Analytic Calculation and Application of the Q-Law Guidance Algorithm Partial Derivatives

The closed-loop Q-Law guidance algorithm has been shown to be a very capable and efficient method for producing low-thrust trajectories. This paper poses a Q-Law optimization problem for computing locally optimal gain values and for enforcing nonlinear constraints on the initial state using nonlinear programming (NLP). Gradient-based optimization methods have been shown to benefit greatly when analytical partial derivatives are supplied to the optimizer. Therefore, we present derivations of the Q-Law thrust vector partial derivatives with respect to the Q-law gains as well as with respect to the spacecraft's state. These partials are leveraged to produce a state transition matrix, which contains exact partial derivatives of the terminal state with respect to the NLP problem decision vector. The Q-Law NLP problem can be coupled with the Sims-Flanagan interplanetary model in the same optimization problem. In this approach, the NLP solver uses a Q-Law model to design the planetocentric capture/escape spirals and a Sims-Flanagan model to design the interplanetary legs, resulting in end-to-end trajectory optimization.