Consider a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Delaunay diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λs(n)), where λs,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites. In the special case of points moving at constant speed along straight lines, we get s = 4, implying an upper bound of O(n32α(n)), where α(n) is the extremely slowly-growing inverse of Ackermann 's function. Our results are a linear-factor improvement over the naive quartic bound on the number of topological events.

Voronoi Diagrams of Moving Points in the Plane

Voronoi diagrams over dynamic scenes

By combining the simplicity of the sphere and the power of the motion of hierarchy of detail, a novel model is proposed with applications to collision avoidance and detection in 3D. The model is based on a double spherical representation for solid bodies. First, each element making up the robot and the obstacles is approximated by a set of exterior spheres which are automatically defined. Second, another set composed of interior spheres is generated. These representations define a hierarchy, since they can be redefined as many times as necessary; starting with two spheres per element, the approximation may be improved until it contains hundreds of spheres. Moreover, they converge to a zero-error representation. The proposed models leads to a simple treatment for the problem of collision detection, and it is further applied to collision-free path planning for manipulators in 3D. >

A new representation for collision avoidance and detection

Swap conditions for dynamic Voronoi diagrams for circles and line segments

Dynamic Voronoi diagrams in motion planning

The path existence problem and the collision detection problem for time-varying objects in a geometric scene are discussed. For a large class of spherical nonrigid objects, exact solutions of the path existence problem are developed based on decomposition techniques and graph traversal. For the collision detection problem of a single moving circle in the plane, efficient data structures are presented for linear/circular and polynomial paths.

http://link.springer.com/content/pdf/10.1007%2FBF01928919.pdf

Collision avoidance for nonrigid objects

The path existence problem and the collision detection problem for time-varying objects in a geometric scene are discussed For a large class of spherical nonrigid objects, exact solutions of the path existence problem are developed based on decomposition techniques and graph traversal

Collision avoidance for nonrigid objects (extended abstract)

Searching connected components in very large grid graphs

Given m circles in the plane, contacts between them can be specified by a system of quadratic distance equalities. An approximative solution for the trajectories of the circles for a system of one degree of freedom is given, by replacing the circles by translationally moving regular k-gons. The approximation yields trajectories that are piecewise linear. The next linear generation of the m trajectories are found by an incremental algorithm in O(m2) time. Further, an algorithm is presented which finds the next collision between m k-gons moving on lines at constant speed in time O(k · m2−x) for a constant x>0 using linear space. Finally, more practical collision detection algorithms are sketched based on neighborhood information which, however, do not guarantee a nontrivial worst-case time bound.

Stephan Abramowski

Papers

Collision avoidance for nonrigid objects

Collision avoidance for nonrigid objects (extended abstract)

Searching connected components in very large grid graphs

Moving Regular k-Gons in Contact