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Research Papers

# Batch Process and Sensitivity Analysis of Collision Detection of Planar Convex Polygons in Motion

[+] Author and Article Information
Cheng-fu Chen

Department of Mechanical Engineering,
P.O. Box 755905,
Fairbanks, AK 99775-5905

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received March 12, 2012; final manuscript received June 28, 2013; published online August 19, 2013. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 13(4), 041001 (Aug 19, 2013) (8 pages) Paper No: JCISE-12-1044; doi: 10.1115/1.4024971 History: Received March 12, 2012; Revised June 28, 2013

## Abstract

A new method for formulation, solution, and sensitivity analysis of collision detection of convex objects in motion is presented. The collision detection problem is formulated as a parametric programming problem governed by the changes in the relative translation and relative rotation between the two objects considered. The two parameters together determine all the possible relative configurations between two moving convex objects. Therefore, solving this parametric problem allows for knowing the proximity information for all the possible configurations of the objects. We develop a two-step decomposition procedure to solve this parametric programming problem, and show that the solution is a convex function of the two parameters. This convexity feature enables an archive of the proximity information and sensitivity analysis for the collision detection problem.

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## Figures

Fig. 1

The growth model and growth distance [12]. The original shape of each object is in gray; and its growth model (similar to its original shape) is in white. Three possible proximal relations between the two objects: (a) separate (d > 1), (b) just in contact (d = 1), or (c) interfered (d < 1).

Fig. 2

A convex polygon can be expressed as the convex combination of the vectors pointing from a seed point pk to each vertex of the polygon. The motion of this object can be uniquely addressed, in the sense of kinematics, by its translational degrees of freedom pk(t) and rotational degree of freedom θK(t).

Fig. 3

The growth distance of two polygons is determined by their relative configuration regardless their global coordinates

Fig. 4

The change in the relative configuration between two convex polygons (in gray) is a kinematic combination of a pure translation Δp˜(t) and a pure rotation Δθ˜(t). (a) The initial configurations of OA and OQ, together with their growth models outlined in a just-contact condition. The change in the relative configuration is kinematically equivalent to either (b) the translation-first and rotation-second sequence Q1 → Q2 → Q3, or (c) the rotation-first and translation-second sequence Q1 →Q3 → Q2.

Fig. 5

Illustration of the two-step decomposition procedure for the change in the relative configuration of two moving convex polygons OA and OQ by a translation-rotation sequence (Q1 → Q2 → Q3). (a) Q1 → Q2 is a kinematic translation Δp˜(t), whereby the growth models OAT and OQT are scaled to be just in contact. The growth distance is l2/l1, where l1 and l2 are the size index to their original and “grown” shapes, respectively. (b) Q2 → Q3 is a kinematic rotation Δθ˜(t). Their growth distance is l3/l2.

Fig. 6

Example. (a) The initial configurations of two square objects. (b) dTrans is an unbounded, convex function of Δp˜ = [Δx˜,Δy˜] and has a minimum 0 at Δp˜ = [4,7] where the only vertex locates. (c) The projection of dTrans. The path ABC is used for the illustration in Fig. 7. (d) dRot has a periodicity of 180 deg for this example.

Fig. 7

The change in the relative translation of two square objects (in gray) follows the path ABC defined in Fig. 6(c). The growth models are shown in white. (a) The growth models of the two objects, subject to the “just in contact” condition, remain unchanged along the path from A to B. The growth models are in point-contact in configurations A and B, while in line-contact between A and B. (b) and (c) As the relative translation follows the path from B to C, the growth models is expanded until they are just in contact to each other.

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