Output-Sensitive Computation of Force-Closure Grasps of a Semi-Algebraic Object

We propose a technique which significantly simplifies the computation of frictionless force-closure grasps of a curved planar part
$P$
. We use a colored projection scheme from the three-dimensional wrench space to two-dimensional screens, which allows us to reduce the problem of identifying combinations of arcs and concave vertices of
$P$
that admit frictionless force-closure grasps, to colored intersection searching problems in the screens. We show how to combine this technique with existing intersection searching algorithms to obtain efficient, output-sensitive algorithms to compute all force-closure grasps of
$P$
, where at most four hard, frictionless point contacts exert exactly four wrenches on
$P$
. If the boundary of
$P$
consists of
$n$
algebraic arcs of constant complexity and
$m$
concave vertices, we show how to compute all force-closure grasps with:
four contacts along four arcs in $O(n^{8/3}\log ^{1/3}n+K)$ time;