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<div id="org-div-home-and-up">
@@ -267,14 +55,14 @@
</li>
<li><a href="#org86b4b35">4. Estimation of the range validity of the approximate inverse kinematics</a>
<ul>
<li><a href="#org7423428">4.1. Stewart architecture definition</a></li>
<li><a href="#orga78aa66">4.1. Stewart architecture definition</a></li>
<li><a href="#orgd83ccf3">4.2. Comparison for &ldquo;pure&rdquo; translations</a></li>
<li><a href="#org4871c83">4.3. Conclusion</a></li>
</ul>
</li>
<li><a href="#org63255f9">5. Estimated required actuator stroke from specified platform mobility</a>
<ul>
<li><a href="#org3e1d400">5.1. Stewart architecture definition</a></li>
<li><a href="#orgadaa219">5.1. Stewart architecture definition</a></li>
<li><a href="#orgde50dd3">5.2. Wanted translations and rotations</a></li>
<li><a href="#org24e45ca">5.3. Needed stroke for &ldquo;pure&rdquo; rotations or translations</a></li>
<li><a href="#orgf6ba90c">5.4. Needed stroke for &ldquo;combined&rdquo; rotations or translations</a></li>
@@ -282,36 +70,41 @@
</li>
<li><a href="#orgbbbf7b3">6. Estimated platform mobility from specified actuator stroke</a>
<ul>
<li><a href="#org53d6532">6.1. Stewart architecture definition</a></li>
<li><a href="#org6a6a5df">6.1. Stewart architecture definition</a></li>
<li><a href="#org2c6819e">6.2. Pure translations</a></li>
</ul>
</li>
<li><a href="#orgc4916dc">7. Functions</a>
<li><a href="#orgad495dd">7. Estimation of the Joint required Stroke</a>
<ul>
<li><a href="#org26e8b28">7.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<li><a href="#orgae20178">7.1. Example of the initialization of a Stewart Platform</a></li>
</ul>
</li>
<li><a href="#orgc4916dc">8. Functions</a>
<ul>
<li><a href="#orgd0d007d">Function description</a></li>
<li><a href="#orge1b5b04">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org26e8b28">8.1. <code>computeJacobian</code>: Compute the Jacobian Matrix</a>
<ul>
<li><a href="#orgcde905e">Function description</a></li>
<li><a href="#org5be121e">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0cd57b5">Compute Jacobian Matrix</a></li>
<li><a href="#orge21dcfc">Compute Stiffness Matrix</a></li>
<li><a href="#orgae76071">Compute Compliance Matrix</a></li>
<li><a href="#org78f18d7">Populate the <code>stewart</code> structure</a></li>
</ul>
</li>
<li><a href="#orgb82066f">7.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<li><a href="#orgb82066f">8.2. <code>inverseKinematics</code>: Compute Inverse Kinematics</a>
<ul>
<li><a href="#org89930b7">Theory</a></li>
<li><a href="#org755b2ae">Function description</a></li>
<li><a href="#org867b3a0">Optional Parameters</a></li>
<li><a href="#org318eb5f">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orgb66d0e9">Function description</a></li>
<li><a href="#org0aeb7ad">Optional Parameters</a></li>
<li><a href="#orga54645b">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#org0d64c23">Compute</a></li>
</ul>
</li>
<li><a href="#orgf5d8f0b">7.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
<li><a href="#orgf5d8f0b">8.3. <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</a>
<ul>
<li><a href="#orgba3bc64">Function description</a></li>
<li><a href="#org7af7974">Optional Parameters</a></li>
<li><a href="#org2ba5e64">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orgc074bc3">Function description</a></li>
<li><a href="#org9a855b1">Optional Parameters</a></li>
<li><a href="#orgdc0187a">Check the <code>stewart</code> structure elements</a></li>
<li><a href="#orge5ade24">Computation</a></li>
</ul>
</li>
@@ -322,7 +115,7 @@
</div>
<p>
The kinematic analysis of a parallel manipulator is well described in <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>:
The kinematic analysis of a parallel manipulator is well described in (<a href="#citeproc_bib_item_1">Taghirad 2013</a>):
</p>
<blockquote>
<p>
@@ -349,7 +142,7 @@ The current document is divided in the following sections:
<a id="orgc45d118"></a>
</p>
<p>
From <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>:
From (<a href="#citeproc_bib_item_1">Taghirad 2013</a>):
</p>
<blockquote>
<p>
@@ -478,6 +271,7 @@ As explain in <a href="stewart-architecture.html">this</a> document, each Actuat
<ul class="org-ul">
<li>A spring with a stiffness \(k_{i}\)</li>
<li>A dashpot with a damping \(c_{i}\)</li>
<li>A force source \(\tau_i\)</li>
</ul>
<p>
@@ -657,15 +451,15 @@ This will also gives us the range for which the approximate forward kinematic is
</p>
</div>
<div id="outline-container-org7423428" class="outline-3">
<h3 id="org7423428"><span class="section-number-3">4.1</span> Stewart architecture definition</h3>
<div id="outline-container-orga78aa66" class="outline-3">
<h3 id="orga78aa66"><span class="section-number-3">4.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-4-1">
<p>
We first define some general Stewart architecture.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStewartPose(stewart);
@@ -692,16 +486,16 @@ The estimate required strut stroke for both the approximate and exact solutions
The relative strut length displacement is shown in Figure <a href="#org02d8e34">2</a>.
</p>
<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(-6, -1, 100); % Wanted X translation of the mobile platform [m]
Ls_approx = zeros(6, length(Xrs));
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>
Xr = Xrs(<span class="org-constant">i</span>);
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-keyword">end</span>
for i = 1:length(Xrs)
Xr = Xrs(i);
L_approx(:, i) = stewart.kinematics.J*[Xr; 0; 0; 0; 0; 0;];
[~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xr; 0; 0]);
end
</pre>
</div>
@@ -758,15 +552,15 @@ This is what is analyzed in this section.
</p>
</div>
<div id="outline-container-org3e1d400" class="outline-3">
<h3 id="org3e1d400"><span class="section-number-3">5.1</span> Stewart architecture definition</h3>
<div id="outline-container-orgadaa219" class="outline-3">
<h3 id="orgadaa219"><span class="section-number-3">5.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let&rsquo;s first define the Stewart platform architecture that we want to study.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStewartPose(stewart);
@@ -787,12 +581,12 @@ stewart = computeJacobian(stewart);
Let&rsquo;s now define the wanted extreme translations and rotations.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Tx_max = 50e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
Ty_max = 50e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
Tz_max = 50e<span class="org-type">-</span>6; <span class="org-comment">% Translation [m]</span>
Rx_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
Ry_max = 30e<span class="org-type">-</span>6; <span class="org-comment">% Rotation [rad]</span>
Rz_max = 0; <span class="org-comment">% Rotation [rad]</span>
<pre class="src src-matlab">Tx_max = 50e-6; % Translation [m]
Ty_max = 50e-6; % Translation [m]
Tz_max = 50e-6; % Translation [m]
Rx_max = 30e-6; % Rotation [rad]
Ry_max = 30e-6; % Rotation [rad]
Rz_max = 0; % Rotation [rad]
</pre>
</div>
</div>
@@ -807,12 +601,12 @@ We do that using either the Inverse Kinematic solution or the Jacobian matrix as
</p>
<div class="org-src-container">
<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.kinematics.J<span class="org-type">*</span>[0 Ty_max 0 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.kinematics.J<span class="org-type">*</span>[0 0 0 Rx_max 0 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.kinematics.J<span class="org-type">*</span>[0 0 0 0 0 Rz_max]<span class="org-type">'</span>;
<pre class="src src-matlab">LTx = stewart.kinematics.J*[Tx_max 0 0 0 0 0]';
LTy = stewart.kinematics.J*[0 Ty_max 0 0 0 0]';
LTz = stewart.kinematics.J*[0 0 Tz_max 0 0 0]';
LRx = stewart.kinematics.J*[0 0 0 Rx_max 0 0]';
LRy = stewart.kinematics.J*[0 0 0 0 Ry_max 0]';
LRz = stewart.kinematics.J*[0 0 0 0 0 Rz_max]';
</pre>
</div>
@@ -845,7 +639,7 @@ To do so, we may estimate the required actuator stroke for all possible combinat
Let&rsquo;s first generate all the possible combination of maximum translation and rotations.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Ps = [2<span class="org-type">*</span>(dec2bin(0<span class="org-type">:</span>5<span class="org-type">^</span>2<span class="org-type">-</span>1,5)<span class="org-type">-</span><span class="org-string">'0'</span>)<span class="org-type">-</span>1, zeros(5<span class="org-type">^</span>2, 1)]<span class="org-type">.*</span>[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
<pre class="src src-matlab">Ps = [2*(dec2bin(0:5^2-1,5)-'0')-1, zeros(5^2, 1)].*[Tx_max Ty_max Tz_max Rx_max Ry_max Rz_max];
</pre>
</div>
@@ -1110,29 +904,29 @@ For all possible combination, we compute the required actuator stroke using the
<pre class="src src-matlab">L_min = 0;
L_max = 0;
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:size(Ps,1)</span>
for i = 1:size(Ps,1)
Rx = [1 0 0;
0 cos(Ps(<span class="org-constant">i</span>, 4)) <span class="org-type">-</span>sin(Ps(<span class="org-constant">i</span>, 4));
0 sin(Ps(<span class="org-constant">i</span>, 4)) cos(Ps(<span class="org-constant">i</span>, 4))];
0 cos(Ps(i, 4)) -sin(Ps(i, 4));
0 sin(Ps(i, 4)) cos(Ps(i, 4))];
Ry = [ cos(Ps(<span class="org-constant">i</span>, 5)) 0 sin(Ps(<span class="org-constant">i</span>, 5));
Ry = [ cos(Ps(i, 5)) 0 sin(Ps(i, 5));
0 1 0;
<span class="org-type">-</span>sin(Ps(<span class="org-constant">i</span>, 5)) 0 cos(Ps(<span class="org-constant">i</span>, 5))];
-sin(Ps(i, 5)) 0 cos(Ps(i, 5))];
Rz = [cos(Ps(<span class="org-constant">i</span>, 6)) <span class="org-type">-</span>sin(Ps(<span class="org-constant">i</span>, 6)) 0;
sin(Ps(<span class="org-constant">i</span>, 6)) cos(Ps(<span class="org-constant">i</span>, 6)) 0;
Rz = [cos(Ps(i, 6)) -sin(Ps(i, 6)) 0;
sin(Ps(i, 6)) cos(Ps(i, 6)) 0;
0 0 1];
ARB = Rz<span class="org-type">*</span>Ry<span class="org-type">*</span>Rx;
[<span class="org-type">~</span>, Ls] = inverseKinematics(stewart, <span class="org-string">'AP'</span>, Ps(<span class="org-constant">i</span>, 1<span class="org-type">:</span>3)<span class="org-type">'</span>, <span class="org-string">'ARB'</span>, ARB);
ARB = Rz*Ry*Rx;
[~, Ls] = inverseKinematics(stewart, 'AP', Ps(i, 1:3)', 'ARB', ARB);
<span class="org-keyword">if</span> min(Ls) <span class="org-type">&lt;</span> L_min
if min(Ls) &lt; L_min
L_min = min(Ls)
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> max(Ls) <span class="org-type">&gt;</span> L_max
end
if max(Ls) &gt; L_max
L_max = max(Ls)
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
end
end
</pre>
</div>
@@ -1178,15 +972,15 @@ However, for small displacements, we can use the Jacobian as an approximate solu
</p>
</div>
<div id="outline-container-org53d6532" class="outline-3">
<h3 id="org53d6532"><span class="section-number-3">6.1</span> Stewart architecture definition</h3>
<div id="outline-container-org6a6a5df" class="outline-3">
<h3 id="org6a6a5df"><span class="section-number-3">6.1</span> Stewart architecture definition</h3>
<div class="outline-text-3" id="text-6-1">
<p>
Let&rsquo;s first define the Stewart platform architecture that we want to study.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, <span class="org-string">'H'</span>, 90e<span class="org-type">-</span>3, <span class="org-string">'MO_B'</span>, 45e<span class="org-type">-</span>3);
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStewartPose(stewart);
@@ -1202,8 +996,8 @@ stewart = computeJacobian(stewart);
Let&rsquo;s now define the actuator stroke.
</p>
<div class="org-src-container">
<pre class="src src-matlab">L_min = <span class="org-type">-</span>50e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
L_max = 50e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
<pre class="src src-matlab">L_min = -50e-6; % [m]
L_max = 50e-6; % [m]
</pre>
</div>
</div>
@@ -1231,21 +1025,21 @@ To obtain the mobility &ldquo;volume&rdquo; attainable by the Stewart platform w
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);
<pre class="src src-matlab">thetas = linspace(0, pi, 50);
phis = linspace(0, 2*pi, 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>));
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.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>
dL = stewart.kinematics.J*[Tx; Ty; Tz; 0; 0; 0;]; % dL required for 1m displacement in theta/phi direction
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>
rs(i, j) = max([dL(dL&lt;0)*L_min; dL(dL&gt;0)*L_max]);
end
end
</pre>
</div>
@@ -1291,16 +1085,77 @@ We can also approximate the mobility by a sphere with a radius equal to the mini
</div>
</div>
<div id="outline-container-orgc4916dc" class="outline-2">
<h2 id="orgc4916dc"><span class="section-number-2">7</span> Functions</h2>
<div id="outline-container-orgad495dd" class="outline-2">
<h2 id="orgad495dd"><span class="section-number-2">7</span> Estimation of the Joint required Stroke</h2>
<div class="outline-text-2" id="text-7">
</div>
<div id="outline-container-orgae20178" class="outline-3">
<h3 id="orgae20178"><span class="section-number-3">7.1</span> Example of the initialization of a Stewart Platform</h3>
<div class="outline-text-3" id="text-7-1">
<p>
Let&rsquo;s first define the Stewart Platform Geometry.
</p>
<div class="org-src-container">
<pre class="src src-matlab">stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 150e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
As_init = stewart.geometry.As;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Tx_max = 50e-6; % Translation [m]
Ty_max = 50e-6; % Translation [m]
Tz_max = 50e-6; % Translation [m]
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Ps = [2*(dec2bin(0:3^2-2,3)-'0')-1].*[Tx_max Ty_max Tz_max];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">flex_ang = zeros(size(Ps, 1), 6);
for Ps_i = 1:size(Ps, 1)
stewart.geometry.FO_M = [0; 0; 90e-3] + Ps(Ps_i, :)';
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
flex_ang(Ps_i, :) = acos(sum(As_init.*stewart.geometry.As));
end
</pre>
</div>
<p>
And the maximum bending of the flexible joints is: (in [mrad])
</p>
<div class="org-src-container">
<pre class="src src-matlab">1e3*max(max(abs(flex_ang)))
</pre>
</div>
<pre class="example">
0.90937
</pre>
</div>
</div>
</div>
<div id="outline-container-orgc4916dc" class="outline-2">
<h2 id="orgc4916dc"><span class="section-number-2">8</span> Functions</h2>
<div class="outline-text-2" id="text-8">
<p>
<a id="orgf9a6042"></a>
</p>
</div>
<div id="outline-container-org26e8b28" class="outline-3">
<h3 id="org26e8b28"><span class="section-number-3">7.1</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
<div class="outline-text-3" id="text-7-1">
<h3 id="org26e8b28"><span class="section-number-3">8.1</span> <code>computeJacobian</code>: Compute the Jacobian Matrix</h3>
<div class="outline-text-3" id="text-8-1">
<p>
<a id="org2387f19"></a>
</p>
@@ -1310,42 +1165,42 @@ This Matlab function is accessible <a href="../src/computeJacobian.m">here</a>.
</p>
</div>
<div id="outline-container-orgd0d007d" class="outline-4">
<h4 id="orgd0d007d">Function description</h4>
<div class="outline-text-4" id="text-orgd0d007d">
<div id="outline-container-orgcde905e" class="outline-4">
<h4 id="orgcde905e">Function description</h4>
<div class="outline-text-4" id="text-orgcde905e">
<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>)
<span class="org-comment">% computeJacobian -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [stewart] = computeJacobian(stewart)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - stewart - With at least the following fields:</span>
<span class="org-comment">% - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}</span>
<span class="org-comment">% - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}</span>
<span class="org-comment">% - actuators.K [6x1] - Total stiffness of the actuators</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - stewart - With the 3 added field:</span>
<span class="org-comment">% - kinematics.J [6x6] - The Jacobian Matrix</span>
<span class="org-comment">% - kinematics.K [6x6] - The Stiffness Matrix</span>
<span class="org-comment">% - kinematics.C [6x6] - The Compliance Matrix</span>
<pre class="src src-matlab">function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
% - stewart - With at least the following fields:
% - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
% - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
% - actuators.K [6x1] - Total stiffness of the actuators
%
% Outputs:
% - stewart - With the 3 added field:
% - kinematics.J [6x6] - The Jacobian Matrix
% - kinematics.K [6x6] - The Stiffness Matrix
% - kinematics.C [6x6] - The Compliance Matrix
</pre>
</div>
</div>
</div>
<div id="outline-container-orge1b5b04" class="outline-4">
<h4 id="orge1b5b04">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-orge1b5b04">
<div id="outline-container-org5be121e" class="outline-4">
<h4 id="org5be121e">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org5be121e">
<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, 'As'), 'stewart.geometry should have attribute As')
As = stewart.geometry.As;
assert(isfield(stewart.geometry, <span class="org-string">'Ab'</span>), <span class="org-string">'stewart.geometry should have attribute Ab'</span>)
assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab')
Ab = stewart.geometry.Ab;
assert(isfield(stewart.actuators, <span class="org-string">'K'</span>), <span class="org-string">'stewart.actuators should have attribute K'</span>)
assert(isfield(stewart.actuators, 'K'), 'stewart.actuators should have attribute K')
Ki = stewart.actuators.K;
</pre>
</div>
@@ -1357,7 +1212,7 @@ Ki = stewart.actuators.K;
<h4 id="org0cd57b5">Compute Jacobian Matrix</h4>
<div class="outline-text-4" id="text-org0cd57b5">
<div class="org-src-container">
<pre class="src src-matlab">J = [As<span class="org-type">'</span> , cross(Ab, As)<span class="org-type">'</span>];
<pre class="src src-matlab">J = [As' , cross(Ab, As)'];
</pre>
</div>
</div>
@@ -1367,7 +1222,7 @@ Ki = stewart.actuators.K;
<h4 id="orge21dcfc">Compute Stiffness Matrix</h4>
<div class="outline-text-4" id="text-orge21dcfc">
<div class="org-src-container">
<pre class="src src-matlab">K = J<span class="org-type">'*</span>diag(Ki)<span class="org-type">*</span>J;
<pre class="src src-matlab">K = J'*diag(Ki)*J;
</pre>
</div>
</div>
@@ -1398,8 +1253,8 @@ stewart.kinematics.C = C;
<div id="outline-container-orgb82066f" class="outline-3">
<h3 id="orgb82066f"><span class="section-number-3">7.2</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
<div class="outline-text-3" id="text-7-2">
<h3 id="orgb82066f"><span class="section-number-3">8.2</span> <code>inverseKinematics</code>: Compute Inverse Kinematics</h3>
<div class="outline-text-3" id="text-8-2">
<p>
<a id="orgb8859d7"></a>
</p>
@@ -1445,57 +1300,57 @@ Otherwise, when the limbs&rsquo; lengths derived yield complex numbers, then the
</div>
</div>
<div id="outline-container-org755b2ae" class="outline-4">
<h4 id="org755b2ae">Function description</h4>
<div class="outline-text-4" id="text-org755b2ae">
<div id="outline-container-orgb66d0e9" class="outline-4">
<h4 id="orgb66d0e9">Function description</h4>
<div class="outline-text-4" id="text-orgb66d0e9">
<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>)
<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">%</span>
<span class="org-comment">% Syntax: [stewart] = inverseKinematics(stewart)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - stewart - A structure with the following fields</span>
<span class="org-comment">% - geometry.Aa [3x6] - The positions ai expressed in {A}</span>
<span class="org-comment">% - geometry.Bb [3x6] - The positions bi expressed in {B}</span>
<span class="org-comment">% - geometry.l [6x1] - Length of each strut</span>
<span class="org-comment">% - args - Can have the following fields:</span>
<span class="org-comment">% - AP [3x1] - The wanted position of {B} with respect to {A}</span>
<span class="org-comment">% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}</span>
<span class="org-comment">% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}</span>
<pre class="src src-matlab">function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - geometry.Aa [3x6] - The positions ai expressed in {A}
% - geometry.Bb [3x6] - The positions bi expressed in {B}
% - geometry.l [6x1] - Length of each strut
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
</pre>
</div>
</div>
</div>
<div id="outline-container-org867b3a0" class="outline-4">
<h4 id="org867b3a0">Optional Parameters</h4>
<div class="outline-text-4" id="text-org867b3a0">
<div id="outline-container-org0aeb7ad" class="outline-4">
<h4 id="org0aeb7ad">Optional Parameters</h4>
<div class="outline-text-4" id="text-org0aeb7ad">
<div class="org-src-container">
<pre class="src src-matlab">arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
<span class="org-keyword">end</span>
end
</pre>
</div>
</div>
</div>
<div id="outline-container-org318eb5f" class="outline-4">
<h4 id="org318eb5f">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org318eb5f">
<div id="outline-container-orga54645b" class="outline-4">
<h4 id="orga54645b">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-orga54645b">
<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, 'Aa'), 'stewart.geometry should have attribute Aa')
Aa = stewart.geometry.Aa;
assert(isfield(stewart.geometry, <span class="org-string">'Bb'</span>), <span class="org-string">'stewart.geometry should have attribute Bb'</span>)
assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb')
Bb = stewart.geometry.Bb;
assert(isfield(stewart.geometry, <span class="org-string">'l'</span>), <span class="org-string">'stewart.geometry should have attribute l'</span>)
assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l')
l = stewart.geometry.l;
</pre>
</div>
@@ -1507,12 +1362,12 @@ l = stewart.geometry.l;
<h4 id="org0d64c23">Compute</h4>
<div class="outline-text-4" id="text-org0d64c23">
<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(Bb<span class="org-type">'*</span>Bb) <span class="org-type">+</span> diag(Aa<span class="org-type">'*</span>Aa) <span class="org-type">-</span> (2<span class="org-type">*</span>args.AP<span class="org-type">'*</span>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>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>Bb)<span class="org-type">'*</span>Aa));
<pre class="src src-matlab">Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">dLi = Li<span class="org-type">-</span>l;
<pre class="src src-matlab">dLi = Li-l;
</pre>
</div>
</div>
@@ -1520,8 +1375,8 @@ l = stewart.geometry.l;
</div>
<div id="outline-container-orgf5d8f0b" class="outline-3">
<h3 id="orgf5d8f0b"><span class="section-number-3">7.3</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
<div class="outline-text-3" id="text-7-3">
<h3 id="orgf5d8f0b"><span class="section-number-3">8.3</span> <code>forwardKinematicsApprox</code>: Compute the Approximate Forward Kinematics</h3>
<div class="outline-text-3" id="text-8-3">
<p>
<a id="orgdb31434"></a>
</p>
@@ -1531,48 +1386,48 @@ This Matlab function is accessible <a href="../src/forwardKinematicsApprox.m">he
</p>
</div>
<div id="outline-container-orgba3bc64" class="outline-4">
<h4 id="orgba3bc64">Function description</h4>
<div class="outline-text-4" id="text-orgba3bc64">
<div id="outline-container-orgc074bc3" class="outline-4">
<h4 id="orgc074bc3">Function description</h4>
<div class="outline-text-4" id="text-orgc074bc3">
<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>)
<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">% the Jacobian Matrix</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - stewart - A structure with the following fields</span>
<span class="org-comment">% - kinematics.J [6x6] - The Jacobian Matrix</span>
<span class="org-comment">% - args - Can have the following fields:</span>
<span class="org-comment">% - dL [6x1] - Displacement of each strut [m]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - P [3x1] - The estimated position of {B} with respect to {A}</span>
<span class="org-comment">% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}</span>
<pre class="src src-matlab">function [P, R] = forwardKinematicsApprox(stewart, args)
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
% the Jacobian Matrix
%
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - kinematics.J [6x6] - The Jacobian Matrix
% - args - Can have the following fields:
% - dL [6x1] - Displacement of each strut [m]
%
% Outputs:
% - P [3x1] - The estimated position of {B} with respect to {A}
% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
</pre>
</div>
</div>
</div>
<div id="outline-container-org7af7974" class="outline-4">
<h4 id="org7af7974">Optional Parameters</h4>
<div class="outline-text-4" id="text-org7af7974">
<div id="outline-container-org9a855b1" class="outline-4">
<h4 id="org9a855b1">Optional Parameters</h4>
<div class="outline-text-4" id="text-org9a855b1">
<div class="org-src-container">
<pre class="src src-matlab">arguments
stewart
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
<span class="org-keyword">end</span>
end
</pre>
</div>
</div>
</div>
<div id="outline-container-org2ba5e64" class="outline-4">
<h4 id="org2ba5e64">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-org2ba5e64">
<div id="outline-container-orgdc0187a" class="outline-4">
<h4 id="orgdc0187a">Check the <code>stewart</code> structure elements</h4>
<div class="outline-text-4" id="text-orgdc0187a">
<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, 'J'), 'stewart.kinematics should have attribute J')
J = stewart.kinematics.J;
</pre>
</div>
@@ -1588,7 +1443,7 @@ position and orientation of {B} with respect to {A} using the following formula:
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">X = J<span class="org-type">\</span>args.dL;
<pre class="src src-matlab">X = J\args.dL;
</pre>
</div>
@@ -1596,7 +1451,7 @@ position and orientation of {B} with respect to {A} using the following formula:
The position vector corresponds to the first 3 elements.
</p>
<div class="org-src-container">
<pre class="src src-matlab">P = X(1<span class="org-type">:</span>3);
<pre class="src src-matlab">P = X(1:3);
</pre>
</div>
@@ -1605,8 +1460,8 @@ The next 3 elements are the orientation of {B} with respect to {A} expressed
using the screw axis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">theta = norm(X(4<span class="org-type">:</span>6));
s = X(4<span class="org-type">:</span>6)<span class="org-type">/</span>theta;
<pre class="src src-matlab">theta = norm(X(4:6));
s = X(4:6)/theta;
</pre>
</div>
@@ -1614,9 +1469,9 @@ s = X(4<span class="org-type">:</span>6)<span class="org-type">/</span>theta;
We then compute the corresponding rotation matrix.
</p>
<div class="org-src-container">
<pre class="src src-matlab">R = [s(1)<span class="org-type">^</span>2<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> cos(theta) , s(1)<span class="org-type">*</span>s(2)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">-</span> s(3)<span class="org-type">*</span>sin(theta), s(1)<span class="org-type">*</span>s(3)<span class="org-type">*</span>(1<span class="org-type">-</span>cos(theta)) <span class="org-type">+</span> s(2)<span class="org-type">*</span>sin(theta);
s<span class="org-type">(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);</span>
s<span class="org-type">(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];</span>
<pre class="src src-matlab">R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];
</pre>
</div>
</div>
@@ -1626,14 +1481,16 @@ We then compute the corresponding rotation matrix.
<p>
<h1 class='org-ref-bib-h1'>Bibliography</h1>
<ul class='org-ref-bib'><li><a id="taghirad13_paral">[taghirad13_paral]</a> <a name="taghirad13_paral"></a>Taghirad, Parallel robots : mechanics and control, CRC Press (2013).</li>
</ul>
</p>
<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>Taghirad, Hamid. 2013. <i>Parallel Robots : Mechanics and Control</i>. Boca Raton, FL: CRC Press.</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-03-02 lun. 17:57</p>
<p class="date">Created: 2020-08-05 mer. 13:27</p>
</div>
</body>
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