Three-Dimensional Trajectory Optimization in Constrained Airspace

The operational airspace of aerospace vehicles, including airplanes and unmanned aerial vehicles, is often restricted so that constraints on three-dimensional climbs, descents, and other maneuvers are necessary. In this paper, the problem of determining constrained, three-dimensional, minimum time-to-climb, and minimum fuel-to-climb trajectories for an aircraft in an airspace defined by a rectangular prism of arbitrary height is considered. The optimal control problem is transformed to a parameter optimization problem. Because a helical geometry appears to be a natural choice for climbing and descending trajectories subject to horizontal constraints, helical curves are chosen as starting trajectories. A procedure for solving the minimum time-to-climb and minimum fuel-to-climb problems by using the direct collocation and nonlinear programming methods including Chebyshev pseudospectral and Gauss pseudospectral discretization is discussed. Results obtained when different constraints are placed on airspace and state variables are presented to show their effect on the performance index. The question or optimality of the numerical results is also considered.