Optimal impulsive control of spacecraft relative motion
Keywords: Mechanical Engineering.

Distributed space systems, or the collective usage of two or more interacting spacecraft, open the door to more complex mission applications, ultimately leading to rapid evolution in fields such as astronomy and astrophysics, planetary science, and heliophysics. In many applications, DSS bring an added layer of fault-tolerance because tasks and payloads can be shared among the spacecraft. This division of both workload and components enables the design of simple on-board systems and ensures that a mission's success is not dependent on a single spacecraft. In other mission applications, DSS increase mission coverage and flexibility, which provides more functionality than a single, monolithic spacecraft alone. However, with these benefits of using DSS comes the inherent challenge of controlling the spacecraft relative orbital motion. This dissertation focuses on the class of spacecraft relative orbit control problems that seek to minimize the delta-v cost of impulsive control actions while achieving a desired relative state in fixed time. The result is an autonomous, robust, and efficient impulsive maneuver planning architecture to solve the relative orbit control problem in a closed orbit of arbitrary eccentricity. The six-dimensional (6D) optimization problem is posed in relative orbit elements (ROE) space, a state representation composed of combinations of the classical orbital elements, which describes the motion of a spacecraft in the DSS relative to a real or virtual reference orbit. Parameterizing the relative motion using the ROE yields insight into relative motion geometry and allows for the straightforward inclusion of orbital perturbations in linear time-variant form. Consequently, the choice of state representation enables solving the control problem in closed-form, leading to delta-v optimal, predictable maneuver schemes for computationally efficient algorithmic implementation in spaceborne processors. The relative motion control architecture put forth in this dissertation makes extensive use of reachable set theory to translate the cost-minimization problem into a geometric minimum length path-planning problem. Reachable set theory is a tool typically used to evaluate achievable states given a control action. When applied to the orbit reconfiguration problem, it greatly simplifies the optimization process without loss of generality. In fact, this dissertation exploits several properties of the reachable sets to prove that the cost of an entire reconfiguration is driven by one 2D projected plane. This geometric intuition is the key to formulating a general methodology to derive the closed-form reachable delta-v minimum. The reachable delta-v minimum is a new metric to quantify the reachability and assess the optimality of a maneuver scheme. An equivalently general methodology follows, which describes how to compute maneuver schemes that achieve a prescribed reconfiguration and meet this new optimality criterion. Though the methodology applies to any Linear Time-Varying (LTV) system, this dissertation applies the methodology to the ROE state representation to derive new globally optimal maneuver schemes in closed orbits of arbitrary eccentricity. The problem is further simplified through the use of a modified ROE state representation, in which the elements are redefined such that relative orbital motion within the reference orbit plane (in-plane) and out of the reference orbit plane (out-of-plane) decouples in any closed orbit regime. The maneuver planning algorithms here are robust to orbit regime and employ combinations of maneuvers in the radial, tangential, and normal directions to achieve an optimal reconfiguration solution, in-plane and out-of-plane alike. However, this dissertation also shows how quantifiably sub-optimal solutions can be generated by relaxing constraints in the general methodology when the optimal solutions are unreachable. For example, restricting the maneuvers to occur only in the tangential direction yields entirely analytic expressions for quantifiably sub-optimal maneuver schemes in eccentric reference orbits. The analytic tangential-only sub-optimal solution requires only a minimal delta-v penalty over the optimal and to be orders of magnitude more computationally efficient. This is just one example of how the general solution methodology can be modified to derive quantifiably sub-optimal solutions with high performance and computational feasibility. Uncertainty in the dynamics model, state knowledge, and maneuver execution can propagate into significant errors in the ROE achieved at the end of a reconfiguration. Therefore, to prepare the control software for on-board implementation, this dissertation analyzes the effects of different error sources on the reachable sets. The analysis focuses on developing both a qualitative understanding of how errors alter the relative motion geometry and a quantitative assessment to mathematically determine the effect that each error has on achieving the desired end state. The analysis, which uses a geometric method based on the non-diagonal covariance matrix of each uncertainty source, shows that errors in maneuver timing and the reference satellite's initial absolute state are negligible. However, errors in the initial relative state and the maneuver magnitudes can propagate significantly and must be mitigated. The culmination of the research presented in this dissertation is the development of the dedicated Guidance, Navigation, and Control (GNC) payload onboard the Demonstration with Nanosatellites of Autonomous Rendezvous and Formation-Flying (DWARF) mission. The DWARF mission is a collaborative development effort between the Stanford Space Rendezvous Laboratory, Gauss S.R.L., and King Abdulaziz City for Science and Technology. The mission seeks to demonstrate novel and state-of-the-art relative navigation and control technology in a sun-synchronous Low Earth Orbit using a pair of identical, autonomous, 3U CubeSats with commercial-off-the-shelf hardware and a cold-gas propulsion system. The many lessons learned during the DWARF mission will facilitate new, more complex, DSS technology such as virtual telescopes, on-orbit servicing, and space structure assembly. This dissertation focuses specifically on the design, implementation, and integration of the control subsystem of the GNC payload, which aims to demonstrate safe, robust, and autonomous formation acquisition, keeping, and reconfiguration at separations between 100 meters and 100 kilometers. This dissertation details the algorithmic implementation of the control subsystem as a regularly called finite state machine that manages on-orbit activity such as optimal maneuver scheme generation, data handling, and error and uncertainty mitigation. Also, to alleviate the inaccuracies that can accumulate from the errors mentioned previously, the DWARF software continuously replans the maneuvers analytically, using a diminishing horizon model predictive control. The functionality and performance of the DWARF prototype flight software are rigorously validated in a high-fidelity software-in-the-loop simulation environment for mission scenarios in near-circular and eccentric orbits. The DWARF simulation environment includes a full-force perturbation model, realistic navigation uncertainty, and maneuver execution errors. It additionally includes a Global Navigation Satellite System (GNSS) signal simulator to emulate on-board navigation with realistic uncertainty, as well as realistic maneuver execution and timing errors. The maneuver schemes are stress-tested further by varying the spacecraft ballistic properties and the orbit geometry to evaluate solution performance in highly-perturbed environments across multiple orbit regimes