Stewart Platform - Static Analysis

Coupling

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.

Stewart Platform - Static Analysis

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Coupling