Efficient Collision Detection for Animation and Robotics Part 5 ppt

F E F N F Voronoi Region P [...]... implementation gives better performance and probably is the fastest implemented collision detection algorithm Please see 65 , 13 , 17 , 40 , 39 , 38 , 72 , and 90  This is especially true in a dynamic environment where the trajectories are not known nor are they in closed form even between impacts, but are given by elliptic integrals The expected constant time performance comes from the fact that... initialization and preprocessing procedures The speed of algorithm is an attribute from the incremental nature of our approach and the geometric coherence between successive, discrete movements 3.6 Dynamic Collision Detection for Convex Polyhedra The distance computation algorithm can be easily used for detecting collision while the objects are moving This is done by iteratively checking the distance 57 between... points and the new candidate features if necessary, since local applicability constraints are all we need for tracking a closest pair of closest features The transformation is done by taking the relative transformation between two objects For example, given two objects moving in space, their motions with respect to the origin of the world frame can be characterized by the transformation matrices TA and. .. then a collision is declared We can set the step size of the algorithm once we obtain the initial position, distance, velocity and acceleration among all object pairs We also use a simple queuing scheme to reduce the frequency of collision checks by relying on the fact that only the object pairs which have a small Chapter 5 Furthermore, we will not need to transform all the Voronoi regions and all.. .55 (a) (d) (b) (e) (c) (f) Figure 3.13: Polytopes Used in Example Computations 56 Objects M1 + M2 = N w o Init w Init a,b 8 + 8 = 16 21 3.2 a,f 8 + 16 = 24 23 3.3 a,c 8 + 24 = 32 26 3 .5 d,e 48 + 21 = 69 41 3 .5 c,c 96 + 48 = 144 67 3.0 Table 3.1: Average CPU Time in Milliseconds number of operations involved in a typical calculation For an average computation... relative transformation TAB = TB TA 1 58 Chapter 4 Extension to Non-Convex Objects and Curved Objects Most of objects in the real world are not simple, convex polyhedral objects Curved corners and composite objects are what we see mostly in the man-made environment However, we can use the union of convex polyhedra to model a non-convex object and reasonably re ned polyhedral approximation for curved boundaries... for curved boundaries as well In this chapter, we will discuss how we extend the collision detection algorithm described in Chapter 3 from convex objects to non-convex objects Then, the extension to the curved objects will be described later in the chapter as well 4.1 Collision Detection for Non-convex Objects 4.1.1 Sub -Part Hierarchical Tree Representation Nonconvex objects can be either composite... For an average computation to track or update a pair of closest features, each run takes roughly 50 -70 usec on a SGI Indigo2 XZ or 92-131 arithmetic operations roughly 12 -51 operations to nd the nearest points, 2 x 16 operations on 2 point transformations, 2 x 4 x 6 operations for two applicability tests and each involves 4 inner product calculations independent of the machines By comparing the number... object is given as a union of convex polyhedra or is composed of several nonconvex subparts, each of these can be further represented as a union of convex polyhedra or a union of non-convex objects We use a sub -part hierarchy tree to represent each nonconvex object At each node of the tree, we store either a convex sub -part at a leave or the union of several convex . y8 w1 h5" alt=""