Curvature-based computation of antipodal grasps

It is well known that antipodal grasps can be achieved on curved objects in the presence of friction. This paper presents an efficient algorithm that finds, up to numerical resolution, all pairs of antipodal points on a closed, simple, and twice continuously differentiable plane curve. Dissecting the curve into segments everywhere convex or everywhere concave, the algorithm marches simultaneously on a pair of such segments with provable convergence and interleaves marching with numerical bisection. It makes use of new insights into the differential geometry at two antipodal points. We have avoided resorting to traditional nonlinear programming which would neither be quite as efficient nor guarantee to find all antipodal points. Dissection and the coupling of marching with bisection introduced in this paper are potentially applicable to many optimization problems involving curves and curved shapes.