This paper describes an efficient algorithm for computing the distance between non-convex objects. Objects are modeled as the union of a set of convex components. From this model we construct a hierarchical bounding representation based on spheres. The distance between objects is determined by computing the distance between pairs of convex components using preexisting techniques. The key to efficiency is a simple search routine that uses the bounding representation to ignore most of the possible pairs of components. The efficiency can further be improved by accepting a relative error in the returned result. Several empirical trials are presented to examine the performance of the algorithm. 1. Introduction Computing the distance between objects is a common problem in robotics. Using a mathematical model of two objects, we find a point on each object such that the distance between the points is minimized. If one object is a robot and the other object is the union of all the obstacles i...

We present efficient algorithms for collision detection and contact determination between geometric models, described by linear or curved boundaries, undergoing rigid motion. The heart of our collision detection algorithm is a simple and fast incremental method to compute the distance between two convex polyhedra. It utilizes convexity to establish some local applicability criteria for verifying the closest features. A preprocessing procedure is used to subdivide each feature's neighboring features to a constant size and thus guarantee expected constant running time for each test. The expected constant time performance is an attribute from exploiting the geometric coherence and locality. Let n be the total number of features, the expected run time is between O( p n) and O(n) ...

solid models

This paper describes a method for estimating the distance between a robot and its surrounding environment using best ellipsoid fit. The method consists of the following two stages. First we approximate the detailed geometry of the robot and its environment by minimum-volume enclosing ellipsoids. The computation of these ellipsoids is a convex optimization problem, for which efficient algorithms are known. Then we compute a conservative distance estimate using an important but little known formula for the distance of a point from and n-dimensional ellipse. The computation of the distance estimate (and its gradient vector) is shown to be an eigenvalue problem, whose solution can be rapidly found using standard techniques. We also present an incremental version of the distance computation, which takes place along a continuous trajectory taken by the robot. We have implemented the proposed approach and present some preliminary results.

This paper introduces a new model for rapid physical simulation called sparse dynamics. The method employs a quick first pass to identify likely object interactions. These are then flagged for more detailed analysis. As actual collisions are rare in a sparsely populated environment, efficiency is greatly increased. The first pass uses deterministic Newtonian mechanics to predict future collisions analytically, obviating the need to simulate small uniform time steps. Timings indicate this provides a large speed improvement over more traditional methods. In particular we were able to simulate full collision detections for 1000 polyhedra at real time speeds. We also discuss how to extend the sparse dynamics model to handle complex effects such as gravity, friction, and user interaction. 1 Motivation The graphics community has a tradition of viewing realtime dynamic simulation as intractable. While large strides have been made in real-time rendering and user interfaces, simulations invol...

A method for detecting potential collisions between three-dimensional moving objects is described in this paper. An object-centered, spherical octree representation is defined and implemented for the localisation of potentially colliding features between polyhedral objects. These features are subsequently tested for intersection in order to calculate precisely the actual collision points. Application of the algorithm for the direct manipulation of objects in a virtual scene is considered, to investigate its real-time behaviour. The performance of the algorithm is found to remain linear with respect to the complexity of the colliding objects. 1 Introduction The detection of collision between two arbitrarily moving objects in a three-dimensional computersimulated environment plays an important role in various research fields: a. In the context of dynamic computer animation, where the sequence of object's position has to be dynamically produced by a physical simulation, as the effect of...

This paper presents fast algorithms for penetration and contact determination between general polyhedral models in dynamic environments. The main contribution is an extension of an earlier expected constant time algorithm between convex polytopes to detect penetrations and contacts. For each pair of non-convex polyhedral models, the algorithm uses the convex hull of each object to determine which regions of the objects are colliding. After identifying these regions, it uses a new dynamic technique, sweep and prune, to overcome the bottleneck of O(n 2 ) pairwise feature checks of these regions. The resulting algorithm has been implemented and in practice its performance in dynamic environments is O(n +m), where m corresponds to the number of feature pairs close to each other. 1 Introduction The problem of determining whether objects are penetrating or separated is of great importance in many areas. It has been extensively studied in robotics, molecular-based modeling, computational ...

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xvii 1 Introduction 2 1.1 Motivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Previous Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2.1 Obstacles Representation : : : : : : : : : : : : : : : : : : : : : 4 1.2.2 Configuration Space : : : : : : : : : : : : : : : : : : : : : : : 5 1.2.3 Dynamic Planning : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2.4 Optimal Planning : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.2.5 Missile Guidance : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.2.6 Differential Game Theory : : : : : : : : : : : : : : : : : : : : 9 1.2.7 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3 Objective : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.4 Outline of the Approach : : : : : : : : : : : : : : : : : : : : : : : : : 12 1.5 Organization : : : : : : : : : : : : : : : : : ...

Distance computation is essential for collision prediction and/or detection in real-world robotic tasks, computer simulation and animation, and CAD/CAM. This paper addresses distance computation to deal with a rarely researched type of collision prediction/detection problem: Given two objects in certain contact, determine if and when a relative rotation constrained by contact will cause a collision (which results in a new contact state) between the two objects. We use the positive angle of rotation as the measure of rotation distance and present a method to compute, given two contacting convex polytopes G and H and a rotation axis containing contact point(s) between them, the shortest rotation distance (SRD) of G which will cause new colllision between G and H . The method is fully implemented, and the algorithm is very efficient. If each vertex of G or H is the intersection of n e=v edges, then the worst-case time complexity of the algorithm is O(n 2 e=v ).

In this paper, we present an interior point approach to exact distance computation between convex objects represented as intersections of implicit surfaces. The implicit surfaces considered include planes (polyhedra), quadrics, and generalizations of quadrics including superquadrics and hyperquadrics, as well as intersections of these surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make contact, such as in dynamics simulations and in contact point prediction for dextrous manipulation. They can also be used in the narrow phase of hierarchical collision detection. In contrast to geometric approaches developed for polyhedral objects, we formulate the distance computation problem as a convex optimization problem; this optimization formulation has been previously described for polyhedral objects. We demonstrate that for general convex objects represented as implicit surfaces, interior point approaches are reasonably fast and in some cases, owing to their global convergence properties, are the only provably good choice for solving proximity query problems. We use a primal-dual interior point algorithm that solves the KKT conditions resulting from the convex programming formulation. We present preliminary implementation results, including distance computation between deforming superquadrics, to demonstrate the merit of the interior point approach.