Manipulating algebraic parts in the plane

When manipulating parts, it is important to determine the orientation of the part with respect to the gripper. This orientation may not be known precisely or may be disturbed by the act of grasping. In some cases, it is possible to use mechanical compliance to orient parts during grasping. Goldberg (1993) showed that any part with polygonal boundary can be oriented and grasped in this manner using a parallel-jaw gripper. Many of the curves currently used in engineering design are algebraic but nonlinear. Although these curves can be approximated as polygons for the purpose of visualization, such approximations can lead to false conclusions about mechanical behavior. In this paper we consider the class of parts whose planar projection has a piecewise algebraic convex hull. Our primary result is a proof that a grasp plan exists for any such part. We give a planning algorithm that produces the shortest plan and runs in time O(n² log n+N), where n is the number of transitions in the grasp function and N is the length of the plan produced. We believe this to be the first paper to address the problem of manipulating algebraic parts when nothing is know about their initial orientation