The problem of tracking the distance between two convex polyhedra is finding applications in many areas of robotics, including intersection detection, collision detection, and path planning. We present new results that confirm an almost-constant time complexity for an enhanced version of Gilbert, Johnson and Keerthi's algorithm, and also describe modifications to the algorithm to compute measures of penetration distance.

solid models

In a geometric context, a collision or proximity query reports information about the relative configuration or placement of two objects. Some of the common examples of such queries include checking whether two objects overlap in space, or whether their boundaries intersect, or computing the minimum Euclidean separation distance between their boundaries. Hundreds of papers have been published on di#erent aspects of these queries in computational geometry and related areas such as robotics, computer graphics, virtual environments, and computer-aided design. These queries arise in di#erent applications including robot motion planning, dynamic simulation, haptic rendering, virtual prototyping, interactive walkthroughs, computer gaming, and molecular modeling. For example, a large-scale virtual environment, e.g., a walkthrough, creates a model of the environment with virtual objects. Such an environment is used to give the user a sense of presence in a synthetic world and it s

with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m + m ) expected time, for any constant " > 0. It also computes a vector t such that ktk = (A; B) and int(A + t) \ B = ;. We show that if the Minkowski sum B ( A) has K facets, then the expected running time of our algorithm is O K , for any " > 0.

Surface texture is among the most salient haptic characteristics of objects; it can induce vibratory contact forces that lead to perception of roughness. In this paper, we present a new algorithm to display haptic texture information resulting from the interaction between two textured objects. We compute contact forces and torques using low-resolution geometric representations along with texture images that encode surface details. We also introduce a novel force model based on directional penetration depth and describe an efficient implementation on programmable graphics hardware that enables interactive haptic texture rendering of complex models. Our force model takes into account important factors identified by psychophysics studies and is able to haptically display interaction due to fine surface textures that previous algorithms do not capture.

We present an incremental algorithm to estimate the penetration depth between convex polytopes in 3D. The algorithm incrementally seeks a "locally optimal solution" by walking on the surface of the Minkowski sums. The surface of the Minkowski sums is computed implicitly by constructing a local Gauss map. In practice, the algorithm works well when there is high motion coherence in the environment and is able to compute the optimal solution in most cases.

We present a fast algorithm to estimate the penetration depth between convex polytopes in 3D. The algorithm incrementally seeks a "locally optimal solution" by walking on the surface of the Minkowski sums. The surface of the Minkowski sums is computed implicitly by constructing a local dual mapping on the Gauss map. We also present three heuristic techniques that are used to estimate the initial features used by the walking algorithm. We have implemented the algorithm and compared its performance with earlier approaches. In our experiments, the algorithm is able to estimate the penetration depth in about a milli-second on an 1 GHz Pentium PC. Moreover, its performance is almost independent of model complexity in environments with high coherence between successive instances.

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 ...

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 ).

Given two sculptured objects, described by a collection of rational Bdzier patches, we propose an algorithm that provides the distance between them if the objects are not intersecting, or a measure of penetration otherwise. The algorithm extends the upper-lower bound subdivision approach to the computation of the nearest points by considering the relative orientations between the subdivided patches. All required geometric constructions can be described as rational Bdzier patches so that their control points can be precomputed from those of the original patches. Additional operations have been designed to exhibit linear complexity with the total number of involved control points. 1