Stewart Platforms
-
-
1 Simscape Model
+
+
-1 Simscape Model
- Model of the Stewart Platform @@ -260,8 +282,8 @@ for the JavaScript code in this tag.
-
2 Architecture Study
+
+
-2 Architecture Study
- Kinematic Study @@ -272,8 +294,8 @@ for the JavaScript code in this tag.
-
diff --git a/index.org b/index.org
index 5155291..222b12a 100644
--- a/index.org
+++ b/index.org
@@ -9,6 +9,16 @@
#+HTML_HEAD:
#+HTML_HEAD:
#+HTML_HEAD:
+
+#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
+#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
+#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
+#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
+#+PROPERTY: header-args:latex+ :results raw replace :buffer no
+#+PROPERTY: header-args:latex+ :eval no-export
+#+PROPERTY: header-args:latex+ :exports both
+#+PROPERTY: header-args:latex+ :mkdirp yes
+#+PROPERTY: header-args:latex+ :output-dir figs
:END:
* Simscape Model
@@ -25,3 +35,73 @@
- Active Damping
- Inertial Control
- Decentralized Control
+* Notes about Stewart platforms
+** Jacobian
+*** Relation to platform parameters
+A Jacobian is defined by:
+- the orientations of the struts $\hat{s}_i$ expressed in a frame $\{A\}$ linked to the fixed platform.
+- the vectors from $O_B$ to $b_i$ expressed in the frame $\{A\}$
+
+Then, the choice of $O_B$ changes the Jacobian.
+
+*** Jacobian for displacement
+\[ \dot{q} = J \dot{X} \]
+With:
+- $q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]$ vector of linear displacement of actuated joints
+- $X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]$ position and orientation of $O_B$ expressed in the frame $\{A\}$
+
+For very small displacements $\delta q$ and $\delta X$, we have $\delta q = J \delta X$.
+
+*** Jacobian for forces
+\[ F = J^T \tau \]
+With:
+- $\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]$ vector of actuator forces
+- $F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]$ force and torque acting on point $O_B$
+
+** Stiffness matrix $K$
+
+\[ K = J^T \text{diag}(k_i) J \]
+
+If all the struts have the same stiffness $k$, then $K = k J^T J$
+
+$K$ 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 $\{B\}$.
+
+\[ F = K X \]
+
+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 $\{A\}$.
+
+\[ C = K^{-1} \]
+
+The compliance element $C_{ij}$ is then the stiffness
+\[ X_i = C_{ij} F_j \]
+
+** Coupling
+What causes the coupling from $F_i$ to $X_i$ ?
+
+#+begin_src latex :file coupling.pdf :post pdf2svg(file=*this*, ext="png") :exports both
+ \begin{tikzpicture}
+ \node[block] (Jt) at (0, 0) {$J^{-T}$};
+ \node[block, right= of Jt] (G) {$G$};
+ \node[block, right= of G] (J) {$J^{-1}$};
+
+ \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$F_i$};
+ \draw[->] (Jt.east) -- (G.west) node[above left]{$\tau_i$};
+ \draw[->] (G.east) -- (J.west) node[above left]{$q_i$};
+ \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$X_i$};
+ \end{tikzpicture}
+#+end_src
+
+#+name: fig:block_diag_coupling
+#+caption: Block diagram to control an hexapod
+#+RESULTS:
+[[file:figs/coupling.png]]
+
+There is no coupling from $F_i$ to $X_j$ if $J^{-1} G J^{-T}$ is diagonal.
+
+If $G$ is diagonal (cubic configuration), then $J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}$
+
+Thus, the system is uncoupled if $G$ and $K$ are diagonal.
+
+* Bibliography :ignore:
+bibliographystyle:unsrt
+bibliography:references.bib
diff --git a/simscape-model.org b/simscape-model.org
index 7a8ff35..fc42f6a 100644
--- a/simscape-model.org
+++ b/simscape-model.org
@@ -20,6 +20,7 @@
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:
+* Introduction :ignore:
Stewart platforms are generated in multiple steps.
First, geometrical parameters are defined:
3 Motion Control
+
+
+3 Motion Control
- Active Damping @@ -282,6 +304,145 @@ for the JavaScript code in this tag.
+
+
+4 Notes about Stewart platforms
+
+
+
+
+
+4.1 Jacobian
+
+
+
+
+
+4.1.1 Relation to platform parameters
+
+
++A Jacobian is defined by: +
+-
+
- the orientations of the struts \(\hat{s}_i\) expressed in a frame \(\{A\}\) linked to the fixed platform. +
- the vectors from \(O_B\) to \(b_i\) expressed in the frame \(\{A\}\) +
+Then, the choice of \(O_B\) changes the Jacobian. +
+
+
+
+4.1.2 Jacobian for displacement
+
+
++\[ \dot{q} = J \dot{X} \] +With: +
+-
+
- \(q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]\) vector of linear displacement of actuated joints +
- \(X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]\) position and orientation of \(O_B\) expressed in the frame \(\{A\}\) +
+For very small displacements \(\delta q\) and \(\delta X\), we have \(\delta q = J \delta X\). +
+
+
+4.1.3 Jacobian for forces
+
+
++\[ F = J^T \tau \] +With: +
+-
+
- \(\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]\) vector of actuator forces +
- \(F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]\) force and torque acting on point \(O_B\) +
+
+
+4.2 Stiffness matrix \(K\)
+
+
++\[ K = J^T \text{diag}(k_i) J \] +
+ ++If all the struts have the same stiffness \(k\), then \(K = k J^T J\) +
+ ++\(K\) 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 \(\{B\}\). +
+ ++\[ F = K X \] +
+ ++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 \(\{A\}\). +
+ ++\[ C = K^{-1} \] +
+ ++The compliance element \(C_{ij}\) is then the stiffness +\[ X_i = C_{ij} F_j \] +
+
+
+4.3 Coupling
+
+
++What causes the coupling from \(F_i\) to \(X_i\) ? +
+ +
+
+
+
+\begin{tikzpicture} + \node[block] (Jt) at (0, 0) {$J^{-T}$}; + \node[block, right= of Jt] (G) {$G$}; + \node[block, right= of G] (J) {$J^{-1}$}; + + \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$F_i$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\tau_i$}; + \draw[->] (G.east) -- (J.west) node[above left]{$q_i$}; + \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$X_i$}; +\end{tikzpicture} ++
+
+
+
+
Figure 1: Block diagram to control an hexapod
++There is no coupling from \(F_i\) to \(X_j\) if \(J^{-1} G J^{-T}\) is diagonal. +
+ ++If \(G\) is diagonal (cubic configuration), then \(J^{-1} G J^{-T} = G J^{-1} J^{-T} = G (J^{T} J)^{-1} = G K^{-1}\) +
+ ++Thus, the system is uncoupled if \(G\) and \(K\) are diagonal. +
++ +references.bib +