used to determine the complex modal constants \\(A\_r\\)
@@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
 
 | ![](/ox-hugo/ewins00_composite_raw.png)   | ![](/ox-hugo/ewins00_composite_sum.png) |
 |-------------------------------------------|-----------------------------------------|
-| <a id="org9ce04ed"></a> Individual curves | <a id="orgf0fa0fe"></a> Composite curve |
+| <a id="org3f9a0d6"></a> Individual curves | <a id="org9ebc973"></a> Composite curve |
 | width=\linewidth                          | width=\linewidth                        |
 
 The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@@ -3949,7 +3949,7 @@ First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possib
 
 Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data.
 In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position.
-If we measure \\(H\_{kk}\\), then by using [eq:modal_model_from_frf](#eq:modal_model_from_frf), we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
+If we measure \\(H\_{kk}\\), then by using \eqref{eq:modal_model_from_frf}, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
 
 \begin{equation}
   H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m
@@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
 Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [fig:static_display](#fig:static_display) (b).
 The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
 
-<a id="orgbbe2809"></a>
+<a id="orge0d2fb3"></a>
 
 {{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
 
@@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
 All the higher modes will be indistinguishable from these first few.
 This is a well known problem of **spatial aliasing**.
 
-<a id="org10e3c8a"></a>
+<a id="org5c16ec7"></a>
 
 {{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
 
@@ -4415,7 +4415,7 @@ In this respect, the demands of the response model are more stringent that those
 
 #### Synthesis of FRF curves {#synthesis-of-frf-curves}
 
-One of the implications of equation [eq:regenerate_full_frf_matrix](#eq:regenerate_full_frf_matrix) is that **it is possible to synthesize the FRF curves which were not measured**.
+One of the implications of equation \eqref{eq:regenerate_full_frf_matrix} is that **it is possible to synthesize the FRF curves which were not measured**.
 This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc.
 
 However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below.
@@ -4440,7 +4440,7 @@ The inclusion of these two additional terms (obtained here only after measuring
 
 | ![](/ox-hugo/ewins00_H22_without_residual.png)         | ![](/ox-hugo/ewins00_H22_with_residual.png)               |
 |--------------------------------------------------------|-----------------------------------------------------------|
-| <a id="org0111dfe"></a> Using measured modal data only | <a id="org2abbdff"></a> After inclusion of residual terms |
+| <a id="orgee3fc43"></a> Using measured modal data only | <a id="org959e2d5"></a> After inclusion of residual terms |
 | width=\linewidth                                       | width=\linewidth                                          |
 
 The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
 In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [fig:transmissibility_plots](#fig:transmissibility_plots).
 This may explain why transmissibilities are not widely used in modal analysis.
 
-<a id="org6f53493"></a>
+<a id="orgb69dd65"></a>
 
 {{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
 
@@ -4516,7 +4516,7 @@ The fact that the excitation force is not measured is responsible for the lack o
 
 | ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
 |---------------------------------------------------------|-------------------------------------------------------|
-| <a id="orgfcd5181"></a> Conventional modal test setup   | <a id="org544b696"></a> Base excitation setup         |
+| <a id="orgfb8d62b"></a> Conventional modal test setup   | <a id="orgb803ff7"></a> Base excitation setup         |
 | height=4cm                                              | height=4cm                                            |
 
 
@@ -4541,7 +4541,7 @@ from which is would appear that we can write
   \end{aligned}
 \end{equation}
 
-However, equation [eq:m_k_from_modes](#eq:m_k_from_modes) is **only applicable when we have available the complete \\(N \times N\\) modal model**.
+However, equation \eqref{eq:m_k_from_modes} is **only applicable when we have available the complete \\(N \times N\\) modal model**.
 
 It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible.
 One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices.

content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md

@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
 
 #### Feedback {#feedback}
 
-<a id="orgcd0067e"></a>
+<a id="orgff2b033"></a>
 
 {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
 
-The principle of feedback is represented on figure [fig:classical_feedback_small](#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](#orgff2b033). 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**.
@@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
 
 #### Feedforward {#feedforward}
 
-<a id="orgf21f883"></a>
+<a id="orgc74bab2"></a>
 
 {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: 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](#orgf21f883).
+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](#orgc74bab2).
 
 The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
 
@@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
 
 ### The Various Steps of the Design {#the-various-steps-of-the-design}
 
-<a id="orgca19f4b"></a>
+<a id="org4c2b243"></a>
 
 {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
 
-The various steps of the design of a controlled structure are shown in figure [3](#orgca19f4b).
+The various steps of the design of a controlled structure are shown in figure [3](#org4c2b243).
 
 The **starting point** is:
 
@@ -154,14 +154,14 @@ 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 [fig:general_plant](#fig:general_plant)):
+From the block diagram of the control system (figure [4](#org1c8100c)):
 
 \begin{align\*}
 y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
 z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
 \end{align\*}
 
-<a id="org06b8843"></a>
+<a id="org1c8100c"></a>
 
 {{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
 
@@ -186,12 +186,12 @@ 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 [fig:cas_plot](#fig:cas_plot).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgbde617f).
 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.
 
-<a id="orgefc00fd"></a>
+<a id="orgbde617f"></a>
 
 {{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
 
@@ -398,11 +398,11 @@ With:
   D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
 \end{equation}
 
-<a id="orgde5a280"></a>
+<a id="orgac9e4c8"></a>
 
 {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: 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 [fig:neglected_modes](#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](#orgac9e4c8)).
 
 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 \\]
@@ -441,9 +441,9 @@ 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 [fig:collocated_control_frf](#fig:collocated_control_frf)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org6374ca8)).
 
-<a id="orgc840265"></a>
+<a id="org6374ca8"></a>
 
 {{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
 
@@ -457,9 +457,9 @@ 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 [fig:collocated_zero](#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](#org55d54ab).
 
-<a id="orgd860ef7"></a>
+<a id="org55d54ab"></a>
 
 {{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
 
@@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
 
 </div>
 
-By looking at figure [fig:collocated_control_frf](#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](#org6374ca8), 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="org245f75f"></a>
+<a id="org8cc5426"></a>
 
 {{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
 
@@ -486,7 +486,7 @@ 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](#org245f75f). 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](#org8cc5426). 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.
 
 
@@ -508,12 +508,12 @@ 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 [fig:voice_coil_schematic](#fig:voice_coil_schematic)):
+The system consists of (see figure [10](#orga882e0c)):
 
 -   A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
 -   A coil which is free to move axially
 
-<a id="org605d681"></a>
+<a id="orga882e0c"></a>
 
 {{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
 
@@ -551,9 +551,9 @@ 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 [fig:proof_mass_actuator](#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](#orgd7b8271)).
 
-<a id="org5fa45df"></a>
+<a id="orgd7b8271"></a>
 
 {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
 
@@ -583,9 +583,9 @@ 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 [fig:proof_mass_tf](#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](#orgc8992eb)).
 
-<a id="org0b93a83"></a>
+<a id="orgc8992eb"></a>
 
 {{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
 
@@ -610,7 +610,7 @@ By using the two equations, we obtain:
 
 Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
 
-<a id="org45de7ce"></a>
+<a id="org2a6e175"></a>
 
 {{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
 
@@ -619,9 +619,9 @@ 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 [fig:electro_mechanical_transducer](#fig:electro_mechanical_transducer).
+The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org12ba88f).
 
-<a id="orgdc255dc"></a>
+<a id="org12ba88f"></a>
 
 {{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
 
@@ -646,7 +646,7 @@ With:
 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 [fig:bridge_circuit](#fig:bridge_circuit) can be used.
+To do so, the bridge circuit as shown on figure [15](#orgdf63f4a) can be used.
 
 We can show that
 
@@ -656,7 +656,7 @@ We can show that
 
 which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
 
-<a id="orgab771fe"></a>
+<a id="orgdf63f4a"></a>
 
 {{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
 
@@ -664,9 +664,9 @@ 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 [fig:smart_materials](#fig:smart_materials) lists various effects that are observed in materials in response to various inputs.
+Figure [16](#orgc0d19b7) lists various effects that are observed in materials in response to various inputs.
 
-<a id="orga77d2f2"></a>
+<a id="orgc0d19b7"></a>
 
 {{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
 
@@ -761,7 +761,7 @@ 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 [fig:piezo_stack](#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:
+If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgc13be77)) 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}
@@ -782,7 +782,7 @@ where
 -   \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
 -   \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
 
-<a id="orge0cb54d"></a>
+<a id="orgc13be77"></a>
 
 {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
 
@@ -802,7 +802,7 @@ Equation \eqref{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 [fig:piezo_discrete](#fig:piezo_discrete).
+Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org24eee83).
 
 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
 
@@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
   dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
 \end{equation}
 
-<a id="orge625816"></a>
+<a id="org24eee83"></a>
 
 {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
 
@@ -844,10 +844,10 @@ 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 [fig:piezo_stack_admittance](#fig:piezo_stack_admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
+Consider the system of figure [19](#org76e4915), 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="org9c326d7"></a>
+<a id="org76e4915"></a>
 
 {{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
 
@@ -866,9 +866,9 @@ And one can see that
   \frac{z^2 - p^2}{z^2} = k^2
 \end{equation}
 
-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 [fig:piezo_admittance_curve](#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](#orgba7797e)).
 
-<a id="orgb11212e"></a>
+<a id="orgba7797e"></a>
 
 {{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
 
@@ -1566,7 +1566,7 @@ 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 [fig:hac_lac_control](#fig:hac_lac_control).
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org278a785).
 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:
 
@@ -1574,7 +1574,7 @@ This approach has the following advantages:
 -   The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
 -   The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
 
-<a id="orgf36086a"></a>
+<a id="org278a785"></a>
 
 {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
 

content/book/schmidt14_desig_high_perfor_mechat_revis_edition.md

@@ -5,7 +5,7 @@ draft = false
 +++
 
 Tags
-: [Reference Books]({{< relref "reference_books" >}})
+: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
 
 Reference
 : <sup id="37468bbe5988cc7f4878fcd664d9cb7f"><a href="#schmidt14_desig_high_perfor_mechat_revis_edition" title="Schmidt, Schitter \&amp; Rankers, The Design of High Performance Mechatronics - 2nd Revised  Edition, Ios Press (2014).">(Schmidt {\it et al.}, 2014)</a></sup>
@@ -29,7 +29,7 @@ Section 2.2.2 Force and Motion
 > One should however be aware that another non-destructive source of non-linearity is found in a tried important field of mechanics, called _kinematics_.
 > The relation between angles and positions is often non-linear in such a mechanism, because of the changing angles, and controlling these often requires special precautions to overcome the inherent non-linearities by linearisation around actual position and adapting the optimal settings of the controller to each position.
 
-<a id="org462f9a1"></a>
+<a id="org8d0a076"></a>
 
 {{< figure src="/ox-hugo/schmidt14_high_low_freq_regions.png" caption="Figure 1: Stabiliby condition and robustness of a feedback controlled system. The desired shape of these curves guide the control design by optimising the lvels and sloppes of the amplitude Bode-plot at low and high frequencies for suppression of the disturbances and of the base Bode-plot in the cross-over frequency region. This is called **loop shaping design**" >}}
 

content/book/skogestad07_multiv_feedb_contr.md

@@ -63,7 +63,7 @@ Year
 
 ## Introduction {#introduction}
 
-<a id="orga7066d6"></a>
+<a id="org5d595bf"></a>
 
 
 ### The Process of Control System Design {#the-process-of-control-system-design}
@@ -234,7 +234,7 @@ Notations used throughout this note are summarized in tables&nbsp;[table:notatio
 
 ## Classical Feedback Control {#classical-feedback-control}
 
-<a id="orgaabb1e5"></a>
+<a id="orgeb70e42"></a>
 
 
 ### Frequency Response {#frequency-response}
@@ -283,7 +283,7 @@ Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
 The objective of control is to manipulate \\(u\\) (design \\(K\\)) such that the control error \\(e\\) remains small in spite of disturbances \\(d\\).
 The control error is defined as \\(e = y-r\\).
 
-<a id="org3b71acb"></a>
+<a id="orgf5eda9f"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}}
 
@@ -595,7 +595,7 @@ 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;[fig:classical_feedback_2dof_alt](#fig:classical_feedback_2dof_alt)), rather than operating on their difference \\(r - y\_m\\).
 
-<a id="org9265d45"></a>
+<a id="orgb16dfc9"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}}
 
@@ -604,7 +604,7 @@ The controller can be slit into two separate blocks (Fig.&nbsp;[fig:classical_fe
 -   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
 
-<a id="org0e3d8d7"></a>
+<a id="org66943f1"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}}
 
@@ -673,7 +673,7 @@ With (see Fig.&nbsp;[fig:performance_weigth](#fig:performance_weigth)):
 
 </div>
 
-<a id="org0656ee4"></a>
+<a id="orgac866a9"></a>
 
 {{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}}
 
@@ -697,7 +697,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
 
 ## Introduction to Multivariable Control {#introduction-to-multivariable-control}
 
-<a id="org25e187e"></a>
+<a id="org62230ab"></a>
 
 
 ### Introduction {#introduction}
@@ -740,7 +740,7 @@ For negative feedback system (Fig.&nbsp;[fig:classical_feedback_bis](#fig:classi
 -   \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
 -   \\(T \triangleq L(I + L)^{-1}\\) is the transfer function from \\(r\\) to \\(y\\)
 
-<a id="org10be303"></a>
+<a id="org409aae2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}}
 
@@ -1055,7 +1055,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
 
 The general control problem formulation is represented in Fig.&nbsp;[fig:general_control_names](#fig:general_control_names).
 
-<a id="org410e618"></a>
+<a id="orgfca0967"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}}
 
@@ -1085,7 +1085,7 @@ We consider:
 -   The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
 -   The weighted or normalized controlled outputs \\(z = W\_z \tilde{z}\\) (where \\(\tilde{z}\\) often consists of the control error \\(y-r\\) and the manipulated input \\(u\\))
 
-<a id="org98354a0"></a>
+<a id="orgea1a525"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}}
 
@@ -1128,7 +1128,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
 
 The general control configuration may be extended to include model uncertainty as shown in Fig.&nbsp;[fig:general_config_model_uncertainty](#fig:general_config_model_uncertainty).
 
-<a id="orgaee1f77"></a>
+<a id="org9ebb3a4"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}}
 
@@ -1156,7 +1156,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
 
 ## Elements of Linear System Theory {#elements-of-linear-system-theory}
 
-<a id="orgb820714"></a>
+<a id="org0d33f26"></a>
 
 
 ### System Descriptions {#system-descriptions}
@@ -1442,7 +1442,7 @@ RHP-zeros therefore imply high gain instability.
 
 ### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems}
 
-<a id="orgb1e4209"></a>
+<a id="org3b4a8f6"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}}
 
@@ -1589,7 +1589,7 @@ 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}
 
-<a id="org76a7a2f"></a>
+<a id="orgaa2db6e"></a>
 
 
 ### Input-Output Controllability {#input-output-controllability}
@@ -1981,7 +1981,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
 
 ### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control}
 
-<a id="org4ab6880"></a>
+<a id="org19dc4dc"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}}
 
@@ -2010,7 +2010,7 @@ In summary:
 Sometimes, the disturbances are so large that we hit input saturation or the required bandwidth is not achievable. To avoid the latter problem, we must at least require that the effect of the disturbance is less than \\(1\\) at frequencies beyond the bandwidth:
 \\[ \abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c \\]
 
-<a id="orga143a9d"></a>
+<a id="orgbee9012"></a>
 
 {{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}}
 
@@ -2032,7 +2032,7 @@ 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}
 
-<a id="org6b25e5b"></a>
+<a id="org5b47882"></a>
 
 
 ### Introduction {#introduction}
@@ -2343,7 +2343,7 @@ We here focus on input and output uncertainty.
 In multiplicative form, the input and output uncertainties are given by (see Fig.&nbsp;[fig:input_output_uncertainty](#fig:input_output_uncertainty)):
 \\[ G^\prime = (I + E\_O) G (I + E\_I) \\]
 
-<a id="org367c804"></a>
+<a id="orgc33ff63"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}}
 
@@ -2479,7 +2479,7 @@ However, the situation is usually the opposite with model uncertainty because fo
 
 ## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
 
-<a id="org80d55a0"></a>
+<a id="org395bba3"></a>
 
 
 ### Introduction to Robustness {#introduction-to-robustness}
@@ -2553,7 +2553,7 @@ which may be represented by the diagram in Fig.&nbsp;[fig:input_uncertainty_set]
 
 </div>
 
-<a id="org865770b"></a>
+<a id="orga0f232e"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}}
 
@@ -2607,7 +2607,7 @@ To illustrate how parametric uncertainty translate into frequency domain uncerta
     In general, these uncertain regions have complicated shapes and complex mathematical descriptions
 -   **Step 2**. We therefore approximate such complex regions as discs, resulting in a **complex additive uncertainty description**
 
-<a id="org168b9ff"></a>
+<a id="org87a3682"></a>
 
 {{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}}
 
@@ -2630,7 +2630,7 @@ At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped regi
 
 </div>
 
-<a id="org46ced8b"></a>
+<a id="org5bedb67"></a>
 
 {{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}}
 
@@ -2687,7 +2687,7 @@ To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_
 
 </div>
 
-<a id="org797da76"></a>
+<a id="org5399ef5"></a>
 
 {{< figure src="/ox-hugo/skogestad07_uncertainty_weight.png" caption="Figure 16: Relative error for 27 combinations of \\(k,\ \tau\\) and \\(\theta\\). Solid and dashed lines: two weights \\(\abs{w\_I}\\)" >}}
 
@@ -2726,7 +2726,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
 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;[fig:neglected_time_delay](#fig:neglected_time_delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
 \\[ 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} \\]
 
-<a id="org8ddf130"></a>
+<a id="org3b8e0bc"></a>
 
 {{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}}
 
@@ -2736,7 +2736,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
 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;[fig:neglected_first_order_lag](#fig:neglected_first_order_lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
 \\[ w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \\]
 
-<a id="orge3ddb3c"></a>
+<a id="orgdce561a"></a>
 
 {{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}}
 
@@ -2753,7 +2753,7 @@ However, as shown in Fig.&nbsp;[fig:lag_delay_uncertainty](#fig:lag_delay_uncert
 
 It is suggested to start with the simple weight and then if needed, to try the higher order weight.
 
-<a id="orgb652b95"></a>
+<a id="org4320c38"></a>
 
 {{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}}
 
@@ -2793,7 +2793,7 @@ We use the Nyquist stability condition to test for robust stability of the close
                   &\Longleftrightarrow \quad L\_p \ \text{should not encircle -1}, \ \forall L\_p
 \end{align\*}
 
-<a id="org0fda45b"></a>
+<a id="orgbd21fcf"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}}
 
@@ -2809,7 +2809,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
                   &\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\
 \end{align\*}
 
-<a id="org4ead586"></a>
+<a id="org68e60a3"></a>
 
 {{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}}
 
@@ -2847,7 +2847,7 @@ And we obtain the same condition as before.
 We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig.&nbsp;[fig:inverse_uncertainty_set](#fig:inverse_uncertainty_set)) in which
 \\[ G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \\]
 
-<a id="orgaad9987"></a>
+<a id="orgaad840d"></a>
 
 {{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}}
 
@@ -2899,7 +2899,7 @@ The condition for nominal performance when considering performance in terms of t
 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;[fig:nyquist_performance_condition](#fig:nyquist_performance_condition).
 
-<a id="org8e66342"></a>
+<a id="org2f13fac"></a>
 
 {{< figure src="/ox-hugo/skogestad07_nyquist_performance_condition.png" caption="Figure 23: Nyquist plot illustration of the nominal performance condition \\(\abs{w\_P} < \abs{1 + L}\\)" >}}
 
@@ -2924,7 +2924,7 @@ Let's consider the case of multiplicative uncertainty as shown on Fig.&nbsp;[fig
 The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
 \\[ L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \\]
 
-<a id="org3ca06cf"></a>
+<a id="orgbf2f9c6"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}}
 
@@ -3090,7 +3090,7 @@ with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
 
 This is illustrated in the block diagram of Fig.&nbsp;[fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
 
-<a id="orgd286b2a"></a>
+<a id="orga439362"></a>
 
 {{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
 
@@ -3108,7 +3108,7 @@ We also derived a condition for robust performance with multiplicative uncertain
 
 ## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
 
-<a id="orgb076a9b"></a>
+<a id="org630c4f9"></a>
 
 
 ### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
@@ -3119,13 +3119,13 @@ where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g.
 
 If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig.&nbsp;[fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
 
-<a id="org0853688"></a>
+<a id="orge8af5ee"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: 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;[fig:general_control_Ndelta](#fig:general_control_Ndelta).
 
-<a id="orgc524251"></a>
+<a id="orgad34338"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
 
@@ -3145,7 +3145,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
 
 To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig.&nbsp;[fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
 
-<a id="orge0e68f2"></a>
+<a id="org5ac3a4a"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
 
@@ -3197,7 +3197,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb
 
 | ![](/ox-hugo/skogestad07_additive_uncertainty.png) | ![](/ox-hugo/skogestad07_input_uncertainty.png)          | ![](/ox-hugo/skogestad07_output_uncertainty.png)          |
 |----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------|
-| <a id="org94556ee"></a> Additive uncertainty       | <a id="org205e138"></a> Multiplicative input uncertainty | <a id="org884d99b"></a> Multiplicative output uncertainty |
+| <a id="org05d42ff"></a> Additive uncertainty       | <a id="orgae1758d"></a> Multiplicative input uncertainty | <a id="org2db9e52"></a> Multiplicative output uncertainty |
 
 In Fig.&nbsp;[fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
 
@@ -3220,7 +3220,7 @@ In Fig.&nbsp;[fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feed
 
 | ![](/ox-hugo/skogestad07_inv_additive_uncertainty.png) | ![](/ox-hugo/skogestad07_inv_input_uncertainty.png)              | ![](/ox-hugo/skogestad07_inv_output_uncertainty.png)              |
 |--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------|
-| <a id="org17a4e6d"></a> Inverse additive uncertainty   | <a id="org2765e1d"></a> Inverse multiplicative input uncertainty | <a id="org33356e1"></a> Inverse multiplicative output uncertainty |
+| <a id="org5ea3793"></a> Inverse additive uncertainty   | <a id="org31e7d89"></a> Inverse multiplicative input uncertainty | <a id="org00c8166"></a> Inverse multiplicative output uncertainty |
 
 
 ##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
@@ -3290,7 +3290,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
 Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig.&nbsp;[fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
 \\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
 
-<a id="org2ebb26f"></a>
+<a id="orge5740dc"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}}
 
@@ -3450,7 +3450,7 @@ Where \\(G = M\_l^{-1} N\_l\\) is a left coprime factorization of the nominal pl
 
 This uncertainty description is surprisingly **general**, it allows both zeros and poles to cross into the right-half plane, and has proven to be very useful in applications.
 
-<a id="org71a706b"></a>
+<a id="org6fb438e"></a>
 
 {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
 
@@ -3482,7 +3482,7 @@ where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same di
 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;[fig:block_diagonal_scalings](#fig:block_diagonal_scalings).
 This clearly has no effect on stability.
 
-<a id="org949fb62"></a>
+<a id="org7b2b472"></a>
 
 {{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
 
@@ -3798,7 +3798,7 @@ with the decoupling controller we have:
 \\[ \bar{\sigma}(N\_{22}) = \bar{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \\]
 and we see from Fig.&nbsp;[fig:mu_plots_distillation](#fig:mu_plots_distillation) that the NP-condition is satisfied.
 
-<a id="orga318a7a"></a>
+<a id="org79285a0"></a>
 
 {{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
 
@@ -3921,7 +3921,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
 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;[fig:weights_distillation](#fig:weights_distillation).
 
-<a id="orgd273607"></a>
+<a id="org8fb66bd"></a>
 
 {{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
 
@@ -3944,11 +3944,11 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
 -   Iteration No. 3.
     Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations.
 
-<a id="org10a3970"></a>
+<a id="org5f826e8"></a>
 
 {{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
 
-<a id="org400285f"></a>
+<a id="orga6732c2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
 
@@ -3956,13 +3956,13 @@ The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\
 The objectives of RS and NP are easily satisfied.
 The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
 
-<a id="org519a9ca"></a>
+<a id="org7cbd772"></a>
 
 {{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(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;[fig:perturb_s_k3](#fig:perturb_s_k3).
 
-<a id="orgfcb21f2"></a>
+<a id="orge989e28"></a>
 
 {{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
 
@@ -4017,7 +4017,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
 
 ## Controller Design {#controller-design}
 
-<a id="orga616dec"></a>
+<a id="orgcb8e2f1"></a>
 
 
 ### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
@@ -4037,7 +4037,7 @@ We have the following important relationships:
   \end{align}
 \end{subequations}
 
-<a id="org8d4f22a"></a>
+<a id="org68d662f"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
 
@@ -4079,7 +4079,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
 
 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;[fig:design_trade_off_mimo_gk](#fig:design_trade_off_mimo_gk).
 
-<a id="org6e3f117"></a>
+<a id="org058fc8c"></a>
 
 {{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
 
@@ -4136,7 +4136,7 @@ The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}
 
 We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig.&nbsp;[fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
 
-<a id="org6f521b9"></a>
+<a id="org7162554"></a>
 
 {{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
 
@@ -4186,7 +4186,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
 
 </div>
 
-<a id="orgf0f14d9"></a>
+<a id="orgaa1f9d2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
 
@@ -4207,7 +4207,7 @@ It has the same degree (number of poles) as the plant.<br />
 
 For the LQG-controller, as shown on Fig.&nbsp;[fig:lqg_kalman_filter](#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;[fig:lqg_integral](#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="orgb7cfb99"></a>
+<a id="orga77c0d2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
 
@@ -4220,18 +4220,18 @@ Their main limitation is that they can only be applied to minimum phase plants.
 
 ### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
 
-<a id="org6da7635"></a>
+<a id="org2f8c60b"></a>
 
 
 #### General Control Problem Formulation {#general-control-problem-formulation}
 
-<a id="org1448cec"></a>
+<a id="orgbe54b98"></a>
 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;[fig:general_control](#fig:general_control).
 
-<a id="org3b91a51"></a>
+<a id="orgf28c426"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
 
@@ -4482,7 +4482,7 @@ The elements of the generalized plant are
   \end{array}
 \end{equation\*}
 
-<a id="org35551f8"></a>
+<a id="org3f11e63"></a>
 
 {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
 
@@ -4491,7 +4491,7 @@ 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;[fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
 
-<a id="org55460a0"></a>
+<a id="org758eceb"></a>
 
 {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
 
@@ -4515,7 +4515,7 @@ The elements of the generalized plant are
   \end{array}
 \end{equation\*}
 
-<a id="org007c976"></a>
+<a id="org68fca49"></a>
 
 {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
 
@@ -4543,7 +4543,7 @@ The focus of attention has moved to the size of signals and away from the size a
 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;[fig:input_uncertainty_hinf](#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="orgabff04a"></a>
+<a id="orge5eb5e5"></a>
 
 {{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
 
@@ -4565,7 +4565,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
   w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\  u = u
 \end{equation\*}
 
-<a id="org5056f35"></a>
+<a id="org896492e"></a>
 
 {{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
 
@@ -4576,7 +4576,7 @@ This problem is a non-standard \\(\hinf\\) optimization.
 It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
 \\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
 
-<a id="org7befd92"></a>
+<a id="org01deb15"></a>
 
 {{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
 
@@ -4634,7 +4634,7 @@ For the perturbed feedback system of Fig.&nbsp;[fig:coprime_uncertainty_bis](#fi
 
 Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
 
-<a id="org4f3b2f4"></a>
+<a id="org5d52156"></a>
 
 {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
 
@@ -4681,7 +4681,7 @@ It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\)
 
 #### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure}
 
-<a id="org929fa3b"></a>
+<a id="orge4e2f7e"></a>
 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.
@@ -4694,7 +4694,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
 
 as shown in Fig.&nbsp;[fig:shaped_plant](#fig:shaped_plant).
 
-<a id="orgbc1e59e"></a>
+<a id="org1f55ae2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
 
@@ -4731,7 +4731,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
     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\\)
 
-<a id="org83cf8d8"></a>
+<a id="org50d434b"></a>
 
 {{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
 
@@ -4757,7 +4757,7 @@ But in cases where stringent time-domain specifications are set on the output re
 A general two degrees-of-freedom feedback control scheme is depicted in Fig.&nbsp;[fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
 The commands and feedbacks enter the controller separately and are independently processed.
 
-<a id="org8f1d974"></a>
+<a id="orgbb5e97f"></a>
 
 {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
 
@@ -4768,7 +4768,7 @@ The design problem is illustrated in Fig.&nbsp;[fig:coprime_uncertainty_hinf](#f
 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**.
 
-<a id="orgd00d786"></a>
+<a id="orgce4d7f1"></a>
 
 {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
 
@@ -4793,7 +4793,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
 
 The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig.&nbsp;[fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
 
-<a id="org3d681ec"></a>
+<a id="orgac8907e"></a>
 
 {{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
 
@@ -4865,7 +4865,7 @@ where \\(u\_a\\) is the **actual plant input**, that is the measurement at the *
 
 The situation is illustrated in Fig.&nbsp;[fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
 
-<a id="org3867b27"></a>
+<a id="orgef9fa56"></a>
 
 {{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
 
@@ -4913,14 +4913,14 @@ Moreover, one should be careful about combining controller synthesis and analysi
 
 ## Controller Structure Design {#controller-structure-design}
 
-<a id="org6fc0469"></a>
+<a id="orgfc18612"></a>
 
 
 ### Introduction {#introduction}
 
 In previous sections, we considered the general problem formulation in Fig.&nbsp;[fig:general_control_names_bis](#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="org366605e"></a>
+<a id="orga10d2bd"></a>
 
 {{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
 
@@ -4955,7 +4955,7 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
 
 Additional layers are possible, as is illustrated in Fig.&nbsp;[fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
 
-<a id="org42e952b"></a>
+<a id="org73996e3"></a>
 
 {{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
 
@@ -4977,7 +4977,7 @@ However, this solution is normally not used for a number a reasons, included the
 
 | ![](/ox-hugo/skogestad07_optimize_control_a.png) | ![](/ox-hugo/skogestad07_optimize_control_b.png)                               | ![](/ox-hugo/skogestad07_optimize_control_c.png)            |
 |--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------|
-| <a id="org6986695"></a> Open loop optimization   | <a id="orgaae7402"></a> Closed-loop implementation with separate control layer | <a id="orge8ee4d7"></a> Integrated optimization and control |
+| <a id="orgcd78f08"></a> Open loop optimization   | <a id="orgd4993ff"></a> Closed-loop implementation with separate control layer | <a id="org0552911"></a> Integrated optimization and control |
 
 
 ### Selection of Controlled Outputs {#selection-of-controlled-outputs}
@@ -5184,7 +5184,7 @@ A cascade control structure results when either of the following two situations
 
 | ![](/ox-hugo/skogestad07_cascade_extra_meas.png)      | ![](/ox-hugo/skogestad07_cascade_extra_input.png) |
 |-------------------------------------------------------|---------------------------------------------------|
-| <a id="org4e7be08"></a> Extra measurements \\(y\_2\\) | <a id="org1a947e7"></a> Extra inputs \\(u\_2\\)   |
+| <a id="org7eda032"></a> Extra measurements \\(y\_2\\) | <a id="org765934a"></a> Extra inputs \\(u\_2\\)   |
 
 
 #### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
@@ -5233,7 +5233,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
 
 </div>
 
-<a id="org664489f"></a>
+<a id="orgcccf6fb"></a>
 
 {{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 59: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
 
@@ -5283,7 +5283,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
 
 </div>
 
-<a id="org12e1e27"></a>
+<a id="org87719b3"></a>
 
 {{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
 
@@ -5398,7 +5398,7 @@ We get:
   \end{aligned}
 \end{equation}
 
-<a id="orgffa343f"></a>
+<a id="orga0f78b2"></a>
 
 {{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
 
@@ -5518,7 +5518,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
 
 In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig.&nbsp;[fig:decentralized_diagonal_control](#fig:decentralized_diagonal_control)).
 
-<a id="org301990a"></a>
+<a id="org7292a61"></a>
 
 {{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
 
@@ -5905,7 +5905,7 @@ The conditions are also useful in an **input-output controllability analysis** f
 
 ## Model Reduction {#model-reduction}
 
-<a id="org7648c32"></a>
+<a id="org98665b1"></a>
 
 
 ### Introduction {#introduction}

content/book/taghirad13_paral.md

@@ -19,7 +19,7 @@ Year
 
 ## Introduction {#introduction}
 
-<a id="org2669f89"></a>
+<a id="orgb4a81bf"></a>
 
 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.
@@ -44,7 +44,7 @@ The control of parallel robots is elaborated in the last two chapters, in which
 
 ## Motion Representation {#motion-representation}
 
-<a id="orge862be5"></a>
+<a id="org9be8358"></a>
 
 
 ### Spatial Motion Representation {#spatial-motion-representation}
@@ -59,7 +59,7 @@ Let us define:
 
 The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
 
-<a id="orgf94f362"></a>
+<a id="org221cb93"></a>
 
 {{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
 
@@ -84,7 +84,7 @@ 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](#org42e4742)).
+Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org1448d5b)).
 
 A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
 
@@ -106,7 +106,7 @@ in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\
 
 The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
 
-<a id="org42e4742"></a>
+<a id="org1448d5b"></a>
 
 {{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
 
@@ -132,7 +132,7 @@ The term screw axis for this axis of rotation has the benefit that a general mot
 The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
 These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
 
-<a id="org4b4688a"></a>
+<a id="org778c755"></a>
 
 {{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
 
@@ -158,7 +158,7 @@ Three other types of Euler angles are consider with respect to a moving frame: t
 
 The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
 
-<a id="orgfdb1f6b"></a>
+<a id="orga52af18"></a>
 
 {{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
 
@@ -260,7 +260,7 @@ 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](#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}
   \tcmbox{{}^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
 Let us consider the motion of a rigid body described at three locations (Figure [fig:consecutive_transformations](#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="orgce1a4bb"></a>
+<a id="org49a98a7"></a>
 
 {{< figure src="/ox-hugo/taghirad13_consecutive_transformations.png" caption="Figure 5: Motion of a rigid body represented at three locations by frame \\(\\{\bm{A}\\}\\), \\(\\{\bm{B}\\}\\) and \\(\\{\bm{C}\\}\\)" >}}
 
@@ -426,7 +426,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
 
 ## Kinematics {#kinematics}
 
-<a id="org4d32ea5"></a>
+<a id="org56e6a14"></a>
 
 
 ### Introduction {#introduction}
@@ -533,7 +533,7 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
   \end{bmatrix}
 \end{equation}
 
-<a id="org6f4d53b"></a>
+<a id="org10ade7e"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="Figure 6: Geometry of a Stewart-Gough platform" >}}
 
@@ -586,7 +586,7 @@ The complexity of the problem depends widely on the manipulator architecture and
 
 ## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
 
-<a id="org58aa31a"></a>
+<a id="org150c88f"></a>
 
 
 ### Introduction {#introduction}
@@ -625,7 +625,7 @@ 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](#eq:angular_velocity_vector).
+The angular velocity vector is related to the screw formalism by equation \eqref{eq:angular_velocity_vector}.
 
 \begin{equation}
   \tcmbox{\bm{\Omega} \triangleq \dot{\theta} \hat{\bm{s}}}
@@ -683,7 +683,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
 
 Now consider the general motion of a rigid body shown in Figure [fig:general_motion](#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="orgbdb62a8"></a>
+<a id="org29fa342"></a>
 
 {{< figure src="/ox-hugo/taghirad13_general_motion.png" caption="Figure 7: Instantaneous velocity of a point \\(P\\) with respect to a moving frame \\(\\{\bm{B}\\}\\)" >}}
 
@@ -700,7 +700,7 @@ The time derivative of the rotation matrix \\({}^A\dot{\bm{R}}\_B\\) is:
   \tcmbox{{}^A\dot{\bm{R}}\_B = {}^A\bm{\Omega}^\times \ {}^A\bm{R}\_B}
 \end{equation}
 
-And we finally obtain equation [eq:absolute_velocity_formula](#eq:absolute_velocity_formula).
+And we finally obtain equation \eqref{eq:absolute_velocity_formula}.
 
 <div class="cbox">
   <div></div>
@@ -760,7 +760,7 @@ The **general Jacobian matrix** is defined as:
   \dot{\bm{q}} = \bm{J} \dot{\bm{\mathcal{X}}}
 \end{equation}
 
-From equation [eq:jacobians](#eq:jacobians), we have:
+From equation \eqref{eq:jacobians}, we have:
 
 \begin{equation}
   \bm{J} = {\bm{J}\_q}^{-1} \bm{J}\_x
@@ -847,7 +847,7 @@ 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](#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\\):
@@ -884,9 +884,9 @@ 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](#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](#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 \\]
 
@@ -942,7 +942,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
 
 As shown in Figure [fig:stewart_static_forces](#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="org6eae91e"></a>
+<a id="org55b47c8"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
 
@@ -1006,7 +1006,7 @@ 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](#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}
   \tcmbox{\bm{\tau} = \mathcal{K} \cdot \Delta \bm{q}}
@@ -1015,7 +1015,7 @@ Re-writing the equation [eq:stiffness_actuator](#eq:stiffness_actuator) for all
 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](#eq:jacobian_disp) for infinitesimal deflection read
+Writing the Jacobian relation given in equation \eqref{eq:jacobian_disp} for infinitesimal deflection read
 
 \begin{equation}
   \Delta \bm{q} = \bm{J} \cdot \Delta \bm{\mathcal{X}}
@@ -1023,19 +1023,19 @@ Writing the Jacobian relation given in equation [eq:jacobian_disp](#eq:jacobian_
 
 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](#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}
   \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
 \end{equation}
 
-Hence, by substituting [eq:stiffness_matrix_relation](#eq:stiffness_matrix_relation) and [eq:jacobian_disp_inf](#eq:jacobian_disp_inf) in [eq:jacobian_force_inf](#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}
   \tcmbox{\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}\_{\bm{K}} \cdot \Delta \bm{\mathcal{X}}}
 \end{equation}
 
-Equation [eq:stiffness_jacobian](#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="cbox">
   <div></div>
@@ -1099,7 +1099,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
 
 In this section, we restrict our analysis to a 3-6 structure (Figure [fig:stewart36](#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="org65471b6"></a>
+<a id="org2921078"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
 
@@ -1129,7 +1129,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
 
 ## Dynamics {#dynamics}
 
-<a id="org5b3c9d5"></a>
+<a id="orgd3cd6ba"></a>
 
 
 ### Introduction {#introduction}
@@ -1213,7 +1213,7 @@ where \\(\\{\theta, \hat{\bm{s}}\\}\\) are the screw parameters representing the
 
 </div>
 
-As shown by [eq:angular_acceleration](#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}
@@ -1260,7 +1260,7 @@ For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\
 
 In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
 
-<a id="orgf3666fb"></a>
+<a id="orgdfab15a"></a>
 
 {{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
 
@@ -1310,7 +1310,7 @@ in which
 
 ##### Principal Axes {#principal-axes}
 
-As seen in equation [eq:moment_inertia](#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}
@@ -1374,7 +1374,7 @@ in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with res
 This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
 This highlights the important of the center of mass in dynamic formulation of rigid bodies.
 
-<a id="org0925a29"></a>
+<a id="org0c8049e"></a>
 
 {{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
 
@@ -1398,10 +1398,10 @@ 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](#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](#eq:angular_momentum) is very analogous to that of linear momentum [eq: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}
@@ -1519,7 +1519,7 @@ The position vector of these two center of masses can be determined by the follo
   \bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
 \end{align}
 
-<a id="org8ab78ec"></a>
+<a id="orgc1e3ded"></a>
 
 {{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
 
@@ -1615,7 +1615,7 @@ 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](#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}
@@ -1673,7 +1673,7 @@ 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](#eq:dyn_form_implicit_trans) and [eq:dyn_form_implicit_rot](#eq:dyn_form_implicit_rot) can be simply converted into a closed form of Equation [eq:close_form_dynamics_platform](#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}
   \begin{aligned}
@@ -1733,11 +1733,11 @@ in which
 
 As shown in Figure [fig:stewart_forward_dynamics](#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="orgd407041"></a>
+<a id="orgff76393"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: 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](#eq:closed_form_dynamic_stewart_wanted), whose terms are given in [eq:close_form_dynamics_stewart_terms](#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}
@@ -1752,11 +1752,11 @@ 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](#eq:closed_form_intermediate_parameters) and [eq:close_form_dynamics_stewart_terms](#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](#eq:jacobian_intermediate), and then compute compatible matrices for the limbs given in [eq:closed_form_stewart_manipulator](#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](#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.
@@ -1766,7 +1766,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
   \bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
 \end{equation}
 
-<a id="org07cbdc9"></a>
+<a id="org077ac35"></a>
 
 {{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
 
@@ -1791,7 +1791,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
 
 ## Motion Control {#motion-control}
 
-<a id="org7a85646"></a>
+<a id="orgdd65dc9"></a>
 
 
 ### Introduction {#introduction}
@@ -1812,7 +1812,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
 
 ### Controller Topology {#controller-topology}
 
-<a id="org5a89036"></a>
+<a id="org5d92117"></a>
 
 <div class="cbox">
   <div></div>
@@ -1861,7 +1861,7 @@ Figure [fig:general_topology_motion_feedback](#fig:general_topology_motion_feedb
 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 />
 
-<a id="orgab9400f"></a>
+<a id="orga16758c"></a>
 
 {{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
 
@@ -1871,7 +1871,7 @@ The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) an
 
 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 [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis) to implement such a controller.
 
-<a id="org212e259"></a>
+<a id="org72c22c0"></a>
 
 {{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
 
@@ -1885,7 +1885,7 @@ To overcome the implementation problem of the control topology in Figure [fig:ge
 In this topology, depicted in Figure [fig:general_topology_motion_feedback_ter](#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="orgc845c97"></a>
+<a id="org65fcdb0"></a>
 
 {{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
 
@@ -1899,7 +1899,7 @@ For the topology in Figure [fig:general_topology_motion_feedback_ter](#fig:gener
 To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [fig:general_topology_motion_feedback_quater](#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="org566f432"></a>
+<a id="orgf53b839"></a>
 
 {{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
 
@@ -1909,7 +1909,7 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
 
 ### Motion Control in Task Space {#motion-control-in-task-space}
 
-<a id="orgf6bab13"></a>
+<a id="org7befb5a"></a>
 
 
 #### Decentralized PD Control {#decentralized-pd-control}
@@ -1918,7 +1918,7 @@ In the control structure in Figure [fig:decentralized_pd_control_task_space](#fi
 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.
 
-<a id="org80664c6"></a>
+<a id="org84d7f6d"></a>
 
 {{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
 
@@ -1941,7 +1941,7 @@ A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the
 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) \\]
 
-<a id="org5a2762c"></a>
+<a id="org11304dd"></a>
 
 {{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
 
@@ -2004,7 +2004,7 @@ Furthermore, mass matrix is added in the forward path in addition to the desired
 
 As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.<br />
 
-<a id="orgbab4b31"></a>
+<a id="org7bcc842"></a>
 
 {{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
 
@@ -2056,9 +2056,9 @@ 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](#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](#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](#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.
@@ -2066,7 +2066,7 @@ 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](#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:
@@ -2104,7 +2104,7 @@ 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](#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}})\\\\\\
@@ -2126,14 +2126,14 @@ in which
 \\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
 is a measure of modeling uncertainty.
 
-<a id="orgabfe014"></a>
+<a id="org3cb985a"></a>
 
 {{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
 
 
 #### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
 
-<a id="org27f4777"></a>
+<a id="org2d5c44d"></a>
 
 {{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
 
@@ -2218,7 +2218,7 @@ In this control structure, depicted in Figure [fig:decentralized_pd_control_join
 
 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 />
 
-<a id="org8778164"></a>
+<a id="orgef87ac5"></a>
 
 {{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
 
@@ -2240,7 +2240,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
 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 [fig:feedforward_pd_control_joint_space](#fig:feedforward_pd_control_joint_space).
 
-<a id="org82bffda"></a>
+<a id="org7dd1247"></a>
 
 {{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
 
@@ -2288,7 +2288,7 @@ Furthermore, the mass matrix is acting in the **forward path**, in addition to t
 Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
 In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.<br />
 
-<a id="orgb3e85c7"></a>
+<a id="orgd592b03"></a>
 
 {{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
 
@@ -2564,7 +2564,7 @@ Hence, it is recommended to design and implement controllers in the task space,
 
 ## Force Control {#force-control}
 
-<a id="org6d7e26a"></a>
+<a id="org7035f2e"></a>
 
 
 ### Introduction {#introduction}
@@ -2620,7 +2620,7 @@ The output control loop is called the **primary loop**, while the inner loop is
 
 </div>
 
-<a id="orgd4ccfa2"></a>
+<a id="orgc775ba2"></a>
 
 {{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
 
@@ -2654,7 +2654,7 @@ As seen in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascad
 
 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 />
 
-<a id="orgd1518f8"></a>
+<a id="orgc0138a8"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: 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" >}}
 
@@ -2662,7 +2662,7 @@ Other alternatives for force control topology may be suggested based on the vari
 If the force is measured in the joint space, the topology suggested in Figure [fig:taghira13_cascade_force_outer_loop_tau](#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="org56763bc"></a>
+<a id="orgf060936"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: 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" >}}
 
@@ -2673,7 +2673,7 @@ However, as the inner loop is constructed in the joint space, the desired motion
 
 Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.<br />
 
-<a id="orgd855e74"></a>
+<a id="orgfe2ecb4"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
 
@@ -2691,7 +2691,7 @@ By this means, when the manipulator is not in contact with a stiff environment,
 However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
 This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.<br />
 
-<a id="org1b62063"></a>
+<a id="org1509ffc"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: 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" >}}
 
@@ -2699,7 +2699,7 @@ Other alternatives for control topology may be suggested based on the variations
 If the force is measured in the joint space, control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau](#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="org9e07f41"></a>
+<a id="org4e33f80"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: 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" >}}
 
@@ -2708,7 +2708,7 @@ The inner loop is based on the measured actuator force vector in the joint space
 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.
 
-<a id="org688e959"></a>
+<a id="org159af61"></a>
 
 {{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
 
@@ -2727,7 +2727,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
 
 ### Direct Force Control {#direct-force-control}
 
-<a id="org7d75d2a"></a>
+<a id="orgb779c33"></a>
 
 {{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
 
@@ -2818,7 +2818,7 @@ The impedance of the system may be found from the Laplace transform of the above
 
 </div>
 
-<a id="org649bf2b"></a>
+<a id="orge490c94"></a>
 
 {{< figure src="/ox-hugo/taghirad13_impedance_control_rlc.png" caption="Figure 35: Analogy of electrical impedance in (a) an electrical RLC circuit to (b) a mechanical mass-spring-damper system" >}}
 
@@ -2877,7 +2877,7 @@ Moreover, direct force-tracking objective is not assigned in this control 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.
 By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.<br />
 
-<a id="org93dc62d"></a>
+<a id="org9902c61"></a>
 
 {{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
 

content/inproceedings/mcinroy99_dynam.md

@@ -0,0 +1,173 @@
++++
+title = "Dynamic modeling of flexure jointed hexapods for control purposes"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: <sup id="5da427f78c552aa92cd64c2a6df961f1"><a class="reference-link" href="#mcinroy99_dynam" title="McInroy, Dynamic modeling of flexure jointed hexapods for control  purposes, nil, in in: {Proceedings of the 1999 IEEE International Conference on
+                      Control Applications (Cat. No.99CH36328)}, edited by (1999)">(McInroy, 1999)</a></sup>
+
+Author(s)
+: McInroy, J.
+
+Year
+: 1999
+
+This conference paper has been further published in a journal as a short note <sup id="8bfe2d2dce902a584fa016e86a899044"><a class="reference-link" href="#mcinroy02_model_desig_flexur_joint_stewar" title="McInroy, Modeling and Design of Flexure Jointed Stewart Platforms  for Control Purposes, {IEEE/ASME Transactions on Mechatronics}, v(1), 95-99 (2002).">(McInroy, 2002)</a></sup>.
+
+
+## Abstract {#abstract}
+
+> This paper presents a new dynamic model suitable for control of flexure jointed hexapods (FJH).
+>
+> Novel contributions include:
+>
+> 1.  Base acceleration inputs are included
+> 2.  The dynamic model is experimentally verified
+> 3.  The model is developed so that it is suitable for control
+> 4.  A decoupled force control is derived
+
+
+## Strut Dynamics {#strut-dynamics}
+
+The actuators for FJHs can be divided into two categories:
+
+1.  soft (voice coil), which employs a spring flexure mount
+2.  hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
+
+<a id="org7016c5c"></a>
+
+{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: 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](#orga202dc3).
+
+In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
+
+<a id="orga202dc3"></a>
+
+{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
+
+<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="#orga202dc3">2</a>
+</div>
+
+| **Symbol**                   | **Meaning**                                |
+|------------------------------|--------------------------------------------|
+| \\(m\_i\\)                   | moving strut mass                          |
+| \\(k\_i\\)                   | spring constant                            |
+| \\(b\_i\\)                   | damping constant                           |
+| \\(f\_m\\)                   | force the actuator applies                 |
+| \\(f\_{p\_i}\\)              | forced exerted by the payload              |
+| \\(p\_i\\)                   | three dimensional position of the top      |
+| \\(q\_i\\)                   | three dimensional position of the bottom   |
+| \\(l\_i\\)                   | strut length                               |
+| \\(l\_{r\_i}\\)              | relaxed strut length                       |
+| \\(v\_i = p\_i - q\_i\\)     | vector pointing from the bottom to the top |
+| \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut                |
+
+It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in <sup id="8bfe2d2dce902a584fa016e86a899044"><a class="reference-link" href="#mcinroy02_model_desig_flexur_joint_stewar" title="McInroy, Modeling and Design of Flexure Jointed Stewart Platforms  for Control Purposes, {IEEE/ASME Transactions on Mechatronics}, v(1), 95-99 (2002).">(McInroy, 2002)</a></sup>).
+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](#orga202dc3) (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
+\end{equation}
+
+The acceleration \\(\hat{u}\_i^T \ddot{p}\_i\\) can be written as:
+\\[ \hat{u}\_i^T \ddot{p}\_i = \ddot{l}\_i + \hat{u}\_i^T \ddot{q}\_i - \dot{\hat{u}}\_i^T \dot{v}\_i \\]
+
+-   [ ] Not sure how the last term is obtained
+
+Separating strut and base accelerations, and putting all six strut equations in a single vector yields:
+
+\begin{equation}
+  f\_p = 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 \label{eq:strut\_dynamics\_vec}
+\end{equation}
+
+where:
+
+-   \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
+-   \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
+-   \\(Ms = \diag([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
+-   \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\)
+
+
+## Payload Dynamics {#payload-dynamics}
+
+The payload is modeled as a rigid body:
+
+\begin{equation}
+\underbrace{\begin{bmatrix}
+m I\_3 & 0\_{3\times 3} \\\\\\
+0\_{3\times 3}   & {}^cI
+\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
+  0\_{3 \times 1} \\ \omega \times {}^cI\omega
+\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
+\end{equation}
+
+where:
+
+-   \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
+-   \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
+-   \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
+-   \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
+-   \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
+
+Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
+
+First, consider the **generalized force due to struts**.
+Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
+
+\begin{equation}
+  \mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
+\end{equation}
+
+where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
+
+The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
+
+\begin{equation}
+  \mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
+\end{equation}
+
+where:
+
+-   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
+-   \\(g\\) is the gravity vector
+
+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\\\\\\
+  & - 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)
+\end{aligned}
+
+Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
+In the next section, a connection between the two will be found to complete the formulation
+
+
+## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
+
+
+## Joint Space Dynamics {#joint-space-dynamics}
+
+
+## Control Example {#control-example}
+
+# Bibliography
+<a class="bibtex-entry" id="mcinroy99_dynam">McInroy, J., *Dynamic modeling of flexure jointed hexapods for control purposes*, In , Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328) (pp. ) (1999). : .</a> [↩](#5da427f78c552aa92cd64c2a6df961f1)
+
+<a class="bibtex-entry" id="mcinroy02_model_desig_flexur_joint_stewar">McInroy, J., *Modeling and design of flexure jointed stewart platforms for control purposes*, IEEE/ASME Transactions on Mechatronics, *7(1)*, 95–99 (2002).  http://dx.doi.org/10.1109/3516.990892</a> [↩](#8bfe2d2dce902a584fa016e86a899044)
+
+
+## Backlinks {#backlinks}
+
+-   [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})

content/paper/butler11_posit_contr_lithog_equip.md

@@ -1,20 +0,0 @@
-+++
-title = "Position control in lithographic equipment"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Multivariable Control]({{< relref "multivariable_control" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
-
-Reference
-: <sup id="6a014e3a2ee3e41d20bd0644654c56f0"><a href="#butler11_posit_contr_lithog_equip" title="Hans Butler, Position Control in Lithographic Equipment, {IEEE Control Systems}, v(5), 28-47 (2011).">(Hans Butler, 2011)</a></sup>
-
-Author(s)
-: Butler, H.
-
-Year
-: 2011
-
-# Bibliography
-<a id="butler11_posit_contr_lithog_equip"></a>Butler, H., *Position control in lithographic equipment*, IEEE Control Systems, *31(5)*, 28–47 (2011).  http://dx.doi.org/10.1109/mcs.2011.941882 [↩](#6a014e3a2ee3e41d20bd0644654c56f0)

content/paper/chen00_ident_decoup_contr_flexur_joint_hexap.md

@@ -1,22 +0,0 @@
-+++
-title = "Identification and decoupling control of flexure jointed hexapods"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
-
-Reference
-: <sup id="ba05ff213f8e5963d91559d95becfbdb"><a href="#chen00_ident_decoup_contr_flexur_joint_hexap" title="Yixin Chen \&amp; McInroy, Identification and Decoupling Control of Flexure Jointed  Hexapods, nil, in in: {Proceedings 2000 ICRA. Millennium Conference. IEEE
-                      International Conference on Robotics and Automation. Symposia
-                      Proceedings (Cat. No.00CH37065)}, edited by (2000)">(Yixin Chen \& McInroy, 2000)</a></sup>
-
-Author(s)
-: Chen, Y., & McInroy, J.
-
-Year
-: 2000
-
-# Bibliography
-<a id="chen00_ident_decoup_contr_flexur_joint_hexap"></a>Chen, Y., & McInroy, J., *Identification and decoupling control of flexure jointed hexapods*, In , Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065) (pp. ) (2000). : . [↩](#ba05ff213f8e5963d91559d95becfbdb)

content/paper/fleming12_estim.md

@@ -1,22 +0,0 @@
-+++
-title = "Estimating the resolution of nanopositioning systems from frequency domain data"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-:
-
-
-Reference
-: <sup id="a1cc9b70316a7dda2f652efd146caf84"><a href="#fleming12_estim" title="Andrew Fleming, Estimating the resolution of nanopositioning systems from  frequency domain data, nil, in in: {2012 IEEE International Conference on Robotics and
-                      Automation}, edited by (2012)">(Andrew Fleming, 2012)</a></sup>
-
-Author(s)
-: Fleming, A. J.
-
-Year
-: 2012
-
-# Bibliography
-<a id="fleming12_estim"></a>Fleming, A. J., *Estimating the resolution of nanopositioning systems from frequency domain data*, In , 2012 IEEE International Conference on Robotics and Automation (pp. ) (2012). : . [↩](#a1cc9b70316a7dda2f652efd146caf84)

content/paper/furqan17_studies_stewar_platf_manip.md

@@ -1,22 +0,0 @@
-+++
-title = "Studies on stewart platform manipulator: a review"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}})
-
-Reference
-: <sup id="cc10fe9545c7c381cc2b610e8f91a071"><a href="#furqan17_studies_stewar_platf_manip" title="Mohd Furqan, Mohd Suhaib \&amp; Nazeer Ahmad, Studies on Stewart Platform Manipulator: a Review, {Journal of Mechanical Science and Technology}, v(9), 4459-4470 (2017).">(Mohd Furqan {\it et al.}, 2017)</a></sup>
-
-Author(s)
-: Furqan, M., Suhaib, M., & Ahmad, N.
-
-Year
-: 2017
-
-Lots of references.
-
-# Bibliography
-<a id="furqan17_studies_stewar_platf_manip"></a>Furqan, M., Suhaib, M., & Ahmad, N., *Studies on stewart platform manipulator: a review*, Journal of Mechanical Science and Technology, *31(9)*, 4459–4470 (2017).  http://dx.doi.org/10.1007/s12206-017-0846-1 [↩](#cc10fe9545c7c381cc2b610e8f91a071)

content/paper/gao15_measur_techn_precis_posit.md

@@ -1,20 +0,0 @@
-+++
-title = "Measurement technologies for precision positioning"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Position Sensors]({{< relref "position_sensors" >}})
-
-Reference
-: <sup id="b820b918ced36901ea0ad4bf653202c6"><a href="#gao15_measur_techn_precis_posit" title="Gao, Kim, Bosse, Haitjema, , Chen, Lu, Knapp, Weckenmann, , Estler \&amp; Kunzmann, Measurement Technologies for Precision Positioning, {CIRP Annals}, v(2), 773-796 (2015).">(Gao {\it et al.}, 2015)</a></sup>
-
-Author(s)
-: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
-
-Year
-: 2015
-
-# Bibliography
-<a id="gao15_measur_techn_precis_posit"></a>Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …, *Measurement technologies for precision positioning*, CIRP Annals, *64(2)*, 773–796 (2015).  http://dx.doi.org/10.1016/j.cirp.2015.05.009 [↩](#b820b918ced36901ea0ad4bf653202c6)

content/paper/hanieh03_activ_stewar.md

@@ -1,34 +0,0 @@
-+++
-title = "Active isolation and damping of vibrations via stewart platform"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}})
-
-Reference
-: <sup id="10e535e895bdcd6b921bff33ef68cd81"><a href="#hanieh03_activ_stewar" title="@phdthesis{hanieh03_activ_stewar,
-      author          = {Hanieh, Ahmed Abu},
-      school          = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
-      title           = {Active isolation and damping of vibrations via Stewart
-                      platform},
-      year            = 2003,
-      tags            = {parallel robot},
-    }">@phdthesis{hanieh03_activ_stewar,
-      author          = {Hanieh, Ahmed Abu},
-      school          = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
-      title           = {Active isolation and damping of vibrations via Stewart
-                      platform},
-      year            = 2003,
-      tags            = {parallel robot},
-    }</a></sup>
-
-Author(s)
-: Hanieh, A. A.
-
-Year
-: 2003
-
-# Bibliography
-<a id="hanieh03_activ_stewar"></a>Hanieh, A. A., *Active isolation and damping of vibrations via stewart platform* (Doctoral dissertation) (2003). Universit{\'e} Libre de Bruxelles, Brussels, Belgium, . [↩](#10e535e895bdcd6b921bff33ef68cd81)

content/paper/holterman05_activ_dampin_based_decoup_colloc_contr.md

@@ -1,20 +0,0 @@
-+++
-title = "Active damping based on decoupled collocated control"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Active Damping]({{< relref "active_damping" >}})
-
-Reference
-: <sup id="cc7836a555fe4dbae791e17008c29bfd"><a href="#holterman05_activ_dampin_based_decoup_colloc_contr" title="Holterman \&amp; deVries, Active Damping Based on Decoupled Collocated Control, {IEEE/ASME Transactions on Mechatronics}, v(2), 135-145 (2005).">(Holterman \& deVries, 2005)</a></sup>
-
-Author(s)
-: Holterman, J., & deVries, T.
-
-Year
-: 2005
-
-# Bibliography
-<a id="holterman05_activ_dampin_based_decoup_colloc_contr"></a>Holterman, J., & deVries, T., *Active damping based on decoupled collocated control*, IEEE/ASME Transactions on Mechatronics, *10(2)*, 135–145 (2005).  http://dx.doi.org/10.1109/tmech.2005.844702 [↩](#cc7836a555fe4dbae791e17008c29bfd)

content/paper/jiao18_dynam_model_exper_analy_stewar.md

@@ -1,20 +0,0 @@
-+++
-title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
-
-Reference
-: <sup id="ee917739f88877d6c2758c1c36565deb"><a href="#jiao18_dynam_model_exper_analy_stewar" title="Jian Jiao, Ying Wu, Kaiping Yu \&amp; Rui Zhao, Dynamic Modeling and Experimental Analyses of Stewart  Platform With Flexible Hinges, {Journal of Vibration and Control}, v(1), 151-171 (2018).">(Jian Jiao {\it et al.}, 2018)</a></sup>
-
-Author(s)
-: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
-
-Year
-: 2018
-
-# Bibliography
-<a id="jiao18_dynam_model_exper_analy_stewar"></a>Jiao, J., Wu, Y., Yu, K., & Zhao, R., *Dynamic modeling and experimental analyses of stewart platform with flexible hinges*, Journal of Vibration and Control, *25(1)*, 151–171 (2018).  http://dx.doi.org/10.1177/1077546318772474 [↩](#ee917739f88877d6c2758c1c36565deb)

content/paper/li01_simul_vibrat_isolat_point_contr.md

@@ -1,23 +0,0 @@
-+++
-title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
-
-Reference
-: <sup id="e3df2691f750617c3995644d056d553a"><a href="#li01_simul_vibrat_isolat_point_contr" title="Xiaochun Li, Jerry Hamann \&amp; John McInroy, Simultaneous Vibration Isolation and Pointing Control of  Flexure Jointed Hexapods, nil, in in: {Smart Structures and Materials 2001: Smart Structures and
-                      Integrated Systems}, edited by (2001)">(Xiaochun Li {\it et al.}, 2001)</a></sup>
-
-Author(s)
-: Li, X., Hamann, J. C., & McInroy, J. E.
-
-Year
-: 2001
-
--   if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
-
-# Bibliography
-<a id="li01_simul_vibrat_isolat_point_contr"></a>Li, X., Hamann, J. C., & McInroy, J. E., *Simultaneous vibration isolation and pointing control of flexure jointed hexapods*, In , Smart Structures and Materials 2001: Smart Structures and Integrated Systems (pp. ) (2001). : . [↩](#e3df2691f750617c3995644d056d553a)

content/paper/oomen18_advan_motion_contr_precis_mechat.md

@@ -1,24 +0,0 @@
-+++
-title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Motion Control]({{< relref "motion_control" >}})
-
-Reference
-: <sup id="73fd325bd20a6ef8972145e535f38198"><a href="#oomen18_advan_motion_contr_precis_mechat" title="Tom Oomen, Advanced Motion Control for Precision Mechatronics:  Control, Identification, and Learning of Complex Systems, {IEEJ Journal of Industry Applications}, v(2), 127-140 (2018).">(Tom Oomen, 2018)</a></sup>
-
-Author(s)
-: Oomen, T.
-
-Year
-: 2018
-
-<a id="org55ab131"></a>
-
-{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
-
-# Bibliography
-<a id="oomen18_advan_motion_contr_precis_mechat"></a>Oomen, T., *Advanced motion control for precision mechatronics: control, identification, and learning of complex systems*, IEEJ Journal of Industry Applications, *7(2)*, 127–140 (2018).  http://dx.doi.org/10.1541/ieejjia.7.127 [↩](#73fd325bd20a6ef8972145e535f38198)

content/paper/poel10_explor_activ_hard_mount_vibrat.md

@@ -1,42 +0,0 @@
-+++
-title = "An exploration of active hard mount vibration isolation for precision equipment"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Vibration Isolation]({{< relref "vibration_isolation" >}})
-
-Reference
-: <sup id="bcab548922e0e1ad6a2c310f63879596"><a href="#poel10_explor_activ_hard_mount_vibrat" title="@phdthesis{poel10_explor_activ_hard_mount_vibrat,
-      author          = {van der Poel, Gerrit Wijnand},
-      doi             = {10.3990/1.9789036530163},
-      isbn            = {978-90-365-3016-3},
-      school          = {University of Twente},
-      title           = {An Exploration of Active Hard Mount Vibration Isolation for
-                      Precision Equipment},
-      url             = {https://doi.org/10.3990/1.9789036530163},
-      year            = 2010,
-      year            = 2010,
-      tags            = {parallel robot},
-    }">@phdthesis{poel10_explor_activ_hard_mount_vibrat,
-      author          = {van der Poel, Gerrit Wijnand},
-      doi             = {10.3990/1.9789036530163},
-      isbn            = {978-90-365-3016-3},
-      school          = {University of Twente},
-      title           = {An Exploration of Active Hard Mount Vibration Isolation for
-                      Precision Equipment},
-      url             = {https://doi.org/10.3990/1.9789036530163},
-      year            = 2010,
-      year            = 2010,
-      tags            = {parallel robot},
-    }</a></sup>
-
-Author(s)
-: van der Poel, G. W.
-
-Year
-: 2010
-
-# Bibliography
-<a id="poel10_explor_activ_hard_mount_vibrat"></a>van der Poel, G. W., *An exploration of active hard mount vibration isolation for precision equipment* (Doctoral dissertation) (2010). University of Twente, . [↩](#bcab548922e0e1ad6a2c310f63879596)

content/paper/schellekens98_desig_precis.md

@@ -1,20 +0,0 @@
-+++
-title = "Design for precision: current status and trends"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Precision Engineering]({{< relref "precision_engineering" >}})
-
-Reference
-: <sup id="89f7d8f4c31f79f83e3666017687f525"><a href="#schellekens98_desig_precis" title="Schellekens, Rosielle, Vermeulen, , Vermeulen, Wetzels \&amp; Pril, Design for Precision: Current Status and Trends, {Cirp Annals}, v(2), 557-586 (1998).">(Schellekens {\it et al.}, 1998)</a></sup>
-
-Author(s)
-: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
-
-Year
-: 1998
-
-# Bibliography
-<a id="schellekens98_desig_precis"></a>Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W., *Design for precision: current status and trends*, Cirp Annals, *(2)*, 557–586 (1998).  http://dx.doi.org/10.1016/s0007-8506(07)63243-0 [↩](#89f7d8f4c31f79f83e3666017687f525)

content/paper/sebastian12_nanop_with_multip_sensor.md

@@ -1,20 +0,0 @@
-+++
-title = "Nanopositioning with multiple sensors: a case study in data storage"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Sensor Fusion]({{< relref "sensor_fusion" >}})
-
-Reference
-: <sup id="eb5a15a8c900d93de0b9bab520e1b6da"><a href="#sebastian12_nanop_with_multip_sensor" title="Abu Sebastian \&amp; Angeliki Pantazi, Nanopositioning With Multiple Sensors: a Case Study in Data  Storage, {IEEE Transactions on Control Systems Technology}, v(2), 382-394 (2012).">(Abu Sebastian \& Angeliki Pantazi, 2012)</a></sup>
-
-Author(s)
-: Sebastian, A., & Pantazi, A.
-
-Year
-: 2012
-
-# Bibliography
-<a id="sebastian12_nanop_with_multip_sensor"></a>Sebastian, A., & Pantazi, A., *Nanopositioning with multiple sensors: a case study in data storage*, IEEE Transactions on Control Systems Technology, *20(2)*, 382–394 (2012).  http://dx.doi.org/10.1109/tcst.2011.2177982 [↩](#eb5a15a8c900d93de0b9bab520e1b6da)

content/paper/tang18_decen_vibrat_contr_voice_coil.md

@@ -1,21 +0,0 @@
-+++
-title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Stewart Platforms]({{< relref "stewart_platforms" >}})
-
-Reference
-: <sup id="85f81ff678aabc195636437548e4234a"><a href="#tang18_decen_vibrat_contr_voice_coil" title="Jie Tang, Dengqing Cao \&amp; Tianhu Yu, Decentralized Vibration Control of a Voice Coil Motor-Based  Stewart Parallel Mechanism: Simulation and Experiments, {Proceedings of the Institution of Mechanical Engineers,
-                      Part C: Journal of Mechanical Engineering Science}, v(1), 132-145 (2018).">(Jie Tang {\it et al.}, 2018)</a></sup>
-
-Author(s)
-: Tang, J., Cao, D., & Yu, T.
-
-Year
-: 2018
-
-# Bibliography
-<a id="tang18_decen_vibrat_contr_voice_coil"></a>Tang, J., Cao, D., & Yu, T., *Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments*, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, *233(1)*, 132–145 (2018).  http://dx.doi.org/10.1177/0954406218756941 [↩](#85f81ff678aabc195636437548e4234a)

content/paper/wang12_autom_marker_full_field_hard.md

@@ -1,29 +0,0 @@
-+++
-title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
-author = ["Thomas Dehaeze"]
-draft = false
-+++
-
-Tags
-: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
-
-Reference
-: <sup id="1bccbe15e35ed02229afbc6528c5057e"><a href="#wang12_autom_marker_full_field_hard" title="Jun Wang, Yu-chen Karen Chen, Qingxi Yuan, Andrei, Tkachuk, Can Erdonmez, Benjamin Hornberger, Michael \&amp; Feser, Automated Markerless Full Field Hard X-Ray Microscopic  Tomography At Sub-50 Nm 3-dimension Spatial Resolution, {Applied Physics Letters}, v(14), 143107 (2012).">(Jun Wang {\it et al.}, 2012)</a></sup>
-
-Author(s)
-: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
-
-Year
-: 2012
-
-**Introduction of Markers**:
-That limits the type of samples that is studied
-
-There is a need for markerless nano-tomography
-=> the key requirement is the precision and stability of the positioning stages.
-
-**Passive rotational run-out error system**:
-It uses calibrated metrology disc and capacitive sensors
-
-# Bibliography
-<a id="wang12_autom_marker_full_field_hard"></a>Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M., *Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution*, Applied Physics Letters, *100(14)*, 143107 (2012).  http://dx.doi.org/10.1063/1.3701579 [↩](#1bccbe15e35ed02229afbc6528c5057e)

content/phdthesis/jabben07_mechat.md

@@ -0,0 +1,224 @@
++++
+title = "Mechatronic design of a magnetically suspended rotating platform"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
+
+Author
+: Jabben, L.
+
+Year
+: 2007
+
+DOI
+:
+
+
+## Dynamic Error Budgeting {#dynamic-error-budgeting}
+
+
+### Introduction {#introduction}
+
+A large class of mechatronic machines have specifications based on their _standstill_ performance.
+The standstill performance is then limited by the (stochastic) disturbances action on the closed loop.
+
+The difficulty in calculation with stochastic signals and Bode plots, is that, instead of calculating with the complex response at one frequency, the **area** over a frequency range should be taken into account.
+
+The **error budgeting** is often used to estimate how much each component contributes to the total error
+
+Since many of the disturbances have a stochastic nature, they can be modelled with their **Power Spectral Densities**.
+
+The PSD of the performance measure in the closed loop system is the weigted sum of PSDs of the contributions of each disturbance to the performance channel.
+This approach allows frequency dependent error budgeting, which is why it is referred to as **Dynamic Error Budgeting**.
+
+
+### Common Mechatronics Disturbances {#common-mechatronics-disturbances}
+
+
+#### Ground vibrations {#ground-vibrations}
+
+
+#### Electronic Noise {#electronic-noise}
+
+**Thermal Noise** (or Johson 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] \\]
+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.
+
+```text
+A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of $0.13 \mu V$ from zero up to one kHz.
+```
+
+**Shot Noise**.
+Seen with junctions in a transistor.
+It has a white spectral density:
+\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
+with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
+
+```text
+An averable current of 1 A will introduce noise with a STD of $10 \cdot 10^{-9}\,[A]$ from zero up to one kHz.
+```
+
+**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.
+
+**Signal to Noise Ration**
+Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
+In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
+The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
+**The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.**
+Usually, the SNR is specified up to a certain cut-off frequency.
+If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
+\\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
+with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
+
+
+#### AD and DA converters {#ad-and-da-converters}
+
+ADC and DAC add quantization noise to the signal.
+The variance can be calculated to be:
+\\[ \sigma^2 = \frac{q^2}{12} \\]
+with \\(q\\) the quantization interval.
+
+The corresponding PSD is white up to the Nyquist frequency:
+\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
+with \\(f\_N\\) the Nyquist frequency [Hz].
+
+```text
+Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
+Let's suppose the ADC is used to measure a position over a range of 1 mm.
+- ADC quantization noise: it has 16 bots over the 1 mm range.
+  The standard diviation from the quantization is:
+  \[ \sigma_{ADq} = \frac{1 \cdot 10^6/2^16}{\sqrt{12}} = 4.4\,[nm] \]
+- ADC electronic noise: the RMS value of a sine that covers to full range is $\frac{0.5}{\sqrt{2}} = 0.354\,[mm]$.
+  With a SNR of 80dB, the electronic noise from the ADC becomes:
+  \[ \sigma_{ADn} = 35\,[nm] \]
+
+Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of $\sigma_{sn} = 17\,[nm]$.
+
+The PSD of this digitalized sensor noise is:
+\[ \sigma_s = \sqrt{\sigma_{sn}^2 + \sigma_{ADq}^2 + \sigma_{ADn}^2} = 39\,[nm]\]
+from which the PSD of the total sensor noise $S_s$ is calculated:
+\[ S_s = \frac{\sigma_s^2}{f_N} = 1.55\,[nm^2/Hz] \]
+with $f_N$ is the Nyquist frequency of 1kHz.
+```
+
+
+#### Acoustic Noise {#acoustic-noise}
+
+This can be a big error source in high precision machines, especially when the surface is big compare to the mass.
+
+The disturbance force acting on a body, is the **difference of pressure between the front and the back times the surface**.
+To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
+For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
+
+```text
+Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal $2.2 \cdot 10^{-2}\,[N^2/Hz]$
+```
+
+
+#### Brownian Noise {#brownian-noise}
+
+This is due to thermal effects and it notable where a small mass needs positioning.
+
+
+#### Turbulence {#turbulence}
+
+Rotation of the spindle introduces and air flow in which turbulence is cause by sharp angles on the rotor and stator.
+
+
+### Optimal Control {#optimal-control}
+
+
+#### The use of Optimal Control in DEB {#the-use-of-optimal-control-in-deb}
+
+Three factors influence the performance:
+
+-   the disturbances: often a given value
+-   the plant: can be costly to redesign
+-   the controller
+
+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](#orgf051865), 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="orgf051865"></a>
+
+{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
+
+
+#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
+
+Since disturbances are generally not white, the system of Figure [1](#orgf051865) 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](#org35c7d66) 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:
+
+-   the performance channels might have different dimensions, which require scaling in order to compare
+-   some performance channels may be of more importance than others
+-   by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
+
+<a id="org35c7d66"></a>
+
+{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
+
+The weighting filters should be stable transfer functions.
+
+**Obtaining the weighting filters**:
+
+If the PSD is given as a function \\(S\_x(j\omega)\\), the disturbance filter can be using **spectral factorization**:
+
+> Given a positive even function \\(S\_x(f)\\) of finite area, find a minimum-phase stable function \\(L(s)\\), such that \\(|L(j2\pi f)|^2 = S(s)\\)
+
+**Harmonic signals** can be approximately modeled by filtering white noise with a badly damped second order system, having a \\(+1\\) slope below the resonance frequency and a \\(-1\\) slope above the resonance frequency:
+\\[ V\_h = \frac{s}{s^2 + 2 \xi \omega\_h + \omega\_h^2} \\]
+with \\(\xi\\) the relative damping and \\(\omega\_h\\) the resonance frequency [rad/s].
+By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of the harmonic signal, the propagation of the disturbance to the performance channel can be well approximated.
+
+
+#### Balancing Control Effort vs Performance {#balancing-control-effort-vs-performance}
+
+IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
+So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
+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](#org742f80a) is obtained.
+
+<a id="org742f80a"></a>
+
+{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
+
+
+## Conclusion {#conclusion}
+
+> Using the DEB analysis during the design helped to formulate the specifications of the several subcomponents, such as:
+>
+> -   The target bandwidth of the decentralized closed loops, which is very important for the mechanical design, as mechanical resonances can severely limit the bandwidth.
+>     This value was also used to specify the current loop bandwidth of the custom designed power amplifiers for the RTAs and other components such as sensors and filters.
+> -   The target value of the stiffness of the actuators was derived at 1000 N/m.
+>     It was shown that the stiffness of a motor with back-iron is too much for the separated frame concept.
+> -   The analysis pinpointed the most limiting component in the final design to be the Analogue-to-Digital Converter (ADC).
+
+<!--quoteend-->
+
+> In the DEB-framework there are three distinct factors which determine the performance.
+> These are the plant, the controller and the disturbances.
+> Synthesizing optimal controllers, such as H2-control, in the design helps to eliminate the controller out of the equation.
+> If the performance specifications are not met with an optimal controller, it is certain that a redesign of the system is required.
+> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
+> To derive such models, spectral factorization is used.
+> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.

content/phdthesis/monkhorst04_dynam_error_budget.md

@@ -0,0 +1,166 @@
++++
+title = "Dynamic error budgeting, a design approach"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
+
+Reference
+: <sup id="651e626e040250ee71a0847aec41b60c"><a class="reference-link" href="#monkhorst04_dynam_error_budget" title="@phdthesis{monkhorst04_dynam_error_budget,
+      author          = {Wouter Monkhorst},
+      school          = {Delft University},
+      title           = {Dynamic Error Budgeting, a design approach},
+      year            = 2004,
+    }">@phdthesis{monkhorst04_dynam_error_budget,
+      author          = {Wouter Monkhorst},
+      school          = {Delft University},
+      title           = {Dynamic Error Budgeting, a design approach},
+      year            = 2004,
+    }</a></sup>
+
+Author(s)
+: Monkhorst, W.
+
+Year
+: 2004
+
+
+## Introduction {#introduction}
+
+Challenge definition of this thesis:
+
+> Develop a tool which enables the designer to account for stochastic disturbances during the design of a mechatronics system.
+
+Develop tools should enable the designer to:
+
+-   Predict the final performance level of a system which is subject to stochastic disturbances
+-   Gain insight in performance limiting factors of the system.
+    This insight should enable the designer to point out critical system components/properties and to improve the performance of the system
+-   Objectively compare the performance of different system designs
+
+
+## Dynamic Error Budgeting {#dynamic-error-budgeting}
+
+
+### Motivations {#motivations}
+
+Main motivations are:
+
+-   Cutting costs in the design phase
+-   Speeding up the design process
+-   Enhancing design insight
+
+
+### DEB design process {#deb-design-process}
+
+Step by step, the process is as follows:
+
+-   design a concept system
+-   model the concept system, such that the closed loop transfer functions can be determined
+-   Identify all significant disturbances.
+    Model them with their _Power Spectral Density_
+-   Define the performance outputs of the system and simulate the output error.
+    Using the theory of _propagation_, the contribution of each disturbance to the output error can be analyzed and the critical disturbance can be pointed out
+-   Make changes to the system that are expected to improve the performance level, and simulate the output error again.
+    Iterate until the error budget is meet.
+
+
+### Assumptions {#assumptions}
+
+The assumptions when applying DEB are:
+
+-   the system can be accurately described with a **linear time invariant model**.
+    This is usually the case as much effort is put in to make systems have a linear behavior and because feedback loops have a " linearizing" effect on the closed loop behavior.
+-   the disturbances action on the system must be **stationary** (their statistical properties are not allowed to change over time).
+-   the disturbances are **uncorrelated** with each other.
+    This is more difficult to satisfy for MIMO systems and the designer must make sure that the separate disturbances all originate from separate independent sources.
+-   the disturbance signals are modeled by their **Power Spectral Density**.
+    This implies that only stochastic disturbances are allowed.
+    Deterministic components like sinusoidal and DC signals are infinite peaks in their PSD and should not be used.
+    For the deterministic part, other techniques can be used to determine their influence to the error.
+-   the calculation method makes no assumption on the distribution of the distribution functions of the disturbances.
+    In practice, many disturbances will have a normal like distribution.
+
+
+### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2--control-maximizing-performance}
+
+
+#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the--mathcal-h-2--norm-and-variance-of-the-output}
+
+The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
+\\[ \\|H\\|\_2^2 = \int\_{-\infty}^\infty |H(j2\pi f)|^2 df \\]
+
+Stochastic interpretation of the \\(\mathcal{H}\_2\\) norm: the squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
+
+
+#### The \\(\mathcal{H}\_2\\) control problem {#the--mathcal-h-2--control-problem}
+
+Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_2\\) norm of the closed loop system \\(H\\):
+\\[ C\_{\mathcal{H}\_2} \in \arg \min\_C \\|H\\|\_2 \\]
+
+
+#### Using weighting filters to model disturbances {#using-weighting-filters-to-model-disturbances}
+
+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](#org7f8d04e)).
+
+<a id="org7f8d04e"></a>
+
+{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
+
+The white noise input \\(w(t)\\) is dimensionless, and when the weighting filter has units [SI], the resulting weighted signal \\(\bar{w}(t)\\) has units [SI].
+The PSD \\(S\_w(f)\\) of the weighted signal is:
+\\[ |S\_w(f)| = V\_w(j 2 \pi f) V\_w^T(-j 2 \pi f) \\]
+
+Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
+However, this can be avoided if the modelling 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](#org4f416df)).
+
+<a id="org4f416df"></a>
+
+{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
+
+
+#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
+
+In this research, the outputs of the closed loop system (Figure [3](#orgc347ae6)) are:
+
+-   the performance (error) signal \\(e\\)
+-   the controller output \\(u\\)
+
+In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
+
+<a id="orgc347ae6"></a>
+
+{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
+
+The resulting problem is a multi-objective control problem: while constraining the variance of the controller output \\(u\\), the variance of the performance channel should be minimized.
+This problem can be solved by scaling the controller output \\(u\\) with a factor \\(\alpha\\) during the \\(\mathcal{H}\_2\\) synthesis.
+When varying \\(\alpha\\), one can plot the amount of control effort at one axis and the achieve performance on the other axis.
+The resulting points lie on the so-called **Pareto curve**.
+
+
+## Conclusions {#conclusions}
+
+\\(\mathcal{H}\_2\\) control strategy is an extension of the DEB approach.
+It offers the designer the opportunity to optimize over the degree of freedom given by the controller, enabling the designer to predict the maximum achievable performance level of a system concept.
+Using this technique, the designer is able to objectively compare the performance potential of different system concepts.
+
+The accuracy of the predicted performance by DEB with respect to the measured results can be improved by using higher order models of the disturbances.
+Increasing of order of the disturbance model might even allow modelling of harmonic disturbances by using inverse notches.
+To achieve the highest degree of prediction accuracy, it is recommended to use to actual measured disturbance spectra in the simulations.
+
+When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, it can give the control designer useful hints about how to control the system best for optimal performance.
+Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
+
+# Bibliography
+<a class="bibtex-entry" id="monkhorst04_dynam_error_budget">Monkhorst, W., *Dynamic error budgeting, a design approach* (2004). Delft University.</a> [↩](#651e626e040250ee71a0847aec41b60c)
+
+
+## Backlinks {#backlinks}
+
+-   [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})

content/posts/homepages.md

@@ -0,0 +1,18 @@
++++
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+## Homepage for Papers {#main}
+
+Here is the list of papers I took note about.
+
+
+## Homepage for Books {#main}
+
+Here is the list of books I took note about.
+
+
+## Homepage for Zettels {#main}
+
+Here is the list of subjects I took note about.

content/zettels/actuator_fusion.md

@@ -0,0 +1,34 @@
++++
+title = "Actuator Fusion"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Complementary Filters]({{< relref "complementary_filters" >}})
+
+<sup id="89b9470055c4d7f0f1957aa4400df9e8"><a href="#beijen19_mixed_feedb_feedf_contr_desig" title="Michiel Beijen, Marcel Heertjes, Hans Butler, \&amp; Maarten Steinbuch, Mixed Feedback and Feedforward Control Design for  Multi-Axis Vibration Isolation Systems, Mechatronics, v(), 106 - 116 (2019).">("Michiel Beijen {\it et al.}, 2019)</a></sup>
+
+<sup id="28a270550e13d2c8d045c9e0a9557945"><a href="#beijen18_distur" title="@phdthesis{beijen18_distur,
+  author          = {Beijen, MA},
+  school          = {Technische Universiteit Eindhoven},
+  title           = {Disturbance feedforward control for vibration isolation
+                  systems: analysis, design, and implementation},
+  year            = 2018,
+}">@phdthesis{beijen18_distur,
+  author          = {Beijen, MA},
+  school          = {Technische Universiteit Eindhoven},
+  title           = {Disturbance feedforward control for vibration isolation
+                  systems: analysis, design, and implementation},
+  year            = 2018,
+}</a></sup> (section 6.3.1)
+
+# Bibliography
+<a id="beijen19_mixed_feedb_feedf_contr_desig"></a>Beijen, M. A., Heertjes, M. F., Butler, H., & Steinbuch, M., *Mixed feedback and feedforward control design for multi-axis vibration isolation systems*, Mechatronics, *61()*, 106–116 (2019).  http://dx.doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005 [↩](#89b9470055c4d7f0f1957aa4400df9e8)
+
+<a id="beijen18_distur"></a>Beijen, M., *Disturbance feedforward control for vibration isolation systems: analysis, design, and implementation* (Doctoral dissertation) (2018). Technische Universiteit Eindhoven, . [↩](#28a270550e13d2c8d045c9e0a9557945)
+
+
+## Backlinks {#backlinks}
+
+-   [Sensor Fusion]({{< relref "sensor_fusion" >}})

content/zettels/actuators.md

@@ -12,7 +12,7 @@ Tags
 
 For vibration isolation:
 
--   In <sup id="aad53368e29e8a519e2f63857044fa46"><a href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators  Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>, the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}}))
+-   In <sup id="aad53368e29e8a519e2f63857044fa46"><a class="reference-link" href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators  Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>, the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}}))
 
 
 ## Piezoelectric {#piezoelectric}
@@ -23,8 +23,29 @@ For vibration isolation:
 | PI           | [link](https://www.physikinstrumente.com/en/)                                      |
 | Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) |
 | Noliac       | [link](http://www.noliac.com/)                                                     |
+| Thorlabs     | [link](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup%5Fid=8700)           |
 
-A model of a multi-layer monolithic piezoelectric stack actuator is described in <sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup> ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
+A model of a multi-layer monolithic piezoelectric stack actuator is described in <sup id="c823f68dd2a72b9667a61b3c046b4731"><a class="reference-link" href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for  High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup> ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
+
+
+### Piezoelectric Stack Actuators {#piezoelectric-stack-actuators}
+
+Typical strain is \\(0.1\%\\).
+
+
+### Mechanically Amplified Piezoelectric actuators {#mechanically-amplified-piezoelectric-actuators}
+
+The Amplified Piezo Actuators principle is presented in <sup id="5decd2b31c4a9842b80c58b56f96590a"><a class="reference-link" href="#claeyssen07_amplif_piezoel_actuat" title="Frank Claeyssen, Le Letty, Barillot, \&amp; Sosnicki, Amplified Piezoelectric Actuators: Static \&amp; Dynamic  Applications, {Ferroelectrics}, v(1), 3-14 (2007).">(Frank Claeyssen {\it et al.}, 2007)</a></sup>:
+
+> The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
+> The flatter is the actuator, the higher is the amplification.
+
+A model of an amplified piezoelectric actuator is described in <sup id="849750850d9986ed326e74bd3c448d03"><a class="reference-link" href="#lucinskis16_dynam_charac" title="@misc{lucinskis16_dynam_charac,
+  author          = {R. Lucinskis and C. Mangeot},
+  title           = {Dynamic Characterization of an amplified piezoelectric
+                  actuator},
+  year            = 2016,
+}">(Lucinskis \& Mangeot, 2016)</a></sup>.
 
 
 ## Voice Coil {#voice-coil}
@@ -55,16 +76,20 @@ A model of a multi-layer monolithic piezoelectric stack actuator is described in
 
 ## Brush-less DC Motor {#brush-less-dc-motor}
 
--   <sup id="d2e68d39d09d7e8e71ff08a6ebd45400"><a href="#yedamale03_brush_dc_bldc_motor_fundam" title="Yedamale, Brushless Dc (BLDC) Motor Fundamentals, {Microchip Technology Inc}, v(), 3--15 (2003).">(Yedamale, 2003)</a></sup>
+-   <sup id="d2e68d39d09d7e8e71ff08a6ebd45400"><a class="reference-link" href="#yedamale03_brush_dc_bldc_motor_fundam" title="Yedamale, Brushless Dc (BLDC) Motor Fundamentals, {Microchip Technology Inc}, v(), 3--15 (2003).">(Yedamale, 2003)</a></sup>
 
 <https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
 
 # Bibliography
-<a id="ito16_compar_class_high_precis_actuat"></a>Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016).  http://dx.doi.org/10.1109/tmech.2015.2478658 [↩](#aad53368e29e8a519e2f63857044fa46)
+<a class="bibtex-entry" id="ito16_compar_class_high_precis_actuat">Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016).  http://dx.doi.org/10.1109/tmech.2015.2478658</a> [↩](#aad53368e29e8a519e2f63857044fa46)
 
-<a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)
+<a class="bibtex-entry" id="fleming10_nanop_system_with_force_feedb">Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422</a> [↩](#c823f68dd2a72b9667a61b3c046b4731)
 
-<a id="yedamale03_brush_dc_bldc_motor_fundam"></a>Yedamale, P., *Brushless dc (bldc) motor fundamentals*, Microchip Technology Inc, *20()*, 3–15 (2003).  [↩](#d2e68d39d09d7e8e71ff08a6ebd45400)
+<a class="bibtex-entry" id="claeyssen07_amplif_piezoel_actuat">Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O., *Amplified piezoelectric actuators: static \& dynamic applications*, Ferroelectrics, *351(1)*, 3–14 (2007).  http://dx.doi.org/10.1080/00150190701351865</a> [↩](#5decd2b31c4a9842b80c58b56f96590a)
+
+<a class="bibtex-entry" id="lucinskis16_dynam_charac">Lucinskis, R., & Mangeot, C. (2016). *Dynamic characterization of an amplified piezoelectric actuator*. Retrieved from [](). .</a> [↩](#849750850d9986ed326e74bd3c448d03)
+
+<a class="bibtex-entry" id="yedamale03_brush_dc_bldc_motor_fundam">Yedamale, P., *Brushless dc (bldc) motor fundamentals*, Microchip Technology Inc, *20()*, 3–15 (2003). </a> [↩](#d2e68d39d09d7e8e71ff08a6ebd45400)
 
 
 ## Backlinks {#backlinks}

content/zettels/complementary_filters.md

@@ -12,4 +12,6 @@ Tags
 
 ## Backlinks {#backlinks}
 
+-   [Actuator Fusion]({{< relref "actuator_fusion" >}})
+-   [Sensor Fusion]({{< relref "sensor_fusion" >}})
 -   [Advances in internal model control technique: a review and future prospects]({{< relref "saxena12_advan_inter_model_contr_techn" >}})

content/zettels/cubic_architecture.md

@@ -7,11 +7,25 @@ draft = false
 Tags
 :
 
-<./biblio/references.bib>
+
+## Description of the Cubic Architecture {#description-of-the-cubic-architecture}
+
+
+## Special Properties {#special-properties}
+
+Cubic Stewart Platforms can be decoupled provided that (from <sup id="ba05ff213f8e5963d91559d95becfbdb"><a href="#chen00_ident_decoup_contr_flexur_joint_hexap" title="Yixin Chen \&amp; McInroy, Identification and Decoupling Control of Flexure Jointed  Hexapods, nil, in in: {Proceedings 2000 ICRA. Millennium Conference. IEEE
+                  International Conference on Robotics and Automation. Symposia
+                  Proceedings (Cat. No.00CH37065)}, edited by (2000)">(Yixin Chen \& McInroy, 2000)</a></sup>)
+
+> 1.  The payload mass-inertia matrix is diagonal
+> 2.  If a mutually orthogonal geometry has been selected, the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
+
+# Bibliography
+<a id="chen00_ident_decoup_contr_flexur_joint_hexap"></a>Chen, Y., & McInroy, J., *Identification and decoupling control of flexure jointed hexapods*, In , Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065) (pp. ) (2000). : . [↩](#ba05ff213f8e5963d91559d95becfbdb)
 
 
 ## Backlinks {#backlinks}
 
 -   [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
--   [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
 -   [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
+-   [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})

content/zettels/dynamic_error_budgeting.md

@@ -0,0 +1,58 @@
++++
+title = "Dynamic Error Budgeting"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+A good introduction to Dynamic Error Budgeting is given in <sup id="651e626e040250ee71a0847aec41b60c"><a class="reference-link" href="#monkhorst04_dynam_error_budget" title="@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}">@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}</a></sup>.
+
+
+## Step by Step process {#step-by-step-process}
+
+Taken from <sup id="651e626e040250ee71a0847aec41b60c"><a class="reference-link" href="#monkhorst04_dynam_error_budget" title="@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}">@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}</a></sup>: ([Notes]({{< relref "monkhorst04_dynam_error_budget" >}}))
+
+> Step by step, the process is as follows:
+>
+> -   design a concept system
+> -   model the concept system, such that the closed loop transfer functions can be determined
+> -   Identify all significant disturbances.
+>     Model them with their _Power Spectral Density_
+> -   Define the performance outputs of the system and simulate the output error.
+>     Using the theory of _propagation_, the contribution of each disturbance to the output error can be analyzed and the critical disturbance can be pointed out
+> -   Make changes to the system that are expected to improve the performance level, and simulate the output error again.
+>     Iterate until the error budget is meet.
+
+# Bibliography
+<a class="bibtex-entry" id="monkhorst04_dynam_error_budget">Monkhorst, W., *Dynamic error budgeting, a design approach* (Doctoral dissertation) (2004). Delft University, .</a> [↩](#651e626e040250ee71a0847aec41b60c)
+
+
+## Backlinks {#backlinks}
+
+-   [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
+-   [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
+-   [Mechatronic design of a magnetically suspended rotating platform]({{< relref "jabben07_mechat" >}})
+-   [Systems and Signals Norms]({{< relref "norms" >}})
+-   [Dynamic error budgeting, a design approach]({{< relref "monkhorst04_dynam_error_budget" >}})

content/zettels/electronics.md

@@ -13,3 +13,4 @@ Tags
 ## Backlinks {#backlinks}
 
 -   [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
+-   [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})

content/zettels/finite_element_model.md

@@ -0,0 +1,31 @@
++++
+title = "Finite Element Model"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+## Matlab State Space Model from FEM on Ansys {#matlab-state-space-model-from-fem-on-ansys}
+
+Some resources:
+
+-   <sup id="484c4fad309f6b0e866a7cacf4653d74"><a class="reference-link" href="#hatch00_vibrat_matlab_ansys" title="Hatch, Vibration simulation using MATLAB and ANSYS, CRC Press (2000).">(Hatch, 2000)</a></sup> ([Notes]({{< relref "hatch00_vibrat_matlab_ansys" >}}))
+-   <sup id="961d4331bc9da7f553368ca6a06cb743"><a class="reference-link" href="#khot11_model_respon_analy_dynam_system" title="Khot \&amp; Yelve, Modeling and Response Analysis of Dynamic Systems By Using  ANSYS{\copyright} and MATLAB{\copyright}, {Journal of Vibration and Control}, v(6), 953--958 (2011).">(Khot \& Yelve, 2011)</a></sup>
+-   <sup id="326e544dd573b7069b69e0ec90fad499"><a class="reference-link" href="#kosarac15_creat_siso_ansys" title="Ko\vsarac, Zeljkovi\'c, , Mla\djenovi\'c \&amp; \vZivkovi\'c, Create SISO state space model of main spindle from ANSYS  model, 37--41, in in: {12th International Scientific Conference, Novi Sad, Serbia}, edited by (2015)">(Ko\vsarac {\it et al.}, 2015)</a></sup>
+
+The idea is to extract reduced state space model from Ansys into Matlab.
+
+# Bibliography
+<a class="bibtex-entry" id="hatch00_vibrat_matlab_ansys">Hatch, M. R., *Vibration simulation using matlab and ansys* (2000), : CRC Press.</a> [↩](#484c4fad309f6b0e866a7cacf4653d74)
+
+<a class="bibtex-entry" id="khot11_model_respon_analy_dynam_system">Khot, S., & Yelve, N. P., *Modeling and response analysis of dynamic systems by using ansys\copyright and matlab\copyright*, Journal of Vibration and Control, *17(6)*, 953–958 (2011). </a> [↩](#961d4331bc9da7f553368ca6a06cb743)
+
+<a class="bibtex-entry" id="kosarac15_creat_siso_ansys">Ko\vsarac, A, Zeljkovi\'c, M, Mla\djenovi\'c, C, & \vZivkovi\'c, A, *Create siso state space model of main spindle from ansys model*, In , 12th International Scientific Conference, Novi Sad, Serbia (pp. 37–41) (2015). : .</a> [↩](#326e544dd573b7069b69e0ec90fad499)
+
+
+## Backlinks {#backlinks}
+
+-   [Vibration simulation using matlab and ansys]({{< relref "hatch00_vibrat_matlab_ansys" >}})

content/zettels/flexible_joints.md

@@ -7,15 +7,49 @@ draft = false
 Tags
 :
 
-<./biblio/references.bib>
+
+## Resources {#resources}
+
+Books:
+
+-   <sup id="7d07367ac4d34d56738dbfe0eb53371f"><a href="#lobontiu02_compl" title="Lobontiu, Compliant mechanisms: design of flexure hinges, CRC press (2002).">(Lobontiu, 2002)</a></sup>
+-   <sup id="53d819004fa64ee1fe2e715469c5991f"><a href="#henein03_concep_guidag_flexib" title="Henein, Conception des Guidages Flexibles, Presses polytechniques et universitaires romandes (2003).">(Henein, 2003)</a></sup>
+-   <sup id="ccc31a1054040cbdbbb28ba9e590af72"><a href="#smith05_found" title="Smith, Foundations of ultra-precision mechanism design, CRC Press (2005).">(Smith, 2005)</a></sup>
+-   <sup id="13540f4d4ba6bb415fdc21c85dde63cc"><a href="#soemers11_desig_princ" title="Soemers, Design Principles for precision mechanisms, T-Pointprint (2011).">(Soemers, 2011)</a></sup>
+-   <sup id="880641d23cd52fb47b40104731883e32"><a href="#cosandier17_flexur_mechan_desig" title="Cosandier, Flexure Mechanism Design, Distributed by CRC Press, 2017EOFL Press (2017).">(Cosandier, 2017)</a></sup>
+
+
+## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms}
+
+From <sup id="ba05ff213f8e5963d91559d95becfbdb"><a href="#chen00_ident_decoup_contr_flexur_joint_hexap" title="Yixin Chen \&amp; McInroy, Identification and Decoupling Control of Flexure Jointed  Hexapods, nil, in in: {Proceedings 2000 ICRA. Millennium Conference. IEEE
+                  International Conference on Robotics and Automation. Symposia
+                  Proceedings (Cat. No.00CH37065)}, edited by (2000)">(Yixin Chen \& McInroy, 2000)</a></sup>:
+
+> To avoid the extremely non-linear micro-dynamics of joint friction and backlash, these hexapods employ flexure joints.
+> 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.
+
+# Bibliography
+<a id="lobontiu02_compl"></a>Lobontiu, N., *Compliant mechanisms: design of flexure hinges* (2002), : CRC press. [↩](#7d07367ac4d34d56738dbfe0eb53371f)
+
+<a id="henein03_concep_guidag_flexib"></a>Henein, S., *Conception des guidages flexibles* (2003), Lausanne, Suisse: Presses polytechniques et universitaires romandes. [↩](#53d819004fa64ee1fe2e715469c5991f)
+
+<a id="smith05_found"></a>Smith, S. T., *Foundations of ultra-precision mechanism design* (2005), : CRC Press. [↩](#ccc31a1054040cbdbbb28ba9e590af72)
+
+<a id="soemers11_desig_princ"></a>Soemers, H., *Design principles for precision mechanisms* (2011), : T-Pointprint. [↩](#13540f4d4ba6bb415fdc21c85dde63cc)
+
+<a id="cosandier17_flexur_mechan_desig"></a>Cosandier, F., *Flexure Mechanism Design* (2017), Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press. [↩](#880641d23cd52fb47b40104731883e32)
+
+<a id="chen00_ident_decoup_contr_flexur_joint_hexap"></a>Chen, Y., & McInroy, J., *Identification and decoupling control of flexure jointed hexapods*, In , Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065) (pp. ) (2000). : . [↩](#ba05ff213f8e5963d91559d95becfbdb)
 
 
 ## Backlinks {#backlinks}
 
--   [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
 -   [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}})
 -   [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}})
--   [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
 -   [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
+-   [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
 -   [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
+-   [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
+-   [Dynamic modeling of flexure jointed hexapods for control purposes]({{< relref "mcinroy99_dynam" >}})
 -   [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})

content/zettels/force_sensors.md

@@ -21,3 +21,9 @@ An analysis the dynamics and noise of a piezoelectric force sensor is done in <s
 
 # Bibliography
 <a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010).  http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)
+
+
+## Backlinks {#backlinks}
+
+-   [Nanopositioning system with force feedback for high-performance tracking and vibration control]({{< relref "fleming10_nanop_system_with_force_feedb" >}})
+-   [Position Sensors]({{< relref "position_sensors" >}})

content/zettels/hac_hac.md

@@ -9,17 +9,33 @@ Tags
 
 High-Authority Control/Low-Authority Control
 
-From <sup id="454500a3af67ef66a7a754d1f2e1bd4a"><a href="#preumont18_vibrat_contr_activ_struc_fourt_edition" title="Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).">(Andre Preumont, 2018)</a></sup>:
+From <sup id="454500a3af67ef66a7a754d1f2e1bd4a"><a class="reference-link" href="#preumont18_vibrat_contr_activ_struc_fourt_edition" title="Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).">(Andre Preumont, 2018)</a></sup>:
 
-> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#org2e37874). 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](#org21fb08d). 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)
 > -   The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
 
-<a id="org2e37874"></a>
+<a id="org21fb08d"></a>
 
 {{< figure src="/ox-hugo/hac_lac_control_architecture.png" caption="Figure 1: HAC-LAC Control Architecture" >}}
 
+Nice papers:
+
+-   <sup id="ef357e45dadd8cc8869beda6e463777b"><a class="reference-link" href="#williams89_limit" title="Williams \&amp; Antsaklis, Limitations of vibration suppression in flexible space  structures, nil, in in: {Proceedings of the 28th IEEE Conference on Decision and
+                      Control}, edited by (1989)">(Williams \& Antsaklis, 1989)</a></sup>
+-   <sup id="df6fde1eeef81966b2c7fb5421adbe8d"><a class="reference-link" href="#aubrun80_theor_contr_struc_by_low_author_contr" title="Aubrun, Theory of the Control of Structures By Low-Authority  Controllers, {Journal of Guidance and Control}, v(5), 444-451 (1980).">(Aubrun, 1980)</a></sup>
+
 # Bibliography
-<a id="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Preumont, A., *Vibration control of active structures - fourth edition* (2018), : Springer International Publishing. [↩](#454500a3af67ef66a7a754d1f2e1bd4a)
+<a class="bibtex-entry" id="preumont18_vibrat_contr_activ_struc_fourt_edition">Preumont, A., *Vibration control of active structures - fourth edition* (2018), : Springer International Publishing.</a> [↩](#454500a3af67ef66a7a754d1f2e1bd4a)
+
+<a class="bibtex-entry" id="williams89_limit">Williams, T., & Antsaklis, P., *Limitations of vibration suppression in flexible space structures*, In , Proceedings of the 28th IEEE Conference on Decision and Control (pp. ) (1989). : .</a> [↩](#ef357e45dadd8cc8869beda6e463777b)
+
+<a class="bibtex-entry" id="aubrun80_theor_contr_struc_by_low_author_contr">Aubrun, J., *Theory of the control of structures by low-authority controllers*, Journal of Guidance and Control, *3(5)*, 444–451 (1980).  http://dx.doi.org/10.2514/3.56019</a> [↩](#df6fde1eeef81966b2c7fb5421adbe8d)
+
+
+## Backlinks {#backlinks}
+
+-   [Control of spacecraft and aircraft]({{< relref "bryson93_contr_spacec_aircr" >}})
+-   [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})

content/zettels/inertial_sensors.md

@@ -25,7 +25,7 @@ Wireless Accelerometers
 
 -   <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
 
-<a id="orgdad9a09"></a>
+<a id="org868d283"></a>
 
 {{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 1: Characteristics of commercially available accelerometers <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration  measurement, cern, (2011).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
 
@@ -37,7 +37,7 @@ Wireless Accelerometers
 | Sercel   | [link](http://www.sercel.com/products/Pages/seismometers.aspx) |
 | Wilcoxon | [link](https://wilcoxon.com/)                                  |
 
-<a id="org8c39d2f"></a>
+<a id="orgbf3a5fe"></a>
 
 {{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 2: Characteristics of commercially available geophones <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration  measurement, cern, (2011).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
 
@@ -45,3 +45,8 @@ Wireless Accelerometers
 <a id="collette12_review"></a>Collette, C., Janssens, S., Fernandez-Carmona, P., Artoos, K., Guinchard, M., Hauviller, C., & Preumont, A., *Review: inertial sensors for low-frequency seismic vibration measurement*, Bulletin of the Seismological Society of America, *102(4)*, 1289–1300 (2012).  http://dx.doi.org/10.1785/0120110223 [↩](#dd5109075933cf543c7eba0979c0ba50)
 
 <a id="collette11_review"></a>Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011). [↩](#642a18d86de4e062c6afb0f5f20501c4)
+
+
+## Backlinks {#backlinks}
+
+-   [Position Sensors]({{< relref "position_sensors" >}})

content/zettels/irr_and_fir_filters.md

@@ -0,0 +1,52 @@
++++
+title = "IRR and FIR Filters"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<div class="table-caption">
+  <span class="table-number">Table 1</span>:
+  Comparison of IRR and FIR Filters
+</div>
+
+|           | **IIR**                                    | **FIR**                               |
+|-----------|--------------------------------------------|---------------------------------------|
+| Phase     | No particular phase                        | Linear phase possible                 |
+| Stability | Can be unstable                            | Always stable (feedback not involved) |
+| Analog    | Derived from analog filter                 | Cannot simulate analog response       |
+| Linearity | Used for applications which are not linear | Linear-phase characteristic           |
+| num/den   | Both numerator and denominator             | Only has numerators                   |
+
+> Digital filters with finite-duration impulse response (all-zero, or FIR filters) have both advantages and disadvantages compared to infinite-duration impulse response (IIR) filters.
+>
+> FIR filters have the following primary advantages:
+>
+> -   They can have exactly linear phase.
+> -   They are always stable.
+> -   The design methods are generally linear.
+> -   They can be realized efficiently in hardware.
+> -   The filter startup transients have finite duration.
+>
+> The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.
+
+From <sup id="d875134273304770f6a0334525ecfa27"><a class="reference-link" href="#shaw90_bandw_enhan_posit_measur_using_measur_accel" title="Shaw \&amp; Srinivasan, Bandwidth Enhancement of Position Measurements Using  Measured Acceleration, {Mechanical Systems and Signal Processing}, v(1), 23-38 (1990).">(Shaw \& Srinivasan, 1990)</a></sup>
+
+> The FIR are capable of realizing filters with linear phase shift characteristics and furthermore are less susceptible to signal input and filter coefficient quantization effects.
+> However, their computational demands are excessively large because of the large number of multiplications and additions to be performed at each sampling interval.
+> The effective time delay corresponding to the linear phase shift is large and would have a destabilizing effect in closed loop applications.
+> IIR filters are computationally less demanding. The fact that their phase shift characteristics do not vary linearly with frequency is not a disadvantage in this application.
+> IIR filters are however, more susceptible to signal input and coefficient quantization effects.
+
+From <https://dsp.stackexchange.com/a/30999>
+
+> FIR filters are fairly common in some areas of control theory. As they usually incur a lot of added phase/time-delay, they are not really usable in the feedback path of regular control systems, but they are useful when the added phase/time-delay is not affecting the system in an adverse way, or when the particular phase response and time-delay is desired.
+>
+> Examples:
+>
+> -   Feed-forward control. FIR filters are useful for producing filters that approximate arbitrary frequency responses, hence they can be used to shape a reference signal. A typical example is to use an FIR filter with the inverse frequency response of the plant -- trying to counteract the dynamics of the plant in order to get a desired output. Phase/time-delay is not interfering with the stability or performance since the computation can be done offline. FIR filters can often produce higher performance than IIR filters, especially where there are non-minimum phase zeros.
+
+# Bibliography
+<a class="bibtex-entry" id="shaw90_bandw_enhan_posit_measur_using_measur_accel">Shaw, F., & Srinivasan, K., *Bandwidth enhancement of position measurements using measured acceleration*, Mechanical Systems and Signal Processing, *4(1)*, 23–38 (1990).  http://dx.doi.org/10.1016/0888-3270(90)90038-m</a> [↩](#d875134273304770f6a0334525ecfa27)

content/zettels/matlab.md

@@ -0,0 +1,66 @@
++++
+title = "Matlab"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+## Resources on Matlab {#resources-on-matlab}
+
+Books:
+
+-   <sup id="88712982e0649b89da706b6abbcbc6c2"><a href="#higham17_matlab" title="Higham, MATLAB guide, Society for Industrial and Applied Mathematics (2017).">(Higham, 2017)</a></sup>
+-   <sup id="15f4380b6ce8a647387d3ccea25711f1"><a href="#attaway18_matlab" title="Attaway, MATLAB : a practical introduction to programming and  problem solving, Butterworth-Heinemann (2018).">(Attaway, 2018)</a></sup>
+-   <sup id="e770e23b0d222a65eb74f036227b13b2"><a href="#overflow18_matlab_notes_profes" title="Stack OverFlow, MATLAB Notes for Professionals, GoalKicker.com (2018).">(Stack OverFlow, 2018)</a></sup>
+-   <sup id="87b279fa5b4ec9b1a73abed2d00b313f"><a href="#johnson10_matlab" title="Johnson, The elements of MATLAB style, Cambridge University Press (2010).">(Johnson, 2010)</a></sup>
+-   <sup id="1b4159c36c5367ee0c92139fb403e7e1"><a href="#hahn16_essen_matlab" title="Hahn \&amp; Valentine, Essential MATLAB for engineers and scientists, Academic Press (2016).">(Hahn \& Valentine, 2016)</a></sup>
+
+
+## Useful Commands {#useful-commands}
+
+| Command                | Description                                                 |
+|------------------------|-------------------------------------------------------------|
+| `desktop`              | Open the Matlab Desktop                                     |
+| `workspace`            | Open the Workspace                                          |
+| `who`                  | List all variables in the workspace                         |
+| `edit <filename>`      | Edit the file using Matlab Desktop (usefully for debugging) |
+| `help <function>`      |                                                             |
+| `doc <function>`       |                                                             |
+| `checkcode <filename>` | Check Matlab code files for possible problems               |
+| `preferences`          | Open Matlab preferences                                     |
+
+
+## Tips {#tips}
+
+-   Folder that starts with a `+` are automatically added to the path.
+    It is useful to add function inside such folder.
+    Then the function is accessible with `folder.function`.
+
+
+## Snippets {#snippets}
+
+
+### Do not show legend for one plot {#do-not-show-legend-for-one-plot}
+
+```matlab
+figure;
+hold on;
+plot(x, y1, 'DisplayName, 'lengendname');
+plot(x, y2, 'HandleVisibility', 'off');
+hold off;
+legend('Location', 'northeast');
+```
+
+# Bibliography
+<a id="higham17_matlab"></a>Higham, D., *Matlab guide* (2017), Philadelphia: Society for Industrial and Applied Mathematics. [↩](#88712982e0649b89da706b6abbcbc6c2)
+
+<a id="attaway18_matlab"></a>Attaway, S., *Matlab : a practical introduction to programming and problem solving* (2018), Amsterdam: Butterworth-Heinemann. [↩](#15f4380b6ce8a647387d3ccea25711f1)
+
+<a id="overflow18_matlab_notes_profes"></a>OverFlow, S., *Matlab notes for professionals* (2018), : GoalKicker.com. [↩](#e770e23b0d222a65eb74f036227b13b2)
+
+<a id="johnson10_matlab"></a>Johnson, R. K., *The elements of matlab style* (2010), : Cambridge University Press. [↩](#87b279fa5b4ec9b1a73abed2d00b313f)
+
+<a id="hahn16_essen_matlab"></a>Hahn, B., & Valentine, D. T., *Essential matlab for engineers and scientists* (2016), : Academic Press. [↩](#1b4159c36c5367ee0c92139fb403e7e1)

content/zettels/multivariable_control.md

@@ -5,15 +5,15 @@ draft = false
 +++
 
 Tags
-:
+: [Norms]({{< relref "norms" >}})
 
 <./biblio/references.bib>
 
 
 ## Backlinks {#backlinks}
 
+-   [Multivariable control systems: an engineering approach]({{< relref "albertos04_multiv_contr_system" >}})
 -   [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
 -   [Implementation challenges for multivariable control: what you did not learn in school!]({{< relref "garg07_implem_chall_multiv_contr" >}})
 -   [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
--   [Multivariable control systems: an engineering approach]({{< relref "albertos04_multiv_contr_system" >}})
 -   [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})

content/zettels/nano_active_stabilization_system.md

@@ -12,6 +12,6 @@ Tags
 
 ## Backlinks {#backlinks}
 
+-   [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})
 -   [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})
 -   [Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution]({{< relref "wang12_autom_marker_full_field_hard" >}})
--   [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})

content/zettels/norms.md

@@ -0,0 +1,64 @@
++++
+title = "Systems and Signals Norms"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+Resources:
+
+-   <sup id="ad6f62e369b7a8d31c21671886adec1f"><a href="#skogestad07_multiv_feedb_contr" title="Skogestad \&amp; Postlethwaite, Multivariable Feedback Control: Analysis and Design, John Wiley (2007).">(Skogestad \& Postlethwaite, 2007)</a></sup>
+-   <sup id="90e96a2c8cdb40b7bdf895cf013c0946"><a href="#toivonen02_robus_contr_method" title="@misc{toivonen02_robus_contr_method,
+      author          = {Hannu T. Toivonen},
+      institution     = {Abo Akademi University},
+      title           = {Robust Control Methods},
+      year            = 2002,
+    }">(Hannu Toivonen, 2002)</a></sup>
+-   <sup id="8db224194542fbd4c7f4fbe56fdd4e73"><a href="#zhang11_quant_proces_contr_theor" title="Zhang, Quantitative Process Control Theory, CRC Press (2011).">(Zhang, 2011)</a></sup>
+
+
+## \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
+
+SISO Systems => absolute value => bode plot
+MIMO Systems => singular value
+Signal
+
+
+## \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
+
+RMS value
+
+The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
+
+As explained in <sup id="651e626e040250ee71a0847aec41b60c"><a href="#monkhorst04_dynam_error_budget" title="@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}">@phdthesis{monkhorst04_dynam_error_budget,
+  author          = {Wouter Monkhorst},
+  school          = {Delft University},
+  title           = {Dynamic Error Budgeting, a design approach},
+  year            = 2004,
+}</a></sup>, the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
+
+> The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
+
+
+## Link between signal and system norms {#link-between-signal-and-system-norms}
+
+# Bibliography
+<a id="skogestad07_multiv_feedb_contr"></a>Skogestad, S., & Postlethwaite, I., *Multivariable feedback control: analysis and design* (2007), : John Wiley. [↩](#ad6f62e369b7a8d31c21671886adec1f)
+
+<a id="toivonen02_robus_contr_method"></a>Toivonen, H. T. (2002). *Robust Control Methods*. Retrieved from [](). . [↩](#90e96a2c8cdb40b7bdf895cf013c0946)
+
+<a id="zhang11_quant_proces_contr_theor"></a>Zhang, W., *Quantitative Process Control Theory* (2011), : CRC Press. [↩](#8db224194542fbd4c7f4fbe56fdd4e73)
+
+<a id="monkhorst04_dynam_error_budget"></a>Monkhorst, W., *Dynamic error budgeting, a design approach* (Doctoral dissertation) (2004). Delft University, . [↩](#651e626e040250ee71a0847aec41b60c)
+
+
+## Backlinks {#backlinks}
+
+-   [Multivariable Control]({{< relref "multivariable_control" >}})

content/zettels/position_sensors.md

@@ -5,18 +5,24 @@ draft = false
 +++
 
 Tags
-: [Inertial Sensors]({{< relref "inertial_sensors" >}})
+: [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
 
 
-## Reviews of position sensors {#reviews-of-position-sensors}
+## Absolute Position Sensors {#absolute-position-sensors}
 
--   <sup id="0b0b67de6dddc4d28031ab2d3b28cd3d"><a href="#collette12_compar" title="Collette, Janssens, Mokrani, Fueyo-Roza, L, Artoos, Esposito, Fernandez-Carmona, , Guinchard \&amp; Leuxe, Comparison of new absolute displacement sensors, in in: {International Conference on Noise and Vibration Engineering
+-   Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement <sup id="dd5109075933cf543c7eba0979c0ba50"><a href="#collette12_review" title="Collette, Janssens, Fernandez-Carmona, , Artoos, Guinchard, Hauviller \&amp; Preumont, Review: Inertial Sensors for Low-Frequency Seismic  Vibration Measurement, {Bulletin of the Seismological Society of America}, v(4), 1289-1300 (2012).">(Collette {\it et al.}, 2012)</a></sup>
+-   Collette, C. et al., Comparison of new absolute displacement sensors <sup id="0b0b67de6dddc4d28031ab2d3b28cd3d"><a href="#collette12_compar" title="Collette, Janssens, Mokrani, Fueyo-Roza, L, Artoos, Esposito, Fernandez-Carmona, , Guinchard \&amp; Leuxe, Comparison of new absolute displacement sensors, in in: {International Conference on Noise and Vibration Engineering
                       (ISMA)}, edited by (2012)">(Collette {\it et al.}, 2012)</a></sup>
--   Fleming, A. J., A review of nanometer resolution position sensors: operation and performance <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors:  Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup> ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}))
+
+<a id="org436fa72"></a>
+
+{{< figure src="/ox-hugo/collette12_absolute_disp_sensors.png" caption="Figure 1: Dynamic range of several types of inertial sensors; Price versus resolution for several types of inertial sensors" >}}
 
 
 ## Relative Position Sensors {#relative-position-sensors}
 
+-   Fleming, A. J., A review of nanometer resolution position sensors: operation and performance <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors:  Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup> ([Notes]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}}))
+
 <a id="table--tab:characteristics-relative-sensor"></a>
 <div class="table-caption">
   <span class="table-number"><a href="#table--tab:characteristics-relative-sensor">Table 1</a></span>:
@@ -111,9 +117,9 @@ Description:
 <sup id="7658b1219a4458a62ae8c6f51b767542"><a href="#jang17_compen_refrac_index_air_laser" title="Yoon-Soo Jang \&amp; Seung-Woo Kim, Compensation of the Refractive Index of Air in Laser  Interferometer for Distance Measurement: a Review, {International Journal of Precision Engineering and
                   Manufacturing}, v(12), 1881-1890 (2017).">(Yoon-Soo Jang \& Seung-Woo Kim, 2017)</a></sup>
 
-<a id="orge1e204f"></a>
+<a id="orgb68b41e"></a>
 
-{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}}
+{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 2: Expected precision of interferometer as a function of measured distance" >}}
 
 
 ### Fiber Optic Displacement Sensor {#fiber-optic-displacement-sensor}
@@ -123,6 +129,8 @@ Description:
 | Unipulse | [link](https://www.unipulse.com/product/atw200-2/) |
 
 # Bibliography
+<a id="collette12_review"></a>Collette, C., Janssens, S., Fernandez-Carmona, P., Artoos, K., Guinchard, M., Hauviller, C., & Preumont, A., *Review: inertial sensors for low-frequency seismic vibration measurement*, Bulletin of the Seismological Society of America, *102(4)*, 1289–1300 (2012).  http://dx.doi.org/10.1785/0120110223 [↩](#dd5109075933cf543c7eba0979c0ba50)
+
 <a id="collette12_compar"></a>Collette, C., Janssens, S., Mokrani, B., Fueyo-Roza, L., Artoos, K., Esposito, M., Fernandez-Carmona, P., …, *Comparison of new absolute displacement sensors*, In , International Conference on Noise and Vibration Engineering (ISMA) (pp. ) (2012). : . [↩](#0b0b67de6dddc4d28031ab2d3b28cd3d)
 
 <a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013).  http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)
@@ -134,5 +142,6 @@ Description:
 
 ## Backlinks {#backlinks}
 
--   [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
 -   [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
+-   [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
+-   [Inertial Sensors]({{< relref "inertial_sensors" >}})

content/zettels/positioning_stations.md

@@ -12,6 +12,6 @@ Tags
 
 ## Backlinks {#backlinks}
 
--   [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})
 -   [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
 -   [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})
+-   [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})

content/zettels/precision_engineering.md

@@ -12,5 +12,5 @@ Tags
 
 ## Backlinks {#backlinks}
 
--   [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}})
 -   [Basics of precision engineering - 1st edition]({{< relref "leach18_basic_precis_engin_edition" >}})
+-   [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}})

content/zettels/reference_books.md

@@ -15,6 +15,6 @@ Tags
 -   [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
 -   [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
 -   [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
+-   [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
 -   [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
 -   [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
--   [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})