367 lines
13 KiB
HTML
367 lines
13 KiB
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<!-- 2020-04-17 ven. 09:35 -->
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<title>Kinematics of the station</title>
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<meta name="author" content="Dehaeze Thomas" />
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<div id="org-div-home-and-up">
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<a accesskey="h" href="./index.html"> UP </a>
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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">Kinematics of the station</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="#org46d4418">1. Micro Hexapod</a>
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<ul>
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<li><a href="#org6cc9e73">1.1. How the Symetrie Hexapod is controlled on the micro station</a></li>
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<li><a href="#orgfcd44a9">1.2. Control of the Micro-Hexapod using Simscape</a>
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<ul>
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<li><a href="#org3924132">1.2.1. Using Bushing Joint</a></li>
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<li><a href="#org949c942">1.2.2. Using Inverse Kinematics and Leg Actuators</a>
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<ul>
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<li><a href="#orgc9eab88">1.2.2.1. Theory</a></li>
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<li><a href="#orge812977">1.2.2.2. Matlab Implementation</a></li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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In this document, we discuss the way the motion of each stage is defined.
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</p>
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<div id="outline-container-org46d4418" class="outline-2">
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<h2 id="org46d4418"><span class="section-number-2">1</span> Micro Hexapod</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org6cc9e73" class="outline-3">
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<h3 id="org6cc9e73"><span class="section-number-3">1.1</span> How the Symetrie Hexapod is controlled on the micro station</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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For the Micro-Hexapod, the convention for the angles are defined in <code>MAN_A_Software API_4.0.150918_EN.pdf</code> on page 13 (section 2.4 - Rotation Vectors):
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</p>
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<blockquote>
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<p>
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The <b>Euler type II convention</b> is used to express the rotation vector.
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This convention is mainly used in the aeronautics field (standard ISO 1151 concerning flight mechanics).
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</p>
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<p>
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This convention uses the concepts of rotation of vehicles (ship, car and plane).
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Generally, we consider that the main movement of the vehicle is following the X-axis and the Z-axis is parallel to the axis of gravity (at the initial position).
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The roll rotation is around the X-axis, the pitch is around the Y-axis and yaw is the rotation around the Z-axis.
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<b>The order of rotation is: Rx, Ry and then Rz.</b>
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</p>
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<p>
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In most case, rotations are related to a reference with fixed axis; thus we say the rotations are around fixed axes.
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The combination of these three rotations enables to write a rotation matrix.
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This writing is unique and equal to:
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\[ \bm{R} = \bm{R}_z(\gamma) \cdot \bm{R}_y(\beta) \cdot \bm{R}_x(\alpha) \]
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</p>
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<p>
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The Euler type II convention corresponding to the <b>succession of rotations with respect to fixed axes</b>: first around X0, then Y0 and Z0.
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This is equivalent to the succession of rotations with respect to mobile axes: first around Z0, then Y1’ and X2’.
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</p>
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</blockquote>
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<p>
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More generally on the Control of the Micro-Hexapod:
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</p>
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<blockquote>
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<p>
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Note that for all control modes, <b>the rotation center coincides with Object coordinate system origin</b>.
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Moreover, the movements are controlled with <b>translation components at first</b> (Tx, Ty, Tz) <b>then rotation components</b> (Rx, Ry, Rz).
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</p>
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</blockquote>
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<p>
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Thus, it does the translations and then the rotation around the new translated frame.
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</p>
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</div>
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</div>
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<div id="outline-container-orgfcd44a9" class="outline-3">
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<h3 id="orgfcd44a9"><span class="section-number-3">1.2</span> Control of the Micro-Hexapod using Simscape</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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We can think of two main ways to position the Micro-Hexapod using Simscape.
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</p>
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<p>
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The first one is to use only one Bushing Joint between the base and the mobile platform.
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The advantage is that it is very easy to impose the wanted displacement, however, we loose the dynamical properties of the Hexapod.
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</p>
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<p>
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The second way is to specify the wanted length of the legs of the Hexapod in order to have the wanted position of the mobile platform.
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This require a little bit more of mathematical derivations but this is the chosen solution.
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</p>
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</div>
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<div id="outline-container-org3924132" class="outline-4">
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<h4 id="org3924132"><span class="section-number-4">1.2.1</span> Using Bushing Joint</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<p>
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In the documentation of the Bushing Joint (<code>doc "Bushing Joint"</code>) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure <a href="#org9af6f4f">1</a>.
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</p>
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<div id="org9af6f4f" class="figure">
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<p><img src="figs/bushing_joint_transform.png" alt="bushing_joint_transform.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Joint Transformation Sequence for the Bushing Joint</p>
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</div>
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<p>
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Basically, it performs the translations, and then the rotation along the X, Y and Z axis of the moving frame.
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The three rotations that we define thus corresponds to the Euler U-V-W angles.
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</p>
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<p>
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We should have the <b>same behavior</b> for the Micro-Hexapod on Simscape (same inputs at least).
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However, the Bushing Joint makes rotations around mobiles axes (X, Y’ and then Z’’) and not fixed axes (X, Y and Z).
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</p>
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</div>
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</div>
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<div id="outline-container-org949c942" class="outline-4">
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<h4 id="org949c942"><span class="section-number-4">1.2.2</span> Using Inverse Kinematics and Leg Actuators</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<p>
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Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform.
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</p>
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<p>
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The advantages are:
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</p>
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<ul class="org-ul">
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<li>we can position the Hexapod as we want by specifying a rotation matrix</li>
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<li>the hexapod keeps its full flexibility as we don’t specify any wanted displacements, only leg’s rest position</li>
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</ul>
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<p>
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However:
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</p>
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<ul class="org-ul">
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<li>even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity</li>
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</ul>
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<p>
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Thus, for this simulation, we <b>remove the gravity</b>.
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</p>
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</div>
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<div id="outline-container-orgc9eab88" class="outline-5">
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<h5 id="orgc9eab88"><span class="section-number-5">1.2.2.1</span> Theory</h5>
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<div class="outline-text-5" id="text-1-2-2-1">
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<p>
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For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
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</p>
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<p>
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From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as
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</p>
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\begin{align*}
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l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
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&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
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\end{align*}
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<p>
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To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:
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</p>
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\begin{equation}
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l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
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\end{equation}
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<p>
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Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:
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</p>
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\begin{equation}
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l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
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\end{equation}
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<p>
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If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
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Otherwise, when the limbs’ lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
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</p>
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</div>
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</div>
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<div id="outline-container-orge812977" class="outline-5">
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<h5 id="orge812977"><span class="section-number-5">1.2.2.2</span> Matlab Implementation</h5>
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<div class="outline-text-5" id="text-1-2-2-2">
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<p>
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We open the Simulink file.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">open(<span class="org-string">'nass_model.slx'</span>)
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</pre>
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</div>
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<p>
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We load the configuration and set a small <code>StopTime</code>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'mat/conf_simulink.mat'</span>);
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<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">'0.1'</span>);
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</pre>
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</div>
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<p>
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We define the wanted position/orientation of the Hexapod under study.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">tx = 0.05; <span class="org-comment">% [rad]</span>
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ty = 0.1; <span class="org-comment">% [rad]</span>
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tz = 0.02; <span class="org-comment">% [rad]</span>
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Rx = [1 0 0;
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0 cos(tx) <span class="org-type">-</span>sin(tx);
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0 sin(tx) cos(tx)];
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Ry = [ cos(ty) 0 sin(ty);
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0 1 0;
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<span class="org-type">-</span>sin(ty) 0 cos(ty)];
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Rz = [cos(tz) <span class="org-type">-</span>sin(tz) 0;
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sin(tz) cos(tz) 0;
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0 0 1];
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ARB = Rz<span class="org-type">*</span>Ry<span class="org-type">*</span>Rx;
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AP = [0.1; 0.005; 0.01]; <span class="org-comment">% [m]</span>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">initializeSimscapeConfiguration(<span class="org-string">'gravity'</span>, <span class="org-constant">false</span>);
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initializeGround(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeGranite(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeTy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeRy(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeRz(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeMicroHexapod(<span class="org-string">'type'</span>, <span class="org-string">'rigid'</span>, <span class="org-string">'AP'</span>, AP, <span class="org-string">'ARB'</span>, ARB);
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initializeAxisc(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeMirror(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeNanoHexapod(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeSample(<span class="org-string">'type'</span>, <span class="org-string">'none'</span>);
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initializeLoggingConfiguration(<span class="org-string">'log'</span>, <span class="org-string">'all'</span>);
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</pre>
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</div>
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<p>
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We run the simulation.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-simulink-keyword">sim</span>(<span class="org-string">'nass_model'</span>);
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</pre>
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</div>
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<p>
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And we verify that we indeed succeed to go to the wanted position.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">[simout.Dhm.x.Data(end) ; simout.Dhm.y.Data(end) ; simout.Dhm.z.Data(end)] <span class="org-type">-</span> AP
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</pre>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="org-right" />
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">8.4655e-16</td>
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</tr>
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<tr>
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<td class="org-right">1.5586e-15</td>
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</tr>
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<tr>
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<td class="org-right">-2.1337e-16</td>
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</tr>
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</tbody>
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</table>
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<div class="org-src-container">
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<pre class="src src-matlab">simout.Dhm.R.Data(<span class="org-type">:</span>, <span class="org-type">:</span>, end)<span class="org-type">-</span>ARB
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</pre>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<tbody>
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<tr>
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<td class="org-right">-1.1102e-16</td>
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<td class="org-right">-1.36e-15</td>
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<td class="org-right">4.2744e-15</td>
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</tr>
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<tr>
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<td class="org-right">1.0651e-15</td>
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<td class="org-right">6.6613e-16</td>
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<td class="org-right">5.1278e-15</td>
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</tr>
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<tr>
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<td class="org-right">-4.2882e-15</td>
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<td class="org-right">-4.9336e-15</td>
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<td class="org-right">1.1102e-16</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-04-17 ven. 09:35</p>
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</div>
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</body>
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</html>
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