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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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||||
<h1 class="title">Stewart Platform - Decentralized Active Damping</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="#orgd59c804">Inertial Control</a>
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<ul>
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<li><a href="#org5f749c8">Identification of the Dynamics</a></li>
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<li><a href="#org543be7a">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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<li><a href="#org9a605b4">Obtained Damping</a></li>
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<li><a href="#org42a74ed">Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org74c7eb4">Integral Force Feedback</a>
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<ul>
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<li><a href="#orgc96f772">Identification of the Dynamics with perfect Joints</a></li>
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<li><a href="#orgd119d8a">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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<li><a href="#org2b5e45a">Obtained Damping</a></li>
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<li><a href="#org39ddf1e">Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org08917d6">Direct Velocity Feedback</a>
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<ul>
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<li><a href="#org243b924">Identification of the Dynamics with perfect Joints</a></li>
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||||
<li><a href="#orgcdb3ee5">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
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||||
<li><a href="#orgff0cbf9">Obtained Damping</a></li>
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<li><a href="#org4027234">Conclusion</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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<p>
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The following decentralized active damping techniques are briefly studied:
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</p>
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<ul class="org-ul">
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<li>Inertial Control (proportional feedback of the absolute velocity): Section <a href="#orgeb37c7d">No description for this link</a></li>
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||||
<li>Integral Force Feedback: Section <a href="#orgab5e6b5">No description for this link</a></li>
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||||
<li>Direct feedback of the relative velocity of each strut: Section <a href="#org0aa816a">No description for this link</a></li>
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</ul>
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||||
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||||
<div id="outline-container-orgd59c804" class="outline-2">
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<h2 id="orgd59c804">Inertial Control</h2>
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||||
<div class="outline-text-2" id="text-orgd59c804">
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||||
<p>
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||||
<a id="orgeb37c7d"></a>
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||||
</p>
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||||
</div>
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||||
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||||
<div id="outline-container-org5f749c8" class="outline-3">
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||||
<h3 id="org5f749c8">Identification of the Dynamics</h3>
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<div class="outline-text-3" id="text-org5f749c8">
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||||
<div class="org-src-container">
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||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
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||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
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stewart = generateGeneralConfiguration(stewart);
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||||
stewart = computeJointsPose(stewart);
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||||
stewart = initializeStrutDynamics(stewart);
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||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
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||||
stewart = initializeCylindricalPlatforms(stewart);
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||||
stewart = initializeCylindricalStruts(stewart);
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||||
stewart = computeJacobian(stewart);
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||||
stewart = initializeStewartPose(stewart);
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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 class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
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||||
options = linearizeOptions;
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options.SampleTime = 0;
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
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||||
mdl = <span class="org-string">'stewart_active_damping'</span>;
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||||
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||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
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||||
clear io; io_i = 1;
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||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
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||||
io(io_i) = linio([mdl, <span class="org-string">'/Vm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute velocity of each leg [m/s]</span>
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<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
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||||
G = linearize(mdl, io, options);
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||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
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||||
G.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
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||||
</pre>
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||||
</div>
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||||
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||||
<p>
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||||
The transfer function from actuator forces to force sensors is shown in Figure <a href="#org834d990">1</a>.
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||||
</p>
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||||
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||||
<div id="org834d990" class="figure">
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||||
<p><img src="figs/inertial_plant_coupling.png" alt="inertial_plant_coupling.png" />
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||||
</p>
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||||
<p><span class="figure-number">Figure 1: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity of the same leg \(v_{m,i}\) and to the absolute velocity of the other legs \(v_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/inertial_plant_coupling.png">png</a>, <a href="./figs/inertial_plant_coupling.pdf">pdf</a>)</p>
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||||
</div>
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||||
</div>
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||||
</div>
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|
||||
<div id="outline-container-org543be7a" class="outline-3">
|
||||
<h3 id="org543be7a">Effect of the Flexible Joint stiffness on the Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-org543be7a">
|
||||
<p>
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
Gf.OutputName = {<span class="org-string">'Vm1'</span>, <span class="org-string">'Vm2'</span>, <span class="org-string">'Vm3'</span>, <span class="org-string">'Vm4'</span>, <span class="org-string">'Vm5'</span>, <span class="org-string">'Vm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The new dynamics from force actuator to force sensor is shown in Figure <a href="#org683c779">2</a>.
|
||||
</p>
|
||||
|
||||
<div id="org683c779" class="figure">
|
||||
<p><img src="figs/inertial_plant_flexible_joint_decentralized.png" alt="inertial_plant_flexible_joint_decentralized.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Transfer function from the Actuator force \(F_{i}\) to the absolute velocity sensor \(v_{m,i}\) (<a href="./figs/inertial_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/inertial_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9a605b4" class="outline-3">
|
||||
<h3 id="org9a605b4">Obtained Damping</h3>
|
||||
<div class="outline-text-3" id="text-org9a605b4">
|
||||
<p>
|
||||
The control is a performed in a decentralized manner.
|
||||
The \(6 \times 6\) control is a diagonal matrix with pure proportional action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
1 & & 0 \\
|
||||
& \ddots & \\
|
||||
0 & & 1
|
||||
\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The root locus is shown in figure <a href="#org9af9e33">3</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org4e6b73b">4</a>.
|
||||
</p>
|
||||
|
||||
<div id="org9af9e33" class="figure">
|
||||
<p><img src="figs/root_locus_inertial_rot_stiffness.png" alt="root_locus_inertial_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/root_locus_inertial_rot_stiffness.png">png</a>, <a href="./figs/root_locus_inertial_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org4e6b73b" class="figure">
|
||||
<p><img src="figs/pole_damping_gain_inertial_rot_stiffness.png" alt="pole_damping_gain_inertial_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_inertial_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_inertial_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org42a74ed" class="outline-3">
|
||||
<h3 id="org42a74ed">Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-org42a74ed">
|
||||
<div class="important">
|
||||
<p>
|
||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74c7eb4" class="outline-2">
|
||||
<h2 id="org74c7eb4">Integral Force Feedback</h2>
|
||||
<div class="outline-text-2" id="text-org74c7eb4">
|
||||
<p>
|
||||
<a id="orgab5e6b5"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc96f772" class="outline-3">
|
||||
<h3 id="orgc96f772">Identification of the Dynamics with perfect Joints</h3>
|
||||
<div class="outline-text-3" id="text-orgc96f772">
|
||||
<p>
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeAmplifiedStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_active_damping'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Fm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Force Sensor Outputs [N]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The transfer function from actuator forces to force sensors is shown in Figure <a href="#org3fca9dd">5</a>.
|
||||
</p>
|
||||
|
||||
<div id="org3fca9dd" class="figure">
|
||||
<p><img src="figs/iff_plant_coupling.png" alt="iff_plant_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Transfer function from the Actuator force \(F_{i}\) to the Force sensor of the same leg \(F_{m,i}\) and to the force sensor of the other legs \(F_{m,j}\) with \(i \neq j\) in grey (<a href="./figs/iff_plant_coupling.png">png</a>, <a href="./figs/iff_plant_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd119d8a" class="outline-3">
|
||||
<h3 id="orgd119d8a">Effect of the Flexible Joint stiffness on the Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-orgd119d8a">
|
||||
<p>
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
Gf.OutputName = {<span class="org-string">'Fm1'</span>, <span class="org-string">'Fm2'</span>, <span class="org-string">'Fm3'</span>, <span class="org-string">'Fm4'</span>, <span class="org-string">'Fm5'</span>, <span class="org-string">'Fm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The new dynamics from force actuator to force sensor is shown in Figure <a href="#org090868b">6</a>.
|
||||
</p>
|
||||
|
||||
<div id="org090868b" class="figure">
|
||||
<p><img src="figs/iff_plant_flexible_joint_decentralized.png" alt="iff_plant_flexible_joint_decentralized.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Transfer function from the Actuator force \(F_{i}\) to the force sensor \(F_{m,i}\) (<a href="./figs/iff_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/iff_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b5e45a" class="outline-3">
|
||||
<h3 id="org2b5e45a">Obtained Damping</h3>
|
||||
<div class="outline-text-3" id="text-org2b5e45a">
|
||||
<p>
|
||||
The control is a performed in a decentralized manner.
|
||||
The \(6 \times 6\) control is a diagonal matrix with pure integration action on the diagonal:
|
||||
\[ K(s) = g
|
||||
\begin{bmatrix}
|
||||
\frac{1}{s} & & 0 \\
|
||||
& \ddots & \\
|
||||
0 & & \frac{1}{s}
|
||||
\end{bmatrix} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The root locus is shown in figure <a href="#orge21bbea">7</a> and the obtained pole damping function of the control gain is shown in figure <a href="#org94d6943">8</a>.
|
||||
</p>
|
||||
|
||||
<div id="orge21bbea" class="figure">
|
||||
<p><img src="figs/root_locus_iff_rot_stiffness.png" alt="root_locus_iff_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/root_locus_iff_rot_stiffness.png">png</a>, <a href="./figs/root_locus_iff_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org94d6943" class="figure">
|
||||
<p><img src="figs/pole_damping_gain_iff_rot_stiffness.png" alt="pole_damping_gain_iff_rot_stiffness.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints (<a href="./figs/pole_damping_gain_iff_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_iff_rot_stiffness.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org39ddf1e" class="outline-3">
|
||||
<h3 id="org39ddf1e">Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-org39ddf1e">
|
||||
<div class="important">
|
||||
<p>
|
||||
The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
|
||||
Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org08917d6" class="outline-2">
|
||||
<h2 id="org08917d6">Direct Velocity Feedback</h2>
|
||||
<div class="outline-text-2" id="text-org08917d6">
|
||||
<p>
|
||||
<a id="org0aa816a"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org243b924" class="outline-3">
|
||||
<h3 id="org243b924">Identification of the Dynamics with perfect Joints</h3>
|
||||
<div class="outline-text-3" id="text-org243b924">
|
||||
<p>
|
||||
We first initialize the Stewart platform without joint stiffness.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, <span class="org-string">'disable'</span>, <span class="org-constant">true</span>);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
And we identify the dynamics from force actuators to force sensors.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_active_damping'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force Inputs [N]</span>
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Dm'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Relative Displacement Outputs [N]</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The transfer function from actuator forces to relative motion sensors is shown in Figure <a href="#orgcc86228">9</a>.
|
||||
</p>
|
||||
|
||||
<div id="orgcc86228" class="figure">
|
||||
<p><img src="figs/dvf_plant_coupling.png" alt="dvf_plant_coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Transfer function from the Actuator force \(F_{i}\) to the Relative Motion Sensor \(D_{m,j}\) with \(i \neq j\) (<a href="./figs/dvf_plant_coupling.png">png</a>, <a href="./figs/dvf_plant_coupling.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orgcdb3ee5" class="outline-3">
|
||||
<h3 id="orgcdb3ee5">Effect of the Flexible Joint stiffness on the Dynamics</h3>
|
||||
<div class="outline-text-3" id="text-orgcdb3ee5">
|
||||
<p>
|
||||
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
Gf.OutputName = {<span class="org-string">'Dm1'</span>, <span class="org-string">'Dm2'</span>, <span class="org-string">'Dm3'</span>, <span class="org-string">'Dm4'</span>, <span class="org-string">'Dm5'</span>, <span class="org-string">'Dm6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The new dynamics from force actuator to relative motion sensor is shown in Figure <a href="#org5a86447">10</a>.
|
||||
</p>
|
||||
|
||||
<div id="org5a86447" class="figure">
|
||||
<p><img src="figs/dvf_plant_flexible_joint_decentralized.png" alt="dvf_plant_flexible_joint_decentralized.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Transfer function from the Actuator force \(F_{i}\) to the relative displacement sensor \(D_{m,i}\) (<a href="./figs/dvf_plant_flexible_joint_decentralized.png">png</a>, <a href="./figs/dvf_plant_flexible_joint_decentralized.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgff0cbf9" class="outline-3">
|
||||
<h3 id="orgff0cbf9">Obtained Damping</h3>
|
||||
<div class="outline-text-3" id="text-orgff0cbf9">
|
||||
<p>
|
||||
The control is a performed in a decentralized manner.
|
||||
The \(6 \times 6\) control is a diagonal matrix with pure derivative action on the diagonal:
|
||||
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|
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|
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|
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|
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
The root locus is shown in figure <a href="#org277d60d">11</a> and the obtained pole damping function of the control gain is shown in figure <a href="#orgd673396">12</a>.
|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
<p><span class="figure-number">Figure 11: </span>Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/root_locus_dvf_rot_stiffness.png">png</a>, <a href="./figs/root_locus_dvf_rot_stiffness.pdf">pdf</a>)</p>
|
||||
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|
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|
||||
|
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|
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|
||||
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|
||||
<p><span class="figure-number">Figure 12: </span>Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints (<a href="./figs/pole_damping_gain_dvf_rot_stiffness.png">png</a>, <a href="./figs/pole_damping_gain_dvf_rot_stiffness.pdf">pdf</a>)</p>
|
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|
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|
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|
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|
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|
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|
||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||
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|
||||
|
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|
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|
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<p class="author">Author: Dehaeze Thomas</p>
|
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<p class="date">Created: 2020-02-11 mar. 15:26</p>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Stewart Platform - Control Study</h1>
|
||||
<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<ul>
|
||||
<li><a href="#orgc1805a8">First Control Architecture</a>
|
||||
<ul>
|
||||
<li><a href="#org066d914">Control Schematic</a></li>
|
||||
<li><a href="#org64f6d6b">Initialize the Stewart platform</a></li>
|
||||
<li><a href="#org4493ec7">Identification of the plant</a></li>
|
||||
<li><a href="#org72dad5c">Plant Analysis</a></li>
|
||||
<li><a href="#orga9fb0f5">Controller Design</a></li>
|
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|
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|
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|
||||
<h3 id="org066d914">Control Schematic</h3>
|
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<div class="outline-text-3" id="text-org066d914">
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<span class="org-font-latex-sedate">\draw</span>[->] (K.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tau</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G.east) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\bm</span></span><span class="org-font-latex-math">{L}$</span>} -| (<span class="org-font-latex-math">$(G.east)+(1, -1)$</span>) -| (subr.south);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div class="figure">
|
||||
<p><img src="figs/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org64f6d6b" class="outline-3">
|
||||
<h3 id="org64f6d6b">Initialize the Stewart platform</h3>
|
||||
<div class="outline-text-3" id="text-org64f6d6b">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4493ec7" class="outline-3">
|
||||
<h3 id="org4493ec7">Identification of the plant</h3>
|
||||
<div class="outline-text-3" id="text-org4493ec7">
|
||||
<p>
|
||||
Let’s identify the transfer function from \(\bm{\tau}}\) to \(\bm{L}\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_control'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/tau'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/L'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'L1'</span>, <span class="org-string">'L2'</span>, <span class="org-string">'L3'</span>, <span class="org-string">'L4'</span>, <span class="org-string">'L5'</span>, <span class="org-string">'L6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org72dad5c" class="outline-3">
|
||||
<h3 id="org72dad5c">Plant Analysis</h3>
|
||||
<div class="outline-text-3" id="text-org72dad5c">
|
||||
<p>
|
||||
Diagonal terms
|
||||
Compare to off-diagonal terms
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orga9fb0f5" class="outline-3">
|
||||
<h3 id="orga9fb0f5">Controller Design</h3>
|
||||
<div class="outline-text-3" id="text-orga9fb0f5">
|
||||
<p>
|
||||
One integrator should be present in the controller.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
A lead is added around the crossover frequency which is set to be around 500Hz.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-comment">% wint = 2*pi*100; % Integrate until [rad]</span>
|
||||
<span class="org-comment">% wlead = 2*pi*500; % Location of the lead [rad]</span>
|
||||
<span class="org-comment">% hlead = 2; % Lead strengh</span>
|
||||
|
||||
<span class="org-comment">% Kl = 1e6 * ... % Gain</span>
|
||||
<span class="org-comment">% (s + wint)/(s) * ... % Integrator until 100Hz</span>
|
||||
<span class="org-comment">% (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead</span>
|
||||
|
||||
wc = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>100;
|
||||
Kl = 1<span class="org-type">/</span>abs(freqresp(G(1,1), wc)) <span class="org-type">*</span> wc<span class="org-type">/</span>s <span class="org-type">*</span> 1<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>(3<span class="org-type">*</span>wc));
|
||||
Kl = Kl <span class="org-type">*</span> eye(6);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-11 mar. 15:23</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
145
docs/css/htmlize.css
Normal file
145
docs/css/htmlize.css
Normal file
@@ -0,0 +1,145 @@
|
||||
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|
||||
.org-bold-italic { /* bold-italic */ font-weight: bold; font-style: italic; }
|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||
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|
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|
||||
.org-comint-highlight-input { /* comint-highlight-input */ font-weight: bold; }
|
||||
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|
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.org-comment { /* font-lock-comment-face */ color: #999988; font-style: italic; }
|
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|
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|
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.org-completions-common-part { /* completions-common-part */ color: #000000; background-color: #ffffff; }
|
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.org-completions-first-difference { /* completions-first-difference */ font-weight: bold; }
|
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.org-constant { /* font-lock-constant-face */ color: #008b8b; }
|
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|
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|
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|
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.org-diff-function { /* diff-function */ background-color: #cccccc; }
|
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|
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|
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|
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|
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|
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.org-dired-header { /* dired-header */ color: #228b22; }
|
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|
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.org-dired-mark { /* dired-mark */ color: #008b8b; }
|
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.org-dired-marked { /* dired-marked */ color: #ff0000; font-weight: bold; }
|
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.org-dired-perm-write { /* dired-perm-write */ color: #b22222; }
|
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.org-dired-symlink { /* dired-symlink */ color: #a020f0; }
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.org-dired-warning { /* dired-warning */ color: #ff0000; font-weight: bold; }
|
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.org-doc { /* font-lock-doc-face */ color: #8b2252; }
|
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|
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|
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|
||||
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|
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.org-fringe { /* fringe */ background-color: #f2f2f2; }
|
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.org-function-name { /* font-lock-function-name-face */ color: teal; }
|
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|
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|
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.org-highlight { /* highlight */ background-color: #b4eeb4; }
|
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.org-holiday { /* holiday */ background-color: #ffc0cb; }
|
||||
.org-isearch { /* isearch */ color: #b0e2ff; background-color: #cd00cd; }
|
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.org-isearch-fail { /* isearch-fail */ background-color: #ffc1c1; }
|
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.org-italic { /* italic */ font-style: italic; }
|
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.org-keyword { /* font-lock-keyword-face */ color: #0086b3; }
|
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.org-lazy-highlight { /* lazy-highlight */ background-color: #afeeee; }
|
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.org-link { /* link */ color: #0000ff; text-decoration: underline; }
|
||||
.org-link-visited { /* link-visited */ color: #8b008b; text-decoration: underline; }
|
||||
.org-log-edit-header { /* log-edit-header */ color: #a020f0; }
|
||||
.org-log-edit-summary { /* log-edit-summary */ color: #0000ff; }
|
||||
.org-log-edit-unknown-header { /* log-edit-unknown-header */ color: #b22222; }
|
||||
.org-match { /* match */ background-color: #ffff00; }
|
||||
.org-next-error { /* next-error */ background-color: #eedc82; }
|
||||
.org-nobreak-space { /* nobreak-space */ color: #a52a2a; text-decoration: underline; }
|
||||
.org-org-archived { /* org-archived */ color: #7f7f7f; }
|
||||
.org-org-block { /* org-block */ color: #7f7f7f; }
|
||||
.org-org-block-begin-line { /* org-block-begin-line */ color: #b22222; }
|
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.org-org-block-end-line { /* org-block-end-line */ color: #b22222; }
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|
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|
||||
.org-org-checkbox-statistics-todo { /* org-checkbox-statistics-todo */ color: #ff0000; font-weight: bold; }
|
||||
.org-org-clock-overlay { /* org-clock-overlay */ background-color: #ffff00; }
|
||||
.org-org-code { /* org-code */ color: #7f7f7f; }
|
||||
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|
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|
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|
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.org-org-footnote { /* org-footnote */ color: #a020f0; text-decoration: underline; }
|
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.org-org-formula { /* org-formula */ color: #b22222; }
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.org-org-level-4 { /* org-level-4 */ color: #b22222; }
|
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|
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|
||||
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|
||||
.org-org-scheduled { /* org-scheduled */ color: #006400; }
|
||||
.org-org-scheduled-previously { /* org-scheduled-previously */ color: #b22222; }
|
||||
.org-org-scheduled-today { /* org-scheduled-today */ color: #006400; }
|
||||
.org-org-sexp-date { /* org-sexp-date */ color: #a020f0; }
|
||||
.org-org-special-keyword { /* org-special-keyword */ color: #a020f0; }
|
||||
.org-org-table { /* org-table */ color: #0000ff; }
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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.org-outline-3 { /* outline-3 */ color: #a020f0; }
|
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.org-outline-4 { /* outline-4 */ color: #b22222; }
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|
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|
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|
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|
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|
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|
||||
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|
||||
.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
|
||||
1095
docs/css/readtheorg.css
Normal file
1095
docs/css/readtheorg.css
Normal file
File diff suppressed because it is too large
Load Diff
1045
docs/cubic-configuration.html
Normal file
1045
docs/cubic-configuration.html
Normal file
File diff suppressed because it is too large
Load Diff
748
docs/dynamics-study.html
Normal file
748
docs/dynamics-study.html
Normal file
@@ -0,0 +1,748 @@
|
||||
<?xml version="1.0" encoding="utf-8"?>
|
||||
<?xml version="1.0" encoding="utf-8"?>
|
||||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-02-11 mar. 15:27 -->
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<a accesskey="h" href="./index.html"> UP </a>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
|
||||
<h1 class="title">Stewart Platform - Dynamics Study</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orgdae5fe1">Some tests</a>
|
||||
<ul>
|
||||
<li><a href="#orga032902">Simscape Model</a></li>
|
||||
<li><a href="#orgdbd3cde">test</a></li>
|
||||
<li><a href="#orgc59e712">Compare external forces and forces applied by the actuators</a></li>
|
||||
<li><a href="#org81ab204">Comparison of the static transfer function and the Compliance matrix</a></li>
|
||||
<li><a href="#orge663148">Transfer function from forces applied in the legs to the displacement of the legs</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdae5fe1" class="outline-2">
|
||||
<h2 id="orgdae5fe1">Some tests</h2>
|
||||
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|
||||
</div>
|
||||
<div id="outline-container-orga032902" class="outline-3">
|
||||
<h3 id="orga032902">Simscape Model</h3>
|
||||
<div class="outline-text-3" id="text-orga032902">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">open(<span class="org-string">'stewart_platform_dynamics.slx'</span>)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdbd3cde" class="outline-3">
|
||||
<h3 id="orgdbd3cde">test</h3>
|
||||
<div class="outline-text-3" id="text-orgdbd3cde">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/J-T'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">G_cart = minreal(G<span class="org-type">*</span>inv(stewart.J<span class="org-type">'</span>));
|
||||
G_cart.InputName = {<span class="org-string">'Fnx'</span>, <span class="org-string">'Fny'</span>, <span class="org-string">'Fnz'</span>, <span class="org-string">'Mnx'</span>, <span class="org-string">'Mny'</span>, <span class="org-string">'Mnz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-type">figure</span>; bode(G, G_cart)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Fext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
Gd = linearize(mdl, io, options);
|
||||
Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
|
||||
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = logspace(0, 3, 1000);
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
bode(Gd, G)
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc59e712" class="outline-3">
|
||||
<h3 id="orgc59e712">Compare external forces and forces applied by the actuators</h3>
|
||||
<div class="outline-text-3" id="text-orgc59e712">
|
||||
<p>
|
||||
Initialization of the Stewart platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Estimation of the transfer function from \(\mathcal{\bm{F}}_{d}\) to \(\mathcal{\bm{X}}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/Fext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
Gd = linearize(mdl, io, options);
|
||||
Gd.InputName = {<span class="org-string">'Fex'</span>, <span class="org-string">'Fey'</span>, <span class="org-string">'Fez'</span>, <span class="org-string">'Mex'</span>, <span class="org-string">'Mey'</span>, <span class="org-string">'Mez'</span>};
|
||||
Gd.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Comparison of the two transfer function matrices.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = logspace(0, 4, 1000);
|
||||
|
||||
<span class="org-type">figure</span>;
|
||||
bode(Gd, G, freqs)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
Seems quite similar.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org81ab204" class="outline-3">
|
||||
<h3 id="org81ab204">Comparison of the static transfer function and the Compliance matrix</h3>
|
||||
<div class="outline-text-3" id="text-org81ab204">
|
||||
<p>
|
||||
Initialization of the Stewart platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/F'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/X'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'Fx'</span>, <span class="org-string">'Fy'</span>, <span class="org-string">'Fz'</span>, <span class="org-string">'Mx'</span>, <span class="org-string">'My'</span>, <span class="org-string">'Mz'</span>};
|
||||
G.OutputName = {<span class="org-string">'Edx'</span>, <span class="org-string">'Edy'</span>, <span class="org-string">'Edz'</span>, <span class="org-string">'Erx'</span>, <span class="org-string">'Ery'</span>, <span class="org-string">'Erz'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Let’s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
|
||||
</p>
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
|
||||
<colgroup>
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
</colgroup>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-right">2.0e-06</td>
|
||||
<td class="org-right">-9.1e-19</td>
|
||||
<td class="org-right">-5.3e-12</td>
|
||||
<td class="org-right">7.3e-18</td>
|
||||
<td class="org-right">1.7e-05</td>
|
||||
<td class="org-right">1.3e-18</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">-1.7e-18</td>
|
||||
<td class="org-right">2.0e-06</td>
|
||||
<td class="org-right">8.6e-19</td>
|
||||
<td class="org-right">-1.7e-05</td>
|
||||
<td class="org-right">-1.5e-17</td>
|
||||
<td class="org-right">6.7e-12</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">3.6e-13</td>
|
||||
<td class="org-right">3.2e-19</td>
|
||||
<td class="org-right">5.0e-07</td>
|
||||
<td class="org-right">-2.5e-18</td>
|
||||
<td class="org-right">8.1e-12</td>
|
||||
<td class="org-right">-1.5e-19</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">1.0e-17</td>
|
||||
<td class="org-right">-1.7e-05</td>
|
||||
<td class="org-right">-5.0e-18</td>
|
||||
<td class="org-right">1.9e-04</td>
|
||||
<td class="org-right">9.1e-17</td>
|
||||
<td class="org-right">-3.5e-11</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">1.7e-05</td>
|
||||
<td class="org-right">-6.9e-19</td>
|
||||
<td class="org-right">-5.3e-11</td>
|
||||
<td class="org-right">6.9e-18</td>
|
||||
<td class="org-right">1.9e-04</td>
|
||||
<td class="org-right">4.8e-18</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">-3.5e-18</td>
|
||||
<td class="org-right">-4.5e-12</td>
|
||||
<td class="org-right">1.5e-18</td>
|
||||
<td class="org-right">7.1e-11</td>
|
||||
<td class="org-right">-3.4e-17</td>
|
||||
<td class="org-right">4.6e-05</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<p>
|
||||
And now at the Compliance matrix.
|
||||
</p>
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
|
||||
<colgroup>
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
</colgroup>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-right">2.0e-06</td>
|
||||
<td class="org-right">2.9e-22</td>
|
||||
<td class="org-right">2.8e-22</td>
|
||||
<td class="org-right">-3.2e-21</td>
|
||||
<td class="org-right">1.7e-05</td>
|
||||
<td class="org-right">1.5e-37</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">-2.1e-22</td>
|
||||
<td class="org-right">2.0e-06</td>
|
||||
<td class="org-right">-1.8e-23</td>
|
||||
<td class="org-right">-1.7e-05</td>
|
||||
<td class="org-right">-2.3e-21</td>
|
||||
<td class="org-right">1.1e-22</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">3.1e-22</td>
|
||||
<td class="org-right">-1.6e-23</td>
|
||||
<td class="org-right">5.0e-07</td>
|
||||
<td class="org-right">1.7e-22</td>
|
||||
<td class="org-right">2.2e-21</td>
|
||||
<td class="org-right">-8.1e-39</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">2.1e-21</td>
|
||||
<td class="org-right">-1.7e-05</td>
|
||||
<td class="org-right">2.0e-22</td>
|
||||
<td class="org-right">1.9e-04</td>
|
||||
<td class="org-right">2.3e-20</td>
|
||||
<td class="org-right">-8.7e-21</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">1.7e-05</td>
|
||||
<td class="org-right">2.5e-21</td>
|
||||
<td class="org-right">2.0e-21</td>
|
||||
<td class="org-right">-2.8e-20</td>
|
||||
<td class="org-right">1.9e-04</td>
|
||||
<td class="org-right">1.3e-36</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-right">3.7e-23</td>
|
||||
<td class="org-right">3.1e-22</td>
|
||||
<td class="org-right">-6.0e-39</td>
|
||||
<td class="org-right">-1.0e-20</td>
|
||||
<td class="org-right">3.1e-22</td>
|
||||
<td class="org-right">4.6e-05</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge663148" class="outline-3">
|
||||
<h3 id="orge663148">Transfer function from forces applied in the legs to the displacement of the legs</h3>
|
||||
<div class="outline-text-3" id="text-orge663148">
|
||||
<p>
|
||||
Initialization of the Stewart platform.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Estimation of the transfer function from \(\bm{\tau}\) to \(\bm{L}\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Options for Linearized</span></span>
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
||||
mdl = <span class="org-string">'stewart_platform_dynamics'</span>;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/J-T'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
io(io_i) = linio([mdl, <span class="org-string">'/L'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
|
||||
G = linearize(mdl, io, options);
|
||||
G.InputName = {<span class="org-string">'F1'</span>, <span class="org-string">'F2'</span>, <span class="org-string">'F3'</span>, <span class="org-string">'F4'</span>, <span class="org-string">'F5'</span>, <span class="org-string">'F6'</span>};
|
||||
G.OutputName = {<span class="org-string">'L1'</span>, <span class="org-string">'L2'</span>, <span class="org-string">'L3'</span>, <span class="org-string">'L4'</span>, <span class="org-string">'L5'</span>, <span class="org-string">'L6'</span>};
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = logspace(1, 3, 1000);
|
||||
<span class="org-type">figure</span>; bode(G, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>freqs)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">bodeFig({G(1,1), G(1,2)}, freqs, struct(<span class="org-string">'phase'</span>, <span class="org-constant">true</span>));
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-11 mar. 15:27</p>
|
||||
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|
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|
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|
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|
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<body>
|
||||
<div id="content">
|
||||
<h1 class="title">Stewart Platforms</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orgff0bfd7">Simulink Project (link)</a></li>
|
||||
<li><a href="#org38b9089">Stewart Platform Architecture Definition (link)</a></li>
|
||||
<li><a href="#orgf1c7b3b">Simscape Model of the Stewart Platform (link)</a></li>
|
||||
<li><a href="#org369c8bb">Kinematic Analysis (link)</a></li>
|
||||
<li><a href="#org2e3169e">Identification of the Stewart Dynamics (link)</a></li>
|
||||
<li><a href="#orgc3a4c87">Active Damping (link)</a></li>
|
||||
<li><a href="#org5b4e9b0">Motion Control of the Stewart Platform (link)</a></li>
|
||||
<li><a href="#org1f468b1">Cubic Configuration (link)</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The goal of this project is to provide a Matlab/Simscape Toolbox to study Stewart platforms.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The project is divided into several section listed below.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-orgff0bfd7" class="outline-2">
|
||||
<h2 id="orgff0bfd7">Simulink Project (<a href="simulink-project.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-orgff0bfd7">
|
||||
<p>
|
||||
The project is managed with a <b>Simulink Project</b>.
|
||||
Such project is briefly presented <a href="simulink-project.html">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org38b9089" class="outline-2">
|
||||
<h2 id="org38b9089">Stewart Platform Architecture Definition (<a href="stewart-architecture.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org38b9089">
|
||||
<p>
|
||||
The way the Stewart Platform is defined <a href="stewart-architecture.html">here</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
All the geometrical parameters are defined including:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Definition of the location of the frames</li>
|
||||
<li>Location/orientation of the limbs</li>
|
||||
<li>Size/inertia of the platforms and the limbs</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Other parameters are also defined such as:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Stiffness and damping of the struts</li>
|
||||
<li>Rest position of the Stewart platform</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf1c7b3b" class="outline-2">
|
||||
<h2 id="orgf1c7b3b">Simscape Model of the Stewart Platform (<a href="simscape-model.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-orgf1c7b3b">
|
||||
<p>
|
||||
The Stewart Platform is then modeled using <a href="https://www.mathworks.com/products/simscape.html">Simscape</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The way to model is build and works is explained <a href="simscape-model.html">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org369c8bb" class="outline-2">
|
||||
<h2 id="org369c8bb">Kinematic Analysis (<a href="kinematic-study.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org369c8bb">
|
||||
<p>
|
||||
From the defined geometry of the Stewart platform, we can perform static analysis such as:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><b>Jacobian Analysis</b> that links the velocity of each limb to the velocity of the mobile platform</li>
|
||||
<li><b>Static Forces Analysis</b> that links the forces applied by each limb to the resulting force/torque applied to the mobile platform</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
From the strut stiffness, we can also perform a <b>Stiffness Analysis</b> that consists of determining the Stiffness matrix and Compliance matrix of the Stewart platform from the geometry.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
All these analysis are described <a href="kinematic-study.html">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2e3169e" class="outline-2">
|
||||
<h2 id="org2e3169e">Identification of the Stewart Dynamics (<a href="identification.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org2e3169e">
|
||||
<p>
|
||||
The Dynamics of the Stewart platform can be identified using the Simscape model.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
It is possible to:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Determine the dynamics from the actuators to the various sensors included in the Stewart platform</li>
|
||||
<li>Extract State Space models for further analysis / control synthesis</li>
|
||||
<li>Extract the Resonant Frequencies, Modal Damping, and associated Mode Shapes</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The code that is used for identification is explained <a href="identification.html">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc3a4c87" class="outline-2">
|
||||
<h2 id="orgc3a4c87">Active Damping (<a href="active-damping.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-orgc3a4c87">
|
||||
<p>
|
||||
The use of different sensors are compared for active damping:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Inertial Sensor in each strut</li>
|
||||
<li>Inertial Sensor fixed to the mobile platform</li>
|
||||
<li>Force Sensor in each strut</li>
|
||||
<li>Relative Motion Sensor in each strut</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The result of the analysis is accessible <a href="active-damping.html">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5b4e9b0" class="outline-2">
|
||||
<h2 id="org5b4e9b0">Motion Control of the Stewart Platform (<a href="control-study.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org5b4e9b0">
|
||||
<p>
|
||||
Some control architecture for motion control of the Stewart platform are applied on the Simscape model and compared in <a href="control-study.html">this</a> document.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f468b1" class="outline-2">
|
||||
<h2 id="org1f468b1">Cubic Configuration (<a href="cubic-configuration.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org1f468b1">
|
||||
<p>
|
||||
The cubic configuration is a special class of Stewart platform that has interesting properties.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
These properties are studied in <a href="cubic-configuration.html">this</a> document.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-11 mar. 15:27</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
7
docs/js/bootstrap.min.js
vendored
Normal file
7
docs/js/bootstrap.min.js
vendored
Normal file
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4
docs/js/jquery.min.js
vendored
Normal file
4
docs/js/jquery.min.js
vendored
Normal file
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1
docs/js/jquery.stickytableheaders.min.js
vendored
Normal file
1
docs/js/jquery.stickytableheaders.min.js
vendored
Normal file
@@ -0,0 +1 @@
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|
||||
85
docs/js/readtheorg.js
Normal file
85
docs/js/readtheorg.js
Normal file
@@ -0,0 +1,85 @@
|
||||
$(function() {
|
||||
$('.note').before("<p class='admonition-title note'>Note</p>");
|
||||
$('.seealso').before("<p class='admonition-title seealso'>See also</p>");
|
||||
$('.warning').before("<p class='admonition-title warning'>Warning</p>");
|
||||
$('.caution').before("<p class='admonition-title caution'>Caution</p>");
|
||||
$('.attention').before("<p class='admonition-title attention'>Attention</p>");
|
||||
$('.tip').before("<p class='admonition-title tip'>Tip</p>");
|
||||
$('.important').before("<p class='admonition-title important'>Important</p>");
|
||||
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
|
||||
$('.error').before("<p class='admonition-title error'>Error</p>");
|
||||
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
|
||||
});
|
||||
|
||||
$( document ).ready(function() {
|
||||
|
||||
// Shift nav in mobile when clicking the menu.
|
||||
$(document).on('click', "[data-toggle='wy-nav-top']", function() {
|
||||
$("[data-toggle='wy-nav-shift']").toggleClass("shift");
|
||||
$("[data-toggle='rst-versions']").toggleClass("shift");
|
||||
});
|
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// Close menu when you click a link.
|
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$(document).on('click', ".wy-menu-vertical .current ul li a", function() {
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$("[data-toggle='wy-nav-shift']").removeClass("shift");
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|
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|
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|
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|
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$("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
|
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|
||||
|
||||
$( document ).ready(function() {
|
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|
||||
// ScrollSpy also requires that we use
|
||||
// a Bootstrap nav component.
|
||||
$('body').scrollspy({target: '#text-table-of-contents'});
|
||||
|
||||
// add sticky table headers
|
||||
$('table').stickyTableHeaders();
|
||||
|
||||
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|
||||
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|
||||
var $tableOfContents = $('#table-of-contents');
|
||||
$tableOfContents.css({paddingBottom: $postamble.outerHeight()});
|
||||
|
||||
// add TOC button
|
||||
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|
||||
$('#content').prepend(toggleSidebar);
|
||||
|
||||
// add close button when sidebar showed in mobile screen
|
||||
var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
|
||||
var tocTitle = $('#table-of-contents').find('h2');
|
||||
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|
||||
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|
||||
|
||||
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|
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|
||||
var navBar,
|
||||
win,
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|
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|
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|
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|
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|
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|
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|
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|
||||
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|
||||
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|
||||
enable : enable
|
||||
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|
||||
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|
||||
return {
|
||||
StickyNav : stickyNav
|
||||
};
|
||||
}($));
|
||||
1622
docs/kinematic-study.html
Normal file
1622
docs/kinematic-study.html
Normal file
File diff suppressed because it is too large
Load Diff
676
docs/simscape-model.html
Normal file
676
docs/simscape-model.html
Normal file
@@ -0,0 +1,676 @@
|
||||
<?xml version="1.0" encoding="utf-8"?>
|
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|
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|
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|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>Stewart Platform - Simscape Model</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
<meta name="author" content="Dehaeze Thomas" />
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
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|
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|
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
|
||||
<h1 class="title">Stewart Platform - Simscape Model</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orgc6e0b93">Parameters used for the Simscape Model</a></li>
|
||||
<li><a href="#org66977e8">Simulation Configuration - Configuration reference</a></li>
|
||||
<li><a href="#orgb2362eb">Subsystem Reference</a></li>
|
||||
<li><a href="#orgdfad86d">Subsystem - Fixed base and Mobile Platform</a></li>
|
||||
<li><a href="#org9d4af75">Subsystem - Struts</a>
|
||||
<ul>
|
||||
<li><a href="#org45d9234">Strut Configuration</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org7e2c432">Other Elements</a>
|
||||
<ul>
|
||||
<li><a href="#org4bdfc33">Z-Axis Geophone</a>
|
||||
<ul>
|
||||
<li><a href="#org60cad49">Working Principle</a></li>
|
||||
<li><a href="#org48bfa57">Initialization function</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org99786f1">Z-Axis Accelerometer</a>
|
||||
<ul>
|
||||
<li><a href="#org631cd1a">Working Principle</a></li>
|
||||
<li><a href="#org514425a">Initialization function</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
In this document is explained how the Simscape model of the Stewart Platform is implemented.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
It is divided in the following sections:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>section <a href="#org8d965c3">No description for this link</a>: is explained how the parameters of the Stewart platform are set for the Simscape model</li>
|
||||
<li>section <a href="#org354bfdb">No description for this link</a>: the Simulink configuration (solver, simulation time, …) is shared among all the Simulink files. It is explain how this is done.</li>
|
||||
<li>section <a href="#org66bbae2">No description for this link</a>: All the elements (platforms, struts, sensors, …) are saved in separate files and imported in Simulink files using “subsystem referenced”.</li>
|
||||
<li>section <a href="#orga4915c4">No description for this link</a>: The simscape model for the fixed base and mobile platform are described in this section.</li>
|
||||
<li>section <a href="#orgdb5206f">No description for this link</a>: The simscape model for the Stewart platform struts is described in this section.</li>
|
||||
</ul>
|
||||
|
||||
<div id="outline-container-orgc6e0b93" class="outline-2">
|
||||
<h2 id="orgc6e0b93">Parameters used for the Simscape Model</h2>
|
||||
<div class="outline-text-2" id="text-orgc6e0b93">
|
||||
<p>
|
||||
<a id="org8d965c3"></a>
|
||||
The Simscape Model of the Stewart Platform is working with the <code>stewart</code> structure generated using the functions described <a href="stewart-architecture.html">here</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
All the geometry and inertia of the mechanical elements are defined in the <code>stewart</code> structure.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
By updating the <code>stewart</code> structure in the workspace, the Simscape model will be automatically updated.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Thus, nothing should be changed by hand inside the Simscape model.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The main advantage to have all the parameters defined in one structure (and not hard-coded in some simulink blocs) it that we can easily change the Stewart architecture/parameters in a Matlab script to perform some parametric study for instance.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org66977e8" class="outline-2">
|
||||
<h2 id="org66977e8">Simulation Configuration - Configuration reference</h2>
|
||||
<div class="outline-text-2" id="text-org66977e8">
|
||||
<p>
|
||||
<a id="org354bfdb"></a>
|
||||
As multiple simulink files will be used for simulation and tests, it is very useful to determine good simulation configuration that will be <b>shared</b> among all the simulink files.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This is done using something called “<b>Configuration Reference</b>” (<a href="https://fr.mathworks.com/help/simulink/ug/more-about-configuration-references.html">documentation</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Basically, the configuration is stored in a mat file <code>conf_simscape.mat</code> and then loaded in the workspace for it to be accessible to all the simulink models.
|
||||
It is automatically loaded when the Simulink project is open. It can be loaded manually with the command:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
It is however possible to modify specific parameters just for one simulation using the <code>set_param</code> command:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, 1);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb2362eb" class="outline-2">
|
||||
<h2 id="orgb2362eb">Subsystem Reference</h2>
|
||||
<div class="outline-text-2" id="text-orgb2362eb">
|
||||
<p>
|
||||
<a id="org66bbae2"></a>
|
||||
Several Stewart platform models are used, for instance one is use to study the dynamics while the other is used to apply active damping techniques.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
However, all the Simscape models share some subsystems using the <b>Subsystem Reference</b> Simulink block (<a href="https://fr.mathworks.com/help/simulink/ug/referenced-subsystem-1.html">documentation</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
These shared subsystems are:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><code>Fixed_Based.slx</code> - Fixed base of the Stewart Platform</li>
|
||||
<li><code>Mobile_Platform.slx</code> - Mobile platform of the Stewart Platform</li>
|
||||
<li><code>stewart_strut.slx</code> - One strut containing two spherical/universal joints, the actuator as well as the included sensors. A parameter <code>i</code> is initialized to determine what it the “number” of the strut.</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
These subsystems are referenced from another subsystem called <code>Stewart_Platform.slx</code> shown in figure <a href="#orgf687c71">1</a>, that basically connect them correctly.
|
||||
This subsystem is then referenced in other simulink models for various purposes (control, analysis, simulation, …).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf687c71" class="figure">
|
||||
<p><img src="figs/simscape_stewart_platform.png" alt="simscape_stewart_platform.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Simscape Subsystem of the Stewart platform. Encapsulate the Subsystems corresponding to the fixed base, mobile platform and all the struts.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdfad86d" class="outline-2">
|
||||
<h2 id="orgdfad86d">Subsystem - Fixed base and Mobile Platform</h2>
|
||||
<div class="outline-text-2" id="text-orgdfad86d">
|
||||
<p>
|
||||
<a id="orga4915c4"></a>
|
||||
Both the fixed base and the mobile platform simscape models share many similarities.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Their are both composed of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>a solid body representing the platform</li>
|
||||
<li>6 rigid transform blocks to go from the frame \(\{F\}\) (resp. \(\{M\}\)) to the location of the joints.
|
||||
These rigid transform are using \({}^F\bm{a}_i\) (resp. \({}^M\bm{b}_i\)) for the position of the joint and \({}^F\bm{R}_{a_i}\) (resp. \({}^M\bm{R}_{b_i}\)) for the orientation of the joint.</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
As always, the parameters that define the geometry are taken from the <code>stewart</code> structure.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org858f0b4" class="figure">
|
||||
<p><img src="figs/simscape_fixed_base.png" alt="simscape_fixed_base.png" width="1000px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Simscape Model of the Fixed base</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org4b31aa3" class="figure">
|
||||
<p><img src="figs/simscape_mobile_platform.png" alt="simscape_mobile_platform.png" width="800px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Simscape Model of the Mobile platform</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9d4af75" class="outline-2">
|
||||
<h2 id="org9d4af75">Subsystem - Struts</h2>
|
||||
<div class="outline-text-2" id="text-org9d4af75">
|
||||
<p>
|
||||
<a id="orgdb5206f"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org45d9234" class="outline-3">
|
||||
<h3 id="org45d9234">Strut Configuration</h3>
|
||||
<div class="outline-text-3" id="text-org45d9234">
|
||||
<p>
|
||||
For the Stewart platform, the 6 struts are identical.
|
||||
Thus, all the struts used in the Stewart platform are referring to the same subsystem called <code>stewart_strut.slx</code> and shown in Figure <a href="#org1dc8fce">4</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This strut as the following structure:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><b>Universal Joint*</b> connected on the Fixed base</li>
|
||||
<li><b>Prismatic Joint*</b> for the actuator</li>
|
||||
<li><b>Spherical Joint*</b> connected on the Mobile platform</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
This configuration is called <b>UPS</b>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The other common configuration <b>SPS</b> has the disadvantage of having additional passive degrees-of-freedom corresponding to the rotation of the strut around its main axis.
|
||||
This is why the <b>UPS</b> configuration is used, but other configuration can be easily implemented.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org1dc8fce" class="figure">
|
||||
<p><img src="figs/simscape_strut.png" alt="simscape_strut.png" width="800px" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Simscape model of the Stewart platform’s strut</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Several sensors are included in the strut that may or may not be used for control:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Relative Displacement sensor: gives the relative displacement of the strut.</li>
|
||||
<li>Force sensor: measure the total force applied by the force actuator, the stiffness and damping forces in the direction of the strut.</li>
|
||||
<li>Inertial sensor: measure the absolute motion (velocity) of the top part of the strut in the direction of the strut.</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
There is two main types of inertial sensor that can be used to measure the absolute motion of the top part of the strut in the direction of the strut:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>a geophone that measures the absolute velocity above some frequency</li>
|
||||
<li>an accelerometer that measures the absolute acceleration below some frequency</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Both inertial sensors are described bellow.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7e2c432" class="outline-2">
|
||||
<h2 id="org7e2c432">Other Elements</h2>
|
||||
<div class="outline-text-2" id="text-org7e2c432">
|
||||
</div>
|
||||
<div id="outline-container-org4bdfc33" class="outline-3">
|
||||
<h3 id="org4bdfc33">Z-Axis Geophone</h3>
|
||||
<div class="outline-text-3" id="text-org4bdfc33">
|
||||
</div>
|
||||
<div id="outline-container-org60cad49" class="outline-4">
|
||||
<h4 id="org60cad49">Working Principle</h4>
|
||||
<div class="outline-text-4" id="text-org60cad49">
|
||||
<p>
|
||||
From the schematic of the Z-axis geophone shown in Figure <a href="#org819fba8">5</a>, we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\):
|
||||
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
||||
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org819fba8" class="figure">
|
||||
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Schematic of a Z-Axis geophone</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We see that at frequencies above \(\omega_0\):
|
||||
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org48bfa57" class="outline-4">
|
||||
<h4 id="org48bfa57">Initialization function</h4>
|
||||
<div class="outline-text-4" id="text-org48bfa57">
|
||||
<p>
|
||||
<a id="orgd31bda9"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This Matlab function is accessible <a href="../src/initializeZAxisGeophone.m">here</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[geophone]</span> = <span class="org-function-name">initializeZAxisGeophone</span>(<span class="org-variable-name">args</span>)
|
||||
arguments
|
||||
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
||||
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 1 <span class="org-comment">% [Hz]</span>
|
||||
<span class="org-keyword">end</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
||||
geophone.m = args.mass;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
||||
geophone.k = geophone.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
||||
geophone.c = 2<span class="org-type">*</span>sqrt(geophone.m <span class="org-type">*</span> geophone.k);
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
||||
save(<span class="org-string">'./mat/geophone_z_axis.mat'</span>, <span class="org-string">'geophone'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org99786f1" class="outline-3">
|
||||
<h3 id="org99786f1">Z-Axis Accelerometer</h3>
|
||||
<div class="outline-text-3" id="text-org99786f1">
|
||||
</div>
|
||||
<div id="outline-container-org631cd1a" class="outline-4">
|
||||
<h4 id="org631cd1a">Working Principle</h4>
|
||||
<div class="outline-text-4" id="text-org631cd1a">
|
||||
<p>
|
||||
From the schematic of the Z-axis accelerometer shown in Figure <a href="#org1274602">6</a>, we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\):
|
||||
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\)</li>
|
||||
<li>\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org1274602" class="figure">
|
||||
<p><img src="figs/inertial_sensor.png" alt="inertial_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Schematic of a Z-Axis geophone</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We see that at frequencies below \(\omega_0\):
|
||||
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Note that there is trade-off between:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>the highest measurable acceleration \(\omega_0\)</li>
|
||||
<li>the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\)</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org514425a" class="outline-4">
|
||||
<h4 id="org514425a">Initialization function</h4>
|
||||
<div class="outline-text-4" id="text-org514425a">
|
||||
<p>
|
||||
<a id="orge91f65f"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This Matlab function is accessible <a href="../src/initializeZAxisAccelerometer.m">here</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[accelerometer]</span> = <span class="org-function-name">initializeZAxisAccelerometer</span>(<span class="org-variable-name">args</span>)
|
||||
arguments
|
||||
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e<span class="org-type">-</span>3 <span class="org-comment">% [kg]</span>
|
||||
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 5e3 <span class="org-comment">% [Hz]</span>
|
||||
<span class="org-keyword">end</span>
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
|
||||
accelerometer.m = args.mass;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% The Stiffness is set to have the damping resonance frequency</span></span>
|
||||
accelerometer.k = accelerometer.m <span class="org-type">*</span> (2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>args.freq)<span class="org-type">^</span>2;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% We set the damping value to have critical damping</span></span>
|
||||
accelerometer.c = 2<span class="org-type">*</span>sqrt(accelerometer.m <span class="org-type">*</span> accelerometer.k);
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Gain correction of the accelerometer to have a unity gain until the resonance</span></span>
|
||||
accelerometer.gain = <span class="org-type">-</span>accelerometer.k<span class="org-type">/</span>accelerometer.m;
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Save</span></span>
|
||||
save(<span class="org-string">'./mat/accelerometer_z_axis.mat'</span>, <span class="org-string">'accelerometer'</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-11 mar. 15:26</p>
|
||||
</div>
|
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</body>
|
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325
docs/simulink-project.html
Normal file
325
docs/simulink-project.html
Normal file
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|
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|
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<a accesskey="h" href="./index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="./index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Simulink Project for the Stewart Simscape folder</h1>
|
||||
|
||||
<p>
|
||||
A Simulink Project is used for the study of Stewart platforms using Simscape.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From the <a href="https://mathworks.com/products/simulink/projects.html">Simulink project</a> mathworks page:
|
||||
</p>
|
||||
<blockquote>
|
||||
<p>
|
||||
Simulink® and Simulink Projects provide a collaborative, scalable environment that enables teams to manage their files and data in one place.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
With Simulink Projects, you can:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><b>Collaborate</b>: Enforce companywide standards such as company tools, libraries, and standard startup and shutdown scripts. Share your work with rich sharing options including MATLAB® toolboxes, email, and archives.</li>
|
||||
<li><b>Automate</b>: Set up your project environment correctly every time by automating steps such as loading the data, managing the path, and opening the models.</li>
|
||||
<li><b>Integrate with source control</b>: Enable easy integration with source control and configuration management tools.</li>
|
||||
</ul>
|
||||
</blockquote>
|
||||
|
||||
<p>
|
||||
The project can be opened using the <code>simulinkproject</code> function:
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">simulinkproject(<span class="org-string">'./'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
When the project opens, a startup script is ran.
|
||||
The startup script is defined below and is exported to the <code>project_startup.m</code> script.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">project = simulinkproject;
|
||||
projectRoot = project.RootFolder;
|
||||
|
||||
myCacheFolder = fullfile(projectRoot, <span class="org-string">'.SimulinkCache'</span>);
|
||||
myCodeFolder = fullfile(projectRoot, <span class="org-string">'.SimulinkCode'</span>);
|
||||
|
||||
Simulink.fileGenControl(<span class="org-string">'set'</span>,...
|
||||
<span class="org-string">'CacheFolder'</span>, myCacheFolder,...
|
||||
<span class="org-string">'CodeGenFolder'</span>, myCodeFolder,...
|
||||
<span class="org-string">'createDir'</span>, <span class="org-constant">true</span>);
|
||||
|
||||
<span class="org-matlab-cellbreak"><span class="org-comment">%% Load the Simscape Configuration</span></span>
|
||||
load(<span class="org-string">'mat/conf_simscape.mat'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
When the project closes, it runs the <code>project_shutdown.m</code> script defined below.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Simulink.fileGenControl(<span class="org-string">'reset'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The project also permits to automatically add defined folder to the path when the project is opened.
|
||||
</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
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323
docs/static-analysis.html
Normal file
323
docs/static-analysis.html
Normal file
@@ -0,0 +1,323 @@
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|
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|
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|
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|
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(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
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|
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|
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|
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|
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|
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#org-div-home-and-up
|
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.org-info-js_search-highlight
|
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|
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|
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|
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|
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<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
|
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<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
|
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<script src="./js/jquery.min.js"></script>
|
||||
<script src="./js/bootstrap.min.js"></script>
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<script src="./js/jquery.stickytableheaders.min.js"></script>
|
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<script src="./js/readtheorg.js"></script>
|
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<script type="text/javascript">
|
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/*
|
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@licstart The following is the entire license notice for the
|
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|
||||
|
||||
Copyright (C) 2012-2020 Free Software Foundation, Inc.
|
||||
|
||||
The JavaScript code in this tag is free software: you can
|
||||
redistribute it and/or modify it under the terms of the GNU
|
||||
General Public License (GNU GPL) as published by the Free Software
|
||||
Foundation, either version 3 of the License, or (at your option)
|
||||
any later version. The code is distributed WITHOUT ANY WARRANTY;
|
||||
without even the implied warranty of MERCHANTABILITY or FITNESS
|
||||
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
|
||||
|
||||
As additional permission under GNU GPL version 3 section 7, you
|
||||
may distribute non-source (e.g., minimized or compacted) forms of
|
||||
that code without the copy of the GNU GPL normally required by
|
||||
section 4, provided you include this license notice and a URL
|
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through which recipients can access the Corresponding Source.
|
||||
|
||||
|
||||
@licend The above is the entire license notice
|
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|
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|
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|
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|
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|
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|
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|
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|
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</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
<a accesskey="h" href="./index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="./index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Stewart Platform - Static Analysis</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orgc502e97">Coupling</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc502e97" class="outline-2">
|
||||
<h2 id="orgc502e97">Coupling</h2>
|
||||
<div class="outline-text-2" id="text-orgc502e97">
|
||||
<p>
|
||||
What causes the coupling from \(F_i\) to \(X_i\) ?
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (J.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$X</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org41430df" class="figure">
|
||||
<p><img src="figs/coupling.png" alt="coupling.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Block diagram to control an hexapod</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
There is no coupling from \(F_i\) to \(X_j\) if \(J^{-1} G J^{-T}\) is diagonal.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
If \(G\) is diagonal (cubic configuration), then \(J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}\)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Thus, the system is uncoupled if \(G\) and \(K\) are diagonal.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-02-11 mar. 15:27</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
2393
docs/stewart-architecture.html
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2393
docs/stewart-architecture.html
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Reference in New Issue
Block a user