Grasping and Fixturing as Submodular Coverage Problems

Grasping and fixturing are concerned with immobilizing objects. Most prior work in this area uses a small number of contacts that is sufficient for force closure. However, for delicate objects or surfaces such as glass or bone (in medical applications), a larger number of contacts can be used to reduce the forces needed at each contact to resist applied wrenches. We focus on the problem of choosing a set of contact points out of a larger set of candidates to optimize grasp quality, which is defined as the size of the ball in wrench space that can be resisted given a constraint on each contact force or the sum of the contact forces. In purely geometric terms, our algorithms select a set of vectors out of a larger set of candidates to maximize the residual radius of the Minkowski sum or convex hull. We provide a method that is guaranteed to find near-optimal solutions in linear time. At the core of our approach are (i) an novel formula for the quality functions, and a discretization technique that evaluates them with bounded error, (ii) the insight that the resulting problem is a submodular coverage problem. This allows us to exploit the submodular saturation algorithm, which has recently been derived for applications in sensor placement. Our alternative formulas also makes it possible relax the optimization problem into a linear program, and we give a branch-and-bound procedure for exactly optimizing the objective. Our approach is applicable in situations with or without friction, for a large class of friction models. We also show that grasp quality metrics are relevant to towing (carrying a heavy object with aerial vehicles), and describe how to use the same methods to find optimal towing configurations.