Benedikt Aigner, Universität der Bundeswehr München; Fabian Dallinger, Universität der Bundeswehr München; Thomas Andert, Universität der Bundeswehr München; Tobias Vorderobermeier, Universität der Bundeswehr München
https://doi.org/10.64861/WSUH9895
Keywords: Passive RF; one-way Doppler; Doppler-only orbit determination; physics-informed neural networks; high-fidelity dynamics; orbit recovery
Abstract:
One-way Doppler (received-frequency) time series from passive RF satellite downlinks provide metric information on relative motion between a spacecraft and a ground receiver. This enables orbit recovery and refinement from short, sparse arcs. Improving Doppler-based orbit determination under these conditions supports Space Situational Awareness (SSA) by helping maintain and update orbits when radar and optical coverage are limited.
This paper studies a physics-informed orbit determination framework for one-way Doppler time series and benchmarks it against a classical batch weighted least squares (WLS) estimator with state transition matrix (STM) integration. The proposed approach follows a physics-informed paradigm in which observation fitting is coupled with explicit enforcement of the equations of motion during training. Rather than relying solely on iterative linearization around an initial orbit guess, the physics-informed formulation estimates a continuous trajectory while regularizing solutions toward dynamically consistent motion. We quantify robustness and convergence behavior for Doppler-only estimation and document where each method succeeds or fails.
We consider a one-way received-frequency measurement model derived from line-of-sight geometry between a known receiver and the spacecraft state. The predicted frequency is computed from the nominal carrier frequency and a Doppler factor determined by the line-of-sight direction and the receiver and spacecraft velocities expressed in a common inertial frame. Each observation is modeled as the predicted frequency plus measurement noise. Because Doppler-only arcs are geometrically weak, this configuration provides a demanding test of estimator robustness under the limited-information conditions that can challenge classical WLS.
All simulations are implemented in an in-house research codebase that provides SSA-relevant measurement models and high-fidelity dynamics. These include spherical-harmonic gravity, atmospheric drag, solar radiation pressure, and third-body perturbations, together with SPICE-based timing and reference-frame transformations. The physics-informed estimator and the WLS baseline use the same force and measurement models, so the main differences between them arise from the trajectory representation and the optimization strategy. The proposed method estimates a continuous Cartesian trajectory using a physics-informed neural network representation. In this work, the representation is realized as a Physics-Informed Extreme Learning Machine, whose hidden layer is drawn at random and kept fixed. The model is trained by minimizing a combined loss that includes weighted measurement residuals and a dynamics residual evaluated at collocation points. Two parameterizations are compared. In the direct form, the second time derivative of the position output is constrained. In the reduced form, the network additionally outputs velocity, and the dynamics are enforced through two first-order residuals. Analytical Jacobians of the full loss are derived and implemented for both formulations.
We conduct two controlled low Earth orbit simulation studies using 1 Hz received-frequency measurements with representative S-band scaling and noise. The first study considers one orbital revolution of the International Space Station observed by four ESA ground stations, and the second considers one revolution of a Sun-synchronous orbit observed by three. In Experiment 1, we evaluate Doppler-only orbit recovery by comparing the physics-informed estimator with a classical batch WLS baseline under identical assumptions. Experiment 2 examines robustness under poor initialization. Because the physics-informed estimator requires no a priori orbit estimate, initialization enters instead through the random draw of its hidden layer, which is fixed once generated. Convergence is therefore assessed over a systematic sweep spanning both scenarios, both parameterizations, a range of hidden-layer sampling scales, and a large ensemble of random draws. The sweep identifies regimes in which recovery becomes unreliable.
The results characterize the accuracy of the physics-informed estimator relative to the WLS baseline when both are applied to the same measurements. Unlike the baseline, the physics-informed estimator requires no a priori orbit estimate. The results also characterize convergence behavior, including the fraction of random draws that converge, the dependence of that fraction on the hidden-layer sampling scale, and the differences between the two parameterizations. The outcomes include a Doppler-only physics-informed orbit determination formulation that is compatible with high-fidelity dynamics, a reproducible benchmark against a classical STM-based batch WLS estimator under identical assumptions, and a practical assessment of performance limits and failure modes. The study also provides guidance on the choice of sampling scale and on the number of random restarts required for reliable operation. The results inform passive RF applications by clarifying when Doppler-only estimation is sufficient and when additional constraints are beneficial.
Date of Conference: September 15-18, 2026
Track: Astrodynamics