Optimization-Based Guidance for Satellite Relative Motion

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Virginia Tech

Spacecraft relative motion modeling and control promises to enable or augment a wide range of missions for scientific research, military applications, and space situational awareness. This dissertation focuses on the development of novel, optimization-based, control design for some representative relative-motion-enabled missions. Spacecraft relative motion refers to two (or more) satellites in nearly identical orbits. We examine control design for relative configurations on the scale of meters (for the purposes of proximity operations) as well as on the scale of tens of kilometers (representative of science gathering missions). Realistic control design for satellites is limited by accurate modeling of the relative orbital perturbations as well as the highly constrained nature of most space systems. We present solutions to several types of optimal orbital maneuvers using a variety of different, realistic assumptions based on the maneuver objectives.

Initially, we assume a perfectly circular orbit with a perfectly spherical Earth and analytically solve the under-actuated, minimum-energy, optimal transfer using techniques from optimal control and linear operator theory. The resulting open-loop control law is guaranteed to be a global optimum. Then, recognizing that very few, if any, orbits are truly circular, the optimal transfer problem is generalized to the elliptical linear and nonlinear systems which describe the relative motion. Solution of the minimum energy transfer for both the linear and nonlinear systems reveals that the resulting trajectories are nearly identical, implying that the nonlinearity has little effect on the relative motion. A continuous-time, nonlinear, sliding mode controller which tracks the linear trajectory in the presence of a higher fidelity orbit model shows that the closed-loop system is both asymptotically stable and robust to disturbances and un-modeled dynamics.

Next, a novel method of computing discrete-time, multi-revolution, finite-thrust, fuel-optimal, relative orbit transfers near an elliptical, perturbed orbit is presented. The optimal control problem is based on the classical, continuous-time, fuel-optimization problem from calculus of variations, and we present the discrete-time analogue of this problem using a transcription-based method. The resulting linear program guarantees a global optimum in terms of fuel consumption, and we validate the results using classical impulsive orbit transfer theory. The new method is shown to converge to classical impulsive orbit transfer theory in the limit that the duration of the zero-order hold discretization approaches zero and the time horizon extends to infinity. Then the fuel/time optimal control problem is solved using a hybrid approach which uses a linear program to solve the fuel optimization, and a genetic algorithm to find the minimizing time-of-flight. The method developed in this work allows mission planners to determine the feasibility for realistic spacecraft and motion models.

Proximity operations for robotic inspection have the potential to aid manned and unmanned systems in space situational awareness and contingency planning in the event of emergency. A potential limiting factor is the large number of constraints imposed on the inspector vehicle due to collision avoidance constraints and limited power and computational resources. We examine this problem and present a solution to the coupled orbit and attitude control problem using model predictive control. This control technique allows state and control constraints to be encoded as a mathematical program which is solved on-line. We present a new thruster constraint which models the minimum-impulse bit as a semi-continuous variable, resulting in a mixed-integer program. The new model, while computationally more expensive, is shown to be more fuel-efficient than a sub-optimal approximation. The result is a fuel efficient, trajectory tracking, model predictive controller with a linear-quadratic attitude regulator which tracks along a pre-computed ``safe'' trajectory in the presence of un-modeled dynamics on a higher fidelity orbital and attitude model.

Model predictive control, astrodynamics, optimal control, Optimization, proximity operations