We present a simple local geometric characterization of the configuration space of two polyhedra in coratact that provides a representation of all infinitesimal motions that separate them. The considered polyhedra are general in the sense that they possibly have nonconvex faces and arbitrary number of holes.The approach presented herein has two main advantages over former ones: (a) it only relies on the classical basic contacts between polyhedra, i.e. the vertexface and edge-edge contacts; and (b) it does not require the local decomposition of non-convem"ties into convex parts.
IntroductionThis paper presents an algorithm for computing the infinitesimal motions separating two polyhedra A4 and S in contact, seen as a moving body and a static environment, respectively.The algorithm is complete, in the sense that it finds all feasible motions, and conceptually simpler than previous approaches.Our algorithm mainly operates in the 6-dimensional configuration space of &f, i.e. SO(3) x w. First, it considers the C-surfaces associated with the basic contacts at a configuration X.. Then, it computes the arrangement of tangent hyperplanes to these surfaces in X., obtaining all 6-dimensional convex cones it induces in !R6. Finally, the output is a classification of each of these cones into occupied or free, depending on whether the configurations it contains correspond to intersecting or separated polyhedra. This classification is efficiently attained by evaluating simple boolean expressions that depend on the geometric configurations of the basic contacts. The problem at hand arises in the assembly partitioning problem given a collection of polyhedral parts in contact, is there some infinitesimal motion that rigidly removes some subset of the parts from the rest? A negative answer means that the parts cannot be assembled by a sequence of rigid motions that fit them together. A positive answer, with the possible directions of motion, does not provide a plan on how to disassemble the parts, since the subsets need to be separated completely -and not only infinitesimally. However, this constitutes a very useful information for an assembly sequence planner [20].The paper is structured as follows. Section 2 summarizes the contributions of this work and its connections with previous research in the field. Section 3 introduces several basic concepts needed throughout the paper. Afterwards, section 4 gives a general description of the algorithm, which is followed by explanations on the role of non-effective contacts (section 5), and details on the main geometric constructions: the computation of tangent hyperplanes (section 6) and the construction of the resulting plane arrangement (section 7). The characterization of free and occupied cones follows in section 8. Finally, section 9 gives the conclusions and points that deserve further attention.
Related workMany different approaches to the motion planning and related problems have been presented over the years. A common topic of many of them is Configuration Space, the spa...