Sean Wolfe, University of Toronto; M. Reza Emami, University of Toronto
https://doi.org/10.64861/EWBF4827
Keywords: Initial Orbit Determination, Orbit Determination, Physics-informed Extreme Learning Machine, Physics-informed Neural Network, Extreme Learning Machine, Temporarily Bound Object, Near-Earth Asteroid
Abstract:
Conventional Orbit Determination (OD) techniques, namely, batch and sequential filters, rely on nontrivially good Initial Orbit Determination (IOD) solutions. Physics-Informed Extreme Learning Machines (PIELM) have recently been introduced for OD, which can converge to a solution even with a random initialization, thus do not need IOD. However, despite the random initialization, it has been shown that PIELM does not significantly outperform the standard batch filer, and that in more complicated scenarios the random initialization of PIELM may cause the OD solution to diverge. In the PIELM framework, a resultant model, achieved through the minimization of a loss function, reconstructs the orbit using the observation arc and adopted orbital dynamics model. The approach is a derivative of the Physics-Informed Neural Network (PINN), which can describe physical regimes given proper initial and boundary conditions, but utilizes Extreme Learning Machine (ELM) instead of multi-layer perceptron, allowing for a faster OD and a more flexible optimization strategy.
The possibility of random initialization with PIELM suggests that it may actually be better suited for IOD rather than OD. The IOD problem generally considers detection scenarios with a limited number of observations that span short arcs, and conventional techniques utilized include the Gauss, Laplace, double-r, and Gooding methods. Assuming that there are at least three Line-of-Sight (LoS) measurements available and the measurements span a well-defined arc, IOD solutions can be obtained for these methods. For many scenarios, these solutions represent an adequate initialization for downstream OD tasks, but in more complex scenarios such solutions become degenerate, as all methods are based on the Keplerian motion. A PIELM framework can easily mitigate this limitation, as the dynamics utilized is user-defined. However, it is unlikely that the existing PIELM proposed for OD will yield acceptable results in challenging IOD scenarios, as PIELM can become trapped in local optima when complex dynamics are adopted.
To address the limitations of both conventional IOD solutions and PIELM for OD, this paper presents a PIELM framework compatible with IOD, which differs from current PIELM implementations as follows: i) basin-hopping is utilized as a search heuristic to avoid solutions that are in fact local minima; ii) range regularization is incorporated into the loss function to mitigate the inherent range ambiguity from angles-only measurements; and iii) a method to artificially increase arc length is formulated, improving IOD performance. Basin-hopping is a global optimization technique that has seen considerable success and utilization in the early stages of trajectory planning, primarily for its ability to adequately explore the optimization landscape. The range regularization term added to the loss function penalizes physically feasible solutions that are at observationally infeasible distances. To enforce the adopted physical regime, a strategic placement of collocation points is required. Typically this equates to distributing the points between the observation epochs, but in this work the points are allowed to lie within some pre-defined time offset before and after the observation arc, essentially artificially increasing the arc length. Further, the various hyperparameters of PIELM, e.g., the size of the model or the number of collocation points to utilize, heavily impact the resultant solution, yet a few research has been conducted on the interdependence of these parameters and how to best set them. For this reason, an extensive hyperparameter search is conducted in the paper leveraging high performance computing. The search first divides hyperparameters into similar groups, e.g., model-related parameters and basin-hopping-related parameters, in order to reduce the dimensionality of the search space. Then, for each group of parameters an analysis of variance is performed, in order to quantify the relative importance of a parameter and its pairwise interactions on the resultant orbit error. Finally, for interdependent parameters, grid searches are performed to establish relationships between parameters.
To verify the basin-hopping PIELM with range regularization and arc extension for IOD, a case study with Temporarily Bound Objects (TBO) is considered. Temporarily bound objects are natural Earth satellites, captured by the Earth-Moon System (EMS) for a period of time, which approach Earth far more closely than other natural bodies. To date, only a few TBO (four) have been discovered, thus researchers have constructed synthetic populations of potential TBO to investigate their characteristics. It has been proposed in the literature that the rate of TBO detection can be increased through spaceborne observations from the Sun-EMS Lagrange point L1. However, the IOD of such bodies with conventional techniques would pose significant challenges, given the chaotic nature of TBO trajectories and short observation arcs from a spaceborne observer. In the case study, a microsatellite equipped with a visible optical payload fabricated from commercial-off-the-shelf components (COTS) is deployed to the Sun-EMS L1, in order to first detect then perform the IOD of TBO. The observation uncertainty and cadence is based on current COTS and detection capabilities for TBO, and the observer orbit is modeled after the Lisa Pathfinder quasi-halo orbit. A synthetic database consisting of tens of thousands of distinct TBO trajectories obtained from N-body integrations is utilized to verify the proposed PIELM framework for IOD, and compare its performance to conventional IOD algorithms.
The case study reveals several key findings. The parameter analysis shows that: i) basin-hopping and collocation point parameters can be tuned and considered separately with minimal expected effect on the IOD solution; and ii) collocation point parameters have a greater impact on the IOD solution as compared to basin-hopping parameters. Further, a major result of the parameter analysis is that artificially extending the observation arc through collocation point placement can significantly decrease IOD solution error, a quality for IOD unique to PIELM. In general, the analysis shows that properly setting hyperparameters, as outlined in this paper, can have noticeable impacts on the resultant orbit error and computation time.
Date of Conference: September 15-18, 2026
Track: Astrodynamics