The Cost of Bounded Curvature

We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations $\sigma, \sigma'$, let $\ell(\sigma, \sigma')$ be the shortest bounded-curvature path from $\sigma$ to $\sigma'$. For $d \geq 0$, let $\ell(d)$ be the supremum of $\ell(\sigma, \sigma')$, over all pairs $(\sigma, \sigma')$ that are at Euclidean distance $d$. We study the function \dub(d) = \ell(d) - d, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that \dub(d) decreases monotonically from \dub(0) = 7\pi/3 to \dub(\ds) = 2\pi, and is constant for d \geq \ds. Here \ds \approx 1.5874. We describe pairs of configurations that exhibit the worst-case of \dub(d) for every distance $d$

Path Planning For Persistent Surveillance Applications Using Fixed-Wing Unmanned Aerial Vehicles

This thesis addresses coordinated path planning for fixed-wing Unmanned Aerial Vehicles (UAVs) engaged in persistent surveillance missions. While uniquely suited to this mission, fixed wing vehicles have maneuver constraints that can limit their performance in this role. Current technology vehicles are capable of long duration flight with a minimal acoustic footprint while carrying an array of cameras and sensors. Both military tactical and civilian safety applications can benefit from this technology. We make three main contributions: C1 A sequential path planner that generates a C2 flight plan to persistently acquire a covering set of data over a user designated area of interest. The planner features the following innovations: • A path length abstraction that embeds kino-dynamic motion constraints to estimate feasible path length • A Traveling Salesman-type planner to generate a covering set route based on the path length abstraction • A smooth path generator that provides C2 routes that satisfy user specified curvature constraints C2 A set of algorithms to coordinate multiple UAVs, including mission commencement from arbitrary locations to the start of a coordinated mission and de-confliction of paths to avoid collisions with other vehicles and fixed obstacles iv C3 A numerically robust toolbox of spline-based algorithms tailored for vehicle routing validated through flight test experiments on multiple platforms. A variety of tests and platforms are discussed. The algorithms presented are based on a technical approach with approximately equal emphasis on analysis, computation, dynamic simulation, and flight test experimentation. Our planner (C1) directly takes into account vehicle maneuverability and agility constraints that could otherwise render simple solutions infeasible. This is especially important when surveillance objectives elevate the importance of optimized paths. Researchers have devel oped a diverse range of solutions for persistent surveillance applications but few directly address dynamic maneuver constraints. The key feature of C1 is a two stage sequential solution that discretizes the problem so that graph search techniques can be combined with parametric polynomial curve generation. A method to abstract the kino-dynamics of the aerial platforms is then presented so that a graph search solution can be adapted for this application. An A* Traveling Salesman Problem (TSP) algorithm is developed to search the discretized space using the abstract distance metric to acquire more data or avoid obstacles. Results of the graph search are then transcribed into smooth paths based on vehicle maneuver constraints. A complete solution for a single vehicle periodic tour of the area is developed using the results of the graph search algorithm. To execute the mission, we present a simultaneous arrival algorithm (C2) to coordinate execution by multiple vehicles to satisfy data refresh requirements and to ensure there are no collisions at any of the path intersections. We present a toolbox of spline-based algorithms (C3) to streamline the development of C2 continuous paths with numerical stability. These tools are applied to an aerial persistent surveillance application to illustrate their utility. Comparisons with other parametric poly nomial approaches are highlighted to underscore the benefits of the B-spline framework. Performance limits with respect to feasibility constraints are documented

Distributed Task Allocation and Task Sequencing for Robots with Motion Constraints

This thesis considers two routing and scheduling problems. The first problem is task allocation and sequencing for multiple robots with differential motion constraints. Each task is defined as visiting a point in a subset of the robot configuration space -- this definition captures a variety of tasks including inspection and servicing, as well as one-in-a-set tasks. Our approach is to transform the problem into a multi-vehicle generalized traveling salesman problem (GTSP). We analyze the GTSP insertion methods presented in literature and we provide bounds on the performance of the three insertion mechanisms. We then develop a combinatorial-auction-based distributed implementation of the allocation and sequencing algorithm. The number of the bids in a combinatorial auction, a crucial factor in the runtime, is shown to be linear in the size of the tasks. Finally, we present extensive benchmarking results to demonstrate the improvement over existing distributed task allocation methods. In the second part of this thesis, we address the problem of computing optimal paths through three consecutive points for the curvature-constrained forward moving Dubins vehicle. Given initial and final configurations of the Dubins vehicle and a midpoint with an unconstrained heading, the objective is to compute the midpoint heading that minimizes the total Dubins path length. We provide a novel geometrical analysis of the optimal path and establish new properties of the optimal Dubins' path through three points. We then show how our method can be used to quickly refine Dubins TSP tours produced using state-of-the-art techniques. We also provide extensive simulation results showing the improvement of the proposed approach in both runtime and solution quality over the conventional method of uniform discretization of the heading at the mid-point, followed by solving the minimum Dubins path for each discrete heading