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**A Survey of Estimator Performance in the Tracking of Chaotic Orbits from the Lunar Surface**

*Marcus Bever, ExoAnalytic Solutions*

**Location:** Beacon B

That the field of aerospace engineering is replete with sufficiently nonlinear problems so as to warrant the study of a diverse array of estimation techniques is perhaps self-evident. From mapping and navigation to structural modeling and orbital analysis, the applications are as varied as the approaches. This study is focused on a problem of emerging interest in the astrodynamics community: statistical estimation in the context of the chaotic dynamics of the circular restricted three-body problem. Notably, the performance of a wide array of linear and nonlinear filtering techniques is to be investigated and assessed primarily against the metric of estimator consistency. The extended and unscented Kalman filters are frequent choices for problems of modest complexity, yet a diffuse prior can pose significant challenges. While the quadrature and cubature Kalman filters offer additional opportunities for comparison, the chief interest for this work lies in a contrast of two nonlinear techniques of a common mixture model form: the probabilistic Gaussian sum filter, of which the AEGIS formulation will be used [4], and the more recent credibilistic Gaussian mixture filter using possibility functions [1].

Three years after the halo orbit was introduced to the community by Farquhar in 1968 [6], Alspach and Sorenson published their work on the Gaussian mixture model [9]. A most wide variety of adaptations have since been established, from the splitting capabilities demonstrated by DeMars [4] to the recent merging operations popularized by Psiaki's resampling algorithm [7]. Such fundamental flexibility to represent more faithfully nonlinear and non-Gaussian systems has made the Gaussian mixture a prime candidate for another branch of uncertainty quantification: that of credibilistic reasoning. Driven by the principle of minimal specificity, an outer probability measure ultimately seeks to represent the uncertainty of systems by differentiating what may be considered possible from that which evidence suggests to be entirely impossible and not warranting further consideration [8]. In more recent years, a Gaussian mixture adaptation was implemented for a class of outer measures, yielding a fully recursive solution for filtering [3]. (A brief, yet more in-depth discussion of the motivation for such an approach was presented in a 2019 paper in which Delande served as the guiding mentor [2].) Implicitly, this work of Delande relied on a publication of the same year by a co-author in which the Kalman filtering equations were shown to hold in an unmodified form [5] for what was later referred to as credibilistic reasoning: uncertainty quantification using the class of outer probability measures defined by a sole possibility function. This forms the basis for the credibilistic approach here used.

The present study on orbit determination will leverage an optical measurement system based on the lunar south pole with the objective to track a rogue satellite along an unstable trajectory. Given a motivation to test the robustness of each filtering algorithm, in the first effort of this paper, exact observation sensor conditions are tuned to extreme values, such that few algorithms repeatably obtain a successful result. This process of refining suggests a hierarchy of the more robust approaches. Continuing, the second portion of the work, indeed the primary interest for the paper, is an empirically driven analysis of estimator consistency over stochastically-chosen initial conditions. This Monte Carlo simulation uses a more modest sensor tuning to avoid continual divergence for a family of algorithms. In all cases, the rogue spacecraft in question performs an impulse maneuver midway through the simulation so as to evaluate a filter's ability to recover an accurate estimate.

Preliminary results suggest that the two nonlinear Gaussian mixture models (being themselves supported by the unscented transform) have a far superior capacity to recover from a poor estimate, surpassing even the extended Kalman filter. This is not entirely unintuitive, given the diverse spread of mixand estimates leveraged to survey a region of the chaotic state space. That said, more detailed studies are needed to assess the practical strengths and weaknesses of the dual approaches, probabilistic and credibilistic, in relation to each other, particularly in the presence of unmodeled maneuvers. It may be expected that the probabilistic machinery generally yields a more precise result, with the credibilistic formulation focusing instead on delivering an increasingly accurate estimate, but this has not yet been confirmed empirically. The scaled version of the unscented Kalman filter has performed poorly, yet additional tuning may improve such results.

A broader interest in extending space situational awareness to the cislunar regime has been motivated by several factors, notably the recent successes of the Chinese Chang'e satellite programs. However, with no current observation architecture deployed to persistently monitor the 4-pi steradians of the cislunar sphere, it may be expected that sparse data will initially be a necessary component of early situational and domain awareness endeavors until such time as additional sensors are fielded. Consequently, understanding the performance characteristics of the classic tools in estimation theory will enable the strategic selection of algorithms, ones that have been proven capable of reliable performance in this highly nonlinear, but crucial problem. This survey of algorithm behavior over a baseline tracking scenario is presented as a means to further the discussion in the community.

References

[1] M. Bever. "On the Application of Credibilistic Filtering to Uncertainty Quantification and Assessment in the Operational Space Domain". MA thesis. The University of Texas at Austin, 2020.

[2] Marcus Bever, Emmanuel Delande, and Moriba Jah. "Outer Probability Measures for Quantitative First- and Second-Order Uncertainty in the Space Domain". In: IAA/AAS SciTech Forum 2019 on Space Flight Mechanics and Space Structures and Materials (Moscow, RU). Vol. 174. Advances in the Astronautical Sciences. American Astronautical Society, 2019, pp. 67-80.

[3] Emmanuel Delande, Jeremie Houssineau, and Moriba Jah. "A New Representation of Uncertainty for Data Fusion in SSA Detection and Tracking Problems". In: 21st International Conference on Information Fusion (Cambridge, UK). IEEE, 2018, pp. 1309-1316.

[4] Kyle J. DeMars, Robert H. Bishop, and Moriba K. Jah. Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems. 2013.

[5] J. Houssineau and A.N. Bishop. "Smoothing and filtering with a class of outer measures". In: SIAM/ASA Journal on Uncertainty Quantification 6.2 (2018), pp. 845-866.

[6] Kathleen Connor Howell. "Three-Dimensional, Periodic, 'Halo' Orbits". In: Celestial Mechanics 32.1 (1984), pp. 53-71.

[7] Mark L. Psiaki. "Gaussian Mixture Nonlinear Filtering with Resampling for Mixand Narrowing". In: IEEE Transactions on Signal Processing 64.21 (2016), pp. 5499-5512.

[8] Branko Ristic, Jeremie Houssineau, and Sanjeev Arulampalam. "Robust Target Motion Analysis Using the Possibility Particle Filter". In: IET Radar, Sonar Navigation 13.1 (2019), pp. 18-22.

[9] H.W Sorenson and D.L Alspach. "Recursive Bayesian Estimation Using Gaussian Sums". In: Automatica 7.4 (1971), pp. 465-479.

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