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< h1 class = "title" > Stewart Platform - Simscape Model< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org3273efc" > 1. Procedure< / a > < / li >
< li > < a href = "#org0cfaf56" > 2. Matlab Code< / a >
< ul >
< li > < a href = "#org47e6dec" > 2.1. Simscape Model< / a > < / li >
< li > < a href = "#org80710ea" > 2.2. Test the functions< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org6005875" > 3. < code > initializeFramesPositions< / code > : Initialize the positions of frames {A}, {B}, {F} and {M}< / a >
< ul >
< li > < a href = "#orga3d2c9e" > 3.1. Function description< / a > < / li >
< li > < a href = "#org61a1e26" > 3.2. Documentation< / a > < / li >
< li > < a href = "#org713ea9a" > 3.3. Optional Parameters< / a > < / li >
< li > < a href = "#org88e3f87" > 3.4. Initialize the Stewart structure< / a > < / li >
< li > < a href = "#org4a2edf9" > 3.5. Compute the position of each frame< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org8eb2e40" > 4. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
< ul >
< li > < a href = "#org04bea4c" > 4.1. Function description< / a > < / li >
< li > < a href = "#org924cbfe" > 4.2. Documentation< / a > < / li >
< li > < a href = "#org335de0c" > 4.3. Optional Parameters< / a > < / li >
< li > < a href = "#orgc1a1209" > 4.4. Position of the Cube< / a > < / li >
< li > < a href = "#orga1cfc06" > 4.5. Compute the pose< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgc137b22" > 5. < code > computeJointsPose< / code > : Compute the Pose of the Joints< / a >
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< ul >
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< li > < a href = "#orge4dc5d9" > 5.1. Function description< / a > < / li >
< li > < a href = "#org5e72282" > 5.2. Documentation< / a > < / li >
< li > < a href = "#orgafa4ad9" > 5.3. Compute the position of the Joints< / a > < / li >
< li > < a href = "#org50bd360" > 5.4. Compute the strut length and orientation< / a > < / li >
< li > < a href = "#org3d0d76f" > 5.5. Compute the orientation of the Joints< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc7ae05f" > 6. < code > initializeStrutDynamics< / code > : Add Stiffness and Damping properties of each strut< / a >
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< ul >
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< li > < a href = "#org234504f" > 6.1. Function description< / a > < / li >
< li > < a href = "#org8edf5d0" > 6.2. Optional Parameters< / a > < / li >
< li > < a href = "#orgf9bb882" > 6.3. Add Stiffness and Damping properties of each strut< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org61b514c" > 7. < code > computeJacobian< / code > : Compute the Jacobian Matrix< / a >
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< ul >
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< li > < a href = "#org2983146" > 7.1. Function description< / a > < / li >
< li > < a href = "#org996fbd8" > 7.2. Compute Jacobian Matrix< / a > < / li >
< li > < a href = "#org9e42bbe" > 7.3. Compute Stiffness Matrix< / a > < / li >
< li > < a href = "#org36e44cd" > 7.4. Compute Compliance Matrix< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org76be477" > 8. < code > inverseKinematics< / code > : Compute Inverse Kinematics< / a >
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< ul >
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< li > < a href = "#org16e3f5f" > 8.1. Function description< / a > < / li >
< li > < a href = "#org5ccd852" > 8.2. Optional Parameters< / a > < / li >
< li > < a href = "#org8896a32" > 8.3. Theory< / a > < / li >
< li > < a href = "#orgf5481aa" > 8.4. Compute< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org1c1fd0d" > 9. < code > forwardKinematicsApprox< / code > : Compute the Forward Kinematics< / a >
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< ul >
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< li > < a href = "#orgcc0b9b5" > 9.1. Function description< / a > < / li >
< li > < a href = "#orgdbe04db" > 9.2. Optional Parameters< / a > < / li >
< li > < a href = "#orgc0a97e0" > 9.3. Computation< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
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< p >
Stewart platforms are generated in multiple steps.
< / p >
< p >
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We define 4 important < b > frames< / b > :
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< / p >
< ul class = "org-ul" >
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< li > \(\{F\}\): Frame fixed to the < b > Fixed< / b > base and located at the center of its bottom surface.
This is used to fix the Stewart platform to some support.< / li >
< li > \(\{M\}\): Frame fixed to the < b > Moving< / b > platform and located at the center of its top surface.
This is used to place things on top of the Stewart platform.< / li >
< li > \(\{A\}\): Frame fixed to the fixed base.
It defined the center of rotation of the moving platform.< / li >
< li > \(\{B\}\): Frame fixed to the moving platform.
The motion of the moving platforms and forces applied to it are defined with respect to this frame \(\{B\}\).< / li >
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< / ul >
< p >
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Then, we define the < b > location of the spherical joints< / b > :
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< / p >
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< ul class = "org-ul" >
< li > \(\bm{a}_{i}\) are the position of the spherical joints fixed to the fixed base< / li >
< li > \(\bm{b}_{i}\) are the position of the spherical joints fixed to the moving platform< / li >
< / ul >
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< p >
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We define the < b > rest position< / b > of the Stewart platform:
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< / p >
< ul class = "org-ul" >
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< li > For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.< / li >
< li > Thus, to define the rest position of the Stewart platform, we just have to defined its total height \(H\).
\(H\) corresponds to the distance from the bottom of the fixed base to the top of the moving platform.< / li >
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< / ul >
< p >
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From \(\bm{a}_{i}\) and \(\bm{b}_{i}\), we can determine the < b > length and orientation of each strut< / b > :
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< / p >
< ul class = "org-ul" >
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< li > \(l_{i}\) is the length of the strut< / li >
< li > \({}^{A}\hat{\bm{s}}_{i}\) is the unit vector align with the strut< / li >
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< / ul >
< p >
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The position of the Spherical joints can be computed using various methods:
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< / p >
< ul class = "org-ul" >
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< li > Cubic configuration< / li >
< li > Circular configuration< / li >
< li > Arbitrary position< / li >
< li > These methods should be easily scriptable and corresponds to specific functions that returns \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\).
The input of these functions are the parameters corresponding to the wanted geometry.< / li >
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< / ul >
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< p >
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For Simscape, we need:
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< / p >
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< ul class = "org-ul" >
< li > The position and orientation of each spherical joint fixed to the fixed base: \({}^{F}\bm{a}_{i}\) and \({}^{F}\bm{R}_{a_{i}}\)< / li >
< li > The position and orientation of each spherical joint fixed to the moving platform: \({}^{M}\bm{b}_{i}\) and \({}^{M}\bm{R}_{b_{i}}\)< / li >
< li > The rest length of each strut: \(l_{i}\)< / li >
< li > The stiffness and damping of each actuator: \(k_{i}\) and \(c_{i}\)< / li >
< li > The position of the frame \(\{A\}\) with respect to the frame \(\{F\}\): \({}^{F}\bm{O}_{A}\)< / li >
< li > The position of the frame \(\{B\}\) with respect to the frame \(\{M\}\): \({}^{M}\bm{O}_{B}\)< / li >
< / ul >
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< div id = "outline-container-org3273efc" class = "outline-2" >
< h2 id = "org3273efc" > < span class = "section-number-2" > 1< / span > Procedure< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< p >
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The procedure to define the Stewart platform is the following:
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< / p >
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< ol class = "org-ol" >
< li > Define the initial position of frames {A}, {B}, {F} and {M}.
We do that using the < code > initializeFramesPositions< / code > function.
We have to specify the total height of the Stewart platform \(H\) and the position \({}^{M}O_{B}\) of {B} with respect to {M}.< / li >
< li > Compute the positions of joints \({}^{F}a_{i}\) and \({}^{M}b_{i}\).
We can do that using various methods depending on the wanted architecture:
< ul class = "org-ul" >
< li > < code > generateCubicConfiguration< / code > permits to generate a cubic configuration< / li >
< / ul > < / li >
< li > Compute the position and orientation of the joints with respect to the fixed base and the moving platform.
This is done with the < code > computeJointsPose< / code > function.< / li >
< li > Define the dynamical properties of the Stewart platform.
The output are the stiffness and damping of each strut \(k_{i}\) and \(c_{i}\).
This can be done we simply choosing directly the stiffness and damping of each strut.
The stiffness and damping of each actuator can also be determine from the wanted stiffness of the Stewart platform for instance.< / li >
< li > Define the mass and inertia of each element of the Stewart platform.< / li >
< / ol >
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< p >
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By following this procedure, we obtain a Matlab structure < code > stewart< / code > that contains all the information for the Simscape model and for further analysis.
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< / p >
< / div >
< / div >
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< div id = "outline-container-org0cfaf56" class = "outline-2" >
< h2 id = "org0cfaf56" > < span class = "section-number-2" > 2< / span > Matlab Code< / h2 >
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< h3 id = "org47e6dec" > < span class = "section-number-3" > 2.1< / span > Simscape Model< / h3 >
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< pre class = "src src-matlab" > open(< span class = "org-string" > 'stewart_platform.slx'< / span > )
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< h3 id = "org80710ea" > < span class = "section-number-3" > 2.2< / span > Test the functions< / h3 >
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< pre class = "src src-matlab" > stewart = initializeFramesPositions(< 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 = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , 60e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FOc'< / span > , 45e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FHa'< / span > , 5e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MHb'< / span > , 5e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , 1e6< span class = "org-type" > *< / span > ones(6,1), < span class = "org-string" > 'Ci'< / span > , 1e2< span class = "org-type" > *< / span > ones(6,1));
stewart = computeJacobian(stewart);
[Li, dLi] = inverseKinematics(stewart, < span class = "org-string" > 'AP'< / span > , [0;0;0.00001], < span class = "org-string" > 'ARB'< / span > , eye(3));
[P, R] = forwardKinematicsApprox(stewart, < span class = "org-string" > 'dL'< / span > , dLi)
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< h2 id = "org6005875" > < span class = "section-number-2" > 3< / span > < code > initializeFramesPositions< / code > : Initialize the positions of frames {A}, {B}, {F} and {M}< / h2 >
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< p >
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< a id = "org1dfe172" > < / a >
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This Matlab function is accessible < a href = "src/initializeFramesPositions.m" > here< / a > .
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< h3 id = "orga3d2c9e" > < span class = "section-number-3" > 3.1< / span > Function description< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > initializeFramesPositions< / span > (< span class = "org-variable-name" > args< / span > )
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< span class = "org-comment" > % initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}< / span >
< span class = "org-comment" > %< / span >
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< span class = "org-comment" > % Syntax: [stewart] = initializeFramesPositions(args)< / span >
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< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
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< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]< / span >
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< span class = "org-comment" > % - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following fields:< / span >
< span class = "org-comment" > % - H [1x1] - Total Height of the Stewart Platform [m]< / span >
< span class = "org-comment" > % - FO_M [3x1] - Position of {M} with respect to {F} [m]< / span >
< span class = "org-comment" > % - MO_B [3x1] - Position of {B} with respect to {M} [m]< / span >
< span class = "org-comment" > % - FO_A [3x1] - Position of {A} with respect to {F} [m]< / span >
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< h3 id = "org61a1e26" > < span class = "section-number-3" > 3.2< / span > Documentation< / h3 >
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< p > < img src = "figs/stewart-frames-position.png" alt = "stewart-frames-position.png" / >
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< p > < span class = "figure-number" > Figure 1: < / span > Definition of the position of the frames< / p >
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< h3 id = "org713ea9a" > < span class = "section-number-3" > 3.3< / span > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > arguments
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e< span class = "org-type" > -< / span > 3
args.MO_B (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
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< span class = "org-keyword" > end< / span >
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< h3 id = "org88e3f87" > < span class = "section-number-3" > 3.4< / span > Initialize the Stewart structure< / h3 >
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< pre class = "src src-matlab" > stewart = struct();
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< h3 id = "org4a2edf9" > < span class = "section-number-3" > 3.5< / span > Compute the position of each frame< / h3 >
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< pre class = "src src-matlab" > stewart.H = args.H; < span class = "org-comment" > % Total Height of the Stewart Platform [m]< / span >
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stewart.FO_M = [0; 0; stewart.H]; < span class = "org-comment" > % Position of {M} with respect to {F} [m]< / span >
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stewart.MO_B = [0; 0; args.MO_B]; < span class = "org-comment" > % Position of {B} with respect to {M} [m]< / span >
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stewart.FO_A = stewart.MO_B < span class = "org-type" > +< / span > stewart.FO_M; < span class = "org-comment" > % Position of {A} with respect to {F} [m]< / span >
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< h2 id = "org8eb2e40" > < span class = "section-number-2" > 4< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h2 >
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< a id = "org7944b80" > < / a >
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This Matlab function is accessible < a href = "src/generateCubicConfiguration.m" > here< / a > .
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< h3 id = "org04bea4c" > < span class = "section-number-3" > 4.1< / span > Function description< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > generateCubicConfiguration< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
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< span class = "org-comment" > % generateCubicConfiguration - Generate a Cubic Configuration< / span >
< span class = "org-comment" > %< / span >
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< span class = "org-comment" > % Syntax: [stewart] = generateCubicConfiguration(stewart, args)< / span >
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< 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" > % - H [1x1] - Total height of the platform [m]< / span >
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< span class = "org-comment" > % - args - Can have the following fields:< / span >
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< span class = "org-comment" > % - Hc [1x1] - Height of the "useful" part of the cube [m]< / span >
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< span class = "org-comment" > % - FOc [1x1] - Height of the center of the cube with respect to {F} [m]< / span >
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< span class = "org-comment" > % - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]< / span >
< span class = "org-comment" > % - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - updated Stewart structure with the added fields:< / span >
< span class = "org-comment" > % - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< h3 id = "org924cbfe" > < span class = "section-number-3" > 4.2< / span > Documentation< / h3 >
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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
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< p > < span class = "figure-number" > Figure 2: < / span > Cubic Configuration< / p >
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< h3 id = "org335de0c" > < span class = "section-number-3" > 4.3< / span > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e< span class = "org-type" > -< / span > 3
args.FOc (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
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< h3 id = "orgc1a1209" > < span class = "section-number-3" > 4.4< / span > Position of the Cube< / h3 >
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We define the useful points of the cube with respect to the Cube’ s center.
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
located at the center of the cube and aligned with {F} and {M}.
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< pre class = "src src-matlab" > sx = [ 2; < span class = "org-type" > -< / span > 1; < span class = "org-type" > -< / span > 1];
sy = [ 0; 1; < span class = "org-type" > -< / span > 1];
sz = [ 1; 1; 1];
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R = [sx, sy, sz]< span class = "org-type" > ./< / span > vecnorm([sx, sy, sz]);
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L = args.Hc< span class = "org-type" > *< / span > sqrt(3);
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Cc = R< span class = "org-type" > '*< / span > [[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] < span class = "org-type" > -< / span > [0;0;1.5< span class = "org-type" > *< / span > args.Hc];
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CCf = [Cc(< span class = "org-type" > :< / span > ,1), Cc(< span class = "org-type" > :< / span > ,3), Cc(< span class = "org-type" > :< / span > ,3), Cc(< span class = "org-type" > :< / span > ,5), Cc(< span class = "org-type" > :< / span > ,5), Cc(< span class = "org-type" > :< / span > ,1)]; < span class = "org-comment" > % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg< / span >
CCm = [Cc(< span class = "org-type" > :< / span > ,2), Cc(< span class = "org-type" > :< / span > ,2), Cc(< span class = "org-type" > :< / span > ,4), Cc(< span class = "org-type" > :< / span > ,4), Cc(< span class = "org-type" > :< / span > ,6), Cc(< span class = "org-type" > :< / span > ,6)]; < span class = "org-comment" > % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg< / span >
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< h3 id = "orga1cfc06" > < span class = "section-number-3" > 4.5< / span > Compute the pose< / h3 >
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< p >
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We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
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< pre class = "src src-matlab" > CSi = (CCm < span class = "org-type" > -< / span > CCf)< span class = "org-type" > ./< / span > vecnorm(CCm < span class = "org-type" > -< / span > CCf);
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We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
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< pre class = "src src-matlab" > stewart.Fa = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc] < span class = "org-type" > +< / span > ((args.FHa< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
stewart.Mb = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc< span class = "org-type" > -< / span > stewart.H] < span class = "org-type" > +< / span > ((stewart.H< span class = "org-type" > -< / span > args.MHb< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
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< h2 id = "orgc137b22" > < span class = "section-number-2" > 5< / span > < code > computeJointsPose< / code > : Compute the Pose of the Joints< / h2 >
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< a id = "org566dd7e" > < / a >
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< p >
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This Matlab function is accessible < a href = "src/computeJointsPose.m" > here< / a > .
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > computeJointsPose< / span > (< span class = "org-variable-name" > stewart< / span > )
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< span class = "org-comment" > % computeJointsPose -< / span >
< span class = "org-comment" > %< / span >
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< span class = "org-comment" > % Syntax: [stewart] = computeJointsPose(stewart)< / span >
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< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following fields< / span >
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< span class = "org-comment" > % - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< span class = "org-comment" > % - FO_A [3x1] - Position of {A} with respect to {F}< / span >
< span class = "org-comment" > % - MO_B [3x1] - Position of {B} with respect to {M}< / span >
< span class = "org-comment" > % - FO_M [3x1] - Position of {M} with respect to {F}< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following added fields< / span >
< span class = "org-comment" > % - Aa [3x6] - The i'th column is the position of ai with respect to {A}< / span >
< span class = "org-comment" > % - Ab [3x6] - The i'th column is the position of bi with respect to {A}< / span >
< span class = "org-comment" > % - Ba [3x6] - The i'th column is the position of ai with respect to {B}< / span >
< span class = "org-comment" > % - Bb [3x6] - The i'th column is the position of bi with respect to {B}< / span >
< span class = "org-comment" > % - l [6x1] - The i'th element is the initial length of strut i< / span >
< span class = "org-comment" > % - As [3x6] - The i'th column is the unit vector of strut i expressed in {A}< / span >
< span class = "org-comment" > % - Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}< / span >
< span class = "org-comment" > % - FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}< / span >
< span class = "org-comment" > % - MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}< / span >
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< / pre >
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< / div >
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< div id = "outline-container-org5e72282" class = "outline-3" >
< h3 id = "org5e72282" > < span class = "section-number-3" > 5.2< / span > Documentation< / h3 >
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< div class = "outline-text-3" id = "text-5-2" >
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< div id = "org8d57a7b" class = "figure" >
< p > < img src = "figs/stewart-struts.png" alt = "stewart-struts.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Position and orientation of the struts< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgafa4ad9" class = "outline-3" >
< h3 id = "orgafa4ad9" > < span class = "section-number-3" > 5.3< / span > Compute the position of the Joints< / h3 >
< div class = "outline-text-3" id = "text-5-3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.Aa = stewart.Fa < span class = "org-type" > -< / span > repmat(stewart.FO_A, [1, 6]);
stewart.Bb = stewart.Mb < span class = "org-type" > -< / span > repmat(stewart.MO_B, [1, 6]);
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stewart.Ab = stewart.Bb < span class = "org-type" > -< / span > repmat(< span class = "org-type" > -< / span > stewart.MO_B< span class = "org-type" > -< / span > stewart.FO_M< span class = "org-type" > +< / span > stewart.FO_A, [1, 6]);
stewart.Ba = stewart.Aa < span class = "org-type" > -< / span > repmat( stewart.MO_B< span class = "org-type" > +< / span > stewart.FO_M< span class = "org-type" > -< / span > stewart.FO_A, [1, 6]);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org50bd360" class = "outline-3" >
< h3 id = "org50bd360" > < span class = "section-number-3" > 5.4< / span > Compute the strut length and orientation< / h3 >
< div class = "outline-text-3" id = "text-5-4" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.As = (stewart.Ab < span class = "org-type" > -< / span > stewart.Aa)< span class = "org-type" > ./< / span > vecnorm(stewart.Ab < span class = "org-type" > -< / span > stewart.Aa); < span class = "org-comment" > % As_i is the i'th vector of As< / span >
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stewart.l = vecnorm(stewart.Ab < span class = "org-type" > -< / span > stewart.Aa)< span class = "org-type" > '< / span > ;
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.Bs = (stewart.Bb < span class = "org-type" > -< / span > stewart.Ba)< span class = "org-type" > ./< / span > vecnorm(stewart.Bb < span class = "org-type" > -< / span > stewart.Ba);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3d0d76f" class = "outline-3" >
< h3 id = "org3d0d76f" > < span class = "section-number-3" > 5.5< / span > Compute the orientation of the Joints< / h3 >
< div class = "outline-text-3" id = "text-5-5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.FRa = zeros(3,3,6);
stewart.MRb = zeros(3,3,6);
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
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stewart.FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [cross([0;1;0], stewart.As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )) , cross(stewart.As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ), cross([0;1;0], stewart.As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))) , stewart.As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )];
stewart.FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = stewart.FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > ./< / span > vecnorm(stewart.FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ));
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stewart.MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [cross([0;1;0], stewart.Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )) , cross(stewart.Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ), cross([0;1;0], stewart.Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))) , stewart.Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )];
stewart.MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = stewart.MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > ./< / span > vecnorm(stewart.MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ));
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< span class = "org-keyword" > end< / span >
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< / pre >
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< / div >
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< div id = "outline-container-orgc7ae05f" class = "outline-2" >
< h2 id = "orgc7ae05f" > < span class = "section-number-2" > 6< / span > < code > initializeStrutDynamics< / code > : Add Stiffness and Damping properties of each strut< / h2 >
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< div class = "outline-text-2" id = "text-6" >
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< p >
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< a id = "orge0b89e3" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "src/initializeStrutDynamics.m" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-org234504f" class = "outline-3" >
< h3 id = "org234504f" > < span class = "section-number-3" > 6.1< / span > Function description< / h3 >
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< div class = "outline-text-3" id = "text-6-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > initializeStrutDynamics< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
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< span class = "org-comment" > % initializeStrutDynamics - Add Stiffness and Damping properties of each strut< / span >
< span class = "org-comment" > %< / span >
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< span class = "org-comment" > % Syntax: [stewart] = initializeStrutDynamics(args)< / span >
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< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
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< span class = "org-comment" > % - args - Structure with the following fields:< / span >
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< span class = "org-comment" > % - Ki [6x1] - Stiffness of each strut [N/m]< / span >
< span class = "org-comment" > % - Ci [6x1] - Damping of each strut [N/(m/s)]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - updated Stewart structure with the added fields:< / span >
< span class = "org-comment" > % - Ki [6x1] - Stiffness of each strut [N/m]< / span >
< span class = "org-comment" > % - Ci [6x1] - Damping of each strut [N/(m/s)]< / span >
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< / pre >
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< div id = "outline-container-org8edf5d0" class = "outline-3" >
< h3 id = "org8edf5d0" > < span class = "section-number-3" > 6.2< / span > Optional Parameters< / h3 >
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< div class = "outline-text-3" id = "text-6-2" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.Ki (6,1) double {mustBeNumeric, mustBePositive} = 1e6< span class = "org-type" > *< / span > ones(6,1)
args.Ci (6,1) double {mustBeNumeric, mustBePositive} = 1e2< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-keyword" > end< / span >
< / pre >
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< div id = "outline-container-orgf9bb882" class = "outline-3" >
< h3 id = "orgf9bb882" > < span class = "section-number-3" > 6.3< / span > Add Stiffness and Damping properties of each strut< / h3 >
< div class = "outline-text-3" id = "text-6-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.Ki = args.Ki;
stewart.Ci = args.Ci;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org61b514c" class = "outline-2" >
< h2 id = "org61b514c" > < span class = "section-number-2" > 7< / span > < code > computeJacobian< / code > : Compute the Jacobian Matrix< / h2 >
< div class = "outline-text-2" id = "text-7" >
< p >
< a id = "org27d5ba7" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "src/computeJacobian.m" > here< / a > .
< / p >
< / div >
< div id = "outline-container-org2983146" class = "outline-3" >
< h3 id = "org2983146" > < span class = "section-number-3" > 7.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-7-1" >
< 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" > % - As [3x6] - The 6 unit vectors for each strut expressed in {A}< / span >
< span class = "org-comment" > % - Ab [3x6] - The 6 position of the joints bi expressed in {A}< / 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" > % - J [6x6] - The Jacobian Matrix< / span >
< span class = "org-comment" > % - K [6x6] - The Stiffness Matrix< / span >
< span class = "org-comment" > % - C [6x6] - The Compliance Matrix< / span >
< / pre >
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< div id = "outline-container-org996fbd8" class = "outline-3" >
< h3 id = "org996fbd8" > < span class = "section-number-3" > 7.2< / span > Compute Jacobian Matrix< / h3 >
< div class = "outline-text-3" id = "text-7-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.J = [stewart.As< span class = "org-type" > '< / span > , cross(stewart.Ab, stewart.As)< span class = "org-type" > '< / span > ];
< / pre >
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< / div >
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< div id = "outline-container-org9e42bbe" class = "outline-3" >
< h3 id = "org9e42bbe" > < span class = "section-number-3" > 7.3< / span > Compute Stiffness Matrix< / h3 >
< div class = "outline-text-3" id = "text-7-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.K = stewart.J< span class = "org-type" > '*< / span > diag(stewart.Ki)< span class = "org-type" > *< / span > stewart.J;
< / pre >
< / div >
< / div >
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< div id = "outline-container-org36e44cd" class = "outline-3" >
< h3 id = "org36e44cd" > < span class = "section-number-3" > 7.4< / span > Compute Compliance Matrix< / h3 >
< div class = "outline-text-3" id = "text-7-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.C = inv(stewart.K);
< / pre >
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< div id = "outline-container-org76be477" class = "outline-2" >
< h2 id = "org76be477" > < span class = "section-number-2" > 8< / span > < code > inverseKinematics< / code > : Compute Inverse Kinematics< / h2 >
< div class = "outline-text-2" id = "text-8" >
< p >
< a id = "orgb8a3565" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "src/inverseKinematics.m" > here< / a > .
< / p >
< / div >
< div id = "outline-container-org16e3f5f" class = "outline-3" >
< h3 id = "org16e3f5f" > < span class = "section-number-3" > 8.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-8-1" >
< 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" > % - Aa [3x6] - The positions ai expressed in {A}< / span >
< span class = "org-comment" > % - Bb [3x6] - The positions bi expressed in {B}< / 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 >
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< div id = "outline-container-org5ccd852" class = "outline-3" >
< h3 id = "org5ccd852" > < span class = "section-number-3" > 8.2< / span > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-8-2" >
< 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 >
< / pre >
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< div id = "outline-container-org8896a32" class = "outline-3" >
< h3 id = "org8896a32" > < span class = "section-number-3" > 8.3< / span > Theory< / h3 >
< div class = "outline-text-3" id = "text-8-3" >
< p >
For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables, namely, \(\bm{L} = [l_1, l_2, \dots, l_6]^T\).
< / 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*}
l_i {}^A\hat{\bm{s}}_i & = {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
& = {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
\end{align*}
< p >
To obtain the length of each actuator and eliminate \(\hat{\bm{s}}_i\), it is sufficient to dot multiply each side by itself:
< / p >
\begin{equation}
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]
\end{equation}
< p >
Hence, for \(i = 1, 2, \dots, 6\), each limb length can be uniquely determined by:
< / p >
\begin{equation}
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}
\end{equation}
< p >
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.
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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< h3 id = "orgf5481aa" > < span class = "section-number-3" > 8.4< / span > Compute< / h3 >
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< pre class = "src src-matlab" > Li = sqrt(args.AP< span class = "org-type" > '*< / span > args.AP < span class = "org-type" > +< / span > diag(stewart.Bb< span class = "org-type" > '*< / span > stewart.Bb) < span class = "org-type" > +< / span > diag(stewart.Aa< span class = "org-type" > '*< / span > stewart.Aa) < span class = "org-type" > -< / span > (2< span class = "org-type" > *< / span > args.AP< span class = "org-type" > '*< / span > stewart.Aa)< span class = "org-type" > '< / span > < span class = "org-type" > +< / span > (2< span class = "org-type" > *< / span > args.AP< span class = "org-type" > '*< / span > (args.ARB< span class = "org-type" > *< / span > stewart.Bb))< span class = "org-type" > '< / span > < span class = "org-type" > -< / span > diag(2< span class = "org-type" > *< / span > (args.ARB< span class = "org-type" > *< / span > stewart.Bb)< span class = "org-type" > '*< / span > stewart.Aa));
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< pre class = "src src-matlab" > dLi = Li< span class = "org-type" > -< / span > stewart.l;
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< h2 id = "org1c1fd0d" > < span class = "section-number-2" > 9< / span > < code > forwardKinematicsApprox< / code > : Compute the Forward Kinematics< / h2 >
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< a id = "org55e7dad" > < / a >
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This Matlab function is accessible < a href = "src/forwardKinematicsApprox.m" > here< / a > .
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< h3 id = "orgcc0b9b5" > < span class = "section-number-3" > 9.1< / span > Function description< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [P, R]< / span > = < span class = "org-function-name" > forwardKinematicsApprox< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< 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" > % - 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 >
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< h3 id = "orgdbe04db" > < span class = "section-number-3" > 9.2< / span > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > arguments
stewart
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
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< h3 id = "orgc0a97e0" > < span class = "section-number-3" > 9.3< / span > Computation< / h3 >
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< p >
From a small displacement of each strut \(d\bm{\mathcal{L}}\), we can compute the
position and orientation of {B} with respect to {A} using the following formula:
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
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< pre class = "src src-matlab" > X = stewart.J< span class = "org-type" > \< / span > args.dL;
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The position vector corresponds to the first 3 elements.
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< pre class = "src src-matlab" > P = X(1< span class = "org-type" > :< / span > 3);
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The next 3 elements are the orientation of {B} with respect to {A} expressed
using the screw axis.
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< 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;
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< p >
We then compute the corresponding rotation matrix.
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< 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 >
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< p class = "author" > Author: Dehaeze Thomas< / p >
< p class = "date" > Created: 2020-01-06 lun. 18:16< / p >
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