TY - JOUR

T1 - Convex hull representations of models for computing collisions between multiple bodies

AU - Sherali, Hanif D.

AU - Smith, J. Cole

AU - Selim, Shokri Z.

N1 - Funding Information:
This material is based upon work supported by the National Science Foundation under Grant Number DMI-9812047. The authors also thank the King Fahd University of Petroleum and Minerals for its support during the performance of this work.

PY - 2001/12/16

Y1 - 2001/12/16

N2 - In this paper, we consider a collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of S.Z. Selim and H.A. Almohamad [European Journal of Operational Research 119 (1) (1999) 121-129], and compare the relative effectiveness in solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear segment. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once. We provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios, as well as the times at which the first collision occurs.

AB - In this paper, we consider a collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of S.Z. Selim and H.A. Almohamad [European Journal of Operational Research 119 (1) (1999) 121-129], and compare the relative effectiveness in solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear segment. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once. We provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios, as well as the times at which the first collision occurs.

KW - Convex hull

KW - Integer programming

KW - Linear programming

KW - Multi-body collision

UR - http://www.scopus.com/inward/record.url?scp=0035900866&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035900866&partnerID=8YFLogxK

U2 - 10.1016/S0377-2217(00)00324-6

DO - 10.1016/S0377-2217(00)00324-6

M3 - Article

AN - SCOPUS:0035900866

SN - 0377-2217

VL - 135

SP - 514

EP - 526

JO - European Journal of Operational Research

JF - European Journal of Operational Research

IS - 3

ER -