A Brief Overview of Collision Detection

Abstract

Collision detection is a fundamental problem in many fields of com- puter science, including physically-based simulation, path-planning and haptic rendering. Many algorithms have been proposed in the last decades to accelerate collision queries. However, there are still some open challenges: For instance, the extremely high frequencies that are required for haptic rendering. In this thesis we present a novel geometric data structure for collision detection at haptic rates between arbitrary rigid objects. The main idea is to bound objects from the inside with a set of non-overlapping spheres. Based on such sphere packings, an “inner bounding volume hierarchy” can be con- structed. Our data structure that we call Inner Sphere Trees supports different kinds of queries; namely proximity queries as well as time of impact computations and a new method to measure the amount of interpenetration, the penetration volume. The penetration volume is related to the water displacement of the overlapping region and thus, corresponds to a physically motivated force. Moreover, these penalty forces and torques are continuous both in direction and magnitude. In order to compute such dense sphere packings, we have devel- oped a new algorithm that extends the idea of space filling Apol- lonian sphere packings to arbitrary objects. Our method relies on prototype-based approaches known from machine learning and leads to a parallel algorithm. As a by-product our algorithm yields an ap- proximation of the object’s medial axis that has applications ranging from path-planning to surface reconstruction. Collision detection for deformable objects is another open chal- lenge, because pre-computed data structures become invalid under deformations. In this thesis, we present novel algorithms for effi- ciently updating bounding volume hierarchies of objects undergoing arbitrary deformations. The event-based approach of the kinetic data structures framework enables us to prove that our algorithms are opti- mal in the number of updates. Additionally, we extend the idea of ki- netic data structures even to the collision detection process itself. Our new acceleration approach, the kinetic Separation-List, supports fast continuous collision detection of deformable objects for both, pair- wise and self-collision detection. In order to guarantee a fair comparison of different collision de- tection algorithms we propose several new methods both in theory and in the real world. This includes a model for the theoretic run- ning time of hierarchical collision detection algorithms and an open source benchmarking suite that evaluates both the performance as well as the quality of the collision response. Finally, our new data structures enabled us to realize some new applications. For instance, we adopted our sphere packings to define a new volume preserving deformation scheme, the sphere-spring system, that extends the classical mass-spring systems. Furthermore, we present an application of our Inner Sphere Trees to real-time obstacle avoidance in dynamic environments for autonomous robots, and last but not least we show the results of a comprehensive user study that evaluates the influence of the degrees of freedom on the users performance in complex bi-manial haptic interaction tasks.