Synthesis and Singularity Analysis of N-$UU$ Parallel Wrists: A Symmetric Space Approach

We report some recent advances in kinematics and singularity analysis of the mirror-symmetric $N$-$UU$ parallel wrists using symmetric space theory. We show that both the finite displacement and infinitesimal singularity kinematics of a $N$-$UU$ wrist are governed by the mirror symmetry property and half-angle property of the underlying motion manifold, which is a symmetric submanifold of the special Euclidean group SE(3). Our result is stronger than and may be considered a closure of Hunt's argument for instantaneous mirror symmetry in his pioneering exposition of constant velocity shaft couplings. Moreover, we show that the wrist can, to some extent, be treated as a spherical mechanism, even though dependent translation exists, and the singularity-free workspace of a $N$-$UU$ wrist may be analytically derived. This leads to a straightforward optimal design for maximal singularity-free workspace.