Presence Of Polygonal Obstacles Research Articles

Bi-level Flight Path Planning of UAV Formations with Collision Avoidance

This paper deals with the problem of generating 3D flight paths for a swarm of cooperating Unmanned Aerial Vechicles (UAVs) flying in a formation having a prespecified shape, in the presence of polygonal obstacles, no-fly zones and other non cooperative aircraft. UAVs are modeled as Dubins flying vehicles with bounds on the turning radius and flight path climb/descent angle. A Reduced Visibility Graph (RVG) based method, connecting selected nodes by means of circular arcs and segments, is adopted to minimize the length of each path. Then, to keep as much as possible the formation shape while flying between obstacles, the RVG is refined with the addition of so called Rendez-Vous Waypoints (RVWs). These are placed between groups of obstacles where it is impossible to maintain the desired formation. Waypoints locations and UAVs paths are optimized using a bi-level game theoretic approach based on the leader-follower Stackelberg model, where the lower level and upper level problems are the search of the shortest paths and the optimal locations of waypoints respectively. Such an approach allows to fly between obstacles, dispersing the formation and forcing UAVs to recompose it at given waypoints (RVWs) beyond groups of obstacles. Collision avoidance among UAVs and possible non-cooperative aircrafts, called intruders, is then achieved solving a set of linear quadratic optimization problems based on an original geometric based formulation. The effectiveness of the proposed approach is shown by means of numerical simulations where RVWs positions are optimized via a genetic algorithm.

AN ALGORITHMIC FRAMEWORK FOR SOLVING GEOMETRIC COVERING PROBLEMS — WITH APPLICATIONS

In geometric covering problems, the aim is to cover a set of given points using a minimum number of shapes with prescribed properties. These problems extend in many ways such as covering in the presence of obstacles and partial covering. Since in most cases these problems are NP-hard, researchers have developed a large number of approximation algorithms to solve them. In this paper, with a more general view on these approaches, we propose an algorithmic framework that gives an approximation algorithm for the covering problem at hand, provided that its prerequisites are satisfied. As the result, we present a general class of geometric covering problems for which one can obtain polynomial-time approximation schemes (PTASs). This class of problems involves covering point sets with bounded shapes in a space with fixed dimension. For a problem to be in this class, it must be possible to find a polynomial number of maximal shapes and it must be possible to cover all the points in a small box with a small number of shapes. Using this framework, we present two new PTASs for the problems of covering with disks in the presence of polygonal obstacles and covering with sectors (pie-shaped wedges). The first problem has never been addressed before and for the second problem, only a logarithmic approximation algorithm was known in general and a 9-approximation algorithm for wide angles.

Bounded-Curvature Shortest Paths through a Sequence of Points Using Convex Optimization

We consider the problem of computing shortest paths having curvature at most one almost everywhere and visiting a sequence of $n$ points in the plane in a given order. This problem is a subproblem of the Dubins traveling salesman problem and also arises naturally in path planning for point car-like robots in the presence of polygonal obstacles. We show that when consecutive waypoints are a distance of at least four apart, this question reduces to a family of convex optimization problems over polyhedra in $\mathbb{R}^n$.

An approximation algorithm for d 1-optimal motion of a rod robot with fixed rotations

Given a translating and rotating rod robot in a plane in the presence of polygonal obstacles with the initial and final placements of the rod known, the d 1-optimal motion planning problem is defined as finding a collision-free motion of the rod such that the orbit length of a fixed but arbitrary point F on the rod is minimized. In this paper we study a special case of this problem in which the rod can translate freely, but can only rotate by some pre-specified given angles around F. We first characterize the d 1-optimal motion of the robot under the given conditions and then present a (1 + ε)–approximation algorithm for finding the optimal path. The running time of the algorithm is bounded by a polynomial in terms of some parameters related to the problem input.

LABELING POINTS WITH RECTANGLES OF VARIOUS SHAPES

We deal with a map-abeling problem, named LOFL (Left-part Ordered Flexible Labeling), to label a set of points in a plane in the presence of polygonal obstacles. The label for each point is selected from a set of rectangles with various shapes satisfying the left-part ordered property, and is placed near to the point after scaled by a scaling factor σ which is common to all points. In this paper, we give an optimal O((n + m) log (n + m)) algorithm to decide the feasibility of LOFL for a fixed scaling factor σ, and an O((n + m) log 2 (n + m)) time algorithm to find the largest feasible scaling factor σ, where n is the number of points and m is the total number of edges of the polygonal obstacles.

Cruise Missile Mission Planning: A Heuristic Algorithm for Automatic Path Generation

This manuscript presents a heuristic algorithm based on geometric concepts for the problem of finding a path composed of line segments from a given origin to a given destination in the presence of polygonal obstacles. The basic idea involves constructing circumscribing triangles around the obstacles to be avoided. Our heuristic algorithm considers paths composed primarily of line segments corresponding to partial edges of these circumscribing triangles, and uses a simple branch-and-bound procedure to find a relatively short path of this type. This work was motivated by the military planning problem of developing mission plans for cruise missiles, but is applicable to any two-dimensional path-planning problem involving obstacles.

An Optimal Algorithm for Euclidean Shortest Paths in the Plane

We propose an optimal-time algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worst-case time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer single-source shortest path queries in O(log n) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources.

SHORTEST PATHS AMONG OBSTACLES IN THE PLANE

We give a subquadratic (O(n3/2+∊) time and O(n) space) algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles; previous time bounds were at least quadratic in n, in the worst case. The method avoids use of visibility graphs, relying instead on the continuous Dijkstra paradigm. The output is a shortest path map (of size O(n)) with respect to a given source point, which allows shortest path length queries to be answered in time O( log n). The algorithm extends to the case of multiple source points, yielding a method to compute a Voronoi diagram with respect to the shortest path metric.

Collision detection of a moving polygon in the presence of polygonal obstacles in the plane

This paper presents a new approach for the following collision detection problem in the plane: Let a simple polygon P rotate at a center /spl ogr/ with constant angular velocity /spl omega/ and translate towards a set of polygonal obstacles S with constant velocity /spl nu/. Given P and S as well as their initial positions, and given also the velocities of P, determine whether or not P will collide with any element of S and report the collided elements of S if collisions occurred. An O(mn) worst-case optimal algorithm is proposed to solve this problem, where n is the number of vertices of P and m is the number of vertices of the obstacles in S. >

A new algorithm for shortest paths among obstacles in the plane

We introduce a new algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles. In particular, for a given start points, we build a planar subdivision (ashortest path map) that supports efficient queries for shortest paths froms to any destination pointt. The worst-case time complexity of our algorithm isO(kn log2n), wheren is the number of vertices describing the polygonal obstacles, andk is a parameter we call the “illumination depth” of the obstacle space. Our algorithm usesO(n) space, avoiding the possibly quadratic space complexity of methods that rely on visibility graphs. The quantityk is frequently significantly smaller thann, especially in some of the cases in which the visibility graph has quadratic size. In particular,k is bounded above by the number of different obstacles that touch any shortest path froms.

Constrained delaunay triangulations

Given a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimalO(n logn) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (non-Delaunay) triagulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.