On the hybrid workpiece localization/envelopment problems

This paper defines a hybrid localization/envelopment problem, develops a formulation of the hybrid localization/envelopment problem, and presents a simple algorithm for computing its solutions. First, we show that when the finished surfaces of a workpiece are inadequate to fully constrain the rigid motions of the workpiece, then the remaining set of free motions must form a subgroup G₀ of the Euclidean group SE(3). This allows us to decompose the hybrid problem into a (symmetric) localization problem on the homogeneous space SE(3)/G₀ and an envelopment problem on G₀. The geometric properties of SE(S)/G₀ are used to convert the envelopment problem into a nonlinear programming problem with a convex objective function, which is then solved using techniques from nonlinear programming. Finally, we present simulation results to illustrate the effectiveness of our method for the hybrid problem