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<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="#org7743c04">1. Compare external forces and forces applied by the actuators</a>
<ul>
<li><a href="#orgc730bef">1.1. Comparison with fixed support</a></li>
<li><a href="#orgefde538">1.2. Comparison with a flexible support</a></li>
<li><a href="#orga9eb2fd">1.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgb6a1ef7">2. Comparison of the static transfer function and the Compliance matrix</a>
<ul>
<li><a href="#org3f1c253">2.1. Analysis</a></li>
<li><a href="#orge261263">2.2. Conclusion</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org7743c04" class="outline-2">
<h2 id="org7743c04"><span class="section-number-2">1</span> Compare external forces and forces applied by the actuators</h2>
<div class="outline-text-2" id="text-1">
<p>
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
</p>
</div>
<div id="outline-container-orgc730bef" class="outline-3">
<h3 id="orgc730bef"><span class="section-number-3">1.1</span> Comparison with fixed support</h3>
<div class="outline-text-3" id="text-1-1">
<p>
Let&rsquo;s generate a Stewart platform.
</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">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
<p>
We don&rsquo;t put any flexibility below the Stewart platform such that <b>its base is fixed to an inertial frame</b>.
We also don&rsquo;t put any payload on top of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
<p>
The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
</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_model'</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">'/Controller'</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">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</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">'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>
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame \(\{B\}\) fixed to the mobile platform:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc.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>
<p>
We also extract the transfer function from external forces \(\bm{\mathcal{F}}_{\text{ext}}\) on the frame \(\{B\}\) fixed to the mobile platform to the relative displacement \(\mathcal{\bm{X}}\) of \(\{B\}\) with respect to frame \(\{A\}\):
</p>
<div class="org-src-container">
<pre class="src src-matlab"><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">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
<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>
The comparison of the two transfer functions is shown in Figure <a href="#org2de43b3">1</a>.
</p>
<div id="org2de43b3" class="figure">
<p><img src="figs/comparison_Fext_F_fixed_base.png" alt="comparison_Fext_F_fixed_base.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_fixed_base.png">png</a>, <a href="./figs/comparison_Fext_F_fixed_base.pdf">pdf</a>)</p>
</div>
<p>
This can be understood from figure <a href="#orgd6db375">2</a> where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\).
</p>
<div id="orgd6db375" class="figure">
<p><img src="figs/1dof_actuator_external_forces.png" alt="1dof_actuator_external_forces.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Schematic representation of the stewart platform on a rigid support</p>
</div>
</div>
</div>
<div id="outline-container-orgefde538" class="outline-3">
<h3 id="orgefde538"><span class="section-number-3">1.2</span> Comparison with a flexible support</h3>
<div class="outline-text-3" id="text-1-2">
<p>
We now add a flexible support under the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'flexible'</span>);
</pre>
</div>
<p>
And we perform again the identification.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><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">'/Controller'</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">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</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">'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>};
Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc.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>};
<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">'/Disturbances'</span>], 1, <span class="org-string">'openinput'</span>, [], <span class="org-string">'F_ext'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces/torques applied on {B}</span>
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</span>
<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>
The comparison between the obtained transfer functions is shown in Figure <a href="#org593368e">3</a>.
</p>
<div id="org593368e" class="figure">
<p><img src="figs/comparison_Fext_F_flexible_base.png" alt="comparison_Fext_F_flexible_base.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (<a href="./figs/comparison_Fext_F_flexible_base.png">png</a>, <a href="./figs/comparison_Fext_F_flexible_base.pdf">pdf</a>)</p>
</div>
<p>
The addition of a flexible support can be schematically represented in Figure <a href="#orga537ded">4</a>.
We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\).
And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly <b>not</b> the same effect on \(\mathcal{X}_{x}\).
</p>
<div id="orga537ded" class="figure">
<p><img src="figs/2dof_actuator_external_forces.png" alt="2dof_actuator_external_forces.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Schematic representation of the stewart platform on top of a flexible support</p>
</div>
</div>
</div>
<div id="outline-container-orga9eb2fd" class="outline-3">
<h3 id="orga9eb2fd"><span class="section-number-3">1.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-3">
<div class="important" id="org4878fef">
<p>
The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform&rsquo;s base is fixed.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb6a1ef7" class="outline-2">
<h2 id="orgb6a1ef7"><span class="section-number-2">2</span> Comparison of the static transfer function and the Compliance matrix</h2>
<div class="outline-text-2" id="text-2">
<p>
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
</p>
</div>
<div id="outline-container-org3f1c253" class="outline-3">
<h3 id="org3f1c253"><span class="section-number-3">2.1</span> Analysis</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Initialization of the Stewart platform.
</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">'type_F'</span>, <span class="org-string">'universal_p'</span>, <span class="org-string">'type_M'</span>, <span class="org-string">'spherical_p'</span>);
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, <span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
</pre>
</div>
<p>
No flexibility below the Stewart platform and no payload.
</p>
<div class="org-src-container">
<pre class="src src-matlab">ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</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_model'</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">'/Controller'</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">'/Relative Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Position/Orientation of {B} w.r.t. {A}</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">'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">Gc = minreal(G<span class="org-type">*</span>inv(stewart.kinematics.J<span class="org-type">'</span>));
Gc.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>
<p>
Let&rsquo;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">4.7e-08</td>
<td class="org-right">-7.2e-19</td>
<td class="org-right">5.0e-18</td>
<td class="org-right">-8.9e-18</td>
<td class="org-right">3.2e-07</td>
<td class="org-right">9.9e-18</td>
</tr>
<tr>
<td class="org-right">4.7e-18</td>
<td class="org-right">4.7e-08</td>
<td class="org-right">-5.7e-18</td>
<td class="org-right">-3.2e-07</td>
<td class="org-right">-1.6e-17</td>
<td class="org-right">-1.7e-17</td>
</tr>
<tr>
<td class="org-right">3.3e-18</td>
<td class="org-right">-6.3e-18</td>
<td class="org-right">2.1e-08</td>
<td class="org-right">4.4e-17</td>
<td class="org-right">6.6e-18</td>
<td class="org-right">7.4e-18</td>
</tr>
<tr>
<td class="org-right">-3.2e-17</td>
<td class="org-right">-3.2e-07</td>
<td class="org-right">6.2e-18</td>
<td class="org-right">5.2e-06</td>
<td class="org-right">-3.5e-16</td>
<td class="org-right">6.3e-17</td>
</tr>
<tr>
<td class="org-right">3.2e-07</td>
<td class="org-right">2.7e-17</td>
<td class="org-right">4.8e-17</td>
<td class="org-right">-4.5e-16</td>
<td class="org-right">5.2e-06</td>
<td class="org-right">-1.2e-19</td>
</tr>
<tr>
<td class="org-right">4.0e-17</td>
<td class="org-right">-9.5e-17</td>
<td class="org-right">8.4e-18</td>
<td class="org-right">4.3e-16</td>
<td class="org-right">5.8e-16</td>
<td class="org-right">1.7e-06</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">4.7e-08</td>
<td class="org-right">-2.0e-24</td>
<td class="org-right">7.4e-25</td>
<td class="org-right">5.9e-23</td>
<td class="org-right">3.2e-07</td>
<td class="org-right">5.9e-24</td>
</tr>
<tr>
<td class="org-right">-7.1e-25</td>
<td class="org-right">4.7e-08</td>
<td class="org-right">2.9e-25</td>
<td class="org-right">-3.2e-07</td>
<td class="org-right">-5.4e-24</td>
<td class="org-right">-3.3e-23</td>
</tr>
<tr>
<td class="org-right">7.9e-26</td>
<td class="org-right">-6.4e-25</td>
<td class="org-right">2.1e-08</td>
<td class="org-right">1.9e-23</td>
<td class="org-right">5.3e-25</td>
<td class="org-right">-6.5e-40</td>
</tr>
<tr>
<td class="org-right">1.4e-23</td>
<td class="org-right">-3.2e-07</td>
<td class="org-right">1.3e-23</td>
<td class="org-right">5.2e-06</td>
<td class="org-right">4.9e-22</td>
<td class="org-right">-3.8e-24</td>
</tr>
<tr>
<td class="org-right">3.2e-07</td>
<td class="org-right">7.6e-24</td>
<td class="org-right">1.2e-23</td>
<td class="org-right">6.9e-22</td>
<td class="org-right">5.2e-06</td>
<td class="org-right">-2.6e-22</td>
</tr>
<tr>
<td class="org-right">7.3e-24</td>
<td class="org-right">-3.2e-23</td>
<td class="org-right">-1.6e-39</td>
<td class="org-right">9.9e-23</td>
<td class="org-right">-3.3e-22</td>
<td class="org-right">1.7e-06</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orge261263" class="outline-3">
<h3 id="orge261263"><span class="section-number-3">2.2</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-2">
<div class="important" id="org2428297">
<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>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-08 ven. 15:52</p>
</div>
</body>
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