Computing Wrench Cones for Planar Rigid Body Contact Tasks

The contribution of this paper is the expansion of the range of possibilities in the analysis, planning, and control of contact tasks. The successful execution of any contact task fundamentally requires the application of wrenches (forces and moments) consistent with the task. We develop an algorithm for computing the entire set of external wrenches consistent with achieving a given augmented contact mode (e.g., sliding at contact 1, rolling at contact 2, and approaching potential contact 3) for one fixed and one moving part in the plane. Unfortunately, because of the problem of frictional indeterminacy, it is usually not enough to determine the set of wrenches consistent with achieving a contact mode. Some of the computed wrenches may also be consistent with other undesirable contact modes. However, set operations on the cones returned by our algorithm can be designed to find the wrenches consistent only with desired contact modes. In this paper we also present some applications of the algorithm to analysis and planning problems. We show how to use set operations to compute the set of external wrenches with respect to which a fixtured part is `strongly stable'. If the applied external wrench is in this set, the part is guaranteed not to move. We also show how the algorithm may be used to create sensorless plans that guarantee that a workpiece will be correctly inserted into a fixture.