1294 lines
53 KiB
HTML
1294 lines
53 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="#org0a89915">1. Introduction</a></li>
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<li><a href="#org123b9ca">2. System</a>
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<ul>
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<li><a href="#org9aa32bf">2.1. System description</a></li>
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<li><a href="#orgc0c2859">2.2. Equations</a></li>
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<li><a href="#org0ff8b08">2.3. Numerical Values for the NASS</a></li>
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<li><a href="#org361651b">2.4. Euler and Coriolis forces - Numerical Result</a></li>
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<li><a href="#orgd2c08cb">2.5. Negative Spring Effect - Numerical Result</a></li>
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<li><a href="#org3d790d8">2.6. Limitations due to coupling</a>
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<ul>
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<li><a href="#org041128a">2.6.1. Numerical Analysis</a></li>
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</ul>
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</li>
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<li><a href="#org11f0d9f">2.7. Limitations due to negative stiffness effect</a></li>
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</ul>
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</li>
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<li><a href="#org6ae3c88">3. Control Strategies</a>
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<ul>
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<li><a href="#orgbf1698b">3.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#orgf37122a">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="#org3b55a8d">4. Multi Body Model - Simscape</a>
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<ul>
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<li><a href="#orgd50d938">4.1. Identification in the rotating referenced frame</a>
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<ul>
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<li><a href="#org2ede09d">4.1.1. Piezo and Voice coil</a></li>
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<li><a href="#orgbb666ab">4.1.2. 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="#org3ecccf8">4.2. Identification in the fixed frame</a></li>
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<li><a href="#orga260738">4.3. Identification from actuator forces to displacement in the fixed frame</a></li>
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<li><a href="#orge0b7f4f">4.4. Effect of the rotating Speed</a>
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<ul>
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<li><a href="#orgd25182f">4.4.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
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<li><a href="#org0368be9">4.4.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
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</ul>
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</li>
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<li><a href="#org8ddc330">4.5. Effect of the X-Y stage stiffness</a>
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<ul>
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<li><a href="#orgd282a96">4.5.1. <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="#orgdfc37fb">5. Control Implementation</a>
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<ul>
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<li><a href="#org5ed4967">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-org0a89915" class="outline-2">
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<h2 id="org0a89915"><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="#org725c7e8">2</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="#orgc886060">3</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="#org889e9c9">4</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="#orgaa1e82e">5</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-org123b9ca" class="outline-2">
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<h2 id="org123b9ca"><span class="section-number-2">2</span> System</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org725c7e8"></a>
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</p>
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</div>
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<div id="outline-container-org9aa32bf" class="outline-3">
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<h3 id="org9aa32bf"><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="#org492a56d">1</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>
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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.
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</p>
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<div id="org492a56d" class="figure">
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<p><img src="./Figures/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
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</p>
|
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<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
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</div>
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<p>
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In the following block diagram:
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</p>
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<ul class="org-ul">
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<li>\(G\) is the transfer function from the forces applied in the actuators to the measurement</li>
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<li>\(K\) is the controller to design</li>
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<li>\(J\) is a Jacobian matrix usually used to change the reference frame</li>
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</ul>
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<p>
|
|
Indices \(x\) and \(y\) corresponds to signals in the fixed reference frame (along \(\vec{i}_x\) and \(\vec{i}_y\)):
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</p>
|
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<ul class="org-ul">
|
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<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-orgc0c2859" class="outline-3">
|
|
<h3 id="orgc0c2859"><span class="section-number-3">2.2</span> Equations</h3>
|
|
<div class="outline-text-3" id="text-2-2">
|
|
<p>
|
|
<a id="org07b8498"></a>
|
|
Based on the figure <a href="#org492a56d">1</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{org97ab84a}
|
|
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{org5b05ded}
|
|
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{orgf971d6e}
|
|
\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{orge8fa8fd}
|
|
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{orge386db1}
|
|
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-org0ff8b08" class="outline-3">
|
|
<h3 id="org0ff8b08"><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-org361651b" class="outline-3">
|
|
<h3 id="org361651b"><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="#org87d7458">1</a>.
|
|
</p>
|
|
|
|
<table id="org87d7458" 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-orgd2c08cb" class="outline-3">
|
|
<h3 id="orgd2c08cb"><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="#orgda1ded2">2</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="orgda1ded2" 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-org3d790d8" class="outline-3">
|
|
<h3 id="org3d790d8"><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{orge8fa8fd} and \eqref{orge386db1}, 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}
|
|
\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-org041128a" class="outline-4">
|
|
<h4 id="org041128a"><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="#org7303972">2</a>).</li>
|
|
<li>with the heavy sample (figure <a href="#org3386e78">3</a>).</li>
|
|
</ul>
|
|
|
|
|
|
<div id="org7303972" class="figure">
|
|
<p><img src="Figures/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="org3386e78" class="figure">
|
|
<p><img src="Figures/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-org11f0d9f" class="outline-3">
|
|
<h3 id="org11f0d9f"><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="#org3a5730b">3</a>.
|
|
</p>
|
|
<table id="org3a5730b" 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 <a href="#org9464a39">4</a>)
|
|
</p>
|
|
|
|
<table id="org9464a39" 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>
|
|
|
|
<div id="outline-container-org6ae3c88" class="outline-2">
|
|
<h2 id="org6ae3c88"><span class="section-number-2">3</span> Control Strategies</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgc886060"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgbf1698b" class="outline-3">
|
|
<h3 id="orgbf1698b"><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="#org8698ee9">4</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org8698ee9" class="figure">
|
|
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </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-orgf37122a" class="outline-3">
|
|
<h3 id="orgf37122a"><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="#org6f71235">5</a>
|
|
</p>
|
|
|
|
|
|
<div id="org6f71235" class="figure">
|
|
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </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-org3b55a8d" class="outline-2">
|
|
<h2 id="org3b55a8d"><span class="section-number-2">4</span> Multi Body Model - Simscape</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org889e9c9"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd50d938" class="outline-3">
|
|
<h3 id="orgd50d938"><span class="section-number-3">4.1</span> Identification in the rotating referenced frame</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
We initialize the inputs and outputs of the system to identify.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fu'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>du'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2ede09d" class="outline-4">
|
|
<h4 id="org2ede09d"><span class="section-number-4">4.1.1</span> Piezo and Voice coil</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
<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">rot_speed = wlight;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
m = mlight;
|
|
|
|
k = kpz;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gpz_light = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
k = kvc;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gvc_light = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
Gpz_light.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gpz_light.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_light.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_light.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_simscape_light.png" alt="coupling_simscape_light.png" />
|
|
</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: #D0372D;">0</span>;
|
|
m = mheavy;
|
|
|
|
k = kpz;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gpz_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
k = kvc;
|
|
c = <span style="color: #D0372D;">1e3</span>;
|
|
Gvc_heavy = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
Gpz_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gpz_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_heavy.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
Gvc_heavy.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_simscape_heavy.png" alt="coupling_simscape_heavy.png" />
|
|
</p>
|
|
</div>
|
|
|
|
<p>
|
|
Plot the ratio between the main transfer function and the coupling term:
|
|
</p>
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_ration_simscape_light.png" alt="coupling_ration_simscape_light.png" />
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div class="figure">
|
|
<p><img src="Figures/coupling_ration_simscape_heavy.png" alt="coupling_ration_simscape_heavy.png" />
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgbb666ab" class="outline-4">
|
|
<h4 id="orgbb666ab"><span class="section-number-4">4.1.2</span> Low rotation speed and High rotation speed</h4>
|
|
<div class="outline-text-4" id="text-4-1-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; angle_e = <span style="color: #D0372D;">0</span>;
|
|
G_low = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; angle_e = <span style="color: #D0372D;">0</span>;
|
|
G_high = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span><span style="color: #707183;">)</span>;
|
|
|
|
G_low.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G_low.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
G_high.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G_high.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Du', 'Dv'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G_low, G_high<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org3ecccf8" class="outline-3">
|
|
<h3 id="org3ecccf8"><span class="section-number-3">4.2</span> Identification in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<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: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</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: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
rot_speed = <span style="color: #D0372D;">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: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
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G.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
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G.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
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</pre>
|
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">0</span>;
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rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
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|
Gr = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
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|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
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|
Gr.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
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|
Gr.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
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|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Run the linearization</span>
|
|
angle_e = <span style="color: #D0372D;">1</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">180</span>;
|
|
rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
Ge = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output names</span>
|
|
Ge.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fx', 'Fy'</span><span style="color: #707183;">}</span>;
|
|
Ge.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G<span style="color: #707183;">)</span>;
|
|
<span style="color: #8D8D84; font-style: italic;">% exportFig('G_x_y', 'wide-tall');</span>
|
|
|
|
<span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>Ge<span style="color: #707183;">)</span>;
|
|
<span style="color: #8D8D84; font-style: italic;">% exportFig('G_x_y_e', 'normal-normal');</span>
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga260738" class="outline-3">
|
|
<h3 id="orga260738"><span class="section-number-3">4.3</span> Identification from actuator forces to displacement in the fixed frame</h3>
|
|
<div class="outline-text-3" id="text-4-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Options for Linearized</span>
|
|
options = linearizeOptions;
|
|
options.SampleTime = <span style="color: #D0372D;">0</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Name of the Simulink File</span>
|
|
mdl = <span style="color: #008000;">'rotating_frame'</span>;
|
|
|
|
<span style="color: #8D8D84; font-weight: bold; font-style: italic; text-decoration: overline;">%% Input/Output definition</span>
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">1</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fu'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">2</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>fv'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'input'<span style="color: #707183;">)</span>;
|
|
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">3</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dx'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
io<span style="color: #707183;">(</span><span style="color: #D0372D;">4</span><span style="color: #707183;">)</span> = linio<span style="color: #707183;">(</span><span style="color: #7388D6;">[</span>mdl, '<span style="color: #6434A3;">/</span>dy'<span style="color: #7388D6;">]</span>, <span style="color: #D0372D;">1</span>, 'output'<span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">0</span><span style="color: #707183;">)</span>;
|
|
|
|
G.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G1 = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">4</span><span style="color: #707183;">)</span>;
|
|
|
|
G1.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G1.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">rot_speed = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>;
|
|
angle_e = <span style="color: #D0372D;">0</span>;
|
|
G2 = linearize<span style="color: #707183;">(</span>mdl, io, <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">8</span><span style="color: #707183;">)</span>;
|
|
|
|
G2.InputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Fu', 'Fv'</span><span style="color: #707183;">}</span>;
|
|
G2.OutputName = <span style="color: #707183;">{</span><span style="color: #008000;">'Dx', 'Dy'</span><span style="color: #707183;">}</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span style="color: #6434A3;">figure</span>;
|
|
bode<span style="color: #707183;">(</span>G, G1, G2<span style="color: #707183;">)</span>;
|
|
exportFig<span style="color: #707183;">(</span><span style="color: #008000;">'G_u_v_to_x_y', 'wide-tall'</span><span style="color: #707183;">)</span>;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge0b7f4f" class="outline-3">
|
|
<h3 id="orge0b7f4f"><span class="section-number-3">4.4</span> Effect of the rotating Speed</h3>
|
|
<div class="outline-text-3" id="text-4-4">
|
|
<p>
|
|
<a id="orgd1e3b9e"></a>
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd25182f" class="outline-4">
|
|
<h4 id="orgd25182f"><span class="section-number-4">4.4.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h4>
|
|
</div>
|
|
<div id="outline-container-org0368be9" class="outline-4">
|
|
<h4 id="org0368be9"><span class="section-number-4">4.4.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h4>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-org8ddc330" class="outline-3">
|
|
<h3 id="org8ddc330"><span class="section-number-3">4.5</span> Effect of the X-Y stage stiffness</h3>
|
|
<div class="outline-text-3" id="text-4-5">
|
|
<p>
|
|
<a id="org072e1f4"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgd282a96" class="outline-4">
|
|
<h4 id="orgd282a96"><span class="section-number-4">4.5.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h4>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgdfc37fb" class="outline-2">
|
|
<h2 id="orgdfc37fb"><span class="section-number-2">5</span> Control Implementation</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="orgaa1e82e"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org5ed4967" class="outline-3">
|
|
<h3 id="org5ed4967"><span class="section-number-3">5.1</span> Measurement in the fixed reference frame</h3>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Thomas Dehaeze</p>
|
|
<p class="date">Created: 2019-01-21 lun. 23:44</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|