Space situational awareness (SSA) relies on accurate and efficient uncertainty realism and propagation (UR&P). In the cislunar domain, there is growing interest in the use of orbits near the collinear libration points, which are relative equilibria in the simplified circular restricted three-body model. These equilibria are hyperbolic and therefore possess dynamical structures in phase space that separate the flow and present difficulties for UR&P. In this paper, we build on prior efforts to define a canonical distribution on a cylindrical space given by the application of normal form theory at a collinear equilibria. The canonical distribution is constructed to have components with disjoint support on the phase space, each of which properly represents the characteristic dynamics for the specific subset of phase space. This is achieved by a product of a multivariate folded normal distribution and conditional normal and von Mises distributions. Comparisons to a regularized distribution, the generalized Bernoulli-Gauss-von Mises, are made.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Most aspects of space situational awareness (SSA) rely on accurate and efficient uncertainty realism, propagation, and nonlinear filtering. A new frontier for SSA is the application to the cislunar realm, which lacks a global orbital element coordinate set. The dynamics of a representative model, the circular restricted three-body problem (CR3BP), for the cislunar domain provides the opportunity to define local orbital elements using dynamical systems techniques such as normal form theory. Motivated the structure of the CR3BP SSA problem, we construct a generalized Bernoulli Gauss von Mises distribution, that is defined on local orbit element coordinates generated from normal form theory at a saddle-center-center equilibrium point, and show its ability to capture what may be a common deformation mode of the CR3BP.

Particle flow methods are effective in resolving the particle degeneracy issue in the standard particle filtering algorithm. However, flow methods have their own difficulties, such as the necessity to solve a Poisson equation in the feedback particle filtering (FPF) method. This is computationally heavy, and we observe a numerical sensitivity and singularity issue dependent on parameter selection when applying to chaotic dynamical systems with limited particle size and coarse integration step size. In this paper, we address the numerical singularity issue by flowing particles in the unstable subspace (UAS), and we name the novel method the projected FPF. It brings the local dynamical information into the assimilation step by using the finite-time Lyapunov exponents and vectors to project observations and particle states to the UAS, where the error diverges. The projected FPF is tested against the Lorenz 1963 model – a nonlinear, low-dimensional, chaotic dynamical system.