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。

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.

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

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