Geometric Design of Spherical Serial Chains With Curvature Constraints in the Environment

This paper builds up on recent results on planar kinematic synthesis with contact direction and curvature constraints on the workpiece. We consider the synthesis of spherical serial chains to guide a rigid body, such that it does not violate normal direction and curvature constraints imposed by contact with objects in the environment. We show how to derive these constraints from the geometry of the task and transform them into conditions on velocity and acceleration of points in the moving body to obtain synthesis equations which can be solved by algebraic elimination. Trajectory interpolation formulas yield the movement of the chain with the desired contact properties in each of the task positions. An example shows the application of the developed theory to the failure recovery of a robot manipulator, using kinematic synthesis techniques.