Dynamics-Based Cislunar Uncertainty Characterization

Accurate characterization of state uncertainty can become challenging when considering nonlinear systems. Initially Gaussian distributions become increasingly more difficult to estimate as Gaussian as a state may be propagated forward in time, an operation which may be necessary in space situational awareness mission scenarios. With the amount of spacecraft operating in cislunar space expected to increase in the near future, wherein spacecraft are governed by nonlinear multi-body dynamical systems, the development of accurate uncertainty estimation methods is essential to safe and practical spacecraft operation.

In this work a robust analysis of various uncertainty characterization methods is performed in order to evaluate estimation accuracy of non-Gaussian propagated state distributions. Motion in cislunar space is simulated by applying the dynamical system defined in the Circular Restricted Three Body Problem to objects in orbits about the L1 and L2 equilibrium points used in real mission scenarios. Emphasis is placed on the alteration of existing axis-aligned or optimal point symmetric sample sets using the principal stretching directions of the state distribution derived from state dynamics. Estimation accuracy is evaluated by observing cumulative distribution function computations across multiple measurement spaces to assess sample set replication of distribution shape and density. Error in statistical moment estimates is also analyzed. Probability of collision accuracy is computed to assess performance in a highly sensitive and practical application.

It is observed that the rotation of axis-aligned sample sets according to stretching directions of propagated distributions generally reduces estimation error for smaller propagation times, whereas optimal point-symmetric sample sets are unaffected. Augmentation of sample sets using initial stretching directions is found to improve estimation error in all scenarios. A Gaussian mixture model splitting algorithm applying stretching direction and magnitude is found to have reduced estimation error in orbits experiencing less pronounced deviations from Gaussianity. Principal stretching directions are found to be a sub-optimal choice of splitting direction. Probability of collision accuracy is found to not be significantly altered by alterations to sample sets, likely due to small sample sizes.