Range of Motion of the iCub Shoulders

The range of motion of the shoulders of iCub 2.x is constrained by the length of the tendons. Unfortunately, these constraints cannot be written in the form \(\left[\theta_{\text{min}}, \; \theta_{\text{max}}\right]\), which are the constraints currently implemented in the firmware.

Warning

Therefore, care must be taken when controlling the arm in the Joint Space to avoid reaching configurations that may break the tendons.

Note

By contrast, when controlling the arm in the Cartesian Space by means of the Cartesian Controller, the software will always find solutions for the reaching task that comply with the cable length constraints.

Correct Limits

The length of the tendons imposes the following constraints on the shoulder's joints:

\[ \begin{equation} \mathbf{A} \cdot \mathbf{q} + \mathbf{b} > 0, \end{equation} \]

where \(\mathbf{q}\) is the vector of the 3 shoulder's joints expressed in radians, whereas:

\[ \begin{equation} \mathbf{A} = \begin{pmatrix} c & -c & 0 \\ -1 & 1 & 0 \\ -c & c & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 0 \\ c & -c & -c \\ -1 & 1 & 1 \\ -c & c & c \\ 1 & -1 & -1 \\ 0 & 1 & 1 \\ 0 & -1 & -1 \end{pmatrix} \quad \mathbf{b} = \frac{\pi}{180} \cdot \begin{pmatrix} 404 \\ 54.3 \\ 46 \\ 305.7 \\ 215.7 \\ 150 \\ 54.3 \\ 210 \\ 431 \\ 101.7 \\ 109 \\ 258.3 \\ 71.7 \\ 228.3 \end{pmatrix}, \end{equation} \]

with \(c = 1.71\).

Remarkably, this set of constraints can be reduced by solving a linear programming problem whose outcome is:

\[ \begin{equation} \mathbf{A^*} = \begin{pmatrix} c & -c & 0 \\ c & -c & -c \\ 0 & 1 & 1 \\ -c & c & c \\ 0 & -1 & -1 \end{pmatrix} \quad \mathbf{b^*} = \frac{\pi}{180} \cdot \begin{pmatrix} 347 \\ 366.57 \\ 66.6 \\ 112.42 \\ 213.3 \end{pmatrix} \end{equation} \]

Note

The Cartesian Controller software component implements the couple \(\left( \mathbf{A^*}, \; \mathbf{b^*}\right)\).