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Based on the results of the study of convex object motion 1 (J. Hopcroft and G. Wilfong, "Motion of objects in contact," Int. J. Robot. Res., 4(4), 32-46 (1986)), this paper addresses the problem of exact collision detection of a pair of scaled convex polyhedra in relative motion, and determines the contact conditions of tangential contact features, arbitrary relative motion involving translation and rotation, and uniform scaling of the objects about a fixed point. We propose a new concept of the decision curve based on analytical contact equations that characterize a continuum of scaling factors (or a single scaling factor), which ensures that a pair of objects undergoing a scaling transformation will maintain the same tangential contact feature pair (or make instantaneous tangential contact feature transitions). We propose a reliable simulation-based approach to construct the decision curve by hybridizing analytical contact equations and conventional collision detection method, called the Fast Collision Detection Method (FCDM). This method can determine whether two scaled objects will make contact at specific tangential contact features (vertices, edges, or faces) under particular uniform scaling factors and after distinctive relative motion with better accuracy and less computational time than the existing collision detection methods. Finally, we demonstrate our approach for solving motion design in simple assembly/disassembly problems.

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-This paper first presents the decision curve as an economical way that describes the correlations between scaling pairs ) , ( 2 1 ρ ρ , small relative motion conditions (type, direction and amount) and contact configurations (feature) of two scaled convex polyhedral objects where i ρ denotes the scaling factor of polyhedral object i=1,2. Via Analytical Equations of contact feature, the new decision curves after an intended relative motion of any type within its coherent range could be immediately obtained by taking advantage of original decision curve. We illustrate via two examples this method allows users to systematically and efficiently search a lot of design of fine motions to reorient two nominally contacting convex or nonconvex polyhedral objects with assigned tolerance of scaling factors that satisfy the requirements.