本論文主要在分析拍翼式微型飛行器(MAVs)之動力學。淡江大學微機電實驗室已開發微飛行器多年。我們以該架飛行器為基礎，用以導出拍翼式微飛行器的動力學模型，並與實飛的飛行數據做比較。雖然目前已有很多文獻探討過類似的議題，但在此篇論文中，我們利用實際的飛行數據來印證我們所推導的動力學模型以及假設之正確性。此外，我們也透過數學推導，來證明空氣動力學領域中的“Advance Ratio”，可以直接應用於飛行動力學的推導。並且透過此轉換關係，可以將定翼機的分析方法，套用至拍翼機的分析。最後本文使用Matlab開發了圖形化介面的工具，歸納了我們的動力學模型，擬應用於未來拍翼式微型飛行器之飛行模擬。

The dynamics of flapping wing micro aerial vehicles (MAVs) is studied in this thesis. The MEMS Laboratory in Tamkang University has been developing flapping wing MAVs for
several years. Based on the developed flapping-wing MAVs we study its dynamics and compare our results with flight test data. Although several papers have discussed similar topics previously, using our flight test data we demonstrate the validity of the assumptions and derivations. We also propose a claim that links the average aerodynamical
forces to the wind tunnel test data, so that a flapping MAV can be analyzed with the same methodology as what we have done to a fixed-wing aircraft. Flight test data and
numerical simulations are also provided to demonstrate the validity of our derivation. Finally, we create a graphic user interface with Matlab as the 1st step to the flight animation of a flapping-MAV.

Contents
Chinses Abstract i
Abstract ii
Acknowledgement iii
Nomenclature iv
1 Introduction 1
1.1 Overall Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Research Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Flapping MAVs in TKU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Four-Bar-Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 Actuating System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Dynamics Model 10
2.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Dynamical Parameters 19
3.1 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Instantaneous Aerodynamics Force . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Averaging Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Other Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Simulation and Flight Test 49
4.1 Integration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Flight Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 The Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Conclusion and Future Work 64
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Bibliography 66
List of Tables
3.1 The parameters in force coefficients for a flapping wing. [1] . . . . . . . . . 25
3.2 The parameters in force coefficients for a tail wing. . . . . . . . . . . . . . 37
List of Figures
1.1 The flapping-wing MAV of TKU MEMS LAB [2]. . . . . . . . . . . . . . . 6
1.2 The structure of the actuator of MAV[2]. . . . . . . . . . . . . . . . . . . . 7
1.3 The concept illustration of four bar linkages. . . . . . . . . . . . . . . . . . 8
1.4 The interface of four-linkage design software. . . . . . . . . . . . . . . . . . 9
1.5 The variation of phase between right and left wings. . . . . . . . . . . . . . 9
2.1 A cartoon showing the definition of xbzb−plane in body-fixed coordinate [2]. 11
2.2 A cartoon showing the definition of ybzb−plane in body-fixed coordinate [2]. 12
2.3 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Relationship between body and inertial axes systems . . . . . . . . . . . . 14
3.1 The aerodynamic force distribution during downstroke. . . . . . . . . . . . 23
3.2 The aerodynamic force distribution during upstroke. . . . . . . . . . . . . . 23
3.3 A cartoon showing the definition of wing parameters. . . . . . . . . . . . . 26
3.4 The variation of lift force during a flapping period [1]. . . . . . . . . . . . . 29
3.5 The variation of thrust force during a flapping period [1]. . . . . . . . . . . 30
3.6 The flight trajectory captured by high speed camera shows that flapping
doesn’t affect the vertical movement of the flapping MAV too much[3]. . . 31
3.7 A cartoon showing the geometric parameters of the fuselage. . . . . . . . . 36
3.8 A simple illustrate for tunnel test. . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Relate the experimental data from tunnel test to the model of MAV. . . . 36
3.10 The lift performance of tail wing in wind tunnel test. . . . . . . . . . . . . 38
3.11 The drag performance of tail wing in wind tunnel test. . . . . . . . . . . . 38
3.12 The curve fitting of the lift of tail wing at −20o. . . . . . . . . . . . . . . . 39
3.13 The curve fitting of the lift of tail wing at −10o. . . . . . . . . . . . . . . . 39
3.14 The curve fitting of the lift of tail wing at 0o. . . . . . . . . . . . . . . . . 40
3.15 The curve fitting of the lift of tail wing at 10o. . . . . . . . . . . . . . . . . 40
3.16 The curve fitting of the lift of tail wing at 20o. . . . . . . . . . . . . . . . . 41
3.17 The curve fitting of the lift of tail wing at 30o. . . . . . . . . . . . . . . . . 41
3.18 The curve fitting of the lift of tail wing at 40o. . . . . . . . . . . . . . . . . 42
3.19 The curve fitting of the lift of tail wing at 50o. . . . . . . . . . . . . . . . . 42
3.20 The curve fitting of the drag of tail wing at −20o. . . . . . . . . . . . . . . 43
3.21 The curve fitting of the drag of tail wing at −10o. . . . . . . . . . . . . . . 43
3.22 The curve fitting of the drag of tail wing at 0o. . . . . . . . . . . . . . . . . 44
3.23 The curve fitting of the drag of tail wing at 10o. . . . . . . . . . . . . . . . 44
3.24 The curve fitting of the drag of tail wing at 20o. . . . . . . . . . . . . . . . 45
3.25 The curve fitting of the drag of tail wing at 30o. . . . . . . . . . . . . . . . 45
3.26 The curve fitting of the drag of tail wing at 40o. . . . . . . . . . . . . . . . 46
3.27 The curve fitting of the drag of tail wing at 50o. . . . . . . . . . . . . . . . 46
4.1 An example of the flight test trajectory. [1] . . . . . . . . . . . . . . . . . . 53
4.2 An example of the flight test trajectory. [1] . . . . . . . . . . . . . . . . . . 54
4.3 An example of the flight test trajectory. [1] . . . . . . . . . . . . . . . . . . 55
4.4 The flight trajectory in case 1. . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 The flight trajectory in case 2. . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 The flight trajectory in case 3. . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7 A graphical user interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 The result of simulation shows in the interface. . . . . . . . . . . . . . . . . 62
4.9 A enlarged image shows in the interface. . . . . . . . . . . . . . . . . . . . 63

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