﻿ 能量最优与燃料最优Lambert交会问题<sup>*</sup>
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Energy-optimal and fuel-optimal problems for Lambert rendezvous
XU Limin, ZHANG Tao, TAO Jiawei
Department of Automation, Tsinghua University, Beijing 100084, China
Received: 2017-11-22; Accepted: 2018-03-09; Published online: 2018-05-17 18:01
Foundation item: National Natural Science Foundation of China (61673239)
Corresponding author. ZHANG Tao, E-mail:taozhang@tsinghua.edu.cn
Abstract: The Lambert two-impulse rendezvous problem is an important problem in orbital-transfer, rendezvous and docking and other fields in space engineering. Fuel-optimal and energy-optimal Lambert rendezvous problems are a kind of Lambert optimization problem that has the typical application background and engineering requirements. In this paper, an analytical calculation method based on vector form is proposed for energy-optimal and fuel-optimal Lambert rendezvous problems, and then the analytic solution in vector form is developed for the energy-optimal and fuel-optimal Lambert rendezvous problems. The nature and characteristics of the two analytic solutions for optimization rendezvous problem are analyzed and contrasted. The simulation results prove the correctness of this method and that fuel consumption of fuel-optimal orbit is less than that of energy-optimal orbit.
Keywords: two-impulse orbital-transfer     Lambert rendezvous     energy-optimal     fuel-optimal     two-point boundary value problem     optimal planning

Lambert问题是航天工程中双脉冲轨道转移的基本问题[1-3]，在航天器交会领域有广泛的应用，近年来仍然属于热点研究范畴[4-7]

Lambert飞行时间定理指出，对同一个平方反比中心引力场的椭圆轨道转移问题，给定转移前后的空间位置P1P2，则转移时间t2-t1仅依赖于轨道半长轴aT，两点离引力中心F的矢径长度之和r1+r2以及连接两点的弦长cT。如果这3个参数aTr1+r2cT给定，则转移时间t2-t1是确定的，单圈Lambert转移情况下通常是2个解(最小能量轨道时是一个解)，与转移椭圆的形状(偏心率eT)无关[8]。基于飞行时间定理，基本Lambert问题是指给定轨道上两点的位置矢量及飞行时间，求连接两点的轨道参数。基本Lambert问题是一个典型的双脉冲变轨问题，其本质是求解微分方程两点边值问题[9]

1 能量最优与燃料最优Lambert问题的定义

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 图 1 位置和速度向量定义 Fig. 1 Definition of location and velocity vectors

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2 能量最优和燃料最优Lambert问题的计算

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2.1 ΔV2 Lambert问题的求解

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2.2 ΔV Lambert问题的求解

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2.3 速度导数

ΔV和ΔV2 Lambert问题的求解中都涉及到变轨起始点和终止点的速度w1w2相对于半通径p的一阶和二阶导数。为计算方便，令

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2.4 求解导数方程的相关结论

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3 仿真算例

 ALSET 1 1 27 559U 02 054A 08 259.526 859 48 -0.000 000 02 00 000-0 84 653-5 0 6 025 2 27 559 097.980 7 137.478 4 0 009 664 216.549 4 143.504 7 14.629 778 973 095 34 ARIANE 44L 1 28 576U 91 075N 08 351.945 684 14 0.000 001 79 00 000-0 64 019-2 0 6 927 2 28 576 006.553 4 128.062 9 6 595 687 237.361 1 042.002 9 02.835 874 63 72 170

 km 位置 rx ry rz r1 3 160.125 4 -3 850.670 7 -5 011.985 2 r2 -16 875.892 6 14 279.183 4 516.039 2

 km/s 速度 vx vy vz v1 -4.458 3.101 2 -5.191 6 v2 -1.276 5 1.799 5 3.043 9

 km/s 速度增量 Δwx Δwy Δwz Δw1 -1.361 22 0.147 84 -1.625 77 Δw2 8.786×10-6 8.841×10-6 10.017×10-6

 km/s 速度增量 Δwx Δwy Δwz Δw1 -1.300 08 0.085 91 -1.670 27 Δw2 -0.076 03 0.066 70 0.012 16

 目标函数 p/km 目标函数值/(km·s-1) ΔVtot 11 360.100 67 291.462 2.125 56 ΔVtot2 11 285.930 67 071.429 4.497 8

ΔV和ΔV2目标函数值随p变化情况如图 2所示(对数坐标)，局部细节情况见图 3。极值点和表 4~表 6计算结果一致。

 图 2 ΔVtot和ΔVtot2目标函数值随p变化情况 Fig. 2 Variation of ΔVtot and ΔVtot2 function with p
 图 3 ΔVtot和ΔVtot2目标函数值随p变化情况(极点附近) Fig. 3 Variation of ΔVtot and ΔVtot2 function with p (near peak point)
4 结论

1) 本文提出一种基于矢量形式的求解能量最优和燃料最优Lambert问题的总体框架，统一了能量最优和燃料最优Lambert问题的分析方法和相关结论，避免了大量三角函数运算和坐标变换等较繁琐的处理方式。

2) 对燃料最优Lambert问题的分析突破了相关研究中轨道共面条件的限制。本文的推导过程相比以往的研究具有更加简洁的表示形式。

3) 仿真算例展示了具体方法的求解结果，并验证了燃料最优比能量最优的最优速度增量多不超过17%的事实。

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文章信息

XU Limin, ZHANG Tao, TAO Jiawei

Energy-optimal and fuel-optimal problems for Lambert rendezvous

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(9): 1888-1893
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0731