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< a accesskey = "h" href = "./index.html" > UP < / a >
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< / div > < div id = "content" >
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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 >
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< li > < a href = "#org95af62e" > 1.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd70418b" > 2. Configuration with the Cube’ s center above the mobile platform< / a >
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< ul >
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< li > < a href = "#org8afa645" > 2.1. Having Cube’ s center above the top platform< / a > < / li >
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< li > < a href = "#org7673551" > 2.2. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#orgcc4ecce" > 3. Cubic size analysis< / a >
< ul >
< li > < a href = "#org0029d8c" > 3.1. Analysis< / a > < / li >
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< li > < a href = "#orgcad29fd" > 3.2. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org5abef15" > 4. Dynamic Coupling< / a >
< ul >
< li > < a href = "#org0d67b92" > 4.1. Cube’ s center at the Center of Mass of the Payload< / a > < / li >
< li > < a href = "#org876e05f" > 4.2. Dynamic decoupling between the actuators and sensors< / a > < / li >
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< li > < a href = "#org213d9b9" > 4.3. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org3044455" > 5. Functions< / a >
< ul >
< li > < a href = "#org56504f1" > 5.1. < 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 Cubic configuration for the Stewart platform was first proposed in < a class = 'org-ref-reference' href = "#geng94_six_degree_of_freed_activ" > geng94_six_degree_of_freed_activ< / a > .
This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube.
This configuration 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 offers the following advantages:
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< / p >
< blockquote >
< p >
This topology provides a uniform control capability and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other).
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< / p >
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< / blockquote >
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< p >
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In this document, the cubic architecture is analyzed:
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< / p >
< ul class = "org-ul" >
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< li > In section < a href = "#orgda0ee50" > 1< / a > , we study the link between the Stiffness matrix and the cubic architecture and we find what are the conditions to obtain a diagonal stiffness matrix< / li >
< li > In section < a href = "#orgb73265d" > 2< / a > , we study cubic configurations where the cube’ s center is located above the mobile platform< / li >
< li > In section < a href = "#org348ec7d" > 3< / a > , we study the effect of the cube’ s size on the Stewart platform properties< / li >
< li > In section < a href = "#orgc379ec8" > 4< / a > , we study the dynamic coupling of the cubic configuration< / li >
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< / ul >
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< p >
To generate and study the Stewart platform with a Cubic configuration, the Matlab function < code > generateCubicConfiguration< / code > is used (described < a href = "#orga8311d3" > here< / a > ).
< / p >
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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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< p >
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< a id = "orgda0ee50" > < / a >
< / p >
< p >
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First, we have to understand what is the physical meaning of the Stiffness matrix \(\bm{K}\).
< / p >
< p >
The Stiffness matrix links forces \(\bm{f}\) and torques \(\bm{n}\) applied on the mobile platform at \(\{B\}\) to the displacement \(\Delta\bm{\mathcal{X}}\) of the mobile platform represented by \(\{B\}\) with respect to \(\{A\}\):
\[ \bm{\mathcal{F}} = \bm{K} \Delta\bm{\mathcal{X}} \]
< / p >
< p >
with:
< / p >
< ul class = "org-ul" >
< li > \(\bm{\mathcal{F}} = [\bm{f}\ \bm{n}]^{T}\)< / li >
< li > \(\Delta\bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_{x}, \delta \theta_{y}, \delta \theta_{z}]^{T}\)< / li >
< / ul >
< p >
If the stiffness matrix is inversible, its inverse is the compliance matrix: \(\bm{C} = \bm{K}^{-1\) and:
\[ \Delta \bm{\mathcal{X}} = C \bm{\mathcal{F}} \]
< / p >
< p >
Thus, if the stiffness matrix is diagonal, the compliance matrix is also diagonal and a force (resp. torque) \(\bm{\mathcal{F}}_i\) applied on the mobile platform at \(\{B\}\) will induce a pure translation (resp. rotation) of the mobile platform represented by \(\{B\}\) with respect to \(\{A\}\).
< / p >
< p >
One has to note that this is only valid in a static way.
< / p >
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< p >
We here study what makes the Stiffness matrix diagonal when using a cubic configuration.
< / p >
< / 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 = "#orgaba20c8" > 1< / a > ) in such a way that the center of the cube (black star) 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 class = "org-src-container" >
< pre class = "src src-matlab" > H = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > H< span class = "org-type" > /< / span > 2; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
Hc = H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , H, < span class = "org-string" > 'MO_B'< / span > , MO_B);
stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , Hc, < span class = "org-string" > 'FOc'< / span > , FOc, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
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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 = "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 >
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< p > < span class = "figure-number" > Figure 1: < / 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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< / div >
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< table id = "org4baf591" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Stiffness Matrix< / caption >
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< 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 >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.5e-16< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2.1e-17< / td >
< td class = "org-right" > 0< / 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" > 2< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -7.8e-19< / td >
< 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" > -2.4e-18< / td >
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 0< / td >
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< / tr >
< tr >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -7.8e-19< / td >
< td class = "org-right" > -2.4e-18< / 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" > 1.8e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -1.1e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 0.015< / td >
< 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" > -3.3e-18< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 1.7e-18< / td >
< td class = "org-right" > 0< / 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-orga88e79a" class = "outline-3" >
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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 = "#org47f8142" > 2< / a > ).
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The Jacobian matrix is not estimated at the location of the center of the cube.
< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > H = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = 20e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
Hc = H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H< span class = "org-type" > /< / span > 2; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , H, < span class = "org-string" > 'MO_B'< / span > , MO_B);
stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , Hc, < span class = "org-string" > 'FOc'< / span > , FOc, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
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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 >
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< p > < span class = "figure-number" > Figure 2: < / 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 >
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< / div >
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< table id = "org5cc2020" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 2:< / span > Stiffness Matrix< / caption >
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< 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 >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.5e-16< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > -0.14< / 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" > 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" > 0.14< / 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" > -5.3e-19< / td >
< td class = "org-right" > 0< / 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" > 0< / td >
< td class = "org-right" > 0.14< / td >
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > 0.025< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 8.7e-19< / td >
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< / tr >
< tr >
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< td class = "org-right" > -0.14< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2.6e-18< / td >
< td class = "org-right" > 1.6e-19< / td >
< td class = "org-right" > 0.025< / 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" > -3.3e-18< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 8.9e-19< / 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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< 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 = "#org0235d3a" > 3< / a > ).
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The Jacobian is estimated at the cube center.
< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > H = 80e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > 30e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
Hc = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , H, < span class = "org-string" > 'MO_B'< / span > , MO_B);
stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , Hc, < span class = "org-string" > 'FOc'< / span > , FOc, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
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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 >
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< p > < span class = "figure-number" > Figure 3: < / 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 >
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< / div >
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< table id = "org6b3d8b1" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 3:< / span > Stiffness Matrix< / caption >
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< 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 >
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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" > 4.9e-17< / 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" > 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" > -2.2e-17< / 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" > 1.1e-18< / 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 >
< tr >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -2.2e-17< / td >
< td class = "org-right" > 1.1e-18< / td >
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< td class = "org-right" > 0.015< / td >
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< td class = "org-right" > 0< / td >
< td class = "org-right" > 3.5e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 4.4e-17< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > -5.7e-20< / td >
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< td class = "org-right" > 0.015< / td >
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< td class = "org-right" > -8.7e-19< / 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" > 3.5e-18< / td >
< td class = "org-right" > -8.7e-19< / td >
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< 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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< div id = "outline-container-org43fd7e4" class = "outline-3" >
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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 >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > H = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > H< span class = "org-type" > /< / span > 2; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
Hc = 1.5< span class = "org-type" > *< / span > H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H< span class = "org-type" > /< / span > 2 < span class = "org-type" > +< / span > 10e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , H, < span class = "org-string" > 'MO_B'< / span > , MO_B);
stewart = generateCubicConfiguration(stewart, < span class = "org-string" > 'Hc'< / span > , Hc, < span class = "org-string" > 'FOc'< / span > , FOc, < span class = "org-string" > 'FHa'< / span > , 0, < span class = "org-string" > 'MHb'< / span > , 0);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
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stewart = computeJacobian(stewart);
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stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 215e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Mpr'< / span > , 195e< 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 >
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< p > < span class = "figure-number" > Figure 4: < / 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 >
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< / div >
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< table id = "org846d51c" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 4:< / span > Stiffness Matrix< / caption >
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< 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 >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 1.5e-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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< / tr >
< 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" > -0.02< / 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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< / tr >
< tr >
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< td class = "org-right" > 1.5e-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" > -3e-18< / td >
< td class = "org-right" > -2.8e-17< / td >
< td class = "org-right" > 0< / 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 >
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< td class = "org-right" > -3e-18< / td >
< td class = "org-right" > 0.034< / td >
< td class = "org-right" > -8.7e-19< / td >
< td class = "org-right" > 5.2e-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" > -2.2e-17< / td >
< td class = "org-right" > -4.4e-19< / td >
< td class = "org-right" > 0.034< / 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" > 5.9e-18< / td >
< td class = "org-right" > -7.5e-18< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 3.5e-18< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.14< / td >
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< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org95af62e" class = "outline-3" >
< h3 id = "org95af62e" > < span class = "section-number-3" > 1.5< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
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< div class = "important" >
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< p >
Here are the conclusion about the Stiffness matrix for the Cubic configuration:
< / p >
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< 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 >
< div id = "outline-container-orgd70418b" class = "outline-2" >
< h2 id = "orgd70418b" > < span class = "section-number-2" > 2< / span > Configuration with the Cube’ s center above the mobile platform< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
< a id = "orgb73265d" > < / a >
< / p >
< p >
We saw in section < a href = "#orgda0ee50" > 1< / a > that in order to have a diagonal stiffness matrix, we need the cube’ s center to be located at frames \(\{A\}\) and \(\{B\}\).
Or, we usually want to have \(\{A\}\) and \(\{B\}\) located above the top platform where forces are applied and where displacements are expressed.
< / p >
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< p >
We here see if the cubic configuration can provide a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are above the mobile platform.
< / p >
< / 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" > 2.1< / span > Having Cube’ s center above the top platform< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
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Let’ s say we want to have a diagonal stiffness matrix when \(\{A\}\) and \(\{B\}\) are located above the top platform.
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Thus, we want the cube’ s center to be located above the top center.
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< / p >
< p >
Let’ s fix the Height of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\):
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > H = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = 20e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
< / pre >
< / div >
< p >
We find the several Cubic configuration for the Stewart platform where the center of the cube is located at frame \(\{A\}\).
The differences between the configuration are the cube’ s size:
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< / p >
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< ul class = "org-ul" >
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< li > Small Cube Size in Figure < a href = "#org105635f" > 5< / a > < / li >
< li > Medium Cube Size in Figure < a href = "#org264ab9c" > 6< / a > < / li >
< li > Large Cube Size in Figure < a href = "#org52254fe" > 7< / a > < / li >
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< / ul >
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< p >
For each of the configuration, the Stiffness matrix is diagonal with \(k_x = k_y = k_y = 2k\) with \(k\) is the stiffness of each strut.
However, the rotational stiffnesses are increasing with the cube’ s size but the required size of the platform is also increasing, so there is a trade-off here.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Hc = 0.4< span class = "org-type" > *< / span > H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
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< / pre >
< / div >
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< div id = "org105635f" class = "figure" >
< p > < img src = "figs/stewart_cubic_conf_type_1.png" alt = "stewart_cubic_conf_type_1.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Cubic Configuration for the Stewart Platform - Small Cube Size (< a href = "./figs/stewart_cubic_conf_type_1.png" > png< / a > , < a href = "./figs/stewart_cubic_conf_type_1.pdf" > pdf< / a > )< / p >
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< / div >
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< table id = "org91f89e4" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 5:< / span > Stiffness Matrix< / caption >
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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" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > -2.8e-16< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 2.4e-17< / td >
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< td class = "org-right" > 0< / td >
< / tr >
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2020-02-06 18:23:01 +01:00
< 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" > -2.3e-17< / td >
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< 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" > -2.8e-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" > -2.1e-19< / td >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > 0< / td >
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< td class = "org-right" > -2.3e-17< / td >
< td class = "org-right" > -2.1e-19< / td >
< td class = "org-right" > 0.0024< / td >
< td class = "org-right" > -5.4e-20< / td >
< td class = "org-right" > 6.5e-19< / td >
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< / tr >
< tr >
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< td class = "org-right" > 2.4e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 4.9e-19< / td >
< td class = "org-right" > -2.3e-20< / td >
< td class = "org-right" > 0.0024< / td >
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< td class = "org-right" > 0< / td >
< / tr >
< tr >
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< td class = "org-right" > -1.2e-18< / td >
< td class = "org-right" > 1.1e-18< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 6.2e-19< / td >
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< td class = "org-right" > 0< / td >
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< td class = "org-right" > 0.0096< / td >
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< / tr >
< / tbody >
< / table >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Hc = 1.5< span class = "org-type" > *< / span > H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
< / div >
< div id = "org264ab9c" class = "figure" >
< p > < img src = "figs/stewart_cubic_conf_type_2.png" alt = "stewart_cubic_conf_type_2.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Cubic Configuration for the Stewart Platform - Medium Cube Size (< a href = "./figs/stewart_cubic_conf_type_2.png" > png< / a > , < a href = "./figs/stewart_cubic_conf_type_2.pdf" > pdf< / a > )< / p >
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< / div >
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< table id = "orgcf84781" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 6:< / span > Stiffness Matrix< / caption >
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< 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" > 0< / td >
< td class = "org-right" > -1.9e-16< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 5.6e-17< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -7.6e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > -1.9e-16< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 2.5e-18< / td >
< td class = "org-right" > 2.8e-17< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -7.6e-17< / td >
< td class = "org-right" > 2.5e-18< / td >
< td class = "org-right" > 0.034< / td >
< td class = "org-right" > 8.7e-19< / td >
< td class = "org-right" > 8.7e-18< / td >
< / tr >
< tr >
< td class = "org-right" > 5.7e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 3.2e-17< / td >
< td class = "org-right" > 2.9e-19< / td >
< td class = "org-right" > 0.034< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > -1e-18< / td >
< td class = "org-right" > -1.3e-17< / td >
< td class = "org-right" > 5.6e-17< / td >
< td class = "org-right" > 8.4e-18< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 0.14< / td >
< / tr >
< / tbody >
< / table >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hc = 2.5< span class = "org-type" > *< / span > H; < span class = "org-comment" > % Size of the useful part of the cube [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
< / pre >
< / div >
< div id = "org52254fe" class = "figure" >
< p > < img src = "figs/stewart_cubic_conf_type_3.png" alt = "stewart_cubic_conf_type_3.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Cubic Configuration for the Stewart Platform - Large Cube Size (< a href = "./figs/stewart_cubic_conf_type_3.png" > png< / a > , < a href = "./figs/stewart_cubic_conf_type_3.pdf" > pdf< / a > )< / p >
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< / div >
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< table id = "org02f7789" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 7:< / span > Stiffness Matrix< / caption >
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< 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" > 0< / td >
< td class = "org-right" > -3e-16< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -8.3e-17< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.2e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 5.6e-17< / td >
< / tr >
< tr >
< td class = "org-right" > -3e-16< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2< / td >
< td class = "org-right" > -9.3e-19< / td >
< td class = "org-right" > -2.8e-17< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.2e-17< / td >
< td class = "org-right" > -9.3e-19< / td >
< td class = "org-right" > 0.094< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 2.1e-17< / td >
< / tr >
< tr >
< td class = "org-right" > -8e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > -3e-17< / td >
< td class = "org-right" > -6.1e-19< / td >
< td class = "org-right" > 0.094< / td >
< td class = "org-right" > 0< / td >
< / tr >
< tr >
< td class = "org-right" > -6.2e-18< / td >
< td class = "org-right" > 7.2e-17< / td >
< td class = "org-right" > 5.6e-17< / td >
< td class = "org-right" > 2.3e-17< / td >
< td class = "org-right" > 0< / td >
< td class = "org-right" > 0.37< / td >
< / tr >
< / tbody >
< / table >
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< / div >
< / div >
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< div id = "outline-container-org7673551" class = "outline-3" >
< h3 id = "org7673551" > < span class = "section-number-3" > 2.2< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< div class = "important" >
< p >
We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform.
Depending on the cube’ s size, we obtain 3 different configurations.
< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgcc4ecce" class = "outline-2" >
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< h2 id = "orgcc4ecce" > < span class = "section-number-2" > 3< / span > Cubic size analysis< / h2 >
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We here study the effect of the size of the cube used for the Stewart Cubic configuration.
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We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames \(\{A\}\) and \(\{B\}\) are also taken at the center of the cube.
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We only vary the size of the cube.
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< h3 id = "org0029d8c" > < span class = "section-number-3" > 3.1< / span > Analysis< / h3 >
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We initialize the wanted cube’ s size.
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< pre class = "src src-matlab" > Hcs = 1e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > [250< span class = "org-type" > :< / span > 20< span class = "org-type" > :< / span > 350]; < span class = "org-comment" > % Heights for the Cube [m]< / span >
Ks = zeros(6, 6, length(Hcs));
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The height of the Stewart platform is fixed:
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< pre class = "src src-matlab" > H = 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
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The frames \(\{A\}\) and \(\{B\}\) are positioned at the Stewart platform center as well as the cube’ s center:
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< pre class = "src src-matlab" > MO_B = < span class = "org-type" > -< / span > 50e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
FOc = H < span class = "org-type" > +< / span > MO_B; < span class = "org-comment" > % Center of the cube with respect to {F}< / span >
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We find that for all the cube’ s size, \(k_x = k_y = k_z = k\) where \(k\) is the strut stiffness.
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We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varying with the cube’ s size (figure < a href = "#orgf5b4a80" > 8< / a > ).
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< p > < img src = "figs/stiffness_cube_size.png" alt = "stiffness_cube_size.png" / >
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< p > < span class = "figure-number" > Figure 8: < / span > \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube< / p >
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< h3 id = "orgcad29fd" > < span class = "section-number-3" > 3.2< / span > Conclusion< / h3 >
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We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
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In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
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< h2 id = "org5abef15" > < span class = "section-number-2" > 4< / span > Dynamic Coupling< / h2 >
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< h3 id = "org0d67b92" > < span class = "section-number-3" > 4.1< / span > Cube’ s center at the Center of Mass of the Payload< / h3 >
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< h3 id = "org876e05f" > < span class = "section-number-3" > 4.2< / span > Dynamic decoupling between the actuators and sensors< / h3 >
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< h3 id = "org213d9b9" > < span class = "section-number-3" > 4.3< / span > Conclusion< / h3 >
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< h2 id = "org3044455" > < span class = "section-number-2" > 5< / span > Functions< / h2 >
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< h3 id = "org56504f1" > < span class = "section-number-3" > 5.1< / 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 = "orga5a9ba8" > 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 >
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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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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
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< p > < span class = "figure-number" > Figure 9: < / 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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< 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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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 = "org3a2f468" > 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" > 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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< 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: 2020-02-12 mer. 11:23< / p >
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