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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 = "#org75c6951" > 1. < span class = "todo TODO" > TODO< / span > Configuration Analysis - Stiffness Matrix< / a >
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< ul >
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< li > < a href = "#orga823f72" > 1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#org4261310" > 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / a > < / li >
< li > < a href = "#orgf297eb8" > 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#orgfeaf9c1" > 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / a > < / li >
< li > < a href = "#org24bdf29" > 1.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org2cb2ab0" > 2. < span class = "todo TODO" > TODO< / span > Cubic size analysis< / a > < / li >
< li > < a href = "#orgeec7b47" > 3. Functions< / a >
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< ul >
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< li > < a href = "#org92224ef" > 3.1. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
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< ul >
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< li > < a href = "#org715472d" > Function description< / a > < / li >
< li > < a href = "#orgbab37f8" > Documentation< / a > < / li >
< li > < a href = "#orgddbe42e" > Optional Parameters< / a > < / li >
< li > < a href = "#org66dd074" > Position of the Cube< / a > < / li >
< li > < a href = "#org388f35d" > Compute the pose< / 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 = "#orgb0ae4eb" > 3.1< / 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-org75c6951" class = "outline-2" >
< h2 id = "org75c6951" > < span class = "section-number-2" > 1< / span > < span class = "todo TODO" > TODO< / 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-orga823f72" class = "outline-3" >
< h3 id = "orga823f72" > < 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 = "#org66ade8d" > 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 >
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< div id = "org66ade8d" 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 >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > opts = struct(...
< span class = "org-string" > 'H_tot'< / span > , 100, ...< span class = "org-comment" > % Total height of the Hexapod [mm]< / span >
< span class = "org-string" > 'L'< / span > , 200< span class = "org-type" > /< / span > sqrt(3), ...< span class = "org-comment" > % Size of the Cube [mm]< / span >
< span class = "org-string" > 'H'< / span > , 60, ...< span class = "org-comment" > % Height between base joints and platform joints [mm]< / span >
< span class = "org-string" > 'H0'< / span > , 200< span class = "org-type" > /< / span > 2< span class = "org-type" > -< / span > 60< span class = "org-type" > /< / span > 2 ...< span class = "org-comment" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
< span class = "org-string" > 'Jd_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 50], ...< span class = "org-comment" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span class = "org-string" > 'Jf_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 50] ...< span class = "org-comment" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
);
stewart = computeGeometricalProperties(stewart, opts);
save(< span class = "org-string" > './mat/stewart.mat'< / span > , < span class = "org-string" > 'stewart'< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K = stewart.Jf< span class = "org-type" > '*< / span > stewart.Jf;
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< div id = "outline-container-org4261310" class = "outline-3" >
< h3 id = "org4261310" > < 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 = "#org66ade8d" > 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" > opts = struct(...
< span class = "org-string" > 'H_tot'< / span > , 100, ...< span class = "org-comment" > % Total height of the Hexapod [mm]< / span >
< span class = "org-string" > 'L'< / span > , 200< span class = "org-type" > /< / span > sqrt(3), ...< span class = "org-comment" > % Size of the Cube [mm]< / span >
< span class = "org-string" > 'H'< / span > , 60, ...< span class = "org-comment" > % Height between base joints and platform joints [mm]< / span >
< span class = "org-string" > 'H0'< / span > , 200< span class = "org-type" > /< / span > 2< span class = "org-type" > -< / span > 60< span class = "org-type" > /< / span > 2 ...< span class = "org-comment" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
< span class = "org-string" > 'Jd_pos'< / span > , [0, 0, 0], ...< span class = "org-comment" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span class = "org-string" > 'Jf_pos'< / span > , [0, 0, 0] ...< span class = "org-comment" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
);
stewart = computeGeometricalProperties(stewart, opts);
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< / pre >
< / div >
< div class = "org-src-container" >
2019-12-20 08:55:55 +01:00
< pre class = "src src-matlab" > K = stewart.Jf< span class = "org-type" > '*< / span > stewart.Jf;
2019-03-25 18:12:43 +01:00
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > -2.3e-17< / td >
< td class = "org-right" > 1.5e-18< / td >
< td class = "org-right" > -0.1< / td >
< td class = "org-right" > -1.5e-17< / td >
< / tr >
< tr >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 6.8e-18< / td >
< td class = "org-right" > 0.1< / td >
< td class = "org-right" > -1.6e-18< / td >
< td class = "org-right" > 4.8e-18< / td >
< / tr >
< tr >
< td class = "org-right" > -2.3e-17< / td >
< td class = "org-right" > 6.8e-18< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -5.1e-19< / td >
< td class = "org-right" > -5.5e-18< / td >
< td class = "org-right" > 5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 1.5e-18< / td >
< td class = "org-right" > 0.1< / td >
< td class = "org-right" > -5.1e-19< / td >
< td class = "org-right" > 0.012< / td >
< td class = "org-right" > -3e-19< / td >
< td class = "org-right" > 3.1e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -0.1< / td >
< td class = "org-right" > -1.6e-18< / td >
< td class = "org-right" > -5.5e-18< / td >
< td class = "org-right" > -3e-19< / td >
< td class = "org-right" > 0.012< / td >
< td class = "org-right" > 1.9e-18< / td >
< / tr >
< tr >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 4.8e-18< / td >
< td class = "org-right" > 5.3e-19< / td >
< td class = "org-right" > 3.1e-19< / td >
< td class = "org-right" > 1.9e-18< / td >
< td class = "org-right" > 0.027< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2020-02-06 17:29:55 +01:00
< div id = "outline-container-orgf297eb8" class = "outline-3" >
< h3 id = "orgf297eb8" > < 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 >
2020-02-06 17:29:55 +01:00
Here, the “ center” of the Stewart platform is not at the cube center (figure < a href = "#org4492663" > 2< / a > ).
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The Jacobian is estimated at the cube center.
< / p >
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< div id = "org4492663" class = "figure" >
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< p > < img src = "./figs/3d-cubic-stewart-misaligned.png" alt = "3d-cubic-stewart-misaligned.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Not centered cubic configuration< / p >
< / div >
< 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" >
2020-01-27 17:42:09 +01:00
< pre class = "src src-matlab" > opts = struct(...
< span class = "org-string" > 'H_tot'< / span > , 100, ...< span class = "org-comment" > % Total height of the Hexapod [mm]< / span >
< span class = "org-string" > 'L'< / span > , 220< span class = "org-type" > /< / span > sqrt(3), ...< span class = "org-comment" > % Size of the Cube [mm]< / span >
< span class = "org-string" > 'H'< / span > , 60, ...< span class = "org-comment" > % Height between base joints and platform joints [mm]< / span >
< span class = "org-string" > 'H0'< / span > , 75 ...< span class = "org-comment" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
< span class = "org-string" > 'Jd_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 65], ...< span class = "org-comment" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span class = "org-string" > 'Jf_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 65] ...< span class = "org-comment" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
);
stewart = computeGeometricalProperties(stewart, opts);
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< / pre >
< / div >
< div class = "org-src-container" >
2019-12-20 08:55:55 +01:00
< pre class = "src src-matlab" > K = stewart.Jf< span class = "org-type" > '*< / span > stewart.Jf;
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 3.3e-18< / td >
< td class = "org-right" > 0.04< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > -0.04< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -8.9e-18< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 3.3e-18< / td >
< td class = "org-right" > -0.04< / td >
< td class = "org-right" > -8.9e-18< / td >
< td class = "org-right" > 0.0089< / td >
< td class = "org-right" > -9.3e-20< / td >
< td class = "org-right" > 9.8e-20< / td >
< / tr >
< tr >
< td class = "org-right" > 0.04< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -9.3e-20< / td >
< td class = "org-right" > 0.0089< / td >
< td class = "org-right" > -2.4e-18< / td >
< / tr >
< tr >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< td class = "org-right" > 9.8e-20< / td >
< td class = "org-right" > -2.4e-18< / td >
< td class = "org-right" > 0.032< / td >
< / 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 >
2020-02-06 17:29:55 +01:00
< div id = "outline-container-orgfeaf9c1" class = "outline-3" >
< h3 id = "orgfeaf9c1" > < 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" >
2020-01-27 17:42:09 +01:00
< pre class = "src src-matlab" > opts = struct(...
< span class = "org-string" > 'H_tot'< / span > , 100, ...< span class = "org-comment" > % Total height of the Hexapod [mm]< / span >
< span class = "org-string" > 'L'< / span > , 220< span class = "org-type" > /< / span > sqrt(3), ...< span class = "org-comment" > % Size of the Cube [mm]< / span >
< span class = "org-string" > 'H'< / span > , 60, ...< span class = "org-comment" > % Height between base joints and platform joints [mm]< / span >
< span class = "org-string" > 'H0'< / span > , 75 ...< span class = "org-comment" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
< span class = "org-string" > 'Jd_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 60], ...< span class = "org-comment" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span class = "org-string" > 'Jf_pos'< / span > , [0, 0, < span class = "org-type" > -< / span > 60] ...< span class = "org-comment" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
);
stewart = computeGeometricalProperties(stewart, opts);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K = stewart.Jf< span class = "org-type" > '*< / span > stewart.Jf;
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > -5.7e-19< / td >
< td class = "org-right" > 0.03< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -1.8e-17< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > -0.03< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 2.6e-17< / td >
< td class = "org-right" > 1.9e-16< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< / tr >
< tr >
< td class = "org-right" > -5.7e-19< / td >
< td class = "org-right" > -0.03< / td >
< td class = "org-right" > -1.5e-17< / td >
< td class = "org-right" > 0.0085< / td >
< td class = "org-right" > 4.9e-20< / td >
< td class = "org-right" > 1.7e-19< / td >
< / tr >
< tr >
< td class = "org-right" > 0.03< / td >
< td class = "org-right" > 2.2e-19< / td >
< td class = "org-right" > 6.5e-19< / td >
< td class = "org-right" > 4.9e-20< / td >
< td class = "org-right" > 0.0085< / td >
< td class = "org-right" > -1.1e-18< / td >
< / tr >
< tr >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > -5.8e-19< / td >
< td class = "org-right" > 1.7e-19< / td >
< td class = "org-right" > -1.1e-18< / td >
< td class = "org-right" > 0.032< / td >
< / 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 >
2020-02-06 17:29:55 +01:00
< div id = "outline-container-org24bdf29" class = "outline-3" >
< h3 id = "org24bdf29" > < 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" >
< 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 Stewart platform and the cube are centered < b > and< / b > the Jacobian is estimated at the cube center< / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
2020-02-06 17:29:55 +01:00
< div id = "outline-container-org2cb2ab0" class = "outline-2" >
< h2 id = "org2cb2ab0" > < span class = "section-number-2" > 2< / span > < span class = "todo TODO" > TODO< / span > Cubic size analysis< / h2 >
< div class = "outline-text-2" id = "text-2" >
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< p >
We here study the effect of the size of the cube used for the Stewart configuration.
< / p >
< p >
We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform.
< / p >
< p >
We only vary the size of the cube.
< / p >
< div class = "org-src-container" >
2020-01-27 17:42:09 +01:00
< pre class = "src src-matlab" > H_cubes = 250< span class = "org-type" > :< / span > 20< span class = "org-type" > :< / span > 350;
stewarts = {zeros(length(H_cubes), 1)};
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(H_cubes)< / span >
H_cube = H_cubes(< span class = "org-constant" > i< / span > );
H_tot = 100;
H = 80;
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opts = struct(...
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< span class = "org-string" > 'H_tot'< / span > , H_tot, ...< span class = "org-comment" > % Total height of the Hexapod [mm]< / span >
2020-01-27 17:42:09 +01:00
< span class = "org-string" > 'L'< / span > , H_cube< span class = "org-type" > /< / span > sqrt(3), ...< span class = "org-comment" > % Size of the Cube [mm]< / span >
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< span class = "org-string" > 'H'< / span > , H, ...< span class = "org-comment" > % Height between base joints and platform joints [mm]< / span >
2020-01-27 17:42:09 +01:00
< span class = "org-string" > 'H0'< / span > , H_cube< span class = "org-type" > /< / span > 2< span class = "org-type" > -< / span > H< span class = "org-type" > /< / span > 2 ...< span class = "org-comment" > % Height between the corner of the cube and the plane containing the base joints [mm]< / span >
);
stewart = initializeCubicConfiguration(opts);
opts = struct(...
< span class = "org-string" > 'Jd_pos'< / span > , [0, 0, H_cube< span class = "org-type" > /< / span > 2< span class = "org-type" > -< / span > opts.H0< span class = "org-type" > -< / span > opts.H_tot], ...< span class = "org-comment" > % Position of the Jacobian for displacement estimation from the top of the mobile platform [mm]< / span >
< span class = "org-string" > 'Jf_pos'< / span > , [0, 0, H_cube< span class = "org-type" > /< / span > 2< span class = "org-type" > -< / span > opts.H0< span class = "org-type" > -< / span > opts.H_tot] ...< span class = "org-comment" > % Position of the Jacobian for force location from the top of the mobile platform [mm]< / span >
);
stewart = computeGeometricalProperties(stewart, opts);
stewarts(< span class = "org-constant" > i< / span > ) = {stewart};
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
The Stiffness matrix is computed for all generated Stewart platforms.
< / p >
< div class = "org-src-container" >
2020-01-27 17:42:09 +01:00
< pre class = "src src-matlab" > Ks = zeros(6, 6, length(H_cube));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(H_cubes)< / span >
Ks(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = stewarts{< span class = "org-constant" > i< / span > }.Jd< span class = "org-type" > '*< / span > stewarts{< span class = "org-constant" > i< / span > }.Jd;
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< span class = "org-keyword" > end< / span >
2019-03-25 18:12:43 +01:00
< / pre >
< / div >
< p >
The only elements of \(K\) that vary are \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\).
< / p >
< p >
Finally, we plot \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\)
< / p >
< div class = "org-src-container" >
2019-08-26 11:58:44 +02:00
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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hold on;
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plot(H_cubes, squeeze(Ks(4, 4, < span class = "org-type" > :< / span > )), < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$k_{\theta_x}$'< / span > );
plot(H_cubes, squeeze(Ks(6, 6, < span class = "org-type" > :< / span > )), < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > '$k_{\theta_z}$'< / span > );
2019-03-25 18:12:43 +01:00
hold off;
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legend(< span class = "org-string" > 'location'< / span > , < span class = "org-string" > 'northwest'< / span > );
xlabel(< span class = "org-string" > 'Cube Size [mm]'< / span > ); ylabel(< span class = "org-string" > 'Rotational stiffnes [normalized]'< / span > );
2019-03-25 18:12:43 +01:00
< / pre >
< / div >
2020-02-06 17:29:55 +01:00
< div id = "org009489e" class = "figure" >
2019-03-25 18:12:43 +01:00
< p > < img src = "figs/stiffness_cube_size.png" alt = "stiffness_cube_size.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube< / p >
< / div >
< p >
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
< / p >
< div class = "important" >
< p >
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
In that case, the legs will the further separated. Size of the cube is then limited by allowed space.
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< h2 id = "orgeec7b47" > < span class = "section-number-2" > 3< / span > Functions< / h2 >
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< h3 id = "org92224ef" > < span class = "section-number-3" > 3.1< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h3 >
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This Matlab function is accessible < a href = "src/generateCubicConfiguration.m" > here< / a > .
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< h4 id = "org715472d" > Function description< / h4 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [stewart]< / span > = < span class = "org-function-name" > generateCubicConfiguration< / span > (< span class = "org-variable-name" > stewart< / span > , < span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateCubicConfiguration - Generate a Cubic Configuration< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [stewart] = generateCubicConfiguration(stewart, args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - stewart - A structure with the following fields< / span >
< span class = "org-comment" > % - H [1x1] - Total height of the platform [m]< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
< span class = "org-comment" > % - Hc [1x1] - Height of the "useful" part of the cube [m]< / span >
< span class = "org-comment" > % - FOc [1x1] - Height of the center of the cube with respect to {F} [m]< / span >
< span class = "org-comment" > % - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]< / span >
< span class = "org-comment" > % - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - stewart - updated Stewart structure with the added fields:< / span >
< span class = "org-comment" > % - Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}< / span >
< span class = "org-comment" > % - Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}< / span >
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< h4 id = "orgbab37f8" > Documentation< / h4 >
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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
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< p > < span class = "figure-number" > Figure 4: < / span > Cubic Configuration< / p >
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< h4 id = "orgddbe42e" > 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
args.FHa (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
args.MHb (1,1) double {mustBeNumeric, mustBePositive} = 15e< span class = "org-type" > -< / span > 3
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< h4 id = "org66dd074" > Position of the Cube< / h4 >
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We define the useful points of the cube with respect to the Cube’ s center.
\({}^{C}C\) are the 6 vertices of the cubes expressed in a frame {C} which is
located at the center of the cube and aligned with {F} and {M}.
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< pre class = "src src-matlab" > sx = [ 2; < span class = "org-type" > -< / span > 1; < span class = "org-type" > -< / span > 1];
sy = [ 0; 1; < span class = "org-type" > -< / span > 1];
sz = [ 1; 1; 1];
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 = "org388f35d" > Compute the pose< / h4 >
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We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
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< pre class = "src src-matlab" > CSi = (CCm < span class = "org-type" > -< / span > CCf)< span class = "org-type" > ./< / span > vecnorm(CCm < span class = "org-type" > -< / span > CCf);
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We now which to compute the position of the joints \(a_{i}\) and \(b_{i}\).
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< pre class = "src src-matlab" > stewart.Fa = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc] < span class = "org-type" > +< / span > ((args.FHa< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
stewart.Mb = CCf < span class = "org-type" > +< / span > [0; 0; args.FOc< span class = "org-type" > -< / span > stewart.H] < span class = "org-type" > +< / span > ((stewart.H< span class = "org-type" > -< / span > args.MHb< span class = "org-type" > -< / span > (args.FOc< span class = "org-type" > -< / span > args.Hc< span class = "org-type" > /< / span > 2))< span class = "org-type" > ./< / span > CSi(3,< span class = "org-type" > :< / span > ))< span class = "org-type" > .*< / span > CSi;
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< 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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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-02-06 jeu. 17:29< / p >
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