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<head> <head>
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Kinematic Study of the Stewart Platform</title> <title>Kinematic Study of the Stewart Platform</title>
@ -268,69 +268,69 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org2ef1540">1. Jacobian Analysis</a> <li><a href="#orgc15a3c4">1. Jacobian Analysis</a>
<ul> <ul>
<li><a href="#orgecc1877">1.1. Jacobian Computation</a></li> <li><a href="#orgfe46499">1.1. Jacobian Computation</a></li>
<li><a href="#org1236ee4">1.2. Jacobian - Velocity loop closure</a></li> <li><a href="#org47a46f9">1.2. Jacobian - Velocity loop closure</a></li>
<li><a href="#org67f6380">1.3. Jacobian - Static Force Transformation</a></li> <li><a href="#org39fe037">1.3. Jacobian - Static Force Transformation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3fca8c2">2. Stiffness Analysis</a> <li><a href="#org022818a">2. Stiffness Analysis</a>
<ul> <ul>
<li><a href="#orgdd0ba48">2.1. Computation of the Stiffness and Compliance Matrix</a></li> <li><a href="#orgd894508">2.1. Computation of the Stiffness and Compliance Matrix</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org436f7d9">3. Forward and Inverse Kinematics</a> <li><a href="#org8111618">3. Forward and Inverse Kinematics</a>
<ul> <ul>
<li><a href="#orgcc359db">3.1. Inverse Kinematics</a></li> <li><a href="#org7985a6a">3.1. Inverse Kinematics</a></li>
<li><a href="#org016b9d2">3.2. Forward Kinematics</a></li> <li><a href="#orgf5c4aa7">3.2. Forward Kinematics</a></li>
<li><a href="#orged25559">3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</a></li> <li><a href="#orgbb81fc8">3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</a></li>
<li><a href="#org3c74917">3.4. Estimation of the range validity of the approximate inverse kinematics</a> <li><a href="#org62d03d6">3.4. Estimation of the range validity of the approximate inverse kinematics</a>
<ul> <ul>
<li><a href="#org3bf1230">3.4.1. Stewart architecture definition</a></li> <li><a href="#orga5bc217">3.4.1. Stewart architecture definition</a></li>
<li><a href="#org89cb768">3.4.2. Comparison for &ldquo;pure&rdquo; translations</a></li> <li><a href="#org871d94a">3.4.2. Comparison for &ldquo;pure&rdquo; translations</a></li>
<li><a href="#org769a9ee">3.4.3. Conclusion</a></li> <li><a href="#orgc70d027">3.4.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org6ec799b">4. Estimated required actuator stroke from specified platform mobility</a> <li><a href="#orgac7950e">4. Estimated required actuator stroke from specified platform mobility</a>
<ul> <ul>
<li><a href="#orgf87c3d4">4.1. Stewart architecture definition</a></li> <li><a href="#org4c2a482">4.1. Stewart architecture definition</a></li>
<li><a href="#org0cde976">4.2. Wanted translations and rotations</a></li> <li><a href="#org0e2cd41">4.2. Wanted translations and rotations</a></li>
<li><a href="#orga78e6b0">4.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li> <li><a href="#org546da0e">4.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li>
<li><a href="#orgc4c5bf4">4.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li> <li><a href="#org782ea0a">4.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org1039168">5. Estimated platform mobility from specified actuator stroke</a> <li><a href="#org531e516">5. Estimated platform mobility from specified actuator stroke</a>
<ul> <ul>
<li><a href="#org672d7ea">5.1. Stewart architecture definition</a></li> <li><a href="#org741de4c">5.1. Stewart architecture definition</a></li>
<li><a href="#org783aeb0">5.2. <span class="todo TODO">TODO</span> Approximate Forward Dynamics</a></li> <li><a href="#orgbc52ab0">5.2. Pure translations</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org75b7ddb">6. Functions</a> <li><a href="#org09341b5">6. Functions</a>
<ul> <ul>
<li><a href="#org73a18e5">6.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a> <li><a href="#orga28bc27">6.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<ul> <ul>
<li><a href="#orgaadc69c">Function description</a></li> <li><a href="#org129055c">Function description</a></li>
<li><a href="#org1f030f2">Compute Jacobian Matrix</a></li> <li><a href="#orgffa7442">Compute Jacobian Matrix</a></li>
<li><a href="#org0940f79">Compute Stiffness Matrix</a></li> <li><a href="#org318cdc7">Compute Stiffness Matrix</a></li>
<li><a href="#org6c60aa8">Compute Compliance Matrix</a></li> <li><a href="#orgeb50736">Compute Compliance Matrix</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org89b61d1">6.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a> <li><a href="#orgccfd192">6.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<ul> <ul>
<li><a href="#org985470b">Function description</a></li> <li><a href="#org14c61e7">Function description</a></li>
<li><a href="#org2b6ddb8">Optional Parameters</a></li> <li><a href="#orgb04bb62">Optional Parameters</a></li>
<li><a href="#orgf420378">Theory</a></li> <li><a href="#orge635491">Theory</a></li>
<li><a href="#orge5ae615">Compute</a></li> <li><a href="#orgbc14960">Compute</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgc366b74">6.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a> <li><a href="#org1f75518">6.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
<ul> <ul>
<li><a href="#orgd6d4e91">Function description</a></li> <li><a href="#org665e0fd">Function description</a></li>
<li><a href="#orgbbf49bb">Optional Parameters</a></li> <li><a href="#org89c6f2b">Optional Parameters</a></li>
<li><a href="#orgad68b5d">Computation</a></li> <li><a href="#org3ae22c5">Computation</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -353,16 +353,17 @@ In this analysis, the relation between the geometrical parameters of the manipul
The current document is divided in the following sections: The current document is divided in the following sections:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Section <a href="#org9eebb1e">1</a>: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform.</li> <li>Section <a href="#org9f3c0b7">1</a>: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform.</li>
<li>Section <a href="#org1b84fbb">2</a>: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut.</li> <li>Section <a href="#orgbd08620">2</a>: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut.</li>
<li>Section <a href="#org4aea27b">3</a>: The Forward and Inverse kinematic problems are presented.</li> <li>Section <a href="#orgde74d25">3</a>: The Forward and Inverse kinematic problems are presented.</li>
<li>Section <a href="#orge3719af">4</a>: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform.</li>
</ul> </ul>
<div id="outline-container-org2ef1540" class="outline-2"> <div id="outline-container-orgc15a3c4" class="outline-2">
<h2 id="org2ef1540"><span class="section-number-2">1</span> Jacobian Analysis</h2> <h2 id="orgc15a3c4"><span class="section-number-2">1</span> Jacobian Analysis</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="org9eebb1e"></a> <a id="org9f3c0b7"></a>
</p> </p>
<p> <p>
From <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>: From <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>:
@ -373,8 +374,8 @@ The Jacobian matrix not only reveals the <b>relation between the joint variable
</p> </p>
</blockquote> </blockquote>
</div> </div>
<div id="outline-container-orgecc1877" class="outline-3"> <div id="outline-container-orgfe46499" class="outline-3">
<h3 id="orgecc1877"><span class="section-number-3">1.1</span> Jacobian Computation</h3> <h3 id="orgfe46499"><span class="section-number-3">1.1</span> Jacobian Computation</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
If we note: If we note:
@ -399,7 +400,7 @@ Then, we can compute the Jacobian with the following equation (the superscript \
\end{equation*} \end{equation*}
<p> <p>
The Jacobian matrix \(\bm{J}\) can be computed using the <code>computeJacobian</code> function (accessible <a href="#orge335390">here</a>). The Jacobian matrix \(\bm{J}\) can be computed using the <code>computeJacobian</code> function (accessible <a href="#orga4eba5e">here</a>).
For instance: For instance:
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -417,8 +418,8 @@ This will add three new matrix to the <code>stewart</code> structure:
</div> </div>
</div> </div>
<div id="outline-container-org1236ee4" class="outline-3"> <div id="outline-container-org47a46f9" class="outline-3">
<h3 id="org1236ee4"><span class="section-number-3">1.2</span> Jacobian - Velocity loop closure</h3> <h3 id="org47a46f9"><span class="section-number-3">1.2</span> Jacobian - Velocity loop closure</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
The Jacobian matrix links the input joint rate \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\) of each strut to the output twist vector of the mobile platform is denoted by \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\): The Jacobian matrix links the input joint rate \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\) of each strut to the output twist vector of the mobile platform is denoted by \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\):
@ -441,13 +442,13 @@ If the Jacobian matrix is inversible, we can also compute \(\dot{\bm{\mathcal{X}
<p> <p>
The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements. The Jacobian matrix can also be used to approximate forward and inverse kinematics for small displacements.
This is explained in section <a href="#org4dd3cac">3.3</a>. This is explained in section <a href="#org907e556">3.3</a>.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org67f6380" class="outline-3"> <div id="outline-container-org39fe037" class="outline-3">
<h3 id="org67f6380"><span class="section-number-3">1.3</span> Jacobian - Static Force Transformation</h3> <h3 id="org39fe037"><span class="section-number-3">1.3</span> Jacobian - Static Force Transformation</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
If we note: If we note:
@ -474,19 +475,19 @@ If the Jacobian matrix is inversible, we also have the following relation:
</div> </div>
</div> </div>
<div id="outline-container-org3fca8c2" class="outline-2"> <div id="outline-container-org022818a" class="outline-2">
<h2 id="org3fca8c2"><span class="section-number-2">2</span> Stiffness Analysis</h2> <h2 id="org022818a"><span class="section-number-2">2</span> Stiffness Analysis</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org1b84fbb"></a> <a id="orgbd08620"></a>
</p> </p>
<p> <p>
Here, we focus on the deflections of the manipulator moving platform that are the result of the external applied wrench to the mobile platform. Here, we focus on the deflections of the manipulator moving platform that are the result of the external applied wrench to the mobile platform.
The amount of these deflections are a function of the applied wrench as well as the manipulator <b>structural stiffness</b>. The amount of these deflections are a function of the applied wrench as well as the manipulator <b>structural stiffness</b>.
</p> </p>
</div> </div>
<div id="outline-container-orgdd0ba48" class="outline-3"> <div id="outline-container-orgd894508" class="outline-3">
<h3 id="orgdd0ba48"><span class="section-number-3">2.1</span> Computation of the Stiffness and Compliance Matrix</h3> <h3 id="orgd894508"><span class="section-number-3">2.1</span> Computation of the Stiffness and Compliance Matrix</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
As explain in <a href="stewart-architecture.html">this</a> document, each Actuator is modeled by 3 elements in parallel: As explain in <a href="stewart-architecture.html">this</a> document, each Actuator is modeled by 3 elements in parallel:
@ -537,24 +538,24 @@ The compliance matrix of a manipulator shows the mapping of the moving platform
\end{equation*} \end{equation*}
<p> <p>
The stiffness and compliance matrices are computed using the <code>computeJacobian</code> function (accessible <a href="#orge335390">here</a>). The stiffness and compliance matrices are computed using the <code>computeJacobian</code> function (accessible <a href="#orga4eba5e">here</a>).
</p> </p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org436f7d9" class="outline-2"> <div id="outline-container-org8111618" class="outline-2">
<h2 id="org436f7d9"><span class="section-number-2">3</span> Forward and Inverse Kinematics</h2> <h2 id="org8111618"><span class="section-number-2">3</span> Forward and Inverse Kinematics</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="org4aea27b"></a> <a id="orgde74d25"></a>
</p> </p>
</div> </div>
<div id="outline-container-orgcc359db" class="outline-3"> <div id="outline-container-org7985a6a" class="outline-3">
<h3 id="orgcc359db"><span class="section-number-3">3.1</span> Inverse Kinematics</h3> <h3 id="org7985a6a"><span class="section-number-3">3.1</span> Inverse Kinematics</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
<a id="org4ecda1c"></a> <a id="orgc35b397"></a>
</p> </p>
<blockquote> <blockquote>
@ -583,16 +584,16 @@ Otherwise, the solution gives complex numbers.
</p> </p>
<p> <p>
This inverse kinematic solution can be obtained using the function <code>inverseKinematics</code> (described <a href="#org1478833">here</a>). This inverse kinematic solution can be obtained using the function <code>inverseKinematics</code> (described <a href="#org38cce35">here</a>).
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org016b9d2" class="outline-3"> <div id="outline-container-orgf5c4aa7" class="outline-3">
<h3 id="org016b9d2"><span class="section-number-3">3.2</span> Forward Kinematics</h3> <h3 id="orgf5c4aa7"><span class="section-number-3">3.2</span> Forward Kinematics</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
<a id="orgf5d098b"></a> <a id="orga5abf69"></a>
</p> </p>
<blockquote> <blockquote>
@ -611,22 +612,22 @@ In a next section, an approximate solution of the forward kinematics problem is
</div> </div>
</div> </div>
<div id="outline-container-orged25559" class="outline-3"> <div id="outline-container-orgbb81fc8" class="outline-3">
<h3 id="orged25559"><span class="section-number-3">3.3</span> Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</h3> <h3 id="orgbb81fc8"><span class="section-number-3">3.3</span> Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p> <p>
<a id="org4dd3cac"></a> <a id="org907e556"></a>
</p> </p>
<p> <p>
For small displacements mobile platform displacement \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around \(\bm{\mathcal{X}}_0\), the associated joint displacement can be computed using the Jacobian: For small displacements mobile platform displacement \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\) around \(\bm{\mathcal{X}}_0\), the associated joint displacement can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
</p> </p>
\begin{equation*} \begin{equation*}
\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
\end{equation*} \end{equation*}
<p> <p>
Similarly, for small joint displacements \(\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T\) around \(\bm{\mathcal{L}}_0\), it is possible to find the induced small displacement of the mobile platform: Similarly, for small joint displacements \(\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T\) around \(\bm{\mathcal{L}}_0\), it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
</p> </p>
\begin{equation*} \begin{equation*}
\delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}
@ -638,13 +639,13 @@ As the inverse kinematic can be easily solved exactly this is not much useful, h
</p> </p>
<p> <p>
The function <code>forwardKinematicsApprox</code> (described <a href="#org5e0bc69">here</a>) can be used to solve the forward kinematic problem using the Jacobian matrix. The function <code>forwardKinematicsApprox</code> (described <a href="#org0199633">here</a>) can be used to solve the forward kinematic problem using the Jacobian matrix.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org3c74917" class="outline-3"> <div id="outline-container-org62d03d6" class="outline-3">
<h3 id="org3c74917"><span class="section-number-3">3.4</span> Estimation of the range validity of the approximate inverse kinematics</h3> <h3 id="org62d03d6"><span class="section-number-3">3.4</span> Estimation of the range validity of the approximate inverse kinematics</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix. As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix.
@ -658,8 +659,8 @@ This will also gives us the range for which the approximate forward kinematic is
</p> </p>
</div> </div>
<div id="outline-container-org3bf1230" class="outline-4"> <div id="outline-container-orga5bc217" class="outline-4">
<h4 id="org3bf1230"><span class="section-number-4">3.4.1</span> Stewart architecture definition</h4> <h4 id="orga5bc217"><span class="section-number-4">3.4.1</span> Stewart architecture definition</h4>
<div class="outline-text-4" id="text-3-4-1"> <div class="outline-text-4" id="text-3-4-1">
<p> <p>
We first define some general Stewart architecture. We first define some general Stewart architecture.
@ -679,8 +680,8 @@ stewart = computeJacobian(stewart);
</div> </div>
</div> </div>
<div id="outline-container-org89cb768" class="outline-4"> <div id="outline-container-org871d94a" class="outline-4">
<h4 id="org89cb768"><span class="section-number-4">3.4.2</span> Comparison for &ldquo;pure&rdquo; translations</h4> <h4 id="org871d94a"><span class="section-number-4">3.4.2</span> Comparison for &ldquo;pure&rdquo; translations</h4>
<div class="outline-text-4" id="text-3-4-2"> <div class="outline-text-4" id="text-3-4-2">
<p> <p>
Let&rsquo;s first compare the perfect and approximate solution of the inverse for pure \(x\) translations. Let&rsquo;s first compare the perfect and approximate solution of the inverse for pure \(x\) translations.
@ -688,8 +689,8 @@ Let&rsquo;s first compare the perfect and approximate solution of the inverse fo
<p> <p>
We compute the approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement. We compute the approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement.
The estimate required strut stroke for both the approximate and exact solutions are shown in Figure <a href="#org3e44240">1</a>. The estimate required strut stroke for both the approximate and exact solutions are shown in Figure <a href="#org9fbaffc">1</a>.
The relative strut length displacement is shown in Figure <a href="#org2b96122">2</a>. The relative strut length displacement is shown in Figure <a href="#org76bebd2">2</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Xrs = logspace(<span class="org-type">-</span>6, <span class="org-type">-</span>1, 100); <span class="org-comment">% Wanted X translation of the mobile platform [m]</span> <pre class="src src-matlab">Xrs = logspace(<span class="org-type">-</span>6, <span class="org-type">-</span>1, 100); <span class="org-comment">% Wanted X translation of the mobile platform [m]</span>
@ -706,14 +707,14 @@ Ls_exact = zeros(6, length(Xrs));
</div> </div>
<div id="org3e44240" class="figure"> <div id="org9fbaffc" class="figure">
<p><img src="figs/inverse_kinematics_approx_validity_x_translation.png" alt="inverse_kinematics_approx_validity_x_translation.png" /> <p><img src="figs/inverse_kinematics_approx_validity_x_translation.png" alt="inverse_kinematics_approx_validity_x_translation.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Comparison of the Approximate solution and True solution for the Inverse kinematic problem (<a href="./figs/inverse_kinematics_approx_validity_x_translation.png">png</a>, <a href="./figs/inverse_kinematics_approx_validity_x_translation.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 1: </span>Comparison of the Approximate solution and True solution for the Inverse kinematic problem (<a href="./figs/inverse_kinematics_approx_validity_x_translation.png">png</a>, <a href="./figs/inverse_kinematics_approx_validity_x_translation.pdf">pdf</a>)</p>
</div> </div>
<div id="org2b96122" class="figure"> <div id="org76bebd2" class="figure">
<p><img src="figs/inverse_kinematics_approx_validity_x_translation_relative.png" alt="inverse_kinematics_approx_validity_x_translation_relative.png" /> <p><img src="figs/inverse_kinematics_approx_validity_x_translation_relative.png" alt="inverse_kinematics_approx_validity_x_translation_relative.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Relative length error by using the Approximate solution of the Inverse kinematic problem (<a href="./figs/inverse_kinematics_approx_validity_x_translation_relative.png">png</a>, <a href="./figs/inverse_kinematics_approx_validity_x_translation_relative.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 2: </span>Relative length error by using the Approximate solution of the Inverse kinematic problem (<a href="./figs/inverse_kinematics_approx_validity_x_translation_relative.png">png</a>, <a href="./figs/inverse_kinematics_approx_validity_x_translation_relative.pdf">pdf</a>)</p>
@ -721,8 +722,8 @@ Ls_exact = zeros(6, length(Xrs));
</div> </div>
</div> </div>
<div id="outline-container-org769a9ee" class="outline-4"> <div id="outline-container-orgc70d027" class="outline-4">
<h4 id="org769a9ee"><span class="section-number-4">3.4.3</span> Conclusion</h4> <h4 id="orgc70d027"><span class="section-number-4">3.4.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-3-4-3"> <div class="outline-text-4" id="text-3-4-3">
<p> <p>
For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct. For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
@ -732,11 +733,11 @@ For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod
</div> </div>
</div> </div>
<div id="outline-container-org6ec799b" class="outline-2"> <div id="outline-container-orgac7950e" class="outline-2">
<h2 id="org6ec799b"><span class="section-number-2">4</span> Estimated required actuator stroke from specified platform mobility</h2> <h2 id="orgac7950e"><span class="section-number-2">4</span> Estimated required actuator stroke from specified platform mobility</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="org8780473"></a> <a id="orge3719af"></a>
</p> </p>
<p> <p>
Let&rsquo;s say one want to design a Stewart platform with some specified mobility (position and orientation). Let&rsquo;s say one want to design a Stewart platform with some specified mobility (position and orientation).
@ -744,8 +745,8 @@ One may want to determine the required actuator stroke required to obtain the sp
This is what is analyzed in this section. This is what is analyzed in this section.
</p> </p>
</div> </div>
<div id="outline-container-orgf87c3d4" class="outline-3"> <div id="outline-container-org4c2a482" class="outline-3">
<h3 id="orgf87c3d4"><span class="section-number-3">4.1</span> Stewart architecture definition</h3> <h3 id="org4c2a482"><span class="section-number-3">4.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
Let&rsquo;s first define the Stewart platform architecture that we want to study. Let&rsquo;s first define the Stewart platform architecture that we want to study.
@ -765,8 +766,8 @@ stewart = computeJacobian(stewart);
</div> </div>
</div> </div>
<div id="outline-container-org0cde976" class="outline-3"> <div id="outline-container-org0e2cd41" class="outline-3">
<h3 id="org0cde976"><span class="section-number-3">4.2</span> Wanted translations and rotations</h3> <h3 id="org0e2cd41"><span class="section-number-3">4.2</span> Wanted translations and rotations</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
Let&rsquo;s now define the wanted extreme translations and rotations. Let&rsquo;s now define the wanted extreme translations and rotations.
@ -783,8 +784,8 @@ Rz_max = 0; <span class="org-comment">% Rotation [rad]</span>
</div> </div>
</div> </div>
<div id="outline-container-orga78e6b0" class="outline-3"> <div id="outline-container-org546da0e" class="outline-3">
<h3 id="orga78e6b0"><span class="section-number-3">4.3</span> Needed stroke for &ldquo;pure&rdquo; rotations or translations</h3> <h3 id="org546da0e"><span class="section-number-3">4.3</span> Needed stroke for &ldquo;pure&rdquo; rotations or translations</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<p> <p>
As a first estimation, we estimate the needed actuator stroke for &ldquo;pure&rdquo; rotations and translation. As a first estimation, we estimate the needed actuator stroke for &ldquo;pure&rdquo; rotations and translation.
@ -815,8 +816,8 @@ This is surely a low estimation of the required stroke.
</div> </div>
</div> </div>
<div id="outline-container-orgc4c5bf4" class="outline-3"> <div id="outline-container-org782ea0a" class="outline-3">
<h3 id="orgc4c5bf4"><span class="section-number-3">4.4</span> Needed stroke for &ldquo;combined&rdquo; rotations or translations</h3> <h3 id="org782ea0a"><span class="section-number-3">4.4</span> Needed stroke for &ldquo;combined&rdquo; rotations or translations</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<p> <p>
We know would like to have a more precise estimation. We know would like to have a more precise estimation.
@ -1136,23 +1137,23 @@ This is probably a much realistic estimation of the required actuator stroke.
</div> </div>
</div> </div>
<div id="outline-container-org1039168" class="outline-2"> <div id="outline-container-org531e516" class="outline-2">
<h2 id="org1039168"><span class="section-number-2">5</span> Estimated platform mobility from specified actuator stroke</h2> <h2 id="org531e516"><span class="section-number-2">5</span> Estimated platform mobility from specified actuator stroke</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="org0ac5fe9"></a> <a id="orgf61fb6c"></a>
</p> </p>
<p> <p>
Here, from some value of the actuator stroke, we would like to estimate the mobility of the Stewart platform. Here, from some value of the actuator stroke, we would like to estimate the mobility of the Stewart platform.
</p> </p>
<p> <p>
As explained in section <a href="#org4aea27b">3</a>, the forward kinematic problem of the Stewart platform is quite difficult to solve. As explained in section <a href="#orgde74d25">3</a>, the forward kinematic problem of the Stewart platform is quite difficult to solve.
However, for small displacements, we can use the Jacobian as an approximate solution. However, for small displacements, we can use the Jacobian as an approximate solution.
</p> </p>
</div> </div>
<div id="outline-container-org672d7ea" class="outline-3"> <div id="outline-container-org741de4c" class="outline-3">
<h3 id="org672d7ea"><span class="section-number-3">5.1</span> Stewart architecture definition</h3> <h3 id="org741de4c"><span class="section-number-3">5.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<p> <p>
Let&rsquo;s first define the Stewart platform architecture that we want to study. Let&rsquo;s first define the Stewart platform architecture that we want to study.
@ -1181,23 +1182,100 @@ L_max = 50e<span class="org-type">-</span>6; <span class="org-comment">% [m]</s
</div> </div>
</div> </div>
<div id="outline-container-org783aeb0" class="outline-3"> <div id="outline-container-orgbc52ab0" class="outline-3">
<h3 id="org783aeb0"><span class="section-number-3">5.2</span> <span class="todo TODO">TODO</span> Approximate Forward Dynamics</h3> <h3 id="orgbc52ab0"><span class="section-number-3">5.2</span> Pure translations</h3>
<div class="outline-text-3" id="text-5-2">
<p>
Let&rsquo;s first estimate the mobility in translation when the orientation of the Stewart platform stays the same.
</p>
<p>
As shown previously, for such small stroke, we can use the approximate Forward Dynamics solution using the Jacobian matrix:
</p>
\begin{equation*}
\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
\end{equation*}
<p>
To obtain the mobility &ldquo;volume&rdquo; attainable by the Stewart platform when it&rsquo;s orientation is set to zero, we use the spherical coordinate \((r, \theta, \phi)\).
</p>
<p>
For each possible value of \((\theta, \phi)\), we compute the maximum radius \(r\) attainable with the constraint that the stroke of each actuator should be between <code>L_min</code> and <code>L_max</code>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">thetas = linspace(0, <span class="org-constant">pi</span>, 50);
phis = linspace(0, 2<span class="org-type">*</span><span class="org-constant">pi</span>, 50);
rs = zeros(length(thetas), length(phis));
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(thetas)</span>
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">j</span></span> = <span class="org-constant">1:length(phis)</span>
Tx = sin(thetas(<span class="org-constant">i</span>))<span class="org-type">*</span>cos(phis(<span class="org-constant">j</span>));
Ty = sin(thetas(<span class="org-constant">i</span>))<span class="org-type">*</span>sin(phis(<span class="org-constant">j</span>));
Tz = cos(thetas(<span class="org-constant">i</span>));
dL = stewart.J<span class="org-type">*</span>[Tx; Ty; Tz; 0; 0; 0;]; <span class="org-comment">% dL required for 1m displacement in theta/phi direction</span>
rs(<span class="org-constant">i</span>, <span class="org-constant">j</span>) = max([dL(dL<span class="org-type">&lt;</span>0)<span class="org-type">*</span>L_min; dL(dL<span class="org-type">&gt;</span>0)<span class="org-type">*</span>L_max]);
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
<p>
Now that we have found the corresponding radius \(r\), we plot the obtained mobility.
We can also approximate the mobility by a sphere with a radius equal to the minimum obtained value of \(r\), this is however a pessimistic estimation of the mobility.
</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" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right"><code>L_min</code> [\(\mu m\)]</th>
<th scope="col" class="org-right"><code>L_max</code> [\(\mu m\)]</th>
<th scope="col" class="org-right"><code>R</code> [\(\mu m\)]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">-50.0</td>
<td class="org-right">50.0</td>
<td class="org-right">31.5</td>
</tr>
</tbody>
</table>
<div id="org16756e7" class="figure">
<p><img src="figs/mobility_translations_null_rotation.png" alt="mobility_translations_null_rotation.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Obtain mobility of the Stewart platform for zero rotations (<a href="./figs/mobility_translations_null_rotation.png">png</a>, <a href="./figs/mobility_translations_null_rotation.pdf">pdf</a>)</p>
</div>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org75b7ddb" class="outline-2"> <div id="outline-container-org09341b5" class="outline-2">
<h2 id="org75b7ddb"><span class="section-number-2">6</span> Functions</h2> <h2 id="org09341b5"><span class="section-number-2">6</span> Functions</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<p> <p>
<a id="orgfd7da28"></a> <a id="org2d73657"></a>
</p> </p>
</div> </div>
<div id="outline-container-org73a18e5" class="outline-3"> <div id="outline-container-orga28bc27" class="outline-3">
<h3 id="org73a18e5"><span class="section-number-3">6.1</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3> <h3 id="orga28bc27"><span class="section-number-3">6.1</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
<div class="outline-text-3" id="text-6-1"> <div class="outline-text-3" id="text-6-1">
<p> <p>
<a id="orge335390"></a> <a id="orga4eba5e"></a>
</p> </p>
<p> <p>
@ -1205,9 +1283,9 @@ This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</p> </p>
</div> </div>
<div id="outline-container-orgaadc69c" class="outline-4"> <div id="outline-container-org129055c" class="outline-4">
<h4 id="orgaadc69c">Function description</h4> <h4 id="org129055c">Function description</h4>
<div class="outline-text-4" id="text-orgaadc69c"> <div class="outline-text-4" id="text-org129055c">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">computeJacobian</span>(<span class="org-variable-name">stewart</span>) <pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[stewart]</span> = <span class="org-function-name">computeJacobian</span>(<span class="org-variable-name">stewart</span>)
<span class="org-comment">% computeJacobian -</span> <span class="org-comment">% computeJacobian -</span>
@ -1229,9 +1307,9 @@ This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org1f030f2" class="outline-4"> <div id="outline-container-orgffa7442" class="outline-4">
<h4 id="org1f030f2">Compute Jacobian Matrix</h4> <h4 id="orgffa7442">Compute Jacobian Matrix</h4>
<div class="outline-text-4" id="text-org1f030f2"> <div class="outline-text-4" id="text-orgffa7442">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">stewart.J = [stewart.As<span class="org-type">'</span> , cross(stewart.Ab, stewart.As)<span class="org-type">'</span>]; <pre class="src src-matlab">stewart.J = [stewart.As<span class="org-type">'</span> , cross(stewart.Ab, stewart.As)<span class="org-type">'</span>];
</pre> </pre>
@ -1239,9 +1317,9 @@ This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org0940f79" class="outline-4"> <div id="outline-container-org318cdc7" class="outline-4">
<h4 id="org0940f79">Compute Stiffness Matrix</h4> <h4 id="org318cdc7">Compute Stiffness Matrix</h4>
<div class="outline-text-4" id="text-org0940f79"> <div class="outline-text-4" id="text-org318cdc7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">stewart.K = stewart.J<span class="org-type">'*</span>diag(stewart.Ki)<span class="org-type">*</span>stewart.J; <pre class="src src-matlab">stewart.K = stewart.J<span class="org-type">'*</span>diag(stewart.Ki)<span class="org-type">*</span>stewart.J;
</pre> </pre>
@ -1249,9 +1327,9 @@ This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org6c60aa8" class="outline-4"> <div id="outline-container-orgeb50736" class="outline-4">
<h4 id="org6c60aa8">Compute Compliance Matrix</h4> <h4 id="orgeb50736">Compute Compliance Matrix</h4>
<div class="outline-text-4" id="text-org6c60aa8"> <div class="outline-text-4" id="text-orgeb50736">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">stewart.C = inv(stewart.K); <pre class="src src-matlab">stewart.C = inv(stewart.K);
</pre> </pre>
@ -1260,11 +1338,11 @@ This Matlab function is accessible <a href="src/computeJacobian.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org89b61d1" class="outline-3"> <div id="outline-container-orgccfd192" class="outline-3">
<h3 id="org89b61d1"><span class="section-number-3">6.2</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3> <h3 id="orgccfd192"><span class="section-number-3">6.2</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
<div class="outline-text-3" id="text-6-2"> <div class="outline-text-3" id="text-6-2">
<p> <p>
<a id="org1478833"></a> <a id="org38cce35"></a>
</p> </p>
<p> <p>
@ -1272,9 +1350,9 @@ This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
</p> </p>
</div> </div>
<div id="outline-container-org985470b" class="outline-4"> <div id="outline-container-org14c61e7" class="outline-4">
<h4 id="org985470b">Function description</h4> <h4 id="org14c61e7">Function description</h4>
<div class="outline-text-4" id="text-org985470b"> <div class="outline-text-4" id="text-org14c61e7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[Li, dLi]</span> = <span class="org-function-name">inverseKinematics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>) <pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[Li, dLi]</span> = <span class="org-function-name">inverseKinematics</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
<span class="org-comment">% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}</span> <span class="org-comment">% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}</span>
@ -1297,9 +1375,9 @@ This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org2b6ddb8" class="outline-4"> <div id="outline-container-orgb04bb62" class="outline-4">
<h4 id="org2b6ddb8">Optional Parameters</h4> <h4 id="orgb04bb62">Optional Parameters</h4>
<div class="outline-text-4" id="text-org2b6ddb8"> <div class="outline-text-4" id="text-orgb04bb62">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">arguments <pre class="src src-matlab">arguments
stewart stewart
@ -1311,9 +1389,9 @@ This Matlab function is accessible <a href="src/inverseKinematics.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-orgf420378" class="outline-4"> <div id="outline-container-orge635491" class="outline-4">
<h4 id="orgf420378">Theory</h4> <h4 id="orge635491">Theory</h4>
<div class="outline-text-4" id="text-orgf420378"> <div class="outline-text-4" id="text-orge635491">
<p> <p>
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\). For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
</p> </p>
@ -1347,9 +1425,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div> </div>
</div> </div>
<div id="outline-container-orge5ae615" class="outline-4"> <div id="outline-container-orgbc14960" class="outline-4">
<h4 id="orge5ae615">Compute</h4> <h4 id="orgbc14960">Compute</h4>
<div class="outline-text-4" id="text-orge5ae615"> <div class="outline-text-4" id="text-orgbc14960">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Li = sqrt(args.AP<span class="org-type">'*</span>args.AP <span class="org-type">+</span> diag(stewart.Bb<span class="org-type">'*</span>stewart.Bb) <span class="org-type">+</span> diag(stewart.Aa<span class="org-type">'*</span>stewart.Aa) <span class="org-type">-</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>stewart.Aa)<span class="org-type">'</span> <span class="org-type">+</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>(args.ARB<span class="org-type">*</span>stewart.Bb))<span class="org-type">'</span> <span class="org-type">-</span> diag(2<span class="org-type">*</span>(args.ARB<span class="org-type">*</span>stewart.Bb)<span class="org-type">'*</span>stewart.Aa)); <pre class="src src-matlab">Li = sqrt(args.AP<span class="org-type">'*</span>args.AP <span class="org-type">+</span> diag(stewart.Bb<span class="org-type">'*</span>stewart.Bb) <span class="org-type">+</span> diag(stewart.Aa<span class="org-type">'*</span>stewart.Aa) <span class="org-type">-</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>stewart.Aa)<span class="org-type">'</span> <span class="org-type">+</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>(args.ARB<span class="org-type">*</span>stewart.Bb))<span class="org-type">'</span> <span class="org-type">-</span> diag(2<span class="org-type">*</span>(args.ARB<span class="org-type">*</span>stewart.Bb)<span class="org-type">'*</span>stewart.Aa));
</pre> </pre>
@ -1363,11 +1441,11 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
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<div id="outline-container-orgc366b74" class="outline-3"> <div id="outline-container-org1f75518" class="outline-3">
<h3 id="orgc366b74"><span class="section-number-3">6.3</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3> <h3 id="org1f75518"><span class="section-number-3">6.3</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
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<p> <p>
<a id="org5e0bc69"></a> <a id="org0199633"></a>
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<p> <p>
@ -1375,9 +1453,9 @@ This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here<
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<div id="outline-container-orgd6d4e91" class="outline-4"> <div id="outline-container-org665e0fd" class="outline-4">
<h4 id="orgd6d4e91">Function description</h4> <h4 id="org665e0fd">Function description</h4>
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<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[P, R]</span> = <span class="org-function-name">forwardKinematicsApprox</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>) <pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[P, R]</span> = <span class="org-function-name">forwardKinematicsApprox</span>(<span class="org-variable-name">stewart</span>, <span class="org-variable-name">args</span>)
<span class="org-comment">% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using</span> <span class="org-comment">% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using</span>
@ -1399,9 +1477,9 @@ This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here<
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<div id="outline-container-orgbbf49bb" class="outline-4"> <div id="outline-container-org89c6f2b" class="outline-4">
<h4 id="orgbbf49bb">Optional Parameters</h4> <h4 id="org89c6f2b">Optional Parameters</h4>
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<pre class="src src-matlab">arguments <pre class="src src-matlab">arguments
stewart stewart
@ -1412,9 +1490,9 @@ This Matlab function is accessible <a href="src/forwardKinematicsApprox.m">here<
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<div id="outline-container-orgad68b5d" class="outline-4"> <div id="outline-container-org3ae22c5" class="outline-4">
<h4 id="orgad68b5d">Computation</h4> <h4 id="org3ae22c5">Computation</h4>
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<p> <p>
From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
position and orientation of {B} with respect to {A} using the following formula: position and orientation of {B} with respect to {A} using the following formula:
@ -1465,7 +1543,7 @@ We then compute the corresponding rotation matrix.
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-01-29 mer. 17:48</p> <p class="date">Created: 2020-01-29 mer. 20:22</p>
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@ -31,7 +31,8 @@ The current document is divided in the following sections:
- Section [[sec:jacobian_analysis]]: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform. - Section [[sec:jacobian_analysis]]: The Jacobian matrix is derived from the geometry of the Stewart platform. Then it is shown that the Jacobian can link velocities and forces present in the system, and thus this matrix can be very useful for both analysis and control of the Stewart platform.
- Section [[sec:stiffness_analysis]]: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut. - Section [[sec:stiffness_analysis]]: The stiffness and compliance matrices are derived from the Jacobian matrix and the stiffness of each strut.
- Section [[sec:forward_inverse_kinematics]]: The Forward and Inverse kinematic problems are presented. - Section [[sec:forward_inverse_kinematics]]: The Forward and Inverse kinematic problems are presented.
- Section [[sec:required_actuator_stroke]]: The Inverse kinematic solution is used to estimate required actuator stroke from the wanted mobility of the Stewart platform.
* Jacobian Analysis * Jacobian Analysis
<<sec:jacobian_analysis>> <<sec:jacobian_analysis>>
** Introduction :ignore: ** Introduction :ignore:
@ -179,12 +180,12 @@ In a next section, an approximate solution of the forward kinematics problem is
** Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix ** Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix
<<sec:approximate_forward_inverse_kinematics>> <<sec:approximate_forward_inverse_kinematics>>
For small displacements mobile platform displacement $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$ around $\bm{\mathcal{X}}_0$, the associated joint displacement can be computed using the Jacobian: For small displacements mobile platform displacement $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$ around $\bm{\mathcal{X}}_0$, the associated joint displacement can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
\begin{equation*} \begin{equation*}
\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}} \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
\end{equation*} \end{equation*}
Similarly, for small joint displacements $\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T$ around $\bm{\mathcal{L}}_0$, it is possible to find the induced small displacement of the mobile platform: Similarly, for small joint displacements $\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T$ around $\bm{\mathcal{L}}_0$, it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
\begin{equation*} \begin{equation*}
\delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}
\end{equation*} \end{equation*}
@ -489,7 +490,69 @@ Let's now define the actuator stroke.
L_max = 50e-6; % [m] L_max = 50e-6; % [m]
#+end_src #+end_src
** TODO Approximate Forward Dynamics ** Pure translations
Let's first estimate the mobility in translation when the orientation of the Stewart platform stays the same.
As shown previously, for such small stroke, we can use the approximate Forward Dynamics solution using the Jacobian matrix:
\begin{equation*}
\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
\end{equation*}
To obtain the mobility "volume" attainable by the Stewart platform when it's orientation is set to zero, we use the spherical coordinate $(r, \theta, \phi)$.
For each possible value of $(\theta, \phi)$, we compute the maximum radius $r$ attainable with the constraint that the stroke of each actuator should be between =L_min= and =L_max=.
#+begin_src matlab
thetas = linspace(0, pi, 50);
phis = linspace(0, 2*pi, 50);
rs = zeros(length(thetas), length(phis));
for i = 1:length(thetas)
for j = 1:length(phis)
Tx = sin(thetas(i))*cos(phis(j));
Ty = sin(thetas(i))*sin(phis(j));
Tz = cos(thetas(i));
dL = stewart.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction
rs(i, j) = max([dL(dL<0)*L_min; dL(dL>0)*L_max]);
end
end
#+end_src
Now that we have found the corresponding radius $r$, we plot the obtained mobility.
We can also approximate the mobility by a sphere with a radius equal to the minimum obtained value of $r$, this is however a pessimistic estimation of the mobility.
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([1e6*L_min, 1e6*L_max, 1e6*(min(min(rs)))], {}, {'=L_min= [$\mu m$]', '=L_max= [$\mu m$]', '=R= [$\mu m$]'}, ' %.1f ');
#+end_src
#+RESULTS:
| =L_min= [$\mu m$] | =L_max= [$\mu m$] | =R= [$\mu m$] |
|-------------------+-------------------+---------------|
| -50.0 | 50.0 | 31.5 |
#+begin_src matlab :exports none
figure;
plot3(reshape(rs.*(sin(thetas)'*cos(phis)), [1, length(thetas)*length(phis)]), ...
reshape(rs.*(sin(thetas)'*sin(phis)), [1, length(thetas)*length(phis)]), ...
reshape(rs.*(cos(thetas)'*ones(1, length(phis))), [1, length(thetas)*length(phis)]))
xlabel('X Translation [m]');
ylabel('Y Translation [m]');
zlabel('Z Translation [m]');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/mobility_translations_null_rotation.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:mobility_translations_null_rotation
#+CAPTION: Obtain mobility of the Stewart platform for zero rotations ([[./figs/mobility_translations_null_rotation.png][png]], [[./figs/mobility_translations_null_rotation.pdf][pdf]])
[[file:figs/mobility_translations_null_rotation.png]]
*** TODO Do that by slice :noexport:
using this function https://fr.mathworks.com/help/matlab/ref/contour3.html
* Functions * Functions
<<sec:functions>> <<sec:functions>>