Computing the Globally Optimal Frictionless Fixture in a Discrete Point Set

This paper presents an algorithm to compute the globally optimal fixture with frictionless contacts in a discrete point set on an object. The capability of a fixture to immobilize the object is evaluated by the minimum of the largest conflict of the object with the contacts over all motion directions, which can be reduced to the radius of the largest origin-centered ball contained in the convex hull of primitive contacts wrenches. All candidate fixtures (combinations of the discrete points of a certain number) are expressed in a tree structure, in which each node contains a discrete point representing a contact location and each path from the roots to a leaf represents a fixture. A preorder walk of the tree is performed to search for the optimal fixture. Necessary conditions are derived to predict whether the subtree of a node contains a better fixture. If any necessary condition is not met, then all the fixtures contained in the subtree can be simply rejected. By this means only a small portion of the tree will be traversed and the globally optimal fixture can be found more quickly. This algorithm has been tested on various three-dimensional objects and has been found to be over ten times faster than the brute-force search.