Add some analysis
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2019-01-18 ven. 17:15 -->
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<!-- 2019-01-18 ven. 17:46 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Control in a rotating frame</title>
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@ -275,47 +275,49 @@ for the JavaScript code in this tag.
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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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<li><a href="#org892e46e">1. Goal of this note</a></li>
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<li><a href="#orgbb4d730">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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<li><a href="#orgf6286ea">2.1. System description</a></li>
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<li><a href="#org6517c3a">2.2. Equations</a></li>
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<li><a href="#org66b66d3">2.3. <span class="todo TODO">TODO</span> Analysis</a>
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<ul>
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<li><a href="#org1aee292">2.3.1. Stiff actuators</a></li>
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<li><a href="#org3d277ca">2.3.2. Negative Stiffness</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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<li><a href="#org1d7bfef">3. Analytical Computation of forces for the NASS</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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<li><a href="#org9862d4d">3.1. Parameters</a></li>
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<li><a href="#orged72531">3.2. Euler and Coriolis forces</a></li>
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<li><a href="#org1ad22a2">3.3. Negative Spring Effect</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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<li><a href="#orgb20d1e2">4. Control Strategies</a>
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<ul>
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<li><a href="#org67681a1">4.1. Measurement in the fixed reference frame</a></li>
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<li><a href="#org358433f">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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<li><a href="#org403dcc8">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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<li><a href="#orga27aa6d">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="#org5b37262">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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<li><a href="#org3eb9f54">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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<li><a href="#org9fbd479">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 id="outline-container-org892e46e" class="outline-2">
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<h2 id="org892e46e"><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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@ -335,18 +337,18 @@ We want to compare the two different approach:
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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 id="outline-container-orgbb4d730" class="outline-2">
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<h2 id="orgbb4d730"><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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<a id="orgfb8b8b0"></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 id="outline-container-orgf6286ea" class="outline-3">
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<h3 id="orgf6286ea"><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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The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#org6527df8">1</a>).
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</p>
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<p>
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@ -358,7 +360,7 @@ The measurement is either the \(x-y\) displacement of the object located on top
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</p>
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<div id="orgeadaa88" class="figure">
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<div id="org6527df8" 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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@ -392,22 +394,22 @@ Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (
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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 id="outline-container-org6517c3a" class="outline-3">
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<h3 id="org6517c3a"><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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<a id="orga34f88d"></a>
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</p>
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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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Based on the figure <a href="#org6527df8">1</a>, we can write the equations of motion of the system.
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</p>
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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}
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\label{org25e6b9c}
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\label{org4d9790f}
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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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@ -416,7 +418,7 @@ V & = \frac{1}{2} k \left( x^2 + y^2 \right)
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The Lagrangian is the kinetic energy minus the potential energy.
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</p>
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\begin{equation}
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\label{orgb6862e5}
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\label{orgb67b4fc}
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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)
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\end{equation}
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@ -425,7 +427,7 @@ L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \le
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The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
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</p>
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\begin{align*}
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\label{org02c8a0d}
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\label{orgcf126c0}
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\frac{\partial L}{\partial x} & = -kx \\
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\frac{\partial L}{\partial y} & = -ky \\
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\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
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@ -441,10 +443,9 @@ F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
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\end{align*}
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<p>
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By appling the Lagrangian equations, we obtain equation \eqref{orgc6b5c7e}.
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By appling the Lagrangian equations, we obtain:
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</p>
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\begin{align}
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\label{orgc6b5c7e}
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m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
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m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
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\end{align}
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@ -468,7 +469,7 @@ We obtain:
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\end{align*}
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<p>
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By injecting the previous result into the Lagrangian equation \eqref{orgc6b5c7e}, we obtain:
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By injecting the previous result into the Lagrangian equation, we obtain:
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</p>
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\begin{align*}
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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}
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@ -504,8 +505,8 @@ We can then subtract and add the previous equations to obtain the following equa
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</div>
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</div>
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<div id="outline-container-org813d43d" class="outline-3">
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<h3 id="org813d43d"><span class="section-number-3">2.3</span> Analysis</h3>
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<div id="outline-container-org66b66d3" class="outline-3">
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<h3 id="org66b66d3"><span class="section-number-3">2.3</span> <span class="todo TODO">TODO</span> Analysis</h3>
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<div class="outline-text-3" id="text-2-3">
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<p>
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We obtain two differential equations that are coupled through:
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@ -520,23 +521,46 @@ Without the coupling terms, each equation is the equation of a one degree of fre
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Thus, the term \(-d_u m\dot{\theta}^2\) acts like a negative stiffness (due to <b>centrifugal forces</b>).
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</p>
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</div>
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<div id="outline-container-org1aee292" class="outline-4">
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<h4 id="org1aee292"><span class="section-number-4">2.3.1</span> Stiff actuators</h4>
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<div class="outline-text-4" id="text-2-3-1">
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<p>
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Let's say we use stiff actuators such that \(m \ddot{d_u} + (k - m\dot{\theta}^2) d_u \approx k d_u\).
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</p>
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<p>
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Let's suppose that \(F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \approx F_u\).
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</p>
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<p>
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Then we obtain \(d_u = \frac{F_u}{k}\) that we can re inject in the other equation to obtain:
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\[ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\frac{\dot{F_u}}{k}\dot{\theta} - m \frac{F_u}{k}\ddot{\theta} \]
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</p>
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</div>
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</div>
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<div id="outline-container-orge651676" class="outline-2">
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<h2 id="orge651676"><span class="section-number-2">3</span> Analytical Computation of forces for the NASS</h2>
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<div id="outline-container-org3d277ca" class="outline-4">
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<h4 id="org3d277ca"><span class="section-number-4">2.3.2</span> Negative Stiffness</h4>
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<div class="outline-text-4" id="text-2-3-2">
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<p>
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If \(\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}\), then the negative spring effect is negligible and \(k - m\dot{\theta}^2 \approx k\).
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</p>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org1d7bfef" class="outline-2">
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<h2 id="org1d7bfef"><span class="section-number-2">3</span> Analytical Computation of forces for the NASS</h2>
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<div class="outline-text-2" id="text-3">
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<p>
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For the NASS, the Euler forces should be less of a problem as \(\ddot{\theta}\) should be very small when conducting an experiment.
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</p>
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<p>
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First we will determine the value for Euler and Coriolis forces during regular experiment.
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</p>
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</div>
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<div id="outline-container-org94f7739" class="outline-3">
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<h3 id="org94f7739"><span class="section-number-3">3.1</span> Euler and Coriolis forces</h3>
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<div id="outline-container-org9862d4d" class="outline-3">
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<h3 id="org9862d4d"><span class="section-number-3">3.1</span> Parameters</h3>
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<div class="outline-text-3" id="text-3-1">
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<p>
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Let's define the parameters for the NASS.
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@ -554,12 +578,21 @@ d = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span
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ddot = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">2</span>; <span style="color: #8D8D84; font-style: italic;">% [m/s]</span>
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-orged72531" class="outline-3">
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<h3 id="orged72531"><span class="section-number-3">3.2</span> Euler and Coriolis forces</h3>
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<div class="outline-text-3" id="text-3-2">
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<p>
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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>.
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First we will determine the value for Euler and Coriolis forces during regular experiment.
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</p>
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<table id="org11a5df8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<p>
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We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table <a href="#orgae713d9">1</a>.
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</p>
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<table id="orgae713d9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
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<colgroup>
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@ -593,16 +626,16 @@ We then compute the corresponding values of the Coriolis and Euler forces, and t
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</div>
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</div>
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<div id="outline-container-orgd457827" class="outline-3">
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<h3 id="orgd457827"><span class="section-number-3">3.2</span> Spring Softening Effect</h3>
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<div class="outline-text-3" id="text-3-2">
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<div id="outline-container-org1ad22a2" class="outline-3">
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<h3 id="org1ad22a2"><span class="section-number-3">3.3</span> Negative Spring Effect</h3>
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<div class="outline-text-3" id="text-3-3">
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<p>
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The values for the spring softening effect are displayed in table <a href="#org76af8f7">2</a>.
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The values for the negative spring effect are displayed in table <a href="#org7244d2d">2</a>.
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This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
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</p>
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<table id="org76af8f7" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Spring Softening effect</caption>
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<table id="org7244d2d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Negative Spring effect</caption>
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<colgroup>
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<col class="org-left" />
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@ -620,7 +653,7 @@ This is definitely negligible when using piezoelectric actuators. It may not be
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</thead>
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<tbody>
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<tr>
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<td class="org-left">Spring Soft.</td>
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<td class="org-left">Neg. Spring</td>
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<td class="org-left">3.5 N/m</td>
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<td class="org-left">8.5 N/m</td>
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</tr>
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@ -630,15 +663,15 @@ This is definitely negligible when using piezoelectric actuators. It may not be
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</div>
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</div>
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<div id="outline-container-org42269b2" class="outline-2">
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<h2 id="org42269b2"><span class="section-number-2">4</span> Control Strategies</h2>
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<div id="outline-container-orgb20d1e2" class="outline-2">
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<h2 id="orgb20d1e2"><span class="section-number-2">4</span> Control Strategies</h2>
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<div class="outline-text-2" id="text-4">
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<p>
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<a id="org25db234"></a>
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<a id="orgec63a1f"></a>
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</p>
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</div>
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<div id="outline-container-org05e6b53" class="outline-3">
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<h3 id="org05e6b53"><span class="section-number-3">4.1</span> Measurement in the fixed reference frame</h3>
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<div id="outline-container-org67681a1" class="outline-3">
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<h3 id="org67681a1"><span class="section-number-3">4.1</span> Measurement in the fixed reference frame</h3>
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<div class="outline-text-3" id="text-4-1">
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<p>
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First, let's consider a measurement in the fixed referenced frame.
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@ -661,11 +694,11 @@ Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon
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</p>
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<p>
|
||||
The block diagram is shown on figure <a href="#orgbae56a5">2</a>.
|
||||
The block diagram is shown on figure <a href="#org4869ac5">2</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbae56a5" class="figure">
|
||||
<div id="org4869ac5" 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>
|
||||
@ -674,31 +707,26 @@ The block diagram is shown on figure <a href="#orgbae56a5">2</a>.
|
||||
<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 \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
|
||||
One question we wish to answer is: 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 id="outline-container-org358433f" class="outline-3">
|
||||
<h3 id="org358433f"><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.
|
||||
Let's consider that the measurement is made in the rotating reference frame.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The corresponding block diagram is shown figure <a href="#org33df600">3</a>
|
||||
The corresponding block diagram is shown figure <a href="#org781b9ae">3</a>
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org33df600" class="figure">
|
||||
<div id="org781b9ae" 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>
|
||||
@ -711,37 +739,38 @@ The loop gain is \(L = G K\).
|
||||
</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 id="outline-container-org403dcc8" class="outline-2">
|
||||
<h2 id="org403dcc8"><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>
|
||||
<a id="org6624b66"></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 id="outline-container-orga27aa6d" class="outline-3">
|
||||
<h3 id="orga27aa6d"><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 id="outline-container-org5b37262" class="outline-3">
|
||||
<h3 id="org5b37262"><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 id="outline-container-org3eb9f54" class="outline-2">
|
||||
<h2 id="org3eb9f54"><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>
|
||||
<a id="org8208f86"></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 id="outline-container-org9fbd479" class="outline-3">
|
||||
<h3 id="org9fbd479"><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="date">Created: 2019-01-18 ven. 17:46</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -109,7 +109,7 @@ We obtain:
|
||||
+ \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*}
|
||||
|
||||
By injecting the previous result into the Lagrangian equation [[eq:lagrangian_eq_inertial]], we obtain:
|
||||
By injecting the previous result into the Lagrangian equation, we obtain:
|
||||
\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}
|
||||
@ -137,7 +137,7 @@ We can then subtract and add the previous equations to obtain the following equa
|
||||
\end{align*}
|
||||
#+end_important
|
||||
|
||||
** Analysis
|
||||
** TODO Analysis
|
||||
We obtain two differential equations that are coupled through:
|
||||
- *Euler forces*: $m d_v \ddot{\theta}$
|
||||
- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
|
||||
@ -145,12 +145,24 @@ We obtain two differential equations that are coupled through:
|
||||
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 *centrifugal forces*).
|
||||
|
||||
*** Stiff actuators
|
||||
Let's say we use stiff actuators such that $m \ddot{d_u} + (k - m\dot{\theta}^2) d_u \approx k d_u$.
|
||||
|
||||
Let's suppose that $F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \approx F_u$.
|
||||
|
||||
Then we obtain $d_u = \frac{F_u}{k}$ that we can re inject in the other equation to obtain:
|
||||
\[ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\frac{\dot{F_u}}{k}\dot{\theta} - m \frac{F_u}{k}\ddot{\theta} \]
|
||||
|
||||
*** Negative Stiffness
|
||||
If $\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}$, then the negative spring effect is negligible and $k - m\dot{\theta}^2 \approx k$.
|
||||
|
||||
* Analytical Computation of forces for the NASS
|
||||
For the NASS, the Euler forces should be less of a problem as $\ddot{\theta}$ should be very small when conducting an experiment.
|
||||
|
||||
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
||||
|
||||
** Euler and Coriolis forces
|
||||
** Parameters
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
Let's define the parameters for the NASS.
|
||||
#+begin_src matlab :exports code :results silent
|
||||
@ -166,6 +178,9 @@ Let's define the parameters for the NASS.
|
||||
ddot = 0.2; % [m/s]
|
||||
#+end_src
|
||||
|
||||
** Euler and Coriolis forces
|
||||
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
||||
|
||||
#+begin_src matlab :exports none :results silent
|
||||
Felight = mlight*d*wdot;
|
||||
Feheavy = mheavy*d*wdot;
|
||||
@ -188,25 +203,25 @@ We then compute the corresponding values of the Coriolis and Euler forces, and t
|
||||
| Coriolis | 44.0 N | 1.8 N |
|
||||
| Euler | 3.5 N | 8.5 N |
|
||||
|
||||
** Spring Softening Effect
|
||||
** Negative Spring Effect
|
||||
#+begin_src matlab :exports none :results silent
|
||||
Klight = mlight*d*wdot^2;
|
||||
Kheavy = mheavy*d*wdot^2;
|
||||
#+end_src
|
||||
|
||||
The values for the spring softening effect are displayed in table [[tab:spring_softening]].
|
||||
The values for the negative spring effect are displayed in table [[tab:negative_spring]].
|
||||
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
||||
|
||||
#+begin_src matlab :results value table :exports results :post addhdr(*this*)
|
||||
ans = sprintf(' | Light | Heavy | \n Spring Soft. | %.1f N/m | %.1f N/m', Klight, Kheavy)
|
||||
ans = sprintf(' | Light | Heavy | \n Neg. Spring | %.1f N/m | %.1f N/m', Klight, Kheavy)
|
||||
#+end_src
|
||||
|
||||
#+NAME: tab:spring_softening
|
||||
#+CAPTION: Spring Softening effect
|
||||
#+NAME: tab:negative_spring
|
||||
#+CAPTION: Negative Spring effect
|
||||
#+RESULTS:
|
||||
| | Light | Heavy |
|
||||
|--------------+---------+---------|
|
||||
| Spring Soft. | 3.5 N/m | 8.5 N/m |
|
||||
|-------------+---------+---------|
|
||||
| Neg. Spring | 3.5 N/m | 8.5 N/m |
|
||||
|
||||
* Control Strategies
|
||||
<<sec:control_strategies>>
|
||||
@ -229,11 +244,10 @@ The block diagram is shown on figure [[fig:control_measure_fixed_2dof]].
|
||||
|
||||
The loop gain is then $L = G(\theta) K J(\theta)$.
|
||||
|
||||
*** QUESTION Is the loop gain is changing with the angle ?
|
||||
Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
|
||||
One question we wish to answer is: is $G(\theta) J(\theta) = G(\theta_0) J(\theta_0)$?
|
||||
|
||||
** Measurement in the rotating frame
|
||||
Let's consider that the measurement is in the rotating reference frame.
|
||||
Let's consider that the measurement is made in the rotating reference frame.
|
||||
|
||||
The corresponding block diagram is shown figure [[fig:control_measure_rotating_2dof]]
|
||||
|
||||
@ -245,6 +259,11 @@ The loop gain is $L = G K$.
|
||||
|
||||
* Effect of the rotating Speed
|
||||
<<sec:effect_rot_speed>>
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
|
||||
|
||||
** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
|
||||
@ -262,7 +281,14 @@ The loop gain is $L = G K$.
|
||||
#+NAME: matlab-init
|
||||
#+BEGIN_SRC matlab :results none :exports none
|
||||
clear; close all; clc;
|
||||
|
||||
%% Add path with some functions
|
||||
addpath('./src/');
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = tf('s');
|
||||
|
||||
%% Initialize ans with org-babel
|
||||
ans = 0;
|
||||
#+END_SRC
|
||||
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user