Stewart Platforms

preumont07_six_axis_singl_stage_activ

1 Simscape Model

2 Architecture Study

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}

coupling.png

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