Displacement Analysis of Spherical Mechanisms Having Three or Fewer Loops

Spherical linkages, having rotational joints whose axes coincide in a common centerpoint, are sometimes used in multi-degree-of-freedom robot manipulators and in one-degree-of-freedom mechanisms. The forward kinematics of parallel-link robots, the inverse kinematics of serial-link robots and the in-put/output motion of single-degree-of-freedom mechanisms are all problems in displacement analysis. In this article, loop equations are formulated and solved for the displacement analysis of all spherical mechanisms up to three loops. We show how to solve each mechanism type using either a formulation in terms of rotation matrices or quaternions. In either formulation, the solution method is a modification of Sylvester’s elimination method, leading directly to numerical calculation via standard eigenvalue routines.