John Gaebler, KBR; Juan Gutierrez, KBR; Paul Billings, KBR; Christopher Craft, KBR; Charles J. Wetterer, KBR; Jason Baldwin, Complex Futures; Micah Dilley, KBR; Jill Bruer, AFRL
Keywords: Uncertainty propagation, adaptive GMM, cislunar
Abstract:
Cislunar space situational awareness involves new challenges for space-object tracking, chief amongst them is uncertainty characterization. Nominal practices and uncertainty propagation for Earth dominated orbits (e.g., covariance propagated with a state transition matrix) are inadequate for cislunar applications due to the highly nonlinear behavior caused by the gravitational interactions from the Earth, Sun, and Moon. A coupling of complex dynamics and extended measurement gaps spanning from days to weeks emphasizes the need for accurate and efficient uncertainty propagation.
The focus of this paper is to develop an efficient and accurate method to propagate the probability density function (PDF) representing the uncertainty in cislunar space. A common method of approximating the PDF is with a Gaussian mixture model (GMM), which is also convenient to use in an estimation filter. GMMs allow for more accurate uncertainty modeling in highly nonlinear scenarios with large observation gaps. A GMM implementation requires tuning for obtaining a tractable balance between computational load and uncertainty accuracy. Several studies have investigated adaptive GMMs, where a measure of nonlinearity is used to trigger splitting a component into several new components. In this work we seek a general implementation that will work for many diverse scenarios with intuitive tuning parameters. Modeling considerations for an adaptive GMM include when to trigger splitting, how to trigger splitting, what direction to split in, how many components to split, and how to merge components.
A key driver in deciding how to implement the GMM is maintaining a tractable number of Gaussian components. For the case where a large number of components are generated, the computational burden of propagating the GMM can become comparable to utilizing a Monte-Carlo method. This paper integrates several algorithms to form an adaptive GMM that is robust while maintaining acceptable computational costs. The proposed approach will maintain the GMM during stable (or near linear) regions of the trajectory, and trigger splitting of components when needed due to highly nonlinear dynamic regions. The trigger considers the cumulative buildup of nonlinearities, triggering when a threshold is reached. When splitting is indicated by the trigger, the component experiencing the most nonlinearity is split in the direction of the maximum nonlinearity. Steps must be taken to limit the growth in the number of components. After the splitting operation, an adaptive merging process is implemented to merge components that are too similar. Adaptive merging is controlled by an optimization routine which minimizes the number of components without excessive loss of information. This prevents the accumulation of overlapping components.
The triggering method utilized is the Mean Square Error (MSE) method. Other triggers were evaluated; however, several were difficult to implement due to a dependence on the dynamic model. An additional complication of some triggering methodologies is in the choice of threshold value. The threshold values can be ill defined and difficult to interpret, potentially requiring a unique value for each scenario. The MSE was chosen because the threshold has an intuitive meaning and can be set to a single value for all scenarios with reasonable success. For merging GMM components, the Gaussian mixture reduction via clustering (GMRC) method was investigated. However, in testing it was found that the optimization steps of the method do not provide significant benefits relative to the added cost of the optimization. Simply using Runnalls method gives valuable results with minimal computation cost relative to the GMRC method. The resulting adaptive GMM splitting/merging method employed is demonstrated using simulated data on several cislunar scenarios, including an Earth-Moon transfer, near-rectilinear halo (NRHO), and a distant retrograde orbit (DRO).
Date of Conference: September 16-19, 2025
Track: Cislunar SDA