Optimal Control of Low-Thrust Satellite Formation-Flying Reconfiguration using a LQR

Low-thrust guidance and control of satellite Formation Flying relative motion is a field of study that is gaining more and more attention by researchers from every part of the world due to the recent development of more and more performing electric propulsion systems and since future missions will follow the trend of miniaturizing spacecraft. The challenge is to design a controller that must address limited thrusting and propellant capabilities while maintaining operational aspects, such as collision safety and time constraints. For the scope of this work, a formation of satellites in LEO is considered, employing thrusters capable of providing thrust in the order of µ-Newtons. The first step of the work is the definition af a dynamical system that describes the Formation-Flying relative motion with respect to the chief spacecraft, for this scope the Mean Relative Orbital Elements (ROE) dynamical system is introduced. The main feature of this model is linearity, thanks to which a Linear Quadratic Regulator can be implemented in order to control the relative motion; another incredible advantage of this dynamic system is the easy visualization of the relative motion with respect to the chief spacecraft, thanks to the eccentricity/inclination vector separation discussed in the work. The Controller based on the Linear Quadratic Regulator Theory is then implemented, proving itself to be very solid and accurate, either in terms of fuel optimality and desired state tracking accuracy, still requesting a limited computational load (only one ODE must be solved). The Controller is then improved for reference trajectory tracking using an elegant Reference Governor method], which performs outstandingly well. Collision Avoidance can be then implemented easily in the model using a powerful mathematical tool as Artificial Potential Functions, in combination with the aforementioned Reference Governor approach. In order to employ this method, parallelism between relative eccentricity and inclination vectors must be assured. This assumption can be easily satisfied by proper mission and maneuver planning, in particular by choosing wisely the desired states. Fuel Balance between spacecraft in a formation is an often disregarded issue in literature but yet very important; in fact, it is vital to avoid that one spacecraft would run out of fuel before the other ones, which is a situation that would cause a loss of controllability for the entire formation. In this work, this issue is thoroughly addressed by proposing two different strategies: - R matrix values are exploited in order to increase the fuel consumption of the less-consuming satellite; - Fuel Balance has been achieved by smart reconfiguration planning, by choosing wisely the desired states in order to satisfy the mission goals and constraints. A discussion about the two strategies is then made, showing how the first one is quite inefficient from a fuel-optimality point of view compared to the second strategy, which is then the preferred one. Finally, the whole model is succesfully validated by importing its results in the GMAT environment in order to perform an high-precision Orbit Propagation simulation, and then by comparing the final results.