文章快速检索 高级检索

Best position to apply an impulse in a planar swing-by
JIA Jianhua , LYU Jing , WANG Qi
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-06-23; Accepted: 2015-10-18; Published online: 2016-06-20 12: 00
Foundation item: National Natural Science Foundation of China (11372018)
Corresponding author. Tel.:010-82313576. E-mail:lvjing@buaa.edu.cn
Abstract: Applying an impulsive thrust during a close encounter with a celestial body can significantly improve the efficiency of the swing-by maneuver. The impulse is usually applied at the periapsis in much literature, but there is no proof that periapsis is the optimal position. In order to study the best position to apply an impulse in a planar swing-by, the powered swing-by is described by the three usual parameters of the standard swing-by maneuver plus the three parameters which specify the magnitude, the direction and the position of the impulse. A set of new analytical equations are derived, including the variation in velocity, energy and angular momentum due to the maneuver as a function of the six parameters. Using these equations, it is possible to find the best position to apply the impulse to maximize the energy change. The results show that periapsis is not the best position, as the application of impulse in the best position is 20% more efficient than the application of impulse at the periapsis in some cases.
Key words: swing-by     impulse     the best position     maneuver     energy change     periapsis

1 在任意位置施加脉冲的引力辅助

 图 1 在任意位置施加脉冲的引力辅助变轨 Fig. 1 Swing-by maneuver applying an impulse at any arbitrary position

1)直角坐标系xOyM2质心为原点；x轴为M1M2的连线；y轴为M2速度方向；V2M2的速度。

2)直角坐标系x1Oy1M2质心为原点；x1轴、y1轴分别与rPVP平行；VD1VD2分别为施加脉冲前后航天器在D点的速度。

3)直角坐标系x2Oy2M2质心为原点；x2轴沿rD方向,y2轴与x2轴垂直。

2 公式推导 2.1 计算航天器在A点的速度和矢径

 图 2 第1段轨道 Fig. 2 The first section of orbit

Step 1 计算vA的大小

 (1)

Step 2 计算vAVP的夹角δ1

 (2)

Step 3 求vA

 (3)

Step 4 计算轨道的偏心率e1、单位相对角动量h1

 (4)
 (5)

Step 5 计算x1轴与rA的夹角θA

 (6)

Step 6 求rA

 (7)

Step 7 求vArA在坐标系xOy中的坐标表示转换矩阵为

 (8)

 (9)
 (10)

Step 8 计算航天器在A点的绝对速度,记为Vin

 (11)
2.2 计算航天器在D点相对于M2的速度

 图 3 施加脉冲之后的航天器速度 Fig. 3 Velocity of spacecraft after impulse is applied

Step 1 计算rD

 (12)

Step 2 计算VrD1VD1VD1的大小和VD1VD1的夹角ξ

 (13)
 (14)
 (15)
 (16)

Step 3 计算VD2

 (17)
2.3 计算航天器在C点的速度和矢径

 图 4 第2段轨道 Fig. 4 The second section of orbit

Step 1 计算vC的大小

 (18)

Step 2 计算第2段双曲线轨道的半长轴a2

 (19)

Step 3 计算角动量h2、半正交弦p2、偏心率e2

 (20)

Step 4 计算施加脉冲之后的双曲线轨道中航天器的真近点角f0

 (21)

f0的正负选取规则：VrD2=VrD1+δVsin(η+ξ),若VrD2＞0,则航天器已经飞过近拱点,所以真近点角为正；若VrD2＜0,航天器正飞向近拱点,所以真近点角为负。

Step 5 由e2计算fLIM(第2段双曲线轨道的渐近线和近拱点方向的夹角)

 (22)

Step 6 得到vC

 (23)

Step 7 计算f1

 (24)

Step 8 得到rC

 (25)

δ2=fLIMf0,θC=f1f0,式(23)和式(25)可以表示为

 (26)
 (27)

Step 9 求vCrC在坐标系xOy中的坐标表示转换矩阵为

 (28)

 (29)
 (30)

Step 10 计算航天器在C点的绝对速度,记为Vout

 (31)
2.4 施加脉冲之后的引力辅助变轨效果

 (32)

 (33)

 (34)

3 施加脉冲的最优位置

 图 5 不同位置处施加脉冲的最优方向 Fig. 5 Optimal directions to apply an impulse in different positions
 图 6 ΔE随θ的变化 Fig. 6 Variation of ΔE with θ

 图 7 ΔE随θ的变化(δV=1.0～2.0) Fig. 7 Variation of ΔE with θ(δV=1.0-2.0)
4 结论

1) 本文通过引入一个参数表征施加脉冲的位置,将带脉冲的引力辅助变轨用6个参数来刻画,导出了变轨之后航天器的能量变化与参数的函数关系。

2) 以能量增加最大为优化目标,对施加脉冲的位置进行了优化。结果表明：施加脉冲的最优位置一般并不在近拱点,而且随其他参数的变化而变化。

3) 在最优位置施加脉冲与在近拱点施加相比,在某些情况下可高效20%以上。

 [1] PRADO A F B A. A study of swing-by trajectories in the Galilean satellites of Jupiter[J]. Journal of Physics:Conference Series,2013, 465 : 012002 . Click to display the text [2] CAI X S, LI J F, GONG S P. Solar sailing trajectory optimization with planetary gravity assist[J]. Science China-Physics Mechanics & Astronomy,2015, 58 (1) : 1 –11. Click to display the text [3] CARNELLI I, DACHWALD B, VASILE M. Evolutionary neurocontrol: A novel method for low-thrust gravity-assist trajectory optimization[J]. Journal of Guidance,Control and Dynamics,2009, 32 (2) : 615 –624. Click to display the text [4] ARMELLIN R, LAVAGNA M, ERCOLI-FINZI A. Aero-gravity assist maneuvers: Controlled dynamics modeling and optimization[J]. Celestial Mechanics and Dynamical Astronomy,2006, 95 (1) : 391 –405. Click to display the text [5] MAZZARACCHIO A. Flight-path angle guidance for aerogravity-assist maneuvers on hyperbolic trajectories[J]. Journal of Guidance,Control and Dynamics,2015, 38 (2) : 238 –248. Click to display the text [6] BROUCKE R A.The celestial mechanics of the gravity assist[C]//AIAA/AAS Astrodynamics Conference.Reston:AIAA,1988:69-78. Click to display the text [7] PRADO A F B A. Close-approach trajectories in the elliptic restricted problem[J]. Journal of Guidance,Control and Dynamics,1997, 20 (4) : 797 –802. Click to display the text [8] 乔栋, 崔平远, 崔祜涛. 基于圆型限制性三体模型的借力飞行机理研究[J]. 宇航学报,2009, 30 (1) : 82 –87. QIAO D, CUI P Y, CUI H T. Research on gravity-assist mechanism in circular restricted three-body model[J]. Journal of Astronautics,2009, 30 (1) : 82 –87. (in Chinese). Cited By in Cnki (0) | Click to display the text [9] 乔栋, 崔平远, 尚海滨. 基于椭圆型限制性三体模型的借力飞行机理研究[J]. 宇航学报,2010, 31 (1) : 36 –43. QIAO D, CUI P Y, SHANG H B. Research on gravity-assist mechanism based on three-dimension elliptic restricted three-body model[J]. Journal of Astronautics,2010, 31 (1) : 36 –43. (in Chinese). Cited By in Cnki (0) | Click to display the text [10] FELIPE G, PRADO A F B A. Classification of out-of-plane swing-by trajectories[J]. Journal of Guidance,Control and Dynamics,1999, 22 (5) : 643 –649. Click to display the text [11] 贾建华, 王琪. 三维引力辅助机理分析[J]. 北京航空航天大学学报,2012, 38 (7) : 981 –986. JIA J H, WANG Q. Research on gravity-assist mechanism based on three-dimension elliptic restricted three-body model[J]. Journal of Beijing University of Aeronautics and Astronautics,2012, 38 (7) : 981 –986. (in Chinese). Cited By in Cnki (0) | Click to display the text [12] 贾建华, 王琪. 三维引力辅助解析分析方法研究[J]. 力学学报,2013, 45 (3) : 412 –420. JIA J H, WANG Q. An analytical study of gravity assist in three dimensions[J]. Chinese Journal of Theoretical and Applied Mechanics,2013, 45 (3) : 412 –420. (in Chinese). Cited By in Cnki (0) | Click to display the text [13] GOBETZ F W. Optimal transfers between hyperbolic asymptotes[J]. AIAA Journal,1963, 1 (9) : 2034 –2041. Click to display the text [14] WALTON J M, MARCHAL C, GULP R D. Synthesis of the types of optimal transfers between hyperbolic asymptotes[J]. AIAA Journal,1975, 13 (8) : 980 –988. Click to display the text [15] PRADO A F B A. Powered swing by[J]. Journal of Guidance,Control and Dynamic,1996, 19 (5) : 1142 –1147. Click to display the text [16] CASALINO L, COLASURDO G, PASTRONE D. Simple strategy for powered swing by[J]. Journal of Guidance,Control and Dynamic,1999, 22 (1) : 156 –159. Click to display the text [17] PRADO A F B A, FELIPE G. An analytical study of the powered swing-by to perform orbital maneuvers[J]. Advances in Space Research,2007, 40 (1) : 102 –112. Click to display the text [18] ZHU K J, LI J F, BAOYI H X. Multi-swing by optimization of mission to Saturn using global optimization algorithms[J]. Acta Mechanica Sinica,2009, 25 (6) : 839 –845. Click to display the text [19] FERREIRA A F D S, PRADO A F B A, WINTER O C. A numerical study of powered swing-bys around the moon[J]. Advances in Space Research,2015, 56 (2) : 252 –272. Click to display the text [20] 侯艳伟, 岳晓奎, 张莹. 基于脉冲机动的引力辅助深空探测轨道设计[J]. 西北工业大学学报,2012, 30 (4) : 491 –496. HOU Y W, YUE X K, ZHANG Y. Design of gravity-assist trajectory based impulsive maneuver[J]. Journal of Northwestern Polytechnical University,2012, 30 (4) : 491 –496. (in Chinese). Cited By in Cnki (0) | Click to display the text

#### 文章信息

JIA Jianhua, LYU Jing, WANG Qi

Best position to apply an impulse in a planar swing-by

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(6): 1156-1161
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0409