Model predictive control in nap-of-Earth flight using polynomial control basis functions

This paper presents analysis using a new set of design tools to compute finite horizon optimal controls specifically for onboard model predictive control based trajectory synthesis. The optimizer must produce a finite-horizon control trajectory that will enable a UAV to track a low-altitude, high-speed (nap-of-Earth flight) reference trajectory. The optimizer must synthesize the finite-horizon controls based on a suitable fidelity plant model. This can rapidly become a high-dimensional, nonconvex optimization search, particularly if the dynamic model is nonlinear and the horizon long compared to the control bandwidth. To reduce the scope of the optimization problem we constrain the finite-horizon controls to a scaleable set of control basis functions (CBF). We also use these CBFs to identify a linear perturbation model around a nominal realizable trajectory. In this paper, we focus on polynomial CBFs such as Laguerre and Legendre. We compare results to a baseline of those obtained without any simplifying approximations, and to a repeating sequence of tent functions that were introduced in previous publications. Our analysis indicates that Laguerre polynomials pose a good choice of CBFs to design the optimal controls for NOE type of experiments; a fourth order Laguerre polynomial supplies 5 distinct and characteristic polynomial terms and is adequate for good tracking performance. The performance is equivalent to using 10 tent functions. However, since only 5 Laguerre polynomials need to be manipulated to form the optimal control, the optimization speed is greatly enhanced. It is significantly faster than solving a full order MPC problem.