Fuel-optimal three-dimensional trajectories from Earth to Mars for spacecraft powered by a low-thrust rocket with variable speci c impulse capability are presented. The problem formulation treats the spacecraft mass as a state variable, thus coupling the spacecraft design to the trajectory optimization. Gravitational effects of the sun, Earth, and Mars are included throughout an entire trajectory. To avoid numerical sensitivity, the trajectory is divided into segments, each de ned with respect to a different central body. These segments are patched at intermediate time points, with proper matching conditions on the states and costates. The optimization problem is solved using an indirect multiple shooting method. Details of trajectories for the outbound legs of crewed missions to Mars, with trip times of 145 and 168 days, are shown. Effects due to variations in the trip time, departure and arrival orbit inclinations, initial fuel mass, and power level are investigated.
The determination of fuel-optimal, planar, Earth-Mars trajectories of spacecraft using low-thrust, variable speci c impulse I sp propulsion is discussed. The characteristics of a plasma thruster currently being developed for crewed/cargo missions to Mars are used. This device can generate variable I sp within the range of 1000-35,000 s, at constant power. The state equations are written in rotating, polar coordinates, and the trajectory is divided into two phases, patched together at an intermediate point between the Earth and Mars. The gravitationaleffects of the sun, Earth, and Mars are included in the two phases. The formulation of the problem treats the spacecraft mass as a state variable, thus, coupling the spacecraft design to the trajectory design. The optimal control problem is solved using an indirect, multiple shooting method. Results for a 144-daycrewed mission to Mars are presented. The variation of the I sp during spacecraft's escape from the Earth's gravitational eld shows an interesting periodic behavior with respect to time. The results obtained are also compared with those obtained by assuming a three-phase trajectory, with the Earth, sun, and Mars, in uencing the spacecraft, one per phase, in sequence.
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