Time-Optimal Paths for a Dubins Car and Dubins Airplane with a Unidirectional Turning Constraint.

The primary goal of an aircraft emergency landing planner is to safely and efficiently land on a runway. One commonly used tool is the solution} of Dubins problem, which defines minimal-time paths for vehicles moving in a plane with constraints on turn rate. The Dubins solution presumes that vehicles can follow straight paths and turn in both directions. A vehicle, however, can be constrained to unidirectional (i.e., either clockwise or counterclockwise) turning motions after experiencing severe structural damage and/or control failure. A unidirectional turning constraint specifies lower and upper bounds on turn rate, both of the same sign. This dissertation addresses, for the first time, the problem of finding time-optimal paths for Dubins vehicles constrained to unidirectional turning motions. This dissertation initially considers a Dubins vehicle in a plane, called Dubins car, with unidirectional turning constraints for which the optimal paths are characterized by employing Pontryagin's minimum principle. A geometric interpretation of the identified extremal paths enables direct identification of the optimal path. To extend these planar results to aircraft emergency landing planning, it is necessary to consider the unidirectional Dubins airplane where the planar motions of this unidirectional Dubins car are supplemented by allowing changes in altitude. Optimal paths for the unidirectional Dubins airplane have one of two transition times: the shortest time for the unidirectional Dubins car or the time equal to the absolute altitude difference required to descend divided by the maximum vertical rate. In some instances of the latter case, a suboptimal path must be constructed to guarantee a feasible solution. Both the unidirectional Dubins car and airplane algorithms developed in this dissertation can be implemented in real-time, thus integrated easily into embedded vehicle management systems. Moreover, these algorithms are complete, thus can be guaranteed to find a feasible solution which in most cases is also time-optimal. Throughout the dissertation, the proposed algorithms are validated through a series of test cases. The dissertation applies the unidirectional Dubins airplane algorithm to aircraft emergency landing at LaGuardia airport to demonstrate its ability to rapidly identify and present a full suite of landing options to the pilot and automation.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108808/1/flyjun_1.pd

Inverse Optimal Planning for Air Traffic Control

We envision a system that concisely describes the rules of air traffic control, assists human operators and supports dense autonomous air traffic around commercial airports. We develop a method to learn the rules of air traffic control from real data as a cost function via maximum entropy inverse reinforcement learning. This cost function is used as a penalty for a search-based motion planning method that discretizes both the control and the state space. We illustrate the methodology by showing that our approach can learn to imitate the airport arrival routes and separation rules of dense commercial air traffic. The resulting trajectories are shown to be safe, feasible, and efficient

On Time-optimal Trajectories for a Car-like Robot with One Trailer

In addition to the theoretical value of challenging optimal control problmes, recent progress in autonomous vehicles mandates further research in optimal motion planning for wheeled vehicles. Since current numerical optimal control techniques suffer from either the curse of dimens ionality, e.g. the Hamilton-Jacobi-Bellman equation, or the curse of complexity, e.g. pseudospectral optimal control and max-plus methods, analytical characterization of geodesics for wheeled vehicles becomes important not only from a theoretical point of view but also from a prac tical one. Such an analytical characterization provides a fast motion planning algorithm that can be used in robust feedback loops. In this work, we use the Pontryagin Maximum Principle to characterize extremal trajectories, i.e. candidate geodesics, for a car-like robot with one trailer. We use time as the distance function. In spite of partial progress, this problem has remained open in the past two decades. Besides straight motion and turn with maximum allowed curvature, we identify planar elastica as the third piece of motion that occurs along our extr emals. We give a detailed characterization of such curves, a special case of which, called \emph{merging curve}, connects maximum curvature turns to straight line segments. The structure of extremals in our case is revealed through analytical integration of the system and adjoint equations

Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance

We consider a two player game, where a first player has to install a surveillance system within an admissible region. The second player needs to enter the the monitored area, visit a target region, and then leave the area, while minimizing his overall probability of detection. Both players know the target region, and the second player knows the surveillance installation details.Optimal trajectories for the second player are computed using a recently developed variant of the fast marching algorithm, which takes into account curvature constraints modeling the second player vehicle maneuverability. The surveillance system optimization leverages a reverse-mode semi-automatic differentiation procedure, estimating the gradient of the value function related to the sensor location in time N log N

Optimizing Steady Turns for Gliding Trajectories

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140649/1/1.g000319.pd

Planning Visual Inspection Tours for a 3D Dubins Airplane Model in an Urban Environment

This paper investigates the problem of planning a minimum-length tour for a three-dimensional Dubins airplane model to visually inspect a series of targets located on the ground or exterior surface of objects in an urban environment. Objects are 2.5D extruded polygons representing buildings or other structures. A visibility volume defines the set of admissible (occlusion-free) viewing locations for each target that satisfy feasible airspace and imaging constraints. The Dubins traveling salesperson problem with neighborhoods (DTSPN) is extended to three dimensions with visibility volumes that are approximated by triangular meshes. Four sampling algorithms are proposed for sampling vehicle configurations within each visibility volume to define vertices of the underlying DTSPN. Additionally, a heuristic approach is proposed to improve computation time by approximating edge costs of the 3D Dubins airplane with a lower bound that is used to solve for a sequence of viewing locations. The viewing locations are then assigned pitch and heading angles based on their relative geometry. The proposed sampling methods and heuristics are compared through a Monte-Carlo experiment that simulates view planning tours over a realistic urban environment.Comment: 18 pages, 10 figures, Presented at 2023 SciTech Intelligent Systems in Guidance Navigation and Control conferenc