Casey Heidrich, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder
Keywords: Space Situational Awareness, Space Domain Awareness, Initial Orbit Determination, Cislunar, Collocation, Trajectory optimization
Abstract:
The future of space situational awareness (SSA) necessitates substantial leaps in observing algorithms and technologies as a growing number of spacecraft operators set their sights on cislunar space. A critical capability in SSA is the initial orbit determination (IOD) problem. IOD seeks to reconstruct a target orbit from optical measurements across several measurement arcs. In two-body Keplerian dynamics, IOD is a well-defined problem yielding an initial state for batch processing or sequential estimation. Examples of two-body IOD approaches include classical methods such as Laplace, Gauss, and double-r [1], as well as Gooding’s method [2] and association of admissible regions (AR) [3]. In contrast, IOD with multi-body dynamics presents a significant challenge due to the lack of analytical solutions (i.e., fewer constants of motion), as well as the chaotic nature of trajectories over long observing periods. Existing efforts in cislunar IOD have largely relied on predictive shooting algorithms to solve for an initial state at epoch [4,5]. Due to the highly nonlinear, unpredictable dynamics in the circular-restricted three body problem (CR3BP), these methods break down for measurements gaps on the order of days or even hours. Further, existing approaches often assume perfect angular rate information, imposing burdensome sensor tasking requirements to produce sequential imaging for measurement tracklet interpolation. The state-of-the-art in cislunar IOD will greatly benefit from the development of advanced algorithms that: a) have a large region of convergence requiring minimal a priori knowledge of the target orbit, and b) can converge to an accurate solution over long measurement gaps.
This work presents a novel angles-only IOD approach in multi-body systems using direct collocation. The method works by discretizing the solution space for a set of uncorrelated measurements across a set of collocation nodes (or “knots”). Transcription of the system dynamics converts the continuous-time problem into a set of interpolating splines at each node. An objective function is designed to minimize the sum-squared measurement residuals at each epoch inversely weighted by the measurement error covariance. Following transcription, the resulting IOD formulation is a large-scale sparse nonlinear programming problem (NLP). For comparison, the number of unknowns with traditional shooting IOD methods is on the order of 10^0, whereas the number of unknown parameters in collocation may be on the order of 10^4 or greater. However, numerical NLP solvers such as IPOPT [6] and SNOPT [7] are widely available and generally meet or exceed capabilities of numerical shooting methods. A major benefit of the collocation approach is that little to no initial guess information is required to successfully converge to an accurate solution, even over a large search domain. While direct collocation is commonplace in trajectory optimization settings such as optimal control [8] and interplanetary transfers [9], its utility in multi-body IOD is largely underutilized.
This paper demonstrates the advantages of collocation in comparison to traditional shooting methods for non-Keplerian orbit determination. Under the assumption of CR3BP dynamics, several periodic observer and target orbits of interest are studied. Results show substantial promise for greatly improving IOD over long measurement gaps. For example, an L1 Northern HALO observer is shown to successfully reconstruct L2 Southern HALO and L4 long period target orbits with measurement gaps on the order of weeks. Further promise is illustrated by converging to these solutions from extremely limited initial guess information for the target orbit (such as initializing the algorithm at one of the Lagrange points). The method is highly flexible, allowing for varying observer geometry using angles-only measurement information. Although the CR3BP dynamics are chosen for study, there are no restrictions that preclude application to more realistic orbits using full ephemerides. When compared to existing shooting methods, the novel IOD approach proves highly reliable in reconstructing multi-body orbits in the absence of dense measurement information over short observing arcs.
SSA is critical to maintaining operational awareness and safety in cislunar space. As the number of objects in the Earth-Moon system continues to expand, operators will require increasing reliability and capability of SSA algorithms. Robust IOD represents a basic requirement for any cislunar observing architecture, as it enables the association of uncorrelated tracks or identification of previously undetected objects from optical measurements. The collocation IOD algorithm presented in this work is a fundamental departure from traditional approaches, demonstrating significant improvements in both solution accuracy and rate of convergence across long measurement intervals with limited initial guess information.
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Date of Conference: September 19-22, 2023
Track: Cislunar SDA