Asymptotic optimization of very long, low thrust propelled inter-orbital maneuvers

Electric propulsion is capable of providing control accelerations of 1 mm/s² or less, four or more decimal orders below orbital gravity accelerations. This implies very long inter-orbital maneuvers counting many hundreds of revolutions. Any straightforward numerical method of thrust optimization, quite successful in deep space trajectory design, is doomed to fail when applied to trajectories this long. A practical approach is use of thrust-to-gravity ratio as a small parameter. When dealing with quadratic quality criteria such as fuel consumption, this approach leads to splitting the problem into a sequence of two Optimal Control problems, corresponding to ‘fast’ and ‘slow’ time. The strategy has much in common with the averaging methods widely used in Celestial Mechanics. Its implementation is based on the use of the two sets of osculating elements, which describe the orbit and the trust program, respectively. The latter one is considered as a control vector. Both sets obey the differential equations derived from the Maximum Principle. Their structure is well adapted to averaging over an orbital period. The procedure eliminates the fast time, i.e., filters out the oscillatory terms with this period from the osculating elements and leaves the slow secular changes untouched. Averaging simplifies the problem so much as to make it solvable numerically. Efficient quadrature formulae were developed supplying exact numerical values as well as convenient semi-analytical representation to the integrals that appear in the averaging process. The averaged (secular, slow time) problem actually constructs an optimal end-to-end trajectory as an optimal concatenation of a set of locally optimal arcs built at the previous phase. Pontryagin's principle reduces the optimization problem to a two-point boundary value problem. An algorithm designed to solve this TPBVP was developed. A representative sample transfer trajectory is presented as an illustration of the proposed technique.