Generalized penetration depth computation

Abstract

Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects.We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are non-convex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6-dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path non-existence.

Global vector field computation for feedback motion planning

“…One of the major issues with approximate cell decomposition is efficient labelling and classification of each cell. We use the cell labelling algorithm proposed by [12], [18], which is based on computing separation distances and penetration depths at different configurations of the robot with the obstacles. Overall, the approximate cell decomposition generates a set of empty cells, which provides an approximate representation of the robot's free space F. A connectivity graph between the cells is extracted from the decomposition.…”

A Fast and Practical Algorithm for Generalized Penetration Depth Computation

“…Ortega et al [25] presented a method to locally minimize the kinetic distance between the configurations of a haptic probe and its proxy using constraintbased dynamics and continuous collision detection. Zhang et al [5] proposed the first rigorous formulation of computing PD g . They presented an efficient algorithm to compute PD g for convex polytopes, and provide bounds on PD g of nonconvex polyhedra.…”

“…Along with collision detection and separation distance, PD is one of the proximity queries that is useful for many applications including dynamics simulation, haptics, motion planning, and CAD/CAM. Specifically, PD is important for computing collision response [1], estimating the time of contact in dynamics simulation [2], sampling for narrow passages in retraction-based motion planing [3], [4], and C-obstacle query in motion planning [5].…”

“…For PD g , it can be formulated by computing the arrangement of contact surfaces, and the combinatorial complexity of the arrangement is O(n 12 ) [7]. As a result, prior algorithms for global PD only compute an approximate solution [5], [8]. Moreover, these algorithms perform convex decomposition on non-convex models and can be rather slow for interactive applications.…”

Inner sphere trees for proximity and penetration queries

“…Another approach, called penalty methods, is to compute repelling forces based on the penetration depth. However, there is no universally accepted definition of the penetration depth between a pair of polygonal models [ZKVM07], [OG96]. Mostly, the minimum translation vector to separate the objects is used, but this may lead to discontinuous forces.…”