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系統識別號 U0002-2307200721060200
中文論文名稱 克普勒型軌道之宇航器編隊飛行探討
英文論文名稱 Spacecraft Formation about a Keplerian Orbit
校院名稱 淡江大學
系所名稱(中) 航空太空工程學系碩士班
系所名稱(英) Department of Aerospace Engineering
學年度 95
學期 2
出版年 96
研究生中文姓名 李欣儒
研究生英文姓名 Shin-Ju Li
學號 694370791
學位類別 碩士
語文別 英文
口試日期 2007-07-17
論文頁數 61頁
口試委員 指導教授-蕭富元
委員-馬德明
委員-曾坤樟
中文關鍵字 編隊飛行  軌道控制  局部時間逼近法  克普勒型軌道 
英文關鍵字 formation flight  orbit control  local time approximation  Keplerian orbits 
學科別分類 學科別應用科學航空太空
中文摘要 本論文探討了克普勒型軌道附近相對運動, 並將之應用於宇航器的編隊飛行上。我們首先討論在克普勒型軌道附近的相對運動的力學模型, 一般在討論此等問題時, 通常是採用Tschauner-Hempel Equation (T-H Equation) 來描述。有了飛行器的運動方程式之後, 我們接著探討了“局部時間逼近法(local timeapproximation)” 在這個問題的可應用性。本文證明此控制法則在近圓軌道上亦能對軌道做有效的控制。然後在這個概念下, 本文提出了兩種可能的控制器設計方法-包括在時域進行設計以及利用T-H Equation進行控制器設計; 我們不僅能穩定原本不穩定的相對運動軌道, 並且能夠重現“縮小版”的主軌道。

由於對軌道施加控制必需額外花費能量, 因此我們也探討了利用自然的週期性相對軌道, 去維持編隊的可能性。我們運用已知的二體問題完整解, 去求出兩條同週期的不同軌道之間的相對位置方程式, 再加以簡化。這個方法可以避免去解線性化之後的常微分方程, 並且得到了一個比直接去積分T-H Equation 更精確的解。我們可以把這個結果應用到宇航器的編隊, 並能降低維持編隊所需要的油耗。最後, 我們也利用數值模擬的方法來驗證我們的結果的正確性。
英文摘要 This thesis investigates relative trajectories about a Keplerian orbit with potential applications to the formation flight of spacecraft. We first consider a spacecraft formation about a nominal Keplerian orbit, whose dynamics is usually described by the Tschauner-Hempel Equation (T-H Equation). Briefly reviewing the results from the T-H Equation, we analytically prove the applicability of the “local time approximation” to the T-H Equation. With the guidance of local time approximation, we propose potential design methods of control law both in the time domain and in the true-anomaly domain. By designing the control law in the true-anomaly domain, we not only stabilize the unstable relative trajectory, but also“re-construct” the “scaled” nominal orbit for our formation of spacecraft.

We also present another methodology for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian orbit. In this method the spacecraft relative dynamics is derived by subtracting two neighborhood orbits, and simplifying the results. Our work avoids solving the linearized differential equations, and provides a more precise approximation to the relative motion than the T-H Equation did. This result can be used for lowering down the fuel usage for relative orbit maintenance. Numerical simulations are also presented to verify our results.
論文目次 Contents


List of Figures iii

Nomenclature vi

1 Introduction 1
1.1 Motivation 1
1.2 Literature Review 2
1.3 Research Methodology 3

2 Dynamics Model 5
2.1 Relative Dynamics 5
2.1.1 General Equations of Motion 5
2.1.2 The Tschauner-Hempel Equation 8
2.1.3 Clohessy-Wiltshire Equation 10
2.2 Local Time Approximation 12
2.2.1 Local Time Approximation 12
2.2.2 Applicability to the Keplerian Formation 14
2.3 System Stability 18

3 Controlled Relative Motion 20
3.1 Control law Design in Time Domain 20
3.2 Position and Velocity Feedback 22
3.3 Numerical Simulation 24

4 Relative Motion Using Natural Periodic Trajectories 28
4.1 General Natural Periodic Solutions 28
4.2 Formation about Elliptic Orbits with Specific Initial Conditions 34
4.3 Formation about Circular Orbits with Specific Initial Conditions 37
4.4 Numerical Simulations for Earth Satellites 38
4.5 Numerical Simulations for Martian Missions 45

5 Conclsuions and Contributions 49
5.1 Conclsuions 49
5.2 Contributions 50

Appendix I 51
Appendix II 54
Appendix III 55

References 59





List of Figures


2.1 A cartoon showing how the coordinates are defined, and the relative positions between the chief and deputy spacecraft. 6

2.2 The relative motion governed by the C-W Equation. The initial conditions are selected to generate a relative circular motion in the y ¡ z plane. In this simulation, the initial excursion is selected as x0 = 1, z0 = 2x0, ˙ y0 =¡2nx0, ˙ z0 = 0 and n = √µe/70003. The simulation is run for one period. 11

3.1 A nominal orbit with semi-major axis of 7000 km, and eccentricity of 0.001 is used to simulate the formation flight. 24

3.2 A spacecraft formation with initial excursion of 50 m from the nominal is simulated. The nominal orbit is shown in Fig. 3.1, and the formation is simulated for one orbit period. 25

3.3 The cost of formation with initial excursion of 50 m from the nominal is simulated. The nominal orbit is shown in Fig. 3.1, and the formation is simulated for one orbit period. 25

3.4 A nominal orbit with semi-major axis of 7000 km, and eccentricity of 0.1 is used to simulate the formation flight. 26

3.5 A spacecraft formation with initial excursion of 50 m from the nominal is simulated. The nominal orbit is shown in Fig. 3.4, and the formation is simulated for one orbit period. 26

3.6 The cost of formation with initial excursion of 50 m from the nominal is simulated. The nominal orbit is shown in Fig. 3.4, and the formation is simulated for one orbit period. 27

4.1 The real trajectories, integrated with original nonlinear model, are shown to compare the validity of the T-H Equation with that of the full approach and Sengupta’s solution. The simulation is run for ten periods. 39

4.2 The drift between exact periodic solution by numerical method and analytic solution in x-direction. 40

4.3 The drift between exact periodic solution by numerical method and analytic solution in y-direction. 40

4.4 The drift between exact periodic solution by numerical method and analytic solution in z-direction. 41

4.5 The real trajectories, integrated with original nonlinear model, are shown to compare the validity of the T-H Equation with that of the full approach and Sengupta’s solution. The simulation is run for ten periods. 41

4.6 The drift between exact periodic solution by numerical method and analytic solution in x-direction. 42

4.7 The drift between exact periodic solution by numerical method and analytic solution in y-direction. 42

4.8 The drift between exact periodic solution by numerical method and analytic solution in y-direction. 43

4.9 The error between predicted trajectories and the real trajectories, integrated with original nonlinear model, are shown to compare the validity of the CW Equation with that of the full approach. The simulation is run for one period. 44

4.10 Relative motion for formation flight. The simulation is run for one period. 44

4.11 Relative motion for formation flight near periapsis. 45

4.12 Spacecraft fleet to Mars via Hoffmann Transfer. The four serving ships moves around the flagship (in the center) in the x¡y plane during the trip. This simulation is run for a full journey from Earth to Mars. The initial separation of spacecraft is about 1 km . 46

4.13 This figure shows the complete trajectory in one full period. All the parameters are the same as the previous figure. 47

4.14 Spacecraft fleet to Mars via Hoffmann Transfer. The two serving ships moves around the flagship (in the center) in the y ¡z plane during the trip. This simulation is run for a full journey from Earth to Mars. The initial separation of spacecraft is about 1 km. 47

4.15 This figure shows the the complete trajectory in one full period. All the parameters are the same as the previous figure. 48
參考文獻 References
[1] Born, M., and Wolf, E., “Principles of optics,” New York: Pergamon Press, 1964

[2] Dyson, J., “Interferometry as a measuring tool,” Brighton: Machinery Publishing, 1970

[3] Hariharan, P., “Basics of interferometry,” Boston: Academic Press, 1992

[4] Hussein, I., Scheeres, D. J., and Hyland D. C.,“Interferometric Observatories in Earth Orbit ,”Journal of Guidance, Control, and Dynamics , Vol. 27, NO. 2, 2004, pp 297-301

[5] Clohessy, W. H., “Terminal Guidance System for Satellite Rendezvous,” Journal of Aerospace Sciences, Vol. 27, No. 9, September 1960, pp. 653-658,674

[6] Tschauner, J. and Hempel, P., “Rendezvous with a Target in Elliptic Orbit,” Astronautica Acta, Vol. 11, No. 2, 1965, pp. 104-109

[7] Vadali, S. R., “An Analytical Solution for Relative Motion of Satellites,” Proceedings of the 5th International Conference on Dynamics and Control of Structures and Systems in Space, Cranfield University Press, 2002, pp.309-316

[8] Melton, R. G., “Time-Explicit Representation of Relative Motion Between Elliptical Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 23, No.4, 2000, pp. 604-610

[9] Vadali, S. R. and Alfriend, K. T., “Modeling and Control of Satellite Formations in High Eccentricity Orbits,” The Journal of the Astronautical Science, Vol. 52, No. 1-2, 2004, pp. 149-168

[10] PRASENJIT SENGUPTA AND VADALI, S. R., “Periodic Relative Motion Near A Keplerian Elliptic Orbit with Nonlinear differential Gravity,” 16th AAS/AIAA Space Flight Mechanics Conference

[11] Hsiao, F. Y., Scheeres, D. J.,“Design of Spacecraft Formation Orbit Relative to a Stabilized Trajectory ,”Journal of Guidance, Control, and Dynamics , Vol. 28, NO. 4, 2005, pp 782-794

[12] Hsiao, F. Y., and Scheeres, D. J., “The Dynamics of Formation Flight About a Stable Trajectory ,”Journal of The Astronautical Sciences,Vol. 50, NO. 3, 2002, pp. 269-287

[13] Scheeres, D. J., Hsiao, F. Y., and Vinh, N. X.,“Stabilizing Motion Relative to an Unstable Orbit: Applications to Spacecraft Formation Flight ,”Journal of Guidance, Control ans Dynamics, Vol. 26, NO. 1, 2003, pp 62-73

[14] Bay, J. S., Fundamentals of Linear State Space Systems, 1999

[15] Nemytskii, V. V, Stepanov, V. V., Qualitative Theory of Differential Equation ,Dover, New York, 1989, pp. 152

[16] Barden, B. T., and Howell, K. C.,“Fundamental Motions Near Collinear Libration Points and Their Transitions ,”Journal of The Astronautical Sciences, Vol. 46, 1998, pp 361-378

[17] Sabol, C. Burns, R., and Mclaughlin, C. A.,“Satellite Formation Flying Design and Evolution ,”Journal of Spacecraft and Rockets, Vol. 38, No. 2, 2001, pp 270-278

[18] Schaub, H., and Alfriend, K. T.,“Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formation ,”Journal of Guidance, Control and Dynamic, Vol. 24, No. 4, 2001, pp 739-745

[19] Battin, R. H., “An introduction to the mathematics and methods of astrodynamics,” (AIAA education series, AIAA, Reston, 1999)
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