Optimal Control of Relative Motion in Arbitrary Fields: Application at Deimos

Abstract

A second-order, general dynamics, relative motion framework is formulated to solve for optimal finite-burn transfers in complex gravity fields that are not amenable to analytic solutions. The second-order variational equations are employed in a Cartesian frame that is general in fidelity and simple to implement. For a passive chief orbit we show that only 16 coefficient functions are necessary to accommodate most dynamical environments of interest. We pre-compute and curve-fit the coefficient functions which represent the time-varying Jacobians and Hessians of the state equations evaluated along the chief orbit. Once the coefficient functions are evaluated, the resulting CUrve-fit quadRatic Variational Equations (CURVE) model is almost transparent to the fidelity level and therefore is well suited for the repeated iterations required by nonlinear optimization. The optimal control problem is solved using a robust, second-order technique that is a variant of differential dynamic programming. The model and optimal rendezvous problems are demonstrated in the highly perturbed dynamical environment of the Martian moon Deimos. The resulting implementation is useful for any relative motion application requiring optimal targeting, particularly in the context of complex force fields. While intended primarily for exotic destinations such as the Moon, asteroids, comets, and planetary satellites, the CURVE model and optimal control framework can also be useful for Earth orbiters, especially in cases of large eccentricity and high fidelity geopotentials.

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