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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 = "#org527cc13" > 1. initializeGeneralConfiguration< / a >
< ul >
< li > < a href = "#orgea5f8f5" > 1.1. Function description< / a > < / li >
< li > < a href = "#org2db42cb" > 1.2. Optional Parameters< / a > < / li >
< li > < a href = "#org2f9279a" > 1.3. Geometry Description< / a > < / li >
< li > < a href = "#org1409cf0" > 1.4. Compute Aa and Ab< / a > < / li >
< li > < a href = "#orgb91c416" > 1.5. Returns Stewart Structure< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgc3aa910" > 2. computeGeometricalProperties< / a >
< ul >
< li > < a href = "#org180196f" > 2.1. Function description< / a > < / li >
< li > < a href = "#org12cee4f" > 2.2. Optional Parameters< / a > < / li >
< li > < a href = "#org0010af5" > 2.3. Rotation matrices< / a > < / li >
< li > < a href = "#org98f4bad" > 2.4. Jacobian matrices< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgb3e53d1" > 3. initializeMechanicalElements< / a >
< ul >
< li > < a href = "#orge7f185e" > 3.1. Function description< / a > < / li >
< li > < a href = "#org6bd219d" > 3.2. Optional Parameters< / a > < / li >
< li > < a href = "#org8d0d9c0" > 3.3. Bottom Plate< / a > < / li >
< li > < a href = "#org23fd88c" > 3.4. Top Plate< / a > < / li >
< li > < a href = "#org96d7dab" > 3.5. Legs< / a > < / li >
< li > < a href = "#org66df86f" > 3.6. Ball Joints< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgf3c4474" > 4. initializeSample< / a >
< ul >
< li > < a href = "#org1ec4152" > 4.1. Function description< / a > < / li >
< li > < a href = "#orgcd3268d" > 4.2. Optional Parameters< / a > < / li >
< li > < a href = "#org29ee9ed" > 4.3. Save the Sample structure< / a > < / li >
< / ul >
< / li >
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< / ul >
< / div >
< / div >
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< p >
Stewart platforms are generated in multiple steps.
< / p >
< p >
First, geometrical parameters are defined:
< / p >
< ul class = "org-ul" >
< li > \({}^Aa_i\) - Position of the joints fixed to the fixed base w.r.t \(\{A\}\)< / li >
< li > \({}^Ab_i\) - Position of the joints fixed to the mobile platform w.r.t \(\{A\}\)< / li >
< li > \({}^Bb_i\) - Position of the joints fixed to the mobile platform w.r.t \(\{B\}\)< / li >
< li > \(H\) - Total height of the mobile platform< / li >
< / ul >
< p >
These parameter are enough to determine all the kinematic properties of the platform like the Jacobian, stroke, stiffness, …
These geometrical parameters can be generated using different functions: < code > initializeCubicConfiguration< / code > for cubic configuration or < code > initializeGeneralConfiguration< / code > for more general configuration.
< / p >
< p >
A function < code > computeGeometricalProperties< / code > is then used to compute:
< / p >
< ul class = "org-ul" >
< li > \(J_f\) - Jacobian matrix for the force location< / li >
< li > \(J_d\) - Jacobian matrix for displacement estimation< / li >
< li > \(R_m\) - Rotation matrices to position the leg vectors< / li >
< / ul >
< p >
Then, geometrical parameters are computed for all the mechanical elements with the function < code > initializeMechanicalElements< / code > :
< / p >
< ul class = "org-ul" >
< li > Shape of the platforms
< ul class = "org-ul" >
< li > External Radius< / li >
< li > Internal Radius< / li >
< li > Density< / li >
< li > Thickness< / li >
< / ul > < / li >
< li > Shape of the Legs
< ul class = "org-ul" >
< li > Radius< / li >
< li > Size of ball joint< / li >
< li > Density< / li >
< / ul > < / li >
< / ul >
< p >
Other Parameters are defined for the Simscape simulation:
< / p >
< ul class = "org-ul" >
< li > Sample mass, volume and position (< code > initializeSample< / code > function)< / li >
< li > Location of the inertial sensor< / li >
< li > Location of the point for the differential measurements< / li >
< li > Location of the Jacobian point for velocity/displacement computation< / li >
< / ul >
< div id = "outline-container-org527cc13" class = "outline-2" >
< h2 id = "org527cc13" > < span class = "section-number-2" > 1< / span > initializeGeneralConfiguration< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< / div >
< div id = "outline-container-orgea5f8f5" class = "outline-3" >
< h3 id = "orgea5f8f5" > < span class = "section-number-3" > 1.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-1-1" >
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< p >
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The < code > initializeGeneralConfiguration< / code > function takes one structure that contains configurations for the hexapod and returns one structure representing the Hexapod.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > function< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #DFAF8F;" > stewart< / span > < span style = "color: #DCDCCC;" > ]< / span > = < span style = "color: #93E0E3;" > initializeGeneralConfiguration< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #DFAF8F;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span >
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org2db42cb" class = "outline-3" >
< h3 id = "org2db42cb" > < span class = "section-number-3" > 1.2< / span > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< p >
Default values for opts.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
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< span style = "color: #CC9393;" > 'H_tot'< / span > , < span style = "color: #BFEBBF;" > 90< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height of the platform [mm]< / span >
< span style = "color: #CC9393;" > 'H_joint'< / span > , < span style = "color: #BFEBBF;" > 15< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Height of the joints [mm]< / span >
< span style = "color: #CC9393;" > 'H_plate'< / span > , < span style = "color: #BFEBBF;" > 10< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Thickness of the fixed and mobile platforms [mm]< / span >
< span style = "color: #CC9393;" > 'R_bot'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Radius where the legs articulations are positionned [mm]< / span >
< span style = "color: #CC9393;" > 'R_top'< / span > , < span style = "color: #BFEBBF;" > 80< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Radius where the legs articulations are positionned [mm]< / span >
< span style = "color: #CC9393;" > 'a_bot'< / span > , < span style = "color: #BFEBBF;" > 10< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Angle Offset [deg]< / span >
< span style = "color: #CC9393;" > 'a_top'< / span > , < span style = "color: #BFEBBF;" > 40< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Angle Offset [deg]< / span >
< span style = "color: #CC9393;" > 'da_top'< / span > , < span style = "color: #BFEBBF;" > 0< / span > < span style = "text-decoration: underline;" > ...< / span > % Angle Offset from < span style = "color: #BFEBBF;" > 0< / span > position [deg]
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< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< p >
Populate opts with input parameters
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > exist< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'opts_param','var'< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > opt< / span > = < span style = "color: #BFEBBF;" > fieldnames< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #BFEBBF;" > '< / span >
opts.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > = opts_param.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2f9279a" class = "outline-3" >
< h3 id = "org2f9279a" > < span class = "section-number-3" > 1.3< / span > Geometry Description< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
< div id = "orgc30ce24" class = "figure" >
< p > < img src = "./figs/stewart_bottom_plate.png" alt = "stewart_bottom_plate.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Schematic of the bottom plates with all the parameters< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org1409cf0" class = "outline-3" >
< h3 id = "org1409cf0" > < span class = "section-number-3" > 1.4< / span > Compute Aa and Ab< / h3 >
< div class = "outline-text-3" id = "text-1-4" >
< p >
We compute \([a_1, a_2, a_3, a_4, a_5, a_6]^T\) and \([b_1, b_2, b_3, b_4, b_5, b_6]^T\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Aa = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
Ab = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
Bb = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 3< / span >
Aa< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = < span style = "color: #DCDCCC;" > [< / span > opts.R_bot< span style = "color: #7CB8BB;" > *< / span > cos< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > opts.a_bot< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.R_bot< span style = "color: #7CB8BB;" > *< / span > sin< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > opts.a_bot< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.H_plate< span style = "color: #7CB8BB;" > +< / span > opts.H_joint< span style = "color: #DCDCCC;" > ]< / span > ;
Aa< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = < span style = "color: #DCDCCC;" > [< / span > opts.R_bot< span style = "color: #7CB8BB;" > *< / span > cos< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.a_bot< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.R_bot< span style = "color: #7CB8BB;" > *< / span > sin< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.a_bot< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.H_plate< span style = "color: #7CB8BB;" > +< / span > opts.H_joint< span style = "color: #DCDCCC;" > ]< / span > ;
Ab< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = < span style = "color: #DCDCCC;" > [< / span > opts.R_top< span style = "color: #7CB8BB;" > *< / span > cos< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.da_top < span style = "color: #7CB8BB;" > -< / span > opts.a_top< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.R_top< span style = "color: #7CB8BB;" > *< / span > sin< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.da_top < span style = "color: #7CB8BB;" > -< / span > opts.a_top< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.H_tot < span style = "color: #7CB8BB;" > -< / span > opts.H_plate < span style = "color: #7CB8BB;" > -< / span > opts.H_joint< span style = "color: #DCDCCC;" > ]< / span > ;
Ab< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = < span style = "color: #DCDCCC;" > [< / span > opts.R_top< span style = "color: #7CB8BB;" > *< / span > cos< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.da_top < span style = "color: #7CB8BB;" > +< / span > opts.a_top< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.R_top< span style = "color: #7CB8BB;" > *< / span > sin< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > pi< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 180< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > 120< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #93E0E3;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #93E0E3;" > )< / span > < span style = "color: #7CB8BB;" > +< / span > opts.da_top < span style = "color: #7CB8BB;" > +< / span > opts.a_top< span style = "color: #D0BF8F;" > )< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "text-decoration: underline;" > ...< / span >
opts.H_tot < span style = "color: #7CB8BB;" > -< / span > opts.H_plate < span style = "color: #7CB8BB;" > -< / span > opts.H_joint< span style = "color: #DCDCCC;" > ]< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
Bb = Ab < span style = "color: #7CB8BB;" > -< / span > opts.H_tot< span style = "color: #7CB8BB;" > *< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > ,< span style = "color: #BFEBBF;" > 0< / span > ,< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > ]< / span > ;
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< div id = "outline-container-orgb91c416" class = "outline-3" >
< h3 id = "orgb91c416" > < span class = "section-number-3" > 1.5< / span > Returns Stewart Structure< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = struct< span style = "color: #DCDCCC;" > ()< / span > ;
stewart.Aa = Aa;
stewart.Ab = Ab;
stewart.Bb = Bb;
stewart.H_tot = opts.H_tot;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
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< / div >
< div id = "outline-container-orgc3aa910" class = "outline-2" >
< h2 id = "orgc3aa910" > < span class = "section-number-2" > 2< / span > computeGeometricalProperties< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< / div >
< div id = "outline-container-org180196f" class = "outline-3" >
< h3 id = "org180196f" > < span class = "section-number-3" > 2.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > function< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #DFAF8F;" > stewart< / span > < span style = "color: #DCDCCC;" > ]< / span > = < span style = "color: #93E0E3;" > computeGeometricalProperties< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #DFAF8F;" > stewart< / span > , < span style = "color: #DFAF8F;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span >
< / pre >
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< / div >
< / div >
< div id = "outline-container-org12cee4f" class = "outline-3" >
< h3 id = "org12cee4f" > < span class = "section-number-3" > 2.2< / span > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
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< p >
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Default values for opts.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'Jd_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 30< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span style = "color: #CC9393;" > 'Jf_pos'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 30< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
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< / pre >
< / div >
< p >
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Populate opts with input parameters
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > exist< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'opts_param','var'< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > opt< / span > = < span style = "color: #BFEBBF;" > fieldnames< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #BFEBBF;" > '< / span >
opts.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > = opts_param.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
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< div id = "outline-container-org0010af5" class = "outline-3" >
< h3 id = "org0010af5" > < span class = "section-number-3" > 2.3< / span > Rotation matrices< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< p >
We initialize \(l_i\) and \(\hat{s}_i\)
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > leg_length = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #DCDCCC;" > )< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
leg_vectors = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
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< p >
We compute \(b_i - a_i\), and then:
< / p >
\begin{align*}
l_i & = \left|b_i - a_i\right| \\
\hat{s}_i & = \frac{b_i - a_i}{l_i}
\end{align*}
< div class = "org-src-container" >
< pre class = "src src-matlab" > legs = stewart.Ab < span style = "color: #7CB8BB;" > -< / span > stewart.Aa;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
leg_length< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > = norm< span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
leg_vectors< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > = legs< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > leg_length< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
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< p >
We compute rotation matrices to have the orientation of the legs.
The rotation matrix transforms the \(z\) axis to the axis of the leg. The other axis are not important here.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.Rm = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'R'< / span > , eye< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
sx = cross< span style = "color: #DCDCCC;" > (< / span > leg_vectors< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sx = sx< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sx< span style = "color: #DCDCCC;" > )< / span > ;
sy = < span style = "color: #7CB8BB;" > -< / span > cross< span style = "color: #DCDCCC;" > (< / span > sx, leg_vectors< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sy = sy< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sy< span style = "color: #DCDCCC;" > )< / span > ;
sz = leg_vectors< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
sz = sz< span style = "color: #7CB8BB;" > /< / span > norm< span style = "color: #DCDCCC;" > (< / span > sz< span style = "color: #DCDCCC;" > )< / span > ;
stewart.Rm< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > < span style = "color: #DCDCCC;" > )< / span > .R = < span style = "color: #DCDCCC;" > [< / span > sx', sy', sz'< span style = "color: #DCDCCC;" > ]< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org98f4bad" class = "outline-3" >
< h3 id = "org98f4bad" > < span class = "section-number-3" > 2.4< / span > Jacobian matrices< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< p >
Compute Jacobian Matrix
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Jd = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
Jd< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > = leg_vectors< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Jd< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 4< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span > < span style = "color: #DCDCCC;" > )< / span > = cross< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 001< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > (< / span > stewart.Bb< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > opts.Jd_pos< span style = "color: #BFEBBF;" > )< / span > , leg_vectors< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
stewart.Jd = Jd;
stewart.Jd_inv = inv< span style = "color: #DCDCCC;" > (< / span > Jd< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Jf = zeros< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 6< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > i< / span > = < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span >
Jf< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > = leg_vectors< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #DCDCCC;" > )< / span > ;
Jf< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #BFEBBF;" > 4< / span > < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > 6< / span > < span style = "color: #DCDCCC;" > )< / span > = cross< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 001< / span > < span style = "color: #7CB8BB;" > *< / span > < span style = "color: #BFEBBF;" > (< / span > stewart.Bb< span style = "color: #D0BF8F;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #D0BF8F;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > opts.Jf_pos< span style = "color: #BFEBBF;" > )< / span > , leg_vectors< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > i< / span > , < span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
stewart.Jf = Jf;
stewart.Jf_inv = inv< span style = "color: #DCDCCC;" > (< / span > Jf< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgb3e53d1" class = "outline-2" >
< h2 id = "orgb3e53d1" > < span class = "section-number-2" > 3< / span > initializeMechanicalElements< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< / div >
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< div id = "outline-container-orge7f185e" class = "outline-3" >
< h3 id = "orge7f185e" > < span class = "section-number-3" > 3.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-3-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > function< / span > < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #DFAF8F;" > stewart< / span > < span style = "color: #DCDCCC;" > ]< / span > = < span style = "color: #93E0E3;" > initializeMechanicalElements< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #DFAF8F;" > stewart< / span > , < span style = "color: #DFAF8F;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org6bd219d" class = "outline-3" >
< h3 id = "org6bd219d" > < span class = "section-number-3" > 3.2< / span > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
< p >
Default values for opts.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'thickness'< / span > , < span style = "color: #BFEBBF;" > 10< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Thickness of the base and platform [mm]< / span >
< span style = "color: #CC9393;" > 'density'< / span > , < span style = "color: #BFEBBF;" > 1000< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Density of the material used for the hexapod [kg/m3]< / span >
< span style = "color: #CC9393;" > 'k_ax'< / span > , < span style = "color: #BFEBBF;" > 1e8< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Stiffness of each actuator [N/m]< / span >
< span style = "color: #CC9393;" > 'c_ax'< / span > , < span style = "color: #BFEBBF;" > 1000< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Damping of each actuator [N/(m/s)]< / span >
< span style = "color: #CC9393;" > 'stroke'< / span > , < span style = "color: #BFEBBF;" > 50e< / span > < span style = "color: #7CB8BB;" > -< / span > < span style = "color: #BFEBBF;" > 6< / span > < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % Maximum stroke of each actuator [m]< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
< / pre >
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< / div >
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< p >
Populate opts with input parameters
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > exist< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'opts_param','var'< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > opt< / span > = < span style = "color: #BFEBBF;" > fieldnames< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #BFEBBF;" > '< / span >
opts.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > = opts_param.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org8d0d9c0" class = "outline-3" >
< h3 id = "org8d0d9c0" > < span class = "section-number-3" > 3.3< / span > Bottom Plate< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
< div id = "org38598b1" class = "figure" >
< p > < img src = "./figs/stewart_bottom_plate.png" alt = "stewart_bottom_plate.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Schematic of the bottom plates with all the parameters< / p >
< / div >
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< p >
The bottom plate structure is initialized.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > BP = struct< span style = "color: #DCDCCC;" > ()< / span > ;
< / pre >
< / div >
< p >
We defined its internal radius (if there is a hole in the bottom plate) and its outer radius.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > BP.Rint = < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "color: #7F9F7F;" > % Internal Radius [mm]< / span >
BP.Rext = < span style = "color: #BFEBBF;" > 150< / span > ; < span style = "color: #7F9F7F;" > % External Radius [mm]< / span >
< / pre >
< / div >
< p >
We define its thickness.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > BP.H = opts.thickness; < span style = "color: #7F9F7F;" > % Thickness of the Bottom Plate [mm]< / span >
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< / pre >
< / div >
< p >
We defined the density of the material of the bottom plate.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > BP.density = opts.density; < span style = "color: #7F9F7F;" > % Density of the material [kg/m3]< / span >
< / pre >
< / div >
< p >
And its color.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > BP.color = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #DCDCCC;" > ]< / span > ; < span style = "color: #7F9F7F;" > % Color [RGB]< / span >
< / pre >
< / div >
< p >
Then the profile of the bottom plate is computed and will be used by Simscape
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > BP.shape = < span style = "color: #DCDCCC;" > [< / span > BP.Rint BP.H; BP.Rint < span style = "color: #BFEBBF;" > 0< / span > ; BP.Rext < span style = "color: #BFEBBF;" > 0< / span > ; BP.Rext BP.H< span style = "color: #DCDCCC;" > ]< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
< / pre >
< / div >
< p >
The structure is added to the stewart structure
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.BP = BP;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org23fd88c" class = "outline-3" >
< h3 id = "org23fd88c" > < span class = "section-number-3" > 3.4< / span > Top Plate< / h3 >
< div class = "outline-text-3" id = "text-3-4" >
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< p >
The top plate structure is initialized.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP = struct< span style = "color: #DCDCCC;" > ()< / span > ;
< / pre >
< / div >
< p >
We defined the internal and external radius of the top plate.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP.Rint = < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
TP.Rext = < span style = "color: #BFEBBF;" > 100< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
< / pre >
< / div >
< p >
The thickness of the top plate.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP.H = < span style = "color: #BFEBBF;" > 10< / span > ; < span style = "color: #7F9F7F;" > % [mm]< / span >
< / pre >
< / div >
< p >
The density of its material.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP.density = opts.density; < span style = "color: #7F9F7F;" > % Density of the material [kg/m3]< / span >
< / pre >
< / div >
< p >
Its color.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP.color = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #DCDCCC;" > ]< / span > ; < span style = "color: #7F9F7F;" > % Color [RGB]< / span >
< / pre >
< / div >
< p >
Then the shape of the top plate is computed
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > TP.shape = < span style = "color: #DCDCCC;" > [< / span > TP.Rint TP.H; TP.Rint < span style = "color: #BFEBBF;" > 0< / span > ; TP.Rext < span style = "color: #BFEBBF;" > 0< / span > ; TP.Rext TP.H< span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< p >
The structure is added to the stewart structure
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.TP = TP;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org96d7dab" class = "outline-3" >
< h3 id = "org96d7dab" > < span class = "section-number-3" > 3.5< / span > Legs< / h3 >
< div class = "outline-text-3" id = "text-3-5" >
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< div id = "orga9ade83" class = "figure" >
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< p > < img src = "./figs/stewart_legs.png" alt = "stewart_legs.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Schematic for the legs of the Stewart platform< / p >
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< / div >
< p >
The leg structure is initialized.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg = struct< span style = "color: #DCDCCC;" > ()< / span > ;
< / pre >
< / div >
< p >
The maximum Stroke of each leg is defined.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.stroke = opts.stroke; < span style = "color: #7F9F7F;" > % [m]< / span >
< / pre >
< / div >
< p >
The stiffness and damping of each leg are defined
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.k_ax = opts.k_ax; < span style = "color: #7F9F7F;" > % Stiffness of each leg [N/m]< / span >
Leg.c_ax = opts.c_ax; < span style = "color: #7F9F7F;" > % Damping of each leg [N/(m/s)]< / span >
< / pre >
< / div >
< p >
The radius of the legs are defined
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.Rtop = < span style = "color: #BFEBBF;" > 10< / span > ; < span style = "color: #7F9F7F;" > % Radius of the cylinder of the top part of the leg[mm]< / span >
Leg.Rbot = < span style = "color: #BFEBBF;" > 12< / span > ; < span style = "color: #7F9F7F;" > % Radius of the cylinder of the bottom part of the leg [mm]< / span >
< / pre >
< / div >
< p >
The density of its material.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.density = opts.density; < span style = "color: #7F9F7F;" > % Density of the material used for the legs [kg/m3]< / span >
< / pre >
< / div >
< p >
Its color.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.color = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 5< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 5< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 5< / span > < span style = "color: #DCDCCC;" > ]< / span > ; < span style = "color: #7F9F7F;" > % Color of the top part of the leg [RGB]< / span >
< / pre >
< / div >
< p >
The radius of spheres representing the ball joints are defined.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.R = < span style = "color: #BFEBBF;" > 1< / span > .< span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #7CB8BB;" > *< / span > Leg.Rbot; < span style = "color: #7F9F7F;" > % Size of the sphere at the extremity of the leg [mm]< / span >
< / pre >
< / div >
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< p >
We estimate the length of the legs.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > legs = stewart.Ab < span style = "color: #7CB8BB;" > -< / span > stewart.Aa;
Leg.lenght = norm< span style = "color: #DCDCCC;" > (< / span > legs< span style = "color: #BFEBBF;" > (< / span > < span style = "color: #BFEBBF;" > 1< / span > ,< span style = "color: #7CB8BB;" > :< / span > < span style = "color: #BFEBBF;" > )< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > /< / span > < span style = "color: #BFEBBF;" > 1< / span > .< span style = "color: #BFEBBF;" > 5< / span > ;
< / pre >
< / div >
< p >
Then the shape of the bottom leg is estimated
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Leg.shape.bot = < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
Leg.Rbot < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "text-decoration: underline;" > ...< / span >
Leg.Rbot Leg.lenght; < span style = "text-decoration: underline;" > ...< / span >
Leg.Rtop Leg.lenght; < span style = "text-decoration: underline;" > ...< / span >
Leg.Rtop < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > Leg.lenght; < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 2< / span > < span style = "color: #7CB8BB;" > *< / span > Leg.lenght< span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
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< p >
The structure is added to the stewart structure
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.Leg = Leg;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org66df86f" class = "outline-3" >
< h3 id = "org66df86f" > < span class = "section-number-3" > 3.6< / span > Ball Joints< / h3 >
< div class = "outline-text-3" id = "text-3-6" >
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< div id = "org250b20b" class = "figure" >
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< p > < img src = "./figs/stewart_ball_joints.png" alt = "stewart_ball_joints.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Schematic of the support for the ball joints< / p >
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< / div >
< p >
< code > SP< / code > is the structure representing the support for the ball joints at the extremity of each leg.
< / p >
< p >
The < code > SP< / code > structure is initialized.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP = struct< span style = "color: #DCDCCC;" > ()< / span > ;
< / pre >
< / div >
< p >
We can define its rotational stiffness and damping. For now, we use perfect joints.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP.k = < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "color: #7F9F7F;" > % [N*m/deg]< / span >
SP.c = < span style = "color: #BFEBBF;" > 0< / span > ; < span style = "color: #7F9F7F;" > % [N*m/deg]< / span >
< / pre >
< / div >
< p >
Its height is defined
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > SP.H = stewart.Aa< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > 1< / span > , < span style = "color: #BFEBBF;" > 3< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #7CB8BB;" > -< / span > BP.H; < span style = "color: #7F9F7F;" > % [mm]< / span >
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< / pre >
< / div >
< p >
Its radius is based on the radius on the sphere at the end of the legs.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP.R = Leg.R; < span style = "color: #7F9F7F;" > % [mm]< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP.section = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > SP.H< span style = "color: #7CB8BB;" > -< / span > SP.R;
< span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > 0< / span > ;
SP.R < span style = "color: #BFEBBF;" > 0< / span > ;
SP.R SP.H< span style = "color: #DCDCCC;" > ]< / span > ;
< / pre >
< / div >
< p >
The density of its material is defined.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP.density = opts.density; % [kg< span style = "color: #7CB8BB;" > /< / span > m< span style = "color: #7CB8BB;" > ^< / span > < span style = "color: #BFEBBF;" > 3< / span > ]
< / pre >
< / div >
< p >
Its color is defined.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > SP.color = < span style = "color: #DCDCCC;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 7< / span > < span style = "color: #DCDCCC;" > ]< / span > ; < span style = "color: #7F9F7F;" > % [RGB]< / span >
< / pre >
< / div >
< p >
The structure is added to the Hexapod structure
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart.SP = SP;
< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf3c4474" class = "outline-2" >
< h2 id = "orgf3c4474" > < span class = "section-number-2" > 4< / span > initializeSample< / h2 >
< div class = "outline-text-2" id = "text-4" >
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< / div >
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< div id = "outline-container-org1ec4152" class = "outline-3" >
< h3 id = "org1ec4152" > < span class = "section-number-3" > 4.1< / span > Function description< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > function< / span > < span style = "color: #DCDCCC;" > []< / span > = < span style = "color: #93E0E3;" > initializeSample< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #DFAF8F;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgcd3268d" class = "outline-3" >
< h3 id = "orgcd3268d" > < span class = "section-number-3" > 4.2< / span > Optional Parameters< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
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< p >
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Default values for opts.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > sample = struct< span style = "color: #DCDCCC;" > (< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #CC9393;" > 'radius'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % radius of the cylinder [mm]< / span >
< span style = "color: #CC9393;" > 'height'< / span > , < span style = "color: #BFEBBF;" > 100< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % height of the cylinder [mm]< / span >
< span style = "color: #CC9393;" > 'mass'< / span > , < span style = "color: #BFEBBF;" > 10< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % mass of the cylinder [kg]< / span >
< span style = "color: #CC9393;" > 'measheight'< / span > , < span style = "color: #BFEBBF;" > 50< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % measurement point z-offset [mm]< / span >
< span style = "color: #CC9393;" > 'offset'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > , < span style = "color: #BFEBBF;" > 0< / span > < span style = "color: #BFEBBF;" > ]< / span > , < span style = "text-decoration: underline;" > ...< / span > < span style = "color: #7F9F7F;" > % offset position of the sample [mm]< / span >
< span style = "color: #CC9393;" > 'color'< / span > , < span style = "color: #BFEBBF;" > [< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 9< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > 0< / span > .< span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > ]< / span > < span style = "text-decoration: underline;" > ...< / span >
< span style = "color: #DCDCCC;" > )< / span > ;
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< / pre >
< / div >
< p >
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Populate opts with input parameters
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > if< / span > exist< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > 'opts_param','var'< / span > < span style = "color: #DCDCCC;" > )< / span >
< span style = "color: #F0DFAF; font-weight: bold;" > for< / span > < span style = "color: #DFAF8F;" > opt< / span > = < span style = "color: #BFEBBF;" > fieldnames< / span > < span style = "color: #DCDCCC;" > (< / span > < span style = "color: #BFEBBF;" > opts_param< / span > < span style = "color: #DCDCCC;" > )< / span > < span style = "color: #BFEBBF;" > '< / span >
sample.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > = opts_param.< span style = "color: #DCDCCC;" > (< / span > opt< span style = "color: #BFEBBF;" > {< / span > < span style = "color: #BFEBBF;" > 1< / span > < span style = "color: #BFEBBF;" > }< / span > < span style = "color: #DCDCCC;" > )< / span > ;
< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
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< span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org29ee9ed" class = "outline-3" >
< h3 id = "org29ee9ed" > < span class = "section-number-3" > 4.3< / span > Save the Sample structure< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > save< span style = "color: #DCDCCC;" > (< / span > < span style = "color: #CC9393;" > './mat/sample.mat', 'sample'< / span > < span style = "color: #DCDCCC;" > )< / span > ;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span style = "color: #F0DFAF; font-weight: bold;" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Thomas Dehaeze< / p >
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< p class = "date" > Created: 2019-03-25 lun. 11:18< / p >
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< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
