Navigation Based Path Planning by Optimal Control Theory
Abstract
Previous studies have shown that implementing trajectory optimization can reduce state estimations errors. These navigation based path planning problems are often difficult to solve being computationally burdensome and exhibiting other numerical issues, so former studies have often used lower-fidelity methods or lacked explanatory power. This work utilizes indirect optimization methods, particularly optimal control theory, to obtain high-quality solutions minimizing state estimation errors approximated by a continuous-time extended Kalman filter. Indirect methods are well-suited to this because necessary conditions of optimality are found prior to discretization and numerical computation. They are also highly parallelizable enabling application to increasingly larger problems. A simple one dimensional problem shows some potential obstacles to solving problems of this type including regions of the trajectory where the control is unimportant. Indirect trajectory optimization is applied to a more complex scenario to minimize location estimation errors of a single cart traveling in a 2-D plane to a goal location and measuring range from a fixed beacon. This resulted in a 96% reduction of the location error variance when compared to the minimum time solution. The single cart problem also highlights the importance of the matrix that encodes the linearization of the vehicle’s measurement with respect to state. It is shown in this case that the vehicle roughly attempts to maximize the magnitude of its elements. Additionally, the cart problem further illustrates problematic regions of a design space where the objective is not significantly affected by the trajectory. An aircraft descent problem demonstrates the applicability of these methods to aerospace problems. In this case, estimation error variance is reduced 28.6% relative to the maximum terminal energy trajectory. Results are shown from two formulations of this problem, one with control constraints and one with control energy cost, to show the benefits and disadvantages of the two methods. Furthermore, the ability to perform trade studies on vehicle and trajectory parameters is shown with this problem by solving for different terminal velocities and different initial locations.
Degree
M.Sc.
Advisors
Grant, Purdue University.
Subject Area
Design|Mathematics|Systems science
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