Certifiably Optimal Satellite Orbit Determination Based on Doppler Measurements for Low-Earth-Orbit Satellites

Low-Earth Orbit (LEO) satellite is promising to provide intelligent services in the future. To fulfill the need, orbit determination (OD) is vital for accurate satellite positions. Doppler-based OD has been widely studied for its low-cost and convenience. Mainstream Doppler-based OD methods rely on local estimation, which require accurate initialization and may converge to suboptimal solution. We address this problem by utilizing convex optimization theory. First, we model Doppler-based OD with quadratic constrained quadratic program (QCQP) and semi-definite program (SDP) for a convex expression. Then, we derive the necessary conditions for solving the globally optimal solution of this OD problem in both noiseless and noise cases. In noise case, parameter noises including Doppler measurement and satellite velocity noise are considered. Specially, we derive an explicit parameter noise bound to guarantee the optimality. Results of simulation show that local optimization based OD converges to local optima when initial point is 1500 km away from the ground truth. In comparison, our proposed method does not require initialization and provides certifiably optimal OD solutions, even under mild noises. Furthermore, a demonstration of explicit noise bound shows 6 m/s for satellite velocity noise and 8 m/s for Doppler measurement noise. Within this predicted noise range, the estimate proves to be optimal.