A guidance law for the aeroassisted plane change maneuver in the presence of atmospheric uncertainties

A stochastic feedback control law for a space vehicle performing an aeroassisted plane change maneuver is developed. The stochastic control law is designed to minimize the energy loss while taking into consideration the uncertainty in the atmospheric density. The solution is based on expansion of the stochastic Hamilton-Jacobi-Bellman equation (or dynamic programming) about a zeroth-order known integrable solution. The zeroth-order optimization problem uses a deterministic dynamical system which neglects inertial terms, assuming that aerodynamic forces dominate. In the stochastic problem, a zero-mean process noise term is added to the exponential density model (both additive noise and multiplicatlve noise, with known spectral densities). Then, the stochastic Hamilton-Jacobi-Bellman equation is expanded using the spectral densities of the noise as the expansion parameters. Each expansion term is determined by a perturbed Hamilton-Jacobi-Bellman equation described by a first-order linear, partial differential equation that can be solved by the method of characteristics. The characteristics are simply the trajectories of the zeroth-order solution. The resulting guidance law is expressed as a series expansion in the noise power spectral densities. A numerical example indicates the potential improvement of this method.