Analytical theory of satellite relative motion with applications to autonomous navigation and control

Accurate and efficient models for the relative motion of two satellites are needed to achieve greater autonomy in an increasingly crowded orbit environment with limited computing power. Analytical models provide a computationally cheap alternative to numerical integration for relative motion propagation, typically at the expense of accuracy. Two avenues to improving relative dynamics modeling accuracy are examined herein. First, higher-order terms in the Keplerian dynamics can be included. This dissertation introduces new, second-order models that are valid for both circular and elliptical orbits in three of the most popular relative state representations: the Cartesian relative position coordinates in the rotating Hill frame, the spherical coordinates in the inertial frame, and the relative orbital elements (ROE). The performance of these models is compared with one another and with several of the most popular models from the relative dynamics literature. The second direction to improve modeling accuracy is the inclusion of non-Keplerian perturbations. A general framework for modeling arbitrary perturbations in the Hill frame is developed and used to derive the equations governing the leading-order corrections for Earth oblateness perturbation, as well as a closed-form solution for the effect of oblateness on the relative motion in near-equatorial orbits. This is compared with Keplerian models and an ROE-based perturbation model in a full-force propagation. In addition to the advancement of relative dynamics models, this dissertation examines two applications of such models. A fast and efficient method for initial relative orbit determination (IROD) from bearing-angle measurements is introduced. The second-order models developed in the course of this research provide a means resolving the range ambiguity problem that arises from linear relative motion models. The IROD problem thereby becomes one of solving a system of polynomial constraint equations linking the line-of-sight measurements to the relative state parameters. An efficient method for solving this system is developed around the insight that these parameters scale with the ratio of the inter-spacecraft separation to the orbit radius and are therefore small for most applications of interest. The method uses a truncated expansion of the quadratic formula to recursively eliminate unknowns, reduce the dimension of the system, and ultimately acquire an approximate solution. Strategies for improving robustness, efficiency, and accuracy are developed and the method is applied to general second-order systems as well as to a broad range of IROD scenarios. Modifications to the constraint equations and solution algorithm are introduced to address the challenge of bias in the bearing-angle measurements. The second application considered is that of low-thrust maneuver planning for formation reconfiguration. The adoption of fuel-efficient electric propulsion systems poses a challenge for relative orbit control schemes, which are typically based on the assumption of impulsive maneuvers. That challenge is met herein with a geometrically intuitive, semi-analytical solution to the low-thrust problem. Beginning with the equations of relative motion of two spacecraft, an unperturbed chief and a continuously-thrusting deputy, a thrust profile is constructed which transforms the equations into a form that is solved analytically. The resulting relative trajectories are the family of sinusoidal spirals, which provide diversity for design and optimization based upon a single thrust parameter. Closed-form expressions are derived for the trajectory shape and time-of-flight for two prescribed relative velocity behaviors, and used to develop a novel patched-spirals trajectory design and optimization method. The example problem of a servicer spacecraft establishing and reconfiguring a formation around a target in geostationary earth orbit is used to demonstrate the application of the patched spirals technique as well as the advantages of the relative spiral trajectories over impulsive maneuvers. The sensitivity of the trajectory solutions to deviations from the underlying assumptions, uncertainties in the state, and errors in thrust are studied through high-fidelity simulation

Keywords: Engineering.
Aeronautics and Astronautics.
Advisor: D'Amico Simone
Barrows Andrew Kevin
Elschot Sigrid