[Paper Review] Performance Bounds for Cooperative Localisation in the Starlink Network

Calum Spring-Turner, Raj Thilak Rajan|arXiv (Cornell University)|Jul 11, 2022

Space Satellite Systems and Control4 citations

TL;DR

This paper evaluates cooperative localisation performance in the Starlink mega-constellation using the Cramér-Rao Bound (CRB) to establish theoretical limits on position estimation accuracy. It analyzes two scenarios—using both inter-satellite optical links and ground station measurements, and using only inter-satellite measurements—finding average RMSEs of 10.15 m and 10.68 m, respectively, over most of a satellite's orbit.

ABSTRACT

Mega-constellations in Low Earth Orbit have the potential to revolutionise worldwide internet access. The concomitant potential of these mega-constellations to impact space sustainability, however, has prompted concern from space actors as well as provoking concern in the ground-based astronomy community. Increasing the knowledge of the orbital state of satellites in mega-constellations improves space situations awareness, reducing the need for collision avoidance manoeuvres and allowing astronomers to prepare better observational mitigation strategies. In this paper, we create a model of Phase 1 of Starlink, one of the more well-studied megaconstellations, and investigate the potential of cooperative localisation using time-ofarrival measurements from the optical inter-satellite links in the constellation. To this end, we study the performance of any unbiased estimator for localisation, by calculating the instantaneous Cram$\acute{ ext{e}}$r-Rao bound for two situations; one in which inter-satellite measurements and measurements from ground stations were considered, and one in which only relative navigation from inter-satellite measurements were considered. Our results show that localisation determined from a combination of inter-satellite measurements and ground stations can have at best an an average RMSE of approximately 10.15 metres over the majority of a satellite's orbit. Relative localisation using only inter-satellite measurements has a slightly poorer performance with an average RMSE of 10.68 metres. The results show that both anchored and anchorless inter-satellite cooperative localisation are dependent on the constellation's geometry and the characteristics of the inter-satellite links, both of which could inform the use of relative navigation in large satellite constellations in future.

Motivation & Objective

• To assess the theoretical performance limits of cooperative localisation in the Starlink mega-constellation using unbiased estimators.
• To investigate how inter-satellite optical links and ground station measurements jointly improve orbital state estimation accuracy.
• To quantify the impact of constellation geometry and inter-satellite link characteristics on localisation performance.
• To compare anchored (ground station-augmented) and anchorless (purely relative) navigation scenarios in terms of position estimation accuracy.
• To provide a benchmark for future development of autonomous navigation systems in large LEO constellations.

Proposed method

• Developed a spatial model of Phase-1 Starlink, including satellite orbital elements, orbital planes, and ground station locations.
• Modeled the inter-satellite network topology based on line-of-sight visibility and optical inter-satellite link (ISL) range constraints.
• Applied the instantaneous Cramé;r-Rao Bound (CRB) to derive the theoretical lower bound on position estimation error for both anchored and anchorless navigation.
• Calculated the Relative Cramé;r-Rao Bound (RCRB) for each satellite to quantify the minimum achievable mean squared error in position estimation.
• Used a signal-to-noise ratio parameter $\gamma$ to model inter-satellite link performance, which directly affects the CRB.
• Performed Monte Carlo-style analysis across multiple orbital arcs to compute average and distributional statistics of the RCRB.

Experimental results

Research questions

• RQ1 What is the theoretical lower bound on position estimation accuracy for Starlink satellites using only inter-satellite optical range measurements?
• RQ2 How does the inclusion of ground station measurements improve the achievable localisation accuracy in a cooperative navigation framework?
• RQ3 How does the time-varying geometry of the Starlink constellation affect the performance of cooperative localisation?
• RQ4 How sensitive is the localisation accuracy to the characteristics of the inter-satellite links, particularly the signal-to-noise ratio parameter $\gamma$?
• RQ5 What is the variability in localisation performance across different satellites and orbital positions within the constellation?

Key findings

• The average RMSE for cooperative localisation using both inter-satellite measurements and ground station data is approximately 10.15 metres over the majority of a satellite's orbit.
• For anchorless relative localisation using only inter-satellite measurements, the average RMSE is 10.68 metres, indicating only a small performance gap compared to the anchored case.
• The RCRB exhibits significant variability, with minimum values as low as 8.87 m and maximum values reaching 36.64 m, primarily due to geometric dilution of precision at high latitudes.
• The performance of both anchored and anchorless navigation clusters closely around 10 metres, with minimal satellite-to-satellite variation across the constellation.
• The results demonstrate that inter-satellite cooperative localisation is highly dependent on the dynamic geometry of the constellation and the signal quality of the optical inter-satellite links.
• The study identifies that future improvements could be achieved by incorporating orbital dynamics and time-varying estimation models such as the Extended Kalman Filter, which were not considered in this instantaneous analysis.