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< h1 class = "title" > Cubic configuration for the Stewart Platform< / 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 = "#org8c6677e" > 1. Configuration Analysis - Stiffness Matrix< / a >
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< ul >
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< li > < a href = "#orgf6f7ad2" > 1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#orga88e79a" > 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / a > < / li >
< li > < a href = "#orge02ec88" > 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#org43fd7e4" > 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / a > < / li >
< li > < a href = "#orgd35acc0" > 1.5. Conclusion< / a > < / li >
< li > < a href = "#org8afa645" > 1.6. Having Cube’ s center above the top platform< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org3044455" > 2. Functions< / a >
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< ul >
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< li > < a href = "#org56504f1" > 2.1. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
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< ul >
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< li > < a href = "#orga5a9ba8" > Function description< / a > < / li >
< li > < a href = "#org3253792" > Documentation< / a > < / li >
< li > < a href = "#org154b5fb" > Optional Parameters< / a > < / li >
< li > < a href = "#orgbb480a6" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#org771c630" > Position of the Cube< / a > < / li >
< li > < a href = "#org3a2f468" > Compute the pose< / a > < / li >
< li > < a href = "#org8c1af4f" > Populate the < code > stewart< / code > structure< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
< p >
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The discovery of the Cubic configuration is done in < a class = 'org-ref-reference' href = "#geng94_six_degree_of_freed_activ" > geng94_six_degree_of_freed_activ< / a > .
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Further analysis is conducted in
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< / p >
< p >
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The specificity of the Cubic configuration is that each actuator is orthogonal with the others.
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< / p >
< p >
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The cubic (or orthogonal) configuration of the Stewart platform is now widely used (< a class = 'org-ref-reference' href = "#preumont07_six_axis_singl_stage_activ" > preumont07_six_axis_singl_stage_activ< / a > ,< a class = 'org-ref-reference' href = "#jafari03_orthog_gough_stewar_platf_microm" > jafari03_orthog_gough_stewar_platf_microm< / a > ).
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< / p >
< p >
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According to < a class = 'org-ref-reference' href = "#preumont07_six_axis_singl_stage_activ" > preumont07_six_axis_singl_stage_activ< / a > , the cubic configuration provides a uniform stiffness in all directions and < b > minimizes the crosscoupling< / b > from actuator to sensor of different legs (being orthogonal to each other).
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< / p >
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< p >
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To generate and study the Cubic configuration, < code > generateCubicConfiguration< / code > is used (description in section < a href = "#orga8311d3" > 2.1< / a > ).
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The goal is to study the benefits of using a cubic configuration:
< / p >
< ul class = "org-ul" >
< li > Equal stiffness in all the degrees of freedom?< / li >
< li > No coupling between the actuators?< / li >
< li > Is the center of the cube an important point?< / li >
< / ul >
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< div id = "outline-container-org8c6677e" class = "outline-2" >
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< h2 id = "org8c6677e" > < span class = "section-number-2" > 1< / span > Configuration Analysis - Stiffness Matrix< / h2 >
< div class = "outline-text-2" id = "text-1" >
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< / div >
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< div id = "outline-container-orgf6f7ad2" class = "outline-3" >
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< h3 id = "orgf6f7ad2" > < span class = "section-number-3" > 1.1< / span > Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / h3 >
< div class = "outline-text-3" id = "text-1-1" >
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< p >
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We create a cubic Stewart platform (figure < a href = "#org9454f54" > 1< / a > ) in such a way that the center of the cube (black dot) is located at the center of the Stewart platform (blue dot).
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The Jacobian matrix is estimated at the location of the center of the cube.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , < span class = "org-type" > -< / span > 50e< span class = "org-type" > -< / span > 3);
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stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FOc'< / span > , 50e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 175e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Mpr'< / span > , 150e< span class = "org-type" > -< / span > 3);
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< / pre >
< / div >
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< div id = "org9454f54" class = "figure" >
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< p > < img src = "./figs/3d-cubic-stewart-aligned.png" alt = "3d-cubic-stewart-aligned.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Centered cubic configuration< / p >
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< / div >
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< div id = "orgaba20c8" class = "figure" >
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< p > < img src = "figs/cubic_conf_centered_J_center.png" alt = "cubic_conf_centered_J_center.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (< a href = "./figs/cubic_conf_centered_J_center.png" > png< / a > , < a href = "./figs/cubic_conf_centered_J_center.pdf" > pdf< / a > )< / p >
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< h3 id = "orga88e79a" > < span class = "section-number-3" > 1.2< / span > Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< p >
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We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure < a href = "#org9454f54" > 1< / a > ).
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The Jacobian matrix is not estimated at the location of the center of the cube.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 0);
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stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FOc'< / span > , 50e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 175e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Mpr'< / span > , 150e< span class = "org-type" > -< / span > 3);
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< / pre >
< / div >
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< div id = "org47f8142" class = "figure" >
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< p > < img src = "figs/cubic_conf_centered_J_not_center.png" alt = "cubic_conf_centered_J_not_center.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (< a href = "./figs/cubic_conf_centered_J_not_center.png" > png< / a > , < a href = "./figs/cubic_conf_centered_J_not_center.pdf" > pdf< / a > )< / p >
< / div >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.5e-16< / td >
< td class = "org-right" > 1.4e-17< / td >
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< td class = "org-right" > -0.1< / td >
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< td class = "org-right" > 0< / td >
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< tr >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.1< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > 0< / td >
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< / tr >
< tr >
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< td class = "org-right" > -2.5e-16< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > 3.4e-18< / td >
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 0< / td >
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< tr >
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< td class = "org-right" > 1.4e-17< / td >
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< td class = "org-right" > 0.1< / td >
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< td class = "org-right" > 3.4e-18< / td >
< td class = "org-right" > 0.02< / td >
< td class = "org-right" > 1.1e-20< / td >
< td class = "org-right" > 3.4e-18< / td >
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< tr >
< td class = "org-right" > -0.1< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 1.4e-19< / td >
< td class = "org-right" > 0.02< / td >
< td class = "org-right" > -1.7e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > -3.3e-18< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 3.6e-18< / td >
< td class = "org-right" > -1.7e-18< / td >
< td class = "org-right" > 0.06< / td >
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< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-orge02ec88" class = "outline-3" >
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< h3 id = "orge02ec88" > < span class = "section-number-3" > 1.3< / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
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< p >
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Here, the “ center” of the Stewart platform is not at the cube center (figure < a href = "#org97b319c" > 4< / a > ).
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The Jacobian is estimated at the cube center.
< / p >
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< div id = "org97b319c" class = "figure" >
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< p > < img src = "./figs/3d-cubic-stewart-misaligned.png" alt = "3d-cubic-stewart-misaligned.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Not centered cubic configuration< / p >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 80e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , < span class = "org-type" > -< / span > 40e< span class = "org-type" > -< / span > 3);
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stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FOc'< / span > , 50e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 175e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Mpr'< / span > , 150e< span class = "org-type" > -< / span > 3);
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< / pre >
< / div >
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< div id = "org0235d3a" class = "figure" >
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< p > < img src = "figs/cubic_conf_not_centered_J_center.png" alt = "cubic_conf_not_centered_J_center.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (< a href = "./figs/cubic_conf_not_centered_J_center.png" > png< / a > , < a href = "./figs/cubic_conf_not_centered_J_center.pdf" > pdf< / a > )< / p >
< / div >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -1.7e-16< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.02< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -0.02< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2.8e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > -1.7e-16< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > 1.2e-19< / td >
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 1.4e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -0.02< / td >
< td class = "org-right" > 1.2e-19< / td >
< td class = "org-right" > 0.015< / td >
< td class = "org-right" > -4.3e-19< / td >
< td class = "org-right" > 1.7e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 0.02< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -7.3e-18< / td >
< td class = "org-right" > -3.3e-21< / td >
< td class = "org-right" > 0.015< / td >
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< td class = "org-right" > 0< / td >
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< / tr >
< tr >
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< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > 2.5e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2e-18< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > 0.06< / td >
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< / tr >
< / tbody >
< / table >
< p >
We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
< / p >
< / div >
< / div >
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< h3 id = "org43fd7e4" > < span class = "section-number-3" > 1.4< / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / h3 >
< div class = "outline-text-3" id = "text-1-4" >
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< p >
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Here, the “ center” of the Stewart platform is not at the cube center.
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The Jacobian is estimated at the center of the Stewart platform.
< / p >
< p >
The center of the cube is at \(z = 110\).
The Stewart platform is from \(z = H_0 = 75\) to \(z = H_0 + H_{tot} = 175\).
The center height of the Stewart platform is then at \(z = \frac{175-75}{2} = 50\).
The center of the cube from the top platform is at \(z = 110 - 175 = -65\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 80e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , < span class = "org-type" > -< / span > 40e< span class = "org-type" > -< / span > 3);
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stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FOc'< / span > , 50e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 175e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Mpr'< / span > , 150e< span class = "org-type" > -< / span > 3);
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< / pre >
< / div >
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< div id = "orgbe766b3" class = "figure" >
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< p > < img src = "figs/cubic_conf_not_centered_J_stewart_center.png" alt = "cubic_conf_not_centered_J_stewart_center.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (< a href = "./figs/cubic_conf_not_centered_J_stewart_center.png" > png< / a > , < a href = "./figs/cubic_conf_not_centered_J_stewart_center.pdf" > pdf< / a > )< / p >
< / div >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.02< / td >
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< td class = "org-right" > 0< / td >
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< tr >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > -0.02< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > 2.8e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > -1.7e-16< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2< / td >
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< td class = "org-right" > 1.2e-19< / td >
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< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 1.4e-17< / td >
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< tr >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -0.02< / td >
< td class = "org-right" > 1.2e-19< / td >
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< td class = "org-right" > 0.015< / td >
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< td class = "org-right" > -4.3e-19< / td >
< td class = "org-right" > 1.7e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 0.02< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -7.3e-18< / td >
< td class = "org-right" > -3.3e-21< / td >
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< td class = "org-right" > 0.015< / td >
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< td class = "org-right" > 0< / td >
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< / tr >
< tr >
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< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > 2.5e-17< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2e-18< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.06< / td >
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< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< h3 id = "orgd35acc0" > < span class = "section-number-3" > 1.5< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
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< div class = "important" >
< ul class = "org-ul" >
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< li > The cubic configuration permits to have \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\)< / li >
< li > The stiffness matrix \(K\) is diagonal for the cubic configuration if the Jacobian is estimated at the cube center.< / li >
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< / ul >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8afa645" class = "outline-3" >
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< h3 id = "org8afa645" > < span class = "section-number-3" > 1.6< / span > Having Cube’ s center above the top platform< / h3 >
< div class = "outline-text-3" id = "text-1-6" >
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< p >
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Let’ s say we want to have a decouple dynamics above the top platform.
Thus, we want the cube’ s center to be located above the top center.
This is possible, to do so:
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< / p >
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< ul class = "org-ul" >
< li > The position of the center of the cube should be positioned at A< / li >
< li > The Height of the “ useful” part of the cube should be at least equal to two times the distance from F to A.
It is possible to have small cube, but then to configuration is a little bit strange.< / li >
< / ul >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 100e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 50e< span class = "org-type" > -< / span > 3);
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FOc = stewart.H < span class = "org-type" > +< / span > stewart.MO_B(3);
Hc = 2< span class = "org-type" > *< / span > (stewart.H < span class = "org-type" > +< / span > stewart.MO_B(3));
stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , Hc, < span class = "org-string" > 'FOc'< / span > , FOc, < span class = "org-string" > 'FHa'< / span > , 10e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MHb'< / span > , 10e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'Ki'< / span > , ones(6,1));
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'disable'< / span > , < span class = "org-constant" > true< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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< / pre >
< / div >
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< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-right" / >
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< col class = "org-right" / >
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< col class = "org-right" / >
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< col class = "org-right" / >
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< col class = "org-right" / >
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< col class = "org-right" / >
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< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > -3.2e-16< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 3.1e-16< / td >
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< td class = "org-right" > 0< / td >
< / tr >
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< tr >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > -1.2e-16< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0< / td >
< / tr >
< tr >
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< td class = "org-right" > -3.2e-16< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
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< td class = "org-right" > 5e-18< / td >
< td class = "org-right" > -5.6e-17< / td >
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< td class = "org-right" > 0< / td >
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< tr >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > -1.2e-16< / td >
< td class = "org-right" > 5e-18< / td >
< td class = "org-right" > 0.14< / td >
< td class = "org-right" > 3.5e-18< / td >
< td class = "org-right" > 1.4e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > 3e-16< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -5.4e-17< / td >
< td class = "org-right" > 2.1e-19< / td >
< td class = "org-right" > 0.14< / td >
< td class = "org-right" > -6.9e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 7.4e-19< / td >
< td class = "org-right" > -2.6e-17< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 1.3e-17< / td >
< td class = "org-right" > -6.9e-18< / td >
< td class = "org-right" > 0.54< / td >
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< p >
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We obtain \(k_x = k_y = k_z\) and \(k_{\theta_x} = k_{\theta_y}\), but the Stiffness matrix is not diagonal.
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< h2 id = "org3044455" > < span class = "section-number-2" > 2< / span > Functions< / h2 >
< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org28ba607" > < / a >
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< h3 id = "org56504f1" > < span class = "section-number-3" > 2.1< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
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< a id = "orga8311d3" > < / a >
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< / p >
< p >
This Matlab function is accessible < a href = "src/generateCubicConfiguration.m" > here< / a > .
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< h4 id = "orga5a9ba8" > Function description< / h4 >
< div class = "outline-text-4" id = "text-orga5a9ba8" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > generateCubicConfiguration< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateCubicConfiguration - Generate a Cubic Configuration< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = generateCubicConfiguration(stewart, args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following fields< / span >
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< span class = "org-comment" > % - geometry.H [1x1] - Total height of the platform [m]< / span >
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< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - Hc [1x1] - Height of the "useful" part of the cube [m]< / span >
< span class = "org-comment" > % - FOc [1x1] - Height of the center of the cube with respect to {F} [m]< / span >
< span class = "org-comment" > % - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]< / span >
< span class = "org-comment" > % - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - updated Stewart structure with the added fields:< / span >
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< span class = "org-comment" > % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< h4 id = "org3253792" > Documentation< / h4 >
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< div id = "org8a7f3d8" class = "figure" >
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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Cubic Configuration< / p >
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< h4 id = "org154b5fb" > Optional Parameters< / h4 >
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< pre class = "src src-matlab" > arguments
stewart
args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e< span class = "org-type" > -< / span > 3
args.FOc (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
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args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
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< span class = "org-keyword" > end< / span >
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< h4 id = "orgbb480a6" > Check the < code > stewart< / code > structure elements< / h4 >
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< pre class = "src src-matlab" > assert(isfield(stewart.geometry, < span class = "org-string" > 'H'< / span > ), < span class = "org-string" > 'stewart.geometry should have attribute H'< / span > )
H = stewart.geometry.H;
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< h4 id = "org771c630" > Position of the Cube< / h4 >
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< p >
We define the useful points of the cube with respect to the Cube’ s center.
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
located at the center of the cube and aligned with {F} and {M}.
< / p >
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< pre class = "src src-matlab" > sx = [ 2; < span class = "org-type" > -< / span > 1; < span class = "org-type" > -< / span > 1];
sy = [ 0; 1; < span class = "org-type" > -< / span > 1];
sz = [ 1; 1; 1];
R = [sx, sy, sz]< span class = "org-type" > ./< / span > vecnorm([sx, sy, sz]);
L = args.Hc< span class = "org-type" > *< / span > sqrt(3);
Cc = R< span class = "org-type" > '*< / span > [[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] < span class = "org-type" > -< / span > [0;0;1.5< span class = "org-type" > *< / span > args.Hc];
CCf = [Cc(< span class = "org-type" > :< / span > ,1), Cc(< span class = "org-type" > :< / span > ,3), Cc(< span class = "org-type" > :< / span > ,3), Cc(< span class = "org-type" > :< / span > ,5), Cc(< span class = "org-type" > :< / span > ,5), Cc(< span class = "org-type" > :< / span > ,1)]; < span class = "org-comment" > % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg< / span >
CCm = [Cc(< span class = "org-type" > :< / span > ,2), Cc(< span class = "org-type" > :< / span > ,2), Cc(< span class = "org-type" > :< / span > ,4), Cc(< span class = "org-type" > :< / span > ,4), Cc(< span class = "org-type" > :< / span > ,6), Cc(< span class = "org-type" > :< / span > ,6)]; < span class = "org-comment" > % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg< / span >
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< h4 id = "org3a2f468" > Compute the pose< / h4 >
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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 >
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< pre class = "src src-matlab" > CSi = (CCm < span class = "org-type" > -< / span > CCf)< span class = "org-type" > ./< / span > vecnorm(CCm < span class = "org-type" > -< / span > CCf);
< / pre >
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< p >
We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
< / p >
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< pre class = "src src-matlab" > Fa = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc] < span class = "org-type" > +< / span > ((args.FHa< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
Mb = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc< span class = "org-type" > -< / span > H] < span class = "org-type" > +< / span > ((H< span class = "org-type" > -< / span > args.MHb< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
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< h4 id = "org8c1af4f" > Populate the < code > stewart< / code > structure< / h4 >
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< pre class = "src src-matlab" > stewart.platform_F.Fa = Fa;
stewart.platform_M.Mb = Mb;
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< p >
< h1 class = 'org-ref-bib-h1' > Bibliography< / h1 >
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< ul class = 'org-ref-bib' > < li > < a id = "geng94_six_degree_of_freed_activ" > [geng94_six_degree_of_freed_activ]< / a > < a name = "geng94_six_degree_of_freed_activ" > < / a > Geng & Haynes, Six Degree-Of-Freedom Active Vibration Control Using the Stewart Platforms, < i > IEEE Transactions on Control Systems Technology< / i > , < b > 2(1)< / b > , 45-53 (1994). < a href = "https://doi.org/10.1109/87.273110" > link< / a > . < a href = "http://dx.doi.org/10.1109/87.273110" > doi< / a > .< / li >
< li > < a id = "preumont07_six_axis_singl_stage_activ" > [preumont07_six_axis_singl_stage_activ]< / a > < a name = "preumont07_six_axis_singl_stage_activ" > < / a > Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, < i > Journal of Sound and Vibration< / i > , < b > 300(3-5)< / b > , 644-661 (2007). < a href = "https://doi.org/10.1016/j.jsv.2006.07.050" > link< / a > . < a href = "http://dx.doi.org/10.1016/j.jsv.2006.07.050" > doi< / a > .< / li >
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< li > < a id = "jafari03_orthog_gough_stewar_platf_microm" > [jafari03_orthog_gough_stewar_platf_microm]< / a > < a name = "jafari03_orthog_gough_stewar_platf_microm" > < / a > Jafari & McInroy, Orthogonal Gough-Stewart Platforms for Micromanipulation, < i > IEEE Transactions on Robotics and Automation< / i > , < b > 19(4)< / b > , 595-603 (2003). < a href = "https://doi.org/10.1109/tra.2003.814506" > link< / a > . < a href = "http://dx.doi.org/10.1109/tra.2003.814506" > doi< / a > .< / li >
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< div id = "postamble" class = "status" >
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-02-11 mar. 15:50< / p >
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