Kinestatic Filtering for Hybrid Control of Constrained Rigid Body Motion

Abstract

The kinestatic filters proposed in this paper are based on the Moore-Penrose generalized inverse of the Jacobian of the mechanism to be manipulated by driving wrenches, whether the mechanism is a robot manipulator or a constrained rigid body. Driving wrenches are wrenches that cannot be expressed as a linear combination of wrenches-ofconstraint. We show that necessary and sufficient conditions for the existence of a wrench filter is that the set of driving wrenches form a subspace. The resultant wrench filter must be an orthogonal projection and equals the product of the Moore-Penrose inverse of the Jacobiantranspose times the Jacobian-transpose. The Moore-Penrose inverse must be independent of any norm defined on the wrench space when the driving wrenches form a subspace. The wrench filter, when it exists, also filters the twists-offreedom. This development extends the work in [7] and [2] and shows that the results in [2] apply only when the Jacobian has full column-rank and the driving wrenches form a subspace. We also demonstrate that when the driving wrenches form a subspace, then the twists-of-constraint, directions in which the mechanism cannot move, also form a subspace and are filtered by the same filter that filters wrenches-of-constraint. Finally, we determine that, under a coordinate transformation from a frame in which the driving wrenches form a subspace to one in which they do not, the wrench and twist filters in the new frame are neither equal nor orthogonal, but are related by a similarity transform.