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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 = "#orgcaca5e0" > 1. < code > initializeStewartPlatform< / code > : Initialize the Stewart Platform structure< / a >
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< ul >
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< li > < a href = "#org8dc2620" > Documentation< / a > < / li >
< li > < a href = "#orgb2b186c" > Function description< / a > < / li >
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< li > < a href = "#org3622825" > Initialize the Stewart structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgac25f89" > 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 = "#org736bb40" > Documentation< / a > < / li >
< li > < a href = "#orgb82d77f" > Function description< / a > < / li >
< li > < a href = "#org8e4bfab" > Optional Parameters< / a > < / li >
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< li > < a href = "#org7d50d54" > Compute the position of each frame< / a > < / li >
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< li > < a href = "#orgecc27b3" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#orgccb31c6" > 3. < code > generateGeneralConfiguration< / code > : Generate a Very General Configuration< / a >
< ul >
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< li > < a href = "#orgd270a86" > Documentation< / a > < / li >
< li > < a href = "#org22aadcf" > Function description< / a > < / li >
< li > < a href = "#orga6492a4" > Optional Parameters< / a > < / li >
< li > < a href = "#org3231a85" > Compute the pose< / a > < / li >
< li > < a href = "#org13d89d2" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org9944c04" > 4. < code > computeJointsPose< / code > : Compute the Pose of the Joints< / a >
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< ul >
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< li > < a href = "#orgf8b573b" > Documentation< / a > < / li >
< li > < a href = "#org85d30c9" > Function description< / a > < / li >
< li > < a href = "#org87cdb4a" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#orge87b302" > Compute the position of the Joints< / a > < / li >
< li > < a href = "#org3a7e3c5" > Compute the strut length and orientation< / a > < / li >
< li > < a href = "#org9e1258f" > Compute the orientation of the Joints< / a > < / li >
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< li > < a href = "#org47a4205" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org1315282" > 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 = "#org9a7c718" > Function description< / a > < / li >
< li > < a href = "#orga7f6cb4" > Optional Parameters< / a > < / li >
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< li > < a href = "#orgbb9abb5" > Use the Inverse Kinematic function< / a > < / li >
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< li > < a href = "#orga35dd52" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org4674203" > 6. < code > initializeCylindricalPlatforms< / code > : Initialize the geometry of the Fixed and Mobile Platforms< / a >
< ul >
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< li > < a href = "#orge53472d" > Function description< / a > < / li >
< li > < a href = "#orgc31cfe7" > Optional Parameters< / a > < / li >
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< li > < a href = "#orgf654de0" > Compute the Inertia matrices of platforms< / a > < / li >
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< li > < a href = "#orga779e2f" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb0a1d7b" > 7. < code > initializeCylindricalStruts< / code > : Define the inertia of cylindrical struts< / a >
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< ul >
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< li > < a href = "#orgf6fcb94" > Function description< / a > < / li >
< li > < a href = "#org8990e71" > Optional Parameters< / a > < / li >
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< li > < a href = "#orgd943059" > Compute the properties of the cylindrical struts< / a > < / li >
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< li > < a href = "#org53395cc" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgae8d0dc" > 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 = "#org15ad3b5" > Documentation< / a > < / li >
< li > < a href = "#org5318aec" > Function description< / a > < / li >
< li > < a href = "#org153d169" > Optional Parameters< / a > < / li >
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< li > < a href = "#org3c2e550" > Add Stiffness and Damping properties of each strut< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org682a09c" > 9. < code > initializeAmplifiedStrutDynamics< / code > : Add Stiffness and Damping properties of each strut for an amplified piezoelectric actuator< / a >
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< ul >
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< li > < a href = "#orgc4169f8" > Documentation< / a > < / li >
< li > < a href = "#org44de918" > Function description< / a > < / li >
< li > < a href = "#org3966929" > Optional Parameters< / a > < / li >
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< li > < a href = "#org2e42182" > Compute the total stiffness and damping< / a > < / li >
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< li > < a href = "#orgfed23b2" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgbc5232e" > 10. < code > initializeJointDynamics< / code > : Add Stiffness and Damping properties for spherical joints< / a >
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< ul >
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< li > < a href = "#orgcc4b26d" > Function description< / a > < / li >
< li > < a href = "#org047804e" > Optional Parameters< / a > < / li >
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< li > < a href = "#orgd5b8278" > Add Actuator Type< / a > < / li >
< li > < a href = "#org51cf135" > Add Stiffness and Damping in Translation of each strut< / a > < / li >
< li > < a href = "#org1e8eceb" > Add Stiffness and Damping in Rotation of each strut< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org3a7f26e" > 11. < code > initializeInertialSensor< / code > : Initialize the inertial sensor in each strut< / a >
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< ul >
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< li > < a href = "#orgcfc37af" > Geophone - Working Principle< / a > < / li >
< li > < a href = "#org986e38f" > Accelerometer - Working Principle< / a > < / li >
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< li > < a href = "#orgcb6bebb" > Function description< / a > < / li >
< li > < a href = "#org5ec99a5" > Optional Parameters< / a > < / li >
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< li > < a href = "#org1c3d7c8" > Compute the properties of the sensor< / a > < / li >
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< li > < a href = "#org5e33aa2" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd6baa46" > 12. < code > displayArchitecture< / code > : 3D plot of the Stewart platform architecture< / a >
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< ul >
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< li > < a href = "#orgc70c9a5" > Function description< / a > < / li >
< li > < a href = "#org63006a7" > Optional Parameters< / a > < / li >
< li > < a href = "#org86735ca" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#orgb11fd92" > Figure Creation, Frames and Homogeneous transformations< / a > < / li >
< li > < a href = "#org7cd8fee" > Fixed Base elements< / a > < / li >
< li > < a href = "#orgacb8eb7" > Mobile Platform elements< / a > < / li >
< li > < a href = "#org7f9320b" > Legs< / a > < / li >
< li > < a href = "#org925a393" > 12.1. Figure parameters< / a > < / li >
< li > < a href = "#org44e536d" > 12.2. Subplots< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgecfd55f" > 13. < code > describeStewartPlatform< / code > : Display some text describing the current defined Stewart Platform< / a >
< ul >
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< li > < a href = "#orgb99230a" > Function description< / a > < / li >
< li > < a href = "#orged88325" > Optional Parameters< / a > < / li >
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< li > < a href = "#org1d49caa" > 13.1. Geometry< / a > < / li >
< li > < a href = "#orgcb66771" > 13.2. Actuators< / a > < / li >
< li > < a href = "#org4630b77" > 13.3. Joints< / a > < / li >
< li > < a href = "#org47a9cf0" > 13.4. Kinematics< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org65fc289" > 14. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
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< ul >
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< li > < a href = "#orgc164e8d" > Function description< / a > < / li >
< li > < a href = "#org0d467b7" > Documentation< / a > < / li >
< li > < a href = "#orgda76f80" > Optional Parameters< / a > < / li >
< li > < a href = "#org4a5a3cf" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#orge94a885" > Position of the Cube< / a > < / li >
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< li > < a href = "#orge8af7a8" > Compute the pose< / a > < / li >
< li > < a href = "#org32b44aa" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org9e8cbfa" > 15. < code > computeJacobian< / code > : Compute the Jacobian Matrix< / a >
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< ul >
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< li > < a href = "#org2c08fbd" > Function description< / a > < / li >
< li > < a href = "#org8071a1b" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#org9bcd9b9" > Compute Jacobian Matrix< / a > < / li >
< li > < a href = "#orgf08eda6" > Compute Stiffness Matrix< / a > < / li >
< li > < a href = "#orgd164132" > Compute Compliance Matrix< / a > < / li >
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< li > < a href = "#orgbf70f7a" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org03168fc" > 16. < code > inverseKinematics< / code > : Compute Inverse Kinematics< / a >
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< ul >
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< li > < a href = "#orgbdc5fb1" > Theory< / a > < / li >
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< li > < a href = "#org18c9841" > Function description< / a > < / li >
< li > < a href = "#org848fb1f" > Optional Parameters< / a > < / li >
< li > < a href = "#org8be974b" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#org8b70a76" > Compute< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org278d55b" > 17. < code > forwardKinematicsApprox< / code > : Compute the Approximate Forward Kinematics< / a >
< ul >
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< li > < a href = "#org07e11bf" > Function description< / a > < / li >
< li > < a href = "#org918179d" > Optional Parameters< / a > < / li >
< li > < a href = "#orgf157791" > Check the < code > stewart< / code > structure elements< / a > < / li >
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< li > < a href = "#orgf17cab9" > 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-orgcaca5e0" class = "outline-2" >
< h2 id = "orgcaca5e0" > < 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 = "orgcac119a" > < / 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-org8dc2620" class = "outline-3" >
< h3 id = "org8dc2620" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-org8dc2620" >
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< div id = "org321fc67" 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-orgb2b186c" class = "outline-3" >
< h3 id = "orgb2b186c" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgb2b186c" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeStewartPlatform()
% initializeStewartPlatform - Initialize the stewart structure
%
% Syntax: [stewart] = initializeStewartPlatform(args)
%
% Outputs:
% - stewart - A structure with the following sub-structures:
% - platform_F -
% - platform_M -
% - joints_F -
% - joints_M -
% - struts_F -
% - struts_M -
% - actuators -
% - geometry -
% - properties -
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< / pre >
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< div id = "outline-container-org3622825" class = "outline-3" >
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< h3 id = "org3622825" > Initialize the Stewart structure< / h3 >
< div class = "outline-text-3" id = "text-org3622825" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = struct();
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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 >
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< h2 id = "orgac25f89" > < 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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< p >
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< a id = "orga56a5c6" > < / 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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< h3 id = "org736bb40" > Documentation< / h3 >
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< p > < img src = "figs/stewart-frames-position.png" alt = "stewart-frames-position.png" / >
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< p > < span class = "figure-number" > Figure 2: < / span > Definition of the position of the frames< / p >
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< pre class = "src src-matlab" > function [stewart] = initializeFramesPositions(stewart, args)
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
%
% Syntax: [stewart] = initializeFramesPositions(stewart, args)
%
% Inputs:
% - args - Can have the following fields:
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
% - stewart - A structure with the following fields:
% - geometry.H [1x1] - Total Height of the Stewart Platform [m]
% - geometry.FO_M [3x1] - Position of {M} with respect to {F} [m]
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]
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< h3 id = "org8e4bfab" > Optional Parameters< / h3 >
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< pre class = "src src-matlab" > arguments
stewart
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args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
end
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< h3 id = "org7d50d54" > Compute the position of each frame< / h3 >
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< pre class = "src src-matlab" > H = args.H; % Total Height of the Stewart Platform [m]
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FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]
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MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]
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FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
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< h3 id = "orgecc27b3" > 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 = "orgccb31c6" > < span class = "section-number-2" > 3< / span > < code > generateGeneralConfiguration< / code > : Generate a Very General Configuration< / h2 >
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< a id = "org32105b0" > < / a >
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< p >
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This Matlab function is accessible < a href = "../src/generateGeneralConfiguration.m" > here< / a > .
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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}.
The radius of the circles can be chosen as well as the angles where the joints are located (see Figure < a href = "#org449c886" > 3< / a > ).
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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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< pre class = "src src-matlab" > function [stewart] = generateGeneralConfiguration(stewart, args)
% generateGeneralConfiguration - Generate a Very General Configuration
%
% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
%
% Inputs:
% - args - Can have the following fields:
% - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
% - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]
% - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
% - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
% - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]
% - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
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< pre class = "src src-matlab" > arguments
stewart
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args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3;
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
end
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< h3 id = "org3231a85" > 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" > for i = 1:6
Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i)); args.FH];
Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
end
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< h3 id = "org13d89d2" > 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 = "org9944c04" > < 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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This Matlab function is accessible < a href = "../src/computeJointsPose.m" > here< / a > .
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< h3 id = "orgf8b573b" > Documentation< / h3 >
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< p > < img src = "figs/stewart-struts.png" alt = "stewart-struts.png" / >
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< p > < span class = "figure-number" > Figure 4: < / span > Position and orientation of the struts< / p >
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< pre class = "src src-matlab" > function [stewart] = computeJointsPose(stewart)
% computeJointsPose -
%
% Syntax: [stewart] = computeJointsPose(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
% - geometry.FO_M [3x1] - Position of {M} with respect to {F}
%
% Outputs:
% - stewart - A structure with the following added fields
% - geometry.Aa [3x6] - The i'th column is the position of ai with respect to {A}
% - geometry.Ab [3x6] - The i'th column is the position of bi with respect to {A}
% - geometry.Ba [3x6] - The i'th column is the position of ai with respect to {B}
% - geometry.Bb [3x6] - The i'th column is the position of bi with respect to {B}
% - geometry.l [6x1] - The i'th element is the initial length of strut i
% - geometry.As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
% - geometry.Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
% - struts_F.l [6x1] - Length of the Fixed part of the i'th strut
% - struts_M.l [6x1] - Length of the Mobile part of the i'th strut
% - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
% - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
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< h3 id = "org87cdb4a" > Check the < code > stewart< / code > structure elements< / h3 >
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< pre class = "src src-matlab" > assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
Fa = stewart.platform_F.Fa;
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assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
Mb = stewart.platform_M.Mb;
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assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
FO_A = stewart.platform_F.FO_A;
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
MO_B = stewart.platform_M.MO_B;
assert(isfield(stewart.geometry, 'FO_M'), 'stewart.geometry should have attribute FO_M')
FO_M = stewart.geometry.FO_M;
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< pre class = "src src-matlab" > Aa = Fa - repmat(FO_A, [1, 6]);
Bb = Mb - repmat(MO_B, [1, 6]);
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Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);
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< pre class = "src src-matlab" > As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As
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l = vecnorm(Ab - Aa)';
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< pre class = "src src-matlab" > Bs = (Bb - Ba)./vecnorm(Bb - Ba);
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< pre class = "src src-matlab" > FRa = zeros(3,3,6);
MRb = zeros(3,3,6);
for i = 1:6
FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));
MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
end
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< h3 id = "org47a4205" > Populate the < code > stewart< / code > structure< / h3 >
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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;
stewart.struts_F.l = l/2;
stewart.struts_M.l = l/2;
stewart.platform_F.FRa = FRa;
stewart.platform_M.MRb = MRb;
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< h2 id = "org1315282" > < 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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< a id = "org05598b5" > < / a >
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< p >
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This Matlab function is accessible < a href = "../src/initializeStewartPose.m" > here< / a > .
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< pre class = "src src-matlab" > function [stewart] = initializeStewartPose(stewart, args)
% initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
% It uses the inverse kinematic
%
% Syntax: [stewart] = initializeStewartPose(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - Aa [3x6] - The positions ai expressed in {A}
% - Bb [3x6] - The positions bi expressed in {B}
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - 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}
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< pre class = "src src-matlab" > arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbb9abb5" class = "outline-3" >
< h3 id = "orgbb9abb5" > Use the Inverse Kinematic function< / h3 >
< div class = "outline-text-3" id = "text-orgbb9abb5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > [Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
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< / pre >
< / div >
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< / div >
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< div id = "outline-container-orga35dd52" class = "outline-3" >
< h3 id = "orga35dd52" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orga35dd52" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.actuators.Leq = dLi;
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< / pre >
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< / div >
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< div id = "outline-container-org4674203" class = "outline-2" >
< h2 id = "org4674203" > < span class = "section-number-2" > 6< / span > < code > initializeCylindricalPlatforms< / code > : Initialize the geometry of the Fixed and Mobile Platforms< / h2 >
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< div class = "outline-text-2" id = "text-6" >
< p >
< a id = "orgac77ed6" > < / a >
< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeCylindricalPlatforms.m" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-orge53472d" class = "outline-3" >
< h3 id = "orge53472d" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orge53472d" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeCylindricalPlatforms(stewart, args)
% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
%
% Syntax: [stewart] = initializeCylindricalPlatforms(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fpm [1x1] - Fixed Platform Mass [kg]
% - Fph [1x1] - Fixed Platform Height [m]
% - Fpr [1x1] - Fixed Platform Radius [m]
% - Mpm [1x1] - Mobile Platform Mass [kg]
% - Mph [1x1] - Mobile Platform Height [m]
% - Mpr [1x1] - Mobile Platform Radius [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platform_F [struct] - structure with the following fields:
% - type = 1
% - M [1x1] - Fixed Platform Mass [kg]
% - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
% - H [1x1] - Fixed Platform Height [m]
% - R [1x1] - Fixed Platform Radius [m]
% - platform_M [struct] - structure with the following fields:
% - M [1x1] - Mobile Platform Mass [kg]
% - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
% - H [1x1] - Mobile Platform Height [m]
% - R [1x1] - Mobile Platform Radius [m]
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< / div >
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< div id = "outline-container-orgc31cfe7" class = "outline-3" >
< h3 id = "orgc31cfe7" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-orgc31cfe7" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
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args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
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args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
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args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
end
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf654de0" class = "outline-3" >
< h3 id = "orgf654de0" > Compute the Inertia matrices of platforms< / h3 >
< div class = "outline-text-3" id = "text-orgf654de0" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
1/2 *args.Fpm * args.Fpr^2]);
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
1/2 *args.Mpm * args.Mpr^2]);
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< / div >
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< div id = "outline-container-orga779e2f" class = "outline-3" >
< h3 id = "orga779e2f" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orga779e2f" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.platform_F.type = 1;
stewart.platform_F.I = I_F;
stewart.platform_F.M = args.Fpm;
stewart.platform_F.R = args.Fpr;
stewart.platform_F.H = args.Fph;
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.platform_M.type = 1;
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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< div id = "outline-container-orgb0a1d7b" class = "outline-2" >
< h2 id = "orgb0a1d7b" > < span class = "section-number-2" > 7< / span > < code > initializeCylindricalStruts< / code > : Define the inertia of cylindrical struts< / h2 >
< div class = "outline-text-2" id = "text-7" >
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< p >
< a id = "orgfd38bf8" > < / a >
< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeCylindricalStruts.m" > here< / a > .
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< / p >
< / div >
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< div id = "outline-container-orgf6fcb94" class = "outline-3" >
< h3 id = "orgf6fcb94" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgf6fcb94" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeCylindricalStruts(stewart, args)
% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
%
% Syntax: [stewart] = initializeCylindricalStruts(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
% - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
% - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
% - Msm [1x1] - Mass of the Mobile part of the struts [kg]
% - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
% - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - struts_F [struct] - structure with the following fields:
% - M [6x1] - Mass of the Fixed part of the struts [kg]
% - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
% - H [6x1] - Height of cylinder for the Fixed part of the struts [m]
% - R [6x1] - Radius of cylinder for the Fixed part of the struts [m]
% - struts_M [struct] - structure with the following fields:
% - M [6x1] - Mass of the Mobile part of the struts [kg]
% - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
% - H [6x1] - Height of cylinder for the Mobile part of the struts [m]
% - R [6x1] - Radius of cylinder for the Mobile part of the struts [m]
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< / div >
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< div id = "outline-container-org8990e71" class = "outline-3" >
< h3 id = "org8990e71" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org8990e71" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
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args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
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args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
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args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
end
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd943059" class = "outline-3" >
< h3 id = "orgd943059" > Compute the properties of the cylindrical struts< / h3 >
< div class = "outline-text-3" id = "text-orgd943059" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Fsm = ones(6,1).*args.Fsm;
Fsh = ones(6,1).*args.Fsh;
Fsr = ones(6,1).*args.Fsr;
Msm = ones(6,1).*args.Msm;
Msh = ones(6,1).*args.Msh;
Msr = ones(6,1).*args.Msr;
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut
I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut
for i = 1:6
I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
1/2 * Fsm(i) * Fsr(i)^2]);
I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
1/2 * Msm(i) * Msr(i)^2]);
end
< / pre >
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< / div >
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< div id = "outline-container-org53395cc" class = "outline-3" >
< h3 id = "org53395cc" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-org53395cc" >
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< div class = "org-src-container" >
< 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;
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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< / div >
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< div id = "outline-container-orgae8d0dc" class = "outline-2" >
< h2 id = "orgae8d0dc" > < span class = "section-number-2" > 8< / span > < code > initializeStrutDynamics< / code > : Add Stiffness and Damping properties of each strut< / h2 >
< div class = "outline-text-2" id = "text-8" >
< p >
< a id = "orgd55e892" > < / a >
< / 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-org15ad3b5" class = "outline-3" >
< h3 id = "org15ad3b5" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-org15ad3b5" >
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< div id = "org99aef3e" class = "figure" >
< 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 = "#orgd4c4025" > 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 = "orgd4c4025" class = "figure" >
< 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-org5318aec" class = "outline-3" >
< h3 id = "org5318aec" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org5318aec" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeStrutDynamics(stewart, args)
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
%
% Syntax: [stewart] = initializeStrutDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - K [6x1] - Stiffness of each strut [N/m]
% - C [6x1] - Damping of each strut [N/(m/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - actuators.type = 1
% - actuators.K [6x1] - Stiffness of each strut [N/m]
% - actuators.C [6x1] - Damping of each strut [N/(m/s)]
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< div id = "outline-container-org153d169" class = "outline-3" >
< h3 id = "org153d169" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org153d169" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
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args.type char {mustBeMember(args.type,{'classical', 'amplified'})} = 'classical'
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args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6*ones(6,1)
args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1*ones(6,1)
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args.k1 (6,1) double {mustBeNumeric} = 1e6
args.ke (6,1) double {mustBeNumeric} = 5e6
args.ka (6,1) double {mustBeNumeric} = 60e6
args.c1 (6,1) double {mustBeNumeric} = 10
args.ce (6,1) double {mustBeNumeric} = 10
args.ca (6,1) double {mustBeNumeric} = 10
args.me (6,1) double {mustBeNumeric} = 0.05
args.ma (6,1) double {mustBeNumeric} = 0.05
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end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org3c2e550" class = "outline-3" >
< h3 id = "org3c2e550" > Add Stiffness and Damping properties of each strut< / h3 >
< div class = "outline-text-3" id = "text-org3c2e550" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > if strcmp(args.type, 'classical')
stewart.actuators.type = 1;
elseif strcmp(args.type, 'amplified')
stewart.actuators.type = 2;
end
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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;
stewart.actuators.ka = args.ka;
stewart.actuators.ca = args.ca;
stewart.actuators.ke = args.ke;
stewart.actuators.ce = args.ce;
stewart.actuators.ma = args.ma;
stewart.actuators.me = args.me;
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< div id = "outline-container-org682a09c" class = "outline-2" >
< h2 id = "org682a09c" > < span class = "section-number-2" > 9< / span > < code > initializeAmplifiedStrutDynamics< / code > : Add Stiffness and Damping properties of each strut for an amplified piezoelectric actuator< / h2 >
< div class = "outline-text-2" id = "text-9" >
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< p >
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< a id = "orga43e091" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/initializeAmplifiedStrutDynamics.m" > here< / a > .
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< / p >
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< / div >
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< div id = "outline-container-orgc4169f8" class = "outline-3" >
< h3 id = "orgc4169f8" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-orgc4169f8" >
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< p >
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An amplified piezoelectric actuator is shown in Figure < a href = "#orgab58ac0" > 7< / a > .
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< / p >
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< div id = "orgab58ac0" class = "figure" >
< p > < img src = "figs/amplified_piezo_with_displacement_sensor.jpg" alt = "amplified_piezo_with_displacement_sensor.jpg" width = "500px" / >
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< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Example of an Amplified piezoelectric actuator with an integrated displacement sensor (Cedrat Technologies)< / p >
< / div >
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< p >
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A simplistic model of such amplified actuator is shown in Figure < a href = "#org7ac3e95" > 8< / a > where:
< / p >
< ul class = "org-ul" >
< li > \(K_{r}\) represent the vertical stiffness when the piezoelectric stack is removed< / li >
< li > \(K_{a}\) is the vertical stiffness contribution of the piezoelectric stack< / li >
< li > \(F_{i}\) represents the part of the piezoelectric stack that is used as a force actuator< / li >
< li > \(F_{m,i}\) represents the remaining part of the piezoelectric stack that is used as a force sensor< / li >
< li > \(v_{m,i}\) represents the absolute velocity of the top part of the actuator< / li >
< li > \(d_{m,i}\) represents the total relative displacement of the actuator< / li >
< / ul >
< div id = "org7ac3e95" class = "figure" >
< p > < img src = "figs/iff_1dof.png" alt = "iff_1dof.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Model of an amplified actuator< / p >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org44de918" class = "outline-3" >
< h3 id = "org44de918" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org44de918" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
% initializeAmplifiedStrutDynamics - Add Stiffness and Damping properties of each strut
%
% Syntax: [stewart] = initializeAmplifiedStrutDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
% - Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
% - Kr [6x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
% - Cr [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - actuators.type = 2
% - actuators.K [6x1] - Total Stiffness of each strut [N/m]
% - actuators.C [6x1] - Total Damping of each strut [N/(m/s)]
% - actuators.Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
% - actuators.Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
% - actuators.Kr [6x1] - Vertical stiffness when the piezoelectric stack is removed [N/m]
% - actuators.Cr [6x1] - Vertical damping when the piezoelectric stack is removed [N/(m/s)]
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org3966929" class = "outline-3" >
< h3 id = "org3966929" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org3966929" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.Kr (6,1) double {mustBeNumeric, mustBeNonnegative} = 5e6*ones(6,1)
args.Cr (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 15e6*ones(6,1)
args.Ca (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org2e42182" class = "outline-3" >
< h3 id = "org2e42182" > Compute the total stiffness and damping< / h3 >
< div class = "outline-text-3" id = "text-org2e42182" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = args.Ka + args.Kr;
C = args.Ca + args.Cr;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfed23b2" class = "outline-3" >
< h3 id = "orgfed23b2" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orgfed23b2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.actuators.type = 2;
stewart.actuators.Ka = args.Ka;
stewart.actuators.Ca = args.Ca;
stewart.actuators.Kr = args.Kr;
stewart.actuators.Cr = args.Cr;
stewart.actuators.K = K;
stewart.actuators.C = K;
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< / div >
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< div id = "outline-container-orgbc5232e" class = "outline-2" >
< h2 id = "orgbc5232e" > < span class = "section-number-2" > 10< / span > < code > initializeJointDynamics< / code > : Add Stiffness and Damping properties for spherical joints< / h2 >
< div class = "outline-text-2" id = "text-10" >
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< p >
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< a id = "orga86aa00" > < / 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-orgcc4b26d" class = "outline-3" >
< h3 id = "orgcc4b26d" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgcc4b26d" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [stewart] = initializeJointDynamics(stewart, args)
% initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
%
% Syntax: [stewart] = initializeJointDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
% - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
% - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
% - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
% - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
% - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
% - Kz_M [6x1] - Translation (Tz) Stiffness for each top joints [N/m]
% - Cz_M [6x1] - Translation (Tz) Damping of each top joint [N/m]
% - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
% - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
% - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
% - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
% - Kz_F [6x1] - Translation (Tz) Stiffness for each bottom joints [N/m]
% - Cz_F [6x1] - Translation (Tz) Damping of each bottom joint [N/m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - stewart.joints_F and stewart.joints_M:
% - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
% - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
% - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
% - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
% - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
% - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
% - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org047804e" class = "outline-3" >
< h3 id = "org047804e" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org047804e" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
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args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof'})} = 'universal'
args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'spherical_3dof'})} = 'spherical'
args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 20*ones(6,1)
args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
args.Kz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
args.Cz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd5b8278" class = "outline-3" >
< h3 id = "orgd5b8278" > Add Actuator Type< / h3 >
< div class = "outline-text-3" id = "text-orgd5b8278" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > switch args.type_F
case 'universal'
stewart.joints_F.type = 1;
case 'spherical'
stewart.joints_F.type = 2;
case 'universal_p'
stewart.joints_F.type = 3;
case 'spherical_p'
stewart.joints_F.type = 4;
case 'universal_3dof'
stewart.joints_F.type = 5;
end
switch args.type_M
case 'universal'
stewart.joints_M.type = 1;
case 'spherical'
stewart.joints_M.type = 2;
case 'universal_p'
stewart.joints_M.type = 3;
case 'spherical_p'
stewart.joints_M.type = 4;
case 'spherical_3dof'
stewart.joints_M.type = 6;
end
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< / div >
< / div >
< / div >
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< div id = "outline-container-org51cf135" class = "outline-3" >
< h3 id = "org51cf135" > Add Stiffness and Damping in Translation of each strut< / h3 >
< div class = "outline-text-3" id = "text-org51cf135" >
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< p >
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Translation Stiffness
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart.joints_M.Kx = zeros(6,1);
stewart.joints_M.Ky = zeros(6,1);
stewart.joints_M.Kz = args.Kz_M;
stewart.joints_F.Kx = zeros(6,1);
stewart.joints_F.Ky = zeros(6,1);
stewart.joints_F.Kz = args.Kz_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.Cx = zeros(6,1);
stewart.joints_M.Cy = zeros(6,1);
stewart.joints_M.Cz = args.Cz_M;
stewart.joints_F.Cx = zeros(6,1);
stewart.joints_F.Cy = zeros(6,1);
stewart.joints_F.Cz = args.Cz_F;
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< / div >
< / div >
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< div id = "outline-container-org1e8eceb" class = "outline-3" >
< h3 id = "org1e8eceb" > Add Stiffness and Damping in Rotation of each strut< / h3 >
< div class = "outline-text-3" id = "text-org1e8eceb" >
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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.Kf_M;
stewart.joints_F.Kf = args.Kf_F;
stewart.joints_F.Kt = args.Kf_F;
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< / div >
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< p >
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Rotational Damping
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< / 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.Cf_M;
stewart.joints_F.Cf = args.Cf_F;
stewart.joints_F.Ct = args.Cf_F;
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< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3a7f26e" class = "outline-2" >
< h2 id = "org3a7f26e" > < span class = "section-number-2" > 11< / span > < code > initializeInertialSensor< / code > : Initialize the inertial sensor in each strut< / h2 >
< div class = "outline-text-2" id = "text-11" >
< p >
< a id = "org10af194" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/initializeInertialSensor.m" > here< / a > .
< / p >
< / div >
< div id = "outline-container-orgcfc37af" class = "outline-3" >
< h3 id = "orgcfc37af" > Geophone - Working Principle< / h3 >
< div class = "outline-text-3" id = "text-orgcfc37af" >
< p >
From the schematic of the Z-axis geophone shown in Figure < a href = "#orgcbee0e9" > 9< / a > , we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\):
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
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 >
< div id = "orgcbee0e9" class = "figure" >
< p > < img src = "figs/inertial_sensor.png" alt = "inertial_sensor.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Schematic of a Z-Axis geophone< / p >
< / 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 >
< div id = "outline-container-org986e38f" class = "outline-3" >
< h3 id = "org986e38f" > Accelerometer - Working Principle< / h3 >
< div class = "outline-text-3" id = "text-org986e38f" >
< p >
From the schematic of the Z-axis accelerometer shown in Figure < a href = "#orgf6281f4" > 10< / a > , we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\):
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
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 >
< div id = "orgf6281f4" class = "figure" >
< p > < img src = "figs/inertial_sensor.png" alt = "inertial_sensor.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Schematic of a Z-Axis geophone< / p >
< / 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-orgcb6bebb" class = "outline-3" >
< h3 id = "orgcb6bebb" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgcb6bebb" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > function [stewart] = initializeInertialSensor(stewart, args)
% initializeInertialSensor - Initialize the inertial sensor in each strut
%
% Syntax: [stewart] = initializeInertialSensor(args)
%
% Inputs:
% - args - Structure with the following fields:
% - type - 'geophone', 'accelerometer', 'none'
% - mass [1x1] - Weight of the inertial mass [kg]
% - freq [1x1] - Cutoff frequency [Hz]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - stewart.sensors.inertial
% - type - 1 (geophone), 2 (accelerometer), 3 (none)
% - K [1x1] - Stiffness [N/m]
% - C [1x1] - Damping [N/(m/s)]
% - M [1x1] - Inertial Mass [kg]
% - G [1x1] - Gain
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5ec99a5" class = "outline-3" >
< h3 id = "org5ec99a5" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org5ec99a5" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.type char {mustBeMember(args.type,{'geophone', 'accelerometer', 'none'})} = 'none'
args.mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e-2
args.freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org1c3d7c8" class = "outline-3" >
< h3 id = "org1c3d7c8" > Compute the properties of the sensor< / h3 >
< div class = "outline-text-3" id = "text-org1c3d7c8" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sensor = struct();
switch args.type
case 'geophone'
sensor.type = 1;
sensor.M = args.mass;
sensor.K = sensor.M * (2*pi*args.freq)^2;
sensor.C = 2*sqrt(sensor.M * sensor.K);
case 'accelerometer'
sensor.type = 2;
sensor.M = args.mass;
sensor.K = sensor.M * (2*pi*args.freq)^2;
sensor.C = 2*sqrt(sensor.M * sensor.K);
sensor.G = -sensor.K/sensor.M;
case 'none'
sensor.type = 3;
end
< / pre >
< / div >
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< / div >
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< div id = "outline-container-org5e33aa2" class = "outline-3" >
< h3 id = "org5e33aa2" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-org5e33aa2" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.sensors.inertial = sensor;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd6baa46" class = "outline-2" >
< h2 id = "orgd6baa46" > < span class = "section-number-2" > 12< / span > < code > displayArchitecture< / code > : 3D plot of the Stewart platform architecture< / h2 >
< div class = "outline-text-2" id = "text-12" >
< p >
< a id = "org455c4f1" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/displayArchitecture.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-orgc70c9a5" class = "outline-3" >
< h3 id = "orgc70c9a5" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgc70c9a5" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > function [] = displayArchitecture(stewart, args)
% displayArchitecture - 3D plot of the Stewart platform architecture
%
% Syntax: [] = displayArchitecture(args)
%
% Inputs:
% - stewart
% - args - Structure with the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
% - F_color [color] - Color used for the Fixed elements
% - M_color [color] - Color used for the Mobile elements
% - L_color [color] - Color used for the Legs elements
% - frames [true/false] - Display the Frames
% - legs [true/false] - Display the Legs
% - joints [true/false] - Display the Joints
% - labels [true/false] - Display the Labels
% - platforms [true/false] - Display the Platforms
% - views ['all', 'xy', 'yz', 'xz', 'default'] -
%
% Outputs:
< / pre >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-org63006a7" class = "outline-3" >
< h3 id = "org63006a7" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org63006a7" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
args.F_color = [0 0.4470 0.7410]
args.M_color = [0.8500 0.3250 0.0980]
args.L_color = [0 0 0]
args.frames logical {mustBeNumericOrLogical} = true
args.legs logical {mustBeNumericOrLogical} = true
args.joints logical {mustBeNumericOrLogical} = true
args.labels logical {mustBeNumericOrLogical} = true
args.platforms logical {mustBeNumericOrLogical} = true
args.views char {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
end
< / pre >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-org86735ca" class = "outline-3" >
< h3 id = "org86735ca" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org86735ca" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
FO_A = stewart.platform_F.FO_A;
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
MO_B = stewart.platform_M.MO_B;
assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
H = stewart.geometry.H;
assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
Fa = stewart.platform_F.Fa;
assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
Mb = stewart.platform_M.Mb;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgb11fd92" class = "outline-3" >
< h3 id = "orgb11fd92" > Figure Creation, Frames and Homogeneous transformations< / h3 >
< div class = "outline-text-3" id = "text-orgb11fd92" >
< p >
The reference frame of the 3d plot corresponds to the frame \(\{F\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if ~strcmp(args.views, 'all')
figure;
else
f = figure('visible', 'off');
end
hold on;
< / 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" >
< pre class = "src src-matlab" > FTa = [eye(3), FO_A; ...
zeros(1,3), 1];
ATb = [args.ARB, args.AP; ...
zeros(1,3), 1];
BTm = [eye(3), -MO_B; ...
zeros(1,3), 1];
FTm = FTa*ATb*BTm;
< / 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" >
< pre class = "src src-matlab" > d_unit_vector = H/4;
< / 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" >
< pre class = "src src-matlab" > d_label = H/20;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org7cd8fee" class = "outline-3" >
< h3 id = "org7cd8fee" > Fixed Base elements< / h3 >
< div class = "outline-text-3" id = "text-org7cd8fee" >
< p >
Let’ s first plot the frame \(\{F\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ff = [0, 0, 0];
if args.frames
quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
if args.labels
text(Ff(1) + d_label, ...
Ff(2) + d_label, ...
Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
end
end
< / pre >
< / div >
< p >
Now plot the frame \(\{A\}\) fixed to the Base.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.frames
quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
if args.labels
text(FO_A(1) + d_label, ...
FO_A(2) + d_label, ...
FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
end
end
< / pre >
< / div >
< p >
Let’ s then plot the circle corresponding to the shape of the Fixed base.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.platforms & & stewart.platform_F.type == 1
theta = [0:0.01:2*pi+0.01]; % Angles [rad]
v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
center = [0; 0; 0]; % Center of the circle
radius = stewart.platform_F.R; % Radius of the circle [m]
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
plot3(points(1,:), ...
points(2,:), ...
points(3,:), '-', 'Color', args.F_color);
end
< / pre >
< / div >
< p >
Let’ s now plot the position and labels of the Fixed Joints
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.joints
scatter3(Fa(1,:), ...
Fa(2,:), ...
Fa(3,:), 'MarkerEdgeColor', args.F_color);
if args.labels
for i = 1:size(Fa,2)
text(Fa(1,i) + d_label, ...
Fa(2,i), ...
Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
end
end
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgacb8eb7" class = "outline-3" >
< h3 id = "orgacb8eb7" > Mobile Platform elements< / h3 >
< div class = "outline-text-3" id = "text-orgacb8eb7" >
< p >
Plot the frame \(\{M\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}
if args.frames
FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)
if args.labels
text(Fm(1) + d_label, ...
Fm(2) + d_label, ...
Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
end
end
< / pre >
< / div >
< p >
Plot the frame \(\{B\}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > FB = FO_A + args.AP;
if args.frames
FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)
if args.labels
text(FB(1) - d_label, ...
FB(2) + d_label, ...
FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
end
end
< / pre >
< / div >
< p >
Let’ s then plot the circle corresponding to the shape of the Mobile platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.platforms & & stewart.platform_M.type == 1
theta = [0:0.01:2*pi+0.01]; % Angles [rad]
v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
center = Fm(1:3); % Center of the circle
radius = stewart.platform_M.R; % Radius of the circle [m]
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
plot3(points(1,:), ...
points(2,:), ...
points(3,:), '-', 'Color', args.M_color);
end
< / pre >
< / div >
< p >
Plot the position and labels of the rotation joints fixed to the mobile platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.joints
Fb = FTm*[Mb;ones(1,6)];
scatter3(Fb(1,:), ...
Fb(2,:), ...
Fb(3,:), 'MarkerEdgeColor', args.M_color);
if args.labels
for i = 1:size(Fb,2)
text(Fb(1,i) + d_label, ...
Fb(2,i), ...
Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
end
end
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org7f9320b" class = "outline-3" >
< h3 id = "org7f9320b" > Legs< / h3 >
< div class = "outline-text-3" id = "text-org7f9320b" >
< p >
Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if args.legs
for i = 1:6
plot3([Fa(1,i), Fb(1,i)], ...
[Fa(2,i), Fb(2,i)], ...
[Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);
if args.labels
text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
(Fa(2,i)+Fb(2,i))/2, ...
(Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
end
end
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org925a393" class = "outline-3" >
< h3 id = "org925a393" > < span class = "section-number-3" > 12.1< / span > Figure parameters< / h3 >
< div class = "outline-text-3" id = "text-12-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > switch args.views
case 'default'
view([1 -0.6 0.4]);
case 'xy'
view([0 0 1]);
case 'xz'
view([0 -1 0]);
case 'yz'
view([1 0 0]);
end
axis equal;
axis off;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org44e536d" class = "outline-3" >
< h3 id = "org44e536d" > < span class = "section-number-3" > 12.2< / span > Subplots< / h3 >
< div class = "outline-text-3" id = "text-12-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > if strcmp(args.views, 'all')
hAx = findobj('type', 'axes');
figure;
s1 = subplot(2,2,1);
copyobj(get(hAx(1), 'Children'), s1);
view([0 0 1]);
axis equal;
axis off;
title('Top')
s2 = subplot(2,2,2);
copyobj(get(hAx(1), 'Children'), s2);
view([1 -0.6 0.4]);
axis equal;
axis off;
s3 = subplot(2,2,3);
copyobj(get(hAx(1), 'Children'), s3);
view([1 0 0]);
axis equal;
axis off;
title('Front')
s4 = subplot(2,2,4);
copyobj(get(hAx(1), 'Children'), s4);
view([0 -1 0]);
axis equal;
axis off;
title('Side')
close(f);
end
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgecfd55f" class = "outline-2" >
< h2 id = "orgecfd55f" > < span class = "section-number-2" > 13< / span > < code > describeStewartPlatform< / code > : Display some text describing the current defined Stewart Platform< / h2 >
< div class = "outline-text-2" id = "text-13" >
< p >
< a id = "org8cc8939" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/describeStewartPlatform.m" > here< / a > .
< / p >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-orgb99230a" class = "outline-3" >
< h3 id = "orgb99230a" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgb99230a" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > function [] = describeStewartPlatform(stewart)
% describeStewartPlatform - Display some text describing the current defined Stewart Platform
%
% Syntax: [] = describeStewartPlatform(args)
%
% Inputs:
% - stewart
%
% Outputs:
< / pre >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-orged88325" class = "outline-3" >
< h3 id = "orged88325" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-orged88325" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
end
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org1d49caa" class = "outline-3" >
< h3 id = "org1d49caa" > < span class = "section-number-3" > 13.1< / span > Geometry< / h3 >
< div class = "outline-text-3" id = "text-13-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('GEOMETRY:\n')
fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)
if stewart.platform_M.MO_B(3) > 0
fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n', 1e3*stewart.platform_M.MO_B(3))
else
fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
end
fprintf('- The initial length of the struts are:\n')
fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
fprintf('\n')
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgcb66771" class = "outline-3" >
< h3 id = "orgcb66771" > < span class = "section-number-3" > 13.2< / span > Actuators< / h3 >
< div class = "outline-text-3" id = "text-13-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('ACTUATORS:\n')
if stewart.actuators.type == 1
fprintf('- The actuators are classical.\n')
fprintf('- The Stiffness and Damping of each actuators is:\n')
fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.K(1), stewart.actuators.C(1))
elseif stewart.actuators.type == 2
fprintf('- The actuators are mechanicaly amplified.\n')
fprintf('- The vertical stiffness and damping contribution of the piezoelectric stack is:\n')
fprintf('\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n', stewart.actuators.Ka(1), stewart.actuators.Ca(1))
fprintf('- Vertical stiffness when the piezoelectric stack is removed is:\n')
fprintf('\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n', stewart.actuators.Kr(1), stewart.actuators.Cr(1))
end
fprintf('\n')
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org4630b77" class = "outline-3" >
< h3 id = "org4630b77" > < span class = "section-number-3" > 13.3< / span > Joints< / h3 >
< div class = "outline-text-3" id = "text-13-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('JOINTS:\n')
< / pre >
< / div >
< p >
Type of the joints on the fixed base.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > switch stewart.joints_F.type
case 1
fprintf('- The joints on the fixed based are universal joints\n')
case 2
fprintf('- The joints on the fixed based are spherical joints\n')
case 3
fprintf('- The joints on the fixed based are perfect universal joints\n')
case 4
fprintf('- The joints on the fixed based are perfect spherical joints\n')
end
< / pre >
< / div >
< p >
Type of the joints on the mobile platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > switch stewart.joints_M.type
case 1
fprintf('- The joints on the mobile based are universal joints\n')
case 2
fprintf('- The joints on the mobile based are spherical joints\n')
case 3
fprintf('- The joints on the mobile based are perfect universal joints\n')
case 4
fprintf('- The joints on the mobile based are perfect spherical joints\n')
end
< / pre >
< / div >
< p >
Position of the fixed joints
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
< / pre >
< / div >
< p >
Position of the mobile joints
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
fprintf('\n')
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org47a9cf0" class = "outline-3" >
< h3 id = "org47a9cf0" > < span class = "section-number-3" > 13.4< / span > Kinematics< / h3 >
< div class = "outline-text-3" id = "text-13-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > fprintf('KINEMATICS:\n')
if isfield(stewart.kinematics, 'K')
fprintf('- The Stiffness matrix K is (in [N/m]):\n')
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
end
if isfield(stewart.kinematics, 'C')
fprintf('- The Damping matrix C is (in [m/N]):\n')
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
end
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org65fc289" class = "outline-2" >
< h2 id = "org65fc289" > < span class = "section-number-2" > 14< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h2 >
< div class = "outline-text-2" id = "text-14" >
< p >
< a id = "org677ea95" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/generateCubicConfiguration.m" > here< / a > .
< / p >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-orgc164e8d" class = "outline-3" >
< h3 id = "orgc164e8d" > Function description< / h3 >
< div class = "outline-text-3" id = "text-orgc164e8d" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > function [stewart] = generateCubicConfiguration(stewart, args)
% generateCubicConfiguration - Generate a Cubic Configuration
%
% Syntax: [stewart] = generateCubicConfiguration(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - geometry.H [1x1] - Total height of the platform [m]
% - args - Can have the following fields:
% - Hc [1x1] - Height of the "useful" part of the cube [m]
% - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
< / pre >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-org0d467b7" class = "outline-3" >
< h3 id = "org0d467b7" > Documentation< / h3 >
< div class = "outline-text-3" id = "text-org0d467b7" >
2020-05-05 11:38:52 +02:00
< div id = "org70070f0" class = "figure" >
< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Cubic Configuration< / p >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-orgda76f80" class = "outline-3" >
< h3 id = "orgda76f80" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-orgda76f80" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
args.FOc (1,1) double {mustBeNumeric} = 50e-3
args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
end
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org4a5a3cf" class = "outline-3" >
< h3 id = "org4a5a3cf" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org4a5a3cf" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
H = stewart.geometry.H;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge94a885" class = "outline-3" >
< h3 id = "orge94a885" > Position of the Cube< / h3 >
< div class = "outline-text-3" id = "text-orge94a885" >
< 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" >
< pre class = "src src-matlab" > sx = [ 2; -1; -1];
sy = [ 0; 1; -1];
sz = [ 1; 1; 1];
R = [sx, sy, sz]./vecnorm([sx, sy, sz]);
L = args.Hc*sqrt(3);
Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];
CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge8af7a8" class = "outline-3" >
< h3 id = "orge8af7a8" > Compute the pose< / h3 >
< div class = "outline-text-3" id = "text-orge8af7a8" >
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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" >
< pre class = "src src-matlab" > CSi = (CCm - CCf)./vecnorm(CCm - CCf);
< / pre >
< / div >
< p >
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org32b44aa" class = "outline-3" >
< h3 id = "org32b44aa" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-org32b44aa" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.platform_F.Fa = Fa;
stewart.platform_M.Mb = Mb;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org9e8cbfa" class = "outline-2" >
< h2 id = "org9e8cbfa" > < span class = "section-number-2" > 15< / span > < code > computeJacobian< / code > : Compute the Jacobian Matrix< / h2 >
< div class = "outline-text-2" id = "text-15" >
< p >
< a id = "orgfb113f5" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/computeJacobian.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org2c08fbd" class = "outline-3" >
< h3 id = "org2c08fbd" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org2c08fbd" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
% - stewart - With at least the following fields:
% - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
% - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
% - actuators.K [6x1] - Total stiffness of the actuators
%
% Outputs:
% - stewart - With the 3 added field:
% - kinematics.J [6x6] - The Jacobian Matrix
% - kinematics.K [6x6] - The Stiffness Matrix
% - kinematics.C [6x6] - The Compliance Matrix
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8071a1b" class = "outline-3" >
< h3 id = "org8071a1b" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org8071a1b" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > assert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As')
As = stewart.geometry.As;
assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab')
Ab = stewart.geometry.Ab;
assert(isfield(stewart.actuators, 'K'), 'stewart.actuators should have attribute K')
Ki = stewart.actuators.K;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org9bcd9b9" class = "outline-3" >
< h3 id = "org9bcd9b9" > Compute Jacobian Matrix< / h3 >
< div class = "outline-text-3" id = "text-org9bcd9b9" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > J = [As' , cross(Ab, As)'];
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgf08eda6" class = "outline-3" >
< h3 id = "orgf08eda6" > Compute Stiffness Matrix< / h3 >
< div class = "outline-text-3" id = "text-orgf08eda6" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = J'*diag(Ki)*J;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd164132" class = "outline-3" >
< h3 id = "orgd164132" > Compute Compliance Matrix< / h3 >
< div class = "outline-text-3" id = "text-orgd164132" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > C = inv(K);
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbf70f7a" class = "outline-3" >
< h3 id = "orgbf70f7a" > Populate the < code > stewart< / code > structure< / h3 >
< div class = "outline-text-3" id = "text-orgbf70f7a" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.kinematics.J = J;
stewart.kinematics.K = K;
stewart.kinematics.C = C;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org03168fc" class = "outline-2" >
< h2 id = "org03168fc" > < span class = "section-number-2" > 16< / span > < code > inverseKinematics< / code > : Compute Inverse Kinematics< / h2 >
< div class = "outline-text-2" id = "text-16" >
< p >
< a id = "org681bcb5" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/inverseKinematics.m" > here< / a > .
< / p >
< / div >
< div id = "outline-container-orgbdc5fb1" class = "outline-3" >
< h3 id = "orgbdc5fb1" > Theory< / h3 >
< div class = "outline-text-3" id = "text-orgbdc5fb1" >
< 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 >
< / div >
< / div >
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< div id = "outline-container-org18c9841" class = "outline-3" >
< h3 id = "org18c9841" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org18c9841" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - geometry.Aa [3x6] - The positions ai expressed in {A}
% - geometry.Bb [3x6] - The positions bi expressed in {B}
% - geometry.l [6x1] - Length of each strut
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org848fb1f" class = "outline-3" >
< h3 id = "org848fb1f" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org848fb1f" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8be974b" class = "outline-3" >
< h3 id = "org8be974b" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-org8be974b" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > assert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa')
Aa = stewart.geometry.Aa;
assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb')
Bb = stewart.geometry.Bb;
assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l')
l = stewart.geometry.l;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org8b70a76" class = "outline-3" >
< h3 id = "org8b70a76" > Compute< / h3 >
< div class = "outline-text-3" id = "text-org8b70a76" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > dLi = Li-l;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org278d55b" class = "outline-2" >
< h2 id = "org278d55b" > < span class = "section-number-2" > 17< / span > < code > forwardKinematicsApprox< / code > : Compute the Approximate Forward Kinematics< / h2 >
< div class = "outline-text-2" id = "text-17" >
< p >
< a id = "org5b15db4" > < / a >
< / p >
< p >
This Matlab function is accessible < a href = "../src/forwardKinematicsApprox.m" > here< / a > .
< / p >
< / div >
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< div id = "outline-container-org07e11bf" class = "outline-3" >
< h3 id = "org07e11bf" > Function description< / h3 >
< div class = "outline-text-3" id = "text-org07e11bf" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > function [P, R] = forwardKinematicsApprox(stewart, args)
% forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
% the Jacobian Matrix
%
% Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - kinematics.J [6x6] - The Jacobian Matrix
% - args - Can have the following fields:
% - dL [6x1] - Displacement of each strut [m]
%
% Outputs:
% - P [3x1] - The estimated position of {B} with respect to {A}
% - R [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org918179d" class = "outline-3" >
< h3 id = "org918179d" > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-org918179d" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
stewart
args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
end
< / pre >
< / div >
< / div >
< / div >
2020-05-20 16:41:34 +02:00
< div id = "outline-container-orgf157791" class = "outline-3" >
< h3 id = "orgf157791" > Check the < code > stewart< / code > structure elements< / h3 >
< div class = "outline-text-3" id = "text-orgf157791" >
2020-05-05 11:38:52 +02:00
< div class = "org-src-container" >
< pre class = "src src-matlab" > assert(isfield(stewart.kinematics, 'J'), 'stewart.kinematics should have attribute J')
J = stewart.kinematics.J;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgf17cab9" class = "outline-3" >
< h3 id = "orgf17cab9" > Computation< / h3 >
< div class = "outline-text-3" id = "text-orgf17cab9" >
< 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:
\[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > X = J\args.dL;
< / pre >
< / div >
< p >
The position vector corresponds to the first 3 elements.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > P = X(1:3);
< / pre >
< / div >
< p >
The next 3 elements are the orientation of {B} with respect to {A} expressed
using the screw axis.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > theta = norm(X(4:6));
s = X(4:6)/theta;
< / pre >
< / div >
< p >
We then compute the corresponding rotation matrix.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > R = [s(1)^2*(1-cos(theta)) + cos(theta) , s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta), s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];
< / pre >
< / div >
< / div >
< / div >
2020-02-25 18:10:20 +01:00
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-05-20 16:41:34 +02:00
< p class = "date" > Created: 2020-05-20 mer. 16:41< / p >
2020-02-25 18:10:20 +01:00
< / div >
< / body >
< / html >
