臺灣博碩士論文加值系統

　　本論文之目的在於研究以連續低推力推進之最省燃料軌道轉換，推力之大小設為固定，其方向則可控制。研究中，係以數學嚴謹之非線性最佳控制理論導出最佳化之必要條件，所形成之最佳控制問題則先以二階坡度演算法並配合向後掃掠矩陣求解，得到最佳控制解之特性後，再設計簡化的控制模式，此簡化之控制模式包含一組未定參數，這些未定參數必須使得軌道轉換所消耗之燃料為最省。設計合適之控制之模式，導出最佳化之必要條件，並分析得其參數，其程序稱為參數最佳化法，此為本論文所提出之主要理論。

　　為了說明此法，本研究分析數個數值例題，包含同平面及非同平面軌道轉換。由二階坡度演算法之分析結果發現，同平面軌道轉換之推力控制角可以用兩個線性時間函數近似之，推力控制角在初始為其中之一線性時間函數，在某一時間點則迅速切換到另一線性時間函數。因此，在分析中，參數最佳化法所選之參數可以為兩線性函數之係數、切換時間及終端時間。在非同平面軌道轉換問題上，推力方向則須要用兩個控制角表示之，其一在原軌道平面內，另一則在與原軌道面垂直之平面上。由二階坡度演算法之分析結果發現，當目標軌道平面與圓軌道間之夾角不大時，推力之兩個控制角亦可以分別用兩個線性時間函數近似之，值得注意的是，兩控制角度分別由其一線性時間函數快速切換到其另一線性時間函數之時間點非常接近。
因此，在參數最佳化法分析中可假設兩控制角度在相同的時間點作切換，以簡化問題。在比較二階坡度法與參數最佳化法對於數個問題的分析後發現，後者之分析結果已可足夠接近前者之分析結果。

　　在廣泛地研究軌道轉換後發現，由於二階坡度演算法相當嚴謹，反復計算收斂的成功率相當低，要成功使用此法必須限制推力大於0.05g，否則將因過長的積分時間而使計算發散。在本論文中所發展之參數最佳化法，則已成功地應用在如0.01g之低推力軌道轉換分析。雖然在本文中並未分析當推力低於0.01g時之軌道轉換，但此法應用之成功，應是可以預期的。

　The objective of this thesis is to investigate the minimum-fuel orbit transfer with a continuous constant low thrust force of which the direction is controllable. In this study, the athematically strict nonlinear optimal control theory is used to formulate the problem. It is first solved by using the second-order gradient method associated with the backward sweep matrix. After obtaining the characteristic of the optimal control, a simplified control pattern is designed with some parameters left to be determined. These arameters are so chosen that the consumed fuel is minimized when the target orbit is reached. The process of designing a proper model of control function, deriving the necessary conditions for parametric optimization, and determining the corresponding optimal parameters is known as the parametric optimization method, which is the principal theory proposed in this thesis.

　To illustrate the methodology, several numerical examples including coplanar and non-coplanar transfers are given. By using the second-order gradient analysis, it is found that, for coplanar transfer, the control angle of the thrust can be approximately represented with two linear time functions. The control angle quickly switches from one linear time function to another at some time during the transfer. ccordingly, in the optimal parametric control analysis, the parameters to be chosen to minimize the consumed fuel are the coefficients of the two linear time functions, the time for the control to switch from one function to another, and the final time. For non-coplanar transfer, the thrust direction must be represented by two control angles, one being in the original orbital plane and the other out of the original orbital plane. By using the second-order gradient analysis, it is found that both control angles can also be approximately represented with two linear functions, respectively. One interesting point is that the times for both control angles to switch respectively from one linear function to another are very close. Therefore, in the parametric optimization analysis, the switching times are assumed to be the same for both angles. Investigation results show that the parametric optimization analysis developed in this thesis is accurate enough as compared with the second-order gradient analysis.

　After an extensive study of orbit transfers, it is found that the second-order gradient analysis is so restrictive that it can be successful only for the thrust being greater than 0.05g, otherwise, the computation will blow up due to a long integration time. With the parametric optimization analysis developed in this thesis, the successful analyses include the cases in which the thrust is as low as 0.01g. The cases with even lower thrust are not analyzed in this thesis but can also be expected to be successful by using the parametric optimization analysis.

摘要　　　　　　　　　　　　　　　　　　　　 i
Abstract　　　　　　　　　　　　　　　　　　ii
Extended Chinese Abstract　　　　　　　　　　iv
第一章之中文摘要　　　　　　　　　　　　　　 v
第二章之中文摘要　　　　　　　　　　　　　　 vii
第三章之中文摘要　　　　　　　　　　　　　　 viii
第四章之中文摘要　　　　　　　　　　　　　　 ix
第五章之中文摘要　　　　　　　　　　　　　　 x
第六章之中文摘要　　　　　　　　　　　　　　 xii
第七章之中文摘要　　　　　　　　　　　　　　 xiii
Contents xiv
List of Tables xvii
List of Figures xix
List of Symbols xxvii
1 Introduction 1
1.1 Impulsive Types of Transfer with High Thrust　　　　　 1
1.2 Continuous Types of Transfer with Low Thrust　　　　　 2
1.3 Review for Methods of Analysis　　　　　 3
1.4 The Method of Analysis Developed in This Thesis　　　　　 11
2 Equations of Motion and Terminal Conditions 　　　　　14
2.1 Coplanar Orbit Transfer　　　　　 14
2.1.1 Equations of Motion in Component Form　　　　　 15
2.1.2 Terminal Conditions for a Circular Target Orbit　　　　　 17
2.1.3 Terminal Conditions for an Elliptic Target Orbit　　　　　 18
2.2 Non-coplanar Orbit Transfer　　　　　 19
2.2.1 Equations of Motion in Component Form　　　　　 19
2.2.2 Terminal Conditions for a Circular Target Orbit　　　　　 22
3 The Necessary Conditions for Optimality　　　　　 25
3.1 Coplanar Orbit Transfer　　　　　 26
3.1.1 Terminal Conditions for a Circular Target Orbit　　　　　 27
3.1.2 Terminal Conditions for an Elliptic Target Orbit　　　　　 28
3.2 Non-coplanar Orbit Transfer　　　　　 29
4 Solutions by Using the Second-Order Gradient Method　　　　　 34
4.1 Initial Guess for the Second-Order Gradient Method　　　　　 34
4.1.1 Release of Some Constraints　　　　　 36
4.1.2 Reenforcement of Previously Released Constraints　　　　　 38
4.1.3 Transfer between Two Very Close Coplanar Circular Orbits　　　　　 42
4.2 Coplanar Transfer from Circular Orbit to Circular Orbit　　　　　 42
4.2.1 Transfer to More Distant Coplanar Circular Target Orbits　　　　　 43
4.2.2 Comparison with the Hohmann Transfer　　　　　 48
4.3 Coplanar Transfer from Circular Orbit to Elliptic Orbit　　　　　 50
4.3.1 Transfer to Elliptic Target Orbits with the Same Orientation Angle but Different Eccentricities 51
4.3.2 Transfer to Elliptic Target Orbits with the Same Eccentricity but Different Orientation Angles　　　　　 55
4.4 Non-coplanar Transfer from Circular Orbit to Circular Orbit　　　　　 60
4.4.1 Transfer To Target Orbits with the Same Longitude but Different Inclination Angles　　　　　 61
4.4.2 Transfer To Target Orbits with the Same Inclination Angle but Different Longitudes　　　　　 68
5 The Parametric Optimization with Formats of Piecewise Continuous Controls　　　　　 78
5.1 Definitions of the Format for Parametric Controls　　　　　 78
5.2 Optimal Parametric Control Methods　　　　　 80
5.3 Coplanar Orbit Transfer to Circular and Elliptic Target Orbits　　　　　 87
5.3.1 Discrete Constant Control of Thrust Direction　　　　　 88
5.3.2 Discrete Linear-Time-Function Control of Thrust Direction　　　　　 89
5.4 Coplanar Orbit Transfer to Circular and Elliptic Target Orbits　　　　　 89
5.4.1 Discrete Constant Control of Thrust Direction　　　　　 90
5.4.2 Discrete Linear-Time-Function Control of Thrust Direction　　　　　 91
6 Solutions by Using the Optimal Parametric Method　　　　　 93
6.1 Coplanar Transfer from a Circular Orbit to Circular Orbits　　　　　 93
6.1.1 Comparison with the Results Obtained by Using the Gradient Method　　 94
6.1.2 A Further Study of Low Thrust Orbit Transfer　　　　　 99
6.2 Coplanar Transfer from a Circular Orbit to Elliptic Orbits　　　　　 103
6.2.1 Transfer to Elliptic Target Orbits with the Same Orientation Angle but Different Eccentricities　　　　　 104
6.2.2 Transfer to Elliptic Target Orbits with the Same Eccentricity but Different Orientation Angles　　　　　 109
6.2.3 A Further Study of Low Thrust Orbit Transfer　　　　　 115
6.3 Non-coplanar Transfer from Circular Orbit to Circular Orbits 121
6.3.1 Transfer To Target Orbits with the Same Longitude but Different Inclination Angles　　　　　 121
6.3.2 Transfer To Target Orbits with the Same Inclination Angle but Different Longitudes　　　　　 129
6.3.3 A Further Study of Low Thrust Orbit Transfer　　　　　 137
7 Conclusions　　　　　 145
Bibliography　　　　　 148
Appendices　　　　　 153
Appendix A Derivatives Used in the Second-Order Gradient Method　　　　　 154
A.1 Coplanar Orbit Transfer　　　　　 154
A.1.1 The First-Order Derivatives　　　　　 154
A.1.2 The Second-Order Derivatives　　　　　 155
A.2 Non-coplanar Orbit Transfer　　　　　 155
A.2.1 The First-Order Derivatives　　　　　 156
A.2.2 The Second-Order Derivatives　　　　　 156
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