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< title > Cubic configuration for the Stewart Platform< / title >
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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 = "#org017fb61" > 1. Stiffness Matrix for the Cubic configuration< / a >
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< ul >
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< li > < a href = "#org469f9dc" > 1.1. Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#org6359f2f" > 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / a > < / li >
< li > < a href = "#org5c37be2" > 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / a > < / li >
< li > < a href = "#org32ac59a" > 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center< / a > < / li >
< li > < a href = "#org4e88cdb" > 1.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org312b7d4" > 2. Configuration with the Cube’ s center above the mobile platform< / a >
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< ul >
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< li > < a href = "#org4983654" > 2.1. Having Cube’ s center above the top platform< / a > < / li >
< li > < a href = "#orge53b7f6" > 2.2. Size of the platforms< / a > < / li >
< li > < a href = "#org561a2bc" > 2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org2387b96" > 3. Cubic size analysis< / a >
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< ul >
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< li > < a href = "#org3647f9f" > 3.1. Analysis< / a > < / li >
< li > < a href = "#org948a425" > 3.2. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org174af3a" > 4. Dynamic Coupling in the Cartesian Frame< / a >
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< ul >
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< li > < a href = "#orgdb33aa6" > 4.1. Cube’ s center at the Center of Mass of the mobile platform< / a > < / li >
< li > < a href = "#org49b330b" > 4.2. Cube’ s center not coincident with the Mass of the Mobile platform< / a > < / li >
< li > < a href = "#org7d50eae" > 4.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org7831cff" > 5. Dynamic Coupling between actuators and sensors of each strut< / a >
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< ul >
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< li > < a href = "#org38e9e8f" > 5.1. Coupling between the actuators and sensors - Cubic Architecture< / a > < / li >
< li > < a href = "#org21d40d3" > 5.2. Coupling between the actuators and sensors - Non-Cubic Architecture< / a > < / li >
< li > < a href = "#orgeb8ae82" > 5.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org3ce1c89" > 6. Functions< / a >
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< ul >
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< li > < a href = "#org9ad761f" > 6.1. < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / a >
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< ul >
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< li > < a href = "#org2cafc68" > Function description< / a > < / li >
< li > < a href = "#org32005ba" > Documentation< / a > < / li >
< li > < a href = "#org4fd2c96" > Optional Parameters< / a > < / li >
< li > < a href = "#orgac26a8b" > Check the < code > stewart< / code > structure elements< / a > < / li >
< li > < a href = "#orgc86b760" > Position of the Cube< / a > < / li >
< li > < a href = "#org1e9ccef" > Compute the pose< / a > < / li >
< li > < a href = "#org153763b" > 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 href = "#citeproc_bib_item_2" > Geng and Haynes 1994< / a > ).
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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.
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This configuration is now widely used ((< a href = "#citeproc_bib_item_5" > Preumont et al. 2007< / a > ; < a href = "#citeproc_bib_item_3" > Jafari and McInroy 2003< / a > )).
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< / p >
< p >
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According to (< a href = "#citeproc_bib_item_5" > Preumont et al. 2007< / a > ), the cubic configuration offers the following advantages:
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< / p >
< blockquote >
< p >
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This topology provides a < b > uniform control capability< / b > and a < b > uniform stiffness< / b > in all directions, and it < b > minimizes the cross-coupling amongst actuators and sensors of different legs< / b > (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 = "#org6bc5f56" > 1< / a > , we study the < b > uniform stiffness< / b > of such configuration and we find the conditions to obtain a diagonal stiffness matrix< / li >
< li > In section < a href = "#org419cdb0" > 2< / a > , we find cubic configurations where the cube’ s center is located above the mobile platform< / li >
< li > In section < a href = "#org53ade24" > 3< / a > , we study the effect of the cube’ s size on the Stewart platform properties< / li >
< li > In section < a href = "#org3507b2b" > 4< / a > , we study the dynamics of the cubic configuration in the cartesian frame< / li >
< li > In section < a href = "#org7b3ed31" > 5< / a > , we study the dynamic < b > cross-coupling< / b > of the cubic configuration from actuators to sensors of each strut< / li >
< li > In section < a href = "#orgef41b92" > 6< / a > , function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function < code > generateCubicConfiguration< / code > is used (described < a href = "#orgd9ae150" > here< / a > ).< / li >
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< / ul >
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< div id = "outline-container-org017fb61" class = "outline-2" >
< h2 id = "org017fb61" > < span class = "section-number-2" > 1< / span > Stiffness Matrix for the Cubic configuration< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< p >
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< a id = "org6bc5f56" > < / a >
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< / p >
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< div class = "note" id = "org6a03293" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/cubic_conf_stiffnessl.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run cubic_conf_stiffness.m< / code > .
< / p >
< / div >
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< 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-org469f9dc" class = "outline-3" >
< h3 id = "org469f9dc" > < span class = "section-number-3" > 1.1< / span > Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center< / h3 >
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< 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 = "#orge8e0501" > 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" >
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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 >
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 >
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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 = "orge8e0501" 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 = "org64c21d6" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< 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-org6359f2f" class = "outline-3" >
< h3 id = "org6359f2f" > < span class = "section-number-3" > 1.2< / span > Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center< / h3 >
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< 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 = "#org58083e8" > 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" >
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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 >
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 >
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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 = "org58083e8" 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 = "org46d41a3" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 2:< / span > Stiffness Matrix< / caption >
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< tr >
< td class = "org-right" > 2< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
< td class = "org-right" > -2.5e-16< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 0< / td >
< td class = "org-right" > -0.14< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 0.14< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > -2.5e-16< / td >
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -5.3e-19< / td >
< td class = "org-right" > 0< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-12 10:22:51 +01:00
< 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 >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-12 10:22:51 +01:00
< 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 >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > -3.3e-18< / td >
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 8.9e-19< / td >
< td class = "org-right" > 0< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0.06< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< / tbody >
< / table >
< / div >
< / div >
2021-01-08 15:34:53 +01:00
< div id = "outline-container-org5c37be2" class = "outline-3" >
< h3 id = "org5c37be2" > < span class = "section-number-3" > 1.3< / span > Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center< / h3 >
2020-02-11 15:50:52 +01:00
< div class = "outline-text-3" id = "text-1-3" >
2019-03-25 18:12:43 +01:00
< p >
2021-01-08 15:34:53 +01:00
Here, the “ center” of the Stewart platform is not at the cube center (figure < a href = "#org45cc38d" > 3< / a > ).
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The Jacobian is estimated at the cube center.
< / p >
2020-02-12 10:22:51 +01:00
< div class = "org-src-container" >
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< 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 >
2020-02-12 10:22:51 +01:00
< / pre >
2019-03-25 18:12:43 +01:00
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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 >
2020-02-07 17:31:52 +01:00
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< div id = "org45cc38d" 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 >
2020-02-07 17:31:52 +01:00
< / div >
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< table id = "orgb0fc90f" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< 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 >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > -1.7e-16< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 4.9e-17< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -2.2e-17< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 2.8e-17< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > -1.7e-16< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 1.1e-18< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > 1.4e-17< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -2.2e-17< / td >
< td class = "org-right" > 1.1e-18< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0.015< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 0< / td >
< td class = "org-right" > 3.5e-18< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 4.4e-17< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -1.4e-17< / td >
< td class = "org-right" > -5.7e-20< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0.015< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -8.7e-19< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > 2.5e-17< / td >
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 3.5e-18< / td >
< td class = "org-right" > -8.7e-19< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0.06< / td >
2019-03-25 18:12:43 +01:00
< / 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-org32ac59a" class = "outline-3" >
< h3 id = "org32ac59a" > < 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 >
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< div class = "outline-text-3" id = "text-1-4" >
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< p >
2020-01-27 17:42:09 +01:00
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" >
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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 >
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 >
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , ones(6,1));
stewart = computeJacobian(stewart);
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);
2019-03-25 18:12:43 +01:00
< / pre >
< / div >
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< div id = "org8c8f97b" 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 >
2020-02-12 11:23:26 +01:00
< 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 >
2020-02-07 17:31:52 +01:00
< / div >
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< table id = "orge20fac2" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< 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 >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 1.5e-16< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0.02< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > -0.02< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 1.5e-16< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< td class = "org-right" > 2< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -3e-18< / td >
< td class = "org-right" > -2.8e-17< / td >
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > -0.02< / td >
2020-02-12 10:22:51 +01:00
< 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 >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0.02< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > -2.2e-17< / td >
< td class = "org-right" > -4.4e-19< / td >
< td class = "org-right" > 0.034< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< tr >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 5.9e-18< / td >
< td class = "org-right" > -7.5e-18< / td >
2020-02-06 18:23:01 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 3.5e-18< / td >
2020-02-07 17:31:52 +01:00
< td class = "org-right" > 0< / td >
2020-02-12 10:22:51 +01:00
< td class = "org-right" > 0.14< / td >
2019-03-25 18:12:43 +01:00
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org4e88cdb" class = "outline-3" >
< h3 id = "org4e88cdb" > < 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" id = "org2fc62c4" >
2020-02-12 10:22:51 +01:00
< p >
Here are the conclusion about the Stiffness matrix for the Cubic configuration:
< / p >
2019-03-25 18:12:43 +01:00
< ul class = "org-ul" >
2020-02-06 18:23:01 +01:00
< 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 >
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< div id = "outline-container-org312b7d4" class = "outline-2" >
< h2 id = "org312b7d4" > < span class = "section-number-2" > 2< / span > Configuration with the Cube’ s center above the mobile platform< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
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< a id = "org419cdb0" > < / a >
2020-02-12 11:18:37 +01:00
< / p >
2020-02-13 16:46:47 +01:00
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< div class = "note" id = "orgd5c0d4d" >
2020-02-13 16:46:47 +01:00
< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/cubic_conf_above_platforml.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run cubic_conf_above_platform.m< / code > .
< / p >
< / div >
2020-02-12 11:18:37 +01:00
< p >
2021-01-08 15:34:53 +01:00
We saw in section < a href = "#org6bc5f56" > 1< / a > that in order to have a diagonal stiffness matrix, we need the cube’ s center to be located at frames \(\{A\}\) and \(\{B\}\).
2020-02-12 11:18:37 +01:00
Or, we usually want to have \(\{A\}\) and \(\{B\}\) located above the top platform where forces are applied and where displacements are expressed.
< / p >
2019-03-25 18:12:43 +01:00
2020-02-12 11:18:37 +01:00
< 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-org4983654" class = "outline-3" >
< h3 id = "org4983654" > < span class = "section-number-3" > 2.1< / span > Having Cube’ s center above the top platform< / h3 >
2020-02-12 11:18:37 +01:00
< div class = "outline-text-3" id = "text-2-1" >
2019-03-25 18:12:43 +01:00
< p >
2020-02-12 10:22:51 +01:00
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" >
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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 >
MO_B = 20e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
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< / 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 = "#orgc21b175" > 5< / a > < / li >
< li > Medium Cube Size in Figure < a href = "#org1a57849" > 6< / a > < / li >
< li > Large Cube Size in Figure < a href = "#org5d21d07" > 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 = "orgc21b175" class = "figure" >
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< 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 = "org4ed8369" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< 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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< 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" >
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< 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 >
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< / pre >
< / div >
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< div id = "org1a57849" class = "figure" >
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< 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 = "org325fd88" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< 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" >
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< 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 >
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< / pre >
< / div >
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< div id = "org5d21d07" class = "figure" >
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< 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 = "org416d72b" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 7:< / span > Stiffness Matrix< / caption >
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< col class = "org-right" / >
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< 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 >
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< div id = "outline-container-orge53b7f6" class = "outline-3" >
< h3 id = "orge53b7f6" > < span class = "section-number-3" > 2.2< / span > Size of the platforms< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
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< p >
The minimum size of the platforms depends on the cube’ s size and the height between the platform and the cube’ s center.
< / p >
< p >
Let’ s denote:
< / p >
< ul class = "org-ul" >
< li > \(H\) the height between the cube’ s center and the considered platform< / li >
< li > \(D\) the size of the cube’ s edges< / li >
< / ul >
< p >
Let’ s denote by \(a\) and \(b\) the points of both ends of one of the cube’ s edge.
< / p >
< p >
Initially, we have:
< / p >
\begin{align}
a & = \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\
b & = \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix}
\end{align}
< p >
We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical.
\[ R = \begin{bmatrix}
\frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\
\frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\
\frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}}
\end{bmatrix} \]
< / p >
< p >
After rotation, the points \(a\) and \(b\) become:
< / p >
\begin{align}
a & = \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\
b & = \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix}
\end{align}
< p >
Points \(a\) and \(b\) define a vector \(u = b - a\) that gives the orientation of one of the Stewart platform strut:
\[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]
< / p >
< p >
Then we want to find the intersection between the line that defines the strut with the plane defined by the height \(H\) from the cube’ s center.
To do so, we first find \(g\) such that:
\[ a_z + g u_z = -H \]
We obtain:
< / p >
\begin{align}
g & = - \frac{H + a_z}{u_z} \\
& = \sqrt{3} \frac{H}{D} - \frac{1}{2}
\end{align}
< p >
Then, the intersection point \(P\) is given by:
< / p >
\begin{align}
P & = a + g u \\
& = \begin{bmatrix}
H \sqrt{2} \\
D \frac{1}{\sqrt{2}} \\
H
\end{bmatrix}
\end{align}
< p >
Finally, the circle can contains the intersection point has a radius \(r\):
< / p >
\begin{align}
r & = \sqrt{P_x^2 + P_y^2} \\
& = \sqrt{2 H^2 + \frac{1}{2}D^2}
\end{align}
< p >
By symmetry, we can show that all the other intersection points will also be on the circle with a radius \(r\).
< / p >
< p >
For a small cube:
\[ r \approx \sqrt{2} H \]
< / p >
< / div >
< / div >
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< div id = "outline-container-org561a2bc" class = "outline-3" >
< h3 id = "org561a2bc" > < span class = "section-number-3" > 2.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< div class = "important" id = "orga24e443" >
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< 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 >
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< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Cube’ s Size< / th >
< th scope = "col" class = "org-left" > Paper with the corresponding cubic architecture< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > Small< / td >
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< td class = "org-left" > (< a href = "#citeproc_bib_item_1" > Furutani, Suzuki, and Kudoh 2004< / a > )< / td >
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< / tr >
< tr >
< td class = "org-left" > Medium< / td >
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< td class = "org-left" > (< a href = "#citeproc_bib_item_6" > Yang et al. 2019< / a > )< / td >
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< / tr >
< tr >
< td class = "org-left" > Large< / td >
< td class = "org-left" >   < / td >
< / tr >
< / tbody >
< / table >
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< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org2387b96" class = "outline-2" >
< h2 id = "org2387b96" > < span class = "section-number-2" > 3< / span > Cubic size analysis< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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< a id = "org53ade24" > < / a >
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< / p >
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< div class = "note" id = "orgfd32e5a" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/cubic_conf_size_analysisl.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run cubic_conf_size_analysis.m< / code > .
< / p >
< / div >
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< p >
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We here study the effect of the size of the cube used for the Stewart Cubic configuration.
< / p >
< p >
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.
< / p >
< p >
We only vary the size of the cube.
< / p >
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< / div >
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< div id = "outline-container-org3647f9f" class = "outline-3" >
< h3 id = "org3647f9f" > < span class = "section-number-3" > 3.1< / span > Analysis< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< p >
We initialize the wanted cube’ s size.
< / p >
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< div class = "org-src-container" >
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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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< / pre >
< / div >
< p >
The height of the Stewart platform is fixed:
< / p >
< div class = "org-src-container" >
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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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< / pre >
< / div >
< p >
The frames \(\{A\}\) and \(\{B\}\) are positioned at the Stewart platform center as well as the cube’ s center:
< / p >
< div class = "org-src-container" >
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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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< / pre >
< / div >
< p >
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 = "#org51805e3" > 8< / a > ).
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< / p >
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< div id = "org51805e3" class = "figure" >
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< p > < img src = "figs/stiffness_cube_size.png" alt = "stiffness_cube_size.png" / >
< / p >
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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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< / div >
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< / div >
< / div >
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< div id = "outline-container-org948a425" class = "outline-3" >
< h3 id = "org948a425" > < span class = "section-number-3" > 3.2< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< p >
We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size.
< / p >
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< div class = "important" id = "org60cc507" >
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< p >
In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.
< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org174af3a" class = "outline-2" >
< h2 id = "org174af3a" > < span class = "section-number-2" > 4< / span > Dynamic Coupling in the Cartesian Frame< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< p >
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< a id = "org3507b2b" > < / a >
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< / p >
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< div class = "note" id = "org5934a9c" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/cubic_conf_coupling_cartesianl.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run cubic_conf_coupling_cartesian.m< / code > .
< / p >
< / div >
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< p >
In this section, we study the dynamics of the platform in the cartesian frame.
< / p >
< p >
We here suppose that there is one relative motion sensor in each strut (\(\delta\bm{\mathcal{L}}\) is measured) and we would like to control the position of the top platform pose \(\delta \bm{\mathcal{X}}\).
< / p >
< p >
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Thanks to the Jacobian matrix, we can use the “ architecture” shown in Figure < a href = "#org2137f5a" > 9< / a > to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.
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< / p >
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< div id = "org2137f5a" class = "figure" >
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< p > < img src = "figs/local_to_cartesian_coordinates.png" alt = "local_to_cartesian_coordinates.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > From Strut coordinate to Cartesian coordinate using the Jacobian matrix< / p >
< / div >
< p >
We here study the dynamics from \(\bm{\mathcal{F}}\) to \(\delta\bm{\mathcal{X}}\).
< / p >
< p >
One has to note that when considering the static behavior:
\[ \bm{G}(s = 0) = \begin{bmatrix}
1/k_1 & & 0 \\
& \ddots & 0 \\
0 & & 1/k_6
\end{bmatrix}\]
< / p >
< p >
And thus:
\[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]
< / p >
< p >
We conclude that the < b > static< / b > behavior of the platform depends on the stiffness matrix.
For the cubic configuration, we have a diagonal stiffness matrix is the frames \(\{A\}\) and \(\{B\}\) are coincident with the cube’ s center.
< / p >
< / div >
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< div id = "outline-container-orgdb33aa6" class = "outline-3" >
< h3 id = "orgdb33aa6" > < span class = "section-number-3" > 4.1< / span > Cube’ s center at the Center of Mass of the mobile platform< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
Let’ s create a Cubic Stewart Platform where the < b > Center of Mass of the mobile platform is located at the center of the cube< / b > .
< / p >
< p >
We define the size of the Stewart platform and the position of frames \(\{A\}\) and \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > H = 200e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > 10e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
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< / pre >
< / div >
< p >
Now, we set the cube’ s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
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< / p >
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< div class = "org-src-container" >
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< 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 >
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< / pre >
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< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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 > , 25e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MHb'< / span > , 25e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , 1e6< span class = "org-type" > *< / span > ones(6,1), < span class = "org-string" > 'C'< / span > , 1e1< span class = "org-type" > *< / span > ones(6,1));
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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< / pre >
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< / div >
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< p >
Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with \(\{A\}\) and \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_F.Fa)), ...
< span class = "org-string" > 'Mpm'< / span > , 10, ...
< span class = "org-string" > 'Mph'< / span > , 20e< span class = "org-type" > -< / span > 3, ...
< span class = "org-string" > 'Mpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_M.Mb)));
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< / pre >
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< / div >
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< p >
And we set small mass for the struts.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeCylindricalStruts(stewart, < span class = "org-string" > 'Fsm'< / span > , 1e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Msm'< / span > , 1e< span class = "org-type" > -< / span > 3);
stewart = initializeInertialSensor(stewart);
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< / pre >
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< / div >
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< p >
No flexibility below the Stewart platform and no payload.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
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< p >
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The obtain geometry is shown in figure < a href = "#orgb6b060a" > 10< / a > .
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< / p >
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< div id = "orgb6b060a" class = "figure" >
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< p > < img src = "figs/stewart_cubic_conf_decouple_dynamics.png" alt = "stewart_cubic_conf_decouple_dynamics.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Geometry used for the simulations - The cube’ s center, the frames \(\{A\}\) and \(\{B\}\) and the Center of mass of the mobile platform are coincident (< a href = "./figs/stewart_cubic_conf_decouple_dynamics.png" > png< / a > , < a href = "./figs/stewart_cubic_conf_decouple_dynamics.pdf" > pdf< / a > )< / p >
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< / div >
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< p >
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > open(< span class = "org-string" > 'stewart_platform_model.slx'< / span > )
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Stewart Platform'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'dLm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Relative Displacement Outputs [m]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
G.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G.OutputName = {< span class = "org-string" > 'Dm1'< / span > , < span class = "org-string" > 'Dm2'< / span > , < span class = "org-string" > 'Dm3'< / span > , < span class = "org-string" > 'Dm4'< / span > , < span class = "org-string" > 'Dm5'< / span > , < span class = "org-string" > 'Dm6'< / span > };
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< / pre >
< / div >
< p >
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Now, thanks to the Jacobian (Figure < a href = "#org2137f5a" > 9< / a > ), we compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc = inv(stewart.kinematics.J)< span class = "org-type" > *< / span > G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > );
Gc = inv(stewart.kinematics.J)< span class = "org-type" > *< / span > G< span class = "org-type" > *< / span > stewart.kinematics.J;
Gc.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
Gc.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< p >
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The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure < a href = "#org12dc231" > 11< / a > .
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< / p >
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< div id = "org12dc231" class = "figure" >
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< p > < img src = "figs/stewart_cubic_decoupled_dynamics_cartesian.png" alt = "stewart_cubic_decoupled_dynamics_cartesian.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (< a href = "./figs/stewart_cubic_decoupled_dynamics_cartesian.png" > png< / a > , < a href = "./figs/stewart_cubic_decoupled_dynamics_cartesian.pdf" > pdf< / a > )< / p >
< / div >
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< p >
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It is interesting to note here that the system shown in Figure < a href = "#org9d84f45" > 12< / a > also yield a decoupled system (explained in section 1.3.3 in (< a href = "#citeproc_bib_item_4" > Li 2001< / a > )).
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< / p >
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< div id = "org9d84f45" class = "figure" >
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< p > < img src = "figs/local_to_cartesian_coordinates_bis.png" alt = "local_to_cartesian_coordinates_bis.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Alternative way to decouple the system< / p >
< / div >
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< div class = "important" id = "org489ed3b" >
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< p >
The dynamics is well decoupled at all frequencies.
< / p >
< p >
We have the same dynamics for:
< / p >
< ul class = "org-ul" >
< li > \(D_x/F_x\), \(D_y/F_y\) and \(D_z/F_z\)< / li >
< li > \(R_x/M_x\) and \(D_y/F_y\)< / li >
< / ul >
< p >
The Dynamics from \(F_i\) to \(D_i\) is just a 1-dof mass-spring-damper system.
< / p >
< p >
This is because the Mass, Damping and Stiffness matrices are all diagonal.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org49b330b" class = "outline-3" >
< h3 id = "org49b330b" > < span class = "section-number-3" > 4.2< / span > Cube’ s center not coincident with the Mass of the Mobile platform< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< p >
Let’ s create a Stewart platform with a cubic architecture where the cube’ s center is at the center of the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > H = 200e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > 100e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
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< / pre >
< / div >
< p >
Now, we set the cube’ s parameters such that the center of the cube is coincident with \(\{A\}\) and \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< 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 >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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 > , 25e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MHb'< / span > , 25e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , 1e6< span class = "org-type" > *< / span > ones(6,1), < span class = "org-string" > 'C'< / span > , 1e1< span class = "org-type" > *< / span > ones(6,1));
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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< / pre >
< / div >
< p >
However, the Center of Mass of the mobile platform is < b > not< / b > located at the cube’ s center.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_F.Fa)), ...
< span class = "org-string" > 'Mpm'< / span > , 10, ...
< span class = "org-string" > 'Mph'< / span > , 20e< span class = "org-type" > -< / span > 3, ...
< span class = "org-string" > 'Mpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_M.Mb)));
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< / pre >
< / div >
< p >
And we set small mass for the struts.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeCylindricalStruts(stewart, < span class = "org-string" > 'Fsm'< / span > , 1e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Msm'< / span > , 1e< span class = "org-type" > -< / span > 3);
stewart = initializeInertialSensor(stewart);
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< / pre >
< / div >
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< p >
No flexibility below the Stewart platform and no payload.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
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< p >
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The obtain geometry is shown in figure < a href = "#orgc57dcd2" > 13< / a > .
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< / p >
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< div id = "orgc57dcd2" class = "figure" >
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< p > < img src = "figs/stewart_cubic_conf_mass_above.png" alt = "stewart_cubic_conf_mass_above.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Geometry used for the simulations - The cube’ s center is coincident with the frames \(\{A\}\) and \(\{B\}\) but not with the Center of mass of the mobile platform (< a href = "./figs/stewart_cubic_conf_mass_above.png" > png< / a > , < a href = "./figs/stewart_cubic_conf_mass_above.pdf" > pdf< / a > )< / p >
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< / div >
< p >
We now identify the dynamics from forces applied in each strut \(\bm{\tau}\) to the displacement of each strut \(d \bm{\mathcal{L}}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > open(< span class = "org-string" > 'stewart_platform_model.slx'< / span > )
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Stewart Platform'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'dLm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Relative Displacement Outputs [m]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
G.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G.OutputName = {< span class = "org-string" > 'Dm1'< / span > , < span class = "org-string" > 'Dm2'< / span > , < span class = "org-string" > 'Dm3'< / span > , < span class = "org-string" > 'Dm4'< / span > , < span class = "org-string" > 'Dm5'< / span > , < span class = "org-string" > 'Dm6'< / span > };
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< / pre >
< / div >
< p >
And we use the Jacobian to compute the transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc = inv(stewart.kinematics.J)< span class = "org-type" > *< / span > G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > );
Gc.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
Gc.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< p >
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The obtain dynamics \(\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}\) is shown in Figure < a href = "#org290b4b3" > 14< / a > .
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< / p >
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< div id = "org290b4b3" class = "figure" >
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< p > < img src = "figs/stewart_conf_coupling_mass_matrix.png" alt = "stewart_conf_coupling_mass_matrix.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Obtained Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (< a href = "./figs/stewart_conf_coupling_mass_matrix.png" > png< / a > , < a href = "./figs/stewart_conf_coupling_mass_matrix.pdf" > pdf< / a > )< / p >
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< / div >
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< div class = "important" id = "org798e5c6" >
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< p >
The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is < b > not< / b > decoupled at all frequencies.
< / p >
< p >
This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame \(\{B\}\)).
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7d50eae" class = "outline-3" >
< h3 id = "org7d50eae" > < span class = "section-number-3" > 4.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
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< div class = "important" id = "org9b1be89" >
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< p >
Some conclusions can be drawn from the above analysis:
< / p >
< ul class = "org-ul" >
< li > Static Decoupling < => Diagonal Stiffness matrix < => {A} and {B} at the cube’ s center< / li >
< li > Dynamic Decoupling < => Static Decoupling + CoM of mobile platform coincident with {A} and {B}.< / li >
< / ul >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7831cff" class = "outline-2" >
< h2 id = "org7831cff" > < span class = "section-number-2" > 5< / span > Dynamic Coupling between actuators and sensors of each strut< / h2 >
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< div class = "outline-text-2" id = "text-5" >
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< p >
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< a id = "org7b3ed31" > < / a >
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< / p >
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< div class = "note" id = "org7a9b096" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/cubic_conf_coupling_strutsl.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run cubic_conf_coupling_struts.m< / code > .
< / p >
< / div >
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< p >
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From (< a href = "#citeproc_bib_item_5" > Preumont et al. 2007< / a > ), the cubic configuration “ < i > minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)< / i > ” .
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< / p >
< p >
In this section, we wish to study such properties of the cubic architecture.
< / p >
< p >
We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).
< / p >
< / div >
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< div id = "outline-container-org38e9e8f" class = "outline-3" >
< h3 id = "org38e9e8f" > < span class = "section-number-3" > 5.1< / span > Coupling between the actuators and sensors - Cubic Architecture< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
< p >
Let’ s generate a Cubic architecture where the cube’ s center and the frames \(\{A\}\) and \(\{B\}\) are coincident.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > H = 200e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > 10e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
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 >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 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 > , 25e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MHb'< / span > , 25e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , 1e6< span class = "org-type" > *< / span > ones(6,1), < span class = "org-string" > 'C'< / span > , 1e1< span class = "org-type" > *< / span > ones(6,1));
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_F.Fa)), ...
< span class = "org-string" > 'Mpm'< / span > , 10, ...
< span class = "org-string" > 'Mph'< / span > , 20e< span class = "org-type" > -< / span > 3, ...
< span class = "org-string" > 'Mpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, < span class = "org-string" > 'Fsm'< / span > , 1e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Msm'< / span > , 1e< span class = "org-type" > -< / span > 3);
stewart = initializeInertialSensor(stewart);
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< / pre >
< / div >
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< p >
No flexibility below the Stewart platform and no payload.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > disturbances = initializeDisturbances();
references = initializeReferences(stewart);
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< / pre >
< / div >
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< div id = "org883dead" class = "figure" >
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< p > < img src = "figs/stewart_architecture_coupling_struts_cubic.png" alt = "stewart_architecture_coupling_struts_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Geometry of the generated Stewart platform (< a href = "./figs/stewart_architecture_coupling_struts_cubic.png" > png< / a > , < a href = "./figs/stewart_architecture_coupling_struts_cubic.pdf" > pdf< / a > )< / p >
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< / div >
< p >
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And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure < a href = "#org8da23ed" > 16< / a > ) and to the force sensors \(\tau_{m,i}\) (Figure < a href = "#orga2981a6" > 17< / a > ).
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< / p >
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< div id = "org8da23ed" class = "figure" >
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< p > < img src = "figs/coupling_struts_relative_sensor_cubic.png" alt = "coupling_struts_relative_sensor_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Dynamics from the force actuators to the relative motion sensors (< a href = "./figs/coupling_struts_relative_sensor_cubic.png" > png< / a > , < a href = "./figs/coupling_struts_relative_sensor_cubic.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orga2981a6" class = "figure" >
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< p > < img src = "figs/coupling_struts_force_sensor_cubic.png" alt = "coupling_struts_force_sensor_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Dynamics from the force actuators to the force sensors (< a href = "./figs/coupling_struts_force_sensor_cubic.png" > png< / a > , < a href = "./figs/coupling_struts_force_sensor_cubic.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org21d40d3" class = "outline-3" >
< h3 id = "org21d40d3" > < span class = "section-number-3" > 5.2< / span > Coupling between the actuators and sensors - Non-Cubic Architecture< / h3 >
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< div class = "outline-text-3" id = "text-5-2" >
< p >
Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > H = 200e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % height of the Stewart platform [m]< / span >
MO_B = < span class = "org-type" > -< / span > 10e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Position {B} with respect to {M} [m]< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , H, < span class = "org-string" > 'MO_B'< / span > , MO_B);
stewart = generateGeneralConfiguration(stewart, < span class = "org-string" > 'FR'< / span > , 250e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MR'< / span > , 150e< span class = "org-type" > -< / span > 3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, < span class = "org-string" > 'K'< / span > , 1e6< span class = "org-type" > *< / span > ones(6,1), < span class = "org-string" > 'C'< / span > , 1e1< span class = "org-type" > *< / span > ones(6,1));
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, < span class = "org-string" > 'Fpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_F.Fa)), ...
< span class = "org-string" > 'Mpm'< / span > , 10, ...
< span class = "org-string" > 'Mph'< / span > , 20e< span class = "org-type" > -< / span > 3, ...
< span class = "org-string" > 'Mpr'< / span > , 1.2< span class = "org-type" > *< / span > max(vecnorm(stewart.platform_M.Mb)));
stewart = initializeCylindricalStruts(stewart, < span class = "org-string" > 'Fsm'< / span > , 1e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'Msm'< / span > , 1e< span class = "org-type" > -< / span > 3);
stewart = initializeInertialSensor(stewart);
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< / pre >
< / div >
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< p >
No flexibility below the Stewart platform and no payload.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
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< div id = "org11aaa58" class = "figure" >
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< p > < img src = "figs/stewart_architecture_coupling_struts_non_cubic.png" alt = "stewart_architecture_coupling_struts_non_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 18: < / span > Geometry of the generated Stewart platform (< a href = "./figs/stewart_architecture_coupling_struts_non_cubic.png" > png< / a > , < a href = "./figs/stewart_architecture_coupling_struts_non_cubic.pdf" > pdf< / a > )< / p >
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< p >
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And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relative motion sensors \(\delta \mathcal{L}_{i}\) (Figure < a href = "#orgf8a39d5" > 19< / a > ) and to the force sensors \(\tau_{m,i}\) (Figure < a href = "#orgf05b7f3" > 20< / a > ).
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< / p >
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< div id = "orgf8a39d5" class = "figure" >
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< p > < img src = "figs/coupling_struts_relative_sensor_non_cubic.png" alt = "coupling_struts_relative_sensor_non_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 19: < / span > Dynamics from the force actuators to the relative motion sensors (< a href = "./figs/coupling_struts_relative_sensor_non_cubic.png" > png< / a > , < a href = "./figs/coupling_struts_relative_sensor_non_cubic.pdf" > pdf< / a > )< / p >
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< div id = "orgf05b7f3" class = "figure" >
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< p > < img src = "figs/coupling_struts_force_sensor_non_cubic.png" alt = "coupling_struts_force_sensor_non_cubic.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Dynamics from the force actuators to the force sensors (< a href = "./figs/coupling_struts_force_sensor_non_cubic.png" > png< / a > , < a href = "./figs/coupling_struts_force_sensor_non_cubic.pdf" > pdf< / a > )< / p >
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< h3 id = "orgeb8ae82" > < span class = "section-number-3" > 5.3< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-5-3" >
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< div class = "important" id = "org74729c6" >
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< p >
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The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control.
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< h2 id = "org3ce1c89" > < span class = "section-number-2" > 6< / span > Functions< / h2 >
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< a id = "orgef41b92" > < / a >
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< h3 id = "org9ad761f" > < span class = "section-number-3" > 6.1< / span > < code > generateCubicConfiguration< / code > : Generate a Cubic Configuration< / h3 >
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< p >
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< a id = "orgd9ae150" > < / a >
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< / p >
< p >
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This Matlab function is accessible < a href = "../src/generateCubicConfiguration.m" > here< / a > .
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< h4 id = "org2cafc68" > 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" > % - geometry.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" > % - 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 = "org32005ba" > Documentation< / h4 >
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< div id = "orgbf3377c" class = "figure" >
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< p > < img src = "figs/cubic-configuration-definition.png" alt = "cubic-configuration-definition.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 21: < / span > Cubic Configuration< / p >
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< h4 id = "org4fd2c96" > Optional Parameters< / h4 >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > stewart< / span >
< span class = "org-variable-name" > args< / span > .Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .FOc (1,1) double {mustBeNumeric} = 50e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-variable-name" > args< / span > .MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e< span class = "org-type" > -< / span > 3
< span class = "org-keyword" > end< / span >
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< h4 id = "orgac26a8b" > 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 = "orgc86b760" > Position of the Cube< / h4 >
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< p >
We define the useful points of the cube with respect to the Cube’ s center.
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\({}^{C}C\) 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];
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R = [sx, sy, sz]< span class = "org-type" > ./< / span > vecnorm([sx, sy, sz]);
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L = args.Hc< span class = "org-type" > *< / span > sqrt(3);
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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];
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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 = "org1e9ccef" > Compute the pose< / h4 >
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< p >
We can compute the vector of each leg \({}^{C}\hat{\bm{s}}_{i}\) (unit vector from \({}^{C}C_{f}\) to \({}^{C}C_{m}\)).
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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 = "org153763b" > 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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< style > . csl-entry { text-indent : -1.5 em ; margin-left : 1.5 em ; } < / style > < h2 class = 'citeproc-org-bib-h2' > Bibliography< / h2 >
< div class = "csl-bib-body" >
< div class = "csl-entry" > < a name = "citeproc_bib_item_1" > < / a > Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” < i > Measurement Science and Technology< / i > 15 (2):467– < a href = "https://doi.org/10.1088/0957-0233/15/2/022" > https://doi.org/10.1088/0957-0233/15/2/022< / a > .< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_2" > < / a > Geng, Z.J., and L.S. Haynes. 1994. “Six Degree-of-Freedom Active Vibration Control Using the Stewart Platforms.” < i > IEEE Transactions on Control Systems Technology< / i > 2 (1):45– < a href = "https://doi.org/10.1109/87.273110" > https://doi.org/10.1109/87.273110< / a > .< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_3" > < / a > Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” < i > IEEE Transactions on Robotics and Automation< / i > 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595– < a href = "https://doi.org/10.1109/tra.2003.814506" > https://doi.org/10.1109/tra.2003.814506< / a > .< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_4" > < / a > Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_5" > < / a > Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” < i > Journal of Sound and Vibration< / i > 300 (3-5):644– < a href = "https://doi.org/10.1016/j.jsv.2006.07.050" > https://doi.org/10.1016/j.jsv.2006.07.050< / a > .< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_6" > < / a > Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” < i > Journal of Sound and Vibration< / i > 439 (January). Elsevier BV:398– < a href = "https://doi.org/10.1016/j.jsv.2018.10.007" > https://doi.org/10.1016/j.jsv.2018.10.007< / a > .< / div >
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-01-08 ven. 15:30< / p >
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