Formation flying in elliptical orbits has gained increasing attention in recent years due to its scientific and operational advantages. Highly elliptical orbits allow spacecraft to remain near apogee for extended durations, potentially enabling extended coverage or communication windows with the ground. At apogee, the relative dynamics between the satellites is also slowing down, which is particularly beneficial for missions requiring long and continuous observation windows, such as solar coronagraphic or coordinated multi-point measurements. A representative example is the ESA’s Proba-3 mission, in which two spacecraft maintain a precisely controlled formation along a highly elliptical orbit to enable high-quality and prolonged solar coronagraph observations.
 
Despite the ongoing success of Proba-3, challenges remain in modelling and controlling the relative dynamics between two satellites in highly eccentric orbits. For instance, linearizing the dynamics near a reference eccentric orbit yields a Linear Time-Varying (LTV) system with periodic coefficients. This periodicity leads to periodic variations in the controllability/observability of the system, its disturbance sensitivity, as well as the required control effort over the course of a single orbital revolution, that makes the direct application of classical Linear Time-Invariant (LTI) control approaches inadequate.
 
To address these inherent challenges and ensure robust stability and performance, this paper introduces a Linear Parameter-Varying (LPV) model that depends on properly defined scheduling variables that capture the time-varying nature of the relative dynamics. Such an LPV representation enables the use of Gain Scheduled (GS) controllers, which can continuously and smoothly adapt to the periodically changing system dynamics and offer improved robust stability and performance guarantees.
 
In GS control design, it is common to approximate the operational domain of the scheduling parameters with a convex polytope defined by the upper and lower bounds of each parameter. The vertices of this polytope serve as representative bounds for controller synthesis. However, the quality of this polytopic approximation has a substantial impact on the achievable control performance. Two factors are well recognized to influence controller effectiveness: (1) the volume of the polytope, which reflects the conservatism introduced in representing the parameter space, and (2) the number of vertices, which affects both computational complexity and the degree of approximation accuracy. Designing a polytope that is unnecessarily large or has an excessive number of vertices often leads to conservative controllers with sub-optimal performance.
 
This research first demonstrates that the relative motion dynamics in elliptical orbits can be rigorously expressed as an LPV system through the selection of appropriate true anomaly-based scheduling parameters. Building on this formulation, we propose a systematic method to construct a convex polytope that simultaneously minimizes the polytope volume and the number of vertices while guaranteeing robustness in the full operational range. The proposed method exploits geometric insights into the structure of the parameter space and provides a more compact and efficient representation compared to more conventional approaches that rely solely on the convex hull of the sampled scheduling parameters.
 
Following the theoretical development of the framework for robust control design, a specific mission will be chosen for validation purposes. There will be specified a clear set of Guidance, Navigation and Control (GNC) requirements for both robust stability and performance against which the performance of the LPV controller will be assessed.
 
Numerical results will show that the controllers designed with the compact polytope achieve significantly improved tracking accuracy and reduced control effort relative to controllers based on traditional convex-hull polytopes. These findings demonstrate that the geometric refinement of the LPV polytope plays a critical role in elevating the practical performance of formation-flying control laws for time-varying orbital environments, making the enhanced LPV approach a promising framework for the robust and efficient maintenance of future elliptic formation flying missions.