Finding effective "force targets" for two-dimensional, multifinger frictional grips

In this paper we apply techniques from computational geometry to solve several problems in grasp planning and control in robotics. We consider the problem of calculating "force targets " for a collection of n fingers which grasp a two-dimensional object at known positions, at which the normals to the surface are also assumed to be known at least approximately. If the points at which the fingers touch the body do not allow a positive grip to be exerted (i.e., a grip in which the fingers hold the body in equilibrium by exerting friction-free forces in the directions of the corresponding inward-directed normals), it is appropriate to find the smallest coefficient of friction for which it is possible to assign a set of forces to be exerted by the fingers (so-called finger-force targets) which hold the object at equilibrium and such that each individual force lies within the corresponding cone of friction. We present an algorithm for this problem which runs in time 0(n log² n log log n). We also present another algorithm for preprocessing the given data so as to allow fast computation of the desired coefficient of friction for the case in which one needs to balance any given "query" external force and torque. Finally, we discuss simpler variants of our techniques which are likely to be more efficient when the problem is solved for a small number of fingers.