火箭垂直回收多阶段最优轨迹规划方法

针对火箭高空再入定点回收，基于凸优化方法提出一种考虑气动力和推力控制的多阶段轨迹优化方法。在气动减速段，通过控制总攻角，实现气动升力和阻力的调制。由于气动力连续变化，使用Legendre-Gauss-Radau伪谱离散方法进行离散化，利用较少的离散点实现较高的数值精度。在动力减速段，推力矢量为控制变量。由于推力调节可能出现不连续，采用等距离散方法进行离散。在此基础上，将发动机开、关机时间也作为优化变量，并考虑各种约束，构建了多阶段离散最优控制模型。使用无损凸化方法对升力约束和推力约束进行松弛，并通过逐次凸化消除由气动力、自由时间变量以及质量引入的非凸约束，最终将问题描述为序列迭代求解的二阶锥规划问题(SOCP)。通过仿真校验，经过少量的逐次凸化迭代，可快速收敛到最优解，且落点调节范围更大，燃料更省。

Multi Stage Trajectory Optimization for Vertical Pin Point Landing of a Reusable Launch Vehicle

 A multi-stage trajectory optimization algorithm is proposed by modulating the aerodynamic force and thrust via the convex optimization. In the aero-braking phase, the trim angle is taken as the control variable, and the Legendre-Gauss-Radau pseudospectral method is adopted to discretize the states and control due to its high numerical precision. During the powered descending phase, the thrust vector is taken as the control variable, and the uniform discretization is applied since the thrust may alter discontinuously. Then, taking the constraints and the start and cut-off time of the engine into account, a multi-stage optimization problem is built up. The lossless convexification is used to relax the lift and thrust constraints, and the successive convexification is used to eliminate the non-convexities, introduced by the aerodynamic, free-time-variables and mass. The problem is finally described as a second-order cone programming problem (SOCP), which can be solved by the sequence iteration. The numerical simulations show that the convergence is obtained after a few successive iterations. As well, the larger landing range and less fuel consumption can be achieved.