Singularity-free Path-planning of Dexterous Pointing Tasks for a Class of Spherical Parallel Mechanisms

The subject of the paper is the optimization of the pose of redundant spherical parallel manipulators (SPMs) performing pointing tasks, i.e., orienting a line in space; a few hints for the synthesis of dextrous machines are also provided. The strengths of parallel mechanisms are often limited by their singular configurations; to overcome this problem, a class of SPMs with 3-RFR topology (Revolute-Planar-Revolute pairs for each of the three limbs) is investigated, with a comprehensive analysis to identify their singular configurations. The geometric parameters of the architectures are chosen in such a way that all kinds of singularity surfaces come to coincide and a single 3 × 3 Jacobian matrix is able to describe all the singular configurations of the machines. Then, singularities are avoided by exploiting the functional redundancy of the manipulator with respect to the task: an optimization algorithm allows the user to find, for each pointing direction, the robot posture that yields the highest dexterity. Finally, an optimal path is traced on the surface of the sphere of motion by finding the Bézier curve that minimizes a task performance index. The kinematic analysis and optimization are worked out by using a robust formulation based on rotation invariants that allows for a straightforward generalization of the results obtained.