Convex Optimization of Relative Orbit Maneuvers Using the Kustaanheimo-Stiefel Transformation

As small-satellite constellations continue to grow in size and complexity, there is an increasing need for autonomous relative navigation and control capabilities. Many small satellites utilize non-impulsive low thrust propulsion or manipulation of perturbation forces such as differential drag for orbit control. These low-acceleration control technologies result in long time horizons over which the control actions must be planned and executed. Currently no dynamics model satisfies the computation, accuracy, and generalizability required for autonomous long-time-horizon control. This paper presents a relative-dynamics model based on the Kustaanheimo-Stiefel transformation. We demonstrate that it achieves equivalent or better accuracy compared to existing relative-orbit models in the literature. In addition, our Kustaanheimo-Stiefel model requires a small number of timesteps per orbit and easily incorporates low-acceleration control inputs. These features make it easily adaptable to convex trajectory optimization methods, which we demonstrate by solving a low-thrust orbital rendezvous problem over a time horizon of 75 orbits with a maximum
$20 \mu \mathrm{m}/\mathrm{s}^{2}$
thrust constraint.