3.2 KiB
Stewart Platform - Static Analysis
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{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}
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.