The Gnomonic Projection for B-Spline parameterized 4-D Trajectory Optimization Problems

Kurzfassung

In aerospace related optimal control problems it is often convenient to parameterize the flight trajectory instead of aircraft control inputs (surface deflections or orientation w.r.t. airflow for instance). This allows to describe the problem with a smaller set of optimizer degrees of freedom, since the kinematical variables representing the trajectory change on a larger timescale and have a smoother shape. One example is the discretization with waypoint positions as parameters, which can then be adjusted by an optimizer to minimize one or multiple criteria. A small parameter set is desirable especially in scenarios with long trajectories (for example intercontinental flights), in order to avoid long computation times and convergence problems. In this work, a new approach for parameterization of such trajectories is presented, which applies the gnomonic projection for generation of shortest-path routes between waypoints. These trajectories feature on the one hand maneouvering phases during departure and approach, and on the other hand long and steady cruise segments, hence necessitating a variable density parameterization approach. Basis splines are first generated in the projected space, subsequently degree-elevated and finally transformed back to ellipsoidal coordinates for computation of derived variables. Ultimately, the method allows to generate realistic 4-D shortest-path trajectories with high continuity, which can be analytically evaluated at every timestep, to serve as reference inputs to a flight control system or an inverse aircraft model.