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<h1 class="title">Identification of the Stewart Platform using Simscape</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orge8b6206">1. Modal Analysis of the Stewart Platform</a>
<ul>
<li><a href="#org40f9c57">1.1. Initialize the Stewart Platform</a></li>
<li><a href="#orgd9529ee">1.2. Identification</a></li>
<li><a href="#orgbdba4a6">1.3. Coordinate transformation</a></li>
<li><a href="#org11e3698">1.4. Analysis</a></li>
<li><a href="#org1db5fc4">1.5. Visualizing the modes</a></li>
</ul>
</li>
<li><a href="#orgfeed9a3">2. Transmissibility Analysis</a>
<ul>
<li><a href="#org7c6996a">2.1. Initialize the Stewart platform</a></li>
<li><a href="#org279dcc8">2.2. Transmissibility</a></li>
</ul>
</li>
<li><a href="#org3ad92e9">3. Compliance Analysis</a>
<ul>
<li><a href="#org5ba3096">3.1. Initialize the Stewart platform</a></li>
<li><a href="#org26cb46a">3.2. Compliance</a></li>
</ul>
</li>
<li><a href="#org51e266f">4. Functions</a>
<ul>
<li><a href="#org25ca725">4.1. Compute the Transmissibility</a>
<ul>
<li><a href="#orgeae7abf">Function description</a></li>
<li><a href="#orge4c0895">Optional Parameters</a></li>
<li><a href="#org17a8811">Identification of the Transmissibility Matrix</a></li>
<li><a href="#orgfd96322">Computation of the Frobenius norm</a></li>
</ul>
</li>
<li><a href="#orgb6e05b3">4.2. Compute the Compliance</a>
<ul>
<li><a href="#orgafb57d0">Function description</a></li>
<li><a href="#orga00af61">Optional Parameters</a></li>
<li><a href="#org2c35042">Identification of the Compliance Matrix</a></li>
<li><a href="#orgbc9a383">Computation of the Frobenius norm</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<p>
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
</p>
<ul class="org-ul">
<li><a href="#orgd142bb4">1</a></li>
<li><a href="#org5213401">2</a></li>
<li><a href="#org39baa25">3</a></li>
</ul>
<div id="outline-container-orge8b6206" class="outline-2">
<h2 id="orge8b6206"><span class="section-number-2">1</span> Modal Analysis of the Stewart Platform</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgd142bb4"></a>
</p>
</div>
<div id="outline-container-org40f9c57" class="outline-3">
<h3 id="org40f9c57"><span class="section-number-3">1.1</span> Initialize the Stewart Platform</h3>
<div class="outline-text-3" id="text-1-1">
<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 = 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);
</pre>
</div>
<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>
</div>
</div>
<div id="outline-container-orgd9529ee" class="outline-3">
<h3 id="orgd9529ee"><span class="section-number-3">1.2</span> Identification</h3>
<div class="outline-text-3" id="text-1-2">
<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>
io(io_i) = linio([mdl, <span class="org-string">'/Relative Motion Sensor'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Velocity 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);
<span class="org-comment">% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};</span>
<span class="org-comment">% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};</span>
</pre>
</div>
<p>
Let’s check the size of <code>G</code>:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> size(G)
</pre>
</div>
<pre class="example">
size(G)
State-space model with 12 outputs, 6 inputs, and 18 states.
'org_babel_eoe'
ans =
'org_babel_eoe'
</pre>
<p>
We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gm = minreal(G);
</pre>
</div>
<pre class="example">
Gm = minreal(G);
6 states removed.
</pre>
<p>
And indeed, we obtain 12 states.
</p>
</div>
</div>
<div id="outline-container-orgbdba4a6" class="outline-3">
<h3 id="orgbdba4a6"><span class="section-number-3">1.3</span> Coordinate transformation</h3>
<div class="outline-text-3" id="text-1-3">
<p>
We can perform the following transformation using the <code>ss2ss</code> command.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gt = ss2ss(Gm, Gm.C);
</pre>
</div>
<p>
Then, the <code>C</code> matrix of <code>Gt</code> is the unity matrix which means that the states of the state space model are equal to the measurements \(\bm{Y}\).
</p>
<p>
The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).
</p>
<p>
We could perform the transformation by hand:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> At = Gm.C<span class="org-type">*</span>Gm.A<span class="org-type">*</span>pinv(Gm.C);
Bt = Gm.C<span class="org-type">*</span>Gm.B;
Ct = eye(12);
Dt = zeros(12, 6);
Gt = ss(At, Bt, Ct, Dt);
</pre>
</div>
</div>
</div>
<div id="outline-container-org11e3698" class="outline-3">
<h3 id="org11e3698"><span class="section-number-3">1.4</span> Analysis</h3>
<div class="outline-text-3" id="text-1-4">
<div class="org-src-container">
<pre class="src src-matlab"> [V,D] = eig(Gt.A);
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Mode Number</th>
<th scope="col" class="org-right">Resonance Frequency [Hz]</th>
<th scope="col" class="org-right">Damping Ratio [%]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">780.6</td>
<td class="org-right">0.4</td>
</tr>
<tr>
<td class="org-right">2.0</td>
<td class="org-right">780.6</td>
<td class="org-right">0.3</td>
</tr>
<tr>
<td class="org-right">3.0</td>
<td class="org-right">903.9</td>
<td class="org-right">0.3</td>
</tr>
<tr>
<td class="org-right">4.0</td>
<td class="org-right">1061.4</td>
<td class="org-right">0.3</td>
</tr>
<tr>
<td class="org-right">5.0</td>
<td class="org-right">1061.4</td>
<td class="org-right">0.2</td>
</tr>
<tr>
<td class="org-right">6.0</td>
<td class="org-right">1269.6</td>
<td class="org-right">0.2</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org1db5fc4" class="outline-3">
<h3 id="org1db5fc4"><span class="section-number-3">1.5</span> Visualizing the modes</h3>
<div class="outline-text-3" id="text-1-5">
<p>
To visualize the i’th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).
</p>
<p>
\[ U(t) = e^{\alpha t} ( ) \]
</p>
<p>
Let’s first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> ws = imag(diag(D));
[ws,I] = sort(ws)
ws = ws(7<span class="org-type">:</span>end); I = I(7<span class="org-type">:</span>end);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(ws)</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> i_mode = I(<span class="org-constant">i</span>); <span class="org-comment">% the argument is the i'th mode</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> lambda_i = D(i_mode, i_mode);
xi_i = V(<span class="org-type">:</span>,i_mode);
a_i = real(lambda_i);
b_i = imag(lambda_i);
</pre>
</div>
<p>
Let do 10 periods of the mode.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> t = linspace(0, 10<span class="org-type">/</span>(imag(lambda_i)<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), 1000);
U_i = pinv(Gt.B) <span class="org-type">*</span> real(xi_i <span class="org-type">*</span> lambda_i <span class="org-type">*</span> (cos(b_i <span class="org-type">*</span> t) <span class="org-type">+</span> 1<span class="org-constant">i</span><span class="org-type">*</span>sin(b_i <span class="org-type">*</span> t)));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> U = timeseries(U_i, t);
</pre>
</div>
<p>
Simulation:
</p>
<div class="org-src-container">
<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simscape.mat'</span>);
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simscape</span>, <span class="org-string">'StopTime'</span>, num2str(t(<span class="org-variable-name">end</span>)));
<span class="org-matlab-simulink-keyword">sim</span>(mdl);
</pre>
</div>
<p>
Save the movie of the mode shape.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> smwritevideo(mdl, sprintf(<span class="org-string">'figs/mode%i'</span>, <span class="org-constant">i</span>), ...
<span class="org-string">'PlaybackSpeedRatio'</span>, 1<span class="org-type">/</span>(b_i<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>), ...
<span class="org-string">'FrameRate'</span>, 30, ...
<span class="org-string">'FrameSize'</span>, [800, 400]);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">end</span>
</pre>
</div>
<div id="orgd5bd1cd" class="figure">
<p><img src="figs/mode1.gif" alt="mode1.gif" />
</p>
<p><span class="figure-number">Figure 1: </span>Identified mode - 1</p>
</div>
<div id="org5c59f9a" class="figure">
<p><img src="figs/mode3.gif" alt="mode3.gif" />
</p>
<p><span class="figure-number">Figure 2: </span>Identified mode - 3</p>
</div>
<div id="org0f2e8c4" class="figure">
<p><img src="figs/mode5.gif" alt="mode5.gif" />
</p>
<p><span class="figure-number">Figure 3: </span>Identified mode - 5</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgfeed9a3" class="outline-2">
<h2 id="orgfeed9a3"><span class="section-number-2">2</span> Transmissibility Analysis</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org5213401"></a>
</p>
</div>
<div id="outline-container-org7c6996a" class="outline-3">
<h3 id="org7c6996a"><span class="section-number-3">2.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-2-1">
<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">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</p>
<div class="org-src-container">
<pre class="src src-matlab"> ground = initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'rot_point'</span>, stewart.platform_F.FO_A);
payload = initializePayload(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org279dcc8" class="outline-3">
<h3 id="org279dcc8"><span class="section-number-3">2.2</span> Transmissibility</h3>
<div class="outline-text-3" id="text-2-2">
<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">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
T = linearize(mdl, io, options);
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
T.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(1, 4, 1000);
<span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
ylim([1e<span class="org-type">-</span>5, 10]);
xlim([freqs(1), freqs(end)]);
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<p>
From (<a href="#citeproc_bib_item_1">Preumont et al. 2007</a>), one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
</p>
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
<div class="org-src-container">
<pre class="src src-matlab"> freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<p>
And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab"> Gamma = T_norm<span class="org-type">/</span>sqrt(6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-type">figure</span>;
plot(freqs, Gamma)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org3ad92e9" class="outline-2">
<h2 id="org3ad92e9"><span class="section-number-2">3</span> Compliance Analysis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org39baa25"></a>
</p>
</div>
<div id="outline-container-org5ba3096" class="outline-3">
<h3 id="org5ba3096"><span class="section-number-3">3.1</span> Initialize the Stewart platform</h3>
<div class="outline-text-3" id="text-3-1">
<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">'accelerometer'</span>, <span class="org-string">'freq'</span>, 5e3);
</pre>
</div>
<p>
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
</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">'rigid'</span>);
controller = initializeController(<span class="org-string">'type'</span>, <span class="org-string">'open-loop'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org26cb46a" class="outline-3">
<h3 id="org26cb46a"><span class="section-number-3">3.2</span> Compliance</h3>
<div class="outline-text-3" id="text-3-2">
<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">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
C = linearize(mdl, io, options);
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
C.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(1, 4, 1000);
<span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
ylim([1e<span class="org-type">-</span>10, 1e<span class="org-type">-</span>3]);
xlim([freqs(1), freqs(end)]);
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<p>
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-type">figure</span>;
plot(freqs, C_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org51e266f" class="outline-2">
<h2 id="org51e266f"><span class="section-number-2">4</span> Functions</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org25ca725" class="outline-3">
<h3 id="org25ca725"><span class="section-number-3">4.1</span> Compute the Transmissibility</h3>
<div class="outline-text-3" id="text-4-1">
<p>
<a id="org78f2be2"></a>
</p>
</div>
<div id="outline-container-orgeae7abf" class="outline-4">
<h4 id="orgeae7abf">Function description</h4>
<div class="outline-text-4" id="text-orgeae7abf">
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">function</span> <span class="org-variable-name">[T, T_norm, freqs]</span> = <span class="org-function-name">computeTransmissibility</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% computeTransmissibility -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - args - Structure with the following fields:</span>
<span class="org-comment">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - T [6x6 ss] - Transmissibility matrix</span>
<span class="org-comment">% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix</span>
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orge4c0895" class="outline-4">
<h4 id="orge4c0895">Optional Parameters</h4>
<div class="outline-text-4" id="text-orge4c0895">
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">arguments</span>
<span class="org-variable-name">args</span>.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
<span class="org-variable-name">args</span>.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-org17a8811" class="outline-4">
<h4 id="org17a8811">Identification of the Transmissibility Matrix</h4>
<div class="outline-text-4" id="text-org17a8811">
<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">'/Disturbances/D_w'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Base Motion [m, rad]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'output'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
T = linearize(mdl, io, options);
T.InputName = {<span class="org-string">'Wdx'</span>, <span class="org-string">'Wdy'</span>, <span class="org-string">'Wdz'</span>, <span class="org-string">'Wrx'</span>, <span class="org-string">'Wry'</span>, <span class="org-string">'Wrz'</span>};
T.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>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> p_handle = zeros(6<span class="org-type">*</span>6,1);
<span class="org-keyword">if</span> args.plots
fig = <span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
linkaxes(p_handle, <span class="org-string">'xy'</span>)
xlim([freqs(1), freqs(end)]);
ylim([1e<span class="org-type">-</span>5, 1e2]);
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
han.XLabel.Visible = <span class="org-string">'on'</span>;
han.YLabel.Visible = <span class="org-string">'on'</span>;
xlabel(han, <span class="org-string">'Frequency [Hz]'</span>);
ylabel(han, <span class="org-string">'Transmissibility [m/m]'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfd96322" class="outline-4">
<h4 id="orgfd96322">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-orgfd96322">
<div class="org-src-container">
<pre class="src src-matlab"> T_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
T_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(T, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> T_norm = T_norm<span class="org-type">/</span>sqrt(6);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">if</span> args.plots
<span class="org-type">figure</span>;
plot(freqs, T_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Transmissibility - Frobenius Norm'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb6e05b3" class="outline-3">
<h3 id="orgb6e05b3"><span class="section-number-3">4.2</span> Compute the Compliance</h3>
<div class="outline-text-3" id="text-4-2">
<p>
<a id="org13d7e8a"></a>
</p>
</div>
<div id="outline-container-orgafb57d0" class="outline-4">
<h4 id="orgafb57d0">Function description</h4>
<div class="outline-text-4" id="text-orgafb57d0">
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">function</span> <span class="org-variable-name">[C, C_norm, freqs]</span> = <span class="org-function-name">computeCompliance</span>(<span class="org-variable-name">args</span>)
<span class="org-comment">% computeCompliance -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [C, C_norm, freqs] = computeCompliance(args)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - args - Structure with the following fields:</span>
<span class="org-comment">% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm</span>
<span class="org-comment">% - freqs [] - Frequency vector to estimate the Frobenius norm</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - C [6x6 ss] - Compliance matrix</span>
<span class="org-comment">% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix</span>
<span class="org-comment">% - freqs [length(freqs)x1] - Frequency vector in [Hz]</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orga00af61" class="outline-4">
<h4 id="orga00af61">Optional Parameters</h4>
<div class="outline-text-4" id="text-orga00af61">
<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">arguments</span>
<span class="org-variable-name">args</span>.plots logical {mustBeNumericOrLogical} = <span class="org-constant">false</span>
<span class="org-variable-name">args</span>.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"> freqs = args.freqs;
</pre>
</div>
</div>
</div>
<div id="outline-container-org2c35042" class="outline-4">
<h4 id="org2c35042">Identification of the Compliance Matrix</h4>
<div class="outline-text-4" id="text-org2c35042">
<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">'/Disturbances/F_ext'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% External forces [N, N*m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/Absolute Motion Sensor'</span>], 1, <span class="org-string">'output'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Absolute Motion [m, rad]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Run the linearization</span></span>
C = linearize(mdl, io, options);
C.InputName = {<span class="org-string">'Fdx'</span>, <span class="org-string">'Fdy'</span>, <span class="org-string">'Fdz'</span>, <span class="org-string">'Mdx'</span>, <span class="org-string">'Mdy'</span>, <span class="org-string">'Mdz'</span>};
C.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>
If wanted, the 6x6 transmissibility matrix is plotted.
</p>
<div class="org-src-container">
<pre class="src src-matlab"> p_handle = zeros(6<span class="org-type">*</span>6,1);
<span class="org-keyword">if</span> args.plots
fig = <span class="org-type">figure</span>;
<span class="org-keyword">for</span> <span class="org-variable-name">ix</span> = <span class="org-constant">1:6</span>
<span class="org-keyword">for</span> <span class="org-variable-name">iy</span> = <span class="org-constant">1:6</span>
p_handle((ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy) = subplot(6, 6, (ix<span class="org-type">-</span>1)<span class="org-type">*</span>6 <span class="org-type">+</span> iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, <span class="org-string">'Hz'</span>))), <span class="org-string">'k-'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
<span class="org-keyword">if</span> ix <span class="org-type"><</span> 6
xticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> iy <span class="org-type">></span> 1
yticklabels({});
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
linkaxes(p_handle, <span class="org-string">'xy'</span>)
xlim([freqs(1), freqs(end)]);
han = <span class="org-type">axes</span>(fig, <span class="org-string">'visible'</span>, <span class="org-string">'off'</span>);
han.XLabel.Visible = <span class="org-string">'on'</span>;
han.YLabel.Visible = <span class="org-string">'on'</span>;
xlabel(han, <span class="org-string">'Frequency [Hz]'</span>);
ylabel(han, <span class="org-string">'Compliance [m/N, rad/(N*m)]'</span>);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgbc9a383" class="outline-4">
<h4 id="orgbc9a383">Computation of the Frobenius norm</h4>
<div class="outline-text-4" id="text-orgbc9a383">
<div class="org-src-container">
<pre class="src src-matlab"> freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(freqs)</span>
C_norm(<span class="org-constant">i</span>) = sqrt(trace(freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">*</span>freqresp(C, freqs(<span class="org-constant">i</span>), <span class="org-string">'Hz'</span>)<span class="org-type">'</span>));
<span class="org-keyword">end</span>
</pre>
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<div class="org-src-container">
<pre class="src src-matlab"> <span class="org-keyword">if</span> args.plots
<span class="org-type">figure</span>;
plot(freqs, C_norm)
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'XScale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'YScale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>);
ylabel(<span class="org-string">'Compliance - Frobenius Norm'</span>);
<span class="org-keyword">end</span>
</pre>
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” <i>Journal of Sound and Vibration</i> 300 (3-5):644–61. <a href="https://doi.org/10.1016/j.jsv.2006.07.050">https://doi.org/10.1016/j.jsv.2006.07.050</a>.</div>
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-08 ven. 15:29</p>
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