An improved Dynamic Z* algorithm for rapid replanning of energy-efficient paths

2015-2016 > Academic research: refereed > Refereed conference paperpreprint_postprin

Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $\gamma$, our metric charges each point of $\gamma$ with its distance to P. The total charge along $\gamma$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam

Finding energy-efficient paths on uneven terrains

Mobile robots are increasingly getting popular in outdoor applications. Long period of continuous operations are common in such applications. Therefore, robot motions need to be optimized to minimize their energy consumption. Shortest paths do not always guarantee minimum energy consumptions of mobile robots. This paper proposes a novel algorithm to generate energy-efficient paths on uneven terrains using an established energycost model for mobile robots. Terrains are represented using grid based elevation maps. Similar to A* algorithm, the energy-cost of traversing through a particular gird depends on a heuristic energy-cost estimation from the current location to the goal. The proposed heuristic energy-cost function makes it possible to generate zigzag-like path patterns on steep hills under the power limitations of the robot. Therefore, the proposed method can find physically feasible energy-efficient paths on any given terrain, provided that such paths exist. Simulation results presented in this paper demonstrate the performance of the proposed algorithm on uneven terrains maps.Department of Electronic and Information EngineeringRefereed conference pape

Optimal grid-free path planning across arbitrarily-contoured terrain with anisotropic friction and gravity effects

This paper appeared in IEEE Transactions on Robotics and Automation, 6, no. 5 (October 1990), 540-553. The equations were redrawn in 2008.Anisotropic (heading-dependent) phenomena arise in the "two-and-one-half-dimensional" path-planning problem of finding minimum-energy routes for a mobile agent across some hilly terrain. We address anisotropic friction and gravity effects, and ranges of impermissible-traversal headings due to overturn danger or power limitations. Our method does not require imposition of a uniform grid, nor average effects in different directions, but reasons about a polyhedral approximation of terrain. It reduces the problem to a finite but provably-optimal set of possibilities, and then uses A* search to find the cost-optimal path. However, the possibilities are not physical locations but path subspaces. Our method exploits the surprising insight that there are only four ways, mathematically simple, to optimally traverse an anisotropic homogeneous region: (1) straight across without braking, the standard isotropic-weighted-region traversal; (2) straight across without braking but as close as possible to a desired impermissible heading; (3) making impermissibility-avoiding switchbacks on the path across a region; and (4) straight across with braking. We prove specific optimality criteria for transitions on the boundaries of regions for each combination of traversal types. These criteria subsume previously published criteria for traversal of isotropic weighted-region terrain. Our method can take considerably less computer time and space than previous methods on some terrain; at the same time, we believe it is the first algorithm to provide truly optimal paths on anisotropic terrain.supported in part by the U. S. Army Combat Developments Experimentation Center under MIPR ATEC 88-86Approved for public release; distribution is unlimited

On approximating shortest paths in weighted triangular tessellations

© 2023 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path , which is a shortest path from s to t in the space; a weighted shortest vertex path , which is an any-angle shortest path; and a weighted shortest grid path , which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. (2021) [6], we prove upper and lower bounds on the ratios , , , which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that in the worst case, and this is tight. As a corollary, for the weighted any-angle path we obtain the approximation result .P. B. is partially supported by NSERC. G. E., D. O. and R. I. S. are partially supported by H2020-MSCA-RISE project 734922 - CONNECT and project PID2019-104129GB-I00 funded by MCIN/AEI/10.13039/501100011033. G. E. and D. O. are also supported by PIUAH21/IA-062 and CM/JIN/2021-004. G. E. is also funded by an FPU of the Universidad de Alcalá.Peer ReviewedPostprint (published version

Obtaining Optimal Mobile-Robot Paths with Non-Smooth Anisotropic Cost Functions Using Qualitative-State Reasoning

This paper appeared in the International Journal of Robotics Research, 16, 3 (June 1997), 375-399. The equations were reconstructed in 2007 for better readability.Realistic path-planning problems frequently show anisotropism, dependency of traversal cost or feasibility on the traversal heading. Gravity, friction, visibility, and safety are often anisotropic for mobile robots. Anisotropism often differs qualitatively with heading, as when a vehicle has insufficient power to go uphill or must brake to avoid accelerating downhill. Modeling qualitative distinctions requires discontinuities in either the cost-per-traversal-distance function or its derivatives, preventing direct application of most results of the calculus of variations. We present a new approach to optimal anisotropic path planning that first identifies qualitative states and permissible transitions between them. If the qualitative states are chosen appropriately, our approach replaces an optimization problem with such discontinuities by a set of subproblems without discontinuities, subproblems for which optimization is likely to be faster and less troublesome. Then the state space in the near neighborhood of any particular state can be partitioned into "behavioral regions" representing states optimally reachable by single qualitative "behaviors", sequences of qualitative states in a finite-state diagram. Simplification of inequalities and other methods can find the behavioral regions. Our ideas solve problems not easily solvable any other way, especially those with what we define as "turn-hostile" anisotropism. We illustrate our methods on two examples, navigation on an arbitrarily curved surface with gravity and friction effects (for which we show much better performance than a previously-published program 22 times longer), and flight of a simple missile.This work was supported in part by the U.S. Army Combat Developments Experimentation Center under MIPR ATEC 88-86. This work was also prepared in part in conjunction with research conducted for the Naval Air Systems Commandfunded by the Naval Postgraduate SchoolApproved for public release; distribution is unlimited

Optimal Path Finding in Direction, Location and Time Dependent Environments.

This dissertation examines optimal path finding problems where cost function and constraints are direction, location and time dependent. Path-finding problems have been studied for decades in various applications; however, the published work introduces numerous assumptions to make the problem more tractable. These assumptions are often so strong as to render the model unrealistic for real life applications. In our research, we relax a number of such restrictive assumptions to create an accurate and yet tractable model suitable for implementation for a large class of problems. We first discuss optimal path finding in an anisotropic (direction-dependent), time and space homogeneous environment. We find a closed form solution for the problems with obstacle-free domain without making any assumptions on the structure of the speed function. We employ our findings to adapt a emph{visibility graph search} method of computational geometry to an anisotropic environment and deliver an optimal obstacle-avoiding path finding algorithm for a direction-dependent medium. Next, we extend our analysis to a set of problems where path curvature is constrained by a direction-dependent minimum turning radius function. We invoke techniques from optimal control theory to demonstrate the problem's controllability (by reducing the problem to Dubins car problem), prove existence of an optimal path (via Filippov's Theorem), and derive a necessary condition for optimality (using Pontryagin's Principle). Further analysis delivers a closed form characterization of an optimal path and presents an algorithm that facilitates the implementation of our results. %the solution of our problem. Finally, the assumption of time and space homogeneity is relaxed, and we develop a dynamic programming model to find an optimal path in a location, direction and time dependent environment. The results for anisotropic homogeneous environment are integrated into the model to improve its accuracy, efficiency and run-time. The path finding model addresses limited information availability, control-feasibility and computational demands of a time-dependent environment. To demonstrate the applicability and performance of our path-finding methods, computational experiments for an optimum vessel performance in evolving wave-field project are presented.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64828/1/dolira_1.pd