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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 = "#orge7f2e4c" > 1. < code > initializeStewartPlatform< / code > : Initialize the Stewart Platform structure< / a >
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< ul >
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< li > < a href = "#org7437904" > Documentation< / a > < / li >
< li > < a href = "#org18108c7" > Function description< / a > < / li >
< li > < a href = "#org69ef310" > Initialize the Stewart structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgbeccb46" > 2. < code > initializeFramesPositions< / code > : Initialize the positions of frames {A}, {B}, {F} and {M}< / a >
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< ul >
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< li > < a href = "#org54ff91d" > Documentation< / a > < / li >
< li > < a href = "#orgae517fc" > Function description< / a > < / li >
< li > < a href = "#org73b6431" > Optional Parameters< / a > < / li >
< li > < a href = "#orgaa23d94" > Compute the position of each frame< / a > < / li >
< li > < a href = "#orgb62946f" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga77e5c5" > 3. < code > generateGeneralConfiguration< / code > : Generate a Very General Configuration< / a >
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< ul >
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< li > < a href = "#org4477edd" > Documentation< / a > < / li >
< li > < a href = "#org79d56c3" > Function description< / a > < / li >
< li > < a href = "#org9f060d3" > Optional Parameters< / a > < / li >
< li > < a href = "#org73e478f" > Compute the pose< / a > < / li >
< li > < a href = "#org35c894b" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc957583" > 4. < code > computeJointsPose< / code > : Compute the Pose of the Joints< / a >
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< ul >
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< li > < a href = "#orga3a832b" > Documentation< / a > < / li >
< li > < a href = "#orgb2d33b9" > Function description< / a > < / li >
< li > < a href = "#org7e7d25f" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#org18c8305" > Compute the position of the Joints< / a > < / li >
< li > < a href = "#orgbce62d5" > Compute the strut length and orientation< / a > < / li >
< li > < a href = "#org1cfa133" > Compute the orientation of the Joints< / a > < / li >
< li > < a href = "#org01435b1" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd545bd5" > 5. < code > initializeStewartPose< / code > : Determine the initial stroke in each leg to have the wanted pose< / a >
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< ul >
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< li > < a href = "#orgffc304e" > Function description< / a > < / li >
< li > < a href = "#orgc2e20c0" > Optional Parameters< / a > < / li >
< li > < a href = "#orgb65c297" > Use the Inverse Kinematic function< / a > < / li >
< li > < a href = "#org900faff" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org3f565b9" > 6. < code > initializeCylindricalPlatforms< / code > : Initialize the geometry of the Fixed and Mobile Platforms< / a >
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< ul >
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< li > < a href = "#orga2ae915" > Function description< / a > < / li >
< li > < a href = "#orge119415" > Optional Parameters< / a > < / li >
< li > < a href = "#org5e2f937" > Compute the Inertia matrices of platforms< / a > < / li >
< li > < a href = "#org6836f86" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org62ecd15" > 7. < code > initializeCylindricalStruts< / code > : Define the inertia of cylindrical struts< / a >
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< ul >
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< li > < a href = "#org06296f6" > Function description< / a > < / li >
< li > < a href = "#org515bf7f" > Optional Parameters< / a > < / li >
< li > < a href = "#org6a8f756" > Compute the properties of the cylindrical struts< / a > < / li >
< li > < a href = "#orgca52b33" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org83fc6ab" > 8. < code > initializeStrutDynamics< / code > : Add Stiffness and Damping properties of each strut< / a >
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< ul >
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< li > < a href = "#org5fc3036" > Documentation< / a > < / li >
< li > < a href = "#orge22b09f" > Function description< / a > < / li >
< li > < a href = "#orgddbcf42" > Optional Parameters< / a > < / li >
< li > < a href = "#orgb7fb7ad" > Add Stiffness and Damping properties of each strut< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb3ee6f9" > 9. < code > initializeJointDynamics< / code > : Add Stiffness and Damping properties for spherical joints< / a >
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< ul >
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< li > < a href = "#org0018bc5" > Function description< / a > < / li >
< li > < a href = "#org5fa604b" > Optional Parameters< / a > < / li >
< li > < a href = "#org914aea7" > Add Actuator Type< / a > < / li >
< li > < a href = "#org4d678a7" > Add Stiffness and Damping in Translation of each strut< / a > < / li >
< li > < a href = "#orgfbfeff9" > Add Stiffness and Damping in Rotation of each strut< / a > < / li >
< li > < a href = "#org11005ad" > Stiffness and Mass matrices for flexible joint< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org311a79b" > 10. < code > initializeInertialSensor< / code > : Initialize the inertial sensor in each strut< / a >
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< ul >
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< li > < a href = "#org62cf4fb" > Geophone - Working Principle< / a > < / li >
< li > < a href = "#orgeccb960" > Accelerometer - Working Principle< / a > < / li >
< li > < a href = "#org4a8ead1" > Function description< / a > < / li >
< li > < a href = "#org1843796" > Optional Parameters< / a > < / li >
< li > < a href = "#orgea5c452" > Compute the properties of the sensor< / a > < / li >
< li > < a href = "#orga99b20d" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org611c75c" > 11. < code > displayArchitecture< / code > : 3D plot of the Stewart platform architecture< / a >
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< ul >
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< li > < a href = "#org6a940aa" > Function description< / a > < / li >
< li > < a href = "#orgdb2c1a7" > Optional Parameters< / a > < / li >
< li > < a href = "#org14706e8" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#org0f76e3b" > Figure Creation, Frames and Homogeneous transformations< / a > < / li >
< li > < a href = "#org9cb7062" > Fixed Base elements< / a > < / li >
< li > < a href = "#orgc272598" > Mobile Platform elements< / a > < / li >
< li > < a href = "#orgfeaf65a" > Legs< / a > < / li >
< li > < a href = "#org8ae9c48" > 11.1. Figure parameters< / a > < / li >
< li > < a href = "#orgb92caec" > 11.2. Subplots< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org334a337" > 12. < code > describeStewartPlatform< / code > : Display some text describing the current defined Stewart Platform< / a >
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< ul >
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< li > < a href = "#orgd7efc3c" > Function description< / a > < / li >
< li > < a href = "#org193f623" > Optional Parameters< / a > < / li >
< li > < a href = "#org316408d" > 12.1. Geometry< / a > < / li >
< li > < a href = "#orgdb5d86c" > 12.2. Actuators< / a > < / li >
< li > < a href = "#org48d5b2e" > 12.3. Joints< / a > < / li >
< li > < a href = "#org62135e5" > 12.4. Kinematics< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org9decc94" > 13. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
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< ul >
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< li > < a href = "#org3485ee4" > Function description< / a > < / li >
< li > < a href = "#orgedfb402" > Documentation< / a > < / li >
< li > < a href = "#org0a637a2" > Optional Parameters< / a > < / li >
< li > < a href = "#org13c71d3" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#org44e0590" > Position of the Cube< / a > < / li >
< li > < a href = "#org92b12fa" > Compute the pose< / a > < / li >
< li > < a href = "#orgc91d6aa" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org10a231c" > 14. < code > computeJacobian< / code > : Compute the Jacobian Matrix< / a >
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< ul >
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< li > < a href = "#org7841d59" > Function description< / a > < / li >
< li > < a href = "#orgf2616df" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#org35ea7eb" > Compute Jacobian Matrix< / a > < / li >
< li > < a href = "#orgf645c75" > Compute Stiffness Matrix< / a > < / li >
< li > < a href = "#org21fadda" > Compute Compliance Matrix< / a > < / li >
< li > < a href = "#orgac75de7" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgcc980cd" > 15. < code > inverseKinematics< / code > : Compute Inverse Kinematics< / a >
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< ul >
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< li > < a href = "#org931c849" > Theory< / a > < / li >
< li > < a href = "#org5e787de" > Function description< / a > < / li >
< li > < a href = "#org2a4efbd" > Optional Parameters< / a > < / li >
< li > < a href = "#org318e667" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#orgeced891" > Compute< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org07fbe58" > 16. < code > forwardKinematicsApprox< / code > : Compute the Approximate Forward Kinematics< / a >
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< ul >
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< li > < a href = "#org5c8b01d" > Function description< / a > < / li >
< li > < a href = "#orgd653374" > Optional Parameters< / a > < / li >
< li > < a href = "#org1699c2d" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#orgf7dc851" > Computation< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
< p >
Stewart platforms are generated in multiple steps.
< / p >
< p >
We define 4 important < b > frames< / b > :
< / p >
< ul class = "org-ul" >
< 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 >
< / ul >
< p >
Then, we define the < b > location of the spherical joints< / b > :
< / p >
< 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 >
< p >
We define the < b > rest position< / b > of the Stewart platform:
< / p >
< ul class = "org-ul" >
< 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 >
< / ul >
< p >
From \(\bm{a}_{i}\) and \(\bm{b}_{i}\), we can determine the < b > length and orientation of each strut< / b > :
< / p >
< ul class = "org-ul" >
< 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 >
< / ul >
< p >
The position of the Spherical joints can be computed using various methods:
< / p >
< ul class = "org-ul" >
< 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 >
< / ul >
< p >
For Simscape, we need:
< / p >
< 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-orge7f2e4c" class = "outline-2" >
< h2 id = "orge7f2e4c" > < span class = "section-number-2" > 1< / span > < code > initializeStewartPlatform< / code > : Initialize the Stewart Platform structure< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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< a id = "orgd31cddc" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeStewartPlatform.m" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-org7437904" class = "outline-3" >
< h3 id = "org7437904" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-org7437904" >
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< div id = "orgd348bb2" class = "figure" >
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< p > < img src = "figs/stewart-frames-position.png" alt = "stewart-frames-position.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Definition of the position of the frames< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org18108c7" class = "outline-3" >
< h3 id = "org18108c7" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org18108c7" >
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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" > initializeStewartPlatform< / span > ()
< span class = "org-comment" > % initializeStewartPlatform - Initialize the stewart structure< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeStewartPlatform(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following sub-structures:< / span >
< span class = "org-comment" > % - platform_F -< / span >
< span class = "org-comment" > % - platform_M -< / span >
< span class = "org-comment" > % - joints_F -< / span >
< span class = "org-comment" > % - joints_M -< / span >
< span class = "org-comment" > % - struts_F -< / span >
< span class = "org-comment" > % - struts_M -< / span >
< span class = "org-comment" > % - actuators -< / span >
< span class = "org-comment" > % - geometry -< / span >
< span class = "org-comment" > % - properties -< / span >
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< / pre >
< / div >
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< / div >
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< div id = "outline-container-org69ef310" class = "outline-3" >
< h3 id = "org69ef310" > Initialize the Stewart structure< / h3 >
< div class = "outline-text-3" id = "text-org69ef310" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = struct();
stewart.platform_F = struct();
stewart.platform_M = struct();
stewart.joints_F = struct();
stewart.joints_M = struct();
stewart.struts_F = struct();
stewart.struts_M = struct();
stewart.actuators = struct();
stewart.sensors = struct();
stewart.sensors.inertial = struct();
stewart.sensors.force = struct();
stewart.sensors.relative = struct();
stewart.geometry = struct();
stewart.kinematics = struct();
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbeccb46" class = "outline-2" >
< h2 id = "orgbeccb46" > < span class = "section-number-2" > 2< / span > < code > initializeFramesPositions< / code > : Initialize the positions of frames {A}, {B}, {F} and {M}< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
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< a id = "org7dba982" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeFramesPositions.m" > here< / a > .
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< / p >
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< h3 id = "org54ff91d" > Documentation< / h3 >
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< div id = "orgb60aeab" class = "figure" >
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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 2: < / span > Definition of the position of the frames< / p >
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< h3 id = "orgae517fc" > Function description< / h3 >
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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" > initializeFramesPositions< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeFramesPositions(stewart, args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< 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 >
< 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" > % - geometry.H [1x1] - Total Height of the Stewart Platform [m]< / span >
< span class = "org-comment" > % - geometry.FO_M [3x1] - Position of {M} with respect to {F} [m]< / span >
< span class = "org-comment" > % - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]< / span >
< span class = "org-comment" > % - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]< / span >
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< h3 id = "org73b6431" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .H (1,1) double {mustBeNumeric, mustBePositive} = 90e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .MO_B (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
< span class = "org-keyword" > end< / span >
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< h3 id = "orgaa23d94" > Compute the position of each frame< / h3 >
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< pre class = "src src-matlab" > H = args.H; < span class = "org-comment" > % Total Height of the Stewart Platform [m]< / span >
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FO_M = [0; 0; H]; < span class = "org-comment" > % Position of {M} with respect to {F} [m]< / span >
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MO_B = [0; 0; args.MO_B]; < span class = "org-comment" > % Position of {B} with respect to {M} [m]< / span >
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FO_A = MO_B < span class = "org-type" > +< / span > FO_M; < span class = "org-comment" > % Position of {A} with respect to {F} [m]< / span >
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< h3 id = "orgb62946f" > Populate the < code > stewart< / code > structure< / h3 >
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< pre class = "src src-matlab" > stewart.geometry.H = H;
stewart.geometry.FO_M = FO_M;
stewart.platform_M.MO_B = MO_B;
stewart.platform_F.FO_A = FO_A;
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< h2 id = "orga77e5c5" > < span class = "section-number-2" > 3< / span > < code > generateGeneralConfiguration< / code > : Generate a Very General Configuration< / h2 >
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< p >
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< a id = "orga3f9dba" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/generateGeneralConfiguration.m" > here< / a > .
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< h3 id = "org4477edd" > Documentation< / h3 >
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< p >
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
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The radius of the circles can be chosen as well as the angles where the joints are located (see Figure < a href = "#org1fedc41" > 3< / a > ).
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< div id = "org1fedc41" class = "figure" >
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< p > < img src = "figs/stewart_bottom_plate.png" alt = "stewart_bottom_plate.png" / >
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< p > < span class = "figure-number" > Figure 3: < / span > Position of the joints< / p >
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< h3 id = "org79d56c3" > 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" > generateGeneralConfiguration< /span>(< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateGeneralConfiguration - Generate a Very General Configuration< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = generateGeneralConfiguration(stewart, args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]< / span >
< span class = "org-comment" > % - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]< / span >
< span class = "org-comment" > % - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]< / span >
< span class = "org-comment" > % - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]< / span >
< span class = "org-comment" > % - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]< / span >
< span class = "org-comment" > % - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]< / 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" > % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< h3 id = "org9f060d3" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .FH (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .FR (1,1) double {mustBeNumeric, mustBePositive} = 115e< span class = "org-type" > -< / span > 3;
< span class = "org-variable-name" > args< / span > .FTh (6,1) double {mustBeNumeric} = [< span class = "org-type" > -< / span > 10, 10, 120< span class = "org-type" > -< / span > 10, 120< span class = "org-type" > +< / span > 10, 240< span class = "org-type" > -< / span > 10, 240< span class = "org-type" > +< / span > 10]< span class = "org-type" > *< / span > (< span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 180);
< span class = "org-variable-name" > args< / span > .MH (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .MR (1,1) double {mustBeNumeric, mustBePositive} = 90e< span class = "org-type" > -< / span > 3;
< span class = "org-variable-name" > args< / span > .MTh (6,1) double {mustBeNumeric} = [< span class = "org-type" > -< / span > 60< span class = "org-type" > +< / span > 10, 60< span class = "org-type" > -< / span > 10, 60< span class = "org-type" > +< / span > 10, 180< span class = "org-type" > -< / span > 10, 180< span class = "org-type" > +< / span > 10, < span class = "org-type" > -< / span > 60< span class = "org-type" > -< / span > 10]< span class = "org-type" > *< / span > (< span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 180);
< span class = "org-keyword" > end< / span >
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< h3 id = "org73e478f" > Compute the pose< / h3 >
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< pre class = "src src-matlab" > Fa = zeros(3,6);
Mb = zeros(3,6);
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< pre class = "src src-matlab" > < 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 >
Fa(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [args.FR< span class = "org-type" > *< / span > cos(args.FTh(< span class = "org-constant" > i< / span > )); args.FR< span class = "org-type" > *< / span > sin(args.FTh(< span class = "org-constant" > i< / span > )); args.FH];
Mb(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [args.MR< span class = "org-type" > *< / span > cos(args.MTh(< span class = "org-constant" > i< / span > )); args.MR< span class = "org-type" > *< / span > sin(args.MTh(< span class = "org-constant" > i< / span > )); < span class = "org-type" > -< / span > args.MH];
< span class = "org-keyword" > end< / span >
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< h3 id = "org35c894b" > Populate the < code > stewart< / code > structure< / h3 >
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< pre class = "src src-matlab" > stewart.platform_F.Fa = Fa;
stewart.platform_M.Mb = Mb;
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< h2 id = "orgc957583" > < span class = "section-number-2" > 4< / span > < code > computeJointsPose< / code > : Compute the Pose of the Joints< / h2 >
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< p >
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< a id = "orga11f6a6" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/computeJointsPose.m" > here< / a > .
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< h3 id = "orga3a832b" > Documentation< / h3 >
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< p > < img src = "figs/stewart-struts.png" alt = "stewart-struts.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Position and orientation of the struts< / p >
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< h3 id = "orgb2d33b9" > 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" > computeJointsPose< / span > (< span class = "org-variable-name" > stewart< / span > )
< span class = "org-comment" > % computeJointsPose -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = computeJointsPose(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" > % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
< span class = "org-comment" > % - platform_F.FO_A [3x1] - Position of {A} with respect to {F}< / span >
< span class = "org-comment" > % - platform_M.MO_B [3x1] - Position of {B} with respect to {M}< / span >
< span class = "org-comment" > % - geometry.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" > % - geometry.Aa [3x6] - The i'th column is the position of ai with respect to {A}< / span >
< span class = "org-comment" > % - geometry.Ab [3x6] - The i'th column is the position of bi with respect to {A}< / span >
< span class = "org-comment" > % - geometry.Ba [3x6] - The i'th column is the position of ai with respect to {B}< / span >
< span class = "org-comment" > % - geometry.Bb [3x6] - The i'th column is the position of bi with respect to {B}< / span >
< span class = "org-comment" > % - geometry.l [6x1] - The i'th element is the initial length of strut i< / span >
< span class = "org-comment" > % - geometry.As [3x6] - The i'th column is the unit vector of strut i expressed in {A}< / span >
< span class = "org-comment" > % - geometry.Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}< / span >
< span class = "org-comment" > % - struts_F.l [6x1] - Length of the Fixed part of the i'th strut< / span >
< span class = "org-comment" > % - struts_M.l [6x1] - Length of the Mobile part of the i'th strut< / span >
< span class = "org-comment" > % - platform_F.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" > % - platform_M.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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< h3 id = "org7e7d25f" > Check the < code > stewart< / code > structure elements< / h3 >
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< pre class = "src src-matlab" > assert(isfield(stewart.platform_F, < span class = "org-string" > 'Fa'< / span > ), < span class = "org-string" > 'stewart.platform_F should have attribute Fa'< / span > )
Fa = stewart.platform_F.Fa;
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assert(isfield(stewart.platform_M, < span class = "org-string" > 'Mb'< / span > ), < span class = "org-string" > 'stewart.platform_M should have attribute Mb'< / span > )
Mb = stewart.platform_M.Mb;
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assert(isfield(stewart.platform_F, < span class = "org-string" > 'FO_A'< / span > ), < span class = "org-string" > 'stewart.platform_F should have attribute FO_A'< / span > )
FO_A = stewart.platform_F.FO_A;
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assert(isfield(stewart.platform_M, < span class = "org-string" > 'MO_B'< / span > ), < span class = "org-string" > 'stewart.platform_M should have attribute MO_B'< / span > )
MO_B = stewart.platform_M.MO_B;
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assert(isfield(stewart.geometry, < span class = "org-string" > 'FO_M'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute FO_M'< / span > )
FO_M = stewart.geometry.FO_M;
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< h3 id = "org18c8305" > Compute the position of the Joints< / h3 >
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< pre class = "src src-matlab" > Aa = Fa < span class = "org-type" > -< / span > repmat(FO_A, [1, 6]);
Bb = Mb < span class = "org-type" > -< / span > repmat(MO_B, [1, 6]);
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Ab = Bb < span class = "org-type" > -< / span > repmat(< span class = "org-type" > -< / span > MO_B< span class = "org-type" > -< / span > FO_M< span class = "org-type" > +< / span > FO_A, [1, 6]);
Ba = Aa < span class = "org-type" > -< / span > repmat( MO_B< span class = "org-type" > +< / span > FO_M< span class = "org-type" > -< / span > FO_A, [1, 6]);
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< / pre >
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< div id = "outline-container-orgbce62d5" class = "outline-3" >
< h3 id = "orgbce62d5" > Compute the strut length and orientation< / h3 >
< div class = "outline-text-3" id = "text-orgbce62d5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > As = (Ab < span class = "org-type" > -< / span > Aa)< span class = "org-type" > ./< / span > vecnorm(Ab < span class = "org-type" > -< / span > Aa); < span class = "org-comment" > % As_i is the i'th vector of As< / span >
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l = vecnorm(Ab < span class = "org-type" > -< / span > Aa)< span class = "org-type" > '< / span > ;
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< / pre >
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< pre class = "src src-matlab" > Bs = (Bb < span class = "org-type" > -< / span > Ba)< span class = "org-type" > ./< / span > vecnorm(Bb < span class = "org-type" > -< / span > Ba);
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< h3 id = "org1cfa133" > Compute the orientation of the Joints< / h3 >
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< pre class = "src src-matlab" > FRa = zeros(3,3,6);
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 >
FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [cross([0;1;0], As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )) , cross(As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ), cross([0;1;0], As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))) , As(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )];
FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > ./< / span > vecnorm(FRa(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ));
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MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [cross([0;1;0], Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )) , cross(Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ), cross([0;1;0], Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ))) , Bs(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )];
MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > ./< / span > vecnorm(MRb(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ));
< span class = "org-keyword" > end< / span >
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< div id = "outline-container-org01435b1" class = "outline-3" >
< h3 id = "org01435b1" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-org01435b1" >
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< pre class = "src src-matlab" > stewart.geometry.Aa = Aa;
stewart.geometry.Ab = Ab;
stewart.geometry.Ba = Ba;
stewart.geometry.Bb = Bb;
stewart.geometry.As = As;
stewart.geometry.Bs = Bs;
stewart.geometry.l = l;
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stewart.struts_F.l = l< span class = "org-type" > /< / span > 2;
stewart.struts_M.l = l< span class = "org-type" > /< / span > 2;
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stewart.platform_F.FRa = FRa;
stewart.platform_M.MRb = MRb;
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< h2 id = "orgd545bd5" > < span class = "section-number-2" > 5< / span > < code > initializeStewartPose< / code > : Determine the initial stroke in each leg to have the wanted pose< / h2 >
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< p >
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< a id = "orgf11f163" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeStewartPose.m" > here< / a > .
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< div id = "outline-container-orgffc304e" class = "outline-3" >
< h3 id = "orgffc304e" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgffc304e" >
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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" > initializeStewartPose< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose< / span >
< span class = "org-comment" > % It uses the inverse kinematic< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeStewartPose(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" > % - 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" > % - stewart - updated Stewart structure with the added fields:< / span >
< span class = "org-comment" > % - actuators.Leq [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 >
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< h3 id = "orgc2e20c0" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .AP (3,1) double {mustBeNumeric} = zeros(3,1)
< span class = "org-variable-name" > args< / span > .ARB (3,3) double {mustBeNumeric} = eye(3)
< span class = "org-keyword" > end< / span >
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< h3 id = "orgb65c297" > Use the Inverse Kinematic function< / h3 >
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< pre class = "src src-matlab" > [Li, dLi] = inverseKinematics(stewart, < span class = "org-string" > 'AP'< / span > , args.AP, < span class = "org-string" > 'ARB'< / span > , args.ARB);
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< h3 id = "org900faff" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-org900faff" >
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< pre class = "src src-matlab" > stewart.actuators.Leq = dLi;
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< h2 id = "org3f565b9" > < span class = "section-number-2" > 6< / span > < code > initializeCylindricalPlatforms< / code > : Initialize the geometry of the Fixed and Mobile Platforms< / h2 >
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< p >
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< a id = "org4eaf7b4" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeCylindricalPlatforms.m" > here< / a > .
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< h3 id = "orga2ae915" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orga2ae915" >
< 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" > initializeCylindricalPlatforms< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeCylindricalPlatforms(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - Fpm [1x1] - Fixed Platform Mass [kg]< / span >
< span class = "org-comment" > % - Fph [1x1] - Fixed Platform Height [m]< / span >
< span class = "org-comment" > % - Fpr [1x1] - Fixed Platform Radius [m]< / span >
< span class = "org-comment" > % - Mpm [1x1] - Mobile Platform Mass [kg]< / span >
< span class = "org-comment" > % - Mph [1x1] - Mobile Platform Height [m]< / span >
< span class = "org-comment" > % - Mpr [1x1] - Mobile Platform Radius [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" > % - platform_F [struct] - structure with the following fields:< / span >
< span class = "org-comment" > % - type = 1< / span >
< span class = "org-comment" > % - M [1x1] - Fixed Platform Mass [kg]< / span >
< span class = "org-comment" > % - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]< / span >
< span class = "org-comment" > % - H [1x1] - Fixed Platform Height [m]< / span >
< span class = "org-comment" > % - R [1x1] - Fixed Platform Radius [m]< / span >
< span class = "org-comment" > % - platform_M [struct] - structure with the following fields:< / span >
< span class = "org-comment" > % - M [1x1] - Mobile Platform Mass [kg]< / span >
< span class = "org-comment" > % - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]< / span >
< span class = "org-comment" > % - H [1x1] - Mobile Platform Height [m]< / span >
< span class = "org-comment" > % - R [1x1] - Mobile Platform Radius [m]< / span >
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< h3 id = "orge119415" > Optional Parameters< / h3 >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
< span class = "org-variable-name" > args< / span > .Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
< span class = "org-variable-name" > args< / span > .Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e< span class = "org-type" > -< / span > 3
< span class = "org-keyword" > end< / span >
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< h3 id = "org5e2f937" > Compute the Inertia matrices of platforms< / h3 >
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< pre class = "src src-matlab" > I_F = diag([1< span class = "org-type" > /< / span > 12< span class = "org-type" > *< / span > args.Fpm < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > args.Fpr< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > args.Fph< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 12< span class = "org-type" > *< / span > args.Fpm < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > args.Fpr< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > args.Fph< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 2 < span class = "org-type" > *< / span > args.Fpm < span class = "org-type" > *< / span > args.Fpr< span class = "org-type" > ^< / span > 2]);
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< pre class = "src src-matlab" > I_M = diag([1< span class = "org-type" > /< / span > 12< span class = "org-type" > *< / span > args.Mpm < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > args.Mpr< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > args.Mph< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 12< span class = "org-type" > *< / span > args.Mpm < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > args.Mpr< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > args.Mph< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 2 < span class = "org-type" > *< / span > args.Mpm < span class = "org-type" > *< / span > args.Mpr< span class = "org-type" > ^< / span > 2]);
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< h3 id = "org6836f86" > Populate the < code > stewart< / code > structure< / h3 >
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< pre class = "src src-matlab" > stewart.platform_F.type = 1;
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stewart.platform_F.I = I_F;
stewart.platform_F.M = args.Fpm;
stewart.platform_F.R = args.Fpr;
stewart.platform_F.H = args.Fph;
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< pre class = "src src-matlab" > stewart.platform_M.type = 1;
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stewart.platform_M.I = I_M;
stewart.platform_M.M = args.Mpm;
stewart.platform_M.R = args.Mpr;
stewart.platform_M.H = args.Mph;
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< h2 id = "org62ecd15" > < span class = "section-number-2" > 7< / span > < code > initializeCylindricalStruts< / code > : Define the inertia of cylindrical struts< / h2 >
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< div class = "outline-text-2" id = "text-7" >
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< p >
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< a id = "org24298b7" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeCylindricalStruts.m" > here< / a > .
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< h3 id = "org06296f6" > Function description< / h3 >
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< 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" > initializeCylindricalStruts< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeCylindricalStruts(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - Fsm [1x1] - Mass of the Fixed part of the struts [kg]< / span >
< span class = "org-comment" > % - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]< / span >
< span class = "org-comment" > % - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]< / span >
< span class = "org-comment" > % - Msm [1x1] - Mass of the Mobile part of the struts [kg]< / span >
< span class = "org-comment" > % - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]< / span >
< span class = "org-comment" > % - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [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" > % - struts_F [struct] - structure with the following fields:< / span >
< span class = "org-comment" > % - M [6x1] - Mass of the Fixed part of the struts [kg]< / span >
< span class = "org-comment" > % - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]< / span >
< span class = "org-comment" > % - H [6x1] - Height of cylinder for the Fixed part of the struts [m]< / span >
< span class = "org-comment" > % - R [6x1] - Radius of cylinder for the Fixed part of the struts [m]< / span >
< span class = "org-comment" > % - struts_M [struct] - structure with the following fields:< / span >
< span class = "org-comment" > % - M [6x1] - Mass of the Mobile part of the struts [kg]< / span >
< span class = "org-comment" > % - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]< / span >
< span class = "org-comment" > % - H [6x1] - Height of cylinder for the Mobile part of the struts [m]< / span >
< span class = "org-comment" > % - R [6x1] - Radius of cylinder for the Mobile part of the struts [m]< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org515bf7f" class = "outline-3" >
< h3 id = "org515bf7f" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org515bf7f" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
< span class = "org-variable-name" > args< / span > .Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
< span class = "org-variable-name" > args< / span > .Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e< span class = "org-type" > -< / span > 3
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6a8f756" class = "outline-3" >
< h3 id = "org6a8f756" > Compute the properties of the cylindrical struts< / h3 >
< div class = "outline-text-3" id = "text-org6a8f756" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Fsm = ones(6,1)< span class = "org-type" > .*< / span > args.Fsm;
Fsh = ones(6,1)< span class = "org-type" > .*< / span > args.Fsh;
Fsr = ones(6,1)< span class = "org-type" > .*< / span > args.Fsr;
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Msm = ones(6,1)< span class = "org-type" > .*< / span > args.Msm;
Msh = ones(6,1)< span class = "org-type" > .*< / span > args.Msh;
Msr = ones(6,1)< span class = "org-type" > .*< / span > args.Msr;
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > I_F = zeros(3, 3, 6); < span class = "org-comment" > % Inertia of the "fixed" part of the strut< / span >
I_M = zeros(3, 3, 6); < span class = "org-comment" > % Inertia of the "mobile" part of the strut< / span >
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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 >
I_F(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = diag([1< span class = "org-type" > /< / span > 12 < span class = "org-type" > *< / span > Fsm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > Fsr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > Fsh(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 12 < span class = "org-type" > *< / span > Fsm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > Fsr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > Fsh(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 2 < span class = "org-type" > *< / span > Fsm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > Fsr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2]);
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I_M(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = diag([1< span class = "org-type" > /< / span > 12 < span class = "org-type" > *< / span > Msm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > Msr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > Msh(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 12 < span class = "org-type" > *< / span > Msm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > (3< span class = "org-type" > *< / span > Msr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > Msh(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2), ...
1< span class = "org-type" > /< / span > 2 < span class = "org-type" > *< / span > Msm(< span class = "org-constant" > i< / span > ) < span class = "org-type" > *< / span > Msr(< span class = "org-constant" > i< / span > )< span class = "org-type" > ^< / span > 2]);
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgca52b33" class = "outline-3" >
< h3 id = "orgca52b33" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orgca52b33" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.struts_M.type = 1;
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stewart.struts_M.I = I_M;
stewart.struts_M.M = Msm;
stewart.struts_M.R = Msr;
stewart.struts_M.H = Msh;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.struts_F.type = 1;
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stewart.struts_F.I = I_F;
stewart.struts_F.M = Fsm;
stewart.struts_F.R = Fsr;
stewart.struts_F.H = Fsh;
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org83fc6ab" class = "outline-2" >
< h2 id = "org83fc6ab" > < span class = "section-number-2" > 8< / span > < code > initializeStrutDynamics< / code > : Add Stiffness and Damping properties of each strut< / h2 >
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< div class = "outline-text-2" id = "text-8" >
< p >
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< a id = "org06fbb6c" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/initializeStrutDynamics.m" > here< / a > .
< / p >
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< / div >
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< div id = "outline-container-org5fc3036" class = "outline-3" >
< h3 id = "org5fc3036" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-org5fc3036" >
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< div id = "org71d2515" class = "figure" >
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< p > < img src = "figs/piezoelectric_stack.jpg" alt = "piezoelectric_stack.jpg" width = "500px" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Example of a piezoelectric stach actuator (PI)< / p >
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< / div >
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< p >
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A simplistic model of such amplified actuator is shown in Figure < a href = "#orge5212d2" > 6< / a > where:
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< / p >
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< ul class = "org-ul" >
< li > \(K\) represent the vertical stiffness of the actuator< / li >
< li > \(C\) represent the vertical damping of the actuator< / li >
< li > \(F\) represents the force applied by the actuator< / li >
< li > \(F_{m}\) represents the total measured force< / li >
< li > \(v_{m}\) represents the absolute velocity of the top part of the actuator< / li >
< li > \(d_{m}\) represents the total relative displacement of the actuator< / li >
< / ul >
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< div id = "orge5212d2" class = "figure" >
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< p > < img src = "figs/actuator_model_simple.png" alt = "actuator_model_simple.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Simple model of an Actuator< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orge22b09f" class = "outline-3" >
< h3 id = "orge22b09f" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orge22b09f" >
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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 > )
< span class = "org-comment" > % initializeStrutDynamics - Add Stiffness and Damping properties of each strut< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeStrutDynamics(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - K [6x1] - Stiffness of each strut [N/m]< / span >
< span class = "org-comment" > % - C [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" > % - actuators.type = 1< / span >
< span class = "org-comment" > % - actuators.K [6x1] - Stiffness of each strut [N/m]< / span >
< span class = "org-comment" > % - actuators.C [6x1] - Damping of each strut [N/(m/s)]< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgddbcf42" class = "outline-3" >
< h3 id = "orgddbcf42" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-orgddbcf42" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .type char {mustBeMember(args.type,{< span class = "org-string" > 'classical'< / span > , < span class = "org-string" > 'amplified'< / span > })} = < span class = "org-string" > 'classical'< / span >
< span class = "org-variable-name" > args< / span > .K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .k1 (6,1) double {mustBeNumeric} = 1e6< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .ke (6,1) double {mustBeNumeric} = 5e6< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .ka (6,1) double {mustBeNumeric} = 60e6< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .c1 (6,1) double {mustBeNumeric} = 10< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .F_gain (6,1) double {mustBeNumeric} = 1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .me (6,1) double {mustBeNumeric} = 0.01< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .ma (6,1) double {mustBeNumeric} = 0.01< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb7fb7ad" class = "outline-3" >
< h3 id = "orgb7fb7ad" > Add Stiffness and Damping properties of each strut< / h3 >
< div class = "outline-text-3" id = "text-orgb7fb7ad" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > strcmp(args.type, < span class = "org-string" > 'classical'< / span > )
stewart.actuators.type = 1;
< span class = "org-keyword" > elseif< / span > strcmp(args.type, < span class = "org-string" > 'amplified'< / span > )
stewart.actuators.type = 2;
< span class = "org-keyword" > end< / span >
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stewart.actuators.K = args.K;
stewart.actuators.C = args.C;
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stewart.actuators.k1 = args.k1;
stewart.actuators.c1 = args.c1;
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stewart.actuators.ka = args.ka;
stewart.actuators.ke = args.ke;
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stewart.actuators.F_gain = args.F_gain;
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stewart.actuators.ma = args.ma;
stewart.actuators.me = args.me;
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< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgb3ee6f9" class = "outline-2" >
< h2 id = "orgb3ee6f9" > < span class = "section-number-2" > 9< / span > < code > initializeJointDynamics< / code > : Add Stiffness and Damping properties for spherical joints< / h2 >
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< div class = "outline-text-2" id = "text-9" >
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< p >
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< a id = "orgbaeb9f9" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeJointDynamics.m" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-org0018bc5" class = "outline-3" >
< h3 id = "org0018bc5" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org0018bc5" >
< 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" > initializeJointDynamics< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeJointDynamics(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'< / span >
< span class = "org-comment" > % - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'< / span >
< span class = "org-comment" > % - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]< / span >
< span class = "org-comment" > % - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]< / span >
< span class = "org-comment" > % - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]< / span >
< span class = "org-comment" > % - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]< / span >
< span class = "org-comment" > % - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]< / span >
< span class = "org-comment" > % - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]< / span >
< span class = "org-comment" > % - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]< / span >
< span class = "org-comment" > % - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/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" > % - stewart.joints_F and stewart.joints_M:< / span >
< span class = "org-comment" > % - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)< / span >
< span class = "org-comment" > % - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]< / span >
< span class = "org-comment" > % - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]< / span >
< span class = "org-comment" > % - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]< / span >
< span class = "org-comment" > % - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]< / span >
< span class = "org-comment" > % - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]< / span >
< span class = "org-comment" > % - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org5fa604b" class = "outline-3" >
< h3 id = "org5fa604b" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org5fa604b" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .type_F char {mustBeMember(args.type_F,{< span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'spherical'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'spherical_p'< / span > , < span class = "org-string" > 'universal_3dof'< / span > , < span class = "org-string" > 'spherical_3dof'< / span > , < span class = "org-string" > 'flexible'< / span > })} = < span class = "org-string" > 'universal'< / span >
< span class = "org-variable-name" > args< / span > .type_M char {mustBeMember(args.type_M,{< span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'spherical'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'spherical_p'< / span > , < span class = "org-string" > 'universal_3dof'< / span > , < span class = "org-string" > 'spherical_3dof'< / span > , < span class = "org-string" > 'flexible'< / span > })} = < span class = "org-string" > 'spherical'< / span >
< span class = "org-variable-name" > args< / span > .Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 33< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e< span class = "org-type" > -< / span > 4< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 236< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 33< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e< span class = "org-type" > -< / span > 4< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 236< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ka_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ca_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Kr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Cr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ka_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Ca_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Kr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .Cr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1< span class = "org-type" > *< / span > ones(6,1)
< span class = "org-variable-name" > args< / span > .K_M double {mustBeNumeric} = zeros(6,6)
< span class = "org-variable-name" > args< / span > .M_M double {mustBeNumeric} = zeros(6,6)
< span class = "org-variable-name" > args< / span > .n_xyz_M double {mustBeNumeric} = zeros(2,3)
< span class = "org-variable-name" > args< / span > .xi_M double {mustBeNumeric} = 0.1
< span class = "org-variable-name" > args< / span > .step_file_M char {} = < span class = "org-string" > ''< / span >
< span class = "org-variable-name" > args< / span > .K_F double {mustBeNumeric} = zeros(6,6)
< span class = "org-variable-name" > args< / span > .M_F double {mustBeNumeric} = zeros(6,6)
< span class = "org-variable-name" > args< / span > .n_xyz_F double {mustBeNumeric} = zeros(2,3)
< span class = "org-variable-name" > args< / span > .xi_F double {mustBeNumeric} = 0.1
< span class = "org-variable-name" > args< / span > .step_file_F char {} = < span class = "org-string" > ''< / span >
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org914aea7" class = "outline-3" >
< h3 id = "org914aea7" > Add Actuator Type< / h3 >
< div class = "outline-text-3" id = "text-org914aea7" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > switch< / span > < span class = "org-constant" > args.type_F< / span >
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal'< / span >
stewart.joints_F.type = 1;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical'< / span >
stewart.joints_F.type = 2;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal_p'< / span >
stewart.joints_F.type = 3;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical_p'< / span >
stewart.joints_F.type = 4;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'flexible'< / span >
stewart.joints_F.type = 5;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal_3dof'< / span >
stewart.joints_F.type = 6;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical_3dof'< / span >
stewart.joints_F.type = 7;
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > switch< / span > < span class = "org-constant" > args.type_M< / span >
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal'< / span >
stewart.joints_M.type = 1;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical'< / span >
stewart.joints_M.type = 2;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal_p'< / span >
stewart.joints_M.type = 3;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical_p'< / span >
stewart.joints_M.type = 4;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'flexible'< / span >
stewart.joints_M.type = 5;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'universal_3dof'< / span >
stewart.joints_M.type = 6;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'spherical_3dof'< / span >
stewart.joints_M.type = 7;
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org4d678a7" class = "outline-3" >
< h3 id = "org4d678a7" > Add Stiffness and Damping in Translation of each strut< / h3 >
< div class = "outline-text-3" id = "text-org4d678a7" >
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< p >
Axial and Radial (shear) Stiffness
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_M.Ka = args.Ka_M;
stewart.joints_M.Kr = args.Kr_M;
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stewart.joints_F.Ka = args.Ka_F;
stewart.joints_F.Kr = args.Kr_F;
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< / pre >
< / div >
< p >
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Translation Damping
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_M.Ca = args.Ca_M;
stewart.joints_M.Cr = args.Cr_M;
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stewart.joints_F.Ca = args.Ca_F;
stewart.joints_F.Cr = args.Cr_F;
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-orgfbfeff9" class = "outline-3" >
< h3 id = "orgfbfeff9" > Add Stiffness and Damping in Rotation of each strut< / h3 >
< div class = "outline-text-3" id = "text-orgfbfeff9" >
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< p >
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Rotational Stiffness
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_M.Kf = args.Kf_M;
stewart.joints_M.Kt = args.Kt_M;
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stewart.joints_F.Kf = args.Kf_F;
stewart.joints_F.Kt = args.Kt_F;
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< / pre >
< / div >
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< p >
Rotational Damping
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_M.Cf = args.Cf_M;
stewart.joints_M.Ct = args.Ct_M;
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stewart.joints_F.Cf = args.Cf_F;
stewart.joints_F.Ct = args.Ct_F;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org11005ad" class = "outline-3" >
< h3 id = "org11005ad" > Stiffness and Mass matrices for flexible joint< / h3 >
< div class = "outline-text-3" id = "text-org11005ad" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_F.M = args.M_F;
stewart.joints_F.K = args.K_F;
stewart.joints_F.n_xyz = args.n_xyz_F;
stewart.joints_F.xi = args.xi_F;
stewart.joints_F.xi = args.xi_F;
stewart.joints_F.step_file = args.step_file_F;
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stewart.joints_M.M = args.M_M;
stewart.joints_M.K = args.K_M;
stewart.joints_M.n_xyz = args.n_xyz_M;
stewart.joints_M.xi = args.xi_M;
stewart.joints_M.step_file = args.step_file_M;
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org311a79b" class = "outline-2" >
< h2 id = "org311a79b" > < span class = "section-number-2" > 10< / span > < code > initializeInertialSensor< / code > : Initialize the inertial sensor in each strut< / h2 >
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< div class = "outline-text-2" id = "text-10" >
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< p >
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< a id = "org79381ea" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/initializeInertialSensor.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org62cf4fb" class = "outline-3" >
< h3 id = "org62cf4fb" > Geophone - Working Principle< / h3 >
< div class = "outline-text-3" id = "text-org62cf4fb" >
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< p >
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From the schematic of the Z-axis geophone shown in Figure < a href = "#orgdaf8258" > 7< / a > , we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\):
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\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0 = \sqrt{\frac{k}{m}}\)< / li >
< li > \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)< / li >
< / ul >
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< div id = "orgdaf8258" class = "figure" >
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< p > < img src = "figs/inertial_sensor.png" alt = "inertial_sensor.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Schematic of a Z-Axis geophone< / p >
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< / div >
< p >
We see that at frequencies above \(\omega_0\):
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
< / p >
< p >
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
< / p >
< p >
We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass.
< / p >
< / div >
< / div >
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< div id = "outline-container-orgeccb960" class = "outline-3" >
< h3 id = "orgeccb960" > Accelerometer - Working Principle< / h3 >
< div class = "outline-text-3" id = "text-orgeccb960" >
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< p >
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From the schematic of the Z-axis accelerometer shown in Figure < a href = "#org4065027" > 8< / a > , we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\):
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\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
< / p >
< ul class = "org-ul" >
< li > \(\omega_0 = \sqrt{\frac{k}{m}}\)< / li >
< li > \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)< / li >
< / ul >
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< div id = "org4065027" class = "figure" >
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< p > < img src = "figs/inertial_sensor.png" alt = "inertial_sensor.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Schematic of a Z-Axis geophone< / p >
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< / div >
< p >
We see that at frequencies below \(\omega_0\):
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
< / p >
< p >
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
< / p >
< p >
Note that there is trade-off between:
< / p >
< ul class = "org-ul" >
< li > the highest measurable acceleration \(\omega_0\)< / li >
< li > the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\)< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org4a8ead1" class = "outline-3" >
< h3 id = "org4a8ead1" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org4a8ead1" >
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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" > initializeInertialSensor< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeInertialSensor - Initialize the inertial sensor in each strut< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = initializeInertialSensor(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - type - 'geophone', 'accelerometer', 'none'< / span >
< span class = "org-comment" > % - mass [1x1] - Weight of the inertial mass [kg]< / span >
< span class = "org-comment" > % - freq [1x1] - Cutoff frequency [Hz]< / 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" > % - stewart.sensors.inertial< / span >
< span class = "org-comment" > % - type - 1 (geophone), 2 (accelerometer), 3 (none)< / span >
< span class = "org-comment" > % - K [1x1] - Stiffness [N/m]< / span >
< span class = "org-comment" > % - C [1x1] - Damping [N/(m/s)]< / span >
< span class = "org-comment" > % - M [1x1] - Inertial Mass [kg]< / span >
< span class = "org-comment" > % - G [1x1] - Gain< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1843796" class = "outline-3" >
< h3 id = "org1843796" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org1843796" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .type char {mustBeMember(args.type,{< span class = "org-string" > 'geophone'< / span > , < span class = "org-string" > 'accelerometer'< / span > , < span class = "org-string" > 'none'< / span > })} = < span class = "org-string" > 'none'< / span >
< span class = "org-variable-name" > args< / span > .mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e< span class = "org-type" > -< / span > 2
< span class = "org-variable-name" > args< / span > .freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgea5c452" class = "outline-3" >
< h3 id = "orgea5c452" > Compute the properties of the sensor< / h3 >
< div class = "outline-text-3" id = "text-orgea5c452" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > sensor = struct();
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< span class = "org-keyword" > switch< / span > < span class = "org-constant" > args.type< / span >
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'geophone'< / span >
sensor.type = 1;
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sensor.M = args.mass;
sensor.K = sensor.M < span class = "org-type" > *< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > args.freq)< span class = "org-type" > ^< / span > 2;
sensor.C = 2< span class = "org-type" > *< / span > sqrt(sensor.M < span class = "org-type" > *< / span > sensor.K);
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'accelerometer'< / span >
sensor.type = 2;
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sensor.M = args.mass;
sensor.K = sensor.M < span class = "org-type" > *< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > args.freq)< span class = "org-type" > ^< / span > 2;
sensor.C = 2< span class = "org-type" > *< / span > sqrt(sensor.M < span class = "org-type" > *< / span > sensor.K);
sensor.G = < span class = "org-type" > -< / span > sensor.K< span class = "org-type" > /< / span > sensor.M;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'none'< / span >
sensor.type = 3;
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga99b20d" class = "outline-3" >
< h3 id = "orga99b20d" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orga99b20d" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.sensors.inertial = sensor;
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org611c75c" class = "outline-2" >
< h2 id = "org611c75c" > < span class = "section-number-2" > 11< / span > < code > displayArchitecture< / code > : 3D plot of the Stewart platform architecture< / h2 >
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< div class = "outline-text-2" id = "text-11" >
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< p >
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< a id = "org3c576cf" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/displayArchitecture.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org6a940aa" class = "outline-3" >
< h3 id = "org6a940aa" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org6a940aa" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > []< / span > = < span class = "org-function-name" > displayArchitecture< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % displayArchitecture - 3D plot of the Stewart platform architecture< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [] = displayArchitecture(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart< / span >
< span class = "org-comment" > % - args - Structure with 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" > % - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}< / span >
< span class = "org-comment" > % - F_color [color] - Color used for the Fixed elements< / span >
< span class = "org-comment" > % - M_color [color] - Color used for the Mobile elements< / span >
< span class = "org-comment" > % - L_color [color] - Color used for the Legs elements< / span >
< span class = "org-comment" > % - frames [true/false] - Display the Frames< / span >
< span class = "org-comment" > % - legs [true/false] - Display the Legs< / span >
< span class = "org-comment" > % - joints [true/false] - Display the Joints< / span >
< span class = "org-comment" > % - labels [true/false] - Display the Labels< / span >
< span class = "org-comment" > % - platforms [true/false] - Display the Platforms< / span >
< span class = "org-comment" > % - views ['all', 'xy', 'yz', 'xz', 'default'] -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgdb2c1a7" class = "outline-3" >
< h3 id = "orgdb2c1a7" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-orgdb2c1a7" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .AP (3,1) double {mustBeNumeric} = zeros(3,1)
< span class = "org-variable-name" > args< / span > .ARB (3,3) double {mustBeNumeric} = eye(3)
< span class = "org-variable-name" > args< / span > .F_color = [0 0.4470 0.7410]
< span class = "org-variable-name" > args< / span > .M_color = [0.8500 0.3250 0.0980]
< span class = "org-variable-name" > args< / span > .L_color = [0 0 0]
< span class = "org-variable-name" > args< / span > .frames logical {mustBeNumericOrLogical} = < span class = "org-constant" > true< / span >
< span class = "org-variable-name" > args< / span > .legs logical {mustBeNumericOrLogical} = < span class = "org-constant" > true< / span >
< span class = "org-variable-name" > args< / span > .joints logical {mustBeNumericOrLogical} = < span class = "org-constant" > true< / span >
< span class = "org-variable-name" > args< / span > .labels logical {mustBeNumericOrLogical} = < span class = "org-constant" > true< / span >
< span class = "org-variable-name" > args< / span > .platforms logical {mustBeNumericOrLogical} = < span class = "org-constant" > true< / span >
< span class = "org-variable-name" > args< / span > .views char {mustBeMember(args.views,{< span class = "org-string" > 'all'< / span > , < span class = "org-string" > 'xy'< / span > , < span class = "org-string" > 'xz'< / span > , < span class = "org-string" > 'yz'< / span > , < span class = "org-string" > 'default'< / span > })} = < span class = "org-string" > 'default'< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org14706e8" class = "outline-3" >
< h3 id = "org14706e8" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org14706e8" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > assert(isfield(stewart.platform_F, < span class = "org-string" > 'FO_A'< / span > ), < span class = "org-string" > 'stewart.platform_F should have attribute FO_A'< / span > )
FO_A = stewart.platform_F.FO_A;
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assert(isfield(stewart.platform_M, < span class = "org-string" > 'MO_B'< / span > ), < span class = "org-string" > 'stewart.platform_M should have attribute MO_B'< / span > )
MO_B = stewart.platform_M.MO_B;
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assert(isfield(stewart.geometry, < span class = "org-string" > 'H'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute H'< / span > )
H = stewart.geometry.H;
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assert(isfield(stewart.platform_F, < span class = "org-string" > 'Fa'< / span > ), < span class = "org-string" > 'stewart.platform_F should have attribute Fa'< / span > )
Fa = stewart.platform_F.Fa;
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assert(isfield(stewart.platform_M, < span class = "org-string" > 'Mb'< / span > ), < span class = "org-string" > 'stewart.platform_M should have attribute Mb'< / span > )
Mb = stewart.platform_M.Mb;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0f76e3b" class = "outline-3" >
< h3 id = "org0f76e3b" > Figure Creation, Frames and Homogeneous transformations< / h3 >
< div class = "outline-text-3" id = "text-org0f76e3b" >
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< p >
The reference frame of the 3d plot corresponds to the frame \(\{F\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > < span class = "org-type" > ~< / span > strcmp(args.views, < span class = "org-string" > 'all'< / span > )
< span class = "org-type" > figure< / span > ;
< span class = "org-keyword" > else< / span >
f = < span class = "org-type" > figure< / span > (< span class = "org-string" > 'visible'< / span > , < span class = "org-string" > 'off'< / span > );
< span class = "org-keyword" > end< / span >
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hold on;
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< / pre >
< / div >
< p >
We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is \(\{F\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > FTa = [eye(3), FO_A; ...
zeros< span class = "org-type" > (1,3), 1];< / span >
ATb = [args.ARB, args.AP; ...
zeros< span class = "org-type" > (1,3), 1];< / span >
BTm = [eye(3), < span class = "org-type" > -< / span > MO_B; ...
zeros< span class = "org-type" > (1,3), 1];< / span >
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FTm = FTa< span class = "org-type" > *< / span > ATb< span class = "org-type" > *< / span > BTm;
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< / pre >
< / div >
< p >
Let’ s define a parameter that define the length of the unit vectors used to display the frames.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > d_unit_vector = H< span class = "org-type" > /< / span > 4;
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< / pre >
< / div >
< p >
Let’ s define a parameter used to position the labels with respect to the center of the element.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > d_label = H< span class = "org-type" > /< / span > 20;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org9cb7062" class = "outline-3" >
< h3 id = "org9cb7062" > Fixed Base elements< / h3 >
< div class = "outline-text-3" id = "text-org9cb7062" >
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< p >
Let’ s first plot the frame \(\{F\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Ff = [0, 0, 0];
< span class = "org-keyword" > if< / span > args.frames
quiver3(Ff(1)< span class = "org-type" > *< / span > ones(1,3), Ff(2)< span class = "org-type" > *< / span > ones(1,3), Ff(3)< span class = "org-type" > *< / span > ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.F_color)
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-type" > text< / span > (Ff(1) < span class = "org-type" > +< / span > d_label, ...
Ff< span class = "org-type" > (2) + d_label, ...< / span >
Ff(3) < span class = "org-type" > +< / span > d_label, < span class = "org-string" > '$\{F\}$'< / span > , < span class = "org-string" > 'Color'< / span > , args.F_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Now plot the frame \(\{A\}\) fixed to the Base.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.frames
quiver3(FO_A(1)< span class = "org-type" > *< / span > ones(1,3), FO_A(2)< span class = "org-type" > *< / span > ones(1,3), FO_A(3)< span class = "org-type" > *< / span > ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.F_color)
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-type" > text< / span > (FO_A(1) < span class = "org-type" > +< / span > d_label, ...
FO_A< span class = "org-type" > (2) + d_label, ...< / span >
FO_A(3) < span class = "org-type" > +< / span > d_label, < span class = "org-string" > '$\{A\}$'< / span > , < span class = "org-string" > 'Color'< / span > , args.F_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Let’ s then plot the circle corresponding to the shape of the Fixed base.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.platforms < span class = "org-type" > & & < / span > stewart.platform_F.type < span class = "org-type" > ==< / span > 1
theta = [0< span class = "org-type" > :< / span > 0.01< span class = "org-type" > :< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > +< / span > 0.01]; < span class = "org-comment" > % Angles [rad]< / span >
v = null([0; 0; 1]< span class = "org-type" > '< / span > ); < span class = "org-comment" > % Two vectors that are perpendicular to the circle normal< / span >
center = [0; 0; 0]; < span class = "org-comment" > % Center of the circle< / span >
radius = stewart.platform_F.R; < span class = "org-comment" > % Radius of the circle [m]< / span >
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points = center< span class = "org-type" > *< / span > ones(1, length(theta)) < span class = "org-type" > +< / span > radius< span class = "org-type" > *< / span > (v(< span class = "org-type" > :< / span > ,1)< span class = "org-type" > *< / span > cos(theta) < span class = "org-type" > +< / span > v(< span class = "org-type" > :< / span > ,2)< span class = "org-type" > *< / span > sin(theta));
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plot3(points(1,< span class = "org-type" > :< / span > ), ...
points< span class = "org-type" > (2,:), ...< / span >
points(3,< span class = "org-type" > :< / span > ), < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.F_color);
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Let’ s now plot the position and labels of the Fixed Joints
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.joints
scatter3(Fa(1,< span class = "org-type" > :< / span > ), ...
Fa< span class = "org-type" > (2,:), ...< / span >
Fa(3,< span class = "org-type" > :< / span > ), < span class = "org-string" > 'MarkerEdgeColor'< / span > , args.F_color);
< span class = "org-keyword" > if< / span > args.labels
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:size(Fa,2)< / span >
< span class = "org-type" > text< / span > (Fa(1,< span class = "org-constant" > i< / span > ) < span class = "org-type" > +< / span > d_label, ...
Fa(2,< span class = "org-constant" > i< / span > ), ...
Fa(3,< span class = "org-constant" > i< / span > ), sprintf(< span class = "org-string" > '$a_{%i}$'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Color'< / span > , args.F_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc272598" class = "outline-3" >
< h3 id = "orgc272598" > Mobile Platform elements< / h3 >
< div class = "outline-text-3" id = "text-orgc272598" >
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< p >
Plot the frame \(\{M\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Fm = FTm< span class = "org-type" > *< / span > [0; 0; 0; 1]; < span class = "org-comment" > % Get the position of frame {M} w.r.t. {F}< / span >
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< span class = "org-keyword" > if< / span > args.frames
FM_uv = FTm< span class = "org-type" > *< / span > [d_unit_vector< span class = "org-type" > *< / span > eye(3); zeros(1,3)]; < span class = "org-comment" > % Rotated Unit vectors< / span >
quiver3(Fm(1)< span class = "org-type" > *< / span > ones(1,3), Fm(2)< span class = "org-type" > *< / span > ones(1,3), Fm(3)< span class = "org-type" > *< / span > ones(1,3), ...
FM_uv(1,1< span class = "org-type" > :< / span > 3), FM_uv(2,1< span class = "org-type" > :< / span > 3), FM_uv(3,1< span class = "org-type" > :< / span > 3), < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.M_color)
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-type" > text< / span > (Fm(1) < span class = "org-type" > +< / span > d_label, ...
Fm< span class = "org-type" > (2) + d_label, ...< / span >
Fm(3) < span class = "org-type" > +< / span > d_label, < span class = "org-string" > '$\{M\}$'< / span > , < span class = "org-string" > 'Color'< / span > , args.M_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Plot the frame \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > FB = FO_A < span class = "org-type" > +< / span > args.AP;
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< span class = "org-keyword" > if< / span > args.frames
FB_uv = FTm< span class = "org-type" > *< / span > [d_unit_vector< span class = "org-type" > *< / span > eye(3); zeros(1,3)]; < span class = "org-comment" > % Rotated Unit vectors< / span >
quiver3(FB(1)< span class = "org-type" > *< / span > ones(1,3), FB(2)< span class = "org-type" > *< / span > ones(1,3), FB(3)< span class = "org-type" > *< / span > ones(1,3), ...
FB_uv(1,1< span class = "org-type" > :< / span > 3), FB_uv(2,1< span class = "org-type" > :< / span > 3), FB_uv(3,1< span class = "org-type" > :< / span > 3), < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.M_color)
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-type" > text< / span > (FB(1) < span class = "org-type" > -< / span > d_label, ...
FB< span class = "org-type" > (2) + d_label, ...< / span >
FB(3) < span class = "org-type" > +< / span > d_label, < span class = "org-string" > '$\{B\}$'< / span > , < span class = "org-string" > 'Color'< / span > , args.M_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Let’ s then plot the circle corresponding to the shape of the Mobile platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.platforms < span class = "org-type" > & & < / span > stewart.platform_M.type < span class = "org-type" > ==< / span > 1
theta = [0< span class = "org-type" > :< / span > 0.01< span class = "org-type" > :< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > +< / span > 0.01]; < span class = "org-comment" > % Angles [rad]< / span >
v = null((FTm(1< span class = "org-type" > :< / span > 3,1< span class = "org-type" > :< / span > 3)< span class = "org-type" > *< / span > [0;0;1])< span class = "org-type" > '< / span > ); < span class = "org-comment" > % Two vectors that are perpendicular to the circle normal< / span >
center = Fm(1< span class = "org-type" > :< / span > 3); < span class = "org-comment" > % Center of the circle< / span >
radius = stewart.platform_M.R; < span class = "org-comment" > % Radius of the circle [m]< / span >
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points = center< span class = "org-type" > *< / span > ones(1, length(theta)) < span class = "org-type" > +< / span > radius< span class = "org-type" > *< / span > (v(< span class = "org-type" > :< / span > ,1)< span class = "org-type" > *< / span > cos(theta) < span class = "org-type" > +< / span > v(< span class = "org-type" > :< / span > ,2)< span class = "org-type" > *< / span > sin(theta));
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plot3(points(1,< span class = "org-type" > :< / span > ), ...
points< span class = "org-type" > (2,:), ...< / span >
points(3,< span class = "org-type" > :< / span > ), < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.M_color);
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Plot the position and labels of the rotation joints fixed to the mobile platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.joints
Fb = FTm< span class = "org-type" > *< / span > [Mb;ones(1,6)];
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scatter3(Fb(1,< span class = "org-type" > :< / span > ), ...
Fb< span class = "org-type" > (2,:), ...< / span >
Fb(3,< span class = "org-type" > :< / span > ), < span class = "org-string" > 'MarkerEdgeColor'< / span > , args.M_color);
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:size(Fb,2)< / span >
< span class = "org-type" > text< / span > (Fb(1,< span class = "org-constant" > i< / span > ) < span class = "org-type" > +< / span > d_label, ...
Fb(2,< span class = "org-constant" > i< / span > ), ...
Fb(3,< span class = "org-constant" > i< / span > ), sprintf(< span class = "org-string" > '$b_{%i}$'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Color'< / span > , args.M_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfeaf65a" class = "outline-3" >
< h3 id = "orgfeaf65a" > Legs< / h3 >
< div class = "outline-text-3" id = "text-orgfeaf65a" >
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< p >
Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.legs
< 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 >
plot3([Fa(1,< span class = "org-constant" > i< / span > ), Fb(1,< span class = "org-constant" > i< / span > )], ...
[Fa(2,< span class = "org-constant" > i< / span > ), Fb(2,< span class = "org-constant" > i< / span > )], ...
[Fa(3,< span class = "org-constant" > i< / span > ), Fb(3,< span class = "org-constant" > i< / span > )], < span class = "org-string" > '-'< / span > , < span class = "org-string" > 'Color'< / span > , args.L_color);
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< span class = "org-keyword" > if< / span > args.labels
< span class = "org-type" > text< / span > ((Fa(1,< span class = "org-constant" > i< / span > )< span class = "org-type" > +< / span > Fb(1,< span class = "org-constant" > i< / span > ))< span class = "org-type" > /< / span > 2 < span class = "org-type" > +< / span > d_label, ...
(Fa(2,< span class = "org-constant" > i< / span > )< span class = "org-type" > +< / span > Fb(2,< span class = "org-constant" > i< / span > ))< span class = "org-type" > /< / span > 2, ...
(Fa(3,< span class = "org-constant" > i< / span > )< span class = "org-type" > +< / span > Fb(3,< span class = "org-constant" > i< / span > ))< span class = "org-type" > /< / span > 2, sprintf(< span class = "org-string" > '$%i$'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Color'< / span > , args.L_color);
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8ae9c48" class = "outline-3" >
< h3 id = "org8ae9c48" > < span class = "section-number-3" > 11.1< / span > Figure parameters< / h3 >
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< div class = "outline-text-3" id = "text-11-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > switch< / span > < span class = "org-constant" > args.views< / span >
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'default'< / span >
view([1 < span class = "org-type" > -< / span > 0.6 0.4]);
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'xy'< / span >
view([0 0 1]);
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'xz'< / span >
view([0 < span class = "org-type" > -< / span > 1 0]);
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'yz'< / span >
view([1 0 0]);
< span class = "org-keyword" > end< / span >
< span class = "org-type" > axis< / span > equal;
< span class = "org-type" > axis< / span > off;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb92caec" class = "outline-3" >
< h3 id = "orgb92caec" > < span class = "section-number-3" > 11.2< / span > Subplots< / h3 >
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< div class = "outline-text-3" id = "text-11-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > strcmp(args.views, < span class = "org-string" > 'all'< / span > )
hAx = findobj(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'axes'< / span > );
< span class = "org-type" > figure< / span > ;
s1 = subplot(2,2,1);
copyobj(< span class = "org-type" > get< / span > (hAx(< span class = "org-variable-name" > 1< / span > ), < span class = "org-string" > 'Children'< / span > ), s1);
view([0 0 1]);
< span class = "org-type" > axis< / span > equal;
< span class = "org-type" > axis< / span > off;
title(< span class = "org-string" > 'Top'< / span > )
s2 = subplot(2,2,2);
copyobj(< span class = "org-type" > get< / span > (hAx(< span class = "org-variable-name" > 1< / span > ), < span class = "org-string" > 'Children'< / span > ), s2);
view([1 < span class = "org-type" > -< / span > 0.6 0.4]);
< span class = "org-type" > axis< / span > equal;
< span class = "org-type" > axis< / span > off;
s3 = subplot(2,2,3);
copyobj(< span class = "org-type" > get< / span > (hAx(< span class = "org-variable-name" > 1< / span > ), < span class = "org-string" > 'Children'< / span > ), s3);
view([1 0 0]);
< span class = "org-type" > axis< / span > equal;
< span class = "org-type" > axis< / span > off;
title(< span class = "org-string" > 'Front'< / span > )
s4 = subplot(2,2,4);
copyobj(< span class = "org-type" > get< / span > (hAx(< span class = "org-variable-name" > 1< / span > ), < span class = "org-string" > 'Children'< / span > ), s4);
view([0 < span class = "org-type" > -< / span > 1 0]);
< span class = "org-type" > axis< / span > equal;
< span class = "org-type" > axis< / span > off;
title(< span class = "org-string" > 'Side'< / span > )
close(f);
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org334a337" class = "outline-2" >
< h2 id = "org334a337" > < span class = "section-number-2" > 12< / span > < code > describeStewartPlatform< / code > : Display some text describing the current defined Stewart Platform< / h2 >
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< div class = "outline-text-2" id = "text-12" >
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< p >
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< a id = "org84ac7ad" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/describeStewartPlatform.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-orgd7efc3c" class = "outline-3" >
< h3 id = "orgd7efc3c" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgd7efc3c" >
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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" > []< / span > = < span class = "org-function-name" > describeStewartPlatform< / span > (< span class = "org-variable-name" > stewart< / span > )
< span class = "org-comment" > % describeStewartPlatform - Display some text describing the current defined Stewart Platform< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [] = describeStewartPlatform(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org193f623" class = "outline-3" >
< h3 id = "org193f623" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org193f623" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org316408d" class = "outline-3" >
< h3 id = "org316408d" > < span class = "section-number-3" > 12.1< / span > Geometry< / h3 >
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< div class = "outline-text-3" id = "text-12-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > 'GEOMETRY:\n'< / span > )
fprintf(< span class = "org-string" > '- The height between the fixed based and the top platform is %.3g [mm].\n'< / span > , 1e3< span class = "org-type" > *< / span > stewart.geometry.H)
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< span class = "org-keyword" > if< / span > stewart.platform_M.MO_B(3) < span class = "org-type" > > < / span > 0
fprintf(< span class = "org-string" > '- Frame {A} is located %.3g [mm] above the top platform.\n'< / span > , 1e3< span class = "org-type" > *< / span > stewart.platform_M.MO_B(3))
< span class = "org-keyword" > else< / span >
fprintf(< span class = "org-string" > '- Frame {A} is located %.3g [mm] below the top platform.\n'< / span > , < span class = "org-type" > -< / span > 1e3< span class = "org-type" > *< / span > stewart.platform_M.MO_B(3))
< span class = "org-keyword" > end< / span >
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fprintf(< span class = "org-string" > '- The initial length of the struts are:\n'< / span > )
fprintf(< span class = "org-string" > '\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n'< / span > , 1e3< span class = "org-type" > *< / span > stewart.geometry.l)
fprintf(< span class = "org-string" > '\n'< / span > )
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgdb5d86c" class = "outline-3" >
< h3 id = "orgdb5d86c" > < span class = "section-number-3" > 12.2< / span > Actuators< / h3 >
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< div class = "outline-text-3" id = "text-12-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > 'ACTUATORS:\n'< / span > )
< span class = "org-keyword" > if< / span > stewart.actuators.type < span class = "org-type" > ==< / span > 1
fprintf(< span class = "org-string" > '- The actuators are classical.\n'< / span > )
fprintf(< span class = "org-string" > '- The Stiffness and Damping of each actuators is:\n'< / span > )
fprintf(< span class = "org-string" > '\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n'< / span > , stewart.actuators.K(1), stewart.actuators.C(1))
< span class = "org-keyword" > elseif< / span > stewart.actuators.type < span class = "org-type" > ==< / span > 2
fprintf(< span class = "org-string" > '- The actuators are mechanicaly amplified.\n'< / span > )
fprintf(< span class = "org-string" > '- The vertical stiffness and damping contribution of the piezoelectric stack is:\n'< / span > )
fprintf(< span class = "org-string" > '\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n'< / span > , stewart.actuators.Ka(1), stewart.actuators.Ca(1))
fprintf(< span class = "org-string" > '- Vertical stiffness when the piezoelectric stack is removed is:\n'< / span > )
fprintf(< span class = "org-string" > '\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n'< / span > , stewart.actuators.Kr(1), stewart.actuators.Cr(1))
< span class = "org-keyword" > end< / span >
fprintf(< span class = "org-string" > '\n'< / span > )
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org48d5b2e" class = "outline-3" >
< h3 id = "org48d5b2e" > < span class = "section-number-3" > 12.3< / span > Joints< / h3 >
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< div class = "outline-text-3" id = "text-12-3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > 'JOINTS:\n'< / span > )
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< / pre >
< / div >
< p >
Type of the joints on the fixed base.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > switch< / span > < span class = "org-constant" > stewart.joints_F.type< / span >
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 1< / span >
fprintf(< span class = "org-string" > '- The joints on the fixed based are universal joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 2< / span >
fprintf(< span class = "org-string" > '- The joints on the fixed based are spherical joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 3< / span >
fprintf(< span class = "org-string" > '- The joints on the fixed based are perfect universal joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 4< / span >
fprintf(< span class = "org-string" > '- The joints on the fixed based are perfect spherical joints\n'< / span > )
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Type of the joints on the mobile platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > switch< / span > < span class = "org-constant" > stewart.joints_M.type< / span >
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 1< / span >
fprintf(< span class = "org-string" > '- The joints on the mobile based are universal joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 2< / span >
fprintf(< span class = "org-string" > '- The joints on the mobile based are spherical joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 3< / span >
fprintf(< span class = "org-string" > '- The joints on the mobile based are perfect universal joints\n'< / span > )
< span class = "org-keyword" > case< / span > < span class = "org-constant" > 4< / span >
fprintf(< span class = "org-string" > '- The joints on the mobile based are perfect spherical joints\n'< / span > )
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Position of the fixed joints
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > '- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n'< / span > )
fprintf(< span class = "org-string" > '\t % .3g \t % .3g \t % .3g\n'< / span > , 1e3< span class = "org-type" > *< / span > stewart.platform_F.Fa)
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< / pre >
< / div >
< p >
Position of the mobile joints
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > '- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n'< / span > )
fprintf(< span class = "org-string" > '\t % .3g \t % .3g \t % .3g\n'< / span > , 1e3< span class = "org-type" > *< / span > stewart.platform_M.Mb)
fprintf(< span class = "org-string" > '\n'< / span > )
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org62135e5" class = "outline-3" >
< h3 id = "org62135e5" > < span class = "section-number-3" > 12.4< / span > Kinematics< / h3 >
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< div class = "outline-text-3" id = "text-12-4" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > fprintf(< span class = "org-string" > 'KINEMATICS:\n'< / span > )
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< span class = "org-keyword" > if< / span > isfield(stewart.kinematics, < span class = "org-string" > 'K'< / span > )
fprintf(< span class = "org-string" > '- The Stiffness matrix K is (in [N/m]):\n'< / span > )
fprintf(< span class = "org-string" > '\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n'< / span > , stewart.kinematics.K)
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > if< / span > isfield(stewart.kinematics, < span class = "org-string" > 'C'< / span > )
fprintf(< span class = "org-string" > '- The Damping matrix C is (in [m/N]):\n'< / span > )
fprintf(< span class = "org-string" > '\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n'< / span > , stewart.kinematics.C)
< span class = "org-keyword" > end< / span >
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< / pre >
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< div id = "outline-container-org9decc94" class = "outline-2" >
< h2 id = "org9decc94" > < span class = "section-number-2" > 13< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h2 >
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< div class = "outline-text-2" id = "text-13" >
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< p >
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< a id = "orgeb6daaf" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/generateCubicConfiguration.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org3485ee4" class = "outline-3" >
< h3 id = "org3485ee4" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org3485ee4" >
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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" > generateCubicConfiguration< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateCubicConfiguration - Generate a Cubic Configuration< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = generateCubicConfiguration(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" > % - geometry.H [1x1] - Total height of the platform [m]< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - Hc [1x1] - Height of the "useful" part of the cube [m]< / span >
< span class = "org-comment" > % - FOc [1x1] - Height of the center of the cube with respect to {F} [m]< / span >
< 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" > % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< / pre >
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< div id = "outline-container-orgedfb402" class = "outline-3" >
< h3 id = "orgedfb402" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-orgedfb402" >
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< div id = "orga5dae44" class = "figure" >
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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Cubic Configuration< / p >
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< div id = "outline-container-org0a637a2" class = "outline-3" >
< h3 id = "org0a637a2" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org0a637a2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .FOc (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-keyword" > end< / span >
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< / pre >
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< / div >
< / div >
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< div id = "outline-container-org13c71d3" class = "outline-3" >
< h3 id = "org13c71d3" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org13c71d3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > assert(isfield(stewart.geometry, < span class = "org-string" > 'H'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute H'< / span > )
H = stewart.geometry.H;
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< div id = "outline-container-org44e0590" class = "outline-3" >
< h3 id = "org44e0590" > Position of the Cube< / h3 >
< div class = "outline-text-3" id = "text-org44e0590" >
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< p >
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}.
< / p >
< div class = "org-src-container" >
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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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< div id = "outline-container-org92b12fa" class = "outline-3" >
< h3 id = "org92b12fa" > Compute the pose< / h3 >
< div class = "outline-text-3" id = "text-org92b12fa" >
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< p >
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
< / p >
< div class = "org-src-container" >
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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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< / pre >
< / div >
< p >
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > 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;
Mb = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc< span class = "org-type" > -< / span > H] < span class = "org-type" > +< / span > ((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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< h3 id = "orgc91d6aa" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orgc91d6aa" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.platform_F.Fa = Fa;
stewart.platform_M.Mb = Mb;
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< h2 id = "org10a231c" > < span class = "section-number-2" > 14< / span > < code > computeJacobian< / code > : Compute the Jacobian Matrix< / h2 >
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< div class = "outline-text-2" id = "text-14" >
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< p >
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< a id = "orgea11d49" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/computeJacobian.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org7841d59" class = "outline-3" >
< h3 id = "org7841d59" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org7841d59" >
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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" > computeJacobian< / span > (< span class = "org-variable-name" > stewart< / span > )
< span class = "org-comment" > % computeJacobian -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = computeJacobian(stewart)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart - With at least the following fields:< / span >
< span class = "org-comment" > % - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}< / span >
< span class = "org-comment" > % - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}< / span >
< span class = "org-comment" > % - actuators.K [6x1] - Total stiffness of the actuators< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - With the 3 added field:< / span >
< span class = "org-comment" > % - kinematics.J [6x6] - The Jacobian Matrix< / span >
< span class = "org-comment" > % - kinematics.K [6x6] - The Stiffness Matrix< / span >
< span class = "org-comment" > % - kinematics.C [6x6] - The Compliance Matrix< / span >
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< div id = "outline-container-orgf2616df" class = "outline-3" >
< h3 id = "orgf2616df" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-orgf2616df" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > assert(isfield(stewart.geometry, < span class = "org-string" > 'As'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute As'< / span > )
As = stewart.geometry.As;
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assert(isfield(stewart.geometry, < span class = "org-string" > 'Ab'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute Ab'< / span > )
Ab = stewart.geometry.Ab;
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assert(isfield(stewart.actuators, < span class = "org-string" > 'K'< / span > ), < span class = "org-string" > 'stewart.actuators should have attribute K'< / span > )
Ki = stewart.actuators.K;
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< h3 id = "org35ea7eb" > Compute Jacobian Matrix< / h3 >
< div class = "outline-text-3" id = "text-org35ea7eb" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > J = [As< span class = "org-type" > '< / span > , cross(Ab, As)< span class = "org-type" > '< / span > ];
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< h3 id = "orgf645c75" > Compute Stiffness Matrix< / h3 >
< div class = "outline-text-3" id = "text-orgf645c75" >
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< pre class = "src src-matlab" > K = J< span class = "org-type" > '*< / span > diag(Ki)< span class = "org-type" > *< / span > J;
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< h3 id = "org21fadda" > Compute Compliance Matrix< / h3 >
< div class = "outline-text-3" id = "text-org21fadda" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > C = inv(K);
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< h3 id = "orgac75de7" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orgac75de7" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.kinematics.J = J;
stewart.kinematics.K = K;
stewart.kinematics.C = C;
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< h2 id = "orgcc980cd" > < span class = "section-number-2" > 15< / span > < code > inverseKinematics< / code > : Compute Inverse Kinematics< / h2 >
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< div class = "outline-text-2" id = "text-15" >
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< p >
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< a id = "org7c6158b" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "../src/inverseKinematics.m" > here< / a > .
< / p >
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< h3 id = "org931c849" > Theory< / h3 >
< div class = "outline-text-3" id = "text-org931c849" >
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< 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 >
< p >
From the geometry of the manipulator, the loop closure for each limb, \(i = 1, 2, \dots, 6\) can be written as
< / p >
\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.
< / p >
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< h3 id = "org5e787de" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org5e787de" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [Li, dLi]< / span > = < span class = "org-function-name" > inverseKinematics< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = inverseKinematics(stewart)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following fields< / span >
< span class = "org-comment" > % - geometry.Aa [3x6] - The positions ai expressed in {A}< / span >
< span class = "org-comment" > % - geometry.Bb [3x6] - The positions bi expressed in {B}< / span >
< span class = "org-comment" > % - geometry.l [6x1] - Length of each strut< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - AP [3x1] - The wanted position of {B} with respect to {A}< / span >
< span class = "org-comment" > % - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}< / span >
< span class = "org-comment" > % - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}< / span >
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< h3 id = "org2a4efbd" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .AP (3,1) double {mustBeNumeric} = zeros(3,1)
< span class = "org-variable-name" > args< / span > .ARB (3,3) double {mustBeNumeric} = eye(3)
< span class = "org-keyword" > end< / span >
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< h3 id = "org318e667" > Check the < code > stewart< / code > structure elements< / h3 >
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< pre class = "src src-matlab" > assert(isfield(stewart.geometry, < span class = "org-string" > 'Aa'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute Aa'< / span > )
Aa = stewart.geometry.Aa;
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assert(isfield(stewart.geometry, < span class = "org-string" > 'Bb'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute Bb'< / span > )
Bb = stewart.geometry.Bb;
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assert(isfield(stewart.geometry, < span class = "org-string" > 'l'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute l'< / span > )
l = stewart.geometry.l;
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< h3 id = "orgeced891" > 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(Bb< span class = "org-type" > '*< / span > Bb) < span class = "org-type" > +< / span > diag(Aa< span class = "org-type" > '*< / span > Aa) < span class = "org-type" > -< / span > (2< span class = "org-type" > *< / span > args.AP< span class = "org-type" > '*< / span > Aa)< span class = "org-type" > '< / span > < span class = "org-type" > +< / span > (2< span class = "org-type" > *< / span > args.AP< span class = "org-type" > '*< / span > (args.ARB< span class = "org-type" > *< / span > Bb))< span class = "org-type" > '< / span > < span class = "org-type" > -< / span > diag(2< span class = "org-type" > *< / span > (args.ARB< span class = "org-type" > *< / span > Bb)< span class = "org-type" > '*< / span > Aa));
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< pre class = "src src-matlab" > dLi = Li< span class = "org-type" > -< / span > l;
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< h2 id = "org07fbe58" > < span class = "section-number-2" > 16< / span > < code > forwardKinematicsApprox< / code > : Compute the Approximate Forward Kinematics< / h2 >
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< p >
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< a id = "orgbaa3db0" > < / a >
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< p >
This Matlab function is accessible < a href = "../src/forwardKinematicsApprox.m" > here< / a > .
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< h3 id = "org5c8b01d" > 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" > % - kinematics.J [6x6] - The Jacobian Matrix< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - dL [6x1] - Displacement of each strut [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - P [3x1] - The estimated position of {B} with respect to {A}< / span >
< span class = "org-comment" > % - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}< / span >
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< h3 id = "orgd653374" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .dL (6,1) double {mustBeNumeric} = zeros(6,1)
< span class = "org-keyword" > end< / span >
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< h3 id = "org1699c2d" > Check the < code > stewart< / code > structure elements< / h3 >
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< pre class = "src src-matlab" > assert(isfield(stewart.kinematics, < span class = "org-string" > 'J'< / span > ), < span class = "org-string" > 'stewart.kinematics should have attribute J'< / span > )
J = stewart.kinematics.J;
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< h3 id = "orgf7dc851" > 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 = 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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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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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-02-20 sam. 23:09< / p >
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