In the space field, to reconstitute an antenna from unit elements for interferometry applications, the frequency band observed imposes the constraints of location and synchronization of the satellites. The positioning constraints is a fraction of the wavelength, which means that an accuracy better than the centimeter is needed as soon as the frequency goes beyond 300 MHz. When using radio frequency links, this kind of precision requires techniques based on carrier phase measurement. These methods are used in the terrestrial environment on GNSS signals. But in the case of GNSS satellites, atomic clocks on board the satellites are used to generate signals of excellent quality and advanced techniques of orbit restitution and correction of various biases are implemented in terrestrial calculation centers. When considering a swarm of satellites, one generally considers satellites of small size and therefore unable to carry atomic clocks as efficient as those used in GNSS satellites. A promising research track to improve the estimation of the absolute position of all satellites is to resort to techniques exploiting the associated Euclidean Distance Matrix (EDM) [1], that is inter-satellites distances. To be effective, these techniques must exploit distance measurement as precise as possible and measurement of the absolute positions of a subset of satellites of the swarm [2]. From the estimation theory viewpoint, two approaches are possible offering two different trade-offs. The first approach consists in exploiting a non-ambiguous phase model, that is, a phase model parameterized by the unknown distance and an unknown global phase misalignment arising from both the transmitter and receiver. Some approaches treat the phase misalignment term as unknown, while others assume ideal calibration and compensation. As a middle ground between these two extremes, a balanced approach is to assume that the phase misalignment can be estimated with a certain degree of uncertainty, a consideration that can be applied to many practical systems. The second approach consists in exploiting an ambiguous phase model which gathers all possible contributors, that is the unknown distance and the unknown global phase misalignment. In that approach, phase measurement inherently provides a measure of the distance modulo the wavelength of the carrier signal. This ambiguity must be removed. 

The first approach, which is addressed in depth in a PhD thesis in the process of finalization [3], allows to have access to precision far lower than centimeter but at the expense of a high signal-to-noise ratio and a computational cost not compatible with most real time applications. 

The second approach has been successfully used for almost 3 decades in RTK GNSS based system but never in the context of a swarm of satellites connected by ISL. Indeed, RTK relies on a smart combination of base-band delay and carrier phase estimates at the receiver level, and of the same estimates but at a georeferenced base-station level, transmitted to the receiver. The output of RTK being the precise position of the receiver relatively to the base-station. A process not suited for a swarm of satellites connected by ISL. 

However, recent works [4] have shown that joint estimation of the phase ambiguities and other real-valued parameters of interest is possible for a linear phase observation model in the presence of bounded real-valued parameters. Thus, the main purpose of the proposed PhD thesis is to explore under which swarm system design constraint (one or several pivot nodes (anchors) for instance) and extension of the findings introduced in [4] (nonlinear phase observation model, and/or combination with a state model to perform a Kalman filtering or particle filtering), a swarm of satellites could obtain, with ambiguous phase measurements, a centimeter level precision (as in RTK) of at least relative positioning (relatively to anchors) and ideally absolute positioning. 

The continuity and availability of the distance measurement will also be part of the thesis. Indeed, it must be considered that the satellites of the swarm are not controlled in attitude and their free rotation could cause periodic discontinuities in the reception of the carrier signal. Consequently, the phase measurement would not be continuous either and the delays for resolving ambiguities after the reacquisition of the signal would also influence the availability of the measurement.

[1] I. Dokmanic et al, Euclidean Distance Matrices. Essential theory, algorithms, and applications, IEEE SP Mag., 32(6), 2015. 

[2] Estimation conjointe de position et écart d’horloge dans un essaim de satellites, CNES-DTN/TPI/STR - 2023.0002896

[3] J. Bernabeu Frias, Use of phase measurement for precise positioning in a swarm of satellites, Phd-Thesis, CNES/IPSA/ENAC/ISAE, 2022-2025 

[4] A. Khodabandeh, Bias-Bounded Estimation of Ambiguity: A Method for Radio Interferometric Positioning, IEEE Trans. on SP, 70, 2022

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For more Information about the topics and the co-financial partner (found by the lab !); 

contact Directeur de thèse - Eric.CHAUMETTE@isae-supaero.fr

Then, prepare a resume, a recent transcript and a reference letter from your M2 supervisor/ engineering school director and you will be ready to apply online  before March 14th, 2025 Midnight Paris time !