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
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 vs onto the line-of-sight unit vector ∥pr−ps∥pr−ps between the receiver position pr and the satellite position ps .
Init. Dist. [km]
GN
DL
SDP
SDP-GN
SDP-DL
1
0.00
0.00
0.71
0.00
0.00
10
0.00
0.00
0.71
0.00
0.00
100
0.00
0.00
0.71
0.00
0.00
580
1839.60
91.65
0.71
0.00
0.00
1000
-
892.23
0.71
0.00
0.00
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 .
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 ∼ 780 km altitude
Frequency
1626270833 Hz
Receiver Location
(22.3045966 ° , 114.180121 ° ,61.384 m)
Visible Satellite
8
Duration
35 minutes
Table 2: Key parameters in the real experiment.
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.
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.
Alex Moody, Penina Axelrad, Rebecca Russell
University of Colorado Boulder Boulder, CO · The Charles Stark Draper Laboratory, Inc. Cambridge, MA
Global navigation satellite system (GNSS) positioning is widely used for urban navigation, but the covariance reported by the GNSS solver is often unreliable in urban canyons. Existing differentiable factor graph optimization (DFGO) methods learn measurement weighting through the solver, but they still use position-only objectives. As a result, the position estimate may improve while the reported covariance remains too small, too large, or incorrectly oriented. We propose CredibleDFGO (CDFGO), a differentiable GNSS factor graph framework that makes covariance credibility an explicit training target. A Weighting Generation Network (WGN) predicts per-satellite reliability weights, and a differentiable Gauss-Newton solver maps these weights to a position estimate and a Hessian-derived posterior covariance. We use proper scoring rules to supervise the East-North predictive distribution end to end. We study negative log-likelihood (NLL), the energy score (ES), and their combination. Results on three UrbanNav test scenes show consistent gains in covariance credibility. Positioning accuracy also improves on the medium-urban and harsh-urban scenes; on the deep-urban scene, both the mean horizontal error and the 95th-percentile error improve. On the harsh-urban Mong Kok (MK) scene, CDFGO-Combined reduces the mean horizontal error from 13.77 m to 11.68 m, reduces NLL from 40.63 to 6.59, and reduces ES from 12.31 to 9.05 relative to DFGO (MAE). Case studies link the MK improvement to better axis-wise consistency, more credible local covariance ellipses, and satellite-level reweighting.
Liang Qian, Penggao Yan, Penghui Xu +1
Department of Aeronautical and Aviation Engineering The Hong Kong Polytechnic University, Hong Kong, China
Future lunar surface missions require reliable positioning for autonomous rover operations. Lunar navigation satellite systems may provide limited vertical positioning accuracy due to satellite availability and observation geometry, particularly during early deployment. This study investigates a radiometric navigation architecture in which a lunar surface station (SS) improves rover positioning through double-differenced satellite observations and additional surface-to-surface ranging. Pseudorange and Doppler measurements from lunar navigation satellites are used for relative positioning between the SS and rover, while dual one-way ranging (DOWR) provides an additional range observable. The radiometric solution is integrated with rover dead-reckoning information. Monte Carlo simulations were conducted using five satellites representative of NASA's Lunar Communications Relay and Navigation Systems (LCRNS) constellation and a rover trajectory near Connecting Ridge at the lunar south pole. The results show that the SS improves horizontal and vertical relative positioning, with further gains from DOWR. In the simulated scenario, the mean 99.7% vertical positioning error decreased from 29.9 m without the SS to 5.5 m with double-differenced pseudorange and Doppler processing, and to 4.8 m with DOWR. An uncorrected DOWR hardware bias significantly degraded positioning, whereas the proposed bias-estimation method achieved a final estimation error of 0.058 +/- 0.057 m. The benefit of DOWR also depended on observation geometry. These results demonstrate the potential of a lunar surface station to improve vertical relative positioning through direct radiometric ranging, complementing its communication and navigation support roles.
Toshiki Tanaka, Satoshi Ikari
Intelligent Space Systems Laboratory, The University of Tokyo, Tokyo, Japan.