Stewart Platforms
preumont07_six_axis_singl_stage_activ
1 Simscape Model
2 Architecture Study
- Kinematic Study
- Stiffness Matrix Study
- Jacobian Study
- Cubic Architecture
3 Motion Control
- Active Damping
- Inertial Control
- Decentralized Control
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.
Bibliography
- [preumont07_six_axis_singl_stage_activ] Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, & Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, Journal of Sound and Vibration, 300(3-5), 644-661 (2007). link. doi.