Casey Heidrich, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder
Keywords: Initial orbit determination, optimization, measurement association, maneuver reconstruction
Abstract:
Initial orbit determination (IOD) has long been considered a critical capability for space situational awareness (SSA). The fundamental objective of IOD is the estimation of a full spacecraft state using electro-optical (EO) or range measurements. Historical IOD methods are built upon analytical solutions from two-body Keplerian mechanics, such as Gauss’s method or double-r [1]. These assumptions fail in complex multi-body systems where classical IOD methods provide little to no practical utility. More recent work has developed IOD methods for non-Keplerian systems using optimization and sequential filtering techniques [2-4]. These methods approach the limitations of classical IOD methods by approximating solutions of nonlinear system trajectories using numerical integration. However, chaotic dynamics such as the circular restricted three-body problem (CR3BP) become difficult to predict over long time scales, and these methods can become unstable with large measurement gaps or insufficient observations.
Recent work has focused on continuous improvements to both the accuracy and reliability of IOD methods in complex orbit environments. The algorithm developed in Ref. [5] demonstrates a novel approach to non-Keplerian IOD using collocation methods. The methodology involves transcription of continuous-time optimization problem to a large-scale sparse nonlinear programing problem (NLP). These methods are considered implicit integrators that enforce system dynamics through collocation constraints (in contrast to time-marching algorithms such as explicit Runge Kutta schemes [6]). The collocation IOD approach provides many benefits, including a large region of convergence from poor initial guesses, as well as improved stability over long observing gaps. Whereas prior work focused on quiescent (non-maneuvering) objects, operational realities will require greater flexibility in terms of dynamical modeling and assumptions for IOD of maneuvering spacecraft.
The measurement association problem has long been an open area of research in SSA, the purpose of which is to identify observations corresponding to the same object at different epochs. The challenge of associating uncorrelated observations has been approached using covariance-based methods [7], Bayesian inference [8], and admissible region (AR) based optimization methods [9], among others. The latter involves the solution of a constrained optimization problem for undetermined quantities satisfying AR constraints at each measurement. Similarly, maneuver reconstruction is an important topic in SSA for inferring spacecraft behavior from sparse measurement information. Classical sequential or batch estimation algorithms often break down when the true system behavior does not match an assumed dynamics model. Prior works use control distance metrics for maneuver detection and object correlation by solving an uncertain two-point boundary value problem (BVP) for the optimal control policy between observations [10,11]. However, these methods require a full reference state and prior, which is typically not available information in IOD problems.
This work develops a robust non-Keplerian IOD algorithm incorporating maneuver reconstruction and observation association without prior information using nonlinear programming. Problem objectives and constraint structures are considered, such as L2-optimal control and minimum fuel thrust profiles. The overall problem structure is retained such that efficient NLP solvers may take advantage of inherent Jacobian sparsity for rapid solution. Preliminary results show significant promise for reconstructing large maneuvers of cislunar spacecraft with limited information, such as angles-only measurements across observation gaps of days or weeks. The algorithm produces accurate results without a close initial guess for the spacecraft state or control history. For example, convergence is demonstrated for initialization at one of the Earth-Moon Lagrange points and null (all zeros) for the control history. This work also develops observability metrics to quantify the accuracy and consistency of IOD estimates.
Reliable and adaptable IOD methods are critical to continued success of operations and safety in cislunar space. While the assumptions of this work do not preclude applications in traditional two-body dynamics (such as near-Earth orbits), they are general enough to allow for rapid IOD with complex multi-body systems and maneuvering targets. Furthermore, optional inclusion of admissible region information as NLP constraints provides a direct approach for implicit measurement association. The results and findings of this work are expected to enhance situational awareness and operational capability in cislunar space for cataloging and tracking agile space objects.
References
[1] Escobal, P. R., Methods of orbit determination, Krieger, Malabar, FL, 1965, pp. 239–292.
[2] Wishnek, S., and Holzinger, M. J, “Robust cislunar initial orbit determination,” in AMOS Conf. Proc., Maui, 2021.
[3] Bolden, Mark, et al. “Probabilistic Initial Orbit Determination and Object Tracking in Cislunar Space Using Optical Sensors.” Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference. 2022.
[4] Chow, C. Channing, et al. “Cislunar Orbit Determination Behavior: Processing Observations of Periodic Orbits with Gaussian Mixture Model Estimation Filters.” The Journal of the Astronautical Sciences 69.5 (2022): 1477-1492.
[5] Heidrich, C., and Holzinger, M. J., “Universical Angles-Only initial Orbit Determination Using Sparse Collocation.” Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference. 2023.
[6] Owren, Brynjulf, and Marino Zennaro. “Derivation of efficient, continuous, explicit Runge–Kutta methods.” SIAM journal on scientific and statistical computing 13.6 (1992): 1488-1501.
[7] Hill, Keric, Kyle Alfriend, and Chris Sabol. “Covariance-based uncorrelated track association.” AIAA/AAS Astrodynamics Specialist Conference and Exhibit. 2008.
[8] Holzinger, Marcus J., et al. “Uncorrelated-track classification, characterization, and prioritization using admissible regions and Bayesian inference.” Journal of Guidance, Control, and Dynamics 39.11 (2016): 2469-2484.
[9] Worthy III, Johnny L., Marcus J. Holzinger, and Daniel J. Scheeres. “An optimization approach for observation association with systemic uncertainty applied to electro-optical systems.” Advances in Space Research 61.11 (2018): 2709-2724.
[10] Holzinger, Marcus J., Daniel J. Scheeres, and Kyle T. Alfriend. “Object correlation, maneuver detection, and characterization using control distance metrics.” Journal of Guidance, Control, and Dynamics 35.4 (2012): 1312-1325.
[11] Lubey, Daniel P., and Daniel J. Scheeres. “An optimal control based estimator for maneuver and natural dynamics reconstruction.” Proceedings of the 2013 Advanced Maui Optical and Space Surveillance Technologies Conference. 2013.
Date of Conference: September 17-20, 2024
Track: Cislunar SDA