Task-space/joint-space damping transformations for passive redundant manipulators

Passive mechanical wrists, capable of imparting a desired damping matrix to a grasped workpiece are discussed. The passive mechanical wrist is to be programmable-it can adopt a wide range of damping matrices-by virtue of a number of tunable dampers which interconnect the joints. The range of damping matrices that such a wrist can adopt is studied, purely by tuning its dampers. It is found that a redundant wrist has a broader range of realizable damping matrices than a non-redundant wrist. A kinematic Jacobian relates the task-space damping matrix to a similar matrix in the hydraulic space of the tunable dampers (joint-space). For redundant wrists, the transformation of damping matrices between task-space and joint-space is not straightforward. The causal directions along which the transformations are linear are identified. It is shown that the joint-space matrices which are obtained as linear transformations of desired task-space matrices are all singular. Many realizable joint-space matrices (corresponding to a desired task-space damping matrix) are shown to exist which are discovered by linear transformations