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25
content/article/alkhatib03_activ_struc_vibrat_contr.md
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@@ -1,6 +1,6 @@
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title = "Active structural vibration control: a review"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@@ -9,10 +9,10 @@ Tags
Reference
: ([Alkhatib and Golnaraghi 2003](#orgdec9959))
: (<a href="#citeproc_bib_item_1">Alkhatib and Golnaraghi 2003</a>)
Author(s)
: Alkhatib, R., & Golnaraghi, M. F.
: Alkhatib, R., &amp; Golnaraghi, M. F.
Year
: 2003
@@ -123,14 +123,14 @@ 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](#orgd2fc896).
A general block diagram of the control system is shown figure [1](#figure--fig:alkhatib03-hinf-control).
A **frequency shaped filter** \\(W(s)\\) coupled to selected inputs and outputs of the plant is included.
The outputs of this frequency shaped filter define the error ouputs used to evaluate the system performance and generate the **cost** that will be used in the design process.
<a id="orgd2fc896"></a>
<a id="figure--fig:alkhatib03-hinf-control"></a>
{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="Figure 1: Block diagram for robust control" >}}
{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="<span class=\"figure-number\">Figure 1: </span>Block diagram for robust control" >}}
The generalized plan \\(G\\) can be partitionned according to the input-output variables. And we have that the transfer function matrix from \\(d\\) to \\(z\\) is:
\\[ H\_{z/d} = G\_{z/d} + G\_{z/u} K (I - G\_{y/u} K)^{-1} G\_{y/d} \\]
@@ -144,7 +144,7 @@ The objective of \\(\mathcal{H}\_\infty\\) control is to design an admissible co
The control \\(u(t)\\) is designed to minimize a cost function \\(J\\), given the initial conditions \\(z(t\_0)\\) and \\(\dot{z}(t\_0)\\) subject to the constraint that:
\begin{align\*}
\dot{z} &= Az + Bu\\\\\\
\dot{z} &= Az + Bu\\\\
y &= Cz
\end{align\*}
@@ -200,11 +200,11 @@ Two different methods
## Active Control Effects on the System {#active-control-effects-on-the-system}
<a id="org4678494"></a>
<a id="figure--fig:alkhatib03-1dof-control"></a>
{{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="Figure 2: 1 DoF control of a spring-mass-damping system" >}}
{{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="<span class=\"figure-number\">Figure 2: </span>1 DoF control of a spring-mass-damping system" >}}
Consider the control system figure [2](#org4678494), the equation of motion of the system is:
Consider the control system figure [2](#figure--fig:alkhatib03-1dof-control), the equation of motion of the system is:
\\[ m\ddot{x} + c\dot{x} + kx = f\_a + f \\]
The controller force can be expressed as: \\(f\_a = -g\_a \ddot{x} + g\_v \dot{x} + g\_d x\\). The equation of motion becomes:
@@ -225,7 +225,8 @@ 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 {#bibliography}
<a id="orgdec9959"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” _The Shock and Vibration Digest_ 35 (5):367–83. <https://doi.org/10.1177/05831024030355002>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” <i>The Shock and Vibration Digest</i> 35 (5): 367–83. doi:<a href="https://doi.org/10.1177/05831024030355002">10.1177/05831024030355002</a>.</div>
</div>

65
content/article/bibel92_guidel_h.md
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+++
title = "Guidelines for the selection of weighting functions for h-infinity control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [H Infinity Control]({{< relref "h_infinity_control" >}})
: [H Infinity Control]({{< relref "h_infinity_control.md" >}})
Reference
: ([Bibel and Malyevac 1992](#org395ccd3))
: (<a href="#citeproc_bib_item_1">Bibel and Malyevac 1992</a>)
Author(s)
: Bibel, J. E., & Malyevac, D. S.
: Bibel, J. E., &amp; Malyevac, D. S.
Year
: 1992
@@ -19,15 +19,15 @@ Year
## Properties of feedback control {#properties-of-feedback-control}
<a id="orgd464a3c"></a>
<a id="figure--fig:bibel92-control-diag"></a>
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="<span class=\"figure-number\">Figure 1: </span>Control System Diagram" >}}
From the figure [1](#orgd464a3c), we have:
From the figure [1](#figure--fig:bibel92-control-diag), we have:
\begin{align\*}
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\\\
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\
e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\
u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s)
\end{align\*}
@@ -38,17 +38,15 @@ With the following definitions
- \\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\\) is the **Transmissibility** function matrix
<div class="cbox">
<div></div>
\\[ S(s) + T(s) = 1 \\]
</div>
<div class="cbox">
<div></div>
- **Command following**: \\(S=0\\) and \\(T=1\\) => large gains
- **Disturbance rejection**: \\(S=0\\) => large gains
- **Command following**: \\(S=0\\) and \\(T=1\\) =&gt; large gains
- **Disturbance rejection**: \\(S=0\\) =&gt; large gains
- **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
- **Control Sensitivity minimization**: \\(K S\\) small
- **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
@@ -68,20 +66,19 @@ We must determine some **tradeoff** between the sensitivity and the complementar
Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \\(|S(j\omega)|\\) small as low frequencies and \\(|T(j\omega)|\\) small at high frequencies.
## \\(H\_\infty\\) and weighting functions {#h-infty--and-weighting-functions}
## \\(H\_\infty\\) and weighting functions {#h-infty-and-weighting-functions}
<div class="cbox">
<div></div>
\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
</div>
<a id="orgf088f75"></a>
<a id="figure--fig:bibel92-general-plant"></a>
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>\\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#orgf088f75)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
New design framework (figure [2](#figure--fig:bibel92-general-plant)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
- \\(w\\): exogenous inputs
- \\(z\\): regulated performance output
@@ -89,7 +86,7 @@ New design framework (figure [2](#orgf088f75)): \\(P(s)\\) is the **generalized
- \\(y\\): measured output variables
The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices:
\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\ P\_{21} & P\_{22} \end{bmatrix}\\]
\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\\ P\_{21} & P\_{22} \end{bmatrix}\\]
## Lower Linear Fractional Transformation {#lower-linear-fractional-transformation}
@@ -97,7 +94,6 @@ The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four
The transformation from the input \\(w\\) to the output \\(z\\), \\(T\_{zw}\\) is called the **Lower Linear Fractional Transformation** \\(F\_l (P, K)\\).
<div class="cbox">
<div></div>
\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
@@ -108,25 +104,24 @@ 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](#orgff0b295)).
Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#figure--fig:bibel92-hinf-weights)).
<a id="orgff0b295"></a>
<a id="figure--fig:bibel92-hinf-weights"></a>
{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="<span class=\"figure-number\">Figure 3: </span>Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
The weights on the input and output variables are selected to reflect the spatial and **frequency dependence** of the respective signals and performance specifications.
These inputs and output weighting functions are defined as rational, stable and **minimum-phase transfer function** (no poles or zero in the right half plane).
## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection--w-s}
## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection-w-s}
\\(W\_S\\) is selected to reflect the desired **performance characteristics**.
The sensitivity function \\(S\\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot.
We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shape of \\(S\\).
<div class="cbox">
<div></div>
- **Low frequency gain**: set to the inverse of the desired steady state tracking error
- **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
@@ -135,12 +130,11 @@ We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shap
</div>
## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection--w-t}
## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection-w-t}
We want \\(T\\) near unity for good tracking of reference and near zero for noise suppresion.
<div class="cbox">
<div></div>
A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
@@ -154,17 +148,17 @@ 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](#orgc150230)).
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#figure--fig:bibel92-unmodeled-dynamics)).
<a id="orgc150230"></a>
<a id="figure--fig:bibel92-unmodeled-dynamics"></a>
{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: Unmodeled dynamics model" >}}
{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="<span class=\"figure-number\">Figure 4: </span>Unmodeled dynamics model" >}}
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#org42e3b7d).
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#figure--fig:bibel92-weight-dynamics).
<a id="org42e3b7d"></a>
<a id="figure--fig:bibel92-weight-dynamics"></a>
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="<span class=\"figure-number\">Figure 5: </span>Example of unmodeled dynamics weight" >}}
## Inputs and Output weighting function {#inputs-and-output-weighting-function}
@@ -182,7 +176,8 @@ 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 {#bibliography}
<a id="org395ccd3"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.</div>
</div>

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content/article/bryson93_contr_spacec_aircr.md
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+++
title = "Control of spacecraft and aircraft"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@@ -9,7 +9,7 @@ Tags
Reference
: ([Bryson 1993](#org14ecce3))
: (<a href="#citeproc_bib_item_1">Bryson 1993</a>)
Author(s)
: Bryson, A. E.
@@ -20,7 +20,7 @@ Year
## 9.2.3 Roll-Off Filters {#9-dot-2-dot-3-roll-off-filters}
[Spillover Effect]({{< relref "spillover_effect" >}})
[Spillover Effect]({{< relref "spillover_effect.md" >}})
> Synthesizing control logic using only one vibration mode means we are consciously **neglecting the higher-order vibration modes**.
> When doing this, it is a good idea to insert "roll-off" into the control logic, so that the loop-transfer gain decreases rapidly with frequency beyond the control bandwidth.
@@ -38,20 +38,21 @@ Year
> If a rate sensor is not co-located with an actuator on a flexible body, ans its signal is fed back to the actuator, some vibration modes are stabilized and others are destabilized, depending on the location of the sensor relative to the actuator.
## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac.md" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
> Figure [fig:bryson93_hac_lac](#fig:bryson93_hac_lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
> Figure <fig:bryson93_hac_lac> shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
> LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
> HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
<a id="orgc9c1915"></a>
<a id="figure--fig:bryson93-hac-lac"></a>
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="Figure 1: HAC-LAC control concept" >}}
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="<span class=\"figure-number\">Figure 1: </span>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 {#bibliography}
<a id="org14ecce3"></a>Bryson, Arthur Earl. 1993. _Control of Spacecraft and Aircraft_. Princeton university press Princeton, New Jersey.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bryson, Arthur Earl. 1993. <i>Control of Spacecraft and Aircraft</i>. Princeton university press Princeton, New Jersey.</div>
</div>

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content/article/butler11_posit_contr_lithog_equip.md
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@@ -1,14 +1,14 @@
+++
title = "Position control in lithographic equipment"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Multivariable Control]({{<relref "multivariable_control.md#" >}}), [Positioning Stations]({{<relref "positioning_stations.md#" >}})
: [Multivariable Control]({{< relref "multivariable_control.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
Reference
: ([Butler 2011](#org9e15931))
: (<a href="#citeproc_bib_item_1">Butler 2011</a>)
Author(s)
: Butler, H.
@@ -17,7 +17,8 @@ Year
: 2011
## Bibliography {#bibliography}
<a id="org9e15931"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” _IEEE Control Systems_ 31 (5):28–47. <https://doi.org/10.1109/mcs.2011.941882>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” <i>Ieee Control Systems</i> 31 (5): 28–47. doi:<a href="https://doi.org/10.1109/mcs.2011.941882">10.1109/mcs.2011.941882</a>.</div>
</div>

32
content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md
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@@ -1,17 +1,17 @@
+++
title = "Identification and decoupling control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Chen and McInroy 2000](#org1c74a9c))
: (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>)
Author(s)
: Chen, Y., & McInroy, J.
: Chen, Y., &amp; McInroy, J.
Year
: 2000
@@ -31,7 +31,7 @@ Year
## Introduction {#introduction}
Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control" >}})) impose two constraints:
Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control.md" >}})) impose two constraints:
- the payload mass/inertia matrix is diagonal
- the geometry of the platform and attachment of the payload must be carefully chosen
@@ -43,11 +43,11 @@ The algorithm derived herein removes these constraints, thus greatly expanding t
## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
The derivation of the dynamic model is done in ([McInroy 1999](#orgebf33dd)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
The derivation of the dynamic model is done in (<a href="#citeproc_bib_item_2">McInroy 1999</a>) ([Notes]({{< relref "mcinroy99_dynam.md" >}})).
<a id="orga594879"></a>
<a id="figure--fig:chen00-flexure-hexapod"></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" >}}
{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="<span class=\"figure-number\">Figure 1: </span>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:
@@ -56,9 +56,9 @@ In the joint space, the dynamics of a flexure jointed hexapod are written as:
\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\_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}
@@ -79,7 +79,7 @@ where:
- \\(\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}\_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}
@@ -100,9 +100,9 @@ where
## Experimental Results {#experimental-results}
## Bibliography {#bibliography}
<a id="org1c74a9c"></a>Chen, Yixin, and J.E. McInroy. 2000. “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)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="orgebf33dd"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 Icra. Millennium Conference. Ieee International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00ch37065)</i>, nil. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
</div>

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title = "Amplified piezoelectric actuators: static & dynamic applications"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
Reference
: ([Claeyssen et al. 2007](#org66395f6))
: (<a href="#citeproc_bib_item_1">Claeyssen et al. 2007</a>)
Author(s)
: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
: Claeyssen, F., Letty, R. L., Barillot, F., &amp; Sosnicki, O.
Year
: 2007
@@ -34,7 +34,8 @@ The maximum dynamic force achievable by the actuator is determined by the prestr
The prestress design allows a peak force equal to half the blocked force.
## Bibliography {#bibliography}
<a id="org66395f6"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static &#38; Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
</div>

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title = "Review of active vibration isolation strategies"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Collette, Janssens, and Artoos 2011](#orgc3712d7))
: (<a href="#citeproc_bib_item_1">Collette, Janssens, and Artoos 2011</a>)
Author(s)
: Collette, C., Janssens, S., & Artoos, K.
: Collette, C., Janssens, S., &amp; Artoos, K.
Year
: 2011
@@ -70,12 +70,13 @@ The general expression of the force delivered by the actuator is \\(f = g\_a \dd
## Conclusions {#conclusions}
<a id="orgdceedb5"></a>
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="Figure 1: Comparison of Active Vibration Isolation Strategies" >}}
<a id="figure--fig:collette11-comp-isolation-strategies"></a>
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="<span class=\"figure-number\">Figure 1: </span>Comparison of Active Vibration Isolation Strategies" >}}
## Bibliography {#bibliography}
<a id="orgc3712d7"></a>Collette, Christophe, Stef Janssens, and Kurt Artoos. 2011. “Review of Active Vibration Isolation Strategies.” _Recent Patents on Mechanical Engineeringe_ 4 (3):212–19. <https://doi.org/10.2174/2212797611104030212>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, Christophe, Stef Janssens, and Kurt Artoos. 2011. “Review of Active Vibration Isolation Strategies.” <i>Recent Patents on Mechanical Engineeringe</i> 4 (3): 212–19. doi:<a href="https://doi.org/10.2174/2212797611104030212">10.2174/2212797611104030212</a>.</div>
</div>

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title = "Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
Reference
: ([Collette and Matichard 2014](#org6b92a7c))
: (<a href="#citeproc_bib_item_1">Collette and Matichard 2014</a>)
Author(s)
: Collette, C., & Matichard, F.
: Collette, C., &amp; Matichard, F.
Year
: 2014
@@ -19,7 +19,7 @@ Year
## Introduction {#introduction}
[Sensor Fusion]({{< relref "sensor_fusion" >}}) is used to combine the benefits of different types of sensors:
[Sensor Fusion]({{< relref "sensor_fusion.md" >}}) is used to combine the benefits of different types of sensors:
- Relative sensor for DC positioning capability at low frequency
- Inertial sensors for isolation at high frequency
@@ -100,7 +100,8 @@ 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 {#bibliography}
<a id="org6b92a7c"></a>Collette, C., and F Matichard. 2014. “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)_.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In <i>International Conference on Noise and Vibration Engineering (Isma2014)</i>.</div>
</div>

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title = "Sensor fusion methods for high performance active vibration isolation systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Collette and Matichard 2015](#orgdf378e9))
: (<a href="#citeproc_bib_item_1">Collette and Matichard 2015</a>)
Author(s)
: Collette, C., & Matichard, F.
: Collette, C., &amp; Matichard, F.
Year
: 2015
@@ -25,7 +25,8 @@ 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 {#bibliography}
<a id="orgdf378e9"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” _Journal of Sound and Vibration_ 342 (nil):1–21. <https://doi.org/10.1016/j.jsv.2015.01.006>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” <i>Journal of Sound and Vibration</i> 342 (nil): 1–21. doi:<a href="https://doi.org/10.1016/j.jsv.2015.01.006">10.1016/j.jsv.2015.01.006</a>.</div>
</div>

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title = "The stewart platform manipulator: a review"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
Reference
: ([Dasgupta and Mruthyunjaya 2000](#org9c198f3))
: (<a href="#citeproc_bib_item_1">Dasgupta and Mruthyunjaya 2000</a>)
Author(s)
: Dasgupta, B., & Mruthyunjaya, T.
: Dasgupta, B., &amp; Mruthyunjaya, T.
Year
: 2000
@@ -34,7 +34,8 @@ Year
The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platform) connected through six extensible legs, each with spherical joints at both ends.
## Bibliography {#bibliography}
<a id="org9c198f3"></a>Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” _Mechanism and Machine Theory_ 35 (1):15–40. <https://doi.org/10.1016/s0094-114x(99)>00006-3.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” <i>Mechanism and Machine Theory</i> 35 (1): 15–40. doi:<a href="https://doi.org/10.1016/s0094-114x(99)00006-3">10.1016/s0094-114x(99)00006-3</a>.</div>
</div>

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title = "A survey of control issues in nanopositioning"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@@ -9,25 +9,26 @@ Tags
Reference
: ([Devasia, Eleftheriou, and Moheimani 2007](#orgfa66307))
: (<a href="#citeproc_bib_item_1">Devasia, Eleftheriou, and Moheimani 2007</a>)
Author(s)
: Devasia, S., Eleftheriou, E., & Moheimani, S. R.
: Devasia, S., Eleftheriou, E., &amp; Moheimani, S. R.
Year
: 2007
- Talks about Scanning Tunneling Microscope (STM) and Scanning Probe Microscope (SPM)
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}): Creep, Hysteresis, Vibrations, Modeling errors
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}): Creep, Hysteresis, Vibrations, Modeling errors
- Interesting analysis about Bandwidth-Precision-Range tradeoffs
- Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
<a id="orgd34b44a"></a>
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="Figure 1: Tradeoffs between bandwidth, precision and range" >}}
<a id="figure--fig:devasia07-piezoelectric-tradeoff"></a>
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="<span class=\"figure-number\">Figure 1: </span>Tradeoffs between bandwidth, precision and range" >}}
## Bibliography {#bibliography}
<a id="orgfa66307"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” _IEEE Transactions on Control Systems Technology_ 15 (5). IEEE:802–23.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” <i>Ieee Transactions on Control Systems Technology</i> 15 (5). IEEE: 802–23.</div>
</div>

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+++
title = "Nanopositioning system with force feedback for high-performance tracking and vibration control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}})
Reference
: ([Fleming 2010](#org21788cf))
: (<a href="#citeproc_bib_item_1">Fleming 2010</a>)
Author(s)
: Fleming, A.
@@ -31,9 +31,9 @@ Year
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
<a id="org699947b"></a>
<a id="figure--fig:fleming10-piezo-model"></a>
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="<span class=\"figure-number\">Figure 1: </span>Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
The actuator experiences an internal stress in response to an applied voltage.
This stress is represented by the voltage dependent force \\(F\_a\\) and is related to free displacement by
@@ -78,7 +78,7 @@ If an **n-layer** piezoelectric transducer is used as a force sensor, the genera
We can use a **charge amplifier** to measure the force \\(F\_s\\).
{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="Figure 2: Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="<span class=\"figure-number\">Figure 2: </span>Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
The output voltage \\(V\_s\\) is equal to
\\[ V\_s = -\frac{q}{C\_s} = -\frac{n d\_{33}F\_s}{C\_s} \\]
@@ -116,12 +116,13 @@ The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and
## Tested feedback control strategies {#tested-feedback-control-strategies}
<a id="orgc6b14a0"></a>
{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="Figure 3: Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}
<a id="figure--fig:fleming10-fb-control-strats"></a>
{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="<span class=\"figure-number\">Figure 3: </span>Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}
## Bibliography {#bibliography}
<a id="org21788cf"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
</div>

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title = "Estimating the resolution of nanopositioning systems from frequency domain data"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming 2012](#org26b3187))
: (<a href="#citeproc_bib_item_1">Fleming 2012</a>)
Author(s)
: Fleming, A. J.
@@ -18,7 +18,8 @@ Year
: 2012
## Bibliography {#bibliography}
<a id="org26b3187"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In _2012 IEEE International Conference on Robotics and Automation_, nil. <https://doi.org/10.1109/icra.2012.6224850>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In <i>2012 Ieee International Conference on Robotics and Automation</i>, nil. doi:<a href="https://doi.org/10.1109/icra.2012.6224850">10.1109/icra.2012.6224850</a>.</div>
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+++
title = "A review of nanometer resolution position sensors: operation and performance"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Position Sensors]({{< relref "position_sensors" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
Reference
: ([Fleming 2013](#org687716f))
: (<a href="#citeproc_bib_item_1">Fleming 2013</a>)
Author(s)
: Fleming, A. J.
@@ -28,28 +28,28 @@ Year
Usually quoted as a percentage of the fill-scale range (FSR):
\begin{equation}
\text{mapping error (\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
\text{mapping error (\\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
\end{equation}
With \\(e\_m(v)\\) is the mapping error.
<a id="org0a1d321"></a>
<a id="figure--fig:mapping-error"></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." >}}
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="<span class=\"figure-number\">Figure 1: </span>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." >}}
### Drift and Stability {#drift-and-stability}
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="orgc781e90"></a>
<a id="figure--fig:drift-stability"></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." >}}
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="<span class=\"figure-number\">Figure 2: </span>The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
### Bandwidth {#bandwidth}
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\,dB\\).
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\\,dB\\).
Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
@@ -57,7 +57,7 @@ The frequency domain position error is
\begin{equation}
\begin{aligned}
e\_{bw}(s) &= x(s) - v(s) \\\\\\
e\_{bw}(s) &= x(s) - v(s) \\\\
&= x(s) (1 - P(s))
\end{aligned}
\end{equation}
@@ -66,7 +66,7 @@ If the actual position is a sinewave of peak amplitude \\(A = \text{FSR}/2\\):
\begin{equation}
\begin{aligned}
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\\\
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\
&\approx \pm A n \frac{f}{f\_c}
\end{aligned}
\end{equation}
@@ -143,15 +143,15 @@ To characterize the resolution, we use the probability that the measured value i
If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
The empirical rule states that there is a \\(99.7\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
The empirical rule states that there is a \\(99.7\\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
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](#org86a5909)).
There is usually a trade-off between bandwidth and resolution (figure [3](#figure--fig:tradeoff-res-bandwidth)).
<a id="org86a5909"></a>
<a id="figure--fig:tradeoff-res-bandwidth"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="<span class=\"figure-number\">Figure 3: </span>The resolution versus banwidth of a position sensor." >}}
Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
@@ -170,19 +170,20 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
</div>
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|----------------|----------------------------------|---------|------------|----------|-----------|
| Metal foil | \\(10-500\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
| Piezoresistive | \\(1-500\,\mu m\\) | 5 ppm | 0.5 nm | >100 kHz | 1% FSR |
| Capacitive | \\(10\,\mu m\\) to \\(10\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
| Electrothermal | \\(10\,\mu m\\) to \\(1\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
| Eddy current | \\(100\,\mu m\\) to \\(80\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
| LVDT | \\(0.5-500\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|----------------|------------------------------------|---------|------------|-------------|-----------|
| Metal foil | \\(10-500\\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
| Piezoresistive | \\(1-500\\,\mu m\\) | 5 ppm | 0.5 nm | &gt;100 kHz | 1% FSR |
| Capacitive | \\(10\\,\mu m\\) to \\(10\\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
| Electrothermal | \\(10\\,\mu m\\) to \\(1\\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
| Eddy current | \\(100\\,\mu m\\) to \\(80\\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
| LVDT | \\(0.5-500\\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
| Interferometer | Meters | | 0.5 nm | &gt;100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | &gt;100kHz | 5 ppm FSR |
## Bibliography {#bibliography}
<a id="org687716f"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190 (nil): 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
</div>

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title = "Low-order damping and tracking control for scanning probe systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,16 +9,17 @@ Tags
Reference
: ([Fleming, Teo, and Leang 2015](#org26aec08))
: (<a href="#citeproc_bib_item_1">Fleming, Teo, and Leang 2015</a>)
Author(s)
: Fleming, A. J., Teo, Y. R., & Leang, K. K.
: Fleming, A. J., Teo, Y. R., &amp; Leang, K. K.
Year
: 2015
## Bibliography {#bibliography}
<a id="org26aec08"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” _Frontiers in Mechanical Engineering_ 1 (nil):nil. <https://doi.org/10.3389/fmech.2015.00014>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” <i>Frontiers in Mechanical Engineering</i> 1 (nil): nil. doi:<a href="https://doi.org/10.3389/fmech.2015.00014">10.3389/fmech.2015.00014</a>.</div>
</div>

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title = "Nanometre-cutting machine using a stewart-platform parallel mechanism"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Furutani, Suzuki, and Kudoh 2004](#org9d14335))
: (<a href="#citeproc_bib_item_1">Furutani, Suzuki, and Kudoh 2004</a>)
Author(s)
: Furutani, K., Suzuki, M., & Kudoh, R.
: Furutani, K., Suzuki, M., &amp; Kudoh, R.
Year
: 2004
@@ -26,7 +26,7 @@ Year
Possible sources of error:
- position error of the link ends in assembly => simulation of position error and it is not significant
- position error of the link ends in assembly =&gt; simulation of position error and it is not significant
- Inaccurate modelling of the links
- insufficient generative force
- unwanted deformation of the links
@@ -35,7 +35,8 @@ 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 {#bibliography}
<a id="org9d14335"></a>Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” _Measurement Science and Technology_ 15 (2):467–74. <https://doi.org/10.1088/0957-0233/15/2/022>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” <i>Measurement Science and Technology</i> 15 (2): 467–74. doi:<a href="https://doi.org/10.1088/0957-0233/15/2/022">10.1088/0957-0233/15/2/022</a>.</div>
</div>

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title = "Measurement technologies for precision positioning"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Position Sensors]({{<relref "position_sensors.md#" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
Reference
: ([Gao et al. 2015](#orgc8ea7ee))
: (<a href="#citeproc_bib_item_1">Gao et al. 2015</a>)
Author(s)
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
@@ -17,7 +17,8 @@ Year
: 2015
## Bibliography {#bibliography}
<a id="orgc8ea7ee"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” _CIRP Annals_ 64 (2):773–96. <https://doi.org/10.1016/j.cirp.2015.05.009>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” <i>Cirp Annals</i> 64 (2): 773–96. doi:<a href="https://doi.org/10.1016/j.cirp.2015.05.009">10.1016/j.cirp.2015.05.009</a>.</div>
</div>

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title = "Implementation challenges for multivariable control: what you did not learn in school!"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
: [Multivariable Control]({{< relref "multivariable_control.md" >}})
Reference
: ([Garg 2007](#org18482cb))
: (<a href="#citeproc_bib_item_1">Garg 2007</a>)
Author(s)
: Garg, S.
@@ -35,7 +35,8 @@ 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 {#bibliography}
<a id="org18482cb"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In _AIAA Guidance, Navigation and Control Conference and Exhibit_, nil. <https://doi.org/10.2514/6.2007-6334>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In <i>Aiaa Guidance, Navigation and Control Conference and Exhibit</i>, nil. doi:<a href="https://doi.org/10.2514/6.2007-6334">10.2514/6.2007-6334</a>.</div>
</div>

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title = "An intelligent control system for multiple degree-of-freedom vibration isolation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Geng et al. 1995](#orgb245b96))
: (<a href="#citeproc_bib_item_1">Geng et al. 1995</a>)
Author(s)
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A.
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., &amp; Garba, J. A.
Year
: 1995
<a id="orgec71c1f"></a>
{{< figure src="/ox-hugo/geng95_control_structure.png" caption="Figure 1: Local force feedback and adaptive acceleration feedback for active isolation" >}}
<a id="figure--fig:geng95-control-structure"></a>
{{< figure src="/ox-hugo/geng95_control_structure.png" caption="<span class=\"figure-number\">Figure 1: </span>Local force feedback and adaptive acceleration feedback for active isolation" >}}
## Bibliography {#bibliography}
<a id="orgb245b96"></a>Geng, Z. Jason, George G. Pan, Leonard S. Haynes, Ben K. Wada, and John A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” _Journal of Intelligent Material Systems and Structures_ 6 (6):787–800. <https://doi.org/10.1177/1045389x9500600607>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geng, Z. Jason, George G. Pan, Leonard S. Haynes, Ben K. Wada, and John A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” <i>Journal of Intelligent Material Systems and Structures</i> 6 (6): 787–800. doi:<a href="https://doi.org/10.1177/1045389x9500600607">10.1177/1045389x9500600607</a>.</div>
</div>

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title = "Sensors and control of a space-based six-axis vibration isolation system"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
Reference
: ([Hauge and Campbell 2004](#org186272b))
: (<a href="#citeproc_bib_item_1">Hauge and Campbell 2004</a>)
Author(s)
: Hauge, G., & Campbell, M.
: Hauge, G., &amp; Campbell, M.
Year
: 2004
@@ -24,22 +24,22 @@ Year
- Vibration isolation using a Stewart platform
- Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
<a id="org37bf22a"></a>
<a id="figure--fig:hauge04-stewart-platform"></a>
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Hexapod for active vibration isolation" >}}
**Stewart platform** (Figure [1](#org37bf22a)):
**Stewart platform** (Figure [1](#figure--fig:hauge04-stewart-platform)):
- Low corner frequency
- Large actuator stroke (\\(\pm5mm\\))
- Sensors in each strut (Figure [2](#org8b97871)):
- Sensors in each strut (Figure [2](#figure--fig:hauge05-struts)):
- three-axis load cell
- base and payload geophone in parallel with the struts
- LVDT
<a id="org8b97871"></a>
<a id="figure--fig:hauge05-struts"></a>
{{< figure src="/ox-hugo/hauge05_struts.png" caption="Figure 2: Strut" >}}
{{< figure src="/ox-hugo/hauge05_struts.png" caption="<span class=\"figure-number\">Figure 2: </span>Strut" >}}
> Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.
@@ -64,9 +64,9 @@ 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="org1bec2a6"></a>
<a id="figure--fig:hauge05-strut-model"></a>
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="Figure 3: Strut model" >}}
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="<span class=\"figure-number\">Figure 3: </span>Strut model" >}}
**Zero Pair when using a Force Sensor**:
@@ -76,8 +76,8 @@ With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and
**Control**:
- Single-axis controllers => combine them into a full six-axis controller => evaluate the full controller in terms of stability and robustness
- Sensitivity weighted LQG controller (SWLQG) => address robustness in flexible dynamic systems
- Single-axis controllers =&gt; combine them into a full six-axis controller =&gt; evaluate the full controller in terms of stability and robustness
- Sensitivity weighted LQG controller (SWLQG) =&gt; address robustness in flexible dynamic systems
- Three type of controller:
- Force feedback (cell-based)
- Inertial feedback (geophone-based)
@@ -126,7 +126,7 @@ And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain
**Inertial feedback**:
- Non-Collocated => multiple phase drops that limit the bandwidth of the controller
- Non-Collocated =&gt; multiple phase drops that limit the bandwidth of the controller
- Good performance, but the transmissibility "pops" due to low phase margin and thus this indicates robustness problems
**Combined force/velocity feedback**:
@@ -136,12 +136,13 @@ 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="org0a496f7"></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)" >}}
<a id="figure--fig:hauge04-obtained-transmissibility"></a>
{{< figure src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="<span class=\"figure-number\">Figure 4: </span>Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)" >}}
## Bibliography {#bibliography}
<a id="org186272b"></a>Hauge, G.S., and M.E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” _Journal of Sound and Vibration_ 269 (3-5):913–31. <https://doi.org/10.1016/s0022-460x(03)>00206-2.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hauge, G.S., and M.E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” <i>Journal of Sound and Vibration</i> 269 (3-5): 913–31. doi:<a href="https://doi.org/10.1016/s0022-460x(03)00206-2">10.1016/s0022-460x(03)00206-2</a>.</div>
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title = "An instrument for 3d x-ray nano-imaging"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
Reference
: ([Holler et al. 2012](#orgacde90c))
: (<a href="#citeproc_bib_item_1">Holler et al. 2012</a>)
Author(s)
: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O.
: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., &amp; Bunk, O.
Year
: 2012
@@ -19,9 +19,9 @@ Year
Instrument similar to the NASS.
Obtain position stability of 10nm (standard deviation).
<a id="org03c494c"></a>
<a id="figure--fig:holler12-station"></a>
{{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
{{< figure src="/ox-hugo/holler12_station.png" caption="<span class=\"figure-number\">Figure 1: </span>Schematic of the tomography setup" >}}
- **Limited resolution due to instrumentation**:
The resolution of ptychographic tomography remains above 100nm due to instabilities and drifts of the scanning systems.
@@ -39,7 +39,8 @@ 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 {#bibliography}
<a id="orgacde90c"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” _Review of Scientific Instruments_ 83 (7):073703. <https://doi.org/10.1063/1.4737624>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” <i>Review of Scientific Instruments</i> 83 (7): 073703. doi:<a href="https://doi.org/10.1063/1.4737624">10.1063/1.4737624</a>.</div>
</div>

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+++
title = "Active damping based on decoupled collocated control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Active Damping](active_damping.md)
: [Active Damping]({{< relref "active_damping.md" >}})
Reference
: ([Holterman and deVries 2005](#org5d6fef0))
: (<a href="#citeproc_bib_item_1">Holterman and deVries 2005</a>)
Author(s)
: Holterman, J., & deVries, T.
: Holterman, J., &amp; deVries, T.
Year
: 2005
## Bibliography {#bibliography}
<a id="org5d6fef0"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” _IEEE/ASME Transactions on Mechatronics_ 10 (2):135–45. <https://doi.org/10.1109/tmech.2005.844702>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 10 (2): 135–45. doi:<a href="https://doi.org/10.1109/tmech.2005.844702">10.1109/tmech.2005.844702</a>.</div>
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title = "Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Actuators]({{< relref "actuators" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Actuators]({{< relref "actuators.md" >}})
Reference
: ([Ito and Schitter 2016](#org3484be8))
: (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>)
Author(s)
: Ito, S., & Schitter, G.
: Ito, S., &amp; Schitter, G.
Year
: 2016
@@ -41,9 +41,9 @@ 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="org7e94abb"></a>
<a id="figure--fig:ito16-low-high-stiffness-actuators"></a>
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="<span class=\"figure-number\">Figure 1: </span>Definition of low-stiffness and high-stiffness actuator" >}}
## Low-Stiffness / High-Stiffness characteristics {#low-stiffness-high-stiffness-characteristics}
@@ -54,9 +54,9 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
## Controller Design {#controller-design}
<a id="org02696ae"></a>
<a id="figure--fig:ito16-transmissibility"></a>
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="<span class=\"figure-number\">Figure 2: </span>Obtained transmissibility" >}}
## Discussion {#discussion}
@@ -67,7 +67,8 @@ 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 {#bibliography}
<a id="org3484be8"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. <https://doi.org/10.1109/tmech.2015.2478658>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>Ieee/Asme Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
</div>

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+++
title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Jiao et al. 2018](#orgfa41a34))
: (<a href="#citeproc_bib_item_1">Jiao et al. 2018</a>)
Author(s)
: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
: Jiao, J., Wu, Y., Yu, K., &amp; Zhao, R.
Year
: 2018
## Bibliography {#bibliography}
<a id="orgfa41a34"></a>Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” _Journal of Vibration and Control_ 25 (1):151–71. <https://doi.org/10.1177/1077546318772474>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” <i>Journal of Vibration and Control</i> 25 (1): 151–71. doi:<a href="https://doi.org/10.1177/1077546318772474">10.1177/1077546318772474</a>.</div>
</div>

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+++
title = "A new isotropic and decoupled 6-dof parallel manipulator"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
Reference
: ([Legnani et al. 2012](#orga1e3bf2))
: (<a href="#citeproc_bib_item_1">Legnani et al. 2012</a>)
Author(s)
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., &amp; Tosi, D.
Year
: 2012
@@ -22,16 +22,17 @@ Year
Example of generated isotropic manipulator (not decoupled).
<a id="org0cc8ba8"></a>
<a id="figure--fig:legnani12-isotropy-gen"></a>
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="<span class=\"figure-number\">Figure 1: </span>Location of the leg axes using an isotropy generator" >}}
<a id="org0474665"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
<a id="figure--fig:legnani12-generated-isotropy"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="<span class=\"figure-number\">Figure 2: </span>Isotropic configuration" >}}
## Bibliography {#bibliography}
<a id="orga1e3bf2"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58 (nil): 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
</div>

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+++
title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: (<a href="#citeproc_bib_item_1">Li, Hamann, and McInroy 2001</a>)
Author(s)
: Li, X., Hamann, J. C., &amp; 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 {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In <i>Smart Structures and Materials 2001: Smart Structures and Integrated Systems</i>, nil. doi:<a href="https://doi.org/10.1117/12.436521">10.1117/12.436521</a>.</div>
</div>

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title = "Disturbance attenuation in precise hexapod pointing using positive force feedback"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,16 +9,17 @@ Tags
Reference
: ([Lin and McInroy 2006](#org5d8be72))
: (<a href="#citeproc_bib_item_1">Lin and McInroy 2006</a>)
Author(s)
: Lin, H., & McInroy, J. E.
: Lin, H., &amp; McInroy, J. E.
Year
: 2006
## Bibliography {#bibliography}
<a id="org5d8be72"></a>Lin, Haomin, and John E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” _Control Engineering Practice_ 14 (11):1377–86. <https://doi.org/10.1016/j.conengprac.2005.10.002>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lin, Haomin, and John E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” <i>Control Engineering Practice</i> 14 (11): 1377–86. doi:<a href="https://doi.org/10.1016/j.conengprac.2005.10.002">10.1016/j.conengprac.2005.10.002</a>.</div>
</div>

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content/article/mcinroy00_desig_contr_flexur_joint_hexap.md
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@@ -1,6 +1,6 @@
+++
title = "Design and control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,16 +9,17 @@ Tags
Reference
: ([McInroy and Hamann 2000](#orgaf3de6d))
: (<a href="#citeproc_bib_item_1">McInroy and Hamann 2000</a>)
Author(s)
: McInroy, J., & Hamann, J.
: McInroy, J., &amp; Hamann, J.
Year
: 2000
## Bibliography {#bibliography}
<a id="orgaf3de6d"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” <i>Ieee Transactions on Robotics and Automation</i> 16 (4): 372–81. doi:<a href="https://doi.org/10.1109/70.864229">10.1109/70.864229</a>.</div>
</div>

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title = "Modeling and design of flexure jointed stewart platforms for control purposes"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy 2002](#org2871bf9))
: (<a href="#citeproc_bib_item_2">McInroy 2002</a>)
Author(s)
: McInroy, J.
@@ -17,7 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to ([McInroy 1999](#org1d169f9)).
This short paper is very similar to (<a href="#citeproc_bib_item_1">McInroy 1999</a>).
> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
@@ -36,15 +36,15 @@ This short paper is very similar to ([McInroy 1999](#org1d169f9)).
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="org4ea1e8b"></a>
<a id="figure--fig:mcinroy02-leg-model"></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" >}}
{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="<span class=\"figure-number\">Figure 1: </span>The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#org4ea1e8b)).
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#figure--fig:mcinroy02-leg-model)).
Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
The model of the strut are shown in Figure [1](#org4ea1e8b) with:
The model of the strut are shown in Figure [1](#figure--fig:mcinroy02-leg-model) with:
- \\(m\_{s\_i}\\) moving strut mass
- \\(k\_i\\) spring constant
@@ -78,10 +78,10 @@ The payload is modeled as a rigid body:
\begin{equation}
\underbrace{\begin{bmatrix}
m I\_3 & 0\_{3\times 3} \\\\\\
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
0\_{3 \times 1} \\\ \omega \times {}^cI\omega
\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
\end{equation}
@@ -107,7 +107,7 @@ where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation mat
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}
\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:
@@ -115,10 +115,10 @@ 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:
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
\begin{equation}
{}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
{}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
\end{equation}
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
@@ -132,21 +132,21 @@ Many prior hexapod dynamic formulations assume that the strut exerts force only
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="org5bc5fa8"></a>
<a id="figure--fig:mcinroy02-model-strut-joint"></a>
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="Figure 2: A simplified dynamic model of a strut and its joint" >}}
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="<span class=\"figure-number\">Figure 2: </span>A simplified dynamic model of a strut and its joint" >}}
Figure [2](#org5bc5fa8) depicts a strut, along with the corresponding force diagram.
Figure [2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
Damping terms are neglected.
\\(k\_r\\) denotes the rotational stiffness of the spherical joint.
From Figure [2](#org5bc5fa8) (b), Newton's second law yields:
From Figure [2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
-f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\\\
m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\\\
-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}
@@ -157,16 +157,16 @@ The force is aligned perfectly with the strut only if \\(m\_s = 0\\) and \\(k\_r
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}
\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 + 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}
@@ -189,7 +189,6 @@ The first part depends on the mechanical terms and the frequency of the movement
\end{equation}
<div class="important">
<div></div>
In order to get dominance at low frequencies, the hexapod must be designed so that:
@@ -201,13 +200,12 @@ In order to get dominance at low frequencies, the hexapod must be designed so th
This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
By satisfying \eqref{eq:cond_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
By satisfying <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 \\]
<div class="important">
<div></div>
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:
@@ -226,16 +224,15 @@ In this case, it is reasonable to use:
\end{equation}
<div class="important">
<div></div>
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.
By designing the flexure jointed hexapod and its controller so that both <eq:cond_stiff> and <eq:cond_bandwidth> are met, the dynamics of the hexapod can be greatly reduced in complexity.
</div>
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
Equation \eqref{eq:eom_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
Equation <eq:eom_fjh> is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
This section will derive this implicit relationship.
Let denote:
@@ -269,9 +266,9 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
\end{equation}
## Bibliography {#bibliography}
<a id="org1d169f9"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="org2871bf9"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>Ieee/Asme Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
</div>

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content/article/mcinroy99_dynam.md
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@@ -1,14 +1,14 @@
+++
title = "Dynamic modeling of flexure jointed hexapods for control purposes"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([McInroy 1999](#orgc5d256d))
: (<a href="#citeproc_bib_item_1">McInroy 1999</a>)
Author(s)
: McInroy, J.
@@ -16,7 +16,7 @@ Author(s)
Year
: 1999
This conference paper has been further published in a journal as a short note ([McInroy 2002](#orge25929e)).
This conference paper has been further published in a journal as a short note (<a href="#citeproc_bib_item_2">McInroy 2002</a>).
## Abstract {#abstract}
@@ -38,22 +38,22 @@ 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="org89aa8b3"></a>
<a id="figure--fig:mcinroy99-general-hexapod"></a>
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}}
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="<span class=\"figure-number\">Figure 1: </span>A general Stewart Platform" >}}
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org0b2b1e5).
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#figure--fig:mcinroy99-strut-model).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="org0b2b1e5"></a>
<a id="figure--fig:mcinroy99-strut-model"></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" >}}
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="<span class=\"figure-number\">Figure 2: </span>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="#org0b2b1e5">2</a>
Definition of quantities on Figure <a href="#org84f1a50">2</a>
</div>
| **Symbol** | **Meaning** |
@@ -70,11 +70,11 @@ In order to provide low frequency passive vibration isolation, the hard actuator
| \\(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 ([McInroy 2002](#orge25929e))).
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in (<a href="#citeproc_bib_item_2">McInroy 2002</a>)).
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](#org0b2b1e5) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
From Figure [2](#figure--fig:mcinroy99-strut-model) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
\begin{equation}
m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
@@ -105,10 +105,10 @@ The payload is modeled as a rigid body:
\begin{equation}
\underbrace{\begin{bmatrix}
m I\_3 & 0\_{3\times 3} \\\\\\
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
0\_{3 \times 1} \\\ \omega \times {}^cI\omega
\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
\end{equation}
@@ -134,7 +134,7 @@ where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation mat
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}
\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:
@@ -142,11 +142,11 @@ 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:
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
\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)
& {}^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.
@@ -162,9 +162,9 @@ In the next section, a connection between the two will be found to complete the
## Control Example {#control-example}
## Bibliography {#bibliography}
<a id="orgc5d256d"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="orge25929e"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>Ieee/Asme Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
</div>

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+++
title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Motion Control]({{<relref "motion_control.md#" >}})
: [Motion Control]({{< relref "motion_control.md" >}})
Reference
: ([Oomen 2018](#org5ed8cf0))
: (<a href="#citeproc_bib_item_1">Oomen 2018</a>)
Author(s)
: Oomen, T.
@@ -16,12 +16,13 @@ Author(s)
Year
: 2018
<a id="orgd73938c"></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." >}}
<a id="figure--fig:oomen18-next-gen-loop-gain"></a>
{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="<span class=\"figure-number\">Figure 1: </span>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 {#bibliography}
<a id="org5ed8cf0"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. <https://doi.org/10.1541/ieejjia.7.127>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” <i>Ieej Journal of Industry Applications</i> 7 (2): 127–40. doi:<a href="https://doi.org/10.1541/ieejjia.7.127">10.1541/ieejjia.7.127</a>.</div>
</div>

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content/article/preumont02_force_feedb_versus_accel_feedb.md
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+++
title = "Force feedback versus acceleration feedback in active vibration isolation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Preumont et al. 2002](#orgbec44eb))
: (<a href="#citeproc_bib_item_1">Preumont et al. 2002</a>)
Author(s)
: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
: Preumont, A., A. Francois, Bossens, F., &amp; Abu-Hanieh, A.
Year
: 2002
@@ -26,16 +26,16 @@ 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](#orga040a9a)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#org1916ab2)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
- For light payload (Figure [1](#figure--fig:preumont02-force-acc-fb-light)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#figure--fig:preumont02-force-acc-fb-heavy)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
<a id="orga040a9a"></a>
<a id="figure--fig:preumont02-force-acc-fb-light"></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" >}}
{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="<span class=\"figure-number\">Figure 1: </span>Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
<a id="org1916ab2"></a>
<a id="figure--fig:preumont02-force-acc-fb-heavy"></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" >}}
{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="<span class=\"figure-number\">Figure 2: </span>Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
Guaranteed stability of the force feedback:
@@ -46,7 +46,8 @@ 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 {#bibliography}
<a id="orgbec44eb"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback Versus Acceleration Feedback in Active Vibration Isolation.” _Journal of Sound and Vibration_ 257 (4):605–13. <https://doi.org/10.1006/jsvi.2002.5047>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback versus Acceleration Feedback in Active Vibration Isolation.” <i>Journal of Sound and Vibration</i> 257 (4): 605–13. doi:<a href="https://doi.org/10.1006/jsvi.2002.5047">10.1006/jsvi.2002.5047</a>.</div>
</div>

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content/article/preumont07_six_axis_singl_stage_activ.md
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+++
title = "A six-axis single-stage active vibration isolator based on stewart platform"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Preumont et al. 2007](#org003735a))
: (<a href="#citeproc_bib_item_1">Preumont et al. 2007</a>)
Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@@ -18,35 +18,36 @@ Year
Summary:
- **Cubic** Stewart platform (Figure [3](#org144c76e))
- **Cubic** Stewart platform (Figure [3](#figure--fig:preumont07-stewart-platform))
- Provides uniform control capability
- Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs
- Flexible joints (Figure [2](#org04bd941))
- Flexible joints (Figure [2](#figure--fig:preumont07-flexible-joints))
- Piezoelectric force sensors
- Voice coil actuators
- Decentralized feedback control approach for vibration isolation
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#org06a63d6))
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#figure--fig:preumont07-iff-effect-stiffness))
- The Stewart platform has 6 suspension modes at different frequencies.
Thus the gain of the IFF controller cannot be optimal for all the modes.
It is better if all the modes of the platform are near to each other.
- 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="org06a63d6"></a>
<a id="figure--fig:preumont07-iff-effect-stiffness"></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" >}}
{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="<span class=\"figure-number\">Figure 1: </span>Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
<a id="org04bd941"></a>
<a id="figure--fig:preumont07-flexible-joints"></a>
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="<span class=\"figure-number\">Figure 2: </span>Flexible joints used for the Stewart platform" >}}
<a id="org144c76e"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
<a id="figure--fig:preumont07-stewart-platform"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="<span class=\"figure-number\">Figure 3: </span>Stewart platform" >}}
## Bibliography {#bibliography}
<a id="org003735a"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. <https://doi.org/10.1016/j.jsv.2006.07.050>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” <i>Journal of Sound and Vibration</i> 300 (3-5): 644–61. doi:<a href="https://doi.org/10.1016/j.jsv.2006.07.050">10.1016/j.jsv.2006.07.050</a>.</div>
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title = "Advances in internal model control technique: a review and future prospects"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}})
: [Complementary Filters]({{< relref "complementary_filters.md" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion.md" >}})
Reference
: ([Saxena and Hote 2012](#org13b6614))
: (<a href="#citeproc_bib_item_1">Saxena and Hote 2012</a>)
Author(s)
: Saxena, S., & Hote, Y.
: Saxena, S., &amp; Hote, Y.
Year
: 2012
## Proposed Filter \\(F(s)\\) {#proposed-filter--fs}
## Proposed Filter \\(F(s)\\) {#proposed-filter-f--s}
\begin{align\*}
F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\\\
F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\
F(s) &= \frac{n \lambda + 1}{(\lambda s + 1)^n}
\end{align\*}
@@ -41,7 +41,7 @@ The structure can then be modified and we obtain a new controller \\(Q(s)\\).
IMC is related to the classical controller through:
\begin{align\*}
Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\\\
Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\
C(s) = \frac{Q(s)}{1-G\_M(s)Q(s)}
\end{align\*}
@@ -85,7 +85,8 @@ 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 {#bibliography}
<a id="org13b6614"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. <https://doi.org/10.4103/0256-4602.105001>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” <i>Iete Technical Review</i> 29 (6): 461. doi:<a href="https://doi.org/10.4103/0256-4602.105001">10.4103/0256-4602.105001</a>.</div>
</div>

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+++
title = "Design for precision: current status and trends"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Precision Engineering]({{<relref "precision_engineering.md#" >}})
: [Precision Engineering]({{< relref "precision_engineering.md" >}})
Reference
: ([Schellekens et al. 1998](#orgc8457bd))
: (<a href="#citeproc_bib_item_1">Schellekens et al. 1998</a>)
Author(s)
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., &amp; Pril, W.
Year
: 1998
## Bibliography {#bibliography}
<a id="orgc8457bd"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” _Cirp Annals_, no. 2:557–86. <https://doi.org/10.1016/s0007-8506(07)63243-0>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” <i>Cirp Annals</i>, no. 2: 557–86. doi:<a href="https://doi.org/10.1016/s0007-8506(07)63243-0">10.1016/s0007-8506(07)63243-0</a>.</div>
</div>

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+++
title = "On compensator design for linear time-invariant dual-input single-output systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,16 +9,17 @@ Tags
Reference
: ([Schroeck, Messner, and McNab 2001](#org722a59f))
: (<a href="#citeproc_bib_item_1">Schroeck, Messner, and McNab 2001</a>)
Author(s)
: Schroeck, S., Messner, W., & McNab, R.
: Schroeck, S., Messner, W., &amp; McNab, R.
Year
: 2001
## Bibliography {#bibliography}
<a id="org722a59f"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. <https://doi.org/10.1109/3516.914391>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” <i>Ieee/Asme Transactions on Mechatronics</i> 6 (1): 50–57. doi:<a href="https://doi.org/10.1109/3516.914391">10.1109/3516.914391</a>.</div>
</div>

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+++
title = "Nanopositioning with multiple sensors: a case study in data storage"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Sensor Fusion]({{<relref "sensor_fusion.md#" >}})
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
Reference
: ([Sebastian and Pantazi 2012](#org22b1de0))
: (<a href="#citeproc_bib_item_1">Sebastian and Pantazi 2012</a>)
Author(s)
: Sebastian, A., & Pantazi, A.
: Sebastian, A., &amp; Pantazi, A.
Year
: 2012
## Bibliography {#bibliography}
<a id="org22b1de0"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. <https://doi.org/10.1109/tcst.2011.2177982>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” <i>Ieee Transactions on Control Systems Technology</i> 20 (2): 382–94. doi:<a href="https://doi.org/10.1109/tcst.2011.2177982">10.1109/tcst.2011.2177982</a>.</div>
</div>

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+++
title = "A concept of active mount for space applications"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Active Damping](active_damping.md)
: [Active Damping]({{< relref "active_damping.md" >}})
Reference
: ([Souleille et al. 2018](#orgdd47abc))
: (<a href="#citeproc_bib_item_1">Souleille et al. 2018</a>)
Author(s)
: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, &amp; Collette, C.
Year
: 2018
@@ -23,25 +23,25 @@ This article discusses the use of Integral Force Feedback with amplified piezoel
## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
Figure [1](#org4d65c6e) shows a picture of the amplified piezoelectric stack.
Figure [1](#figure--fig:souleille18-model-piezo) shows a picture of the amplified piezoelectric stack.
The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
<a id="org4d65c6e"></a>
<a id="figure--fig:souleille18-model-piezo"></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" >}}
{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="<span class=\"figure-number\">Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
<div class="table-caption">
<span class="table-number">Table 1</span>:
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 |
| | 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:
@@ -61,39 +61,40 @@ and the control force is given by:
f = F\_s G(s) = F\_s \frac{g}{s}
\end{equation}
The effect of the controller are shown in Figure [2](#org3336e8f):
The effect of the controller are shown in Figure [2](#figure--fig:souleille18-tf-iff-result):
- the resonance peak is almost critically damped
- the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
- 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="org3336e8f"></a>
<a id="figure--fig:souleille18-tf-iff-result"></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)" >}}
{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="<span class=\"figure-number\">Figure 2: </span>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="org20a69be"></a>
<a id="figure--fig:souleille18-root-locus"></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" >}}
{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="<span class=\"figure-number\">Figure 3: </span>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](#orga310d92)).
A heavy payload is mounted on a set of three isolators (Figure [4](#figure--fig:souleille18-setup-flexible-payload)).
The payload consists of two masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
<a id="orga310d92"></a>
<a id="figure--fig:souleille18-setup-flexible-payload"></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" >}}
{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="<span class=\"figure-number\">Figure 4: </span>Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
As shown in Figure [5](#org3c2e029), both the suspension modes and the flexible modes of the payload can be critically damped.
As shown in Figure [5](#figure--fig:souleille18-result-damping-transmissibility), both the suspension modes and the flexible modes of the payload can be critically damped.
<a id="org3c2e029"></a>
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
<a id="figure--fig:souleille18-result-damping-transmissibility"></a>
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="<span class=\"figure-number\">Figure 5: </span>Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
## Bibliography {#bibliography}
<a id="orgdd47abc"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>Ceas Space Journal</i> 10 (2). Springer: 157–65.</div>
</div>

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content/article/spanos95_soft_activ_vibrat_isolat.md
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+++
title = "A soft 6-axis active vibration isolator"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Spanos, Rahman, and Blackwood 1995](#org2800cc5))
: (<a href="#citeproc_bib_item_1">Spanos, Rahman, and Blackwood 1995</a>)
Author(s)
: Spanos, J., Rahman, Z., & Blackwood, G.
: Spanos, J., Rahman, Z., &amp; Blackwood, G.
Year
: 1995
**Stewart Platform** (Figure [1](#orgcac471d)):
**Stewart Platform** (Figure [1](#figure--fig:spanos95-stewart-platform)):
- Voice Coil
- Flexible joints (cross-blades)
- Force Sensors
- Cubic Configuration
<a id="orgcac471d"></a>
<a id="figure--fig:spanos95-stewart-platform"></a>
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Stewart Platform" >}}
Total mass of the paylaod: 30kg
Center of gravity is 9cm above the geometry center of the mount (cube's center?).
@@ -41,9 +41,9 @@ 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="org5cb89c4"></a>
<a id="figure--fig:spanos95-iff-plant"></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" >}}
{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
The controller used consisted of:
@@ -52,14 +52,15 @@ 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](#orgd8726b9).
The results in terms of transmissibility are shown in Figure [3](#figure--fig:spanos95-results).
<a id="orgd8726b9"></a>
{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
<a id="figure--fig:spanos95-results"></a>
{{< figure src="/ox-hugo/spanos95_results.png" caption="<span class=\"figure-number\">Figure 3: </span>Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
## Bibliography {#bibliography}
<a id="org2800cc5"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In _Proceedings of 1995 American Control Conference - ACC’95_, nil. <https://doi.org/10.1109/acc.1995.529280>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In <i>Proceedings of 1995 American Control Conference - Acc’95</i>, nil. doi:<a href="https://doi.org/10.1109/acc.1995.529280">10.1109/acc.1995.529280</a>.</div>
</div>

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content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
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+++
title = "Interferometric characterization of rotation stages for x-ray nanotomography"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
Reference
: ([Stankevic et al. 2017](#org125690d))
: (<a href="#citeproc_bib_item_1">Stankevic et al. 2017</a>)
Author(s)
: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
@@ -19,18 +19,19 @@ Year
- Similar Station than the NASS
- Similar Metrology with fiber based interferometers and cylindrical reference mirror
<a id="org5481c46"></a>
<a id="figure--fig:stankevic17-station"></a>
{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
{{< figure src="/ox-hugo/stankevic17_station.png" caption="<span class=\"figure-number\">Figure 1: </span>Positioning Station" >}}
- **Thermal expansion**: Stabilized down to \\(5mK/h\\) using passive water flow through the baseplate below the sample stage and in the interferometry reference frame.
- **Controller**: Two Independant PID loops
- Repeatable errors => feedforward (Look Up Table)
- Non-repeatable errors => feedback
- Repeatable errors =&gt; feedforward (Look Up Table)
- Non-repeatable errors =&gt; feedback
- Result: 40nm runout error
## Bibliography {#bibliography}
<a id="org125690d"></a>Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” _Review of Scientific Instruments_ 88 (5):053703. <https://doi.org/10.1063/1.4983405>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” <i>Review of Scientific Instruments</i> 88 (5): 053703. doi:<a href="https://doi.org/10.1063/1.4983405">10.1063/1.4983405</a>.</div>
</div>

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content/article/tang18_decen_vibrat_contr_voice_coil.md
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+++
title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
Reference
: ([Tang, Cao, and Yu 2018](#org2c23b98))
: (<a href="#citeproc_bib_item_1">Tang, Cao, and Yu 2018</a>)
Author(s)
: Tang, J., Cao, D., & Yu, T.
: Tang, J., Cao, D., &amp; Yu, T.
Year
: 2018
## Bibliography {#bibliography}
<a id="org2c23b98"></a>Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “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):132–45. <https://doi.org/10.1177/0954406218756941>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” <i>Proceedings of the Institution of Mechanical Engineers, Part c: Journal of Mechanical Engineering Science</i> 233 (1): 132–45. doi:<a href="https://doi.org/10.1177/0954406218756941">10.1177/0954406218756941</a>.</div>
</div>

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content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
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+++
title = "Sensor fusion for active vibration isolation in precision equipment"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Tjepkema, Dijk, and Soemers 2012](#org06c1cb7))
: (<a href="#citeproc_bib_item_1">Tjepkema, van Dijk, and Soemers 2012</a>)
Author(s)
: Tjepkema, D., Dijk, J. v., & Soemers, H.
: Tjepkema, D., Dijk, J. v., &amp; Soemers, H.
Year
: 2012
@@ -43,11 +43,12 @@ Control law: \\(f = -Gx\\)
## Design constraints and control bandwidth {#design-constraints-and-control-bandwidth}
Heavier sensor => lower noise but it is harder to maintain collocation with the actuator => that limits the bandwidth.
Heavier sensor =&gt; lower noise but it is harder to maintain collocation with the actuator =&gt; that limits the bandwidth.
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 {#bibliography}
<a id="org06c1cb7"></a>Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” _Journal of Sound and Vibration_ 331 (4):735–49. <https://doi.org/10.1016/j.jsv.2011.09.022>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” <i>Journal of Sound and Vibration</i> 331 (4): 735–49. doi:<a href="https://doi.org/10.1016/j.jsv.2011.09.022">10.1016/j.jsv.2011.09.022</a>.</div>
</div>

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content/article/wang12_autom_marker_full_field_hard.md
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+++
title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}})
Reference
: ([Wang et al. 2012](#orgf2371c9))
: (<a href="#citeproc_bib_item_1">Wang et al. 2012</a>)
Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., &amp; Feser, M.
Year
: 2012
@@ -20,13 +20,14 @@ Year
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.
=&gt; 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 {#bibliography}
<a id="orgf2371c9"></a>Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. <https://doi.org/10.1063/1.3701579>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” <i>Applied Physics Letters</i> 100 (14): 143107. doi:<a href="https://doi.org/10.1063/1.3701579">10.1063/1.3701579</a>.</div>
</div>

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content/article/wang16_inves_activ_vibrat_isolat_stewar.md
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+++
title = "Investigation on active vibration isolation of a stewart platform with piezoelectric actuators"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Wang et al. 2016](#org89f2008))
: (<a href="#citeproc_bib_item_1">Wang et al. 2016</a>)
Author(s)
: Wang, C., Xie, X., Chen, Y., & Zhang, Z.
: Wang, C., Xie, X., Chen, Y., &amp; Zhang, Z.
Year
: 2016
@@ -25,23 +25,23 @@ 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="org0d482b7"></a>
<a id="figure--fig:wang16-stewart-platform"></a>
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Stewart Platform" >}}
**Control**:
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
- the FxLMS-based adaptive inverse control =&gt; suppress transmission of periodic vibrations
- direct feedback of integrated forces =&gt; dampen vibration of inherent modes and thus reduce random vibrations
Force Feedback (Figure [2](#org1b645a1)).
Force Feedback (Figure [2](#figure--fig:wang16-force-feedback)).
- the force sensor is mounted **between the base and the strut**
<a id="org1b645a1"></a>
<a id="figure--fig:wang16-force-feedback"></a>
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="<span class=\"figure-number\">Figure 2: </span>Feedback of integrated forces in the platform" >}}
Sorts of HAC-LAC control:
@@ -54,7 +54,8 @@ Sorts of HAC-LAC control:
- Effectiveness of control methods are shown
## Bibliography {#bibliography}
<a id="org89f2008"></a>Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. <https://doi.org/10.1016/j.jsv.2016.07.021>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” <i>Journal of Sound and Vibration</i> 383 (November). Elsevier BV: 1–19. doi:<a href="https://doi.org/10.1016/j.jsv.2016.07.021">10.1016/j.jsv.2016.07.021</a>.</div>
</div>

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content/article/yang19_dynam_model_decoup_contr_flexib.md
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+++
title = "Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
Reference
: ([Yang et al. 2019](#orgb15122e))
: (<a href="#citeproc_bib_item_1">Yang et al. 2019</a>)
Author(s)
: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
: Yang, X., Wu, H., Chen, B., Kang, S., &amp; Cheng, S.
Year
: 2019
@@ -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](#org479da8d)):
**Stewart platform** (Figure [1](#figure--fig:yang19-stewart-platform)):
- piezoelectric actuators
- flexible joints (Figure [2](#org83afe99))
- flexible joints (Figure [2](#figure--fig:yang19-flexible-joints))
- force sensors (used for vibration isolation)
- displacement sensors (used to decouple the dynamics)
- cubic (even though not said explicitly)
<a id="org479da8d"></a>
<a id="figure--fig:yang19-stewart-platform"></a>
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Stewart Platform" >}}
<a id="org83afe99"></a>
<a id="figure--fig:yang19-flexible-joints"></a>
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="<span class=\"figure-number\">Figure 2: </span>Flexible Joints" >}}
The stiffness of the flexible joints (Figure [2](#org83afe99)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
The stiffness of the flexible joints (Figure [2](#figure--fig:yang19-flexible-joints)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
<a id="table--tab:yang19-stiffness-flexible-joints"></a>
<div class="table-caption">
@@ -105,11 +105,11 @@ 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](#orgd526d94).
The block diagram of the control strategy is represented in Figure [3](#figure--fig:yang19-control-arch).
<a id="orgd526d94"></a>
<a id="figure--fig:yang19-control-arch"></a>
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="Figure 3: Control Architecture used" >}}
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="<span class=\"figure-number\">Figure 3: </span>Control Architecture used" >}}
\\(H(s)\\) is designed as a proportional plus integral compensator:
\\[ H(s) = k\_p + k\_i/s \\]
@@ -121,12 +121,12 @@ 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](#orge73e046).
The results are shown in Figure [4](#figure--fig:yang19-results).
In theory, the vibration performance can be improved, however in practice, increasing the gain causes saturation of the piezoelectric actuators and then the instability occurs.
<a id="orge73e046"></a>
<a id="figure--fig:yang19-results"></a>
{{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
{{< figure src="/ox-hugo/yang19_results.png" caption="<span class=\"figure-number\">Figure 4: </span>Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
> A model-based controller is then designed based on the leg’s force and position feedback.
> The position feedback compensates the effect of parasitic bending and torsional stiffness of the flexible joints.
@@ -134,7 +134,8 @@ 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 {#bibliography}
<a id="orgb15122e"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412. <https://doi.org/10.1016/j.jsv.2018.10.007>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” <i>Journal of Sound and Vibration</i> 439 (January). Elsevier BV: 398–412. doi:<a href="https://doi.org/10.1016/j.jsv.2018.10.007">10.1016/j.jsv.2018.10.007</a>.</div>
</div>

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content/article/yun20_inves_two_stage_vibrat_suppr.md
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+++
title = "Investigation on two-stage vibration suppression and precision pointing for space optical payloads"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@@ -9,16 +9,17 @@ Tags
Reference
: ([Yun et al. 2020](#org7bb249c))
: (<a href="#citeproc_bib_item_1">Yun et al. 2020</a>)
Author(s)
: Yun, H., Liu, L., Li, Q., & Yang, H.
: Yun, H., Liu, L., Li, Q., &amp; Yang, H.
Year
: 2020
## Bibliography {#bibliography}
<a id="org7bb249c"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. <https://doi.org/10.1016/j.ast.2019.105543>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” <i>Aerospace Science and Technology</i> 96 (nil): 105543. doi:<a href="https://doi.org/10.1016/j.ast.2019.105543">10.1016/j.ast.2019.105543</a>.</div>
</div>

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content/article/zhang11_six_dof.md
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@@ -1,17 +1,17 @@
+++
title = "Six dof active vibration control using stewart platform with non-cubic configuration"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Zhang et al. 2011](#org293b885))
: (<a href="#citeproc_bib_item_1">Zhang et al. 2011</a>)
Author(s)
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
: Zhang, Z., Liu, J., Mao, J., Guo, Y., &amp; Ma, Y.
Year
: 2011
@@ -20,17 +20,18 @@ Year
- **Flexible** joints
- Magnetostrictive actuators
- Strong coupled motions along different axes
- Non-cubic architecture => permits to have larger workspace which was required
- Non-cubic architecture =&gt; permits to have larger workspace which was required
- Structure parameters (radius of plates, length of struts) are determined by optimization of the condition number of the Jacobian matrix
- **Accelerometers** for active isolation
- Adaptive FIR filters for active isolation control
<a id="orgf49a13c"></a>
{{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
<a id="figure--fig:zhang11-platform"></a>
{{< figure src="/ox-hugo/zhang11_platform.png" caption="<span class=\"figure-number\">Figure 1: </span>Prototype of the non-cubic stewart platform" >}}
## Bibliography {#bibliography}
<a id="org293b885"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In _2011 6th IEEE Conference on Industrial Electronics and Applications_, nil. <https://doi.org/10.1109/iciea.2011.5975679>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six Dof Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In <i>2011 6th Ieee Conference on Industrial Electronics and Applications</i>, nil. doi:<a href="https://doi.org/10.1109/iciea.2011.5975679">10.1109/iciea.2011.5975679</a>.</div>
</div>