Controlling Hamiltonian Integral Invariants of Spacecraft Phase Space Distributions

Phase space distributions outline regions of plausible spacecraft states in phase space given some degree of uncertainty. Under Hamiltonian dynamical flow, geometric features of these distributions become time-invariant and are referred to as Hamiltonian integral invariants. The shape and size of the distributions are correlated to our state uncertainty, so these geometric invariants provide unique, analytical insight into problems of uncertainty propagation. When a Hamiltonian system is perturbed by dissipative forces, the geometric features associated with integral invariants are no longer constant. Rather, they change in time, and the insight into problems of uncertainty propagation also changes. This paper explores how the geometric features of phase space distributions can be targeted with orbital and attitude guidance laws. For example, volume-collapsing guidance laws are designed to restrict both the translational and rotational spacecraft state. Alternative representations for regions of uncertainty are also considered via Monte Carlo and covariance matrix analysis. With these alternative representations, the targeted geometric behavior is validated.