基于A*算法的摄动Lambert同伦迭代方法

doi:10.7527/S1000-6893.2024.30780

Abstract

Abstract:

Perturbed Lambert Problem forms the fundamental basis for tasks such as spacecraft rendezvous and on-orbit servicing. Due to the lack of an analytical solution for the perturbed Lambert problem， numerical solutions can only be obtained through iterative computations. Consequently， existing methods primarily focus on improving iteration convergence and computational efficiency. Building upon current homotopic iteration methods， this paper introduces the general concept of the A* algorithm for graph search and proposes a pruned homotopic iteration approach. Firstly， by combining the gradient direction of the state transition matrix from the terminal position error to the initial velocity increment with the target direction， an iteration direction is derived， and a homotopic mapping based on the A* algorithm is designed. Secondly， utilizing Taylor expansion， a linear perturbation matrix is devised， enabling pruning of divergent end-position paths. This addresses the issue of ineffective iterations due to initial velocity divergence that conventional methods such as Newton’s shooting method and quasi-linearization fail to exclude. Simulation results demonstrate that while ensuring the attainment of optimal solutions， our proposed method improves computational efficiency by over 25%， compared to existing homotopic methods， featuring a broader convergence range. Moreover， the method exhibits favorable characteristics regarding initial value convergence. In the case of orbit transfer in the Earth-Moon three-body system， it achieves more than a 30% increase in computational efficiency compared to quasi-linearization techniques.

Key words: perturbation lambert problem, homotopic iteration method, orbital transfer, A* algorithm, three-body problem

CLC Number:

V412.4

Cong XIE, Zhen YANG, Yangang LIANG. Homotopic perturbed lambert algorithm based on A* algorithm[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(4): 330780.

Figures/Tables 12

Fig.1

Fig.2

Fig.3

Table 1

Initial orbit element of three cases

轨道根数	低轨算例1		中轨算例2		高轨算例3
轨道根数	目标器	追踪器	目标器	追踪器	目标器	追踪器
$a$ /m	$6 716 300$	$6 636 004$	$7 636 004$	$8 591 632$	$42 232 144$	$42 232 144$
$e$ / $10 - 3$	$0.6$	$9.8$	$359.3$	$0.6$	$70.0$	$0.4$
$i$ /（ $°$ ）	$42.854 5$	$42.837 6$	$32.837 6$	$42.921 7$	$28.000 0$	$28.100 2$
$Ω$ /（ $°$ ）	$55.751 7$	$55.916 5$	$110.726$	$55.751 7$	$0$	$0.000 2$
$ω$ /（ $°$ ）	$185.488$	$125.488$	$205.488$	$185.488$	$0$	$14.764 6$
$f$ /（ $°$ ）	$0$	$0$	$164.878$	$0$	$0$	$137.832 0$

Table 1

Table 2

ATK-based simulation verification of the presented method

算例	转移时间/d	本方法解 $Δ v 1$ /（m·s^-1）	本方法解 $Δ v 2$ /（m·s^-1）	ATK仿真末位置误差/m
1	1	$[- 1.567, - 118.057, - 549.499]$	$[115.036,52.842,936.874]$	$39.148 0$
2	7	$[- 189.175, - 708.278,843.818 9]$	$[- 4 340.547, - 1 852.738,5 153.136]$	$93.264 3$
3	100	$[- 136.257,1 444.317 0, - 1 343.967]$	$[- 10.897,1 486.507, - 1 365.847]$	$34.632 5$

Table 2

Table 3

Comparisons of three methods

对比参数	低轨算例1			中轨算例2			高轨算例3
对比参数	牛顿法	同伦法	本方法	牛顿法	同伦法	本方法	牛顿法	同伦法	本方法
同伦迭代次数	$1$	$5$	$5$	$1$	$10$	$10$	$1$	$15$	$15$
最优路径迭代次数	$8$	$30$	$30$	发散	发散	$56$	发散	$88$	$85$
计算时间/s	$0.200$	$0.303$	$0.195$	发散	发散	$5.801$	发散	$4.556$	$3.279$
增量和 $Δ v 1 + Δ v 2$ /（m·s^-1）	$2 598.094$	$1 507.428$	$1 507.427$	发散	发散	$4 400.198$	发散	$3 996.343$	$3 996.343$

Table 3

Fig.4

Fig.5

Table 4

General transfer orbit parameters

初位置 $r 1$ /［L］	末位置 $r 2$ /［L］	转移时间/［T］	末位置误差限/［L］
［ $1.185 0$ ， $0$ ， $0.01$ ］	［ $1.477 6$ ， $- 0.754 5$ ， $- 0.016 5$ ］	$5.5$	$1 × 10 - 6$

Table 4

Fig.6

Table 5

Comparison results between the method in Ref. ［13］ and presented in this paper

方法	计算时间/s	末位置误差/（ $10 - 7$ ［L］）
文献［13］方法	$4.21$	$1.931$
本方法	$2.89$	$1.970$

Table 5

Fig.7

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Homotopic perturbed lambert algorithm based on A* algorithm

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