Admittance Control¶
Basic Equation¶
\[M\ddot{x}+B\dot{x}+Kx=F\]
- \(F\in\mathbb R^N\)
- input, external force
- \(x,\dot{x},\ddot{x}\in\mathbb R^N\)
- output, controled movement variables
- \(M\in\mathbb R\)
- mass
- \(B\in\mathbb R^{N\times N}\)
- damp
- \(K\in\mathbb R^{N\times N}\)
- stiffness
State-space Model¶
\[\begin{split}\begin{split}
A&=\begin{bmatrix} -\frac BM & -\frac KM \\ 1 & 0 \end{bmatrix}\\
B&=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\\
C&=\begin{bmatrix} 0 & \frac 1M \end{bmatrix}\\
D&=\begin{bmatrix} 0 \end{bmatrix}
\end{split}\end{split}\]
let \(T_s\ll T\) and convert it to discrete form,
\[\begin{split}\begin{split}
A&=\begin{bmatrix} 1-\frac BMT_s & -\frac KMT_s \\ T_s & 1 \end{bmatrix}\\
B&=\begin{bmatrix} T_s \\ 0 \end{bmatrix}\\
C&=\begin{bmatrix} 0 & \frac 1M \end{bmatrix}\\
D&=\begin{bmatrix} 0 \end{bmatrix}
\end{split}\end{split}\]