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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-02-11 mar. 17:50 --> <!-- 2020-02-12 mer. 10:22 -->
<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>
@ -287,7 +287,7 @@ for the JavaScript code in this tag.
<li><a href="#org5a3ce80">3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</a></li> <li><a href="#org5a3ce80">3.3. Approximate solution of the Forward and Inverse Kinematic problem for small displacement using the Jacobian matrix</a></li>
<li><a href="#org86b4b35">3.4. Estimation of the range validity of the approximate inverse kinematics</a> <li><a href="#org86b4b35">3.4. Estimation of the range validity of the approximate inverse kinematics</a>
<ul> <ul>
<li><a href="#org8be231b">3.4.1. Stewart architecture definition</a></li> <li><a href="#orgc3c7024">3.4.1. Stewart architecture definition</a></li>
<li><a href="#orgd83ccf3">3.4.2. Comparison for &ldquo;pure&rdquo; translations</a></li> <li><a href="#orgd83ccf3">3.4.2. Comparison for &ldquo;pure&rdquo; translations</a></li>
<li><a href="#org4871c83">3.4.3. Conclusion</a></li> <li><a href="#org4871c83">3.4.3. Conclusion</a></li>
</ul> </ul>
@ -296,7 +296,7 @@ for the JavaScript code in this tag.
</li> </li>
<li><a href="#org63255f9">4. Estimated required actuator stroke from specified platform mobility</a> <li><a href="#org63255f9">4. Estimated required actuator stroke from specified platform mobility</a>
<ul> <ul>
<li><a href="#org4d97075">4.1. Stewart architecture definition</a></li> <li><a href="#orgea4978b">4.1. Stewart architecture definition</a></li>
<li><a href="#orgde50dd3">4.2. Wanted translations and rotations</a></li> <li><a href="#orgde50dd3">4.2. Wanted translations and rotations</a></li>
<li><a href="#org24e45ca">4.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li> <li><a href="#org24e45ca">4.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li>
<li><a href="#orgf6ba90c">4.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li> <li><a href="#orgf6ba90c">4.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li>
@ -304,7 +304,7 @@ for the JavaScript code in this tag.
</li> </li>
<li><a href="#orgbbbf7b3">5. Estimated platform mobility from specified actuator stroke</a> <li><a href="#orgbbbf7b3">5. Estimated platform mobility from specified actuator stroke</a>
<ul> <ul>
<li><a href="#orgc3c7024">5.1. Stewart architecture definition</a></li> <li><a href="#orgbecdc67">5.1. Stewart architecture definition</a></li>
<li><a href="#org2c6819e">5.2. Pure translations</a></li> <li><a href="#org2c6819e">5.2. Pure translations</a></li>
</ul> </ul>
</li> </li>
@ -312,8 +312,8 @@ for the JavaScript code in this tag.
<ul> <ul>
<li><a href="#org26e8b28">6.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a> <li><a href="#org26e8b28">6.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<ul> <ul>
<li><a href="#orgdfb3b70">Function description</a></li> <li><a href="#org1a620fe">Function description</a></li>
<li><a href="#org14aeafc">Check the <code>stewart</code> structure elements</a></li> <li><a href="#org9c3aa33">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0cd57b5">Compute Jacobian Matrix</a></li> <li><a href="#org0cd57b5">Compute Jacobian Matrix</a></li>
<li><a href="#orge21dcfc">Compute Stiffness Matrix</a></li> <li><a href="#orge21dcfc">Compute Stiffness Matrix</a></li>
<li><a href="#orgae76071">Compute Compliance Matrix</a></li> <li><a href="#orgae76071">Compute Compliance Matrix</a></li>
@ -323,17 +323,17 @@ for the JavaScript code in this tag.
<li><a href="#orgb82066f">6.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a> <li><a href="#orgb82066f">6.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<ul> <ul>
<li><a href="#org89930b7">Theory</a></li> <li><a href="#org89930b7">Theory</a></li>
<li><a href="#orgfe68a5b">Function description</a></li> <li><a href="#orgd97252a">Function description</a></li>
<li><a href="#org7a9a427">Optional Parameters</a></li> <li><a href="#org7be1dc4">Optional Parameters</a></li>
<li><a href="#org27c8a3f">Check the <code>stewart</code> structure elements</a></li> <li><a href="#orga7b29d7">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0d64c23">Compute</a></li> <li><a href="#org0d64c23">Compute</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf5d8f0b">6.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a> <li><a href="#orgf5d8f0b">6.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
<ul> <ul>
<li><a href="#org1a620fe">Function description</a></li> <li><a href="#org1a280b0">Function description</a></li>
<li><a href="#org7be1dc4">Optional Parameters</a></li> <li><a href="#orge4b0020">Optional Parameters</a></li>
<li><a href="#org9c3aa33">Check the <code>stewart</code> structure elements</a></li> <li><a href="#org08a6ec5">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orge5ade24">Computation</a></li> <li><a href="#orge5ade24">Computation</a></li>
</ul> </ul>
</li> </li>
@ -666,8 +666,8 @@ This will also gives us the range for which the approximate forward kinematic is
</p> </p>
</div> </div>
<div id="outline-container-org8be231b" class="outline-4"> <div id="outline-container-orgc3c7024" class="outline-4">
<h4 id="org8be231b"><span class="section-number-4">3.4.1</span> Stewart architecture definition</h4> <h4 id="orgc3c7024"><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.
@ -708,7 +708,7 @@ Ls_exact = zeros(6, length(Xrs));
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Xrs)</span> <span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(Xrs)</span>
Xr = Xrs(<span class="org-constant">i</span>); Xr = Xrs(<span class="org-constant">i</span>);
L_approx(<span class="org-type">:</span>, <span class="org-constant">i</span>) = stewart.J<span class="org-type">*</span>[Xr; 0; 0; 0; 0; 0;]; L_approx(<span class="org-type">:</span>, <span class="org-constant">i</span>) = stewart.kinematics.J<span class="org-type">*</span>[Xr; 0; 0; 0; 0; 0;];
[<span class="org-type">~</span>, L_exact(<span class="org-type">:</span>, <span class="org-constant">i</span>)] = inverseKinematics(stewart, <span class="org-string">'AP'</span>, [Xr; 0; 0]); [<span class="org-type">~</span>, L_exact(<span class="org-type">:</span>, <span class="org-constant">i</span>)] = inverseKinematics(stewart, <span class="org-string">'AP'</span>, [Xr; 0; 0]);
<span class="org-keyword">end</span> <span class="org-keyword">end</span>
</pre> </pre>
@ -733,9 +733,12 @@ Ls_exact = zeros(6, length(Xrs));
<div id="outline-container-org4871c83" class="outline-4"> <div id="outline-container-org4871c83" class="outline-4">
<h4 id="org4871c83"><span class="section-number-4">3.4.3</span> Conclusion</h4> <h4 id="org4871c83"><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">
<div class="important">
<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.
</p> </p>
</div>
</div> </div>
</div> </div>
</div> </div>
@ -753,8 +756,8 @@ 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-org4d97075" class="outline-3"> <div id="outline-container-orgea4978b" class="outline-3">
<h3 id="org4d97075"><span class="section-number-3">4.1</span> Stewart architecture definition</h3> <h3 id="orgea4978b"><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.
@ -767,7 +770,7 @@ stewart = computeJointsPose(stewart);
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart); stewart = initializeCylindricalStruts(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1)); stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart); stewart = initializeJointDynamics(stewart);
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
</pre> </pre>
@ -802,12 +805,12 @@ We do that using either the Inverse Kinematic solution or the Jacobian matrix as
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">LTx = stewart.J<span class="org-type">*</span>[Tx_max 0 0 0 0 0]<span class="org-type">'</span>; <pre class="src src-matlab">LTx = stewart.kinematics.J<span class="org-type">*</span>[Tx_max 0 0 0 0 0]<span class="org-type">'</span>;
LTy = stewart.J<span class="org-type">*</span>[0 Ty_max 0 0 0 0]<span class="org-type">'</span>; LTy = stewart.kinematics.J<span class="org-type">*</span>[0 Ty_max 0 0 0 0]<span class="org-type">'</span>;
LTz = stewart.J<span class="org-type">*</span>[0 0 Tz_max 0 0 0]<span class="org-type">'</span>; LTz = stewart.kinematics.J<span class="org-type">*</span>[0 0 Tz_max 0 0 0]<span class="org-type">'</span>;
LRx = stewart.J<span class="org-type">*</span>[0 0 0 Rx_max 0 0]<span class="org-type">'</span>; LRx = stewart.kinematics.J<span class="org-type">*</span>[0 0 0 Rx_max 0 0]<span class="org-type">'</span>;
LRy = stewart.J<span class="org-type">*</span>[0 0 0 0 Ry_max 0]<span class="org-type">'</span>; LRy = stewart.kinematics.J<span class="org-type">*</span>[0 0 0 0 Ry_max 0]<span class="org-type">'</span>;
LRz = stewart.J<span class="org-type">*</span>[0 0 0 0 0 Rz_max]<span class="org-type">'</span>; LRz = stewart.kinematics.J<span class="org-type">*</span>[0 0 0 0 0 Rz_max]<span class="org-type">'</span>;
</pre> </pre>
</div> </div>
@ -1161,8 +1164,8 @@ As explained in section <a href="#orgca82bb8">3</a>, the forward kinematic probl
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-orgc3c7024" class="outline-3"> <div id="outline-container-orgbecdc67" class="outline-3">
<h3 id="orgc3c7024"><span class="section-number-3">5.1</span> Stewart architecture definition</h3> <h3 id="orgbecdc67"><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.
@ -1175,7 +1178,7 @@ stewart = computeJointsPose(stewart);
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart); stewart = initializeCylindricalStruts(stewart);
stewart = initializeStrutDynamics(stewart, <span class="org-string">'Ki'</span>, 1e6<span class="org-type">*</span>ones(6,1), <span class="org-string">'Ci'</span>, 1e2<span class="org-type">*</span>ones(6,1)); stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart); stewart = initializeJointDynamics(stewart);
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
</pre> </pre>
@ -1224,7 +1227,7 @@ rs = zeros(length(thetas), length(phis));
Ty = sin(thetas(<span class="org-constant">i</span>))<span class="org-type">*</span>sin(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>)); 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> dL = stewart.kinematics.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]); 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>
@ -1293,9 +1296,9 @@ This Matlab function is accessible <a href="../src/computeJacobian.m">here</a>.
</p> </p>
</div> </div>
<div id="outline-container-orgdfb3b70" class="outline-4"> <div id="outline-container-org1a620fe" class="outline-4">
<h4 id="orgdfb3b70">Function description</h4> <h4 id="org1a620fe">Function description</h4>
<div class="outline-text-4" id="text-orgdfb3b70"> <div class="outline-text-4" id="text-org1a620fe">
<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>
@ -1318,9 +1321,9 @@ This Matlab function is accessible <a href="../src/computeJacobian.m">here</a>.
</div> </div>
</div> </div>
<div id="outline-container-org14aeafc" class="outline-4"> <div id="outline-container-org9c3aa33" class="outline-4">
<h4 id="org14aeafc">Check the <code>stewart</code> structure elements</h4> <h4 id="org9c3aa33">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org14aeafc"> <div class="outline-text-4" id="text-org9c3aa33">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'As'</span>), <span class="org-string">'stewart.geometry should have attribute As'</span>) <pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'As'</span>), <span class="org-string">'stewart.geometry should have attribute As'</span>)
As = stewart.geometry.As; As = stewart.geometry.As;
@ -1428,9 +1431,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div> </div>
</div> </div>
<div id="outline-container-orgfe68a5b" class="outline-4"> <div id="outline-container-orgd97252a" class="outline-4">
<h4 id="orgfe68a5b">Function description</h4> <h4 id="orgd97252a">Function description</h4>
<div class="outline-text-4" id="text-orgfe68a5b"> <div class="outline-text-4" id="text-orgd97252a">
<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>
@ -1454,9 +1457,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div> </div>
</div> </div>
<div id="outline-container-org7a9a427" class="outline-4"> <div id="outline-container-org7be1dc4" class="outline-4">
<h4 id="org7a9a427">Optional Parameters</h4> <h4 id="org7be1dc4">Optional Parameters</h4>
<div class="outline-text-4" id="text-org7a9a427"> <div class="outline-text-4" id="text-org7be1dc4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">arguments <pre class="src src-matlab">arguments
stewart stewart
@ -1468,9 +1471,9 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div> </div>
</div> </div>
<div id="outline-container-org27c8a3f" class="outline-4"> <div id="outline-container-orga7b29d7" class="outline-4">
<h4 id="org27c8a3f">Check the <code>stewart</code> structure elements</h4> <h4 id="orga7b29d7">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org27c8a3f"> <div class="outline-text-4" id="text-orga7b29d7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'Aa'</span>), <span class="org-string">'stewart.geometry should have attribute Aa'</span>) <pre class="src src-matlab">assert(isfield(stewart.geometry, <span class="org-string">'Aa'</span>), <span class="org-string">'stewart.geometry should have attribute Aa'</span>)
Aa = stewart.geometry.Aa; Aa = stewart.geometry.Aa;
@ -1514,9 +1517,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</p> </p>
</div> </div>
<div id="outline-container-org1a620fe" class="outline-4"> <div id="outline-container-org1a280b0" class="outline-4">
<h4 id="org1a620fe">Function description</h4> <h4 id="org1a280b0">Function description</h4>
<div class="outline-text-4" id="text-org1a620fe"> <div class="outline-text-4" id="text-org1a280b0">
<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">[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>
@ -1538,9 +1541,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</div> </div>
</div> </div>
<div id="outline-container-org7be1dc4" class="outline-4"> <div id="outline-container-orge4b0020" class="outline-4">
<h4 id="org7be1dc4">Optional Parameters</h4> <h4 id="orge4b0020">Optional Parameters</h4>
<div class="outline-text-4" id="text-org7be1dc4"> <div class="outline-text-4" id="text-orge4b0020">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">arguments <pre class="src src-matlab">arguments
stewart stewart
@ -1551,9 +1554,9 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</div> </div>
</div> </div>
<div id="outline-container-org9c3aa33" class="outline-4"> <div id="outline-container-org08a6ec5" class="outline-4">
<h4 id="org9c3aa33">Check the <code>stewart</code> structure elements</h4> <h4 id="org08a6ec5">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org9c3aa33"> <div class="outline-text-4" id="text-org08a6ec5">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">assert(isfield(stewart.kinematics, <span class="org-string">'J'</span>), <span class="org-string">'stewart.kinematics should have attribute J'</span>) <pre class="src src-matlab">assert(isfield(stewart.kinematics, <span class="org-string">'J'</span>), <span class="org-string">'stewart.kinematics should have attribute J'</span>)
J = stewart.kinematics.J; J = stewart.kinematics.J;
@ -1616,7 +1619,7 @@ We then compute the corresponding rotation matrix.
</div> </div>
<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: 2020-02-11 mar. 17:50</p> <p class="date">Created: 2020-02-12 mer. 10:22</p>
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22
org/kinematic-study.org
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@ -270,7 +270,7 @@ The relative strut length displacement is shown in Figure [[fig:inverse_kinemati
for i = 1:length(Xrs) for i = 1:length(Xrs)
Xr = Xrs(i); Xr = Xrs(i);
L_approx(:, i) = stewart.J*[Xr; 0; 0; 0; 0; 0;]; L_approx(:, i) = stewart.kinematics.J*[Xr; 0; 0; 0; 0; 0;];
[~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]); [~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]);
end end
#+end_src #+end_src
@ -321,7 +321,9 @@ The relative strut length displacement is shown in Figure [[fig:inverse_kinemati
[[file:figs/inverse_kinematics_approx_validity_x_translation_relative.png]] [[file:figs/inverse_kinematics_approx_validity_x_translation_relative.png]]
*** Conclusion *** Conclusion
#+begin_important
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.
#+end_important
* Estimated required actuator stroke from specified platform mobility * Estimated required actuator stroke from specified platform mobility
:PROPERTIES: :PROPERTIES:
@ -357,7 +359,7 @@ Let's first define the Stewart platform architecture that we want to study.
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart); stewart = initializeCylindricalStruts(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1)); stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart); stewart = initializeJointDynamics(stewart);
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
#+end_src #+end_src
@ -378,12 +380,12 @@ As a first estimation, we estimate the needed actuator stroke for "pure" rotatio
We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation. We do that using either the Inverse Kinematic solution or the Jacobian matrix as an approximation.
#+begin_src matlab #+begin_src matlab
LTx = stewart.J*[Tx_max 0 0 0 0 0]'; LTx = stewart.kinematics.J*[Tx_max 0 0 0 0 0]';
LTy = stewart.J*[0 Ty_max 0 0 0 0]'; LTy = stewart.kinematics.J*[0 Ty_max 0 0 0 0]';
LTz = stewart.J*[0 0 Tz_max 0 0 0]'; LTz = stewart.kinematics.J*[0 0 Tz_max 0 0 0]';
LRx = stewart.J*[0 0 0 Rx_max 0 0]'; LRx = stewart.kinematics.J*[0 0 0 Rx_max 0 0]';
LRy = stewart.J*[0 0 0 0 Ry_max 0]'; LRy = stewart.kinematics.J*[0 0 0 0 Ry_max 0]';
LRz = stewart.J*[0 0 0 0 0 Rz_max]'; LRz = stewart.kinematics.J*[0 0 0 0 0 Rz_max]';
#+end_src #+end_src
The obtain required stroke is: The obtain required stroke is:
@ -514,7 +516,7 @@ Let's first define the Stewart platform architecture that we want to study.
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart); stewart = initializeCylindricalStruts(stewart);
stewart = initializeStrutDynamics(stewart, 'Ki', 1e6*ones(6,1), 'Ci', 1e2*ones(6,1)); stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart); stewart = initializeJointDynamics(stewart);
stewart = computeJacobian(stewart); stewart = computeJacobian(stewart);
#+end_src #+end_src
@ -547,7 +549,7 @@ For each possible value of $(\theta, \phi)$, we compute the maximum radius $r$ a
Ty = sin(thetas(i))*sin(phis(j)); Ty = sin(thetas(i))*sin(phis(j));
Tz = cos(thetas(i)); Tz = cos(thetas(i));
dL = stewart.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction dL = stewart.kinematics.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]); rs(i, j) = max([dL(dL<0)*L_min; dL(dL>0)*L_max]);
end end