Finding Optimal-Path Maps for Path Planning Across Weighted Regions

This paper appeared in the International Journal of Robotics Research, 19, 2 (February 2000), pp. 83-95, with elaborating additions from [Rowe and Alexander, 1997]. The equations were redone for greatly improved clarity in 2008.Optimal-path maps tell robots or people the best way to reach a goal point from anywhere in a known terrain area, eliminating most of the need to plan during travel. We address the construction of optimal-path maps for twodimensional polygonal weighted-region terrain, terrain partitioned into polygonal areas such that the cost per unit distance traveled is homogeneous and isotropic within each polygon. This is useful for overland route planning across varied ground surfaces and vegetation. We propose a new algorithm that recursively partitions terrain into regions of similar optimal-path behavior, and defines corresponding "path subspaces" for these regions. This process constructs a piecewise-smooth function of terrain position whose gradient direction is everywhere the optimal-path direction, permitting quick finding of optimal paths. Our algorithm is more complicated than the current path-caching and wavefront-propagation algorithms, but gives more accurate maps requiring less space to represent. Experiments with an implementation confirm the practicality of our algorithm.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 Command and funded by the Naval Postgraduate School.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 Command and funded by the Naval Postgraduate School.Approved for public release; distribution is unlimited

Timely Near-Optimal Path Generation for an Unmanned Aerial System in a Highly Constrained Environment

A current challenge in path planning is the ability to efficiently calculate a near-optimum path solution through a highly-constrained environment in near-real time. In addition, computing performance on a small unmanned aerial vehicle is typically limited due to size and weight restrictions. The proposed method determines a solution quickly by first mapping a highly constrained three-dimensional environment to a two-dimensional weighted node surface in which the weighting accounts for both the terrain gradient and the vehicle\u27s performance. The 2D surface is then discretized into triangles which are sized based upon the vehicle maneuverability and terrain gradient. The shortest feasible path between the nodes of the two-dimensional triangulated surface is determined using an A* algorithm. An optimal path is then chosen through the unconstrained corridor to yield a quick near-optimal path solution in three-dimensional space. This technique requires prior knowledge of the terrain map and vehicle performance. The cost to traverse each segment of the map is independent of the starting position on the map and can be pre-calculated once the goal position is known. The proposed method allows for a rapid path solution from any start position to a goal position while satisfying all constraints. It was shown that employing the methodology herein resulted in near-optimal solutions in less than a couple seconds for the scenarios tested. The future work section proposes methods for improving the algorithms efficiency even further

Bayesian Active Edge Evaluation on Expensive Graphs

Robots operate in environments with varying implicit structure. For instance, a helicopter flying over terrain encounters a very different arrangement of obstacles than a robotic arm manipulating objects on a cluttered table top. State-of-the-art motion planning systems do not exploit this structure, thereby expending valuable planning effort searching for implausible solutions. We are interested in planning algorithms that actively infer the underlying structure of the valid configuration space during planning in order to find solutions with minimal effort. Consider the problem of evaluating edges on a graph to quickly discover collision-free paths. Evaluating edges is expensive, both for robots with complex geometries like robot arms, and for robots with limited onboard computation like UAVs. Until now, this challenge has been addressed via laziness i.e. deferring edge evaluation until absolutely necessary, with the hope that edges turn out to be valid. However, all edges are not alike in value - some have a lot of potentially good paths flowing through them, and some others encode the likelihood of neighbouring edges being valid. This leads to our key insight - instead of passive laziness, we can actively choose edges that reduce the uncertainty about the validity of paths. We show that this is equivalent to the Bayesian active learning paradigm of decision region determination (DRD). However, the DRD problem is not only combinatorially hard, but also requires explicit enumeration of all possible worlds. We propose a novel framework that combines two DRD algorithms, DIRECT and BISECT, to overcome both issues. We show that our approach outperforms several state-of-the-art algorithms on a spectrum of planning problems for mobile robots, manipulators and autonomous helicopters