Initialization-Free LEO Doppler Positioning via Semidefinite Relaxation: A Cold-Start Approach

Baoshan Song; Weisong Wen; Qi Zhang; Bing Xu; Penggao Yan; Yuan Li; Li-Ta Hsu

cs.ROSep 21, 2025

Initialization-Free LEO Doppler Positioning via Semidefinite Relaxation: A Cold-Start Approach

Authors: Baoshan Song, Weisong Wen, Qi Zhang, Bing Xu, Penggao Yan, Yuan Li, Li-Ta Hsu

Organizations: The Hong Kong Polytechnic University, Hong Kong SAR, China

Abstract

Low Earth Orbit (LEO) satellite Doppler measurements offer an alternative source of positioning information for Internet of Things devices when Global Navigation Satellite System signals are unavailable. Conventional local optimisation methods depend on an accurate initial position and can fail during a complete cold start. We propose an initialization-free LEO Doppler positioning framework based on semidefinite relaxation. The method reformulates the nonlinear positioning problem into a semidefinite programme to obtain a receiver position estimate without prior location knowledge. We analyse relaxation tightness under noiseless conditions and derive sufficient noise bounds under parameter disturbances. Simulations and field experiments using opportunistic Iridium-NEXT signals evaluate initialization sensitivity, positioning accuracy and computational cost. In the field experiment, the proposed method achieves a three-dimensional positioning error of 140 m without an initial position, whereas local solvers converge to poor solutions under large initial offsets. Using the semidefinite estimate to initialise Gauss-Newton or Dog-Leg reduces the positioning error to 130 m. The framework provides a cold-start initialization stage for subsequent local refinement in opportunistic LEO positioning. The implementation and datasets are publicly available to support reproducibility and further research.

Figures & tables

Panel of Figure 1 — Figure 1: Framework and measurement geometry. (a) Overview of the proposed initialization-free LEO Doppler positioning framework. Section 2 presents the Doppler positioning problem. Section 3 formulates the SDP relaxation that solves the problem without initial estimates. Section 4 analyzes the tight relaxation conditions to guarantee exact recovery. (b) Illustration of LEO Doppler positioning geometry. The carrier Doppler shift is determined by the projection of the satellite velocity vector $\mathbf{v}^{s}$ onto the line-of-sight unit vector $\frac{\mathbf{p}_{r}-\mathbf{p}^{s}}{\|\mathbf{p}_{r}-\mathbf{p}^{s}\|}$ between the receiver position $\mathbf{p}_{r}$ and the satellite position $\mathbf{p}^{s}$ .

Init. Dist. [km]	GN	DL	SDP
1	0.00	0.00	0.71
10	0.00	0.00	0.71
100	0.00	0.00	0.71
580	1839.60	91.65	0.71
1000	-	892.23	0.71

Table 1: 3D positioning error with initial points ranged from different initial distance in the simulation test.

Figure 2: Estimated position with the initial distance of 580 km, where ‘GT’ denotes ground-truth and other legends can be referred to Section 5.1 .

Panel of Figure 3 — Figure 3: Positioning accuracy under parameter disturbances. (a) 3D position errors with various Doppler measurement noises. (b) 3D positioning errors with various satellite velocity noises. (c) 3D positioning errors with various satellite position noises.

Figure 4: Demonstration of explicit noise bound, including satellite velocity and Doppler measurement noises, where grey cells denote rank-tightness and black cells denote explicit bounded noise.

Dataset	LEO Doppler positioning
Constellation	Iridium (66 satellites in operation)
Orbit	Polar orbit $\sim$ 780 km altitude
Frequency	1626270833 Hz
Receiver Location	(22.3045966 $\degree$ , 114.180121 $\degree$ ,61.384 m)
Visible Satellite	8
Duration	35 minutes

Table 2: Key parameters in the real experiment.

Panel of Figure 5 — Figure 5: Real-world Iridium-NEXT experiment. (a) Illustration of satellite orbit determination using Two-Line Element (TLE) data and the Simplified General Perturbations Satellite Orbit Model 4 (SGP4) propagation model. The SGP4 model calculates time-varying satellite positions and velocities from TLE sets to provide reference orbital parameters for LEO Doppler positioning. (b) Doppler shift measurements in the real test. (c) Comparison between local minima and global minimum on Google Earth under a 1000 km initial offset. Local search methods get trapped in local optima due to large initial errors, whereas the proposed SDP solver reliably calculates a cold-start initial position with a 3D error of 140 m and a 2D horizontal error of 122 m.

Init dist [km]	GN	DL	SDP	SDP-GN	SDP-DL
10	0.13	0.13	0.14	0.13	0.13
100	0.13	0.13	0.14	0.13	0.13
500	0.13	0.13	0.14	0.13	0.13
1000	2342.50	2069.00	0.14	0.13	0.13
5000	-	8445.10	0.14	0.13	0.13

Table 3: 3D positioning errors of the compared methods using different initial distances in the real test.

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University of Colorado Boulder Boulder, CO · The Charles Stark Draper Laboratory, Inc. Cambridge, MA

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Low Earth orbit (LEO) satellites are leveraged to support new position, navigation, and timing (PNT) service alternatives to GNSS. These alternatives require accurate propagation of satellite position and velocity with a realistic quantification of uncertainty. It is commonly assumed that the propagated uncertainty distribution is Gaussian; however, the validity of this assumption can be quickly compromised by the mismodeling of atmospheric drag. We develop a machine learning approach that corrects error growth in the argument of latitude for a diverse set of LEO satellites. The improved orbit propagation accuracy extends the applicability of the Gaussian assumption and modeling of the errors with a corrected mean and covariance. We compare the performance of a time-conditioned neural network and a Gaussian Process on datasets computed with an open source orbit propagator and publicly available Vector Covariance Message (VCM) ephemerides. The learned models predict the argument of latitude error as a Gaussian distribution given parameters from a single VCM epoch and reverse propagation errors. We show that this one-dimensional model captures the effect of mismodeled drag, which can be mapped to the Cartesian state space. The correction method only updates information along the dimensions of dominant error growth, while maintaining the physics-based propagation of VCM covariance in the remaining dimensions. We therefore extend the utility of VCM ephemerides to longer time horizons without modifying the functionality of the existing propagator.

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