[...]... time for the rst pair on top of the heap Wakeup time Wi for each object pair Pi is de ned as Wi = tiw + t 0 where tiw is the lower bound on the time to collision for each pair Pi for most situations with exceptions described in the next sub-section and t is the current time 0 5.1.1 Bounding Time to Collision Given a trajectory that each moving object will travel, we can determine the exact collision. .. surfaces e 76 can have more than one closest feature at any time As a result, local optimization methods highlighted in the previous sections may not work for the non-convex objects The problem of collision detection between two B zier surfaces is solved by e nding all the solutions to the equations 4.5 and 4.7 A real solution in the domain to those equations implies a geometric collision and a precise... object will travel, we can determine the exact collision time Please refer to 17 for more details If the path that each object travels is not known in advance, then we can calculate a lower bound on collision time This lower bound on collision time is calculated adaptively to speed up the performance of dynamic collision detection ... queue The algorithm rst has to compute the initial separation and the possible collision time among all pairs of objects and the obstacles, assuming that the magnitude of relative initial velocity, relative maximum acceleration and velocity limits among them are given After initialization, at each step it only computes the closest feature pair and the distance between one object pair of our interests,... attached to each other will not appear in the queue They are sorted by lower bound on time to collision; with the one most likely to collide i.e the one that has the smallest approximate time to collision appearing at the top of the heap The approximation is a lower bound on the time to collision, so no collisions are missed Non-convex objects, which are represented as hierarchy trees as described... is similar for surfaces represented algebraically Objects which can be represented as a union of convex surfaces, we use the hierarchical representation and the algorithm highlighted above on the leaf nodes 4.3.3 Non-Convex Curved Objects In this section we outline the algorithm for non-convex surfaces which cannot be represented as a union of convex surfaces A common example is a torus and even a... especially in a large environment In order to avoid unnecessary computations and to speed up the run time, we present two methods: one assumes the knowledge of maximum acceleration and velocity, the other purely exploits the spatial arrangement without any other information to reduce the number of pairwise interference tests For the scheduling scheme in which we assume the dynamics velocities, accelerations,... a small separation are likely to have an impact within the next few time instances, and those object pairs which are far apart from each other cannot possibly come to interfere with each other until certain time For spatial tests to reduce the number of pairwise comparisons, we assume the environment is rather sparse and the objects move in such a way that the geometric coherence can be preserved, i.e... resultants and eigenvalues is used to nd all the solutions to the equations 4.5 and 4.7 55 This global root nder is used when the control polytopes of two B zier surfaces collide At that instant the two surfaces e may or may not have a geometric contact It is possible that all the solutions to these equations are complex The set of equations in 4.5 represents an overconstrained system and may have... equations 77 Chapter 5 Interference Tests for Multiple Objects In a typical environment, there are moving objects and stationary obstacles as well Assume that there are N objects moving around in an environment of M stationary obstacles Each of N moving objects is also considered as a1 0 moving obstacle N to the other moving objects Keeping track of the distance for @ A + NM pairs of 2 objects at all . a ) Tangent Plane Contact Point Intersection is a boundary point ( b )