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25
content/article/alkhatib03_activ_struc_vibrat_contr.md
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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>

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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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content/article/dasgupta00_stewar_platf_manip.md
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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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content/article/fleming10_nanop_system_with_force_feedb.md
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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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content/article/fleming15_low_order_dampin_track_contr.md
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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>
</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>
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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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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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@ -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>
</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>

19
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>

13
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>

15
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>

15
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>

19
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>

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content/book/du10_model_contr_vibrat_mechan_system.md
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@ -1,17 +1,17 @@
+++
title = "Modeling and control of vibration in mechanical systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Du and Xie 2010](#orga475b60))
: (<a href="#citeproc_bib_item_1">Du and Xie 2010</a>)
Author(s)
: Du, C., & Xie, L.
: Du, C., &amp; Xie, L.
Year
: 2010
@ -110,7 +110,7 @@ Year
### 2.5 Conclusion {#2-dot-5-conclusion}
## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms.md" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
### 3.1 Introduction {#3-dot-1-introduction}
@ -152,7 +152,7 @@ Year
#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension}
#### 4.2.5 An application example &#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
#### 4.2.5 An application example &amp;#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems}
@ -179,13 +179,13 @@ Year
### 5.1 Introduction {#5-dot-1-introduction}
### 5.2 H2 and H&#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
### 5.2 H2 and H&amp;#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm}
#### 5.2.2 H&#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
#### 5.2.2 H&amp;#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control}
@ -197,7 +197,7 @@ Year
#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case}
### 5.4 H&#8734; control {#5-dot-4-h-and-8734-control}
### 5.4 H&amp;#8734; control {#5-dot-4-h-and-8734-control}
#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case}
@ -227,13 +227,13 @@ Year
### 5.8 Conclusion {#5-dot-8-conclusion}
## 6. Mixed H2/H&#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
## 6. Mixed H2/H&amp;#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
### 6.1 Introduction {#6-dot-1-introduction}
### 6.2 Mixed H2/H&#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
### 6.2 Mixed H2/H&amp;#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach}
@ -266,7 +266,7 @@ Year
### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain}
#### 7.3.1 H&#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
#### 7.3.1 H&amp;#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples}
@ -395,7 +395,7 @@ Year
### 10.5 Conclusion {#10-dot-5-conclusion}
## 11. H&#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
## 11. H&amp;#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
### 11.1 Introduction {#11-dot-1-introduction}
@ -533,7 +533,8 @@ Year
### 15.6 Conclusion {#15-dot-6-conclusion}
## Bibliography {#bibliography}
<a id="orga475b60"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.
<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>Du, Chunling, and Lihua Xie. 2010. <i>Modeling and Control of Vibration in Mechanical Systems</i>. Automation and Control Engineering. CRC Press. doi:<a href="https://doi.org/10.1201/9781439817995">10.1201/9781439817995</a>.</div>
</div>

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content/book/du19_multi_actuat_system_contr.md
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@ -1,6 +1,6 @@
+++
title = "Multi-stage actuation systems and control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Proposes a way to combine multiple actuators (short stroke and long stroke) for control."
keywords = ["Control", "Mechatronics"]
draft = false
@ -11,10 +11,10 @@ Tags
Reference
: ([Du and Pang 2019](#org2403f17))
: (<a href="#citeproc_bib_item_1">Du and Pang 2019</a>)
Author(s)
: Du, C., & Pang, C. K.
: Du, C., &amp; Pang, C. K.
Year
: 2019
@ -43,11 +43,11 @@ When high bandwidth, high position accuracy and long stroke are required simulta
Popular choices for coarse actuator are:
- DC motor
- [Voice Coil Motors]({{< relref "voice_coil_actuators" >}}) (VCM)
- [Voice Coil Motors]({{< relref "voice_coil_actuators.md" >}}) (VCM)
- Permanent magnet stepper motor
- Permanent magnet linear synchronous motor
As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) are used.
As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) are used.
In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
@ -75,19 +75,19 @@ which includes the resonance model
and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
\begin{align\*}
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\end{align\*}
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org4cc1c22)) and the microactuator.
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#figure--fig:pzt-actuator)) and the microactuator.
<a id="org4cc1c22"></a>
<a id="figure--fig:pzt-actuator"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class=\"figure-number\">Figure 1: </span>A PZT-actuator suspension" >}}
There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
There characteristics are shown on table [1](#table--tab:microactuator).
@ -107,11 +107,11 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#org3b2af5e), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
A typical closed-loop control system is shown on figure [2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="org3b2af5e"></a>
<a id="figure--fig:single-stage-control"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="<span class=\"figure-number\">Figure 2: </span>Block diagram of a single-stage actuation system" >}}
### Dual-Stage Actuation Systems {#dual-stage-actuation-systems}
@ -119,9 +119,9 @@ A typical closed-loop control system is shown on figure [2](#org3b2af5e), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
<a id="org9af6d44"></a>
<a id="figure--fig:dual-stage-control"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="<span class=\"figure-number\">Figure 3: </span>Block diagram of dual-stage actuation system" >}}
### Three-Stage Actuation Systems {#three-stage-actuation-systems}
@ -145,7 +145,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org0221f39).
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#figure--fig:decoupled-control).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@ -153,9 +153,9 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
<a id="org0221f39"></a>
<a id="figure--fig:decoupled-control"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="<span class=\"figure-number\">Figure 4: </span>Decoupled control structure for the dual-stage actuation system" >}}
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) + P\_v(z) C\_v(z) \hat{P}\_p(z) C\_p(z) \\]
@ -175,16 +175,16 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#org9edcb9b).
Another type of control scheme is the **parallel structure** as shown in figure [5](#figure--fig:parallel-control-structure).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
<a id="org9edcb9b"></a>
<a id="figure--fig:parallel-control-structure"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="<span class=\"figure-number\">Figure 5: </span>Parallel control structure for the dual-stage actuator system" >}}
Because of the limited displacement range of the secondary actuator, the control efforts for the two actuators should be distributed properly when designing respective controllers to meet the required performance, make the actuators not conflict with each other, as well as prevent the saturation of the secondary actuator.
@ -192,7 +192,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org24873cb) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@ -202,11 +202,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org24873cb"></a>
<a id="figure--fig:h-inf-diagram"></a>
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#orga734f85) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
Equation [1](#org563f2ec) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@ -219,18 +219,18 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#orgb5c1410), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
As seen in figure [7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="orgb5c1410"></a>
<a id="figure--fig:dual-stage-loop-gain"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class=\"figure-number\">Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#orgd91ec4c), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
The sensitivity functions are shown in figure [8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="orgd91ec4c"></a>
<a id="figure--fig:dual-stage-sensitivity"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="<span class=\"figure-number\">Figure 8: </span>Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
A basic requirement of the dual-stage actuation control system is to make the individual primary and secondary loops stable.
It also required that the primary actuator path has a higher gain than the secondary actuator path at low frequency range and the secondary actuator path has a higher gain than the primary actuator path in high-frequency range.
@ -261,15 +261,15 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#org4bda714) shows the control structure for the three-stage actuation system.
Figure [9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
<a id="org4bda714"></a>
<a id="figure--fig:three-stage-control"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="<span class=\"figure-number\">Figure 9: </span>Control system for the three-stage actuation system" >}}
The open-loop transfer function of the three-stage actuation system is derived as
@ -280,8 +280,8 @@ The open-loop transfer function of the three-stage actuation system is derived a
with
\begin{align\*}
G\_v(z) &= P\_v(z) C\_v(z) \\\\\\
G\_p(z) &= P\_p(z) C\_p(z) \\\\\\
G\_v(z) &= P\_v(z) C\_v(z) \\\\
G\_p(z) &= P\_p(z) C\_p(z) \\\\
G\_m(z) &= P\_m(z) C\_m(z)
\end{align\*}
@ -296,17 +296,17 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#orgded6e76).
The open-loop frequency responses of the three stages are shown on figure [10](#figure--fig:open-loop-three-stage).
<a id="orgded6e76"></a>
<a id="figure--fig:open-loop-three-stage"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class=\"figure-number\">Figure 10: </span>Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#orgde9819c).
The obtained sensitivity function is shown on figure [11](#figure--fig:sensitivity-three-stage).
<a id="orgde9819c"></a>
<a id="figure--fig:sensitivity-three-stage"></a>
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="<span class=\"figure-number\">Figure 11: </span>Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
### Performance Evaluation {#performance-evaluation}
@ -319,13 +319,13 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#org4bda714), the position error is
For the three-stage control architecture as shown on figure [9](#figure--fig:three-stage-control), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
\begin{align\*}
u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\\\
u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\\\
u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\
u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\
u\_v &= C\_v ( 1 + \hat{P}\_pC\_p ) e,\ y\_v = P\_v ( u\_v + d\_1 )
\end{align\*}
@ -335,11 +335,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orga3b472d)).
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#figure--fig:three-stage-decoupled)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org442b5f7) and
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org9bf2b8d) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@ -348,23 +348,23 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
<a id="orga3b472d"></a>
<a id="figure--fig:three-stage-decoupled"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="<span class=\"figure-number\">Figure 12: </span>Decoupled control structure for the three-stage actuation system" >}}
The control signals and the positions of the three actuators are
\begin{align\*}
u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\\\
u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\\\
u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\
u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org5311716)).
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#figure--fig:three-stage-decoupled-loop-gain)).
<a id="org5311716"></a>
<a id="figure--fig:three-stage-decoupled-loop-gain"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="<span class=\"figure-number\">Figure 13: </span>Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
### Conclusion {#conclusion}
@ -671,7 +671,8 @@ Using PZT elements as a sensor to deal with high-frequency vibration beyond the
As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.
## Bibliography {#bibliography}
<a id="org2403f17"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Chee Khiang Pang. 2019. <i>Multi-Stage Actuation Systems and Control</i>. Boca Raton, FL: CRC Press.</div>
</div>

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@ -1,19 +1,19 @@
+++
title = "Design, modeling and control of nanopositioning systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Talks about various topics related to nano-positioning systems."
keywords = ["Control", "Metrology", "Flexible Joints"]
draft = false
+++
Tags
: [Piezoelectric Actuators]({{<relref "piezoelectric_actuators.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}})
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Fleming and Leang 2014](#orgd16fb21))
: (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>)
Author(s)
: Fleming, A. J., & Leang, K. K.
: Fleming, A. J., &amp; Leang, K. K.
Year
: 2014
@ -728,16 +728,15 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="orgb084203"></a>
<a id="figure--fig:fleming14-amplifier-model"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#orgb084203) where a voltage source is connected to a piezoelectric actuator.
Consider the electrical circuit shown in Figure [1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
<div class="exampl">
<div></div>
Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
@ -810,7 +809,6 @@ For sinusoidal signals, the amplifiers slew rate must exceed:
where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
<div class="exampl">
<div></div>
If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
@ -853,7 +851,8 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
### References {#references}
## Bibliography {#bibliography}
<a id="orgd16fb21"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
<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>Fleming, Andrew J., and Kam K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-06617-2">10.1007/978-3-319-06617-2</a>.</div>
</div>

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@ -1,16 +1,16 @@
+++
title = "Vibration Simulation using Matlab and ANSYS"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Nice techniques to analyze resonant systems with Ansys and Matlab."
keywords = ["Modal Analysis", "FEM"]
draft = false
+++
Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
: [Finite Element Model]({{< relref "finite_element_model.md" >}})
Reference
: ([Hatch 2000](#org4036e02))
: (<a href="#citeproc_bib_item_1">Hatch 2000</a>)
Author(s)
: Hatch, M. R.
@ -23,17 +23,16 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<a id="org96f8e54"></a>
<span class="org-target" id="org-target--sec:introduction"></span>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org97c03ca) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
<div></div>
The steps are:
@ -48,9 +47,9 @@ The steps are:
</div>
<a id="org97c03ca"></a>
<a id="figure--fig:hatch00-modal-analysis-flowchart"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="<span class=\"figure-number\">Figure 1: </span>Modal analysis method flowchart" >}}
### Model Size Reduction {#model-size-reduction}
@ -58,9 +57,8 @@ The steps are:
Because finite element models usually have a very large number of states, an important step is the reduction of the number of states while still providing correct responses for the forcing function input and desired output points.
<div class="important">
<div></div>
Figure [2](#orgdbb9ffa) shows such process, the steps are:
Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -73,14 +71,14 @@ Figure [2](#orgdbb9ffa) shows such process, the steps are:
</div>
<a id="orgdbb9ffa"></a>
<a id="figure--fig:hatch00-model-reduction-chart"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="<span class=\"figure-number\">Figure 2: </span>Model size reduction flowchart" >}}
### Notations {#notations}
Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
@ -129,22 +127,21 @@ Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orgca1a04d"></a>
<span class="org-target" id="org-target--sec:zeros_siso_systems"></span>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org4819d7f) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
Figure [3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org4819d7f"></a>
<a id="figure--fig:hatch00-n-dof-zeros"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="<span class=\"figure-number\">Figure 3: </span>n dof system showing various SISO input/output configurations" >}}
<div class="important">
<div></div>
([Miu 1993](#orgcda3e53)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
(<a href="#citeproc_bib_item_2">Miu 1993</a>) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros.
@ -153,17 +150,16 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<a id="orgc4e6e06"></a>
<span class="org-target" id="org-target--sec:state_space_analysis"></span>
## Modal Analysis {#modal-analysis}
<a id="orge1af07f"></a>
<span class="org-target" id="org-target--sec:modal_analysis"></span>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
<div class="important">
<div></div>
The modal method allows one to replace the n-coupled differential equations with n-uncoupled equations, where each uncoupled equation represents the motion of the system for that mode of vibration.
@ -175,7 +171,6 @@ Heavily damped structures or structures which explicit damping elements, such as
Thus, the present methods only works for lightly damped structures.
<div class="important">
<div></div>
Summarizing the modal analysis method of analyzing linear mechanical systems and the benefits derived:
@ -198,34 +193,34 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#orgde2ed42) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in Figure [4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="orgde2ed42"></a>
<a id="figure--fig:hatch00-undamped-tdof-model"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="<span class=\"figure-number\">Figure 4: </span>Undamped tdof model" >}}
The equations of motions are:
\begin{equation}
\begin{bmatrix}
m & 0 & 0 \\\\\\
0 & m & 0 \\\\\\
m & 0 & 0 \\\\
0 & m & 0 \\\\
0 & 0 & m
\end{bmatrix} \begin{bmatrix}
\ddot{z}\_1 \\\\\\
\ddot{z}\_2 \\\\\\
\ddot{z}\_1 \\\\
\ddot{z}\_2 \\\\
\ddot{z}\_3
\end{bmatrix} + \begin{bmatrix}
k & -k & 0 \\\\\\
-k & 2k & -k \\\\\\
k & -k & 0 \\\\
-k & 2k & -k \\\\
0 & -k & k
\end{bmatrix} \begin{bmatrix}
z\_1 \\\\\\
z\_2 \\\\\\
z\_1 \\\\
z\_2 \\\\
z\_3
\end{bmatrix} = \begin{bmatrix}
0 \\\\\\
0 \\\\\\
0 \\\\
0 \\\\
0
\end{bmatrix} \label{eq:tdof\_eom}
\end{equation}
@ -236,7 +231,6 @@ The equations of motions are:
Since the system is conservative (it has no damping), normal modes of vibration will exist.
<div class="important">
<div></div>
Having normal modes means that at certain frequencies all points in the system will vibrate at the same frequency and in phase, i.e., **all points in the system will reach their minimum and maximum displacements at the same point in time**.
@ -258,7 +252,7 @@ where:
#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
Re-injecting normal modes \eqref{eq:principal_mode} into the equation of motion \eqref{eq:tdof_eom} gives the eigenvalue problem:
Re-injecting normal modes <eq:principal_mode> into the equation of motion <eq:tdof_eom> gives the eigenvalue problem:
\begin{equation}
(\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
@ -285,45 +279,45 @@ One then find:
\begin{equation}
\bm{z}\_1 = \begin{bmatrix}
1 \\\\\\
1 \\\\\\
1 \\\\
1 \\\\
1
\end{bmatrix}, \quad \bm{z}\_2 = \begin{bmatrix}
1 \\\\\\
0 \\\\\\
1 \\\\
0 \\\\
-1
\end{bmatrix}, \quad \bm{z}\_3 = \begin{bmatrix}
1 \\\\\\
-2 \\\\\\
1 \\\\
-2 \\\\
1
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc0f09b0), [6](#org88e7153) and [7](#org8225e3c).
Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig:hatch00-tdof-mode-1), [6](#figure--fig:hatch00-tdof-mode-2) and [7](#figure--fig:hatch00-tdof-mode-3).
<a id="orgc0f09b0"></a>
<a id="figure--fig:hatch00-tdof-mode-1"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="<span class=\"figure-number\">Figure 5: </span>Rigid-Body Mode, 0rad/s" >}}
<a id="org88e7153"></a>
<a id="figure--fig:hatch00-tdof-mode-2"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="<span class=\"figure-number\">Figure 6: </span>Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="org8225e3c"></a>
<a id="figure--fig:hatch00-tdof-mode-3"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="<span class=\"figure-number\">Figure 7: </span>Third Mode, 1.7rad/s" >}}
#### Modal Matrix {#modal-matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal_matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. <eq:modal_matrix>
\begin{equation}
\bm{z}\_m = \begin{bmatrix}
\bm{z}\_1 & \bm{z}\_2 & \bm{z}\_3
\end{bmatrix} = \begin{bmatrix}
z\_{m11} & z\_{m12} & z\_{m13} \\\\\\
z\_{m21} & z\_{m22} & z\_{m23} \\\\\\
z\_{m11} & z\_{m12} & z\_{m13} \\\\
z\_{m21} & z\_{m22} & z\_{m23} \\\\
z\_{m31} & z\_{m32} & z\_{m33}
\end{bmatrix} \label{eq:modal\_matrix}
\end{equation}
@ -339,7 +333,6 @@ It is thus useful to **transform the n-coupled second order differential equatio
In linear algebra terms, the transformation from physical to principal coordinates is known as a **change of basis**.
<div class="important">
<div></div>
There are many options for change of basis, but we will show that **when eigenvectors are used for the transformation, the principal coordinate system has a physical meaning: each of the uncoupled sdof systems represents the motion of a specific mode of vibration**.
@ -348,11 +341,11 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#org0f0be39).
This procedure is schematically shown in Figure [8](#figure--fig:hatch00-schematic-modal-solution).
<a id="org0f0be39"></a>
<a id="figure--fig:hatch00-schematic-modal-solution"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="<span class=\"figure-number\">Figure 8: </span>Roadmap for Modal Solution" >}}
The condition to guarantee diagonalization is the existence of n-linearly independent eigenvectors, which is always the case if either:
@ -407,12 +400,12 @@ One method is to normalize with respect to unity, making the **largest** element
\begin{equation}
\bm{z}\_m = \begin{bmatrix}
1 & 1 & 1 \\\\\\
1 & 0 & -2 \\\\\\
1 & 1 & 1 \\\\
1 & 0 & -2 \\\\
1 & -1 & 1
\end{bmatrix} \Longrightarrow \bm{z}\_n \begin{bmatrix}
1 & 1 & -0.5 \\\\\\
1 & 0 & 1 \\\\\\
1 & 1 & -0.5 \\\\
1 & 0 & 1 \\\\
1 & -1 & -0.5
\end{bmatrix}
\end{equation}
@ -423,12 +416,12 @@ Transforming the mass and stiffness matrices give:
\begin{equation}
\bm{m}\_n = \bm{z}\_n^T \bm{m} \bm{z}\_n = \begin{bmatrix}
3m & 0 & 0 \\\\\\
0 & 2m & 0 \\\\\\
3m & 0 & 0 \\\\
0 & 2m & 0 \\\\
0 & 0 & 1.5m
\end{bmatrix}; \quad \bm{k}\_n = \bm{z}\_n^T \bm{k} \bm{z}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 2k & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 2k & 0 \\\\
0 & 0 & 4.5k
\end{bmatrix}
\end{equation}
@ -455,12 +448,12 @@ And the normalized mass and stiffness matrices are:
\begin{equation}
\bm{m}\_n = \begin{bmatrix}
1 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
1 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 1
\end{bmatrix}; \quad \bm{k}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 3
\end{bmatrix} \frac{k}{m}
\end{equation}
@ -471,7 +464,6 @@ The normalized stiffness matrix is known as the **spectral matrix**.
Normalizing with respect to mass results in an identify principal mass matrix and squares of the eigenvalues on the diagonal in the principal stiffness matrix, this normalization technique is thus very useful for the following reason.
<div class="important">
<div></div>
Since we know the form of the principal matrices when normalizing with respect to mass, no multiplying of modal matrices is actually required: **the homogeneous principal equations of motion can be written by inspection knowing only the eigenvalues**.
@ -498,7 +490,6 @@ Pre-multiplying by \\(\bm{z}\_n^T\\) and inserting \\(I = \bm{z}\_n \bm{z}\_n^{-
Which is re-written in the following form:
<div class="important">
<div></div>
\begin{equation}
\bm{m}\_p \ddot{\bm{z}}\_p + \bm{k}\_p \bm{z}\_p = \bm{F}\_p
@ -517,7 +508,7 @@ where:
The vectors of initial displacements \\(\bm{z}\_{op}\\) and velocities \\(\dot{\bm{z}}\_{op}\\) in the principal coordinate system can be expressed as:
\begin{align}
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\\\
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\
\dot{\bm{z}}\_{op} &= \bm{z}\_n^{-1} \dot{\bm{z}}\_0
\end{align}
@ -529,7 +520,6 @@ where \\(\bm{z}\_0\\) and \\(\dot{\bm{z}}\_0\\) are the vectors of initial displ
We have now everything required to solve the equations in the principal coordinate system.
<div class="important">
<div></div>
The variables in physical coordinates are the positions and velocities of the masses.
The variables in principal coordinates are the displacements and velocities of each mode of vibration.
@ -568,12 +558,12 @@ Let's first examine the force transformation from physical to principal coordina
\begin{equation}
\bm{F}\_p = \bm{z}\_n^T \bm{F} = \begin{bmatrix}
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
z\_{n11} & z\_{n12} & z\_{n13} \\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\
z\_{n31} & z\_{n32} & z\_{n33}
\end{bmatrix}^T \begin{bmatrix}
F\_1 \\\\\\
F\_2 \\\\\\
F\_1 \\\\
F\_2 \\\\
F\_3
\end{bmatrix}
\end{equation}
@ -584,12 +574,12 @@ Let's now examine the displacement transformation from principal to physical coo
\begin{equation}
\bm{z} = \bm{z}\_n \bm{z}\_p = \begin{bmatrix}
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
z\_{n11} & z\_{n12} & z\_{n13} \\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\
z\_{n31} & z\_{n32} & z\_{n33}
\end{bmatrix} \begin{bmatrix}
z\_{p1} \\\\\\
z\_{p2} \\\\\\
z\_{p1} \\\\
z\_{p2} \\\\
z\_{p3}
\end{bmatrix}
\end{equation}
@ -597,7 +587,6 @@ Let's now examine the displacement transformation from principal to physical coo
And thus, if we are only interested in the physical displacement of the mass 2 (\\(z\_2 = z\_{n21} z\_{p1} + z\_{n22} z\_{p2} + z\_{n23} z\_{p3}\\)), only the second row of the modal matrix is required to transform the three displacements \\(z\_{p1}\\), \\(z\_{p2}\\), \\(z\_{p3}\\) in principal coordinates to \\(z\_2\\).
<div class="important">
<div></div>
**Only the rows of the modal matrix that correspond to degrees of freedom to which forces are applied and/or for which displacements are desired are required to complete the model.**
@ -698,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="org027da35"></a>
<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
The procedure to obtain the frequency response from a modal form is as follow:
@ -706,11 +695,11 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#orgafc54fa).
This will be applied to the model shown in Figure [9](#figure--fig:hatch00-tdof-model).
<a id="orgafc54fa"></a>
<a id="figure--fig:hatch00-tdof-model"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="<span class=\"figure-number\">Figure 9: </span>tdof undamped model for modal analysis" >}}
### Review from Previous Results {#review-from-previous-results}
@ -725,8 +714,8 @@ From previous analysis, we know the eigenvalues and eigenvectors normalized with
\begin{equation}
\bm{z}\_n = \frac{1}{\sqrt{m}} \begin{bmatrix}
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\\\
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\\\
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\
\frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{6}}
\end{bmatrix}
\end{equation}
@ -735,13 +724,13 @@ Knowing that in principal coordinates the mass matrix is the identify matrix and
\begin{equation}
\bm{m}\_n = \begin{bmatrix}
1 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
1 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 1
\end{bmatrix}, \quad
\bm{k}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 3
\end{bmatrix} \frac{k}{m}
\end{equation}
@ -761,8 +750,8 @@ The equations of motion in principal coordinates are then:
which give:
\begin{align}
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\\\
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\\\
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\
\ddot{z}\_{p3} + \frac{3k}{m} z\_{p3} &= (F\_1 - 2 F\_2 + F\_3) \frac{1}{\sqrt{6m}}
\end{align}
@ -773,48 +762,48 @@ Taking the Laplace transform of each equation gives:
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{1}} \\\\\\
\frac{z\_{p2}}{F\_{1}} \\\\\\
\frac{z\_{p1}}{F\_{1}} \\\\
\frac{z\_{p2}}{F\_{1}} \\\\
\frac{z\_{p3}}{F\_{1}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p11} \\\\\\
z\_{p21} \\\\\\
z\_{p11} \\\\
z\_{p21} \\\\
z\_{p31}
\end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{2}} \\\\\\
\frac{z\_{p2}}{F\_{2}} \\\\\\
\frac{z\_{p1}}{F\_{2}} \\\\
\frac{z\_{p2}}{F\_{2}} \\\\
\frac{z\_{p3}}{F\_{2}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
0 \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
0 \\\\
\frac{-2}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p12} \\\\\\
z\_{p22} \\\\\\
z\_{p12} \\\\
z\_{p22} \\\\
z\_{p32}
\end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{3}} \\\\\\
\frac{z\_{p2}}{F\_{3}} \\\\\\
\frac{z\_{p1}}{F\_{3}} \\\\
\frac{z\_{p2}}{F\_{3}} \\\\
\frac{z\_{p3}}{F\_{3}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p13} \\\\\\
z\_{p23} \\\\\\
z\_{p13} \\\\
z\_{p23} \\\\
z\_{p33}
\end{bmatrix}
\end{equation}
@ -839,7 +828,7 @@ And the transfer functions \\(\frac{z\_i}{F\_j}\\) can be computed.
For instance, the contributions to the transfer function \\(\frac{z\_1}{F\_1}\\) are:
\begin{align}
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\\\
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\
& = \frac{\frac{1}{3m}}{s^2} + \frac{\frac{1}{2m}}{s^2 + \omega\_2^2} + \frac{\frac{1}{6m}}{s^2 + \omega\_3^2}
\end{align}
@ -858,7 +847,6 @@ The forces transform in the principal coordinates using:
\end{equation}
<div class="important">
<div></div>
Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized eigenvector), then \\(\bm{F}\_p\\) will be null except for its i'th term and only the i'th mode will be excited.
@ -870,7 +858,6 @@ Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized ei
Any transfer function derived from the modal analysis is an additive combination of sdof systems.
<div class="important">
<div></div>
Each single degree of freedom system has a gain determined by the appropriate eigenvector entries and a resonant frequency given by the appropriate eigenvalue.
@ -886,33 +873,33 @@ If modes have some damping:
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
\end{equation}
Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
</div>
Figure [10](#orgf64b6e5) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="orgf64b6e5"></a>
<a id="figure--fig:hatch00-z11-tf-example"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 10: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
The zeros for SISO transfer functions are the roots of the numerator, however, from modal analysis we can see that the zeros arise when modes combine with appropriate phase such that the resulting motion is null.
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="org39bd7f2"></a>
<span class="org-target" id="org-target--sec:siso_state_space"></span>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgc285575).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="orgc285575"></a>
<a id="figure--fig:hatch00-cantilever-beam"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="<span class=\"figure-number\">Figure 11: </span>Cantilever beam with forcing function at midpoint" >}}
The steps to define the smallest model are:
@ -952,7 +939,7 @@ We will discuss in this section two methods of sorting, one which is applicable
The general equation for the overall transfer function of undamped and damped systems are:
\begin{align}
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\\\
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\end{align}
@ -989,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="org658f39a"></a>
<span class="org-target" id="org-target--sec:ground_acceleration"></span>
### Model Description {#model-description}
@ -1003,25 +990,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="orgcd094f5"></a>
<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org97a4ded)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-model)).
### Actuator Description {#actuator-description}
<a id="org97a4ded"></a>
<a id="figure--fig:hatch00-disk-drive-siso-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orga92b66d)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
<a id="orga92b66d"></a>
<a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#orga92b66d) shows the nodes used for the reduced Matlab model.
Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@ -1045,10 +1032,8 @@ A recommended sequence for analyzing dynamic finite element models is:
A small section of the exported `.eig` file from ANSYS is shown bellow..
<div class="exampl">
<div></div>
<div class="monoblock">
<div></div>
LOAD STEP= 1 SUBSTEP= 1
FREQ= 8.1532 LOAD CASE= 0
@ -1089,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<a id="org58a3a47"></a>
<span class="org-target" id="org-target--sec:balanced_reduction"></span>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1117,7 +1102,7 @@ A mode which cannot be excited by the applied force is said to be **uncontrollab
For a state space system described by:
\begin{align\*}
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\\\
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\
\bm{y} &= \bm{C} \bm{x}
\end{align\*}
@ -1159,7 +1144,7 @@ A similar set of definitions can be made for observability:
\begin{equation}
\bm{\mathcal{O}} = \begin{bmatrix}
\bm{C} \\ \bm{C} \bm{A} \\ \bm{C} \bm{A}^{2} \\ \vdots \\ \bm{C} \bm{A}^{n-1}
\bm{C} \\\ \bm{C} \bm{A} \\\ \bm{C} \bm{A}^{2} \\\ \vdots \\\ \bm{C} \bm{A}^{n-1}
\end{bmatrix}
\end{equation}
@ -1204,16 +1189,16 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="orgf33e1dd"></a>
<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org59e7525)).
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
### Actuator Description {#actuator-description}
<a id="org59e7525"></a>
<a id="figure--fig:hatch00-disk-drive-mimo-schematic"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="<span class=\"figure-number\">Figure 14: </span>Drawing of actuator/suspension system" >}}
A piezo-actuator is now bounded into one side of each of the arms.
The piezo actuator consists of a ceramic element that changes size when a voltage is applied.
@ -1221,7 +1206,6 @@ The piezo actuator consists of a ceramic element that changes size when a voltag
Then the fine positioning motion of the piezo is used in conjunction with VCM's coarse positioning motion, higher servo bandwidth is possible.
<div class="important">
<div></div>
Instead of applying voltage as the input into the piezo elements, we will assume that we have calculated an equivalent set of forces which can be applied at the ends of the element that will replicate the voltage force function.
In this model, we will be applying forces to multiple nodes at the ends of both piezo elements.
@ -1233,11 +1217,11 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#org5f31090) are shown the principal nodes used for the model.
In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
<a id="org5f31090"></a>
<a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="<span class=\"figure-number\">Figure 15: </span>Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
### Matlab Model {#matlab-model}
@ -1354,13 +1338,13 @@ And we note:
G = zn * Gp;
```
<a id="orgbe6df95"></a>
<a id="figure--fig:hatch00-z13-tf"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="<span class=\"figure-number\">Figure 16: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="orgcec939e"></a>
<a id="figure--fig:hatch00-z11-tf"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 17: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
## Matlab with ANSYS {#matlab-with-ansys}
@ -1456,15 +1440,15 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#org1183b44)).
Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
<a id="org1183b44"></a>
<a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="<span class=\"figure-number\">Figure 18: </span>Frequency of the modes" >}}
<a id="org350c1cb"></a>
<a id="figure--fig:hatch00-cant-beam-unsorted-dc-gains"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 19: </span>Unsorted DC Gains" >}}
Let's keep only the first 10 modes.
@ -1531,9 +1515,9 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
<a id="orgd64190f"></a>
<a id="figure--fig:hatch00-cant-beam-sorted-dc-gains"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 20: </span>Sorted DC Gains" >}}
Let's keep only the first 10 **sorted** modes.
@ -1875,9 +1859,9 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms
<a id="orgbdc6b3b"></a>
<a id="figure--fig:hatch00-gramians"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="<span class=\"figure-number\">Figure 21: </span>Observability and Controllability Gramians" >}}
We use `balreal` to rank oscillatory states.
@ -1893,9 +1877,9 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
<a id="org2787898"></a>
<a id="figure--fig:hatch00-cant-beam-gramian-balanced"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="<span class=\"figure-number\">Figure 22: </span>Sorted values of the Gramian of the balanced realization" >}}
Now we can choose the number of states to keep.
@ -2136,9 +2120,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
```
## Bibliography {#bibliography}
<a id="org4036e02"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="orgcda3e53"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
</div>

11
content/book/horowitz15_art_of_elect_third_edition.md
View File

@ -1,16 +1,16 @@
+++
title = "The Art of Electronics - Third Edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "One of the best book in electronics. Cover most topics (both analog and digital)."
keywords = ["electronics"]
draft = false
+++
Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
: [Reference Books]({{< relref "reference_books.md" >}}), [Electronics]({{< relref "electronics.md" >}})
Reference
: ([Horowitz 2015](#org8eab88c))
: (<a href="#citeproc_bib_item_1">Horowitz 2015</a>)
Author(s)
: Horowitz, P.
@ -19,7 +19,8 @@ Year
: 2015
## Bibliography {#bibliography}
<a id="org8eab88c"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Horowitz, Paul. 2015. <i>The Art of Electronics - Third Edition</i>. New York, NY, USA: Cambridge University Press.</div>
</div>

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@ -1,15 +1,15 @@
+++
title = "Fundamental principles of engineering nanometrology"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
keywords = ["Metrology"]
draft = false
+++
Tags
: [Metrology]({{<relref "metrology.md#" >}})
: [Metrology]({{< relref "metrology.md" >}})
Reference
: ([Leach 2014](#org27b4df3))
: (<a href="#citeproc_bib_item_1">Leach 2014</a>)
Author(s)
: Leach, R.
@ -64,8 +64,8 @@ The second order nature means that cosine error quickly diminish as the alignmen
## Latest advances in displacement interferometry {#latest-advances-in-displacement-interferometry}
Commercial interferometers
=> fused silica optics housed in Invar mounts
=> all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
=&gt; fused silica optics housed in Invar mounts
=&gt; all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
One advantage that homodyme systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer.
@ -88,7 +88,8 @@ The measurement of angles is then relative.
This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
## Bibliography {#bibliography}
<a id="org27b4df3"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<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>Leach, Richard. 2014. <i>Fundamental Principles of Engineering Nanometrology</i>. Elsevier. doi:<a href="https://doi.org/10.1016/c2012-0-06010-3">10.1016/c2012-0-06010-3</a>.</div>
</div>

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@ -1,24 +1,25 @@
+++
title = "Basics of precision engineering - 1st edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
keywords = ["Metrology", "Mechatronics"]
draft = true
+++
Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
: [Precision Engineering]({{< relref "precision_engineering.md" >}})
Reference
: ([Leach and Smith 2018](#org02e139c))
: (<a href="#citeproc_bib_item_1">Leach and Smith 2018</a>)
Author(s)
: Leach, R., & Smith, S. T.
: Leach, R., &amp; Smith, S. T.
Year
: 2018
## Bibliography {#bibliography}
<a id="org02e139c"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard, and Stuart T. Smith. 2018. <i>Basics of Precision Engineering - 1st Edition</i>. CRC Press.</div>
</div>

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@ -1,16 +1,16 @@
+++
title = "Grounding and Shielding: Circuits and Interference"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Explains in a clear manner what is grounding and shielding and what are the fundamental physics behind these terms."
keywords = ["Electronics"]
draft = false
+++
Tags
: [Electronics]({{< relref "electronics" >}})
: [Electronics]({{< relref "electronics.md" >}})
Reference
: ([Morrison 2016](#org7a49345))
: (<a href="#citeproc_bib_item_1">Morrison 2016</a>)
Author(s)
: Morrison, R.
@ -22,7 +22,6 @@ Year
## Voltage and Capacitors {#voltage-and-capacitors}
<div class="sum">
<div></div>
This first chapter described the electric field that is basic to all electrical activity.
The electric or \\(E\\) field represents forces between charges.
@ -53,9 +52,9 @@ This displacement current flows when charges are added or removed from the plate
### Field representation {#field-representation}
<a id="orga3615d0"></a>
<a id="figure--fig:morrison16-E-field-charge"></a>
{{< figure src="/ox-hugo/morrison16_E_field_charge.svg" caption="Figure 1: The force field lines around a positively chaged conducting sphere" >}}
{{< figure src="/ox-hugo/morrison16_E_field_charge.svg" caption="<span class=\"figure-number\">Figure 1: </span>The force field lines around a positively chaged conducting sphere" >}}
### The definition of voltage {#the-definition-of-voltage}
@ -64,22 +63,22 @@ This displacement current flows when charges are added or removed from the plate
### Equipotential surfaces {#equipotential-surfaces}
### The force field or \\(E\\) field between two conducting plates {#the-force-field-or--e--field-between-two-conducting-plates}
### The force field or \\(E\\) field between two conducting plates {#the-force-field-or-e-field-between-two-conducting-plates}
<a id="org82b88ec"></a>
<a id="figure--fig:morrison16-force-field-plates"></a>
{{< figure src="/ox-hugo/morrison16_force_field_plates.svg" caption="Figure 2: The force field between two conducting plates with equal and opposite charges and spacing distance \\(h\\)" >}}
{{< figure src="/ox-hugo/morrison16_force_field_plates.svg" caption="<span class=\"figure-number\">Figure 2: </span>The force field between two conducting plates with equal and opposite charges and spacing distance \\(h\\)" >}}
### Electric field patterns {#electric-field-patterns}
<a id="org16f20a9"></a>
<a id="figure--fig:morrison16-electric-field-ground-plane"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_ground_plane.svg" caption="Figure 3: The electric field pattern of one circuit trace and two circuit traces over a ground plane" >}}
{{< figure src="/ox-hugo/morrison16_electric_field_ground_plane.svg" caption="<span class=\"figure-number\">Figure 3: </span>The electric field pattern of one circuit trace and two circuit traces over a ground plane" >}}
<a id="org38210cb"></a>
<a id="figure--fig:morrison16-electric-field-shielded-conductor"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_shielded_conductor.svg" caption="Figure 4: Field configuration around a shielded conductor" >}}
{{< figure src="/ox-hugo/morrison16_electric_field_shielded_conductor.svg" caption="<span class=\"figure-number\">Figure 4: </span>Field configuration around a shielded conductor" >}}
### The energy stored in an electric field {#the-energy-stored-in-an-electric-field}
@ -88,11 +87,11 @@ This displacement current flows when charges are added or removed from the plate
### Dielectrics {#dielectrics}
### The \\(D\\) field {#the--d--field}
### The \\(D\\) field {#the-d-field}
<a id="org5a4329e"></a>
<a id="figure--fig:morrison16-E-D-fields"></a>
{{< figure src="/ox-hugo/morrison16_E_D_fields.svg" caption="Figure 5: The electric field pattern in the presence of a dielectric" >}}
{{< figure src="/ox-hugo/morrison16_E_D_fields.svg" caption="<span class=\"figure-number\">Figure 5: </span>The electric field pattern in the presence of a dielectric" >}}
### Capacitance {#capacitance}
@ -122,7 +121,6 @@ This displacement current flows when charges are added or removed from the plate
## Magnetics {#magnetics}
<div class="sum">
<div></div>
This chapter discusses magnetic fields.
As in the electric field, there are two measures of the same magnetic field.
@ -150,11 +148,11 @@ In a few elements, the atomic structure is such that atoms align to generate a n
The flow of electrons is another way to generate a magnetic field.
The letter \\(H\\) is reserved for the magnetic field generated by a current.
Figure [6](#org9b0e888) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
Figure [6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
<a id="org9b0e888"></a>
<a id="figure--fig:morrison16-H-field"></a>
{{< figure src="/ox-hugo/morrison16_H_field.svg" caption="Figure 6: The \\(H\\) field around a current-carrying conductor" >}}
{{< figure src="/ox-hugo/morrison16_H_field.svg" caption="<span class=\"figure-number\">Figure 6: </span>The \\(H\\) field around a current-carrying conductor" >}}
The magnetic field is a force field.
This force can only be exerted on another magnetic field.
@ -169,7 +167,7 @@ Ampere's law states that the integral of the \\(H\\) field intensity in a closed
\boxed{\oint H dl = I}
\end{equation}
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#org9b0e888), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
Solving for \\(H\\), we obtain
\begin{equation}
@ -181,29 +179,29 @@ And we see that \\(H\\) has units of amperes per meter.
### The solenoid {#the-solenoid}
The magnetic field of a solenoid is shown in Figure [7](#orgd3a9cf9).
The magnetic field of a solenoid is shown in Figure [7](#figure--fig:morrison16-solenoid).
The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
Using Ampere's law \eqref{eq:ampere_law}:
Using Ampere's law <eq:ampere_law>:
\begin{equation}
\oint H dl \approx n I l
\end{equation}
<a id="orgd3a9cf9"></a>
<a id="figure--fig:morrison16-solenoid"></a>
{{< figure src="/ox-hugo/morrison16_solenoid.svg" caption="Figure 7: The \\(H\\) field around a solenoid" >}}
{{< figure src="/ox-hugo/morrison16_solenoid.svg" caption="<span class=\"figure-number\">Figure 7: </span>The \\(H\\) field around a solenoid" >}}
### Faraday's law and the induction field {#faraday-s-law-and-the-induction-field}
When a conducting coil is moved through a magnetic field, a voltage appears at the open ends of the coil.
This is illustrated in Figure [8](#org4b2f5c1).
This is illustrated in Figure [8](#figure--fig:morrison16-voltage-moving-coil).
The voltage depends on the number of turns in the coil and the rate at which the flux is changing.
<a id="org4b2f5c1"></a>
<a id="figure--fig:morrison16-voltage-moving-coil"></a>
{{< figure src="/ox-hugo/morrison16_voltage_moving_coil.svg" caption="Figure 8: A voltage induced into a moving coil" >}}
{{< figure src="/ox-hugo/morrison16_voltage_moving_coil.svg" caption="<span class=\"figure-number\">Figure 8: </span>A voltage induced into a moving coil" >}}
The magnetic field has two measured.
The \\(H\\) or magnetic field that is proportional to current flow.
@ -232,14 +230,13 @@ The inverse is also true.
### The definition of inductance {#the-definition-of-inductance}
<div class="definition">
<div></div>
Inductance is defined as the ratio of magnetic flux generated per unit current.
The unit of inductance if the henry.
</div>
For the coil in Figure [7](#orgd3a9cf9):
For the coil in Figure [7](#figure--fig:morrison16-solenoid):
\begin{equation} \label{eq:inductance\_coil}
V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
@ -247,12 +244,12 @@ V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
where \\(k\\) relates to the geometry of the coil.
Equation \eqref{eq:inductance_coil} states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
Equation <eq:inductance_coil> states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
### The energy stored in an inductance {#the-energy-stored-in-an-inductance}
One way to calculate the work stored in a magnetic field is to use Eq. \eqref{eq:inductance_coil}.
One way to calculate the work stored in a magnetic field is to use Eq. <eq:inductance_coil>.
The voltage \\(V\\) applied to a coil results in a linearly increasing current.
At any time \\(t\\), the power \\(P\\) supplied is equal to \\(VI\\).
Power is the rate of change of energy or \\(P = d\bm{E}/dt\\) where \\(\bm{E}\\) is the stored energy in the inductance.
@ -263,7 +260,6 @@ We then have the stored energy in an inductance:
\end{equation}
<div class="important">
<div></div>
An inductor stores field energy.
It does not dissipate energy.
@ -275,7 +271,6 @@ The movement of energy into the inductor thus requires both an electric and a ma
This is due to the Faraday's law that requires a voltage when changing magnetic flux couples to a coil.
<div class="exampl">
<div></div>
Consider a 1mH inductor carrying a current of 0.1A.
The stored energy is \\(5 \times 10^{-4} J\\).
@ -309,7 +304,6 @@ In a typical circuit, conductor carrying current, the average electron velocity
## Digital Electronics {#digital-electronics}
<div class="sum">
<div></div>
This chapter shows that both electric and magnetic field are needed to move energy over pairs of conductors.
The idea of transporting electrical energy in field is extended to traces and conducting planes on printed circuit boards.
@ -415,7 +409,6 @@ Radiation occurs at the leading edge of a wave as it moves down the transmission
## Analog Circuits {#analog-circuits}
<div class="sum">
<div></div>
This chapter treats the general problem of analog instrumentation.
The signals of interest are often generated while testing functioning hardware.
@ -451,7 +444,6 @@ There are many transducers that can measure temperature, strain, stress, positio
The signals generated are usually in the milli-volt range and must be amplified, conditioned, and then recorded for later analysis.
<div class="important">
<div></div>
It can be very difficult to verify that the measurement is valid.
For example, signals that overload an input stage can produce noise that may look like signal.
@ -459,7 +451,6 @@ For example, signals that overload an input stage can produce noise that may loo
</div>
<div class="definition">
<div></div>
1. **Reference Conductor**.
Any conductor used as the zero of voltage.
@ -485,39 +476,39 @@ For example, signals that overload an input stage can produce noise that may loo
### The basic shield enclosure {#the-basic-shield-enclosure}
Consider the simple amplifier circuit shown in Figure [9](#org3286d62) with:
Consider the simple amplifier circuit shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
- \\(V\_1\\) the input lead
- \\(V\_2\\) the output lead
- \\(V\_3\\) the conducting enclosure which is floating and taken as the reference conductor
- \\(V\_4\\) a signal common or reference conductor
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#org3286d62) (b).
The equivalent circuit is shown in Figure [9](#org3286d62) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
The equivalent circuit is shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
<a id="org3286d62"></a>
<a id="figure--fig:morrison16-parasitic-capacitance-amp"></a>
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="Figure 9: Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="<span class=\"figure-number\">Figure 9: </span>Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#org9f3c9db)).
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#figure--fig:morrison16-grounding-shield-amp)).
When this connection is made, the feedback is removed and the enclosure no longer couples signals into the feedback structure.
The conductive enclosure is called a **shield**.
Connecting the signal common to the conductive enclosure is called "**grounding the shield**".
This "grounding" usually removed "hum" from the circuit.
<a id="org9f3c9db"></a>
<a id="figure--fig:morrison16-grounding-shield-amp"></a>
{{< figure src="/ox-hugo/morrison16_grounding_shield_amp.svg" caption="Figure 10: Grounding the shield to limit feedback" >}}
{{< figure src="/ox-hugo/morrison16_grounding_shield_amp.svg" caption="<span class=\"figure-number\">Figure 10: </span>Grounding the shield to limit feedback" >}}
Most practical circuits provide connections to external points.
To see the effect of making a _single_ external connection, open the conductive enclosure and connect the input circuit common to an external ground.
Figure [11](#orgc4242ae) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
A problem can be caused by an incorrect location of the connection between the cable shield and the enclosure.
In Figure [11](#orgc4242ae) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\,\Omega\\), this current induced voltage that it added to the transmitted signal.
In Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\\,\Omega\\), this current induced voltage that it added to the transmitted signal.
Our goal in this chapter is to find ways of keeping interference currents from flowing in any input signal conductor.
To remove this coupling, the shield connection to circuit common must be made at the point, where the circuit common connects to the external ground.
This connection is shown in Figure [11](#orgc4242ae) (b).
This connection is shown in Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
This connection keeps the circulation of interference current on the outside of the shield.
There is only one point of zero signal potential external to the enclosure and that is where the signal common connects to an external hardware ground.
@ -527,7 +518,6 @@ If there is an external electromagnetic field, there will be current flow in the
A voltage gradient will couple interference capacitively to the signal conductors.
<div class="important">
<div></div>
An input circuit shield should connect to the circuit common, where the signal common makes its connection to the source of signal.
Any other shield connection will introduce interference.
@ -535,16 +525,15 @@ Any other shield connection will introduce interference.
</div>
<div class="important">
<div></div>
Shielding is not an issue of finding a "really good ground".
It is an issue of using the _right_ ground.
</div>
<a id="orgc4242ae"></a>
<a id="figure--fig:morrison16-enclosure-shield-1-2-leads"></a>
{{< figure src="/ox-hugo/morrison16_enclosure_shield_1_2_leads.png" caption="Figure 11: (a) The problem of bringing one lead out of a shielded region. Unwanted current circulates in the signal lead 2. (b) The \\(E\\) field circulate current in the shield, not in the signal conductor." >}}
{{< figure src="/ox-hugo/morrison16_enclosure_shield_1_2_leads.png" caption="<span class=\"figure-number\">Figure 11: </span>(a) The problem of bringing one lead out of a shielded region. Unwanted current circulates in the signal lead 2. (b) The \\(E\\) field circulate current in the shield, not in the signal conductor." >}}
### The enclosure and utility power {#the-enclosure-and-utility-power}
@ -554,9 +543,9 @@ The power transformer couples fields from the external environment into the encl
The obvious coupling results from capacitance between the primary coil and the secondary coil.
Note that the secondary coil is connected to the circuit common conductor.
<a id="org5995e31"></a>
<a id="figure--fig:morrison16-power-transformer-enclosure"></a>
{{< figure src="/ox-hugo/morrison16_power_transformer_enclosure.png" caption="Figure 12: A power transformer added to the circuit enclosure" >}}
{{< figure src="/ox-hugo/morrison16_power_transformer_enclosure.png" caption="<span class=\"figure-number\">Figure 12: </span>A power transformer added to the circuit enclosure" >}}
### The two-ground problem {#the-two-ground-problem}
@ -566,9 +555,9 @@ Note that the secondary coil is connected to the circuit common conductor.
The basic analog problem is to condition a signal associated with one ground reference potential and transport this signal to a second ground reference potential without adding interference.
<a id="org3228c82"></a>
<a id="figure--fig:morrison16-two-ground-problem"></a>
{{< figure src="/ox-hugo/morrison16_two_ground_problem.svg" caption="Figure 13: The two-circuit enclosures used to transport signals between grounds" >}}
{{< figure src="/ox-hugo/morrison16_two_ground_problem.svg" caption="<span class=\"figure-number\">Figure 13: </span>The two-circuit enclosures used to transport signals between grounds" >}}
### Strain-gauge instrumentation {#strain-gauge-instrumentation}
@ -582,9 +571,9 @@ The basic analog problem is to condition a signal associated with one ground ref
### The basic low-gain differential amplifier (forward referencing amplifier) {#the-basic-low-gain-differential-amplifier--forward-referencing-amplifier}
<a id="org4f33add"></a>
<a id="figure--fig:morrison16-low-gain-diff-amp"></a>
{{< figure src="/ox-hugo/morrison16_low_gain_diff_amp.svg" caption="Figure 14: The low-gain differential amplifier applied to the two-ground problem" >}}
{{< figure src="/ox-hugo/morrison16_low_gain_diff_amp.svg" caption="<span class=\"figure-number\">Figure 14: </span>The low-gain differential amplifier applied to the two-ground problem" >}}
### Shielding in power transformers {#shielding-in-power-transformers}
@ -599,7 +588,6 @@ The basic analog problem is to condition a signal associated with one ground ref
### Signal flow paths in analog circuits {#signal-flow-paths-in-analog-circuits}
<div class="important">
<div></div>
Here are a few rule that will help in analog board layout:
@ -625,13 +613,13 @@ Here are a few rule that will help in analog board layout:
### Feedback theory {#feedback-theory}
<a id="org4a09d89"></a>
<a id="figure--fig:morrison16-basic-feedback-circuit"></a>
{{< figure src="/ox-hugo/morrison16_basic_feedback_circuit.svg" caption="Figure 15: The basic feedback circuit" >}}
{{< figure src="/ox-hugo/morrison16_basic_feedback_circuit.svg" caption="<span class=\"figure-number\">Figure 15: </span>The basic feedback circuit" >}}
<a id="orgf414d06"></a>
<a id="figure--fig:morrison16-LR-stabilizing-network"></a>
{{< figure src="/ox-hugo/morrison16_LR_stabilizing_network.svg" caption="Figure 16: An LR-stabilizing network" >}}
{{< figure src="/ox-hugo/morrison16_LR_stabilizing_network.svg" caption="<span class=\"figure-number\">Figure 16: </span>An LR-stabilizing network" >}}
### Output loads and circuit stability {#output-loads-and-circuit-stability}
@ -667,27 +655,26 @@ If the resistors are replaced by capacitors, the gain is the ratio of reactances
This feedback circuit is called a **charge converter**.
The charge on the input capacitor is transferred to the feedback capacitor.
If the feedback capacitor is smaller than the transducer capacitance by a factor of 100, then the voltage across the feedback capacitor will be 100 times greater than the open-circuit transducer voltage.
This feedback arrangement is shown in Figure [17](#org74f6090).
This feedback arrangement is shown in Figure [17](#figure--fig:morrison16-charge-amplifier).
The open-circuit input signal voltage is \\(Q/C\_T\\).
The output voltage is \\(Q/C\_{FB}\\).
The voltage gain is therefore \\(C\_T/C\_{FB}\\).
Note that there is essentially no voltage at the summing node \\(s\_p\\).
<div class="important">
<div></div>
A charge converter does not amplifier charge.
It converts a charge signal to a voltage.
</div>
<a id="org74f6090"></a>
<a id="figure--fig:morrison16-charge-amplifier"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier.svg" caption="Figure 17: A basic charge amplifier" >}}
{{< figure src="/ox-hugo/morrison16_charge_amplifier.svg" caption="<span class=\"figure-number\">Figure 17: </span>A basic charge amplifier" >}}
<a id="orgb9f996c"></a>
<a id="figure--fig:morrison16-charge-amplifier-feedback-resistor"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier_feedback_resistor.svg" caption="Figure 18: The resistor feedback arrangement to control the low-frequency response" >}}
{{< figure src="/ox-hugo/morrison16_charge_amplifier_feedback_resistor.svg" caption="<span class=\"figure-number\">Figure 18: </span>The resistor feedback arrangement to control the low-frequency response" >}}
### DC power supplies {#dc-power-supplies}
@ -705,7 +692,6 @@ It converts a charge signal to a voltage.
## Utility Power and Facility Grounding {#utility-power-and-facility-grounding}
<div class="sum">
<div></div>
This chapter discusses the relationship between utility power and the performance of electrical circuits.
Utility installations in facilities are controller by the NEC (National Electrical Code).
@ -798,7 +784,7 @@ Listed equipment
### Neutral conductors {#neutral-conductors}
### \\(k\\) factor in transformers {#k--factor-in-transformers}
### \\(k\\) factor in transformers {#k-factor-in-transformers}
### Power factor correction {#power-factor-correction}
@ -858,7 +844,6 @@ Listed equipment
## Radiation {#radiation}
<div class="sum">
<div></div>
This chapter discusses radiation from circuit boards, transmission lines, conductor loops, and antennas.
The frequency spectrum of square waves and pulses is presented.
@ -917,7 +902,6 @@ Simple tools for locating sources of radiation are suggested.
## Shielding from Radiation {#shielding-from-radiation}
<div class="sum">
<div></div>
Cable shields are often made of aluminum foil or tinned copper braid.
Drain wires make it practical to connect to the foil.
@ -1033,7 +1017,8 @@ To transport RF power without reflections, the source impedance and the terminat
### Shielded and screen rooms {#shielded-and-screen-rooms}
## Bibliography {#bibliography}
<a id="org7a49345"></a>Morrison, Ralph. 2016. _Grounding and Shielding: Circuits and Interference_. John Wiley & Sons.
<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>Morrison, Ralph. 2016. <i>Grounding and Shielding: Circuits and Interference</i>. John Wiley &#38; Sons.</div>
</div>

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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@ -1,16 +1,16 @@
+++
title = "Vibration Control of Active Structures - Fourth Edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Gives a broad overview of vibration control."
keywords = ["Control", "Vibration"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Reference Books]({{< relref "reference_books.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [HAC-HAC]({{< relref "hac_hac.md" >}})
Reference
: ([Preumont 2018](#orgf75c814))
: (<a href="#citeproc_bib_item_1">Preumont 2018</a>)
Author(s)
: Preumont, A.
@ -63,11 +63,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
<a id="org30e8b62"></a>
<a id="figure--fig:classical-feedback-small"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#org30e8b62). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@ -89,12 +89,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
<a id="org0cb2cac"></a>
<a id="figure--fig:feedforward-adaptative"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0cb2cac).
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#figure--fig:feedforward-adaptative).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@ -125,11 +125,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="org5fed023"></a>
<a id="figure--fig:design-steps"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org5fed023).
The various steps of the design of a controlled structure are shown in figure [3](#figure--fig:design-steps).
The **starting point** is:
@ -156,21 +156,20 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgc558cd1)):
From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
\begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*}
<a id="orgc558cd1"></a>
<a id="figure--fig:general-plant"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Block diagram of the control System" >}}
The frequency content of the disturbance \\(w\\) is usually described by its **power spectral density** \\(\Phi\_w (\omega)\\) which describes the frequency distribution of the meas-square value.
<div class="cbox">
<div></div>
\\[\sigma\_w = \sqrt{\int\_0^\infty \Phi\_w(\omega) d\omega}\\]
@ -179,7 +178,6 @@ The frequency content of the disturbance \\(w\\) is usually described by its **p
Even more interesting for the design is the **Cumulative Mean Square** response defined by the integral of the PSD in the frequency range \\([\omega, \infty[\\).
<div class="cbox">
<div></div>
\\[\sigma\_z^2(\omega) = \int\_\omega^\infty \Phi\_z(\nu) d\nu = \int\_\omega^\infty |T\_{zw}|^2 \Phi\_w(\nu) d\nu \\]
@ -188,14 +186,14 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgd0ed9cf).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="orgd0ed9cf"></a>
<a id="figure--fig:cas-plot"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="<span class=\"figure-number\">Figure 5: </span>Error budget distribution in OL and CL for increasing gains" >}}
### Pseudo-inverse {#pseudo-inverse}
@ -254,7 +252,6 @@ This will have usually little impact of the fitting error while reducing conside
The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a discrete, flexible structure with a finite number \\(n\\) of degrees of freedom is
<div class="cbox">
<div></div>
\begin{equation}
M \ddot{x} + C \dot{x} + K x = f
@ -271,7 +268,6 @@ With:
The damping matrix \\(C\\) represents the various dissipation mechanisms in the structure, which are usually poorly known. One of the popular hypotheses is the Rayleigh damping.
<div class="cbox">
<div></div>
\begin{equation}
C = \alpha M + \beta K
@ -299,14 +295,14 @@ The number of mode shapes is equal to the number of degrees of freedom \\(n\\).
The mode shapes are orthogonal with respect to the stiffness and mass matrices:
\begin{align}
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\\\
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\
\phi\_i^T K \phi\_j &= \mu\_i \omega\_i^2 \delta\_{ij}
\end{align}
With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode \\(i\\).
### [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
### [Modal Decomposition]({{< relref "modal_decomposition.md" >}}) {#modal-decomposition--modal-decomposition-dot-md}
#### Structure Without Rigid Body Modes {#structure-without-rigid-body-modes}
@ -314,7 +310,6 @@ With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode
Let perform a change of variable from physical coordinates \\(x\\) to modal coordinates \\(z\\).
<div class="cbox">
<div></div>
\begin{equation}
x = \Phi z
@ -336,12 +331,11 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
One can verify that the Rayleigh damping \eqref{eq:rayleigh_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
One can verify that the Rayleigh damping <eq:rayleigh_damping> complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
And we obtain decoupled modal equations \eqref{eq:modal_eom}.
And we obtain decoupled modal equations <eq:modal_eom>.
<div class="cbox">
<div></div>
\begin{equation}
\ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
@ -355,7 +349,7 @@ with:
</div>
Typical values of the modal damping ratio are summarized on table [tab:damping_ratio](#tab:damping_ratio).
Typical values of the modal damping ratio are summarized on table <tab:damping_ratio>.
<a id="table--tab:damping-ratio"></a>
<div class="table-caption">
@ -372,15 +366,15 @@ Typical values of the modal damping ratio are summarized on table [tab:damping_r
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general_eom} and \eqref{eq:modal_eom} lies in the change of coordinates.
If one accepts the assumption of classical damping, the only difference between equation <eq:general_eom> and <eq:modal_eom> lies in the change of coordinates.
However, in physical coordinates, the number of degrees of freedom is usually very large.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal_eom} can often be restricted to theses modes.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation <eq:modal_eom> can often be restricted to theses modes.
Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
#### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
If we consider the steady-state response of equation \eqref{eq:general_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
If we consider the steady-state response of equation <eq:general_eom> to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
\\[ X = G(\omega) F \\]
Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@ -400,11 +394,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
<a id="orgeec9f86"></a>
<a id="figure--fig:neglected-modes"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orgeec9f86)).
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@ -418,7 +412,6 @@ The quasi-static correction of the high frequency modes \\(R\\) is called the **
### Collocated Control System {#collocated-control-system}
<div class="cbox">
<div></div>
A **collocated control system** is a control system where:
@ -443,30 +436,28 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org2389144)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
<a id="org2389144"></a>
<a id="figure--fig:collocated-control-frf"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="<span class=\"figure-number\">Figure 7: </span>Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
The amplitude of the FRF goes from \\(-\infty\\) at the resonance frequencies \\(\omega\_i\\) to \\(+\infty\\) at the next resonance frequency \\(\omega\_{i+1}\\). Therefore, in every interval, there is a frequency \\(z\_i\\) such that \\(\omega\_i < z\_i < \omega\_{i+1}\\) where the amplitude of the FRF vanishes. The frequencies \\(z\_i\\) are called **anti-resonances**.
<div class="cbox">
<div></div>
Undamped **collocated control systems** have **alternating poles and zeros** on the imaginary axis.
For lightly damped structure, the poles and zeros are just moved a little bit in the left-half plane, but they are still interlacing.
</div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org9a738f7).
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#figure--fig:collocated-zero).
<a id="org9a738f7"></a>
<a id="figure--fig:collocated-zero"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="<span class=\"figure-number\">Figure 8: </span>Structure with collocated actuator and sensor" >}}
<div class="cbox">
<div></div>
The frequency of the transmission zero \\(z\_i\\) and the mode shape associated are the **natural frequency** and the **mode shape** of the system obtained by **constraining the d.o.f. on which the control systems acts**.
@ -476,11 +467,11 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#org2389144), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
By looking at figure [7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org52c26c5"></a>
<a id="figure--fig:alternating-p-z"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="<span class=\"figure-number\">Figure 9: </span>Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
@ -488,7 +479,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in figure [9](#org52c26c5). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
The corresponding Bode plot is represented in figure [9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@ -510,14 +501,14 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#orga1a9b67)):
The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
<a id="orga1a9b67"></a>
<a id="figure--fig:voice-coil-schematic"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="<span class=\"figure-number\">Figure 10: </span>Physical principle of a voice coil transducer" >}}
We note:
@ -527,7 +518,6 @@ We note:
- \\(i\\) the current into the coil
<div class="cbox">
<div></div>
**Faraday's law**:
@ -553,11 +543,11 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orgc439137)).
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#figure--fig:proof-mass-actuator)).
<a id="orgc439137"></a>
<a id="figure--fig:proof-mass-actuator"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="<span class=\"figure-number\">Figure 11: </span>Proof-mass actuator" >}}
If we apply the second law of Newton on the mass:
\\[ m\ddot{x} + c\dot{x} + kx = f = Ti \\]
@ -571,7 +561,6 @@ The total force applied on the support is:
The transfer function between the total force and the current \\(i\\) applied to the coil is :
<div class="cbox">
<div></div>
\begin{equation}
\frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi\_p \omega\_p s + \omega\_p^2}
@ -585,11 +574,11 @@ with:
</div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org3b93a8e)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#figure--fig:proof-mass-tf)).
<a id="org3b93a8e"></a>
<a id="figure--fig:proof-mass-tf"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="<span class=\"figure-number\">Figure 12: </span>Bode plot \\(F/i\\) of the proof-mass actuator" >}}
#### Geophone {#geophone}
@ -600,7 +589,7 @@ The voltage \\(e\\) of the coil is used as the sensor output.
If \\(x\_0\\) is the displacement of the support and if the voice coil is open (\\(i=0\\)), the governing equations are:
\begin{align\*}
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\\\
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\
T(\dot{x}-\dot{x\_0}) &= e
\end{align\*}
@ -612,25 +601,25 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="org7ded49f"></a>
<a id="figure--fig:geophone"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="<span class=\"figure-number\">Figure 13: </span>Model of a geophone based on a voice coil transducer" >}}
Designing geophones with very low corner frequency is in general difficult. Active geophones where the frequency is lowered electronically may constitute a good alternative option.
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org82c090c).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
<a id="org82c090c"></a>
<a id="figure--fig:electro-mechanical-transducer"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="<span class=\"figure-number\">Figure 14: </span>Electrical analog representation of an electromechanical transducer" >}}
In Laplace form the constitutive equations read:
\begin{align}
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\\\
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\
f & = T\_{em} i + Z\_m v \label{eq:gen\_trans\_f}
\end{align}
@ -645,10 +634,10 @@ With:
- \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
- \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#org8e1c5fb) can be used.
To do so, the bridge circuit as shown on figure [15](#figure--fig:bridge-circuit) can be used.
We can show that
@ -658,19 +647,19 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="org8e1c5fb"></a>
<a id="figure--fig:bridge-circuit"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="<span class=\"figure-number\">Figure 15: </span>Bridge circuit for self-sensing actuation" >}}
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org29efe87) lists various effects that are observed in materials in response to various inputs.
Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
<a id="org29efe87"></a>
<a id="figure--fig:smart-materials"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="<span class=\"figure-number\">Figure 16: </span>Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
### Piezoelectric Transducer {#piezoelectric-transducer}
@ -678,14 +667,12 @@ Figure [16](#org29efe87) lists various effects that are observed in materials in
Piezoelectric materials exhibits two effects described below.
<div class="cbox">
<div></div>
Ability to generate an electrical charge in proportion to an external applied force.
</div>
<div class="cbox">
<div></div>
An electric filed parallel to the direction of polarization induces an expansion of the material.
@ -696,11 +683,10 @@ The most popular piezoelectric materials are Lead-Zirconate-Titanate (PZT) which
We here consider a transducer made of one-dimensional piezoelectric material.
<div class="cbox">
<div></div>
\begin{subequations}
\begin{align}
D & = \epsilon^T E + d\_{33} T\\\\\\
D & = \epsilon^T E + d\_{33} T\\\\
S & = d\_{33} E + s^E T
\end{align}
\end{subequations}
@ -720,16 +706,16 @@ With:
#### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
The set of equations \eqref{eq:piezo_eq} can be written in a matrix form:
The set of equations <eq:piezo_eq> can be written in a matrix form:
\begin{equation}
\begin{bmatrix}D\\S\end{bmatrix}
\begin{bmatrix}D\\\S\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T & d\_{33}\\\\\\
\epsilon^T & d\_{33}\\\\
d\_{33} & s^E
\end{bmatrix}
\begin{bmatrix}E\\T\end{bmatrix}
\begin{bmatrix}E\\\T\end{bmatrix}
\end{equation}
Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the dependent variable.
@ -737,13 +723,13 @@ Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the depend
If \\((E, S)\\) are taken as independant variables:
\begin{equation}
\begin{bmatrix}D\\T\end{bmatrix}
\begin{bmatrix}D\\\T\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T(1-k^2) & e\_{33}\\\\\\
\epsilon^T(1-k^2) & e\_{33}\\\\
-e\_{33} & c^E
\end{bmatrix}
\begin{bmatrix}E\\S\end{bmatrix}
\begin{bmatrix}E\\\S\end{bmatrix}
\end{equation}
With:
@ -752,7 +738,6 @@ With:
- \\(e\_{33} = \frac{d\_{33}}{s^E}\\) is the constant relating the electric displacement to the strain for short-circuited electrodes \\([C/m^2]\\)
<div class="cbox">
<div></div>
\begin{equation}
k^2 = \frac{{d\_{33}}^2}{s^E \epsilon^T} = \frac{{e\_{33}}^2}{c^E \epsilon^T}
@ -763,16 +748,16 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org226015b)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
=
\begin{bmatrix}
C & nd\_{33}\\\\\\
C & nd\_{33}\\\\
nd\_{33} & 1/K\_a
\end{bmatrix}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
\end{equation}
where
@ -784,27 +769,27 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="org226015b"></a>
<a id="figure--fig:piezo-stack"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}}
Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
Equation <eq:piezo_stack_eq> can be inverted to obtain
\begin{equation}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
=
\frac{K\_a}{C(1-k^2)}
\begin{bmatrix}
1/K\_a & -nd\_{33}\\\\\\
1/K\_a & -nd\_{33}\\\\
-nd\_{33} & C
\end{bmatrix}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
\end{equation}
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org4316115).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@ -812,11 +797,11 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation}
<a id="org4316115"></a>
<a id="figure--fig:piezo-discrete"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}}
By integrating equation \eqref{eq:piezo_work} and using the constitutive equations \eqref{eq:piezo_stack_eq_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
By integrating equation <eq:piezo_work> and using the constitutive equations <eq:piezo_stack_eq_inv>, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
\begin{equation}
W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@ -830,7 +815,7 @@ The constitutive equations can be recovered by differentiate the stored energy:
\\[ f = \frac{\partial W\_e}{\partial \Delta}, \quad V = \frac{\partial W\_e}{\partial Q} \\]
#### Interpretation of \\(k^2\\) {#interpretation-of--k-2}
#### Interpretation of \\(k^2\\) {#interpretation-of-k-2}
Consider a piezoelectric transducer subjected to the following mechanical cycle: first, it is loaded with a force \\(F\\) with short-circuited electrodes; the resulting extension is \\(\Delta\_1 = F/K\_a\\) where \\(K\_a = A/(s^El)\\) is the stiffness with short-circuited electrodes.
The energy stored in the system is:
@ -846,12 +831,12 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#orgcdbb831), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="orgcdbb831"></a>
<a id="figure--fig:piezo-stack-admittance"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="<span class=\"figure-number\">Figure 19: </span>Elementary dynamical model of the piezoelectric transducer" >}}
From the constitutive equations, one finds
@ -868,11 +853,11 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2
\end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org15dd7b6)).
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
<a id="org15dd7b6"></a>
<a id="figure--fig:piezo-admittance-curve"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="<span class=\"figure-number\">Figure 20: </span>Typical admittance FRF of the transducer" >}}
## Piezoelectric Beam, Plate and Truss {#piezoelectric-beam-plate-and-truss}
@ -1004,13 +989,12 @@ Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the el
#### Equivalent Damping Ratio {#equivalent-damping-ratio}
## Collocated Versus Non-collocated Control {#collocated-versus-non-collocated-control}
## BKMK Collocated Versus Non-collocated Control {#bkmk-collocated-versus-non-collocated-control}
### Pole-Zero Flipping {#pole-zero-flipping}
<div class="cbox">
<div></div>
The Root Locus shows, in a graphical form, the evolution of the poles of the closed-loop system as a function of the scalar gain \\(g\\) applied to the compensator.
The Root Locus is the locus of the solution \\(s\\) of the closed loop characteristic equation \\(1 + gG(s)H(s) = 0\\) when \\(g\\) goes from zero to infinity.
@ -1380,7 +1364,7 @@ Weakness of LQG:
- use frequency independant cost function
- use noise statistics with uniform distribution
To overcome the weakness => frequency shaping either by:
To overcome the weakness =&gt; frequency shaping either by:
- considering a frequency dependant cost function
- using colored noise statistics
@ -1568,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org0c9fed0).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
@ -1576,9 +1560,9 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org0c9fed0"></a>
<a id="figure--fig:hac-lac-control"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="<span class=\"figure-number\">Figure 21: </span>Principle of the dual-loop HAC/LAC control" >}}
#### Wide-Band Position Control {#wide-band-position-control}
@ -1818,7 +1802,8 @@ This approach has the following advantages:
### Problems {#problems}
## Bibliography {#bibliography}
<a id="orgf75c814"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<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>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
</div>

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@ -1,6 +1,6 @@
+++
title = "Properties of orthogonal stewart platform"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy 2003](#orgb7c0811))
: (<a href="#citeproc_bib_item_1">McInroy 2003</a>)
Author(s)
: McInroy, J. E.
@ -20,4 +20,6 @@ Year
## Bibliography {#bibliography}
<a id="orgb7c0811"></a>McInroy, John E. 2003. “Properties of Orthogonal Stewart Platform.” In _Smart Structures and Materials 2003: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.483460>.
<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, John E. 2003. “Properties of Orthogonal Stewart Platform.” In <i>Smart Structures and Materials 2003: Smart Structures and Integrated Systems</i>, nil. doi:<a href="https://doi.org/10.1117/12.483460">10.1117/12.483460</a>.</div>
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@ -1,16 +1,16 @@
+++
title = "Active isolation and damping of vibrations via stewart platform"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
ref_author = "Hanieh, A. A."
ref_year = 2003
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Active Damping]({{<relref "active_damping.md#" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Active Damping]({{< relref "active_damping.md" >}})
Reference
: ([Hanieh 2003](#orgf310fe8))
: (<a href="#citeproc_bib_item_1">Hanieh 2003</a>)
Author(s)
: Hanieh, A. A.
@ -19,7 +19,8 @@ Year
: 2003
## Bibliography {#bibliography}
<a id="orgf310fe8"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.
<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>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.</div>
</div>

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@ -1,16 +1,16 @@
+++
title = "Mechatronic design of a magnetically suspended rotating platform"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
ref_author = "Jabben, L."
ref_year = 2007
+++
Tags
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
Reference
: ([Jabben 2007](#org6250919))
: (<a href="#citeproc_bib_item_1">Jabben 2007</a>)
Author
: Jabben, L.
@ -49,10 +49,9 @@ This approach allows frequency dependent error budgeting, which is why it is ref
This noise can be modeled as a voltage source in series with the system impedance.
The noise source has a PSD given by:
\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
with \\(k = 1.38 \cdot 10^{-23} \,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
with \\(k = 1.38 \cdot 10^{-23} \\,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
<div class="exampl">
<div></div>
A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \mu V\\) from zero up to one kHz.
@ -62,12 +61,11 @@ A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \m
Seen with junctions in a transistor.
It has a white spectral density:
\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
<div class="exampl">
<div></div>
An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\,[A]\\) from zero up to one kHz.
An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
</div>
@ -100,24 +98,23 @@ The corresponding PSD is white up to the Nyquist frequency:
with \\(f\_N\\) the Nyquist frequency [Hz].
<div class="exampl">
<div></div>
Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
Let's suppose the ADC is used to measure a position over a range of 1 mm.
- ADC quantization noise: it has 16 bots over the 1 mm range.
The standard diviation from the quantization is:
\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^16}{\sqrt{12}} = 4.4\,[nm] \\]
- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\,[mm]\\).
- ADC quantization noise: it has 16 bits over the 1 mm range.
The standard deviation from the quantization is:
\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^{16}}{\sqrt{12}} = 4.4\\,[nm] \\]
- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\\,[mm]\\).
With a SNR of 80dB, the electronic noise from the ADC becomes:
\\[ \sigma\_{ADn} = 35\,[nm] \\]
\\[ \sigma\_{ADn} = 35\\,[nm] \\]
Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\,[nm]\\).
Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\\,[nm]\\).
The PSD of this digitalized sensor noise is:
\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\,[nm]\\]
\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\\,[nm]\\]
from which the PSD of the total sensor noise \\(S\_s\\) is calculated:
\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\,[nm^2/Hz] \\]
\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\\,[nm^2/Hz] \\]
with \\(f\_N\\) is the Nyquist frequency of 1kHz.
</div>
@ -132,9 +129,8 @@ To have a pressure difference, the body must have a certain minimum dimension, d
For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
<div class="exampl">
<div></div>
Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\,[N^2/Hz]\\)
Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\\,[N^2/Hz]\\)
</div>
@ -163,21 +159,21 @@ Three factors influence the performance:
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
In Figure [1](#orgcc56194), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
<a id="orgcc56194"></a>
<a id="figure--fig:jabben07-general-plant"></a>
{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#orgcc56194) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
This is illustrated in Figure [2](#org772dfb7) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
This is illustrated in Figure [2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@ -186,9 +182,9 @@ This is useful for three reasons:
- some performance channels may be of more importance than others
- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
<a id="org772dfb7"></a>
<a id="figure--fig:jabben07-weighting-functions"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="<span class=\"figure-number\">Figure 2: </span>Control system with the generalized plant \\(G\\) and weighting functions" >}}
The weighting filters should be stable transfer functions.
@ -209,13 +205,13 @@ By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of
IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#orgeab38dd) is obtained.
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
<a id="orgeab38dd"></a>
<a id="figure--fig:jabben07-pareto-curve-H2"></a>
{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
## Conclusion {#conclusion}
@ -237,8 +233,3 @@ By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha
> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
> To derive such models, spectral factorization is used.
> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
## Bibliography {#bibliography}
<a id="org6250919"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.

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@ -1,16 +1,16 @@
+++
title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
ref_author = "Li, X."
ref_year = 2001
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Cubic Architecture]({{<relref "cubic_architecture.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}}), [Multivariable Control]({{<relref "multivariable_control.md#" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Multivariable Control]({{< relref "multivariable_control.md" >}})
Reference
: ([Li 2001](#org8036ec7))
: (<a href="#citeproc_bib_item_1">Li 2001</a>)
Author(s)
: Li, X.
@ -24,17 +24,17 @@ Year
### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
A general flexible jointed hexapod is shown in Figure [1](#orgd9d105c).
A general flexible jointed hexapod is shown in Figure [1](#figure--fig:li01-flexure-hexapod-model).
<a id="orgd9d105c"></a>
<a id="figure--fig:li01-flexure-hexapod-model"></a>
{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="Figure 1: A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orgaa02e76).
Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#figure--fig:li01-quet-dirty-box).
<a id="orgaa02e76"></a>
<a id="figure--fig:li01-quet-dirty-box"></a>
{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="Figure 2: (left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="<span class=\"figure-number\">Figure 2: </span>(left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
Since only small movements are considered in flexure jointed hexapod, the Jacobian matrix, which relates changes in the Cartesian pose to changes in the strut lengths, can be considered constant.
Thus a static kinematic decoupling algorithm can be implemented for both vibration isolation and pointed controls on flexible jointed hexapods.
@ -43,14 +43,14 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
Moreover, base and/or payload vibrations become significant contributors to the motion.
The University of Wyoming hexapods (example in Figure [3](#orgf80b696)) are:
The University of Wyoming hexapods (example in Figure [3](#figure--fig:li01-stewart-platform)) are:
- Cubic (mutually orthogonal)
- Flexure Jointed
<a id="orgf80b696"></a>
<a id="figure--fig:li01-stewart-platform"></a>
{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 3: Flexure jointed Stewart platform used for analysis and control" >}}
{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="<span class=\"figure-number\">Figure 3: </span>Flexure jointed Stewart platform used for analysis and control" >}}
The objectives of the hexapods are:
@ -81,13 +81,13 @@ p\_x & p\_y & p\_z & \theta\_x & \theta\_y & \theta\_z
\begin{equation}
J = \begin{bmatrix}
{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\\\
\vdots & \vdots \\\\\\
{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\
\vdots & \vdots \\\\
{}^B\hat{u}\_6^T & [({}^B\_PR^P p\_6) \times {}^B\hat{u}\_6]^T
\end{bmatrix}
\end{equation}
where (see Figure [1](#orgd9d105c)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
where (see Figure [1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
@ -98,7 +98,7 @@ The dynamics of a flexure jointed hexapod can be written in joint space:
\begin{equation} \label{eq:hexapod\_eq\_motion}
\begin{split}
& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\\\
& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\
&\quad f\_m - \left( M\_s + J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + J^{-T} \cdot {}^U\_BR^T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C})
\end{split}
\end{equation}
@ -131,20 +131,20 @@ Define a new input and a new output:
u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
\end{equation}
Equation \eqref{eq:hexapod_eq_motion} can be rewritten as:
Equation <eq:hexapod_eq_motion> can be rewritten as:
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
\begin{split}
& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\\\
& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\
&\quad u\_1 - \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
\end{split}
\end{equation}
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#org493f606)).
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#figure--fig:li01-decoupling-conf)).
<a id="org493f606"></a>
<a id="figure--fig:li01-decoupling-conf"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="<span class=\"figure-number\">Figure 4: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
Alternatively, a new set of inputs and outputs can be defined:
@ -152,21 +152,20 @@ Alternatively, a new set of inputs and outputs can be defined:
u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
\end{equation}
And another decoupled plant is found (Figure [5](#orgbeff72d)):
And another decoupled plant is found (Figure [5](#figure--fig:li01-decoupling-conf-bis)):
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
\begin{split}
& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\\\
& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\
&\quad u\_2 - J^{-1} \cdot J^{-T} \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
\end{split}
\end{equation}
<a id="orgbeff72d"></a>
<a id="figure--fig:li01-decoupling-conf-bis"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="<span class=\"figure-number\">Figure 5: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
<div class="important">
<div></div>
These decoupling algorithms have two constraints:
@ -201,17 +200,17 @@ The control bandwidth is divided as follows:
### Vibration Isolation {#vibration-isolation}
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#orgd7c310d).
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#figure--fig:li01-vibration-isolation-control).
<a id="orgd7c310d"></a>
<a id="figure--fig:li01-vibration-isolation-control"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Vibration isolation control strategy" >}}
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="<span class=\"figure-number\">Figure 6: </span>Vibration isolation control strategy" >}}
One of the subsystem plant transfer function is shown in Figure [6](#orgd7c310d)
One of the subsystem plant transfer function is shown in Figure [6](#figure--fig:li01-vibration-isolation-control)
<a id="org1d9e762"></a>
<a id="figure--fig:li01-vibration-isolation-control"></a>
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 7: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="<span class=\"figure-number\">Figure 7: </span>Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
Each compensator is designed using simple loop-shaping techniques.
A typical compensator consists of the following elements:
@ -225,7 +224,6 @@ A typical compensator consists of the following elements:
The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**, so the vibration isolation loop works as a band-pass filter.
<div class="important">
<div></div>
Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
@ -233,7 +231,6 @@ Only about 20 dB separation is achieve between the diagonal and off-diagonal res
</div>
<div class="note">
<div></div>
Severe phase delay exists in the actual transfer function.
This is due to the limited sample frequency and sensor bandwidth limitation.
@ -246,20 +243,20 @@ The reason is not explained.
### Pointing Control Techniques {#pointing-control-techniques}
A block diagram of the pointing control system is shown in Figure [8](#orge6a2624).
A block diagram of the pointing control system is shown in Figure [8](#figure--fig:li01-pointing-control).
<a id="orge6a2624"></a>
<a id="figure--fig:li01-pointing-control"></a>
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 8: Figure caption" >}}
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="<span class=\"figure-number\">Figure 8: </span>Figure caption" >}}
The plant is decoupled into two independent SISO subsystems.
The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
Figure [9](#org54b4cd4) shows the measured transfer function of the \\(\theta\_x\\) axis.
Figure [9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
<a id="org54b4cd4"></a>
<a id="figure--fig:li01-transfer-function-angle"></a>
{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="Figure 9: Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="<span class=\"figure-number\">Figure 9: </span>Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
A typical compensator consists of the following elements:
@ -271,13 +268,13 @@ A typical compensator consists of the following elements:
The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
A feedforward control is added as shown in Figure [10](#orga527171).
A feedforward control is added as shown in Figure [10](#figure--fig:li01-feedforward-control).
\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
Ideally, the feedforward compensator is an invert of the plant dynamics.
<a id="orga527171"></a>
<a id="figure--fig:li01-feedforward-control"></a>
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 10: Feedforward control" >}}
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="<span class=\"figure-number\">Figure 10: </span>Feedforward control" >}}
### Simultaneous Control {#simultaneous-control}
@ -287,14 +284,13 @@ The simultaneous vibration isolation and pointing control is approached in two w
1. **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
2. **Closing the pointing loop first**: Reverse order.
Figure [11](#orge85d506) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
Figure [11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="orge85d506"></a>
<a id="figure--fig:li01-parallel-control"></a>
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 11: A parallel scheme" >}}
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="<span class=\"figure-number\">Figure 11: </span>A parallel scheme" >}}
<div class="important">
<div></div>
The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled.
The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
@ -306,24 +302,23 @@ However, the interaction between loops may affect the transfer functions of the
The dynamic interaction effect:
- Only happens in the unity bandwidth of the loop transmission of the first closed loop.
- Affect the closed loop transmission of the loop first closed (see Figures [12](#org1065b18) and [13](#orgba389c3))
- Affect the closed loop transmission of the loop first closed (see Figures [12](#figure--fig:li01-closed-loop-pointing) and [13](#figure--fig:li01-closed-loop-vibration))
As shown in Figure [12](#org1065b18), the peak resonance of the pointing loop increase after the isolation loop is closed.
As shown in Figure [12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
<a id="org1065b18"></a>
<a id="figure--fig:li01-closed-loop-pointing"></a>
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 12: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="<span class=\"figure-number\">Figure 12: </span>Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#orgba389c3)).
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#figure--fig:li01-closed-loop-vibration)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="orgba389c3"></a>
<a id="figure--fig:li01-closed-loop-vibration"></a>
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 13: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="<span class=\"figure-number\">Figure 13: </span>Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
<div class="important">
<div></div>
From the analysis above, it is hard to say which loop has more significant affect on the other loop, but the isolation loop adds a second resonance peak at its high frequency crossover in the pointing closed loop transfer function, which may cause instability.
Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
@ -333,38 +328,37 @@ Thus, it is recommended to design and implement the isolation control system fir
### Experimental results {#experimental-results}
Two hexapods are stacked (Figure [14](#orgc3b1ba9)):
Two hexapods are stacked (Figure [14](#figure--fig:li01-test-bench)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
<a id="orgc3b1ba9"></a>
<a id="figure--fig:li01-test-bench"></a>
{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 14: Stacked Hexapods" >}}
{{< figure src="/ox-hugo/li01_test_bench.png" caption="<span class=\"figure-number\">Figure 14: </span>Stacked Hexapods" >}}
First, the vibration isolation and pointing controls were implemented separately.
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org933bc12).
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#figure--fig:li01-vibration-isolation-control-results).
<a id="org933bc12"></a>
<a id="figure--fig:li01-vibration-isolation-control-results"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 15: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="<span class=\"figure-number\">Figure 15: </span>Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
The simultaneous control is of dual use:
- it provide simultaneous pointing and isolation control
- it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
The results of simultaneous control is shown in Figure [16](#org3618406) where the bandwidth of the isolation control is expanded to very low frequency.
The results of simultaneous control is shown in Figure [16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="org3618406"></a>
<a id="figure--fig:li01-simultaneous-control-results"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 16: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="<span class=\"figure-number\">Figure 16: </span>Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
### Summary and Conclusion {#summary-and-conclusion}
<div class="sum">
<div></div>
A parallel control scheme is proposed in this chapters.
This scheme is suitable for simultaneous vibration isolation and pointing control.
@ -380,7 +374,6 @@ Experiments show that this scheme takes advantage of the bandwidths of both poin
## Future research areas {#future-research-areas}
<div class="sum">
<div></div>
Proposed future research areas include:
@ -406,7 +399,8 @@ Proposed future research areas include:
</div>
## Bibliography {#bibliography}
<a id="org8036ec7"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
<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. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.</div>
</div>

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@ -7,10 +7,10 @@ ref_year = 2004
+++
Tags
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
Reference
: <monkhorst04_dynam_error_budget>
: (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>)
Author(s)
: Monkhorst, W.
@ -106,11 +106,11 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#orgfce1d5b)).
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#figure--fig:monkhorst04-weighting-filter)).
<a id="orgfce1d5b"></a>
<a id="figure--fig:monkhorst04-weighting-filter"></a>
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>The use of a weighting filter \\(V\_w(f)\\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
The white noise input \\(w(t)\\) is dimensionless, and when the weighting filter has units [SI], the resulting weighted signal \\(\bar{w}(t)\\) has units [SI].
The PSD \\(S\_w(f)\\) of the weighted signal is:
@ -119,25 +119,25 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
However, this can be avoided if the modeling of the disturbances is directly done in terms of weighting filters.
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgd937879)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#figure--fig:monkhorst04-general-weighted-plant)).
<a id="orgd937879"></a>
<a id="figure--fig:monkhorst04-general-weighted-plant"></a>
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#orgf4dc585)) are:
In this research, the outputs of the closed loop system (Figure [3](#figure--fig:monkhorst04-closed-loop-H2)) are:
- the performance (error) signal \\(e\\)
- the controller output \\(u\\)
In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
<a id="orgf4dc585"></a>
<a id="figure--fig:monkhorst04-closed-loop-H2"></a>
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
The resulting problem is a multi-objective control problem: while constraining the variance of the controller output \\(u\\), the variance of the performance channel should be minimized.
This problem can be solved by scaling the controller output \\(u\\) with a factor \\(\alpha\\) during the \\(\mathcal{H}\_2\\) synthesis.
@ -157,3 +157,10 @@ To achieve the highest degree of prediction accuracy, it is recommended to use t
When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, it can give the control designer useful hints about how to control the system best for optimal performance.
Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
## Bibliography {#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>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
</div>

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@ -1,16 +1,16 @@
+++
title = "An exploration of active hard mount vibration isolation for precision equipment"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
ref_author = "van der Poel, G. W."
ref_year = 2010
+++
Tags
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Poel 2010](#org4dd001c))
: (<a href="#citeproc_bib_item_1">Van der Poel 2010</a>)
Author(s)
: van der Poel, G. W.
@ -19,7 +19,8 @@ Year
: 2010
## Bibliography {#bibliography}
<a id="org4dd001c"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.
<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>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036530163">10.3990/1.9789036530163</a>.</div>
</div>

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@ -1,16 +1,16 @@
+++
title = "Element and system design for active and passive vibration isolation"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
ref_author = "Zuo, L."
ref_year = 2004
+++
Tags
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Zuo 2004](#org05cd1c8))
: (<a href="#citeproc_bib_item_1">Zuo 2004</a>)
Author(s)
: Zuo, L.
@ -28,24 +28,25 @@ Year
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
> Note that reaction force actuators can also work with soft mounts or hard mounts.
<a id="orgdaec88b"></a>
<a id="figure--fig:zuo04-piezo-spring-series"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 1: </span>PZT actuator and spring in series" >}}
<a id="org84417be"></a>
<a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 2: </span>Voice coil actuator and spring in parallel" >}}
<a id="orge3c9205"></a>
<a id="figure--fig:zuo04-piezo-plant"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 3: </span>Transmission from PZT voltage to geophone output" >}}
<a id="orge26e6a6"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
<a id="figure--fig:zuo04-voice-coil-plant"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Transmission from voice coil voltage to geophone output" >}}
## Bibliography {#bibliography}
<a id="org05cd1c8"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.</div>
</div>

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@ -1,6 +1,6 @@
+++
title = "First Blog Post"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
date = 2021-04-23T00:00:00+02:00
tags = ["hugo", "org"]
categories = ["emacs", "test"]
@ -87,17 +87,17 @@ Here is some inline mathematics: \\(z = 2\\).
Unumbered equation:
\\[ F(x) = \int\_0^x f(t) dt \\]
Using the `equation` environment in Eq. \eqref{eq:numbered}.
Using the `equation` environment in Eq. <eq:numbered>.
\begin{equation}
F(s) = \int\_0^\infty f(t) e^{-st} dt \label{eq:numbered}
\end{equation}
Using the `align` environment Equations \eqref{eq:align_1} and \eqref{eq:align_2}.
Using the `align` environment Equations <eq:align_1> and <eq:align_2>.
\begin{align}
\mathcal{F}(a) &= \frac{1}{2\pi i}\oint\_\gamma \frac{f(z)}{z - a}\,dz \label{eq:align\_1} \\\\\\
\int\_D (\nabla\cdot \mathcal{F})\,dV &=\int\_{\partial D}\mathcal{F}\cdot n\, dS \label{eq:align\_2}
\mathcal{F}(a) &= \frac{1}{2\pi i}\oint\_\gamma \frac{f(z)}{z - a}\\,dz \label{eq:align\_1} \\\\
\int\_D (\nabla\cdot \mathcal{F})\\,dV &=\int\_{\partial D}\mathcal{F}\cdot n\\, dS \label{eq:align\_2}
\end{align}
@ -106,11 +106,13 @@ Using the `align` environment Equations \eqref{eq:align_1} and \eqref{eq:align_2
Below is a verse.
<p class="verse">
Great clouds overhead<br />
Tiny black birds rise and fall<br />
Snow covers Emacs<br />
<br />
&nbsp;&nbsp;&nbsp;---AlexSchroeder<br />
</p>
Below is a quote.
@ -127,7 +129,6 @@ Below is a quote.
An aside block can be used as shown below.
<aside>
<aside></aside>
This is a note about the text using the `aside` environment.
This can be as long as wanted

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@ -1,8 +1,8 @@
+++
title = "Second Blog Post"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
date = 2021-05-01T00:00:00+02:00
lastmod = 2021-05-02T22:09:31+02:00
lastmod = 2022-03-15T16:31:43+01:00
tags = ["hugo", "org"]
categories = ["emacs"]
draft = false
@ -11,7 +11,6 @@ draft = false
## Heading {#heading}
<div class="important">
<div></div>
this is an important block

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+++
title = "Control Bootcamp"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -122,3 +122,9 @@ Tags
## Control systems with non-minimum phase dynamics {#control-systems-with-non-minimum-phase-dynamics}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Data-Driven Dynamical Systems with Machine Learning"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -112,7 +112,7 @@ Use powerful optimization techniques from machine learning to learn what are goo
### Observer Kalman Filter Identification {#observer-kalman-filter-identification}
### ERA\_OKID Example in Matlab {#era-okid-example-in-matlab}
### ERA_OKID Example in Matlab {#era-okid-example-in-matlab}
## System Identification {#system-identification}
@ -179,3 +179,9 @@ Use powerful optimization techniques from machine learning to learn what are goo
### Applications {#applications}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Active Damping"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -9,9 +9,12 @@ Tags
There are two main control architecture to actively damp structures:
- [Integral Force Feedback]({{< relref "integral_force_feedback" >}})
- [Direct Velocity Feedback]({{< relref "direct_velocity_feedback" >}})
- [Integral Force Feedback]({{< relref "integral_force_feedback.md" >}})
- [Direct Velocity Feedback]({{< relref "direct_velocity_feedback.md" >}})
The idea is to apply a force proportional to the velocity (either relative or inertial) of the structure.
These are usually applied in a collocated way, meaning that the actuator and sensors are collocated (fixed to the same DoF), in order to have guaranteed stability.
## Bibliography {#bibliography}

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@ -1,12 +1,12 @@
+++
title = "Active Isolation Platforms"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
## Manufacturers {#manufacturers}
@ -15,14 +15,14 @@ Tags
|-----------------------------------------------------------------------------------------------|-------------|
| [TMC](https://www.techmfg.com/) | USA |
| [Newport](https://www.newport.com/c/optical-tables-%26-isolation-systems) | USA |
| [Thorlabs](https://www.thorlabs.com/navigation.cfm?guide%5FID=42) | USA |
| [IDE](https://www.ideworld.com/en/active%5Fvibration%5Fisolation.html) | Germany |
| [Thorlabs](https://www.thorlabs.com/navigation.cfm?guide_ID=42) | USA |
| [IDE](https://www.ideworld.com/en/active_vibration_isolation.html) | Germany |
| [Harvard Apparatus](https://www.warneronline.com/labmate-vibraplane-workstations-9100-series) | USA |
| [Herzan](https://www.herzan.com/products/active-vibration-control/avi-series.html) | USA |
| [Standa](http://www.standa.lt/products/catalog/optical%5Ftables?item=335) | Lithuania |
| [Standa](http://www.standa.lt/products/catalog/optical_tables?item=335) | Lithuania |
| [Table Stable](http://www.tablestable.com/en/products/list/2/) | Switzerland |
| [Accurion](https://www.halcyonics.com/active-vibration-isolation-products) | Germany |
| [Vibiso](https://vibiso.com/?page%5Fid=3433) | USA |
| [Vibiso](https://vibiso.com/?page_id=3433) | USA |
## Vibration Isolating Pads {#vibration-isolating-pads}
@ -30,3 +30,9 @@ Tags
| Manufacturer | links | Country |
|--------------|----------------------------------|---------|
| ACE | [link](https://www.ace-ace.com/) | Germany |
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Actuator Fusion"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Complementary Filters]({{< relref "complementary_filters" >}})
: [Complementary Filters]({{< relref "complementary_filters.md" >}})
([Beijen et al. 2019](#orgc6f7554))
([Beijen 2018](#org8e8fef4)) (section 6.3.1)
(<a href="#citeproc_bib_item_2">Beijen et al. 2019</a>)
(<a href="#citeproc_bib_item_1">Beijen 2018</a>) (section 6.3.1)
## Bibliography {#bibliography}
<a id="org8e8fef4"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.
<a id="orgc6f7554"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” _Mechatronics_ 61:106–16. <https://doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” <i>Mechatronics</i> 61: 106–16. doi:<a href="https://doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005">https://doi.org/10.1016/j.mechatronics.2019.06.005</a>.</div>
</div>

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@ -9,22 +9,30 @@ Tags
Links to specific actuators:
- [Voice Coil Actuators]({{<relref "voice_coil_actuators.md#" >}})
- [Piezoelectric Actuators]({{<relref "piezoelectric_actuators.md#" >}})
- [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application}
For vibration isolation:
- In <ito16_compar_class_high_precis_actuat>, the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{<relref "ito16_compar_class_high_precis_actuat.md#" >}}))
- In (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}}))
## Brush-less DC Motor {#brush-less-dc-motor}
- <yedamale03_brush_dc_bldc_motor_fundam>
- (<a href="#citeproc_bib_item_2">Yedamale 2003</a>)
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
## [Stepper Motor]({{<relref "stepper_motor.md#" >}}) {#stepper-motor--stepper-motor-dot-md}
## [Stepper Motor]({{< relref "stepper_motor.md" >}}) {#stepper-motor--stepper-motor-dot-md}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>Ieee/Asme Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (Bldc) Motor Fundamentals.” <i>Microchip Technology Inc</i> 20: 3–15.</div>
</div>

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@ -14,7 +14,7 @@ Tags
<https://dewesoft.com/daq/types-of-adc-converters>
- Delta Sigma <baker11_how_delta_sigma_adcs_work_part>
- Delta Sigma (<a href="#citeproc_bib_item_1">Baker 2011</a>)
- Successive Approximation
@ -86,3 +86,10 @@ The quantization is:
## Oversampling {#oversampling}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” <i>Analog Applications</i> 7.</div>
</div>

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@ -1,12 +1,18 @@
+++
title = "Bipolar Transistor"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
<a id="org67aca6e"></a>
<a id="figure--fig:bipolar-transistor-basic-circuits"></a>
{{< figure src="/ox-hugo/bipolar_transistor_basic_circuits.svg" caption="Figure 1: 5 basic circuits using the bipolar transistor" >}}
{{< figure src="/ox-hugo/bipolar_transistor_basic_circuits.svg" caption="<span class=\"figure-number\">Figure 1: </span>5 basic circuits using the bipolar transistor" >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Cables"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Connectors]({{<relref "connectors.md#" >}})
: [Connectors]({{< relref "connectors.md" >}})
## Typical Cables {#typical-cables}
@ -30,3 +30,9 @@ Tags
## Software {#software}
- [WireViz](https://github.com/formatc1702/WireViz) is a nice software to easily document cables and wiring harnesses
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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@ -1,12 +1,12 @@
+++
title = "Capacitive Sensors"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Position Sensors]({{<relref "position_sensors.md#" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
## Description of Capacitive Sensors {#description-of-capacitive-sensors}
@ -29,3 +29,9 @@ Tags
| [Capacitec](https://www.capacitec.com/Displacement-Sensing-Systems) | USA |
| [MTIinstruments](https://www.mtiinstruments.com/products/non-contact-measurement/capacitance-sensors/) | USA |
| [Althen](https://www.althensensors.com/sensors/linear-position-sensors/capacitive-position-sensors/) | Netherlands |
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Charge Amplifiers"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Electronics]({{<relref "electronics.md#" >}})
: [Electronics]({{< relref "electronics.md" >}})
## Description {#description}
@ -18,19 +18,19 @@ This can be typically used to interface with piezoelectric sensors.
## Basic Circuit {#basic-circuit}
Two basic circuits of charge amplifiers are shown in Figure [1](#org0d411fa) (taken from ([Fleming 2010](#org7834496))) and Figure [2](#org1c3e25d) (taken from ([Schmidt, Schitter, and Rankers 2014](#orgd26dd11)))
Two basic circuits of charge amplifiers are shown in Figure [1](#figure--fig:charge-amplifier-circuit) (taken from (<a href="#citeproc_bib_item_1">Fleming 2010</a>)) and Figure [2](#figure--fig:charge-amplifier-circuit-bis) (taken from (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2014</a>))
<a id="org0d411fa"></a>
<a id="figure--fig:charge-amplifier-circuit"></a>
{{< figure src="/ox-hugo/charge_amplifier_circuit.png" caption="Figure 1: Electrical model of a piezoelectric force sensor is shown in gray. The op-amp charge amplifier is shown on the right. The output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
{{< figure src="/ox-hugo/charge_amplifier_circuit.png" caption="<span class=\"figure-number\">Figure 1: </span>Electrical model of a piezoelectric force sensor is shown in gray. The op-amp charge amplifier is shown on the right. The output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
<a id="org1c3e25d"></a>
<a id="figure--fig:charge-amplifier-circuit-bis"></a>
{{< figure src="/ox-hugo/charge_amplifier_circuit_bis.png" caption="Figure 2: A piezoelectric accelerometer with a charge amplifier as signal conditioning element" >}}
{{< figure src="/ox-hugo/charge_amplifier_circuit_bis.png" caption="<span class=\"figure-number\">Figure 2: </span>A piezoelectric accelerometer with a charge amplifier as signal conditioning element" >}}
The input impedance of the charge amplifier is very small (unlike when using a voltage amplifier).
The gain of the charge amplified (Figure [1](#org0d411fa)) is equal to:
The gain of the charge amplified (Figure [1](#figure--fig:charge-amplifier-circuit)) is equal to:
\\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\]
@ -41,16 +41,16 @@ The gain of the charge amplified (Figure [1](#org0d411fa)) is equal to:
| [PCB](https://www.pcb.com/sensors-for-test-measurement/electronics/line-powered-multi-channel-signal-conditioners) | USA |
| [HBM](https://www.hbm.com/en/2660/paceline-cma-charge-amplifier-analogamplifier/) | Germany |
| [Kistler](https://www.kistler.com/fr/produits/composants/conditionnement-de-signal/) | Swiss |
| [MMF](https://www.mmf.de/signal%5Fconditioners.htm) | Germany |
| [MMF](https://www.mmf.de/signal_conditioners.htm) | Germany |
| [DJB](https://www.djbinstruments.com/products/instrumentation/view/9-Channel-Charge-Voltage-Amplifier-IEPE-Signal-Conditioning-Rack-Mounted) | UK |
| [MTI Instruments](https://www.mtiinstruments.com/products/turbine-balancing-vibration-analysis/charge-amplifiers/ca1800/) | USA |
| [Sinocera](http://www.china-yec.net/instruments/signal-conditioner/multi-channels-charge-amplifier.html) | China |
| [L-Card](https://en.lcard.ru/products/accesories/le-41) | Rusia |
## Bibliography {#bibliography}
<a id="org7834496"></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>.
<a id="orgd26dd11"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. <i>The Design of High Performance Mechatronics - 2nd Revised Edition</i>. Ios Press.</div>
</div>

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+++
title = "Collocated Control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Actuators]({{< relref "actuators" >}}), [Force Sensors]({{< relref "force_sensors" >}}), [Position Sensors]({{< relref "position_sensors" >}}), [Inertial Sensors]({{< relref "inertial_sensors" >}})
: [Actuators]({{< relref "actuators.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}}), [Position Sensors]({{< relref "position_sensors.md" >}}), [Inertial Sensors]({{< relref "inertial_sensors.md" >}})
## Collocated/Dual actuator and sensor {#collocated-dual-actuator-and-sensor}
According to ([Preumont 2018](#orgbf8f4c5)):
According to (<a href="#citeproc_bib_item_1">Preumont 2018</a>):
> A **collocated** control system is a control system where the actuator and the sensor are attached to the same degree of freedom.
>
@ -19,26 +19,25 @@ According to ([Preumont 2018](#orgbf8f4c5)):
## Nearly Collocated Actuator Sensor Pair {#nearly-collocated-actuator-sensor-pair}
From Figure [1](#org5d460f9), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
From Figure [1](#figure--fig:preumont18-nearly-collocated-schematic), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
<a id="org5d460f9"></a>
<a id="figure--fig:preumont18-nearly-collocated-schematic"></a>
{{< figure src="/ox-hugo/preumont18_nearly_collocated_schematic.png" caption="Figure 1: Mode shapes for a uniform beam. \\(u\\) and \\(y\\) are not collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_nearly_collocated_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Mode shapes for a uniform beam. \\(u\\) and \\(y\\) are not collocated actuator and sensor" >}}
## Piezoelectric Stack as a sensor/actuator pair {#piezoelectric-stack-as-a-sensor-actuator-pair}
One can use on part of a piezoelectric stack as an actuator and the other part as a sensor.
One can use on part of a [Piezoelectric Stack]({{< relref "piezoelectric_actuators.md" >}}) as an actuator and the other part as a sensor.
At some frequency, the sensor/actuator pair will not be collocated anymore.
If we want to be collocated up to the highest possible frequency, the sensor part should be made small.
Of course, this will reduce the sensibility.
- [ ] What happens is small pieces of actuators are mixed with small pieces of sensors?
## Bibliography {#bibliography}
<a id="orgbf8f4c5"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<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>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
</div>

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title = "Complementary Filters"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -10,10 +10,11 @@ Tags
## Complementary Filters Synthesis {#complementary-filters-synthesis}
The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#org066e272)).
The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis (<a href="#citeproc_bib_item_1">Dehaeze, Vermat, and Collette 2019</a>).
## Bibliography {#bibliography}
<a id="org066e272"></a>Dehaeze, Thomas, Mohit Vermat, and Collette Christophe. 2019. “Complementary Filters Shaping Using \\(mathcalH\_Infty\\) Synthesis.” In _7th International Conference on Control, Mechatronics and Automation (ICCMA)_, 459–64. <https://doi.org/10.1109/ICCMA46720.2019.8988642>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dehaeze, Thomas, Mohit Vermat, and Christophe Collette. 2019. “Complementary Filters Shaping Using $h_\Infty$ Synthesis.” In <i>7th International Conference on Control, Mechatronics and Automation (Iccma)</i>, 459–64. doi:<a href="https://doi.org/10.1109/ICCMA46720.2019.8988642">10.1109/ICCMA46720.2019.8988642</a>.</div>
</div>

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content/zettels/connectors.md
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+++
title = "Connectors"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Cables]({{<relref "cables.md#" >}})
: [Cables]({{< relref "cables.md" >}})
## Manufacturers {#manufacturers}
@ -20,8 +20,14 @@ Tags
## BNC {#bnc}
BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in Figure [1](#orgf757f74).
BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in Figure [1](#figure--fig:bnc-50-75-ohms).
<a id="orgf757f74"></a>
<a id="figure--fig:bnc-50-75-ohms"></a>
{{< figure src="/ox-hugo/bnc_50_75_ohms.jpg" caption="Figure 1: 75Ohms and 50Ohms BNC connectors" >}}
{{< figure src="/ox-hugo/bnc_50_75_ohms.jpg" caption="<span class=\"figure-number\">Figure 1: </span>75Ohms and 50Ohms BNC connectors" >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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title = "Cubic Architecture"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -13,13 +13,14 @@ Tags
## Special Properties {#special-properties}
Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org2ea9cff)))
Cubic Stewart Platforms can be decoupled provided that (from (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>))
> 1. The payload mass-inertia matrix is diagonal
> 2. If a mutually orthogonal geometry has been selected, the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
## Bibliography {#bibliography}
<a id="org2ea9cff"></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>.
<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>

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+++
title = "Digital to Analog Converters"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Electronics]({{<relref "electronics.md#" >}})
: [Electronics]({{< relref "electronics.md" >}})
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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+++
title = "Direct Velocity Feedback"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Active Damping]({{< relref "active_damping" >}})
: [Active Damping]({{< relref "active_damping.md" >}})
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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title = "Dynamic Error Budgeting"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
A good introduction to Dynamic Error Budgeting is given in ([Monkhorst 2004](#orgda61e4e)).
A good introduction to Dynamic Error Budgeting is given in (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>).
## Step by Step process {#step-by-step-process}
Taken from ([Monkhorst 2004](#orgda61e4e)): ([Notes]({{< relref "monkhorst04_dynam_error_budget" >}}))
Taken from (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>): ([Notes]({{< relref "monkhorst04_dynam_error_budget.md" >}}))
> Step by step, the process is as follows:
>
@ -26,7 +26,8 @@ Taken from ([Monkhorst 2004](#orgda61e4e)): ([Notes]({{< relref "monkhorst04_dyn
> Iterate until the error budget is meet.
## Bibliography {#bibliography}
<a id="orgda61e4e"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<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>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
</div>

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+++
title = "Eddy Current Sensors"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Position Sensors]({{<relref "position_sensors.md#" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
## Manufacturers {#manufacturers}
@ -17,5 +17,11 @@ Tags
| [Lion Precision](https://www.lionprecision.com/products/eddy-current-sensors) | USA |
| [Cedrat](https://www.cedrat-technologies.com/en/products/sensors/eddy-current-sensors.html) | France |
| [Kaman](https://www.kamansensors.com/product/smt-9700/) | USA |
| [Keyence](https://www.keyence.com/ss/products/measure/measurement%5Flibrary/type/inductive/) | USA |
| [Keyence](https://www.keyence.com/ss/products/measure/measurement_library/type/inductive/) | USA |
| [Althen](https://www.althensensors.com/sensors/linear-position-sensors/eddy-current-sensors/) | Netherlands |
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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+++
title = "Electronic Active Filters"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Operational Amplifiers]({{< relref "operational_amplifiers" >}})
: [Operational Amplifiers]({{< relref "operational_amplifiers.md" >}})
TODOS:
@ -29,14 +29,14 @@ With:
- \\(\omega\_0 = \frac{1}{R\sqrt{C\_1 C\_2}}\\)
- \\(\xi = \frac{C\_2}{C\_1}\\)
<a id="orgb2c3453"></a>
<a id="figure--fig:elec-active-second-order-low-pass-filter"></a>
{{< figure src="/ox-hugo/elec_active_second_order_low_pass_filter.png" caption="Figure 1: Second Order Low Pass Filter" >}}
{{< figure src="/ox-hugo/elec_active_second_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>Second Order Low Pass Filter" >}}
## High Pass Filter {#high-pass-filter}
Same as [1](#orgb2c3453) but by exchanging R1 with C1 and R2 with C2
Same as [1](#figure--fig:elec-active-second-order-low-pass-filter) but by exchanging R1 with C1 and R2 with C2
\begin{equation}
\frac{V\_o}{V\_i}(s) = \frac{R^2 C\_1 C\_2 s^2}{R^2 C\_1 C\_2 s^2 + 2 R C\_2 s + 1}
@ -46,3 +46,9 @@ With:
- \\(\omega\_0 = \frac{1}{R\sqrt{C\_1 C\_2}}\\)
- \\(\xi = \frac{C\_2}{C\_1}\\)
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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+++
title = "Electronic Passive Filters"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -16,27 +16,33 @@ TODOS:
## First Order Low Pass Filter {#first-order-low-pass-filter}
<a id="org1c6b488"></a>
<a id="figure--fig:elec-passive-first-order-low-pass-filter"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="Figure 1: First Order Low Pass Filter using an RC circuit" >}}
{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>First Order Low Pass Filter using an RC circuit" >}}
## First Order High Pass Filter {#first-order-high-pass-filter}
<a id="orgecf7617"></a>
<a id="figure--fig:elec-passive-first-order-high-pass-filter"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="Figure 2: First Order High Pass Filter using an RC circuit" >}}
{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 2: </span>First Order High Pass Filter using an RC circuit" >}}
## Second Order Low Pass Filter {#second-order-low-pass-filter}
<a id="orgcfc4c15"></a>
<a id="figure--fig:elec-passive-second-order-low-pass-filter"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="Figure 3: Second Order Low Pass Filter using an RLC circuit" >}}
{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 3: </span>Second Order Low Pass Filter using an RLC circuit" >}}
## Second Order High Pass Filter {#second-order-high-pass-filter}
<a id="org0b32ffe"></a>
<a id="figure--fig:elec-passive-second-order-high-pass-filter"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="Figure 4: Second Order High Pass Filter using an RLC circuit" >}}
{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 4: </span>Second Order High Pass Filter using an RLC circuit" >}}
## Bibliography {#bibliography}
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+++
title = "Electronics"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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+++
title = "Encoders"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Position Sensors]({{<relref "position_sensors.md#" >}})
: [Position Sensors]({{< relref "position_sensors.md" >}})
There are two main types of encoders: optical encoders, and magnetic encoders.
@ -15,7 +15,13 @@ There are two main types of encoders: optical encoders, and magnetic encoders.
| Manufacturers | Country |
|---------------------------------------------------------------------------------|---------|
| [Heidenhain](https://www.heidenhain.com/en%5FUS/products/linear-encoders/) | Germany |
| [Heidenhain](https://www.heidenhain.com/en_US/products/linear-encoders/) | Germany |
| [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA |
| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| [Celera Motion](https://www.celeramotion.com/microe/) | USA |
## Bibliography {#bibliography}
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title = "Finite Element Model"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -12,18 +12,17 @@ Tags
Some resources:
- ([Hatch 2000](#orgddee845)) ([Notes]({{< relref "hatch00_vibrat_matlab_ansys" >}}))
- ([Khot and Yelve 2011](#orgb0a5955))
- ([Kovarac et al. 2015](#org7660da4))
- (<a href="#citeproc_bib_item_1">Hatch 2000</a>) ([Notes]({{< relref "hatch00_vibrat_matlab_ansys.md" >}}))
- (<a href="#citeproc_bib_item_2">Khot and Yelve 2011</a>)
- (<a href="#citeproc_bib_item_3">Kovarac et al. 2015</a>)
The idea is to extract reduced state space model from Ansys into Matlab.
## Bibliography {#bibliography}
<a id="orgddee845"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="orgb0a5955"></a>Khot, SM, and Nitesh P Yelve. 2011. “Modeling and Response Analysis of Dynamic Systems by Using ANSYS and MATLAB.” _Journal of Vibration and Control_ 17 (6). SAGE Publications Sage UK: London, England:953–58.
<a id="org7660da4"></a>Kovarac, A, M Zeljkovic, C Mladjenovic, and A Zivkovic. 2015. “Create SISO State Space Model of Main Spindle from ANSYS Model.” In _12th International Scientific Conference, Novi Sad, Serbia_, 37–41.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Khot, SM, and Nitesh P Yelve. 2011. “Modeling and Response Analysis of Dynamic Systems by Using Ansys and Matlab.” <i>Journal of Vibration and Control</i> 17 (6). SAGE Publications Sage UK: London, England: 953–58.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Kovarac, A, M Zeljkovic, C Mladjenovic, and A Zivkovic. 2015. “Create Siso State Space Model of Main Spindle from Ansys Model.” In <i>12th International Scientific Conference, Novi Sad, Serbia</i>, 37–41.</div>
</div>

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+++
title = "Flexible Joints"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -12,16 +12,16 @@ Tags
Books:
- ([Lobontiu 2002](#orgb45af18))
- ([Henein 2003](#org8ce4916))
- ([Smith 2005](#orgccbed32))
- ([Soemers 2011](#org772b663))
- ([Cosandier 2017](#org7ebf41f))
- (<a href="#citeproc_bib_item_4">Lobontiu 2002</a>)
- (<a href="#citeproc_bib_item_3">Henein 2003</a>)
- (<a href="#citeproc_bib_item_6">Smith 2005</a>)
- (<a href="#citeproc_bib_item_7">Soemers 2011</a>)
- (<a href="#citeproc_bib_item_2">Cosandier 2017</a>)
## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms}
From ([Chen and McInroy 2000](#org64f8175)):
From (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>):
> To avoid the extremely non-linear micro-dynamics of joint friction and backlash, these hexapods employ flexure joints.
> A flexure joint bends material to achieve motion, rather than sliding of rolling across two surfaces.
@ -30,25 +30,20 @@ From ([Chen and McInroy 2000](#org64f8175)):
## Materials {#materials}
- ([Smith 2000](#org299921c))
- ([Lobontiu 2002](#orgb45af18))
- ([Henein 2003](#org8ce4916))
- ([Cosandier 2017](#org7ebf41f))
- (<a href="#citeproc_bib_item_5">Smith 2000</a>)
- (<a href="#citeproc_bib_item_4">Lobontiu 2002</a>)
- (<a href="#citeproc_bib_item_3">Henein 2003</a>)
- (<a href="#citeproc_bib_item_2">Cosandier 2017</a>)
## Bibliography {#bibliography}
<a id="org64f8175"></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="org7ebf41f"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org8ce4916"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="orgb45af18"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="org299921c"></a>Smith, Stuart T. 2000. _Flexures: Elements of Elastic Mechanisms_. Crc Press.
<a id="orgccbed32"></a>———. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press.
<a id="org772b663"></a>Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint.
<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>Cosandier, Florent. 2017. <i>Flexure Mechanism Design</i>. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Henein, Simon. 2003. <i>Conception Des Guidages Flexibles</i>. Lausanne, Suisse: Presses polytechniques et universitaires romandes.</div>
<div class="csl-entry"><a id="citeproc_bib_item_4"></a>Lobontiu, Nicolae. 2002. <i>Compliant Mechanisms: Design of Flexure Hinges</i>. CRC press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_5"></a>Smith, Stuart T. 2000. <i>Flexures: Elements of Elastic Mechanisms</i>. Crc Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_6"></a>———. 2005. <i>Foundations of Ultra-Precision Mechanism Design</i>. Vol. 2. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_7"></a>Soemers, Herman. 2011. <i>Design Principles for Precision Mechanisms</i>. T-Pointprint.</div>
</div>

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content/zettels/force_sensors.md
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@ -1,12 +1,12 @@
+++
title = "Force Sensors"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
Tags
: [Signal Conditioner]({{<relref "signal_conditioner.md#" >}}), [Modal Analysis]({{<relref "modal_analysis.md#" >}})
: [Signal Conditioner]({{< relref "signal_conditioner.md" >}}), [Modal Analysis]({{< relref "modal_analysis.md" >}})
## Technologies {#technologies}
@ -18,11 +18,11 @@ There are two main technique for force sensors:
The choice between the two is usually based on whether the measurement is static (strain gauge) or dynamics (piezoelectric).
Main differences between the two are shown in Figure [1](#orgc9e9a88).
Main differences between the two are shown in Figure [1](#figure--fig:force-sensor-piezo-vs-strain-gauge).
<a id="orgc9e9a88"></a>
<a id="figure--fig:force-sensor-piezo-vs-strain-gauge"></a>
{{< figure src="/ox-hugo/force_sensor_piezo_vs_strain_gauge.png" caption="Figure 1: Piezoelectric Force sensor VS Strain Gauge Force sensor" >}}
{{< figure src="/ox-hugo/force_sensor_piezo_vs_strain_gauge.png" caption="<span class=\"figure-number\">Figure 1: </span>Piezoelectric Force sensor VS Strain Gauge Force sensor" >}}
## Piezoelectric Force Sensors {#piezoelectric-force-sensors}
@ -30,23 +30,23 @@ Main differences between the two are shown in Figure [1](#orgc9e9a88).
### Dynamics and Noise of a piezoelectric force sensor {#dynamics-and-noise-of-a-piezoelectric-force-sensor}
An analysis the dynamics and noise of a piezoelectric force sensor is done in ([Fleming 2010](#org024e377)) ([Notes]({{<relref "fleming10_nanop_system_with_force_feedb.md#" >}})).
An analysis the dynamics and noise of a piezoelectric force sensor is done in (<a href="#citeproc_bib_item_1">Fleming 2010</a>) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb.md" >}})).
### Manufacturers {#manufacturers}
| Manufacturers | Country |
|---------------------------------------------------------------------------------------------------|---------|
| [PCB](https://www.pcb.com/products/productfinder.aspx?tx=17) | USA |
| [HBM](https://www.hbm.com/en/6107/force-sensors-with-flange-mounting/) | Germany |
| [Kistler](https://www.kistler.com/fr/produits/composants/capteurs-de-force/?pfv%5Fmetrics=metric) | Swiss |
| [MMF](https://www.mmf.de/force%5Ftransducers.htm) | Germany |
| [Sinocera](http://www.china-yec.net/sensors/) | China |
| Manufacturers | Country |
|-------------------------------------------------------------------------------------------------|---------|
| [PCB](https://www.pcb.com/products/productfinder.aspx?tx=17) | USA |
| [HBM](https://www.hbm.com/en/6107/force-sensors-with-flange-mounting/) | Germany |
| [Kistler](https://www.kistler.com/fr/produits/composants/capteurs-de-force/?pfv_metrics=metric) | Swiss |
| [MMF](https://www.mmf.de/force_transducers.htm) | Germany |
| [Sinocera](http://www.china-yec.net/sensors/) | China |
### Signal Conditioner {#signal-conditioner}
The voltage generated by the piezoelectric material generally needs to be amplified using a [Signal Conditioner]({{<relref "signal_conditioner.md#" >}}).
The voltage generated by the piezoelectric material generally needs to be amplified using a [Signal Conditioner]({{< relref "signal_conditioner.md" >}}).
Either **charge** amplifiers or **voltage** amplifiers can be used.
@ -76,7 +76,8 @@ However, if a charge conditioner is used, the signal will be doubled.
| [Althen](https://www.althensensors.com/sensors/weighing-sensors-load-cells/) | Netherlands |
## Bibliography {#bibliography}
<a id="org024e377"></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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content/zettels/fractional_order_transfer_functions.md
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@ -1,11 +1,11 @@
+++
title = "Fractional Order Transfer Functions"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
: [Digital Filters]({{< relref "digital_filters.md" >}})
## Example Using the FOMCON toolbox {#example-using-the-fomcon-toolbox}
@ -21,7 +21,7 @@ Here are the parameters that are used to define the wanted properties of the fra
r = 0.5; % Wanted slope, The corresponding phase will be pi*r
```
Then, to create an approximation of a fractional-order operator \\(s^r\\) of order \\(n\\) which is valid in the frequency range \\([\omega\_b\, \omega\_h]\\), the `oustafod` function can be used:
Then, to create an approximation of a fractional-order operator \\(s^r\\) of order \\(n\\) which is valid in the frequency range \\([\omega\_b\\, \omega\_h]\\), the `oustafod` function can be used:
```matlab
G = oustafod(r,n,wb,wh);
@ -37,8 +37,14 @@ G =
Continuous-time transfer function.
```
Few examples of different slopes are shown in Figure [1](#org9241d6d).
Few examples of different slopes are shown in Figure [1](#figure--fig:approximate-deriv-int).
<a id="org9241d6d"></a>
<a id="figure--fig:approximate-deriv-int"></a>
{{< figure src="/ox-hugo/approximate_deriv_int.png" caption="Figure 1: Example of fractional approximations" >}}
{{< figure src="/ox-hugo/approximate_deriv_int.png" caption="<span class=\"figure-number\">Figure 1: </span>Example of fractional approximations" >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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content/zettels/granite.md
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@ -1,6 +1,6 @@
+++
title = "Granite"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
@ -15,3 +15,9 @@ Tags
|--------------------------------------------------|---------|
| [Microplan](https://www.microplan-group.com/fr/) | France |
| [Zali](http://zali-precision.it/en/products/) | Italy |
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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content/zettels/h_infinity_control.md
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@ -1,6 +1,6 @@
+++
title = "H Infinity Control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -15,3 +15,9 @@ From _Rosenbrock, H. H. (1974). Computer-Aided Control System Design, Academic P
> Solutions are constrained by so many requirements that it is virtually impossible to list them all.
> The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on.
> A good design usually has strong aesthetic appeal to those who are competent in the subject.
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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content/zettels/hac_hac.md
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@ -1,6 +1,6 @@
+++
title = "HAC-HAC"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@ -9,29 +9,28 @@ Tags
High-Authority Control/Low-Authority Control
From ([Preumont 2018](#org4171546)):
From (<a href="#citeproc_bib_item_2">Preumont 2018</a>):
> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#org5a821d8). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:
> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#figure--fig:hac-lac-control-architecture). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:
>
> - The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth
> - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
> - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org5a821d8"></a>
<a id="figure--fig:hac-lac-control-architecture"></a>
{{< figure src="/ox-hugo/hac_lac_control_architecture.png" caption="Figure 1: HAC-LAC Control Architecture" >}}
{{< figure src="/ox-hugo/hac_lac_control_architecture.png" caption="<span class=\"figure-number\">Figure 1: </span>HAC-LAC Control Architecture" >}}
Nice papers:
- ([Williams and Antsaklis 1989](#orgb65b217))
- ([Aubrun 1980](#org9a935c0))
- (<a href="#citeproc_bib_item_3">Williams and Antsaklis 1989</a>)
- (<a href="#citeproc_bib_item_1">Aubrun 1980</a>)
## Bibliography {#bibliography}
<a id="org9a935c0"></a>Aubrun, J.N. 1980. “Theory of the Control of Structures by Low-Authority Controllers.” _Journal of Guidance and Control_ 3 (5):444–51. <https://doi.org/10.2514/3.56019>.
<a id="org4171546"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<a id="orgb65b217"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In _Proceedings of the 28th IEEE Conference on Decision and Control_, nil. <https://doi.org/10.1109/cdc.1989.70563>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Aubrun, J.N. 1980. “Theory of the Control of Structures by Low-Authority Controllers.” <i>Journal of Guidance and Control</i> 3 (5): 444–51. doi:<a href="https://doi.org/10.2514/3.56019">10.2514/3.56019</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In <i>Proceedings of the 28th Ieee Conference on Decision and Control</i>, nil. doi:<a href="https://doi.org/10.1109/cdc.1989.70563">10.1109/cdc.1989.70563</a>.</div>
</div>

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content/zettels/inertial_sensors.md
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@ -11,8 +11,8 @@ Tags
## Review of Absolute (inertial) Position Sensors {#review-of-absolute--inertial--position-sensors}
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement (<a href="#citeproc_bib_item_1">Collette, Janssens, Fernandez-Carmona, et al. 2012</a>)
- Collette, C. et al., Comparison of new absolute displacement sensors (<a href="#citeproc_bib_item_2">Collette, Janssens, Mokrani, et al. 2012</a>)
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement (<a href="#citeproc_bib_item_2">Collette, Janssens, Fernandez-Carmona, et al. 2012</a>)
- Collette, C. et al., Comparison of new absolute displacement sensors (<a href="#citeproc_bib_item_3">Collette, Janssens, Mokrani, et al. 2012</a>)
<a id="figure--fig:collette12-absolute-disp-sensors"></a>
@ -36,9 +36,11 @@ Wireless Accelerometers
- <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
Several commercial accelerometers are compared in Table [2](#figure--fig:characteristics-accelerometers) (see (<a href="#citeproc_bib_item_1">Collette et al. 2011</a>)).
<a id="figure--fig:characteristics-accelerometers"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="<span class=\"figure-number\">Figure 2: </span>Characteristics of commercially available accelerometers <collette11_review>" >}}
{{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="<span class=\"figure-number\">Figure 2: </span>Characteristics of commercially available accelerometers" >}}
## Geophones and Seismometers {#geophones-and-seismometers}
@ -53,6 +55,17 @@ Wireless Accelerometers
| [Guralp](https://www.guralp.com/products/surface) | UK |
| [Nanometric](https://www.nanometrics.ca/products/seismometers) | Canada |
(<a href="#citeproc_bib_item_1">Collette et al. 2011</a>)
<a id="figure--fig:characteristics-geophone"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="<span class=\"figure-number\">Figure 3: </span>Characteristics of commercially available geophones <collette11_review>" >}}
{{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="<span class=\"figure-number\">Figure 3: </span>Characteristics of commercially available geophones" >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C, K Artoos, M Guinchard, S Janssens, P Carmona Fernandez, and C Hauviller. 2011. “Review of Sensors for Low Frequency Seismic Vibration Measurement.” CERN.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” <i>Bulletin of the Seismological Society of America</i> 102 (4): 1289–1300. doi:<a href="https://doi.org/10.1785/0120110223">10.1785/0120110223</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In <i>International Conference on Noise and Vibration Engineering (Isma)</i>.</div>
</div>

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