Practical methods for approximating shortest paths on a convex polytope in R3

AbstractWe propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p, q on its surface, let dP(p, q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2dP(p, q) in time O(n). Extending this result, we can also compute an approximation of the shortest path tree rooted at an arbitrary point x ∈ P in time O(n log n). In the approximate tree, the distance between a vertex v ∈ P and x is at most cdP(x, v), where c = 2.38(1 + ε) for any fixed ε > 0. The best algorithms for computing an exact shortest path on a convex polytope take Ω(n2) time in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in O(n) time our algorithm gives a path of length at most 2k times the optimal

Robustness and Randomness

Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust

Improving the Air Mobility Command\u27s Air Refueler Route Building Capabilities

We consider the problem of routing an aircraft (receiver) from a starting location to a target and back to an ending location while maintaining a fuel level above a predetermined level during all stages of the route and avoiding threat and no-fly zones. The receiver is routed to air refueling locations to refuel as required. The development of the network requires the processing of threat and no-fly zones to create the set of nodes that includes the bases (starting and end locations), the targets, and air refueling locations in addition to the restricted zone nodes. We develop a greedy heuristic that builds the route using arc paths and the on board fuel level to determine the termination of each sequential arc path. Post processing of the routes reduces the fuel remaining on board by shifting the time at target or reversing the route. The results from the greedy heuristic are compared to the results from the current methodology and show that the heuristic requires less time to produce routes that require less fuel

Structuring a Wayfinder\u27s Dynamic and Uncertain Environment

Wayfinders typically travel in dynamic environments where barriers and requirements change over time. In many cases, uncertainty exists about the future state of this changing environment. Current geographic information systems lack tools to assist wayfinders in understanding the travel possibilities and path selection options in these dynamic and uncertain settings. The goal of this research is a better understanding of the impact of dynamic and uncertain environments on wayfinding travel possibilities. An integrated spatio-temporal framework, populated with barriers and requirements, models wayfinding scenarios by generating four travel possibility partitions based on the wayfinder\u27s maximum travel speed. Using these partitions, wayfinders select paths to meet scenario requirements. When uncertainty exists, wayfinders often cannot discern the future state of barriers and requirements. The model to address indiscemibility employs a threevalued logic to indicate accessible space, inaccessible space, and possibly inaccessible space. Uncertain scenarios generate up to fifteen distinct travel possibility categories. These fifteen categories generalize into three-valued travel possible partitions based on where travel can occur and where travel is successful. Path selection in these often-complex environments is explored through a specific uncertain scenario that includes a well-defined initial requirement and the possibility of an additional requirement somewhere beforehand. Observations from initial path selection tests with this scenario provide the motivation for the hypothesis that paths arriving as soon as possible to well-defined requirements also maximize the probability of success in meeting possible additional requirements. The hypothesis evaluation occurs within a prototype Travel Possibility Calculator application that employs a set of metrics to test path accessibility in various linear and planar scenarios. The results did not support the hypothesis, but showed instead that path accessibility to possible additional requirements is greatly influenced by the spatio-temporal characteristics of the scenario\u27s barriers

INTELLIGENT VISION-BASED NAVIGATION SYSTEM

This thesis presents a complete vision-based navigation system that can plan and follow an obstacle-avoiding path to a desired destination on the basis of an internal map updated with information gathered from its visual sensor. For vision-based self-localization, the system uses new floor-edges-specific filters for detecting floor edges and their pose, a new algorithm for determining the orientation of the robot, and a new procedure for selecting the initial positions in the self-localization procedure. Self-localization is based on matching visually detected features with those stored in a prior map. For planning, the system demonstrates for the first time a real-world application of the neural-resistive grid method to robot navigation. The neural-resistive grid is modified with a new connectivity scheme that allows the representation of the collision-free space of a robot with finite dimensions via divergent connections between the spatial memory layer and the neuro-resistive grid layer. A new control system is proposed. It uses a Smith Predictor architecture that has been modified for navigation applications and for intermittent delayed feedback typical of artificial vision. A receding horizon control strategy is implemented using Normalised Radial Basis Function nets as path encoders, to ensure continuous motion during the delay between measurements. The system is tested in a simplified environment where an obstacle placed anywhere is detected visually and is integrated in the path planning process. The results show the validity of the control concept and the crucial importance of a robust vision-based self-localization process

Modelling Acoustic Propagation with Dominant Paths

Hearing is one of our major senses. In addition to being, arguably, our primary form of communication, it affects how we perceive the world around us. Accurate acoustic simulation can complement graphical simulation systems, providing users with more immersive and believable environments. Furthermore, sound can assist users in localizing objects within space, and it has been shown to improve the perceived quality of visuals. Many of the audio properties that our brain subconsciously keys upon are artifacts inscribed into the signal by the environment as the sound propagates from the source to the listener. Diffraction, or the bending of sound around objects, is an important mode of transmission and a key source of reverberation in an audio signal. However, many current acoustic propagation simulation systems do not properly model diffraction. This work proposes a new method for performing the simulation of diffraction. Our formulation aims to provide accurate results across the frequency spectrum. Accordingly, we avoid using approximative models of diffraction, such as the Unified Theory of Diffraction, which are ubiquitous in the field, but lead to inaccurate results particularly in the lower frequencies. Furthermore, our algorithm can provide a time-domain solution for reverberant sounds and is controllable for stylistic adjustments. In order to evaluate our method, we compare our results against measured and experimental results reported in the literature

Approximate Euclidean Shortest Path in 3-Space

Papadimitriou's approximation approach to the Euclidean shortest path (ESP) in 3-space is revisited. As this problem is NP-hard, his approach represents an important step towards practical algorithms. However, there are several gaps in the original description. Besides giving a complete treatment in the framework of bit complexity, we also improve on his subdivision method. Among the tools needed are root-separation bounds and non-trivial applications of Brent's complexity bounds on evaluation of elementary functions using floating point numbers. 1 Introduction The Euclidean shortest path (ESP) problem can be formulated as follows: given a collection of polyhedral obstacles in physical space S, and source and target points s 0 ; t 0 2 S, find a shortest obstacle-avoiding path between s 0 and t 0 . Here S is typically E 2 or E 3 . It is evident that this is a basic problem in applications such as robotics. The case S = E 2 has recently seen a major breakthrough, with the O(..

Constrained Shortest Paths in Terrains and Graphs

Finding a shortest path is one of the most well-studied optimization problems. In this thesis we focus on shortest paths in geometric and graph theoretic settings subject to different feasibility constraints that arise in practical applications of such paths. One of the most fundamental problems in computational geometry is finding shortest paths in terrains, which has many applications in robotics, computer graphics and Geographic Information Systems (GISs). There are many variants of the problem in which the feasibility of a path is determined by some geometric property of the terrain. One such variant is the shortest descending path (SDP) problem, where the feasible paths are those that always go downhill. We need to compute an SDP, for example, for laying a canal of minimum length from the source of water at the top of a mountain to fields for irrigation purpose, and for skiing down a mountain along a shortest route. The complexity of finding SDPs is open. We give a full characterization of the bend angles of an SDP, showing that they follow a generalized form of Snell's law of refraction of light. We also reduce the SDP problem to the problem of finding an SDP through a given sequence of faces, by adapting the sequence tree approach of Chen and Han for our problem. Our results have two implications. First, we isolate the difficult aspect of SDPs. The difficulty is not in deciding which face sequence to use, but in finding the SDP through a given face sequence. Secondly, our results help us identify some classes of terrains for which the SDP problem is solvable in polynomial time. We give algorithms for two such classes. The difficulty of finding an exact SDP motivates the study of approximation algorithms for the problem. We devise two approximation algorithms for SDPs in general terrains---these are the first two algorithms to handle the SDP problem in such terrains. The algorithms are robust and easy-to-implement. We also give two approximation algorithms for the case when a face sequence is given. The first one solves the problem by formulating it as a convex optimization problem. The second one uses binary search together with our characterization of the bend angles of an SDP to locate an approximate path. We introduce a generalization of the SDP problem, called the shortest gently descending path (SGDP) problem, where a path descends but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. For example, a vehicle cannot follow a too steep descent---this is why a mountain road has hairpin bends. We give two easy-to-implement approximation algorithms for SGDPs, both using the Steiner point approach. Between a pair of points there can be many SGDPs with different number of bends. In practice an SGDP with fewer bends or smaller total turn-angle is preferred. We show using a reduction from 3-SAT that finding an SGDP with a limited number of bends or a limited total turn-angle is hard. The hardness result applies to a generalization of the SGDP problem called the shortest anisotropic path problem, which is a well-studied computational geometry problem with many practical applications (e.g., robot motion planning), yet of unknown complexity. Besides geometric shortest paths, we also study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an easy-to-implement algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The algorithm handles a forbidden path using vertex replication, i.e., replicating vertices and judiciously deleting edges so as to remove the forbidden path but keep all of its subpaths. The main challenge is that vertex replication can result in an exponential number of copies of any forbidden path that overlaps the current one. The algorithm couples vertex replication with the "growth" of a shortest path tree in such a way that the extra copies of forbidden paths produced during vertex replication are immaterial