[...]... children, and so on In this recursive manner, the algorithm will only signal a collision if there is actually an impact between the sub-parts of two objects; otherwise, there is no collision between the two objects For instance, assume for simplicity that one nonconvex object is composed of m convex polyhedra and the other is composed of n convex polyhedra, the algorithm will rst check for collision. .. terminates and reports no collision In the next step, the algorithm continues to track the motion of the two planes and nds their convex hulls overlap Then, the algorithm goes down the tree to the next level of tree nodes and check whether there is a collision among the subparts the wings, the bodies, and the union of subparts, i.e the union of tail 62 pieces If not, the algorithm will not report a collision. .. union of its children For instance, the root of the sub -part hierarchy tree for a composite solid non-convex object is the nonconvex object with its convex hull in its data structure; each convex sub -part is stored at a leave The root of the sub -part hierarchical tree for a Puma arm is the bounding box of the entire arm which consists of 6 links; and each leave stores a convex link For example, in Fig... curved surfaces and the root of the tree is the entire curved object and the convex hull of its polyhedral approximation For more treatments on the non-convex curved objects, please refer to Sec 4.3.3 4.1.2 Detection for Non-Convex Polyhedra Given the subpart tree representation, we examine the possible interference by a recursive algorithm at each time step The algorithm will rst check for collision between.. .59 Figure 4.1: An example of sub -part hierarchy tree 60 subparts Depending on the applications, we rst construct the convex hull of a rigid non-convex object or dynamic bounding box for articulated bodies at each node and work up the tree as part of preprocessing computation We also include the convex hull or dynamic bounding box of the union of sub-parts in the data structure... two parents, there is no collision and the algorithm updates the closest-feature pair and the distance between them If there is a collision, then it will expand their children All children of one parent node are checked against all children of the other parent node If there is also a collision between the children, then the algorithm will recursively call upon itself to check for possible impacts among... stores a convex link For example, in Fig 4.1, the root of this tree is the aircraft and the convex hull of the aircraft outlined by the wire frames The nodes on the rst level of the tree store the subparts and the union of the subparts of the aircraft, i.e the left and right wings both convex, the convex body of airplane, and the union of the tail pieces Please note that at the node where the union of... possible subpart pairs at the children nodes as well This can lead to On  of comparisons, if each object is consisted of n subparts and the trees only have two levels of leave nodes However, we can achieve a better performance if we have a good hierarchy of the model For complex objects, using a deep hierarchical tree with lower branching factor will keep down the number of nodes which need to be expanded... of the subpart hierarchical tree When we perform the collision checks, we only keep track of the closest feature pairs for the convex hulls of parent nodes, unless a collision has taken place between the convex hulls of two non-convex objects or the bounding boxes of articulated bodies Under such a situation, we have to track the closest pair of features for not only the convex hulls or bounding boxes... penetration of subparts such as two wings interpenetrating each other, then a collision is reported We would like to clarify one fact that we never need to construct the Voronoi regions of the non-convex objects, since it is absolutely unnecessary given the subpart hierarchical tree Instead, at the preprocessing step we only construct the Voronoi regions for each convex piece at each node of the subpart hierarchical . microseconds