In this section, we formulate the problem of the optimal round-trip trajectory as a mathematical programming problem. The complete problem can be decomposed into three separate problems to be solved in sequence: (i) determination of the optimal trajectory for the outgoing trip; (ii) determination of the optimal trajectory for the return trip; (iii) determination of the waiting time in the low Mars orbit.
The optimization of a LEO-to-LMO transfer can be reduced to a mathematical programming problem involving the following performance index, constraints, and parameters.
Performance Index. The most obvious performance is the characteristic velocity,
which is the sum of the terminal velocity impulses: ? V LEO is the accelerating velocity impulse at LEO (Earth coordinates) and ? V LMO is the decelerating velocity impulse at LMO (Mars coordinates).
Constraints. The departure conditions include the radius condition (11a), (9a), decircularization condition (11b), (9c), and tangency condition (11c) [for brevity, constraints (11)]. Satisfaction of the departure conditions is trivial for any choice of the parameters ? V LEO and ? PE(0). By the same token, the differential system (7) is never violated if a forward integration is performed with SCS initial conditions consistent with (11) and (21). The only constraints to be enforced are the final conditions, which include the radius condition (14a), (12a), circularization condition (14b), (12c), and tangency condition (14c) [for brevity, constraints (14)].
Parameters. Let a,b,c denote the following vector parameters:
The 7 1 vector
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