Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces

This paper presents a general method for exact distance computation between convex objects represented as intersections of implicit surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make intermittent contact, such as in dynamic simulat...

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Vydáno v:	IEEE transactions on robotics Ročník 24; číslo 1; s. 211 - 220
Hlavní autoři:	Chakraborty, N., Jufeng Peng, Akella, S., Mitchell, J.E.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.02.2008 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Algorithms Application software Applied sciences Closest points collision detection Computation Computational complexity Computational modeling Computer science; control theory; systems Constraint theory Control theory. Systems Exact sciences and technology Haptic interfaces implicit surfaces Information systems. Data bases interior point algorithms Intersections Mathematical programming Memory organisation. Data processing Object detection Optimization Optimization algorithms Optimization methods Phase detection Polyhedra Polyhedrons Polynomials Problem solving Programming profession Proximity proximity query Queries Robotics Software Intersection Quadric Database query Barrier function Linear time Convex programming Geometrical model Superquadrics User interface Implicit theory Kuhn Tucker condition Dynamic model Mathematical programming Proximity Interior point method Computational complexity Intermittency Polynomial time Polyhedron Phase detector Karush Kuhn Tucker method Primal dual method Collision detection Time complexity Collision avoidance
ISSN:	1552-3098, 1941-0468
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Shrnutí:	This paper presents a general method for exact distance computation between convex objects represented as intersections of implicit surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make intermittent contact, such as in dynamic simulations and in haptic interactions. 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. We use an interior point method to solve the optimization problem and demonstrate that, for general convex objects represented as implicit surfaces, interior point approaches are globally convergent, and fast in practice. Further, they provide polynomial-time guarantees for implicit surface objects when the implicit surfaces have self-concordant barrier functions. We use a primal-dual interior point algorithm that solves the Karush-Kuhn-Tucker (KKT) conditions obtained from the convex programming formulation. For the case of polyhedra and quadrics, we establish a theoretical time complexity of O(n 1.5 ), where n is the number of constraints. We present implementation results for example implicit surface objects, including polyhedra, quadrics, and generalizations of quadrics such as superquadrics and hyperquadrics, as well as intersections of these surfaces. We demonstrate that in practice, the algorithm takes time linear in the number of constraints, and that distance computation rates of about 1 kHz can be achieved. We also extend the approach to proximity queries between deforming convex objects. Finally, we show that continuous collision detection for linearly translating objects can be performed by solving two related convex optimization problems. For polyhedra and quadrics, we establish that the computational complexity of this problem is also O(n 1.5 ).
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23
ISSN:	1552-3098 1941-0468
DOI:	10.1109/TRO.2007.914851