Theory and Methodology
Fast collision detection in four-dimensional space

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Abstract

In this paper, we consider the collision detection problem for general objects. A four-dimensional approach is proposed for this problem which detects exactly and in one-step when and where the earliest collison will occur between the objects. This is done by using four-dimensional sets to represent the objects in both space and time. The problem is then posed as a nonlinear programming problem. The algorithm can handle the case of a rigid body moving on a general path in R2 or R3 with simultaneous translation and rotation. Simulation results on some example problems are given, and show that the algorithm is superior to those available in the literature.

Introduction

The collision detection problem has been extensively studied in the robotics literature and various classes of algorithms are available 1, 2, 3, 5, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26. For a survey of these various algorithms, see Refs. 1, 2. In [1], two algorithms that use the multiple interference methodology [10] are presented. The algorithms apply to objects that can be represented as polyhedral sets defined as intersections of finite halfspaces or as convex hulls of a finite number of vertices. The algorithms can give in principle exactly the collision points (in both time and space) for objects moving on a general path in R2 or R3 with simultaneous translation and rotation. However, since the algorithms essentially reduce the dynamic interference detection problem (collision detection) into a static interference checking by discretizing the space-time frame into finite grids at which intersection is tested, the decisive choice of the time step length is very crucial to the reliability of the algorithms. Furthermore, polytopic representation of objects may be very poor in certain applications especially when smooth objects are involved.

In this paper, we remove some of the limitations of the algorithms in [1], and perform collision detection in four-dimensional space directly. A formal notion of four-dimensional intersection testing was introduced by Cameron 10, 11 using an “extrusion” operation. However, constructive solid geometry was used in the implementation of the algorithm, and various techniques for null set detection were employed. Therefore in this paper, we follow up the development in 10, 11 and present analytical methods for four-dimensional intersection testing which are more amenable to computations, and make the algorithm computationally efficient. The algorithm also finds the collision point in one-step without discretization, and can be applied to both polytopic and smooth objects (that may be nonconvex in nature) moving on a general path in R3 (not necessarily linear) with simultaneous translation and rotation.

In Section 2, we review methods of representation of objects in four-dimensional space as a basis for the new algorithm. Then in Section 3, we formally introduce the problem and propose the new algorithm. We also discuss some of the ways that the algorithm can be extended to nonconvex objects in Section 4. This is followed by computational results from simulations obtained using the algorithm in Section 5. Finally, in Section 6, we give conclusions and suggestions for future work.

Section snippets

Four-dimensional representation of moving objects

Cameron 10, 11 used the concepts of sweeping and extrusion to represent the four-dimensional sets generated by moving objects. While the former refers to the volume swept by the moving object over time, the latter refers to the set of all points occupied by the object at a particular time t. For an object A represented by a compact set KA, with a location function ΛA(t) which describes its position at a given time t, its swept volume over a finite time interval is represented by the setSw(KAA

Collision detection in four-dimensional space

A formal definition of the collision detection problem is as follows:

Definition 3.1. Given representations of N+1 convex objects A,B1,B2,…,BN, whose locations in space at any time t are represented by the sets GA(t),GB1(t),GB2(t),…,GBN(t), respectively, over a time interval [ts,tf], determine whether any pair of the objects occupy some common space at the same time during this interval.

Without any loss of generality, we can assume A in the above definition to be a robot (or robot link) moving

Extension to nonconvex objects

In the previous sections, we have presented an approach for representing complex objects in three-dimensional space using convex sets, and developed an algorithm for detecting possible collisions between them when their motion is parameterized in time. In this section, we seek to extend the representation in Section 2to cover nonconvex objects. Such objects abound everywhere and a complete solution of the collision detection problem should certainly encompass nonconvex objects. It has always

Simulation results

In this section we give results of simulation with the algorithm on various example problems to show its efficiency. We use a standard subroutine “constr” for solving constrained optimization problems from the MATLAB Optimization Toolbox [27] to implement the algorithm.

The first example we consider is from [10].

Example 5.1. Fig. 2 shows a sphere of radius 4 centered at (5, 5, 5) at time 0 moving with velocity (1, 1, 1) unit/s and a cube of sides 4 units centered at (44, 54, 5) at time 0 moving with

Conclusion

We have presented a collision detection algorithm for detecting possible collisions between moving objects in three-dimensional space. The algorithm can deal with both polyhedral and smooth objects that can be represented by systems of linear or nonlinear inequalities. Moreover, it can handle the case of an object moving on a general path (parameterized in time) in three-dimensional space with simultaneous translation and rotation. To the best of our knowledge, this is the first of its kind

Acknowledgements

The authors wish to acknowledge the assistance of Mr. M. Atif Memun in preparing some of the figures, and king Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia for supporting this work.

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