Multi-Satellite Based Passive Localization of Spaceborne Cooperative RF Emitters: Simulation Framework and Impact of Constellation Geometry

This paper presents a simulation framework for space-based cooperative target localization using Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) measurements from Low Earth Orbit (LEO) satellite constellations. The study addresses how geometric constellation configurations affect the accuracy of passive Radio-Frequency (RF) tracking for cooperative spacecraft. The simulation methodology is described through its constituent functional blocks: scenario configuration establishing orbital parameters and RF system specifications, orbit propagation, Signal-to-Noise Ratio (SNR) evaluation, detectability analysis with Line-of-Sight (LOS) verification, synthetic measurement generation, and a robust three-stage position and velocity estimation. This estimation algorithm integrates multi-start initialization to ensure probabilistic coverage of the global minimum, Trust-Region-Reflective (TRR) optimization with analytical Jacobians for computational efficiency, and final validation to reject impossible solutions. The framework is validated through three case studies employing four-satellite cluster architectures at 850-950 km altitude, tracking a cooperative target in Sun-Synchronous orbit at 550 km: a configuration combining co-orbital and cross-track sensors, a purely co-orbital arrangement, and a cross-track configuration with symmetric geometry. Results demonstrate that constellation geometry, quantified by the Geometric Dilution of Precision (GDOP), is the dominant factor determining localization performance for a given target. For the assumed system parameters, the first and most favorable geometry configuration achieves mean position and velocity accuracy of 16.93 m and 0.80 m/s, respectively, when GDOP remains below 5, with progressively degrading performance at higher GDOP values. Conversely, the co-orbital configuration produces GDOP values exceeding 300, yielding mean position and velocity errors above 16 km and 320 m/s, while the symmetric cross-track architecture exhibits bimodal error distributions with 46% of solutions converging to geometrically ambiguous positions displaced by approximately 90° in right ascension. These findings establish that spatial diversity in satellite distribution is essential for accurate TDOA-FDOA-based localization, providing design guidelines for constellation architectures supporting cooperative tracking in congested LEO environments.