Selected Problems in Many-Revolution Trajectory Optimization Using Q-Law

Abstract

Q-Law is a Lyapunov guidance law for low-thrust trajectory design. Most prior implementations of Q-Law were limited to relatively simple low-thrust transfers. This work aims to improve the optimality, usability, and efficiency of Q-Law for better application to the mission design process. To accomplish this, Q-Law is combined with direct collocation to form an efficient hybrid method for high-fidelity, many-revolution trajectory design. Additionally, forward and backward Q-Law propagation are combined to form a novel method for Lunar transfer trajectories. This technique rapidly produces spiral trajectories to the Moon and provides mission designers with a means for efficient trade space exploration. Additionally, backward propagated Q-Law is combined with heritage trajectory design software to produce spiral escape trajectories as well as single and double Lunar swingby trajectories for interplanetary rideshare mission scenarios. Lastly, analytical partial derivatives of the Q-Law thrust vector calculation are derived, and the Q-Law algorithm is wrapped in a nonlinear programming problem. When these derivatives are used to generate the trajectory state transition matrix, the efficiency and accuracy of the optimization is superior to finite difference solutions. Using this approach, a novel Q-Law multiple shooting method is formulated and tested on various low-thrust transfer problems. These enhancements to the standard Q-Law algorithm enable efficient trade space exploration for more complex low-thrust trajectories, with a specific emphasis on the needs of SmallSat rideshare missions.