Singularities of the Euler wrist

We investigate the singular behavior of a simple 3 degrees of freedom rotational mechanism whose kinematics are represented by the Z-Y-Z Euler angles, called the Euler wrist. Our main result states that all singularities of the Euler wrist have corank 1, and that around these singularities the wrist kinematics are locally equivalent to the hyperbolic normal form. An analogous result holds for the Roll-Pitch-Yaw wrist. By the equivalence of all 3-axis wrist mechanisms, this result extends further to any 3-axis wrist. The hyperbolic normal form explains completely the singular behavior of the wrist mechanisms and plays a principal role in solving the singular inverse kinematic problem for these mechanisms within the normal form approach.