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Description
Small spacecraft, which a quarter of a century ago were only taking their first steps in the low-Earth orbit, today are becoming a trend in deep space exploration. Two prospective, albeit challenging, destinations for small spacecraft missions are Mars and Venus. In this light, a group of Russian scientific and industrial enterprises recently proposed a universal small spacecraft platform that can be employed in missions to both these and other destinations. The enabling technology of the proposed platform, as well of many other small spacecraft missions, is an electric propulsion system (EPS), because of its high specific impulse and fuel efficiency. The use of EPS significantly enhances trajectory design flexibility and extends possible launch and arrival dates. However, designing a trajectory of a spacecraft with an EPS poses a rather complex mathematical task, especially if technological peculiarities are taken into account.
This work addresses the problem of designing optimal low-thrust interplanetary trajectories with consideration of real-world features of electric propulsion and energy systems. The focus is on missions to Mars and Venus. The considered features are:
- Extra 30% fuel consumption to compensate for the parasitic torque that appears due to the thrust vector misalignment
- Nonlinear dependence of thrust magnitude on the available electrical power
- Electrical power reservation for other spacecraft systems
- Electrical power dependence on the distance to the Sun
- Solar panels’ degradation
The mission is assumed to consist of three phases. The first phase is the departure from Earth parking orbit. It is assumed to be done by a high-thrust boost stage that provides necessary value and direction of the hyperbolic excess velocity. The second phase is the interplanetary cruise, during which the trajectory is controlled by the EPS. The final phase is a ballistic capture at the target planet, that requires the spacecraft to achieve a negative Keplerian energy relative to the planet by the means of the EPS. The purpose of optimization is to determine the optimal control law for the EPS during the cruise and capture phases.
Both phases are modeled within the framework of two-body problem. The trajectory optimization is done via the Pontryagin maximum principle, with the EPS working mass expenditure being optimized. All the above-mentioned features are incorporated into the optimization procedure and noticeably complicate it. To alleviate the optimization, a modification of the extended system of equations of motion is proposed, which allows the optimization problem to be successfully solved.
As a result of this study, optimal low-thrust trajectories to Mars and Venus were obtained and propellant requirements across 2028-2031 launch dates were evaluated. Dependencies between launch date and (a) time of flight, (b) departure v∞ were established. The impact of planetary arrival conditions on fuel costs were characterized.