求解中途飞越燃料最优转移轨道的同伦方法

Homotopy Method for Fuel-Optimal Trajectory Design in Flyby Mission

提出一种新的同伦方法,用于求解深空探测中对其他天体进行中途飞越的小推力燃料最优转移轨道,克服由于其存在内点约束及不连续Bang-Bang控制所导致的数值优化方法的求解困难。该同伦方法将同伦参数同时嵌入到性能指标和内点约束方程中,将容易求解的无内点约束且控制量连续变化的最优控制问题作为初始问题,求解一系列同伦参数递增所对应的同伦迭代子问题,直到得到原问题的解。该方法能够有效地解决中途飞越所导致的优化变量增加、求解难度增大等难题,能够快速、稳定地求解考虑中途飞越的小推力燃料最优转移轨道。最后,以地球到火星交会并中途飞越小行星和地球到木星交会并中途飞越火星两个任务为例进行数值仿真验证该同伦方法在求解中途飞越的燃料最优问题中的有效性和优越性。

A new homotopy method is proposed to solve the fuel-optimal transfer trajectory with flyby of other celestial bodies in the deep space exploration,overcoming the difficulty due to the interior point constraints and discontinuous structure of the Bang-Bang control.The homotopic parameter is embedded in both the performance index and interior point constraint equations.By starting from a related and easier energy-optimal problem without inner constraints,and solving a series of iterative subproblems with homotopic parameter increased,the solution of the original problem is obtained when the parameter reaches one.With the approach,the difficulty of the optimization variables increased caused by the flyby inner constraints can be resolved,and the low-thrust fuel-optimal transfers can be completed effectively.At last,two examples about the Jupiter rendezvous with the Mars flyby and the Mars rendezvous with an asteroid flyby are given to substantiate the effectiveness and optimality of the homotopy method.