749 lines
27 KiB
HTML
749 lines
27 KiB
HTML
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</head>
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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="#org8bd71db">1. Goal of this note</a></li>
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<li><a href="#org6b844c0">2. System</a>
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<ul>
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<li><a href="#org0e477c9">2.1. System description</a></li>
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<li><a href="#org9d3f997">2.2. Equations</a></li>
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<li><a href="#org813d43d">2.3. Analysis</a></li>
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</ul>
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</li>
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<li><a href="#orge651676">3. Analytical Computation of forces for the NASS</a>
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<ul>
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<li><a href="#org94f7739">3.1. Euler and Coriolis forces</a></li>
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<li><a href="#orgd457827">3.2. Spring Softening Effect</a></li>
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</ul>
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</li>
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<li><a href="#org42269b2">4. Control Strategies</a>
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<ul>
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<li><a href="#org05e6b53">4.1. Measurement in the fixed reference frame</a>
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<ul>
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<li><a href="#org8c638be">4.1.1. <span class="todo QUESTION">QUESTION</span> Is the loop gain is changing with the angle ?</a></li>
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</ul>
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</li>
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<li><a href="#orgb6cb87b">4.2. Measurement in the rotating frame</a></li>
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</ul>
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</li>
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<li><a href="#org502caa6">5. Effect of the rotating Speed</a>
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<ul>
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<li><a href="#org2770fda">5.1. <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="#orgc76e417">5.2. <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="#org07c8778">6. Effect of the X-Y stage stiffness</a>
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<ul>
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<li><a href="#org038f2b3">6.1. <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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</div>
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</div>
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<div id="outline-container-org8bd71db" class="outline-2">
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<h2 id="org8bd71db"><span class="section-number-2">1</span> Goal of this note</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
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</p>
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<p>
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The actuators are also rotating with the object.
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</p>
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<p>
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We want to compare the two different approach:
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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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</div>
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</div>
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<div id="outline-container-org6b844c0" class="outline-2">
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<h2 id="org6b844c0"><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="orgc894f87"></a>
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</p>
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</div>
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<div id="outline-container-org0e477c9" class="outline-3">
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<h3 id="org0e477c9"><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="#orgeadaa88">1</a>).
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</p>
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<p>
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The control inputs are the forces applied in the translation stage (\(F_u\) and \(F_v\)). 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 actuators.
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</p>
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|
|
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<div id="orgeadaa88" 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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|
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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>
|
|
<ul class="org-ul">
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|
<li>\(D_x\) is the measured position of the sample</li>
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|
<li>\(r_x\) is the reference signal which corresponds to the wanted \(D_x\)</li>
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|
<li>\(\epsilon_x\) is the position error</li>
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</ul>
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|
|
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<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">
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|
<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>
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</ul>
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</div>
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</div>
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<div id="outline-container-org9d3f997" class="outline-3">
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<h3 id="org9d3f997"><span class="section-number-3">2.2</span> Equations</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
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<a id="org52a8ecb"></a>
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</p>
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|
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<p>
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|
Based on the figure <a href="#orgeadaa88">1</a>, we can write the equations of motion of the system.
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</p>
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|
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<p>
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|
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
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</p>
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|
\begin{align}
|
|
\label{org25e6b9c}
|
|
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
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|
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
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\end{align}
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<p>
|
|
The Lagrangian is the kinetic energy minus the potential energy.
|
|
</p>
|
|
\begin{equation}
|
|
\label{orgb6862e5}
|
|
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{org02c8a0d}
|
|
\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 equation \eqref{orgc6b5c7e}.
|
|
</p>
|
|
\begin{align}
|
|
\label{orgc6b5c7e}
|
|
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 \eqref{orgc6b5c7e}, 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{align*}
|
|
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} \\
|
|
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{align*}
|
|
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org813d43d" class="outline-3">
|
|
<h3 id="org813d43d"><span class="section-number-3">2.3</span> Analysis</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<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 \(k-d_u m\dot{\theta}^2\).
|
|
Thus, the term \(-d_u m\dot{\theta}^2\) acts like a negative stiffness (due to <b>centrifugal forces</b>).
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orge651676" class="outline-2">
|
|
<h2 id="orge651676"><span class="section-number-2">3</span> Analytical Computation of forces for the NASS</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
For the NASS, the Euler forces should be less of a problem as \(\ddot{\theta}\) should be very small when conducting an experiment.
|
|
</p>
|
|
|
|
<p>
|
|
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org94f7739" class="outline-3">
|
|
<h3 id="org94f7739"><span class="section-number-3">3.1</span> Euler and Coriolis forces</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
Let's define the parameters for the NASS.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">mlight = <span style="color: #D0372D;">35</span>; <span style="color: #8D8D84; font-style: italic;">% [kg]</span>
|
|
mheavy = <span style="color: #D0372D;">85</span>; <span style="color: #8D8D84; font-style: italic;">% [kg]</span>
|
|
|
|
wlight = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
|
wheavy = <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>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
|
|
|
wdot = <span style="color: #D0372D;">1</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s2]</span>
|
|
|
|
d = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span>; <span style="color: #8D8D84; font-style: italic;">% [m]</span>
|
|
ddot = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">2</span>; <span style="color: #8D8D84; font-style: italic;">% [m/s]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table <a href="#org11a5df8">1</a>.
|
|
</p>
|
|
|
|
<table id="org11a5df8" 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">44.0 N</td>
|
|
<td class="org-left">1.8 N</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Euler</td>
|
|
<td class="org-left">3.5 N</td>
|
|
<td class="org-left">8.5 N</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd457827" class="outline-3">
|
|
<h3 id="orgd457827"><span class="section-number-3">3.2</span> Spring Softening Effect</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The values for the spring softening effect are displayed in table <a href="#org76af8f7">2</a>.
|
|
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
|
</p>
|
|
|
|
<table id="org76af8f7" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> Spring Softening 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">Spring Soft.</td>
|
|
<td class="org-left">3.5 N/m</td>
|
|
<td class="org-left">8.5 N/m</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org42269b2" class="outline-2">
|
|
<h2 id="org42269b2"><span class="section-number-2">4</span> Control Strategies</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org25db234"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org05e6b53" class="outline-3">
|
|
<h3 id="org05e6b53"><span class="section-number-3">4.1</span> Measurement in the fixed reference frame</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
<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="#orgbae56a5">2</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgbae56a5" class="figure">
|
|
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 2: </span>Control with a measure from fixed frame</p>
|
|
</div>
|
|
|
|
<p>
|
|
The loop gain is then \(L = G(\theta) K J(\theta)\).
|
|
</p>
|
|
</div>
|
|
|
|
<div id="outline-container-org8c638be" class="outline-4">
|
|
<h4 id="org8c638be"><span class="section-number-4">4.1.1</span> <span class="todo QUESTION">QUESTION</span> Is the loop gain is changing with the angle ?</h4>
|
|
<div class="outline-text-4" id="text-4-1-1">
|
|
<p>
|
|
Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgb6cb87b" class="outline-3">
|
|
<h3 id="orgb6cb87b"><span class="section-number-3">4.2</span> Measurement in the rotating frame</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<p>
|
|
Let's consider that the measurement is in the rotating reference frame.
|
|
</p>
|
|
|
|
<p>
|
|
The corresponding block diagram is shown figure <a href="#org33df600">3</a>
|
|
</p>
|
|
|
|
|
|
<div id="org33df600" class="figure">
|
|
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 3: </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-org502caa6" class="outline-2">
|
|
<h2 id="org502caa6"><span class="section-number-2">5</span> Effect of the rotating Speed</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org389e858"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org2770fda" class="outline-3">
|
|
<h3 id="org2770fda"><span class="section-number-3">5.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h3>
|
|
</div>
|
|
|
|
<div id="outline-container-orgc76e417" class="outline-3">
|
|
<h3 id="orgc76e417"><span class="section-number-3">5.2</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)</h3>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org07c8778" class="outline-2">
|
|
<h2 id="org07c8778"><span class="section-number-2">6</span> Effect of the X-Y stage stiffness</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="org99e68c1"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-org038f2b3" class="outline-3">
|
|
<h3 id="org038f2b3"><span class="section-number-3">6.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h3>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div id="postamble" class="status">
|
|
<p class="author">Author: Thomas Dehaeze</p>
|
|
<p class="date">Created: 2019-01-18 ven. 17:15</p>
|
|
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|