Low-energy transfer from Earth into DRO with hybrid gravity assist and numerical continuation
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摘要:
远距离逆行轨道(DRO)是地月空间中的一类周期轨道，这类轨道具有长期稳定、入轨能量低的特点，可作为未来载人月球和载人火星任务的中转站。对于地球至DRO的两脉冲入轨任务，采用日月综合借力(即同时使用弱稳定边界(WSB)和月球借力(LGA))可以最大化入轨质量，但是这类轨道对初值非常敏感。使用日月综合借力拓展DRO入轨脉冲包络，改进构造方法和提供解析梯度大幅提高多步打靶收敛率，提出2层伪弧长延拓方法进一步降低任务总脉冲。数值仿真采用共振比为2∶1的DRO，脉冲最低解采用“LGA+WSB+2LGA”的飞行模式，飞行时间为123天，近地轨道发射脉冲为3.125 km/s，DRO入轨脉冲仅为19.7 m/s。
Abstract:Distant retrograde orbits (DRO) are well-known trajectory types in cislunar space, such orbits have long-term stability and low insertion cost. In cislunar space, DRO are well-known trajectory types with minimal insertion costs and long-term stability. A cislunar station deployed on DRO might be expected to deliver a crew to the moon or Mars for exploration missions in the future. For low-energy transfer from Earth into DRO, the maximum delivery mass can be achieved by utilizing a weak stability boundary (WSB) and multiple lunar gravity assist (LGA) simultaneously, but this kind of transfer is very sensitive to initial values. A novel two-level pseudo-arc continuation method was proposed to explore local solution space, and this paper aims to improve both computational and transfer efficiency when leveraging hybrid gravity assist in cislunar space. Additionally, a modified problem description with an analytical gradient is used to improve multiple shooting efficiency under a bicircular restricted four-body problem. In the numerical simulation, the minimum cost solution shows “LGA+WSB+2LGA”, where the time of flight is 123 days, the LEO launching cost is 3.125 km/s and the 2:1 DRO insertion maneuver only needs 19.7 m/s.
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图 3 WSB转移示意图（日地旋转系）
Figure 3. Schematic diagram of WSB transfer(Sun-Earth rotation system)
图 4 “LGA+WSB+NLGA”的DRO入轨示意图
Figure 4. Schematic diagram of “LGA+WSB+NLGA” transfer from Earth into DRO
图 5 月球轨道站支持的载人月球和火星任务示意图
Figure 5. Schematic diagram of cislunar orbital station supporting manned missions to the Moon and Mars
图 10 数值延拓后各区间轨道数的变化
Figure 10. Changes in number of orbits after numerical continuation
表 1 有限差分和解析梯度的对比
Table 1. Comparison between finite difference and analytical gradient
名称 收敛数 收敛率/% 计算时间/s 有限差分 309 8.44 18.01 解析梯度 2361 64.54 20.24 
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表 2 数值延拓前后轨道对比
Table 2. Comparison of orbits before and after numerical continuation
任务名称 延拓前 延拓后 LEO发射时间/TU −31.944 945 682 3 −32.992 737 946 4 DRO入轨时间/TU −4.422 676 457 9 −4.699 448 716 6 LEO发射位置/LU [−0.025 926 418 9,
−0.010 152 072 7][−0.020 484 378 0,
−0.014 946 105 5]LEO发射速度/VU [6.340 104 171 1,
−8.603 190 754 6][9.288 605 287 2,
−5.179 229 375 6]DRO相位因子$\sigma $ 0.071 769 660 2 0.976 618 137 7 LEO发射脉冲/（km·s−1） 3.178 4 3.125 2 DRO入轨脉冲/(km·s−1) 0.040 9 0.019 7 任务总脉冲/(km·s−1) 3.219 4 3.144 9 任务总时间/d 119.669 9 123.022 4
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表 3 3种任务场景对比
Table 3. Comparison between 3 mission scenarios
任务名称 发射脉冲/(km·s−1) 入轨脉冲/(km·s−1) 入轨质量/103 kg LEO至GEO 2.454 1.477 6.571 LEO至LLO
（综合借力）3.125 0.650 6.929 LEO至DRO
（综合借力）3.125 0.019 8.584
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C1 数值仿真参数
C1. Numerical simulation parameters
名称 数值 太阳质量常数 μs/(km3·s−2) 132712440017.9870 地球质量常数 μe/(km3·s−2) 398600.4328969392 月球质量常数 μm/(km3·s−2) 4902.800582147764 重力加速度 g0/(m·s−2) 9.80665 地球平均半径 Re/km 6378.137 月球平均半径 Rm/km 1738 LEO地心距约束 r*leo/km 6578.137 月球影响球半径 rsoi,m/km 66100 归一化距离/LU 1 归一化时间/TU 1 归一化速度/VU 1 归一化地月质量参数 μ 0.012151 归一化太阳质量 ms 3.28901e5 归一化太阳距离 ρs 388.811 归一化太阳角速度 ωs −0.92519 多步打靶离散点 n 20 伪弧长延拓步长 Δs 0.1 DRO入轨太阳相位角 θs(τN) [0,2π) DRO入轨相位因子 σ [0,1) DRO入轨脉冲模 ΔV/(m·s−1) [10,100] DRO入轨脉冲方向角 φ/rad [0,2π) 2∶1共振DRO位置初值$r_{\mathrm{dro}}^0 $/LU [0.808936204186,0,] 2∶1共振DRO速度初值$v_{\mathrm{dro}}^0 $/VU [0,0.515632164331] 注：1 LU=384 400 km, 1 TU=4.348 113 05 d, 1 VU=1.023 232 81 km/s。
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