Menelaos I. Karavelas

Scalable nonlinear dynamical systems for agent steering and crowd simulation

Abstract We present a new methodology for agent modeling that is scalable and efficient. It is based on the integration of nonlinear dynamical systems and kinetic data structures. The method consists of three layers, which together model 3D agent steering, crowds and flocks among moving and static obstacles. The first layer, the local layer employs nonlinear dynamical systems theory to models low-level behaviors. It is fast and efficient, and it does not depend on the total number of agents in the environment. This dynamical systems-based approach also allows us to establish continuous numerical parameters for modifying each agents behavior. The second layer, a global environment layer consists of a specifically designed kinetic data structure to track efficiently the immediate environment of each agent and know which obstacles/agents are near or visible to the given agent. This layer reduces the complexity in the local layer. In the third layer, a global planning laye r, the problem of target tracking is generalized in a way that allows navigation in maze-like terrains, avoidance of local minima and cooperation between agents. We implement this layer based on two approaches that are suitable for different applications: One approach is to track the closest single moving or static target; the second is to use a pre-specified vector field, which may be generated automatically (with harmonic functions, for example) or based on user input to achieve the desired output. We also discuss how hybrid systems concepts for global planning can capitalize on both our layered approach and the continuous, reactive nature of our agent steering. We demonstrate the power of the approach through a series of experiments simulating single/multiple agents and crowds moving towards moving/static targets in complex environments.

The predicates of the Apollonius diagram: Algorithmic analysis and implementation

We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient implementation of the algorithm, that handles all special cases. Among our tools we distinguish an inversion transformation and an infinitesimal perturbation for handling degeneracies.The implementation of the predicates requires certain algebraic operations. In studying the latter, we aim at minimizing the algebraic degree of the predicates and the number of arithmetic operations; this twofold optimization corresponds to reducing bit complexity. The proposed algorithms are based on static Sturm sequences. Multivariate resultants provide a deeper understanding of the predicates and are compared against our methods. We expect that our algebraic techniques are sufficiently powerful and general to be applied to a number of analogous geometric problems on curved objects. Their efficiency, and that of the overall implementation, are illustrated by a series of numerical experiments. Our approach can be immediately extended to the incremental construction of abstract Voronoi diagrams for various classes of objects.

Dynamic Additively Weighted Voronoi Diagrams in 2D

In this paper we present a dynamic algorithm for the construction of the additively weighted Voronoi diagram of a set of weighted points in the plane. The novelty in our approach is that we use the dual of the additively weighted Voronoi diagram to represent it. This permits us to perform both insertions and deletions of sites easily. Given a set B of n sites, among which h sites have a non-empty cell, our algorithm constructs the additively weighted Voronoi diagram of B in O(nT(h) + h log h) expected time, where T(k) is the time to locate the nearest neighbor of a query site within a set of k sites. Deletions can be performed for all sites whether or not their cell is empty. The space requirements for the presented algorithm is O(n). Our algorithm is simple to implement and experimental results suggest an O(n log h) behavior.

The Voronoi Diagram of Planar Convex Objects

This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudo-circles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively O(log2 n) and O(log3 n). Our algorithm can easily be adapted to the case of pseudo-circles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects’ union.

Bounding the distance between 2D parametric Bézier curves and their control polygon

Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control polygon based on the Greville abscissae. Several of the norms appearing in these bounds are orientation dependent. We next present algorithms for finding the optimal orientation angle for which two of these norms become minimal. The use of these bounds and algorithms for constructing polygonal envelopes of planar polynomial curves, is illustrated for an open and a closed composite Bézier curve.

Interval methods for kinetic simulations

We propose a speed-up method for discrete-event simulations, including sweep-line or -plane techniques, requiring the repeated calculation of the times at which certain discrete events occur. Instead of calculating these event times precisely, we use interval methods to obtain less expensive approximations that may still be adequate for the simulation. This can happen because some events get descheduled before they actually happen, or because event time comparisons can be resolved using only information about their bounding intervals. The computed intervals are refined as necessary, when greater accuracy is needed. For geometric objects described by polynomials, including moving objects on polynomial trajectories, the proposed method is shown to give a speed-up roughly proportional to the degree of the polynomials.

A package for exact kinetic data structures and sweepline algorithms

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability.

Spatial shape‐preserving interpolation using ν‐splines

We present a global iterative algorithm for constructing spatial G2‐continuous interpolating ν‐splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of ν‐splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.

The maximum number of faces of the minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of k-faces, 0≤k≤d-1, of the Minkowski sum, P₁+P₂+P₃, of three d-dimensional convex polytopes P₁, P₂ and P₃ in R^d, as a function of the number of vertices of the polytopes, for any d≥2. Expressing the Minkowski sum as a section of the Cayley polytope C of its summands, counting the k-faces of P₁+P₂+P₃ reduces to counting the (k+2)-faces of C which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes in R^d, where r≥d. For d≥4, the maximum values are attained when P₁, P₂ and P₃ are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.

Voronoi Diagrams for Moving Disks and Applications

In this paper we discuss the kinetic maintenance of the Euclidean Voronoi diagram and its dual, the Delaunay triangulation, for a set of moving disks. The most important aspect in our approach is that we can maintain the Voronoi diagram even in the case of intersecting disks. We achieve that by augmenting the Delaunay triangulation with some edges associated with the disks that do not contribute to the Voronoi diagram. Using the augmented Delaunay triangulation of the set of disks as the underlying structure, we discuss how to maintain, as the disks move, (1) the closest pair, (2) the connectivity of the set of disks and (3) in the case of non-intersecting disks, the near neighbors of a disk.