A Multi-stage Robust Optimization Guidance Method for Endoatmospheric Powered Descent of Reusable Rockets
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摘要: 针对大气层内可回收火箭的动力下降问题, 提出一种多阶段的鲁棒优化(Robust optimization, RO)方法. 由于大气层内存在未知风场, 如何在火箭下降段考虑这种不确定性具有十分重要的意义. 首先, 建立一个关于高度的不确定风场模型, 在该风场下给出火箭动力下降的鲁棒最优控制问题. 为了求解该问题, 使用一种对不等式约束采取一阶近似并将一阶项作为安全裕量加入约束的鲁棒优化方法, 得到一个可以求解的单阶段鲁棒优化算法. 其次, 定量给出安全裕量的上界, 基于该上界提出一种多阶段鲁棒优化算法, 避免单阶段鲁棒优化算法中安全裕量可能过大导致无法求解的问题. 最后, 通过仿真对比各个算法在多个实际风场下的性能, 结果表明所提出的多阶段鲁棒优化方法同时具有较高的落点精度和对于不同风场的鲁棒性.Abstract: This paper proposes a multi-stage robust optimization (RO) method for the endoatmospheric powered descent problem of reusable rockets. Due to the unknown wind field in the atmosphere, it is of great significance to consider this uncertainty during the rocket descent phase. Firstly, a model of uncertain wind field with respect to altitude is established, and a robust optimal control problem for rocket powered descent is formulated under this wind field. To solve this problem, a tractable single-stage RO algorithm is developed by approximating the inequality constraints using a first-order expansion and incorporating the first-order term as a safety margin. Secondly, an upper bound on the safety margin is quantitatively derived. Based on this upper bound, a multi-stage RO algorithm is proposed, which avoids the infeasibility problem caused by the excessively large safety margin in the single-stage RO algorithm. Finally, simulation results are presented to compare the performance of each algorithm under various actual wind fields. The results demonstrate that the proposed multi-stage RO method achieves both high landing accuracy and robustness against different wind fields.
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图 2 $ y$轴的六个实际风场和一个标称风场
Fig. 2 Six actual wind fields and one nominal wind field on the y-axis
图 3 实际风场参数与标称风场参数的距离
Fig. 3 Distances between parameters of actual wind fields and parameters of nominal wind field
图 5 单阶段RO算法和多阶段RO算法在不同风场下的最终位置误差
Fig. 5 The terminal position errors of single-stage RO algorithm and multi-stage RO algorithm under different wind fields
图 6 单阶段RO算法和多阶段RO算法在不同风场下的最终速度误差
Fig. 6 The terminal velocity errors of single-stage RO algorithm and multi-stage RO algorithm under different wind fields
图 7 单阶段RO算法和多阶段RO算法在各个实际风场下的轨迹
Fig. 7 The trajectories of single-stage RO algorithm and multi-stage RO algorithm under each actual wind field
图 10 单阶段RO算法在不同风场下的最终位置与约束范围
Fig. 10 The terminal position and its constraint range of single-stage RO algorithm under different wind fields
图 11 单阶段RO算法在不同风场下的最终速度与约束范围
Fig. 11 The terminal velocity and its constraint range of single-stage RO algorithm under different wind fields
图 12 多阶段RO算法在不同风场下的最终位置与约束范围
Fig. 12 The terminal position and its constraint range of multi-stage RO algorithm under different wind fields
图 13 多阶段RO算法在不同风场下的最终速度与约束范围
Fig. 13 The terminal velocity and its constraint range of multi-stage RO algorithm under different wind fields
图 14 单阶段RO算法与NO算法在不同风场下的最终位置误差
Fig. 14 The terminal position errors of single-stage RO algorithm and NO algorithm under different wind fields
表 1 仿真参数值
Table 1 Simulation parameter values
参数 取值 初始状态 ${\boldsymbol{r}}_0=\left[2~968~{\rm{m}},263~{\rm{m}},4~326~{\rm{m}}\right]^{\rm{T}} $, ${\boldsymbol{v}}_0 = \left[-295~{\rm{m/s}},27~{\rm{m/s}},-296~{\rm{m/s}}\right]^{\rm{T}}$, $m_0=48~185$ kg 期望落点状态 ${\boldsymbol{r}}_f =\left[0~{\rm{m}},0~{\rm{m}},0~{\rm{m}}\right]^{\rm{T}}$, $ {\boldsymbol{v}}_f =\left[0~{\rm{m/s}},0~{\rm{m/s}},0~{\rm{m/s}}\right]^{\rm{T}}$ 火箭燃料消耗常数 $\kappa=2~975$ 火箭参考面积 $S_{{{\rm{ref}}}}=8.814\ {\rm{m}}^2 $ 空气动力学阻力系数 $C_D=4.5$ 标称风场参数 $\hat{{\boldsymbol{s}}}=\left[-0.073,4.460,5.230\right]^{\rm{T}},\ \mu=2~234,\ \sigma=1~635$, $ k=2.16\times{10}^{-3}, b=0.35$ ${\boldsymbol{S}}_\tau $中的参数 $\tau=1,\ {\boldsymbol{D}}=\text{ diag}\left\{0.83,\ 1.20,\ 2.66\right\}$ 终端约束范围 单阶段RO算法${\boldsymbol{L}}=\left[L_{rx},L_{ry},L_{rz},L_{vx},L_{vy},L_{vz}\right]=\left[7,35,7,35,35,35\right]$
多阶段RO算法${\boldsymbol{L}}=\left[4,20,4,4,4,4 \right]$多阶段RO算法的阶段数 2 
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表 2 单阶段RO算法和多阶段RO算法的运行时间
Table 2 Running time of single-stage RO algorithm and multi-stage RO algorithm
算法 运行时间(s) 单阶段RO算法 6.61 多阶段RO算法 13.14
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