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<a accesskey="H" href="../../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Computation of the Positioning Error with respect to the nano-hexapod</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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||||
<li><a href="#orgf8376ef">1. How do we measure the position of the sample with respect to the granite</a></li>
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<li><a href="#orgb3d760a">2. Verify that the function to compute the reference pose is correct</a>
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||||
<li><a href="#org90cd0e3">1. How do we measure the position of the sample with respect to the granite</a></li>
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<li><a href="#org49398f4">2. Verify that the function to compute the reference pose is correct</a>
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||||
<ul>
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||||
<li><a href="#org824599b">2.1. Prepare the Simulation</a></li>
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||||
<li><a href="#orgd2fa379">2.2. Verify that the pose of the sample is the same as the computed one</a></li>
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||||
<li><a href="#org127e189">2.3. Conclusion</a></li>
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||||
<li><a href="#org9d64b37">2.1. Prepare the Simulation</a></li>
|
||||
<li><a href="#orgcaa1e72">2.2. Verify that the pose of the sample is the same as the computed one</a></li>
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||||
<li><a href="#orgfee0cbc">2.3. Conclusion</a></li>
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||||
</ul>
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||||
</li>
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||||
<li><a href="#orgf6fa84d">3. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a>
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||||
<li><a href="#org7815d37">3. Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</a>
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||||
<ul>
|
||||
<li><a href="#orgf76b12a">3.1. Prepare the Simulation</a></li>
|
||||
<li><a href="#org4e6a2dc">3.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li>
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||||
<li><a href="#orga048dbb">3.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li>
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||||
<li><a href="#org1528ce4">3.4. Conclusion</a></li>
|
||||
<li><a href="#org7632729">3.1. Prepare the Simulation</a></li>
|
||||
<li><a href="#org0b1c527">3.2. Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</a></li>
|
||||
<li><a href="#org3623362">3.3. Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</a></li>
|
||||
<li><a href="#orgdb2de31">3.4. Conclusion</a></li>
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||||
</ul>
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||||
</li>
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||||
</ul>
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||||
@@ -56,11 +60,11 @@
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</div>
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||||
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<p>
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The global measurement and control schematic is shown in figure <a href="#org0939a33">1</a>.
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The global measurement and control schematic is shown in figure <a href="#org2f2bc3c">1</a>.
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</p>
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<div id="org0939a33" class="figure">
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<div id="org2f2bc3c" class="figure">
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<p><img src="figs/control-schematic-nass.png" alt="control-schematic-nass.png" />
|
||||
</p>
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<p><span class="figure-number">Figure 1: </span>Global Control Schematic for the Station</p>
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@@ -82,25 +86,25 @@ Also, all the stages can be perfectly positioned.
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</p>
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<p>
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||||
First, in section <a href="#org82a4cbd">1</a>, is explained how the measurement of the position of the sample with respect to the granite is performed (using Simscape blocs).
|
||||
First, in section <a href="#org1f9249f">1</a>, is explained how the measurement of the position of the sample with respect to the granite is performed (using Simscape blocs).
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</p>
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||||
|
||||
<p>
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||||
In section <a href="#orgbdef361">2</a>, we verify that the function developed to compute the wanted pose (translation and orientation) of the sample with respect to the granite can be determined from the wanted position of each stage (translation stage, tilt stage, spindle and micro-hexapod). This corresponds to the bloc “Compute Wanted Sample Position w.r.t. Granite” in figure <a href="#org0939a33">1</a>.
|
||||
In section <a href="#org0dafc5f">2</a>, we verify that the function developed to compute the wanted pose (translation and orientation) of the sample with respect to the granite can be determined from the wanted position of each stage (translation stage, tilt stage, spindle and micro-hexapod). This corresponds to the bloc “Compute Wanted Sample Position w.r.t. Granite” in figure <a href="#org2f2bc3c">1</a>.
|
||||
To do so, we impose a perfect displacement and all the stage, we perfectly measure the position of the sample with respect to the granite, and we verify that this measured position corresponds to the computed wanted pose of the sample.
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||||
</p>
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||||
|
||||
<p>
|
||||
Then, in section <a href="#orgb2f3e9c">3</a>, we introduce some positioning error in the micro-station’s stages.
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||||
The positioning error of the sample expressed with respect to the granite frame (the one measured) is expressed in a frame connected to the NASS top platform (corresponding to the green bloc “Compute Sample Position Error w.r.t. NASS” in figure <a href="#org0939a33">1</a>).
|
||||
Then, in section <a href="#orge9bc9ee">3</a>, we introduce some positioning error in the micro-station’s stages.
|
||||
The positioning error of the sample expressed with respect to the granite frame (the one measured) is expressed in a frame connected to the NASS top platform (corresponding to the green bloc “Compute Sample Position Error w.r.t. NASS” in figure <a href="#org2f2bc3c">1</a>).
|
||||
Then, we move the NASS such that it compensate for the positioning error that are expressed in the frame of the NASS, and we verify that the positioning error of the sample is well compensated.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-orgf8376ef" class="outline-2">
|
||||
<h2 id="orgf8376ef"><span class="section-number-2">1</span> How do we measure the position of the sample with respect to the granite</h2>
|
||||
<div id="outline-container-org90cd0e3" class="outline-2">
|
||||
<h2 id="org90cd0e3"><span class="section-number-2">1</span> How do we measure the position of the sample with respect to the granite</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="org82a4cbd"></a>
|
||||
<a id="org1f9249f"></a>
|
||||
A transform sensor block gives the translation and orientation of the follower frame with respect to the base frame.
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||||
</p>
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||||
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||||
@@ -126,25 +130,25 @@ We can then determine extract other orientation conventions such that Euler angl
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||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-orgb3d760a" class="outline-2">
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||||
<h2 id="orgb3d760a"><span class="section-number-2">2</span> Verify that the function to compute the reference pose is correct</h2>
|
||||
<div id="outline-container-org49398f4" class="outline-2">
|
||||
<h2 id="org49398f4"><span class="section-number-2">2</span> Verify that the function to compute the reference pose is correct</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgbdef361"></a>
|
||||
<a id="org0dafc5f"></a>
|
||||
</p>
|
||||
<p>
|
||||
The goal here is to perfectly move the station and verify that there is no mismatch between the metrology measurement and the computation of the reference pose.
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||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org824599b" class="outline-3">
|
||||
<h3 id="org824599b"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
|
||||
<div id="outline-container-org9d64b37" class="outline-3">
|
||||
<h3 id="org9d64b37"><span class="section-number-3">2.1</span> Prepare the Simulation</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
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||||
<p>
|
||||
We set a small <code>StopTime</code>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
||||
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
|
||||
<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
||||
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -152,16 +156,16 @@ We set a small <code>StopTime</code>.
|
||||
We initialize all the stages.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 50);
|
||||
<pre class="src src-matlab"> initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 50);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -169,8 +173,8 @@ initializeSample(<span class="org-string">'type'</span>, <span class="org-string
|
||||
No disturbance and no gravity.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -178,24 +182,24 @@ initializeDisturbances(<span class="org-string">'enable'</span>, <span class="or
|
||||
We setup the reference path to be constant.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 5e<span class="org-type">-</span>3, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Dy_period'</span>, 1, ...<span class="org-comment"> % Period of the displacement [s]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, <span class="org-type">-</span>1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Ry_period'</span>, 10, ...<span class="org-comment"> % Period of the displacement [s]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, <span class="org-type">-</span>135<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Rz_period'</span>, 1, ...<span class="org-comment"> % Period of the rotating [s]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0.01; 0.02; <span class="org-type">-</span>0.03; <span class="org-type">-</span>3<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 3<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [1e<span class="org-type">-</span>3; 2e<span class="org-type">-</span>3; 3e<span class="org-type">-</span>3; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 0; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
<pre class="src src-matlab"> initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 5e<span class="org-type">-</span>3, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Dy_period'</span>, 1, ...<span class="org-comment"> % Period of the displacement [s]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, <span class="org-type">-</span>1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Ry_period'</span>, 10, ...<span class="org-comment"> % Period of the displacement [s]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, <span class="org-type">-</span>135<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Rz_period'</span>, 1, ...<span class="org-comment"> % Period of the rotating [s]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0.01; 0.02; <span class="org-type">-</span>0.03; <span class="org-type">-</span>3<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 3<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [1e<span class="org-type">-</span>3; 2e<span class="org-type">-</span>3; 3e<span class="org-type">-</span>3; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180; 0; 1<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -203,12 +207,12 @@ We setup the reference path to be constant.
|
||||
No position error for now (perfect positioning).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Dye = 0; <span class="org-comment">% [m]</span>
|
||||
Rye = 0; <span class="org-comment">% [rad]</span>
|
||||
Rze = 0; <span class="org-comment">% [rad]</span>
|
||||
Dhe = zeros(6,1); <span class="org-comment">% [m,m,m,rad,rad,rad]</span>
|
||||
Dhle = zeros(6,1); <span class="org-comment">% [m,m,m,m,m,m]</span>
|
||||
Dne = zeros(6,1); <span class="org-comment">% [m,m,m,rad,rad,rad]</span>
|
||||
<pre class="src src-matlab"> Dye = 0; <span class="org-comment">% [m]</span>
|
||||
Rye = 0; <span class="org-comment">% [rad]</span>
|
||||
Rze = 0; <span class="org-comment">% [rad]</span>
|
||||
Dhe = zeros(6,1); <span class="org-comment">% [m,m,m,rad,rad,rad]</span>
|
||||
Dhle = zeros(6,1); <span class="org-comment">% [m,m,m,m,m,m]</span>
|
||||
Dne = zeros(6,1); <span class="org-comment">% [m,m,m,rad,rad,rad]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -216,7 +220,7 @@ Dne = zeros(6,1); <span class="org-comment">% [m,m,m,rad,rad,rad]</span>
|
||||
We want to log the signals
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
<pre class="src src-matlab"> initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -224,14 +228,14 @@ We want to log the signals
|
||||
And we run the simulation.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd2fa379" class="outline-3">
|
||||
<h3 id="orgd2fa379"><span class="section-number-3">2.2</span> Verify that the pose of the sample is the same as the computed one</h3>
|
||||
<div id="outline-container-orgcaa1e72" class="outline-3">
|
||||
<h3 id="orgcaa1e72"><span class="section-number-3">2.2</span> Verify that the pose of the sample is the same as the computed one</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
Let’s denote:
|
||||
@@ -254,11 +258,11 @@ We have then computed:
|
||||
We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix \({}^W\bm{T}_R\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = length(simout.r.Dy.Time);
|
||||
WTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
WTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = computeReferencePose(simout.r.Dy.Data(<span class="org-constant">i</span>), simout.r.Ry.Data(<span class="org-constant">i</span>), simout.r.Rz.Data(<span class="org-constant">i</span>), simout.r.Dh.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>), simout.r.Dn.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
<pre class="src src-matlab"> n = length(simout.r.Dy.Time);
|
||||
WTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
WTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = computeReferencePose(simout.r.Dy.Data(<span class="org-constant">i</span>), simout.r.Ry.Data(<span class="org-constant">i</span>), simout.r.Rz.Data(<span class="org-constant">i</span>), simout.r.Dh.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>), simout.r.Dn.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -267,11 +271,11 @@ As the displacement is perfect, we also measure in simulation the pose of the sa
|
||||
From that we can compute the homogeneous transformation matrix \({}^W\bm{T}_M\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
<pre class="src src-matlab"> n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -287,12 +291,12 @@ Or are least:
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">WTr(1<span class="org-type">:</span>3, 4, end)<span class="org-type">-</span>WTm(1<span class="org-type">:</span>3, 4, end)
|
||||
WTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)<span class="org-type">-</span>eye(3)
|
||||
<pre class="src src-matlab"> WTr(1<span class="org-type">:</span>3, 4, end)<span class="org-type">-</span>WTm(1<span class="org-type">:</span>3, 4, end)
|
||||
WTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)<span class="org-type">-</span>eye(3)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example">
|
||||
<pre class="example" id="orgee15e5c">
|
||||
WTr(1:3, 4, end)-WTm(1:3, 4, end)
|
||||
ans =
|
||||
8.53830894875784e-15
|
||||
@@ -307,10 +311,10 @@ ans =
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org127e189" class="outline-3">
|
||||
<h3 id="org127e189"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||||
<div id="outline-container-orgfee0cbc" class="outline-3">
|
||||
<h3 id="orgfee0cbc"><span class="section-number-3">2.3</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<div class="important">
|
||||
<div class="important" id="org91210ce">
|
||||
<p>
|
||||
We are able to compute the wanted position and orientation of the sample.
|
||||
Both the measurement and the theory gives the same result.
|
||||
@@ -321,11 +325,11 @@ Both the measurement and the theory gives the same result.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf6fa84d" class="outline-2">
|
||||
<h2 id="orgf6fa84d"><span class="section-number-2">3</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2>
|
||||
<div id="outline-container-org7815d37" class="outline-2">
|
||||
<h2 id="org7815d37"><span class="section-number-2">3</span> Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="orgb2f3e9c"></a>
|
||||
<a id="orge9bc9ee"></a>
|
||||
</p>
|
||||
<p>
|
||||
We now introduce some positioning error in the stage.
|
||||
@@ -336,15 +340,15 @@ This will induce a global positioning error of the sample with respect to the de
|
||||
We want to verify that we are able to measure this positioning error and convert it in the frame attached to the Nano-hexapod.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-orgf76b12a" class="outline-3">
|
||||
<h3 id="orgf76b12a"><span class="section-number-3">3.1</span> Prepare the Simulation</h3>
|
||||
<div id="outline-container-org7632729" class="outline-3">
|
||||
<h3 id="org7632729"><span class="section-number-3">3.1</span> Prepare the Simulation</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
We set a small <code>StopTime</code>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
||||
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
|
||||
<pre class="src src-matlab"> load(<span class="org-string">'mat/conf_simulink.mat'</span>);
|
||||
<span class="org-matlab-simulink-keyword">set_param</span>(<span class="org-variable-name">conf_simulink</span>, <span class="org-string">'StopTime'</span>, <span class="org-string">'0.5'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -352,16 +356,16 @@ We set a small <code>StopTime</code>.
|
||||
We initialize all the stages.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 50);
|
||||
<pre class="src src-matlab"> initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>);
|
||||
initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'actuator'</span>, <span class="org-string">'piezo'</span>);
|
||||
initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'mass'</span>, 50);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -369,8 +373,8 @@ initializeSample(<span class="org-string">'type'</span>, <span class="org-string
|
||||
No disturbance and no gravity.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
<pre class="src src-matlab"> initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
|
||||
initializeDisturbances(<span class="org-string">'enable'</span>, <span class="org-constant">false</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -378,21 +382,21 @@ initializeDisturbances(<span class="org-string">'enable'</span>, <span class="or
|
||||
We setup the reference path to be constant.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, 0<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0; 0; 0; 0; 0; 0], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [0; 0; 0; 0; 0; 0] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
<pre class="src src-matlab"> initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, 0<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0; 0; 0; 0; 0; 0], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [0; 0; 0; 0; 0; 0] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -400,10 +404,10 @@ We setup the reference path to be constant.
|
||||
Now we introduce some positioning error.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Dye = 1e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
|
||||
Rye = 2e<span class="org-type">-</span>4; <span class="org-comment">% [rad]</span>
|
||||
Rze = 1e<span class="org-type">-</span>5; <span class="org-comment">% [rad]</span>
|
||||
initializePosError(<span class="org-string">'error'</span>, <span class="org-constant">true</span>, <span class="org-string">'Dy'</span>, Dye, <span class="org-string">'Ry'</span>, Rye, <span class="org-string">'Rz'</span>, Rze);
|
||||
<pre class="src src-matlab"> Dye = 1e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
|
||||
Rye = 2e<span class="org-type">-</span>4; <span class="org-comment">% [rad]</span>
|
||||
Rze = 1e<span class="org-type">-</span>5; <span class="org-comment">% [rad]</span>
|
||||
initializePosError(<span class="org-string">'error'</span>, <span class="org-constant">true</span>, <span class="org-string">'Dy'</span>, Dye, <span class="org-string">'Ry'</span>, Rye, <span class="org-string">'Rz'</span>, Rze);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -411,14 +415,14 @@ initializePosError(<span class="org-string">'error'</span>, <span class="org-con
|
||||
And we run the simulation.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e6a2dc" class="outline-3">
|
||||
<h3 id="org4e6a2dc"><span class="section-number-3">3.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3>
|
||||
<div id="outline-container-org0b1c527" class="outline-3">
|
||||
<h3 id="org0b1c527"><span class="section-number-3">3.2</span> Compute the wanted pose of the sample in the NASS Base from the metrology and the reference</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
Now that we have introduced some positioning error, the computed wanted pose and the measured pose will not be the same.
|
||||
@@ -444,11 +448,11 @@ The top platform of the nano-hexapod is considered to be rigidly connected to th
|
||||
We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix \({}^W\bm{T}_R\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = length(simout.r.Dy.Time);
|
||||
WTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
WTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = computeReferencePose(simout.r.Dy.Data(<span class="org-constant">i</span>), simout.r.Ry.Data(<span class="org-constant">i</span>), simout.r.Rz.Data(<span class="org-constant">i</span>), simout.r.Dh.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>), simout.r.Dn.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
<pre class="src src-matlab"> n = length(simout.r.Dy.Time);
|
||||
WTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
WTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = computeReferencePose(simout.r.Dy.Data(<span class="org-constant">i</span>), simout.r.Ry.Data(<span class="org-constant">i</span>), simout.r.Rz.Data(<span class="org-constant">i</span>), simout.r.Dh.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>), simout.r.Dn.Data(<span class="org-constant">i</span>,<span class="org-type">:</span>));
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -457,11 +461,11 @@ We also measure in simulation the pose of the sample with respect to the granite
|
||||
From that we can compute the homogeneous transformation matrix \({}^W\bm{T}_M\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
<pre class="src src-matlab"> n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -483,10 +487,10 @@ The <b>inverse of the transformation matrix</b> can be obtain by (it is less com
|
||||
Finally, we compute \({}^M\bm{T}_R\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">MTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
MTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4,<span class="org-constant">i</span>) ; 0 0 0 1]<span class="org-type">*</span>WTr(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
<pre class="src src-matlab"> MTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
MTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4,<span class="org-constant">i</span>) ; 0 0 0 1]<span class="org-type">*</span>WTr(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -494,18 +498,18 @@ Finally, we compute \({}^M\bm{T}_R\).
|
||||
Verify that the pose error corresponds to the positioning error of the stages.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">MTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)
|
||||
Rx = [1 0 0;
|
||||
0 cos(Erx) <span class="org-type">-</span>sin(Erx);
|
||||
0 sin(Erx) cos(Erx)];
|
||||
<pre class="src src-matlab"> MTr(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, end)
|
||||
Rx = [1 0 0;
|
||||
0 cos(Erx) <span class="org-type">-</span>sin(Erx);
|
||||
0 sin(Erx) cos(Erx)];
|
||||
|
||||
Ry = [ cos(Ery) 0 sin(Ery);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Ery) 0 cos(Ery)];
|
||||
Ry = [ cos(Ery) 0 sin(Ery);
|
||||
0 1 0;
|
||||
<span class="org-type">-</span>sin(Ery) 0 cos(Ery)];
|
||||
|
||||
Rz = [cos(Erz) <span class="org-type">-</span>sin(Erz) 0;
|
||||
sin(Erz) cos(Erz) 0;
|
||||
0 0 1];
|
||||
Rz = [cos(Erz) <span class="org-type">-</span>sin(Erz) 0;
|
||||
sin(Erz) cos(Erz) 0;
|
||||
0 0 1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -553,28 +557,28 @@ Rz = [cos(Erz) <span class="org-type">-</span>sin(Erz) 0;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga048dbb" class="outline-3">
|
||||
<h3 id="orga048dbb"><span class="section-number-3">3.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3>
|
||||
<div id="outline-container-org3623362" class="outline-3">
|
||||
<h3 id="org3623362"><span class="section-number-3">3.3</span> Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
We now keep the wanted pose but we impose a displacement of the nano hexapod corresponding to the measured position error.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, 0<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0; 0; 0; 0; 0; 0], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [Edx; Edy; Edz; Erx; Ery; Erz] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
<pre class="src src-matlab"> initializeReferences(...
|
||||
<span class="org-string">'Ts'</span>, 1e<span class="org-type">-</span>3, ...<span class="org-comment"> % Sampling Frequency [s]</span>
|
||||
<span class="org-string">'Dy_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Dy_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude of the displacement [m]</span>
|
||||
<span class="org-string">'Ry_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "triangular" / "sinusoidal"</span>
|
||||
<span class="org-string">'Ry_amplitude'</span>, 0, ...<span class="org-comment"> % Amplitude [rad]</span>
|
||||
<span class="org-string">'Rz_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % Either "constant" / "rotating"</span>
|
||||
<span class="org-string">'Rz_amplitude'</span>, 0<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">/</span>180, ...<span class="org-comment"> % Initial angle [rad]</span>
|
||||
<span class="org-string">'Dh_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dh_pos'</span>, [0; 0; 0; 0; 0; 0], ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
<span class="org-string">'Rm_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Rm_pos'</span>, [0, <span class="org-constant">pi</span>]<span class="org-type">'</span>, ...<span class="org-comment"> % Initial position of the two masses</span>
|
||||
<span class="org-string">'Dn_type'</span>, <span class="org-string">'constant'</span>, ...<span class="org-comment"> % For now, only constant is implemented</span>
|
||||
<span class="org-string">'Dn_pos'</span>, [Edx; Edy; Edz; Erx; Ery; Erz] ...<span class="org-comment"> % Initial position [m,m,m,rad,rad,rad] of the top platform</span>
|
||||
);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -582,7 +586,7 @@ We now keep the wanted pose but we impose a displacement of the nano hexapod cor
|
||||
And we run the simulation.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
<pre class="src src-matlab"> <span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -595,11 +599,11 @@ As the displacement is perfect, we also measure in simulation the pose of the sa
|
||||
From that we can compute the homogeneous transformation matrix \({}^W\bm{T}_M\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
<pre class="src src-matlab"> n = length(simout.y.R.Time);
|
||||
WTm = zeros(4, 4, n);
|
||||
WTm(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3, <span class="org-type">:</span>) = simout.y.R.Data;
|
||||
WTm(1<span class="org-type">:</span>3, 4, <span class="org-type">:</span>) = [simout.y.x.Data<span class="org-type">'</span> ; simout.y.y.Data<span class="org-type">'</span> ; simout.y.z.Data<span class="org-type">'</span>];
|
||||
WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -607,10 +611,10 @@ WTm(4, 4, <span class="org-type">:</span>) = 1;
|
||||
Finally, we compute \({}^M\bm{T}_R\).
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">MTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
MTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4,<span class="org-constant">i</span>) ; 0 0 0 1]<span class="org-type">*</span>WTr(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
<pre class="src src-matlab"> MTr = zeros(4, 4, n);
|
||||
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:n</span>
|
||||
MTr(<span class="org-type">:</span>, <span class="org-type">:</span>, <span class="org-constant">i</span>) = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3,<span class="org-constant">i</span>)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4,<span class="org-constant">i</span>) ; 0 0 0 1]<span class="org-type">*</span>WTr(<span class="org-type">:</span>,<span class="org-type">:</span>,<span class="org-constant">i</span>);
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@@ -661,10 +665,10 @@ Verify that the pose error is small.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1528ce4" class="outline-3">
|
||||
<h3 id="org1528ce4"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div id="outline-container-orgdb2de31" class="outline-3">
|
||||
<h3 id="orgdb2de31"><span class="section-number-3">3.4</span> Conclusion</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<div class="important">
|
||||
<div class="important" id="orge11a7e8">
|
||||
<p>
|
||||
Indeed, we are able to convert the position error in the frame of the NASS and then compensate these errors with the NASS.
|
||||
</p>
|
||||
@@ -676,7 +680,7 @@ Indeed, we are able to convert the position error in the frame of the NASS and t
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-04-17 ven. 09:35</p>
|
||||
<p class="date">Created: 2021-02-20 sam. 23:08</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user