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< h1 class = "title" > Stewart Platforms< / h1 >
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< p >
< a class = 'org-ref-reference' href = "#preumont07_six_axis_singl_stage_activ" > preumont07_six_axis_singl_stage_activ< / a >
< / p >
< div id = "outline-container-org9cd44e0" class = "outline-2" >
< h2 id = "org9cd44e0" > < span class = "section-number-2" > 1< / span > Simscape Model< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< ul class = "org-ul" >
< li > < a href = "simscape-model.html" > Model of the Stewart Platform< / a > < / li >
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< li > < a href = "identification.html" > Identification of the Simscape Model< / a > < / li >
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< h2 id = "org7a44762" > < span class = "section-number-2" > 2< / span > Architecture Study< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< ul class = "org-ul" >
< li > < a href = "kinematic-study.html" > Kinematic Study< / a > < / li >
< li > < a href = "stiffness-study.html" > Stiffness Matrix Study< / a > < / li >
< li > Jacobian Study< / li >
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< li > < a href = "cubic-configuration.html" > Cubic Architecture< / a > < / li >
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< h2 id = "org77767cc" > < span class = "section-number-2" > 3< / span > Motion Control< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< ul class = "org-ul" >
< li > Active Damping< / li >
< li > Inertial Control< / li >
< li > Decentralized Control< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org9d06c58" class = "outline-2" >
< h2 id = "org9d06c58" > < span class = "section-number-2" > 4< / span > Notes about Stewart platforms< / h2 >
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< h3 id = "orgffe6651" > < span class = "section-number-3" > 4.1< / span > Jacobian< / h3 >
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< h4 id = "org6b92660" > < span class = "section-number-4" > 4.1.1< / span > Relation to platform parameters< / h4 >
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< div class = "outline-text-4" id = "text-4-1-1" >
< p >
A Jacobian is defined by:
< / p >
< ul class = "org-ul" >
< li > the orientations of the struts \(\hat{s}_i\) expressed in a frame \(\{A\}\) linked to the fixed platform.< / li >
< li > the vectors from \(O_B\) to \(b_i\) expressed in the frame \(\{A\}\)< / li >
< / ul >
< p >
Then, the choice of \(O_B\) changes the Jacobian.
< / p >
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< / div >
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< div id = "outline-container-orgcec2e05" class = "outline-4" >
< h4 id = "orgcec2e05" > < span class = "section-number-4" > 4.1.2< / span > Jacobian for displacement< / h4 >
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< div class = "outline-text-4" id = "text-4-1-2" >
< p >
\[ \dot{q} = J \dot{X} \]
With:
< / p >
< ul class = "org-ul" >
< li > \(q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]\) vector of linear displacement of actuated joints< / li >
< li > \(X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]\) position and orientation of \(O_B\) expressed in the frame \(\{A\}\)< / li >
< / ul >
< p >
For very small displacements \(\delta q\) and \(\delta X\), we have \(\delta q = J \delta X\).
< / p >
< / div >
< / div >
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< div id = "outline-container-orgbf33a4e" class = "outline-4" >
< h4 id = "orgbf33a4e" > < span class = "section-number-4" > 4.1.3< / span > Jacobian for forces< / h4 >
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< div class = "outline-text-4" id = "text-4-1-3" >
< p >
\[ F = J^T \tau \]
With:
< / p >
< ul class = "org-ul" >
< li > \(\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]\) vector of actuator forces< / li >
< li > \(F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]\) force and torque acting on point \(O_B\)< / li >
< / ul >
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< div id = "outline-container-org3710914" class = "outline-3" >
< h3 id = "org3710914" > < span class = "section-number-3" > 4.2< / span > Stiffness matrix \(K\)< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< p >
\[ K = J^T \text{diag}(k_i) J \]
< / p >
< p >
If all the struts have the same stiffness \(k\), then \(K = k J^T J\)
< / p >
< p >
\(K\) only depends of the geometry of the stewart platform: it depends on the Jacobian, that is on the orientations of the struts, position of the joints and choice of frame \(\{B\}\).
< / p >
< p >
\[ F = K X \]
< / p >
< p >
With \(F\) forces and torques applied to the moving platform at the origin of \(\{B\}\) and \(X\) the translations and rotations of \(\{B\}\) with respect to \(\{A\}\).
< / p >
< p >
\[ C = K^{-1} \]
< / p >
< p >
The compliance element \(C_{ij}\) is then the stiffness
\[ X_i = C_{ij} F_j \]
< / p >
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< div id = "outline-container-orgd5d2c08" class = "outline-3" >
< h3 id = "orgd5d2c08" > < span class = "section-number-3" > 4.3< / span > Coupling< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
< p >
What causes the coupling from \(F_i\) to \(X_i\) ?
< / p >
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< span class = "org-font-latex-sedate" > \draw< / span > [-> ] (Jt.east) -- (G.west) node[above left]{< span class = "org-font-latex-math" > $< / span > < span class = "org-font-latex-sedate" > < span class = "org-font-latex-math" > \tau< / span > < / span > < span class = "org-font-latex-math" > < span class = "org-font-latex-script-char" > _i< / span > < / span > < span class = "org-font-latex-math" > $< / span > };
< span class = "org-font-latex-sedate" > \draw< / span > [-> ] (G.east) -- (J.west) node[above left]{< span class = "org-font-latex-math" > $q< / span > < span class = "org-font-latex-math" > < span class = "org-font-latex-script-char" > _i< / span > < / span > < span class = "org-font-latex-math" > $< / span > };
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< div id = "orgc118680" class = "figure" >
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< p > < img src = "figs/coupling.png" alt = "coupling.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Block diagram to control an hexapod< / p >
< / div >
< p >
There is no coupling from \(F_i\) to \(X_j\) if \(J^{-1} G J^{-T}\) is diagonal.
< / p >
< p >
If \(G\) is diagonal (cubic configuration), then \(J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}\)
< / p >
< p >
Thus, the system is uncoupled if \(G\) and \(K\) are diagonal.
< / p >
< / div >
< / div >
< / div >
< p >
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< h1 class = 'org-ref-bib-h1' > Bibliography< / h1 >
< ul class = 'org-ref-bib' > < li > < a id = "preumont07_six_axis_singl_stage_activ" > [preumont07_six_axis_singl_stage_activ]< / a > < a name = "preumont07_six_axis_singl_stage_activ" > < / a > Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, < i > Journal of Sound and Vibration< / i > , < b > 300(3-5)< / b > , 644-661 (2007). < a href = "https://doi.org/10.1016/j.jsv.2006.07.050" > link< / a > . < a href = "http://dx.doi.org/10.1016/j.jsv.2006.07.050" > doi< / a > .< / li >
< / ul >
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