Analytic Gradient Computation for Bounded-Impulse Trajectory Models Using Two-Sided Shooting

Abstract

Many optimization methods require accurate partial derivative information in order to ensure efficient, robust, and accurate convergence. This work outlines analytic methods for computing the problem Jacobian for two different bounded-impulse spacecraft trajectory models solved using twosided shooting. The specific two-body Keplerian propagation method used by both of these models is described. Methods for incorporating realistic operational constraints and hardware models at the preliminary stage of a trajectory design effort are also demonstrated and the analytic methods derived are tested for accuracy using automatic differentiation. A companion paper will solve several relevant problems that show the utility of employing analytic derivatives, i.e. compared to using derivatives found using finite differences.

Application and Analysis of Bounded-Impulse Trajectory Models with Analytic Gradients

“…The calculation of STMs and MTMs and the construction of the matrix chains represents, by far, the majority of the computational effort required to evaluate a twopoint shooting phase as depicted in Figure 1. In order to assist implementation, a sample numerical STM-MTM chain multiplication is provided in Appendix (D), which makes use of the analytic expressions presented in the companion paper [8] as well as the hardware modeling that will be discussed in section III of this paper.…”

“…The analytic techniques that were the focus of the companion paper were developed with the multiple gravity assist low-thrust (MGALT) and multiple gravity assist with n deep space maneuvers using shooting (MGAnDSMs) transcriptions in mind, but can be extended to any bounded impulse trajectory model [8]. The optimization of the trajectories encoded with these transcriptions is typically performed with a nonlinear programming (NLP) solver (such as SNOPT [9] or IPOPT [10]), and one of the critical pieces of information required by such a tool are the partial derivatives obtained by taking the partial derivative of the constraint vector c with respect to the vector of decision variables x that constitute the problem to be solved:…”

Indirect Optimization of Fuel-Optimal Many-Revolution Low-Thrust Transfers With Eclipses

“…The higher gradient accuracy is achieved since it elegantly eliminates the subtractive cancellation error [26,28]. However, both AD and CSD requires extensive implementation and the execution time could be high [29]. The variational method is a promising alternative to offer accurate gradients with short computational time [25].…”

Optimal Two-Impulse Interplanetary Trajectories: Earth-Venus and Earth-Mars Missions

“…Abdelkhalik and Mortari (2007) utilize a genetic algorithm to solve the transfer between noncoplanar elliptical orbits utilizing impulsive maneuvers. On the other hand, Ellison et al (2018) develop analytical methods to compute partial-derivatives of two boundedimpulsive trajectories with multiple swing-by maneuvers. Gagg Filho and da Silva Fernandes (2018) build patched-conic approximations to obtain geometrical details of interplanetary missions.…”