Three-Dimensional Trajectory Optimization in Constrained Airspace

Abstract

This dissertation deals with the generation of three-dimensional optimized trajectory in constrained airspace. It expands the previously used two-dimensional aircraft model to a three-dimensional model and includes the consideration of complex airspace constraints not included in previous trajectory optimization studies. Two major branches of optimization methods, indirect and direct methods, are introduced and compared. Both of the methods are applied to solve a two-dimensional minimumtime- to-climb (MTTC) problem. The solution procedure is described in detail. Two traditional problems, the Brachistochrone problem and Zermelo’s problem, are solved using the direct collocation and nonlinear programming method. Because analytical solutions to these problems are known. These solutions provide verification of the numerical methods. Three discretization methods, trapezoidal, Hermite-Simpson and Chebyshev Pseudospectral (CP) are introduced and applied to solve the Brachistochrone problem. The solutions obtained using these discretization methods are compared with the analytical results. An 3-D aircraft model with six state variables and two control variables are presented. Two primary trajectory optimization problems are considered using this model in the dissertation. One is to assume that the aircraft climbs up from sea level to a desired altitude in a square cross section cylinder of arbitrary height. Another is to intercept a constant velocity, constant altitude target in minimum time starting from sea level. Results of the optimal trajectories are compared with the results from the proportional navigation guidance law. Field of View constraint is finally considered in this interception problem. The CP discretization and nonlinear programming method is shown to have advantages over indirect methods in solving three-dimensional (3-D) trajectory optimization problems with multiple controls and complex constraints. Conclusions from both problems are presented and properties of each one are discussed. Finally, suggestions for future research are addressed.