1975 lines
92 KiB
HTML
1975 lines
92 KiB
HTML
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<body>
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<div id="content">
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<h1 class="title">Control in a rotating frame</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org6ad4e56">1. Introduction</a></li>
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<li><a href="#org6f41e95">2. System Description and Analysis</a>
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<ul>
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<li><a href="#org66b5521">2.1. System description</a></li>
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<li><a href="#orgcc922e2">2.2. Equations</a></li>
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<li><a href="#org41d2dca">2.3. Numerical Values for the NASS</a></li>
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<li><a href="#orgb4b3160">2.4. Euler and Coriolis forces - Numerical Result</a></li>
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<li><a href="#orgda7dc96">2.5. Negative Spring Effect - Numerical Result</a></li>
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<li><a href="#org4a4708b">2.6. Limitations due to coupling</a>
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<ul>
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<li><a href="#orgd744f26">2.6.1. Numerical Analysis</a></li>
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</ul>
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</li>
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<li><a href="#orgc0f34f1">2.7. Limitations due to negative stiffness effect</a></li>
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<li><a href="#orgef972c9">2.8. Effect of rotation speed on the plant</a>
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<ul>
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<li><a href="#org1994776">2.8.1. Voice coil actuator</a></li>
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<li><a href="#org39924fc">2.8.2. Piezoelectric actuator</a></li>
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<li><a href="#org3da7ee8">2.8.3. Analysis</a></li>
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<li><a href="#orge7538b5">2.8.4. Campbell diagram</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgc37c9ce">3. Control Strategies</a>
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<ul>
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<li><a href="#org02e89b1">3.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#orgde61f08">3.2. Measurement in the rotating frame</a></li>
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</ul>
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</li>
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<li><a href="#orgcc0ed4a">4. Multi Body Model - Simscape</a>
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<ul>
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<li><a href="#org17650a2">4.1. Initialization</a></li>
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<li><a href="#orga127b88">4.2. Identification in the rotating referenced frame</a></li>
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<li><a href="#org5522059">4.3. Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</a></li>
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<li><a href="#org475c38e">4.4. Plant Control - SISO approach</a>
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<ul>
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<li><a href="#orgdd00e7d">4.4.1. Plant identification</a></li>
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<li><a href="#orgf820010">4.4.2. Effect of rotation speed</a></li>
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<li><a href="#orgedc1f2a">4.4.3. Controller design</a></li>
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<li><a href="#orga14d702">4.4.4. Controlling the rotating system</a></li>
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<li><a href="#orgd419973">4.4.5. Close loop performance</a></li>
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</ul>
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</li>
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<li><a href="#org75899d3">4.5. Plant Control - MIMO approach</a>
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<ul>
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<li><a href="#orgf7efb03">4.5.1. <span class="todo TODO">TODO</span> Analysis - SVD</a></li>
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<li><a href="#orgaec63d4">4.5.2. Closed loop SVD</a></li>
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</ul>
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</li>
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<li><a href="#orgaddeb0f">4.6. test</a>
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<ul>
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<li><a href="#orgb3b8bc1">4.6.1. Low rotation speed and High rotation speed</a></li>
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</ul>
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</li>
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<li><a href="#orgb69d273">4.7. Identification in the fixed frame</a></li>
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<li><a href="#org6480b90">4.8. Identification from actuator forces to displacement in the fixed frame</a></li>
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<li><a href="#org2ce7065">4.9. Effect of the X-Y stage stiffness</a>
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<ul>
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<li><a href="#org2c17f71">4.9.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><a href="#orgf6cc550">5. Control Implementation</a>
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<ul>
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<li><a href="#orgea3a8d3">5.1. Measurement in the fixed reference frame</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org6ad4e56" class="outline-2">
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<h2 id="org6ad4e56"><span class="section-number-2">1</span> Introduction</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame.
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</p>
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<p>
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In section <a href="#sec:system">sec:system</a>, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
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The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
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This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
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</p>
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<p>
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Then, in section <a href="#sec:control_strategies">sec:control_strategies</a>, two different control approach are compared where:
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</p>
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<ul class="org-ul">
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<li>the measurement is made in the fixed frame</li>
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<li>the measurement is made in the rotating frame</li>
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</ul>
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<p>
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In section <a href="#sec:simscape">sec:simscape</a>, the analytical study will be validated using a multi body model of the studied system.
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</p>
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<p>
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Finally, in section <a href="#sec:control">sec:control</a>, the control strategies are implemented using Simulink and Simscape and compared.
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</p>
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</div>
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</div>
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<div id="outline-container-org6f41e95" class="outline-2">
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<h2 id="org6f41e95"><span class="section-number-2">2</span> System Description and Analysis</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org8c50be1"></a>
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</p>
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</div>
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<div id="outline-container-org66b5521" class="outline-3">
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<h3 id="org66b5521"><span class="section-number-3">2.1</span> System description</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#fig:rotating_frame_2dof">fig:rotating_frame_2dof</a>).
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</p>
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<p>
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The control inputs are the forces applied by the actuators of the translation stage (\(F_u\) and \(F_v\)).
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As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(u\) and \(v\).
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</p>
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|
|
<p>
|
|
The measurement is either the \(x-y\) displacement of the object located on top of the translation stage or the \(u-v\) displacement of the sample with respect to a fixed reference frame.
|
|
</p>
|
|
|
|
|
|
<div id="org907b94e" class="figure">
|
|
<p><img src="./figs/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
|
|
</div>
|
|
|
|
<p>
|
|
In the following block diagram:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(G\) is the transfer function from the forces applied in the actuators to the measurement</li>
|
|
<li>\(K\) is the controller to design</li>
|
|
<li>\(J\) is a Jacobian matrix usually used to change the reference frame</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Indices \(x\) and \(y\) corresponds to signals in the fixed reference frame (along \(\vec{i}_x\) and \(\vec{i}_y\)):
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(D_x\) is the measured position of the sample</li>
|
|
<li>\(r_x\) is the reference signal which corresponds to the wanted \(D_x\)</li>
|
|
<li>\(\epsilon_x\) is the position error</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (\(\vec{i}_u\) and \(\vec{i}_v\)):
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(F_u\) and \(F_v\) are forces applied by the actuators</li>
|
|
<li>\(\epsilon_u\) and \(\epsilon_v\) are position error of the sample along \(\vec{i}_u\) and \(\vec{i}_v\)</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc922e2" class="outline-3">
|
|
<h3 id="orgcc922e2"><span class="section-number-3">2.2</span> Equations</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
<a id="orgf8ac75e"></a>
|
|
Based on the figure <a href="#fig:rotating_frame_2dof">fig:rotating_frame_2dof</a>, we can write the equations of motion of the system.
|
|
</p>
|
|
|
|
<p>
|
|
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
|
|
</p>
|
|
\begin{align}
|
|
\label{org1db8b0c}
|
|
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
|
|
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
|
|
\end{align}
|
|
|
|
<p>
|
|
The Lagrangian is the kinetic energy minus the potential energy.
|
|
</p>
|
|
\begin{equation}
|
|
\label{org5a534f3}
|
|
L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
|
|
\end{equation}
|
|
|
|
|
|
<p>
|
|
The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
|
|
</p>
|
|
\begin{align*}
|
|
\label{org8364910}
|
|
\frac{\partial L}{\partial x} & = -kx \\
|
|
\frac{\partial L}{\partial y} & = -ky \\
|
|
\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
|
|
\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} & = m\ddot{y}
|
|
\end{align*}
|
|
|
|
<p>
|
|
The external forces applied to the mass are:
|
|
</p>
|
|
\begin{align*}
|
|
F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
|
|
F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
By appling the Lagrangian equations, we obtain:
|
|
</p>
|
|
\begin{align}
|
|
m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
|
|
m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align}
|
|
|
|
<p>
|
|
We then change coordinates from \((x, y)\) to \((d_x, d_y, \theta)\).
|
|
</p>
|
|
\begin{align*}
|
|
x & = d_u \cos{\theta} - d_v \sin{\theta}\\
|
|
y & = d_u \sin{\theta} + d_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
We obtain:
|
|
</p>
|
|
\begin{align*}
|
|
\ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta}
|
|
- \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\
|
|
\ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta}
|
|
+ \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
|
|
\end{align*}
|
|
|
|
<p>
|
|
By injecting the previous result into the Lagrangian equation, we obtain:
|
|
</p>
|
|
\begin{align*}
|
|
m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
|
|
-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
|
|
+ k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\
|
|
m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta}
|
|
+ m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta}
|
|
+ k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta}
|
|
\end{align*}
|
|
|
|
<p>
|
|
Which is equivalent to:
|
|
</p>
|
|
\begin{align*}
|
|
m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2
|
|
-m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}}
|
|
+ k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\
|
|
m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2
|
|
+ m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}}
|
|
+ k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}}
|
|
\end{align*}
|
|
|
|
<p>
|
|
We can then subtract and add the previous equations to obtain the following equations:
|
|
</p>
|
|
<div class="important">
|
|
\begin{equation}
|
|
\label{org5253d4e}
|
|
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{org0021062}
|
|
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta}
|
|
\end{equation}
|
|
|
|
</div>
|
|
|
|
<p>
|
|
We obtain two differential equations that are coupled through:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Euler forces</b>: \(m d_v \ddot{\theta}\)</li>
|
|
<li><b>Coriolis forces</b>: \(2 m \dot{d_v} \dot{\theta}\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k- m\dot{\theta}^2\).
|
|
Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to <b>centrifugal forces</b>).
|
|
</p>
|
|
|
|
<p>
|
|
The forces induced by the rotating reference frame are independent of the stiffness of the actuator.
|
|
The resulting effect of those forces should then be higher when using softer actuators.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org41d2dca" class="outline-3">
|
|
<h3 id="org41d2dca"><span class="section-number-3">2.3</span> Numerical Values for the NASS</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
Let's define the parameters for the NASS.
|
|
</p>
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Light sample mass [kg]</td>
|
|
<td class="org-right">3.5e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Heavy sample mass [kg]</td>
|
|
<td class="org-right">8.5e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. speed - light [rpm]</td>
|
|
<td class="org-right">6.0e+01</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. speed - heavy [rpm]</td>
|
|
<td class="org-right">1.0e+00</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Voice Coil Stiffness [N/m]</td>
|
|
<td class="org-right">1.0e+03</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Piezo Stiffness [N/m]</td>
|
|
<td class="org-right">1.0e+08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max rot. acceleration [rad/s2]</td>
|
|
<td class="org-right">1.0e+00</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max mass excentricity [m]</td>
|
|
<td class="org-right">1.0e-02</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Max Horizontal speed [m/s]</td>
|
|
<td class="org-right">2.0e-01</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb4b3160" class="outline-3">
|
|
<h3 id="orgb4b3160"><span class="section-number-3">2.4</span> Euler and Coriolis forces - Numerical Result</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li><b>Euler forces</b>: \(m d_v \ddot{\theta}\)</li>
|
|
<li><b>Coriolis forces</b>: \(2 m \dot{d_v} \dot{\theta}\)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The obtained values are displayed in table <a href="#tab:euler_coriolis">tab:euler_coriolis</a>.
|
|
</p>
|
|
|
|
<table id="org0e81c55" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Light</th>
|
|
<th scope="col" class="org-left">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Coriolis</td>
|
|
<td class="org-left">88.0N</td>
|
|
<td class="org-left">3.6N</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Euler</td>
|
|
<td class="org-left">0.4N</td>
|
|
<td class="org-left">0.8N</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgda7dc96" class="outline-3">
|
|
<h3 id="orgda7dc96"><span class="section-number-3">2.5</span> Negative Spring Effect - Numerical Result</h3>
|
|
<div class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).
|
|
</p>
|
|
|
|
<p>
|
|
The values for the negative spring effect are displayed in table <a href="#tab:negative_spring">tab:negative_spring</a>.
|
|
</p>
|
|
|
|
<p>
|
|
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
|
</p>
|
|
|
|
<table id="org98e02da" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> Negative Spring effect</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Light</th>
|
|
<th scope="col" class="org-left">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Neg. Spring</td>
|
|
<td class="org-left">1381.7[N/m]</td>
|
|
<td class="org-left">0.9[N/m]</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org4a4708b" class="outline-3">
|
|
<h3 id="org4a4708b"><span class="section-number-3">2.6</span> Limitations due to coupling</h3>
|
|
<div class="outline-text-3" id="text-2-6">
|
|
<p>
|
|
To simplify, we consider a constant rotating speed \(\dot{\theta} = \omega_0\) and thus \(\ddot{\theta} = 0\).
|
|
</p>
|
|
|
|
<p>
|
|
From equations \eqref{eq:du_coupled} and \eqref{eq:dv_coupled}, we obtain:
|
|
</p>
|
|
\begin{align*}
|
|
(m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\
|
|
(m s^2 + (k - m{\omega_0}^2)) d_v &= F_v - 2 m {\omega_0} s d_u \\
|
|
\end{align*}
|
|
|
|
<p>
|
|
From second equation:
|
|
\[ d_v = \frac{1}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} d_u \]
|
|
</p>
|
|
|
|
<p>
|
|
And we re-inject \(d_v\) into the first equation:
|
|
</p>
|
|
\begin{equation*}
|
|
(m s^2 + (k - m{\omega_0}^2)) d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{(2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u
|
|
\end{equation*}
|
|
|
|
\begin{equation*}
|
|
\frac{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v
|
|
\end{equation*}
|
|
|
|
<p>
|
|
Finally we obtain \(d_u\) function of \(F_u\) and \(F_v\).
|
|
\[ d_u = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u + \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v \]
|
|
</p>
|
|
|
|
<p>
|
|
Similarly we can obtain \(d_v\) function of \(F_u\) and \(F_v\):
|
|
\[ d_v = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v - \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u \]
|
|
</p>
|
|
|
|
<p>
|
|
The two previous equations can be written in a matrix form:
|
|
</p>
|
|
<div class="important">
|
|
\begin{equation}
|
|
\label{orgce910a4}
|
|
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
|
|
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
|
|
\begin{bmatrix}
|
|
ms^2 + (k-m{\omega_0}^2) & 2 m \omega_0 s \\
|
|
-2 m \omega_0 s & ms^2 + (k-m{\omega_0}^2) \\
|
|
\end{bmatrix}
|
|
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
|
\end{equation}
|
|
|
|
</div>
|
|
|
|
<p>
|
|
Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {\omega_0} \omega|\).
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd744f26" class="outline-4">
|
|
<h4 id="orgd744f26"><span class="section-number-4">2.6.1</span> Numerical Analysis</h4>
|
|
<div class="outline-text-4" id="text-2-6-1">
|
|
<p>
|
|
We plot on the same graph \(\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}\) for the voice coil and the piezo:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>with the light sample (figure <a href="#fig:coupling_light">fig:coupling_light</a>).</li>
|
|
<li>with the heavy sample (figure <a href="#fig:coupling_heavy">fig:coupling_heavy</a>).</li>
|
|
</ul>
|
|
|
|
|
|
<div id="orgb4ae1de" class="figure">
|
|
<p><img src="figs/coupling_light.png" alt="coupling_light.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Relative Coupling for light mass and high rotation speed</p>
|
|
</div>
|
|
|
|
<div id="org4fb4d9c" class="figure">
|
|
<p><img src="figs/coupling_heavy.png" alt="coupling_heavy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </span>Relative Coupling for heavy mass and low rotation speed</p>
|
|
</div>
|
|
|
|
<div class="important">
|
|
<p>
|
|
Coupling is higher for actuators with small stiffness.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc0f34f1" class="outline-3">
|
|
<h3 id="orgc0f34f1"><span class="section-number-3">2.7</span> Limitations due to negative stiffness effect</h3>
|
|
<div class="outline-text-3" id="text-2-7">
|
|
<p>
|
|
If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
|
|
</p>
|
|
|
|
<p>
|
|
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible (\(\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}\)). Results are shown table <a href="#tab:negative_stiffness">tab:negative_stiffness</a>.
|
|
</p>
|
|
<table id="org679f76e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</span> Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-left">Voice Coil</th>
|
|
<th scope="col" class="org-left">Piezo</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Light</td>
|
|
<td class="org-left">5[rpm]</td>
|
|
<td class="org-left">1614[rpm]</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Heavy</td>
|
|
<td class="org-left">3[rpm]</td>
|
|
<td class="org-left">1036[rpm]</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
The negative spring effect is proportional to the rotational speed \(\omega\).
|
|
The system dynamics will be much more affected when using soft actuator.
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
Negative stiffness effect has very important effect when using soft actuators.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
The system can even goes unstable when \(m \omega^2 > k\), that is when the centrifugal forces are higher than the forces due to stiffness.
|
|
</p>
|
|
|
|
<p>
|
|
From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table sec:tab:min<sub>k</sub>)
|
|
</p>
|
|
|
|
<table id="orgf263e78" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Minimum possible stiffness</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<thead>
|
|
<tr>
|
|
<th scope="col" class="org-left"> </th>
|
|
<th scope="col" class="org-right">Light</th>
|
|
<th scope="col" class="org-right">Heavy</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">k min [N/m]</td>
|
|
<td class="org-right">2199</td>
|
|
<td class="org-right">89</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgef972c9" class="outline-3">
|
|
<h3 id="orgef972c9"><span class="section-number-3">2.8</span> Effect of rotation speed on the plant</h3>
|
|
<div class="outline-text-3" id="text-2-8">
|
|
<p>
|
|
As shown in equation \eqref{eq:coupledplant}, the plant changes with the rotation speed \(\omega_0\).
|
|
</p>
|
|
|
|
<p>
|
|
Then, we compute the bode plot of the direct term and coupling term for multiple rotating speed.
|
|
</p>
|
|
|
|
<p>
|
|
Then we compare the result between voice coil and piezoelectric actuators.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org1994776" class="outline-4">
|
|
<h4 id="org1994776"><span class="section-number-4">2.8.1</span> Voice coil actuator</h4>
|
|
<div class="outline-text-4" id="text-2-8-1">
|
|
|
|
<div id="org0a1428d" class="figure">
|
|
<p><img src="figs/G_ws_vc.png" alt="G_ws_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Voice coil</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org7e170f8" class="figure">
|
|
<p><img src="figs/Gc_ws_vc.png" alt="Gc_ws_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>caption</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org39924fc" class="outline-4">
|
|
<h4 id="org39924fc"><span class="section-number-4">2.8.2</span> Piezoelectric actuator</h4>
|
|
<div class="outline-text-4" id="text-2-8-2">
|
|
|
|
<div id="org4cafe43" class="figure">
|
|
<p><img src="figs/G_ws_pz.png" alt="G_ws_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Bode plot of the direct transfer function term (from \(F_u\) to \(D_u\)) for multiple rotation speed - Piezoelectric actuator</p>
|
|
</div>
|
|
|
|
|
|
<div id="org360ff28" class="figure">
|
|
<p><img src="figs/Gc_ws_pz.png" alt="Gc_ws_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Bode plot of the coupling transfer function term (from \(F_u\) to \(D_v\)) for multiple rotation speed - Piezoelectric actuator</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3da7ee8" class="outline-4">
|
|
<h4 id="org3da7ee8"><span class="section-number-4">2.8.3</span> Analysis</h4>
|
|
<div class="outline-text-4" id="text-2-8-3">
|
|
<p>
|
|
When the rotation speed is null, the coupling terms are equal to zero and the diagonal terms corresponds to one degree of freedom mass spring system.
|
|
</p>
|
|
|
|
<p>
|
|
When the rotation speed in not null, the resonance frequency is duplicated into two pairs of complex conjugate poles.
|
|
</p>
|
|
|
|
<p>
|
|
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies.
|
|
</p>
|
|
|
|
<p>
|
|
The poles of the coupling terms are the same as the poles of the diagonal terms. The magnitude of the coupling terms are increasing with the rotation speed.
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
As shown in the previous figures, the system with voice coil is much more sensitive to rotation speed.
|
|
</p>
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge7538b5" class="outline-4">
|
|
<h4 id="orge7538b5"><span class="section-number-4">2.8.4</span> Campbell diagram</h4>
|
|
<div class="outline-text-4" id="text-2-8-4">
|
|
<p>
|
|
The poles of the system are computed for multiple values of the rotation frequency. To simplify the computation of the poles, we add some damping to the system.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = mlight;
|
|
k = kvc;
|
|
c = <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">*</span>sqrt<span style="color: #DCDCCC;">(</span>k<span style="color: #7CB8BB;">*</span>m<span style="color: #DCDCCC;">)</span>;
|
|
|
|
wsvc = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">10</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [rad/s]</span>
|
|
|
|
polesvc = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, length<span style="color: #BFEBBF;">(</span>wsvc<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">wsvc</span><span style="color: #DCDCCC;">)</span>
|
|
polei = pole<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">(</span>m<span style="color: #7CB8BB;">*</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> c<span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #93E0E3;">(</span>k <span style="color: #7CB8BB;">-</span> m<span style="color: #7CB8BB;">*</span>wsvc<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #9FC59F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>m<span style="color: #7CB8BB;">*</span>wsvc<span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">*</span>s<span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
polesvc<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = sort<span style="color: #DCDCCC;">(</span>polei<span style="color: #BFEBBF;">(</span>imag<span style="color: #D0BF8F;">(</span>polei<span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">></span> <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = mlight;
|
|
k = kpz;
|
|
c = <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">*</span>sqrt<span style="color: #DCDCCC;">(</span>k<span style="color: #7CB8BB;">*</span>m<span style="color: #DCDCCC;">)</span>;
|
|
|
|
wspz = linspace<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">0</span>, <span style="color: #BFEBBF;">1000</span>, <span style="color: #BFEBBF;">100</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% [rad/s]</span>
|
|
|
|
polespz = zeros<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span>, length<span style="color: #BFEBBF;">(</span>wspz<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #F0DFAF; font-weight: bold;">for</span> <span style="color: #DFAF8F;">i</span> = <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">:length</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">wspz</span><span style="color: #DCDCCC;">)</span>
|
|
polei = pole<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span><span style="color: #D0BF8F;">(</span>m<span style="color: #7CB8BB;">*</span>s<span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> c<span style="color: #7CB8BB;">*</span>s <span style="color: #7CB8BB;">+</span> <span style="color: #93E0E3;">(</span>k <span style="color: #7CB8BB;">-</span> m<span style="color: #7CB8BB;">*</span>wspz<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #9FC59F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span> <span style="color: #7CB8BB;">+</span> <span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span>m<span style="color: #7CB8BB;">*</span>wspz<span style="color: #93E0E3;">(</span><span style="color: #BFEBBF;">i</span><span style="color: #93E0E3;">)</span><span style="color: #7CB8BB;">*</span>s<span style="color: #D0BF8F;">)</span><span style="color: #7CB8BB;">^</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
polespz<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">i</span><span style="color: #DCDCCC;">)</span> = sort<span style="color: #DCDCCC;">(</span>polei<span style="color: #BFEBBF;">(</span>imag<span style="color: #D0BF8F;">(</span>polei<span style="color: #D0BF8F;">)</span> <span style="color: #7CB8BB;">></span> <span style="color: #BFEBBF;">0</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #F0DFAF; font-weight: bold;">end</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then plot the real and imaginary part of the poles as a function of the rotation frequency (figures <a href="#fig:poles_w_vc">fig:poles_w_vc</a> and <a href="#fig:poles_w_pz">fig:poles_w_pz</a>).
|
|
</p>
|
|
|
|
<p>
|
|
When the real part of one pole becomes positive, the system goes unstable.
|
|
</p>
|
|
|
|
<p>
|
|
For the voice coil (figure <a href="#fig:poles_w_vc">fig:poles_w_vc</a>), the system is unstable when the rotation speed is above 5 rad/s. The real and imaginary part of the poles of the system with piezoelectric actuators are changing much less (figure <a href="#fig:poles_w_pz">fig:poles_w_pz</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org6904013" class="figure">
|
|
<p><img src="figs/poles_w_vc.png" alt="poles_w_vc.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Voice Coil and light sample</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org69ed710" class="figure">
|
|
<p><img src="figs/poles_w_pz.png" alt="poles_w_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Real and Imaginary part of the poles of the system as a function of the rotation speed - Piezoelectric actuator and light sample</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc37c9ce" class="outline-2">
|
|
<h2 id="orgc37c9ce"><span class="section-number-2">3</span> Control Strategies</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org2ae6770"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org02e89b1" class="outline-3">
|
|
<h3 id="org02e89b1"><span class="section-number-3">3.1</span> Measurement in the fixed reference frame</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
First, let's consider a measurement in the fixed referenced frame.
|
|
</p>
|
|
|
|
<p>
|
|
The transfer function from actuator \([F_u, F_v]\) to sensor \([D_x, D_y]\) is then \(G(\theta)\).
|
|
</p>
|
|
|
|
<p>
|
|
Then the measurement is subtracted to the reference signal \([r_x, r_y]\) to obtain the position error in the fixed reference frame \([\epsilon_x, \epsilon_y]\).
|
|
</p>
|
|
|
|
<p>
|
|
The position error \([\epsilon_x, \epsilon_y]\) is then express in the rotating frame corresponding to the actuators \([\epsilon_u, \epsilon_v]\).
|
|
</p>
|
|
|
|
<p>
|
|
Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon_v]\) to the actuator forces \([F_u, F_v]\).
|
|
</p>
|
|
|
|
<p>
|
|
The block diagram is shown on figure <a href="#fig:control_measure_fixed_2dof">fig:control_measure_fixed_2dof</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org20f2cd4" class="figure">
|
|
<p><img src="./figs/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Control with a measure from fixed frame</p>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is then \(L = G(\theta) K J(\theta)\).
|
|
</p>
|
|
|
|
<p>
|
|
One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\theta_0)\)?
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgde61f08" class="outline-3">
|
|
<h3 id="orgde61f08"><span class="section-number-3">3.2</span> Measurement in the rotating frame</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
Let's consider that the measurement is made in the rotating reference frame.
|
|
</p>
|
|
|
|
<p>
|
|
The corresponding block diagram is shown figure <a href="#fig:control_measure_rotating_2dof">fig:control_measure_rotating_2dof</a>
|
|
</p>
|
|
|
|
|
|
<div id="org171edfa" class="figure">
|
|
<p><img src="./figs/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Control with a measure from rotating frame</p>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is \(L = G K\).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgcc0ed4a" class="outline-2">
|
|
<h2 id="orgcc0ed4a"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org641fab8"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org17650a2" class="outline-3">
|
|
<h3 id="org17650a2"><span class="section-number-3">4.1</span> Initialization</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
First we define the parameters that must be defined in order to run the Simscape simulation.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>; <span style="color: #7F9F7F;">% Rotation speed [rad/s]</span>
|
|
|
|
theta_e = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% Static measurement error on the angle theta [rad]</span>
|
|
|
|
m = <span style="color: #BFEBBF;">5</span>; <span style="color: #7F9F7F;">% mass of the sample [kg]</span>
|
|
|
|
mTuv = <span style="color: #BFEBBF;">30</span>;<span style="color: #7F9F7F;">% Mass of the moving part of the Tuv stage [kg]</span>
|
|
kTuv = <span style="color: #BFEBBF;">1e8</span>; <span style="color: #7F9F7F;">% Stiffness of the Tuv stage [N/m]</span>
|
|
cTuv = <span style="color: #BFEBBF;">0</span>; <span style="color: #7F9F7F;">% Damping of the Tuv stage [N/(m/s)]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, we defined parameters that will be used in the following analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mlight = <span style="color: #BFEBBF;">5</span>; <span style="color: #7F9F7F;">% Mass for light sample [kg]</span>
|
|
mheavy = <span style="color: #BFEBBF;">55</span>; <span style="color: #7F9F7F;">% Mass for heavy sample [kg]</span>
|
|
|
|
wlight = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>; <span style="color: #7F9F7F;">% Max rot. speed for light sample [rad/s]</span>
|
|
wheavy = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">60</span>; <span style="color: #7F9F7F;">% Max rot. speed for heavy sample [rad/s]</span>
|
|
|
|
kvc = <span style="color: #BFEBBF;">1e3</span>; <span style="color: #7F9F7F;">% Voice Coil Stiffness [N/m]</span>
|
|
kpz = <span style="color: #BFEBBF;">1e8</span>; <span style="color: #7F9F7F;">% Piezo Stiffness [N/m]</span>
|
|
|
|
d = <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">01</span>; <span style="color: #7F9F7F;">% Maximum excentricity from rotational axis [m]</span>
|
|
|
|
freqs = logspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">1000</span><span style="color: #DCDCCC;">)</span>; <span style="color: #7F9F7F;">% Frequency vector for analysis [Hz]</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga127b88" class="outline-3">
|
|
<h3 id="orga127b88"><span class="section-number-3">4.2</span> Identification in the rotating referenced frame</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
We initialize the inputs and outputs of the system to identify:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Inputs: \(f_u\) and \(f_v\)</li>
|
|
<li>Outputs: \(d_u\) and \(d_v\)</li>
|
|
</ul>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #BFEBBF;">0</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #CC9393;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fu'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fv'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>du'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>dv'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We start we identify the transfer functions at high speed with the light sample.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = wlight; <span style="color: #7F9F7F;">% Rotation speed [rad/s]</span>
|
|
m = mlight; <span style="color: #7F9F7F;">% mass of the sample [kg]</span>
|
|
|
|
kTuv = kpz;
|
|
Gpz_light = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
Gpz_light.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gpz_light.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
|
|
kTuv = kvc;
|
|
Gvc_light = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
Gvc_light.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gvc_light.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then we identify the system with an heavy mass and low speed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w = wheavy; <span style="color: #7F9F7F;">% Rotation speed [rad/s]</span>
|
|
m = mheavy; <span style="color: #7F9F7F;">% mass of the sample [kg]</span>
|
|
|
|
kTuv = kpz;
|
|
Gpz_heavy = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
Gpz_heavy.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gpz_heavy.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
|
|
kTuv = kvc;
|
|
Gvc_heavy = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
Gvc_heavy.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gvc_heavy.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org5522059" class="outline-3">
|
|
<h3 id="org5522059"><span class="section-number-3">4.3</span> Coupling ratio between \(f_{uv}\) and \(d_{uv}\)</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<p>
|
|
In order to validate the equations written, we can compute the coupling ratio using the simscape model and compare with the equations.
|
|
</p>
|
|
|
|
<p>
|
|
From the previous identification, we plot the coupling ratio in both case (figure <a href="#fig:coupling_ratio_light_heavy">fig:coupling_ratio_light_heavy</a>).
|
|
</p>
|
|
|
|
<p>
|
|
We obtain the same result than the analytical case (figures <a href="#fig:coupling_light">fig:coupling_light</a> and <a href="#fig:coupling_heavy">fig:coupling_heavy</a>).
|
|
</p>
|
|
|
|
<div id="orgf89eab7" class="figure">
|
|
<p><img src="figs/coupling_ratio_light_heavy.png" alt="coupling_ratio_light_heavy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Coupling ratio obtained with the Simscape model</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org475c38e" class="outline-3">
|
|
<h3 id="org475c38e"><span class="section-number-3">4.4</span> Plant Control - SISO approach</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
</div>
|
|
<div id="outline-container-orgdd00e7d" class="outline-4">
|
|
<h4 id="orgdd00e7d"><span class="section-number-4">4.4.1</span> Plant identification</h4>
|
|
<div class="outline-text-4" id="text-4-4-1">
|
|
<p>
|
|
The goal is to study the control problems due to the coupling that appears because of the rotation.
|
|
</p>
|
|
|
|
<p>
|
|
First, we identify the system when the rotation speed is null and then when the rotation speed is equal to 60rpm.
|
|
</p>
|
|
|
|
<p>
|
|
The actuators are voice coil with some damping added.
|
|
</p>
|
|
|
|
<p>
|
|
The bode plot of the system not rotating and rotating at 60rpm is shown figure <a href="#fig:Gvc_speed">fig:Gvc_speed</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org76b8371" class="figure">
|
|
<p><img src="figs/Gvc_speed.png" alt="Gvc_speed.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Change of transfer functions due to rotating speed</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf820010" class="outline-4">
|
|
<h4 id="orgf820010"><span class="section-number-4">4.4.2</span> Effect of rotation speed</h4>
|
|
<div class="outline-text-4" id="text-4-4-2">
|
|
<p>
|
|
We first identify the system (voice coil and light mass) for multiple rotation speed.
|
|
Then we compute the bode plot of the diagonal element (figure <a href="#fig:Guu_ws">fig:Guu_ws</a>) and of the coupling element (figure <a href="#fig:Guv_ws">fig:Guv_ws</a>).
|
|
</p>
|
|
|
|
<p>
|
|
As the rotation frequency increases:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>one pole goes to lower frequencies while the other goes to higher frequencies</li>
|
|
<li>one zero appears between the two poles</li>
|
|
<li>the zero disappears when \(\omega > \sqrt{\frac{k}{m}}\) and the low frequency pole becomes unstable (positive real part)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
To stabilize the unstable pole, we need a control bandwidth of at least twice of frequency of the unstable pole.
|
|
</p>
|
|
|
|
|
|
<div id="org5ecd99d" class="figure">
|
|
<p><img src="figs/Guu_ws.png" alt="Guu_ws.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Diagonal term as a function of the rotation frequency</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="orgfb3944c" class="figure">
|
|
<p><img src="figs/Guv_ws.png" alt="Guv_ws.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Couplin term as a function of the rotation frequency</p>
|
|
</div>
|
|
|
|
<p>
|
|
Then, we can look at the same plots for the piezoelectric actuator (figure <a href="#fig:Guu_ws_pz">fig:Guu_ws_pz</a>). The effect of the rotation frequency has very little effect on the dynamics of the system to control.
|
|
</p>
|
|
|
|
|
|
<div id="orgea6c407" class="figure">
|
|
<p><img src="figs/Guu_ws_pz.png" alt="Guu_ws_pz.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Diagonal term as a function of the rotation frequency</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgedc1f2a" class="outline-4">
|
|
<h4 id="orgedc1f2a"><span class="section-number-4">4.4.3</span> Controller design</h4>
|
|
<div class="outline-text-4" id="text-4-4-3">
|
|
<p>
|
|
We design a controller based on the identification when the system is not rotating.
|
|
</p>
|
|
|
|
<p>
|
|
The obtained controller is a lead-lag controller with the following transfer function.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Kll = <span style="color: #BFEBBF;">2</span>.<span style="color: #BFEBBF;">0698e09</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">40</span>.<span style="color: #BFEBBF;">45</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #DCDCCC;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">1</span>.<span style="color: #BFEBBF;">181</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">01</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">198</span>.<span style="color: #BFEBBF;">4</span><span style="color: #BFEBBF;">)</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">(</span>s<span style="color: #7CB8BB;">+</span><span style="color: #BFEBBF;">2790</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
K = <span style="color: #DCDCCC;">[</span>Kll <span style="color: #BFEBBF;">0</span>;
|
|
<span style="color: #BFEBBF;">0</span> Kll<span style="color: #DCDCCC;">]</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is displayed figure <a href="#fig:Gvc_loop_gain">fig:Gvc_loop_gain</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgd876a20" class="figure">
|
|
<p><img src="figs/Gvc_loop_gain.png" alt="Gvc_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Loop gain obtained for a lead-lag controller on the system with a voice coil</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga14d702" class="outline-4">
|
|
<h4 id="orga14d702"><span class="section-number-4">4.4.4</span> Controlling the rotating system</h4>
|
|
<div class="outline-text-4" id="text-4-4-4">
|
|
<p>
|
|
We here want to see if the system is robust with respect to the rotation speed. We then use the controller based on the non-rotating system, and see if the system is stable and its dynamics.
|
|
</p>
|
|
|
|
<p>
|
|
We can then plot the same loop gain with the rotating system using the same controller (figure <a href="#fig:Gtvc_loop_gain">fig:Gtvc_loop_gain</a>). The result obtained is unstable.
|
|
</p>
|
|
|
|
|
|
<div id="orgf34271c" class="figure">
|
|
<p><img src="figs/Gtvc_loop_gain.png" alt="Gtvc_loop_gain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Loop gain with the rotating system</p>
|
|
</div>
|
|
|
|
<p>
|
|
We can look at the poles of the system where we control only one direction (\(u\) for instance). We obtain a pole with a positive real part.
|
|
</p>
|
|
|
|
<div class="important">
|
|
<p>
|
|
The system is then unstable when controlling only one direction.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pole<span style="color: #DCDCCC;">(</span>feedback<span style="color: #BFEBBF;">(</span>Gtvc, blkdiag<span style="color: #D0BF8F;">(</span>Kll, <span style="color: #BFEBBF;">0</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">-2798</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-58.916+94.248i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-58.916-94.248i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-71.644</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.1647</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3.3034</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1.1901</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
However, when we look at the poles of the closed loop with a diagonal controller, all the poles have negative real part and the system is stable.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pole<span style="color: #DCDCCC;">(</span>feedback<span style="color: #BFEBBF;">(</span>Gtvc, blkdiag<span style="color: #D0BF8F;">(</span>Kll, Kll<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">-2798+0.035765i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-2798-0.035765i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-56.414+105.34i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-56.414-105.34i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-64.495+79.314i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-64.495-79.314i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-68.509+13.499i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-68.509-13.499i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-1.1837+0.0041422i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">-1.1837-0.0041422i</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<p>
|
|
Check stability of MIMO system.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">isstable<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">+</span>K<span style="color: #7CB8BB;">*</span>Gtvc<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>
|
|
isstable<span style="color: #DCDCCC;">(</span>Gtvc<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">+</span>K<span style="color: #7CB8BB;">*</span>Gtvc<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>
|
|
isstable<span style="color: #DCDCCC;">(</span>Gtvc<span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">+</span>K<span style="color: #7CB8BB;">*</span>Gtvc<span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
0
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd419973" class="outline-4">
|
|
<h4 id="orgd419973"><span class="section-number-4">4.4.5</span> Close loop performance</h4>
|
|
<div class="outline-text-4" id="text-4-4-5">
|
|
<p>
|
|
First, we create the closed loop systems. Then, we plot the transfer function from the reference signals \([\epsilon_u, \epsilon_v]\) to the output \([d_u, d_v]\) (figure <a href="#fig:perfcomp">fig:perfcomp</a>).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">S = eye<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">+</span> Gvc<span style="color: #7CB8BB;">*</span>K<span style="color: #DCDCCC;">)</span>;
|
|
T = Gvc<span style="color: #7CB8BB;">*</span>K <span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">+</span> Gvc<span style="color: #7CB8BB;">*</span>K<span style="color: #DCDCCC;">)</span>;
|
|
|
|
St = eye<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">+</span> Gtvc<span style="color: #7CB8BB;">*</span>K<span style="color: #DCDCCC;">)</span>;
|
|
Tt = Gtvc<span style="color: #7CB8BB;">*</span>K<span style="color: #7CB8BB;">/</span><span style="color: #DCDCCC;">(</span>eye<span style="color: #BFEBBF;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #BFEBBF;">)</span> <span style="color: #7CB8BB;">+</span> Gtvc<span style="color: #7CB8BB;">*</span>K<span style="color: #DCDCCC;">)</span>;
|
|
|
|
freqs = logspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">3</span>, <span style="color: #BFEBBF;">1000</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>S, St, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">*</span>freqs, bode_opts<span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>T, Tt, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">*</span>freqs, bode_opts<span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqs = logspace<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">2</span>, <span style="color: #BFEBBF;">1000</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #7CB8BB;">figure</span>;
|
|
ax1 = subplot<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span>,<span style="color: #BFEBBF;">2</span>,<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
hold on;
|
|
plot<span style="color: #DCDCCC;">(</span>freqs, abs<span style="color: #BFEBBF;">(</span>squeeze<span style="color: #D0BF8F;">(</span>freqresp<span style="color: #93E0E3;">(</span>Tvc<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #9FC59F;">)</span>, freqs, <span style="color: #CC9393;">'Hz'</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
plot<span style="color: #DCDCCC;">(</span>freqs, abs<span style="color: #BFEBBF;">(</span>squeeze<span style="color: #D0BF8F;">(</span>freqresp<span style="color: #93E0E3;">(</span>Ttvc<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">1</span><span style="color: #9FC59F;">)</span>, freqs, <span style="color: #CC9393;">'Hz'</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
hold off;
|
|
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>freqs<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span>, freqs<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
|
|
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; ylabel</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Magnitude </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">m/N</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
|
|
legend<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">{</span>'w = <span style="color: #BFEBBF;">0</span> <span style="color: #D0BF8F;">[</span>rpm<span style="color: #D0BF8F;">]</span>', 'w = <span style="color: #BFEBBF;">60</span> <span style="color: #D0BF8F;">[</span>rpm<span style="color: #D0BF8F;">]</span>'<span style="color: #BFEBBF;">}</span>, 'Location', 'southwest'<span style="color: #DCDCCC;">)</span>
|
|
title<span style="color: #DCDCCC;">(</span>'$G_<span style="color: #BFEBBF;">{</span>r_u <span style="color: #7CB8BB;">\</span>to d_u<span style="color: #BFEBBF;">}</span>$'<span style="color: #DCDCCC;">)</span>
|
|
|
|
ax2 = subplot<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span>,<span style="color: #BFEBBF;">2</span>,<span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span>;
|
|
hold on;
|
|
plot<span style="color: #DCDCCC;">(</span>freqs, abs<span style="color: #BFEBBF;">(</span>squeeze<span style="color: #D0BF8F;">(</span>freqresp<span style="color: #93E0E3;">(</span>Tvc<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">2</span><span style="color: #9FC59F;">)</span>, freqs, <span style="color: #CC9393;">'Hz'</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
plot<span style="color: #DCDCCC;">(</span>freqs, abs<span style="color: #BFEBBF;">(</span>squeeze<span style="color: #D0BF8F;">(</span>freqresp<span style="color: #93E0E3;">(</span>Ttvc<span style="color: #9FC59F;">(</span><span style="color: #BFEBBF;">1</span>, <span style="color: #BFEBBF;">2</span><span style="color: #9FC59F;">)</span>, freqs, <span style="color: #CC9393;">'Hz'</span><span style="color: #93E0E3;">)</span><span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
hold off;
|
|
xlim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>freqs<span style="color: #D0BF8F;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #D0BF8F;">)</span>, freqs<span style="color: #D0BF8F;">(</span>end<span style="color: #D0BF8F;">)</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
ylim<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span><span style="color: #BFEBBF;">1e</span><span style="color: #7CB8BB;">-</span><span style="color: #BFEBBF;">5</span>, <span style="color: #BFEBBF;">1</span><span style="color: #BFEBBF;">]</span><span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #7CB8BB;">set</span><span style="color: #DCDCCC;">(</span><span style="color: #DFAF8F;">gca</span>, <span style="color: #CC9393;">'XScale', 'log'</span><span style="color: #DCDCCC;">)</span><span style="color: #CC9393;">; set</span><span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">gca, 'YScale', 'log'</span><span style="color: #DCDCCC;">)</span>;
|
|
xlabel<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'Frequency </span><span style="color: #BFEBBF;">[</span><span style="color: #CC9393;">Hz</span><span style="color: #BFEBBF;">]</span><span style="color: #CC9393;">'</span><span style="color: #DCDCCC;">)</span>;
|
|
title<span style="color: #DCDCCC;">(</span>'$G_<span style="color: #BFEBBF;">{</span>r_u <span style="color: #7CB8BB;">\</span>to d_v<span style="color: #BFEBBF;">}</span>$'<span style="color: #DCDCCC;">)</span>
|
|
|
|
linkaxes<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>ax1,ax2<span style="color: #BFEBBF;">]</span>,<span style="color: #CC9393;">'x'</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgf5812be" class="figure">
|
|
<p><img src="figs/perfconp.png" alt="perfconp.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Close loop performance for \(\omega = 0\) and \(\omega = 60 rpm\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org75899d3" class="outline-3">
|
|
<h3 id="org75899d3"><span class="section-number-3">4.5</span> Plant Control - MIMO approach</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
</div>
|
|
<div id="outline-container-orgf7efb03" class="outline-4">
|
|
<h4 id="orgf7efb03"><span class="section-number-4">4.5.1</span> <span class="todo TODO">TODO</span> Analysis - SVD</h4>
|
|
<div class="outline-text-4" id="text-4-5-1">
|
|
<p>
|
|
\[ G = U \Sigma V^H \]
|
|
</p>
|
|
|
|
<p>
|
|
With:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(\Sigma\) is an \(2 \times 2\) matrix with 2 non-negative <b>singular values</b> \(\sigma_i\), arranged in descending order along its main diagonal</li>
|
|
<li>\(U\) is an \(2 \times 2\) unitary matrix. The columns vectors of \(U\), denoted \(u_i\), represent the <b>output directions</b> of the plant. They are orthonomal</li>
|
|
<li>\(V\) is an \(2 \times 2\) unitary matrix. The columns vectors of \(V\), denoted \(v_i\), represent the <b>input directions</b> of the plant. They are orthonomal</li>
|
|
</ul>
|
|
|
|
<p>
|
|
We first look at the evolution of the singular values as a function of frequency (figure <a href="#fig:G_sigma">fig:G_sigma</a>).
|
|
</p>
|
|
|
|
<div id="org2596e78" class="figure">
|
|
<p><img src="figs/G_sigma.png" alt="G_sigma.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>caption Evolution of the singular values with frequency</p>
|
|
</div>
|
|
|
|
<p>
|
|
We compute
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #DCDCCC;">[</span>U,S,V<span style="color: #DCDCCC;">]</span> = svd<span style="color: #DCDCCC;">(</span>freqresp<span style="color: #BFEBBF;">(</span>Gtvc, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">10</span><span style="color: #BFEBBF;">)</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
U, S, V
|
|
U =
|
|
-0.707101109012986 - 0.00283224868340902i -0.707104254409621 - 0.00189034277692295i
|
|
0.00283224868340845 - 0.707101109012987i -0.00189034277692242 + 0.70710425440962i
|
|
S =
|
|
9.01532756059351e-06 0
|
|
0 6.01714794171208e-06
|
|
V =
|
|
0.707106781186547 + 0i 0.707106781186548 + 0i
|
|
-1.57009245868378e-16 + 0.707106781186548i 1.57009245868377e-16 - 0.707106781186547i
|
|
</pre>
|
|
|
|
<p>
|
|
The input and output directions are related through the singular values
|
|
\[ G v_i = \sigma_i u_i \]
|
|
</p>
|
|
|
|
<p>
|
|
So, if we consider an input in the direction \(v_i\), then the output is in the direction \(u_i\). Furthermore, since \(\normtwo{v_i}=1\) and \(\normtwo{u_i}=1\), we see that <b>the singular value \(\sigma_i\) directly gives the gain of the matrix \(G\) in this direction</b>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">freqresp<span style="color: #DCDCCC;">(</span>Gtvc, <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">10</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>V<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">-6.3747e-06-2.5534e-08i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">2.5534e-08-6.3747e-06i</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">S<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span><span style="color: #7CB8BB;">*</span>U<span style="color: #DCDCCC;">(</span><span style="color: #7CB8BB;">:</span>, <span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">-6.3747e-06-2.5534e-08i</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">2.5534e-08-6.3747e-06i</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgaec63d4" class="outline-4">
|
|
<h4 id="orgaec63d4"><span class="section-number-4">4.5.2</span> Closed loop SVD</h4>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgaddeb0f" class="outline-3">
|
|
<h3 id="orgaddeb0f"><span class="section-number-3">4.6</span> test</h3>
|
|
<div class="outline-text-3" id="text-4-6">
|
|
|
|
<div id="org96711d5" class="figure">
|
|
<p><img src="figs/coupling_simscape_light.png" alt="coupling_simscape_light.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>caption</p>
|
|
</div>
|
|
|
|
<p>
|
|
And then with the heavy sample.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = wheavy;
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
m = mheavy;
|
|
|
|
k = kpz;
|
|
c = <span style="color: #BFEBBF;">1e3</span>;
|
|
Gpz_heavy = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
k = kvc;
|
|
c = <span style="color: #BFEBBF;">1e3</span>;
|
|
Gvc_heavy = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
Gpz_heavy.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gpz_heavy.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gvc_heavy.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gvc_heavy.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org579e187" class="figure">
|
|
<p><img src="figs/coupling_simscape_heavy.png" alt="coupling_simscape_heavy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>caption</p>
|
|
</div>
|
|
|
|
<p>
|
|
Plot the ratio between the main transfer function and the coupling term:
|
|
</p>
|
|
|
|
<div id="orgdbc733d" class="figure">
|
|
<p><img src="figs/coupling_ratio_simscape_light.png" alt="coupling_ratio_simscape_light.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 23: </span>caption</p>
|
|
</div>
|
|
|
|
|
|
<div id="orge505290" class="figure">
|
|
<p><img src="figs/coupling_ratio_simscape_heavy.png" alt="coupling_ratio_simscape_heavy.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 24: </span>caption</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb3b8bc1" class="outline-4">
|
|
<h4 id="orgb3b8bc1"><span class="section-number-4">4.6.1</span> Low rotation speed and High rotation speed</h4>
|
|
<div class="outline-text-4" id="text-4-6-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">60</span>; angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
G_low = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>; angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
G_high = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
G_low.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G_low.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G_high.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G_high.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Du', 'Dv'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>G_low, G_high<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb69d273" class="outline-3">
|
|
<h3 id="orgb69d273"><span class="section-number-3">4.7</span> Identification in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-7">
|
|
<p>
|
|
Let's define some options as well as the inputs and outputs for linearization.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #BFEBBF;">0</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #CC9393;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fx'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fy'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>dx'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>dy'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then define the error estimation of the error and the rotational speed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
rot_speed = <span style="color: #BFEBBF;">0</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Finally, we run the linearization.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output names</span>
|
|
G.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fx', 'Fy'</span><span style="color: #DCDCCC;">}</span>;
|
|
G.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>;
|
|
Gr = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output names</span>
|
|
Gr.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fx', 'Fy'</span><span style="color: #DCDCCC;">}</span>;
|
|
Gr.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #BFEBBF;">1</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span><span style="color: #7CB8BB;">/</span><span style="color: #BFEBBF;">180</span>;
|
|
rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>;
|
|
Ge = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output names</span>
|
|
Ge.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fx', 'Fy'</span><span style="color: #DCDCCC;">}</span>;
|
|
Ge.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>G<span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #7F9F7F;">% exportFig('G_x_y', 'wide-tall');</span>
|
|
|
|
<span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>Ge<span style="color: #DCDCCC;">)</span>;
|
|
<span style="color: #7F9F7F;">% exportFig('G_x_y_e', 'normal-normal');</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org6480b90" class="outline-3">
|
|
<h3 id="org6480b90"><span class="section-number-3">4.8</span> Identification from actuator forces to displacement in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-8">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #BFEBBF;">0</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #CC9393;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #7F9F7F; font-weight: bold; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">1</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fu'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">2</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>fv'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'input'<span style="color: #DCDCCC;">)</span>;
|
|
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">3</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>dx'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
io<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span> = linio<span style="color: #DCDCCC;">(</span><span style="color: #BFEBBF;">[</span>mdl, '<span style="color: #7CB8BB;">/</span>dy'<span style="color: #BFEBBF;">]</span>, <span style="color: #BFEBBF;">1</span>, 'output'<span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>;
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
G = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">0</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
G.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>;
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
G1 = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">4</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
G1.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G1.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #BFEBBF;">2</span><span style="color: #7CB8BB;">*</span><span style="color: #BFEBBF;">pi</span>;
|
|
angle_e = <span style="color: #BFEBBF;">0</span>;
|
|
G2 = linearize<span style="color: #DCDCCC;">(</span>mdl, io, <span style="color: #BFEBBF;">0</span>.<span style="color: #BFEBBF;">8</span><span style="color: #DCDCCC;">)</span>;
|
|
|
|
G2.InputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Fu', 'Fv'</span><span style="color: #DCDCCC;">}</span>;
|
|
G2.OutputName = <span style="color: #DCDCCC;">{</span><span style="color: #CC9393;">'Dx', 'Dy'</span><span style="color: #DCDCCC;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #7CB8BB;">figure</span>;
|
|
bode<span style="color: #DCDCCC;">(</span>G, G1, G2<span style="color: #DCDCCC;">)</span>;
|
|
exportFig<span style="color: #DCDCCC;">(</span><span style="color: #CC9393;">'G_u_v_to_x_y', 'wide-tall'</span><span style="color: #DCDCCC;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2ce7065" class="outline-3">
|
|
<h3 id="org2ce7065"><span class="section-number-3">4.9</span> Effect of the X-Y stage stiffness</h3>
|
|
<div class="outline-text-3" id="text-4-9">
|
|
<p>
|
|
<a id="orgb8a13f6"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org2c17f71" class="outline-4">
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<h4 id="org2c17f71"><span class="section-number-4">4.9.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
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</div>
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</div>
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</div>
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<div id="outline-container-orgf6cc550" class="outline-2">
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<h2 id="orgf6cc550"><span class="section-number-2">5</span> Control Implementation</h2>
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<div class="outline-text-2" id="text-5">
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<p>
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<a id="org52e5a2f"></a>
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</p>
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</div>
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<div id="outline-container-orgea3a8d3" class="outline-3">
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<h3 id="orgea3a8d3"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Thomas Dehaeze</p>
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<p class="date">Created: 2019-03-20 mer. 15:33</p>
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<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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</div>
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</body>
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</html>
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