Computation of optimal Mars trajectories via combined chemical/electrical propulsion, Part 2: Minimum time solutions with bounded thrust direction

摘要

The success of the solar-electric ion engine powering the DS1 spacecraft has paved the way toward the use of low-thrust electrical engines in future planetary/interplanetary missions. Vis-a-vis a chemical engine, an electrical engine has a higher specific impulse, implying a decrease in propellant mass; however, the low-thrust aspect discourages the use of an electrical engine in the near-planet phases of a trip, since this might result in an increase in flight time. Therefore, a fundamental design problem is to find the best combination of chemical propulsion and electrical propulsion for a given mission, for example, a mission from Earth to Mars. With this in mind, this paper is the second of a series dealing with the optimization of Earth-Mars missions via the use of hybrid engines, namely the combination of high-thrust chemical engines for planetary flight and low-thrust electrical engines for interplanetary flight. We look at the deep-space interplanetary portion of the trajectory under rather idealized conditions. We study minimum time trajectories with bounded thrust direction and bounded thrust magnitude: the thrust direction α is subject to the inequality —α_(max)≤ α ≤ α _(max) while the thrust setting β is subject to the inequality 0≤ β ≤ 1. We use α _(max) as a parameter to generate a family of minimum time trajectories. Numerical results show that, as α _(max) decreases, the flight time increases, while the propellant consumption decreases. Generally speaking, the thrust profile of the optimal trajectory includes three subarcs: the first subarc is characterized by maximum thrust in conjunction with positive (upward) thrust direction; the second subarc is characterized by zero thrust (coasting flight, thrust direction irrelevant); the third subarc is characterized by maximum thrust in conjunction with negative (downward) thrust direction. Two limiting cases have a particular interest. The first limiting case occurs for α _(max) = 180° and has the following properties: (ⅰ) the time length of the coasting subarc reduces to zero and the three-subarc trajectory degenerates into a two-subarc trajectory; (ⅱ) maximum thrust is applied at all times and the thrust direction switches from positive to negative just beyond midway; (ⅲ) the minimum time trajectory for α constrained is the same as the minimum time trajectory for α unconstrained. The second limiting case occurs for α_(max) = 0° and has the following properties: (ⅰ) the thrust magnitude has a bang-zero-bang profile; (ⅱ for the powered subarcs, the thrust direction is tangent to the flight path at all times; (ⅲ) the minimum time trajectory for α constrained is the same as the minimum propellant consumption trajectory for α unconstrained.