Fundamental limits on spacecraft orbit uncertainty and distribution propagation

Abstract

In this paper we present and review a number of fundamental constraints that exist on the propagation of orbit uncertainty and phase volume flows in astrodynamics. These constraints arise due to the Hamiltonian nature of spacecraft dynamics. First we review the role of integral invariants and their connection to orbit uncertainty, and show how they can be used to formally solve the diffusion-less Fokker-Plank equation for a spacecraft probability density function. Then, we apply Gromov's Non-Squeezing Theorem, a recent advance in symplectic topology, to find a previously unrecognized fundamental constraint that exists on general, nonlinear mappings of orbit distributions. Specifically, for a given orbit distribution, it can be shown that the projection of future orbit uncertainties in each coordinate-momentum pair describing the system must be greater than or equal to a fundamental limit, called the symplectic width. This implies that there is always a fundamental limit to which we can know a spacecraft's future location in its coordinate and conjugate momentum space when mapped forward in time from an initial covariance distribution. This serves as an "uncertainty" principle for spacecraft uncertainty distributions.

Global symplectic uncertainty propagation on SO(3)

“…In [17], it is shown that the probability density function is preserved along a Hamiltonian flow on Euclidean space. In this subsection, we generalize this result to a Hamiltonian system evolving on a general symplectic manifold.…”

“…When the flow advecting the probability density is Hamiltonian, the Liouville equation reduces to an ordinary differential equation [17]. Thus, a probability density function can be propagated using the flow map of the Hamiltonian system.…”

Dynamics of symplectic subvolumes

“…Different symplectic constraints arise on such sets, including conservation of the signed 2k-volume projections on the coupled symplectic planes as well as the constraints implied from Gromov's Nonsqueezing Theorem (see Scheeres et al [13] for a discussion of these constraints in relation to orbit uncertainty evolution). We will further present an additional constraint for a minimal obtainable volume that exists on certain classes of 2k-dimensional symplectic sets and show how such a constraint leads to the local collapse of phase space along solution curves in Hamiltonian phase space.…”

Applications of Symplectic Topology to Orbit Uncertainty and Spacecraft Navigation

“…In [1][2][3] applications of a deep topological property (Gromov's symplectic nonsqueezing theorem) are made to the study of uncertainty analysis of Hamiltonian systems. These articles study the implications of the nonsqueezing theorem for probability density functions that define uncertainty distributions for particle trajectories in space, with specific applications to spacecraft [1] and orbit debris predictions [3].…”