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<h1 class="title">Stewart Platform - Decentralized Active Damping</h1>
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<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<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>
<li><a href="#org543be7a">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
<li><a href="#org9a605b4">Obtained Damping</a></li>
<li><a href="#org42a74ed">Conclusion</a></li>
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</ul>
</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>
<li><a href="#orgd119d8a">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
<li><a href="#org2b5e45a">Obtained Damping</a></li>
<li><a href="#org39ddf1e">Conclusion</a></li>
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</ul>
</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>
<li><a href="#orgcdb3ee5">Effect of the Flexible Joint stiffness on the Dynamics</a></li>
<li><a href="#orgff0cbf9">Obtained Damping</a></li>
<li><a href="#org4027234">Conclusion</a></li>
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</ul>
</li>
</ul>
</div>
</div>
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<p>
The following decentralized active damping techniques are briefly studied:
</p>
<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>
<li>Integral Force Feedback: Section <a href="#orgab5e6b5">No description for this link</a></li>
<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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<div id="outline-container-orgd59c804" class="outline-2">
<h2 id="orgd59c804">Inertial Control</h2>
<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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<div id="outline-container-org5f749c8" class="outline-3">
<h3 id="org5f749c8">Identification of the Dynamics</h3>
<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();
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);
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>
<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">'/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>
<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">'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>
</div>
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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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<div id="org834d990" class="figure">
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<p><img src="figs/inertial_plant_coupling.png" alt="inertial_plant_coupling.png" />
</p>
<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>
</div>
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</div>
</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">
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<p>
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
</p>
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<div class="org-src-container">
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<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>};
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</pre>
</div>
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<p>
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The new dynamics from force actuator to force sensor is shown in Figure <a href="#org683c779">2</a>.
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</p>
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<div id="org683c779" class="figure">
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<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>
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</div>
</div>
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<div id="outline-container-org9a605b4" class="outline-3">
<h3 id="org9a605b4">Obtained Damping</h3>
<div class="outline-text-3" id="text-org9a605b4">
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<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>
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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>.
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</p>
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<div id="org9af9e33" class="figure">
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<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>
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<div id="org4e6b73b" class="figure">
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<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>
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<div id="outline-container-org42a74ed" class="outline-3">
<h3 id="org42a74ed">Conclusion</h3>
<div class="outline-text-3" id="text-org42a74ed">
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<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>
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<div id="outline-container-org74c7eb4" class="outline-2">
<h2 id="org74c7eb4">Integral Force Feedback</h2>
<div class="outline-text-2" id="text-org74c7eb4">
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<p>
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<a id="orgab5e6b5"></a>
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</p>
</div>
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<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">
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<p>
We first initialize the Stewart platform without joint stiffness.
</p>
<div class="org-src-container">
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<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);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeAmplifiedStrutDynamics(stewart);
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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>
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<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;
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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>
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>
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<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>};
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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>};
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</pre>
</div>
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<p>
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The transfer function from actuator forces to force sensors is shown in Figure <a href="#org3fca9dd">5</a>.
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</p>
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<div id="org3fca9dd" class="figure">
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<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>
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</div>
</div>
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<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">
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<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>
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The new dynamics from force actuator to force sensor is shown in Figure <a href="#org090868b">6</a>.
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</p>
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<div id="org090868b" class="figure">
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<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>
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<div id="outline-container-org2b5e45a" class="outline-3">
<h3 id="org2b5e45a">Obtained Damping</h3>
<div class="outline-text-3" id="text-org2b5e45a">
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<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>
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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>.
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</p>
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<div id="orge21bbea" class="figure">
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<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>
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<div id="org94d6943" class="figure">
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<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>
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<div id="outline-container-org39ddf1e" class="outline-3">
<h3 id="org39ddf1e">Conclusion</h3>
<div class="outline-text-3" id="text-org39ddf1e">
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<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>
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<div id="outline-container-org08917d6" class="outline-2">
<h2 id="org08917d6">Direct Velocity Feedback</h2>
<div class="outline-text-2" id="text-org08917d6">
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<p>
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<a id="org0aa816a"></a>
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</p>
</div>
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<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">
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<p>
We first initialize the Stewart platform without joint stiffness.
</p>
<div class="org-src-container">
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<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);
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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>
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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>
options = linearizeOptions;
options.SampleTime = 0;
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<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>;
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<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>};
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</pre>
</div>
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<p>
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The transfer function from actuator forces to relative motion sensors is shown in Figure <a href="#orgcc86228">9</a>.
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</p>
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<div id="orgcc86228" class="figure">
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<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>
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<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">
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<p>
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
</p>
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<div class="org-src-container">
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<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>};
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</pre>
</div>
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<p>
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The new dynamics from force actuator to relative motion sensor is shown in Figure <a href="#org5a86447">10</a>.
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</p>
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<div id="org5a86447" class="figure">
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<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>
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</div>
</div>
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<div id="outline-container-orgff0cbf9" class="outline-3">
<h3 id="orgff0cbf9">Obtained Damping</h3>
<div class="outline-text-3" id="text-orgff0cbf9">
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<p>
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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:
\[ K(s) = g
\begin{bmatrix}
s & & \\
& \ddots & \\
& & s
\end{bmatrix} \]
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</p>
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<p>
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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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</p>
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<div id="org277d60d" class="figure">
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<p><img src="figs/root_locus_dvf_rot_stiffness.png" alt="root_locus_dvf_rot_stiffness.png" />
</p>
<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>
</div>
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<div id="orgd673396" class="figure">
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<p><img src="figs/pole_damping_gain_dvf_rot_stiffness.png" alt="pole_damping_gain_dvf_rot_stiffness.png" />
</p>
<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>
</div>
</div>
</div>
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<div id="outline-container-org4027234" class="outline-3">
<h3 id="org4027234">Conclusion</h3>
<div class="outline-text-3" id="text-org4027234">
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<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>
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</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<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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</div>
</body>
</html>