Output-Sensitive Computation of All Form-Closure Grasps of a Semi-Algebraic Set

We propose the first efficient output-sensitive algorithms for computing all form-closure grasps of a planar curved part P with at most four frictionless point contacts. The boundary of P consists of m concave vertices and n algebraic arcs with a constant degree. All our algorithms are output-sensitive, which means that their running times largely depend on the actual output size K rather than the (often much larger) maximum size of the output. More specifically, we show how to determine • all form-closure grasps with four points along four arcs in O(n^8/3log^1/3n + K) time, • all form-closure grasps with four points along three arcs in O(n^5/2+ε+ K) time, • all form-closure grasps with one point at a concave vertex and two points along two arcs in O(n²m^1/2+ε+ K) time, • all form-closure grasps with one point at a concave vertex and two points along a single arc in O(nm) or O(n^3/2+ε+ K) time (depending on the size of m), • all form-closure grasps with two points at concave vertices and one point along arc in O(nm²) or O (n^2+ε+ K) time (depending on the size of m), where ε is an arbitrarily small positive constant. All our algorithms rely on the geometric condition in three-dimensional wrench space, which is transformed into two-dimensional geometric intersection problems.