Flight dynamics and control of flapping-wing MAVs: A review

Nonlinear Dynamics

Volumn 70, Issue 2, 2012, Pages 907-939

  Taha, Haithem E ^a   Hajj, Muhammad R ^a   Nayfeh, Ali H ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Aerodynamic modeling; Averaging; Flapping MAVs insects; Floquet theory; High frequency periodic control and differential flatness; Kinematic design; Leading edge vortex; Stability analysis

Indexed keywords

AERODYNAMIC MODELING; AVERAGING; FLAPPING MAVS/INSECTS; FLOQUET THEORY; KINEMATIC DESIGNS; LEADING-EDGE VORTICES; PERIODIC CONTROL; STABILITY ANALYSIS;

DESIGN; DIFFERENTIAL EQUATIONS; KINEMATICS; MODELS;

FLIGHT DYNAMICS;

EID: 84871010400     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-012-0529-5     Document Type: Review

References (148)

1. Andersen, A., Pesavento, U., Wang, Z.: Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65-90 (2005)
2. Andersen, A., Pesavento, U.,Wang, Z.J.: Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 91-104 (2005)
3. Anderson, T.J., Balachandran, B., Nayfeh, A.H.: Nonlinear resonances in a flexible cantilever beam. J. Vib. Acoust. 116, 480-484 (1994)
4. Ansari, S.A., Zbikowski, R., Knowles, K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: Methodology and analysis. J. Aerosp. Eng. 220, 61-83 (2006)
5. Ansari, S.A., Zbikowski, R., Knowles, K.: Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: Implementation and validation. J. Aerosp. Eng. 220, 169-186 (2006)
6. Van den Berg, C., Ellington, C.P.: The three-dimensional leading-edge vortex of a hovering model hawkmoth. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 352, 329-340 (1997)
7. Van den Berg, C., Ellington, C.P.: The vortex wake of a hovering model hawkmoth. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 352, 317-328 (1997)
8. Berman, G.J., Wang, Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153 (2007). 168
9. Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Dover Publications, New York (1996)
10. Bloch, A.M.: Nonholonomic Mechanics and Control. Applied Mathematics. Springer, Berlin (2003)
11. Bolender, M.A.: Rigid multi-body equations-of-motion for flapping wing MAVs using Kane's equations. In: AIAA Guidance, Navigation, and Control Conference, Chicago, IL (2009)
12. Bullo, F.: Averaging and vibrational control of mechanical systems. J. Control Optim. 41(2), 542-562 (2002)
13. Cebeci, T., Platzer, M., Chen, H., Chang, K.C., Shao, J.P.: Analysis of Low Speed Unsteady Airfoil Flows. Horizons Publishing, Berlin (2005)
14. Cesnik, C.E.S., Brown, E.L.: Modeling of high aspect ratio active flexible wings for roll control. In: AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado (2002)
15. Cesnik, C.E.S., Brown, E.L.: Active wing warping control of a joined-wing airplane configuration. In: AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Norfolk, Virginia (2003)
16. Chadwick, L.E.: Insect Physiology.Wiley, London (1953)
17. Chung, S.J., Dorothy,M.: Neurobiologically inspired control of engineered flapping flight. J. Guid. Control Dyn. 33(2), 440-452 (2010)
18. Chung, S.J., Dorothy, M., Stoner, J.R.: Neurobiologically inspired control of engineered flapping flight. In: AIAA Infotech at Aerospace and Unmanned Unlimited Conference and Exhibit, Seattle, WA (2009)
19. Cook, M.V.: Flight Dynamics Principles. Elsevier, Amsterdam (2007)
20. DeLaurier, J.D.: An aerodynamic model for flapping wing flight. Aeronautical J. 97, 125-130 (1993)
21. Demoll, R.: Der Flug der Insekten und der Vogel. G. Fischer, Jena (1918)
22. Demoll, R.: Der Flug der Insekten. Naturwissenschaften 7, 480-481 (1919)
23. Deng, X., Schenato, L., Wu, W.C., Sastry, S.S.: Flapping flight for biomemetic robotic insects: Part I System modeling. IEEE Trans. Robot. 22(4), 776-788 (2006)
24. Deng, X., Schenato, L., Wu, W.C., Sastry, S.S.: Flapping flight for biomimetic robotic insects: Part II Flight control design. IEEE Trans. Robot. 22(4), 789-803 (2006)
25. Dickinson, M.H., Gotz, K.C.: Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biol. 174, 45-64 (1993)
26. Dickinson, M.H., Lehmann, F.O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284, 1954-1960 (1999)
27. Dickson, W.B., Dickinson, M.H.: The effect of advance ratio on the aerodynamics of revolving wings. J. Exp. Biol. 207, 4269-4281 (2004)
28. Dietl, J.M., Garcia, E.: Stability in ornithopter longitudinal flight dynamics. J. Guid. Control Dyn. 31(4), 1157- 1162 (2008)
29. Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O.: Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions: Part II: Controller. AIAA, Washington. 2010-1024
30. Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O.: Dynamics and control of a minimally actuated biomimetic vehicle, Part I - Aerodynamic model. In: AIAA Guidance, Navigation, and Control Conference. Chicago, Illinois 2009-6160
31. Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O.: Wingbeat shape modulation for flapping-wing micro-airvehicle control during hover. J. Guid. Control Dyn. 33(3), 724-739 (2010)
32. Ellington, C.P.: The aerodynamics of hovering insect flight: I. the quasi-steady analysis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 1-15 (1984)
33. Ellington, C.P.: The aerodynamics of hovering insect flight: II. Morphological parameters. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 17-40 (1984)
34. Ellington, C.P.: The aerodynamics of hovering insect flight: III. kinematics. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 41-78 (1984)
35. Ellington, C.P.: The aerodynamics of hovering insect flight: IV. aerodynamic mechanisms. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 79-113 (1984)
36. Ellington, C.P.: The aerodynamics of hovering insect flight: V. a vortex theory. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 115-144 (1984)
37. Ellington, C.P.: The aerodynamics of hovering insect flight: VI. lift and power requirements. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 305, 145-181 (1984)
38. Ellington, C.P.: Unsteady aerodynamics of insect flight. Soc. Exp. Biol. 49, 109-129 (1995)
39. Ellington, C.P., den Berg, C.V., Willmott, A.P., Thomas, A.L.R.: Leading-edge vortices in insect flight. Nature 384, 626-630 (1996)
40. Faruque, I., Humbert, J.S.: Dipterian insect flight dynamics. Part 1 Longitudinal motion about hover. J. Theor. Biol. 264(2), 538-552 (2010)
41. Fritz, T.E., Long, N.L.: Object-oriented unsteady vortex lattice method for flapping flight. J. Aircr. 41(6), 464-474 (2004)
42. Fry, S., Sayman, R., Dickinson, M.H.: The aerodynamics of free-flight maneuvers in drosophila. Science 300(5618), 495-498 (2003)
43. Gao, N., Aono, H., Liu, H.: A numerical analysis of dynamic flight stability of hawkmoth hovering. J. Biomech. Sci. Eng. 4(1), 105-116 (2009)
44. Gebert, G., Gallmeier, P.: Equations of motion for flapping flight. In: AIAA Atmospheric Flight Mechanics Conference (2001)
45. Ghommem, M.: Modeling and analysis for optimization of aeroelastic systems, PhD Thesis, Virginia Tech, Blacksburg, VA (2011)
46. Ghommem, M., Hajj, M.R., Mook, D.T., Stanford, B.K., Beran, P.S., Snyder, R.D., Watson, L.T.: Global optimization of actively morphing flapping wings. J. Fluids Struct. (2012). doi:10.1016/j.jfluidstructs.2012.04.013
47. Gogulapati, A., Friedmann, P.P., Shyy, W.: Nonlinear aeroelastic effects in flapping wing micro air vehicles. In: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, Illinois (2008)
48. Gogulapati, A., Friedmann, P.P., Shyy, W.: Approximate aeroelastic analysis of flapping wings in hover. In: IFASD, CEAS/AIAA/DGLR International Forum on Aeroelasticity and Structural Dynamics, Seattle, Washington (2009)
49. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, San Diego (2000)
50. Hedrick, T.L., Cheng, B., Deng, X.: Wingbeat time and the scaling of passive rotational damping in flapping flight. Science 324, 252 (2009)
51. Hengstenberg, R.: Mechanosensory control of compensatory head roll during flight in the blowfly calliphora erythrocephala. J. Comp. Physiol. A, Sensory Neural Behav. Physiol. 163, 151-165 (1988)
52. Holst, E.V., Kuchemann, D.: Biologische und aerodynamische probleme des tierfluges. Naturwissenschaften 29, 348-362 (1941)
53. Hooper, S.L.: Central pattern generators. In: Nature Encyclopedia of Life Sciences (2001)
54. Jones, R.T.: Operational treatment of the nonuniform lift theory to airplane dynamics. Tech. Rep. 667, NACA (1938)
55. Jones, R.T.: The unsteady lift of a finite wing. Tech. Rep. 682, NACA (1939)
56. Jones, R.T.: The unsteady lift of a wing of finite aspect ratio. Tech. Rep. 681, NACA (1940)
57. Kastberger, R.: The ocelli control the flight course in honeybees. Physiol. Entomol. 15, 337-346 (1990)
58. Kato, N., Kamimura, S.: Bio-Mechanisms of Swimming and Flying: Fluid Dynamics, Biomimetic Robots, and Sports Science. Springer, Berlin (2008)
59. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New York (2002)
60. Khan, Z.A., Agrawal, S.K.: Force and moment characterization of flapping wings for micro air vehicle application. In: American Control Conference (2005)
61. Khan, Z.A., Agrawal, S.K.: Control of longitudinal flight dynamics of a flapping wing micro air vehicle using time averaged model and differential flatness based controller. In: American Control Conference (2007)
62. Lanchester, F.W.: Aerodenetics. Constable, London (1908)
63. Liberzon, D.: Switching in Systems and Control. Birkhauser, Boston (2003)
64. Lohmiller, W., Slotine, J.-J.E.: Contraction analysis for nonlinear systems. Automatica 34(6), 683-696 (1998)
65. Lust, K.: Improved numerical Floquet multipliers. Int. J. Bifurc. Chaos Appl. Sci. Eng. 11, 2389-2410 (2001)
66. Mueller, T.H.: Fixed and Flapping Wing Aerodynamics for Micro Air Vehicles Applications. AIAA, Washington (2001)
67. Nalbach, G.: The halteres of the blowfly calliphora: I. Kinematics and dynamics. J. Comp. Physiol. A 173, 293- 300 (1993)
68. Nalbach, G., Hengstenberg, R.: The halteres of the blowfly calliphora: I. Three-dimensional organization of compensatory reactions to real and simulated rotations. J. Comp. Physiol. A 175, 695-708 (1994)
69. Nayfeh, A.H.: Nonlinear Interactions. Wiley, New York (2000)
70. Nayfeh, A.H., Chin, C.M.: Nonlinear interactions in a parametrically excited system with widely spaced frequencies. Nonlinear Dyn. 7, 195-216 (1995)
71. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations.Wiley, New York (1995)
72. Nayfeh, A.H., Mook D, T.: Energy transfer from highfrequency to low-frequency modes in structures. J. Mech. Des. J. Vib. Acoust. 117, 186-195 (1995). ASME Special Combined Issue
73. Nayfeh, S.A., Nayfeh, A.H.: Nonlinear interactions between two widely spaced modes - external excitation. Int. J. Bifurc. Chaos 3, 417-427 (1993)
74. Nayfeh, S.A., Nayfeh, A.H.: Energy transfer from highto low-frequency modes in a flexible structure via modulation. J. Vib. Acoust. 116, 203-207 (1994)
75. Nelson, R.C.: Flight Stability and Automatic Control. McGraw-Hill, New York (1989)
76. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)
77. Oh, K., Nayfeh, A.H.: High- to low-frequency modal interactions in a cantilever composite plate. J. Vib. Acoust. 120, 579-587 (1998)
78. Oppenheimer, M.W., Doman, D.B., Sigthorsson, D.O.: Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions: Part I: Aerodynamic Model. AIAA, Washington. 2010-1023
79. Oppenheimer, M.W., Doman, D.B., Sigthorsson, D.O.: Dynamics and control of a minimally actuated biomimetic vehicle, Part II - Control. In: AIAA Guidance, Navigation, and Control Conference. Chicago, Illinois 2009- 6161
80. Oppenheimer, M.W., Doman, D.B., Sigthorsson, D.O.: Dynamics and control of a biomimetic vehicle using biased wingbeat forcing functions. J. Guid. Control Dyn. 34(1), 204-217 (2011)
81. Orlowski, C.T., Girard, A.R.: Averaging of the nonlinear dynamics of flapping wing MAV for symmetrical flapping. In: Aerospace Sciences Meeting, Orlando, Florida (2011)
82. Orlowski, C.T., Girard, A.R.: Modeling and simulation of the nonlinear dynamics of flapping wing MAVs. AIAA J. 49(5), 969-981 (2011)
83. Orlowski, C.T., Girard, A.R., Shyy, W.: Derivation and simulation of the nonlinear dynamics of a flappingwing micro-air vehicle. In: European Micro-Air Vehicle Conference and Competition, Delft, The Netherlands (2009)
84. Orlowski, C.T., Girard, A.R., Shyy, W.: Open loop pitch control of a flappingwing micro airvehicle using a tail and controlmass. In: Institute of Electrical and Electronics Engineers, American Control Conference, Baltimore, MD, pp. 536-541 (2010)
85. Osborne, M.F.M.: Aerodynamics of flapping flight with application to insects. J. Exp. Biol. 28, 221-245 (1951)
86. Parry, A.D.: The swimming of whales and a discussion of Gray's paradox. J. Exp. Biol. 26, 24-34 (1949)
87. Pesavento, U., Wang, Z.J.: Navier-Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93, 116164 (2004)
88. Peters, D.A.: Two-dimensional incompressible unsteady airfoil theory: an overview. J. Fluids Struct. 24, 295-312 (2008)
89. Peters, D.A., Barwey, D., Johnson, M.J.: Finite-state airloads modeling with compressibility and unsteady freestream. In: Sixth International Workshop on Dynamics and Aeroelastic Stability Modelling of Rotorcraft Systems, UCLA, Los Angeles, CA (1995)
90. Peters, D.A., Johnson, M.J.: Finite-state airloads for deformable airfoils on fixed and rotating wings. In: ASME Winter Annual Meeting, Aeroelasticity and Fluid/Structures Interaction Problems, a Mini- Symposium, Chicago, vol. 44, pp. 1-28 (1994)
91. Peters, D.A., Karunamoorthy, S.: State-space inflow models for rotor aeroelasticity. In: Proceedings of the Holt Ashley 70th Anniversary Symposium, Stanford University (1993)
92. Peters, D.A., Karunamoorthy, S.: State-space inflow models for rotor aeroelasticity. In: AIAA, 12th Applied Aerodynamics Conference, Colorado Springs (1994)
93. Peters, D.A., Karunamoorthy, S., Cao, W.: Finite-state induced flow models, Part I: two-dimensional thin airfoil. J. Aircr. 32, 313-322 (1995)
94. Pham, Q.C., Slotine, J.J.E.: Stable concurrent synchronization in dynamic system networks. Neural Netw. 20(1), 62-77 (2007)
95. Phlips, P.J., East, R.A., Pratt, N.H.: An unsteady lifting line theory of flapping wings with application to the forward flight of birds. J. Fluid Mech. 112, 97-125 (1981)
96. Polhamus, E.C.: A concept of the vortex lift of sharpedge delta wings based on a leading-edge-suction analogy. Tech. Rep. NASA TN D-3767, Langely Research Center, Langely Station, Hampton, VA (1966)
97. Popovic, P., Nayfeh, A.H., Oh, K., Nayfeh, S.A.: An experimental investigation of energy transfer from a highfrequency mode to a low frequency mode in a flexible structure. J. Vib. Control 1, 115-128 (1995)
98. Ramamurti, R., Sandberg, W.: A three-dimensional computational study of the aerodynamic mechanisms of insect flight. J. Exp. Biol. 205(10), 1507-1518 (2002)
99. Rayner, J.M.V.: A new approach to animal flight mechanics. J. Exp. Biol. 80, 17-54 (1979)
100. Rayner, J.M.V.: A vortex theory of animal flight. Part 1. The vortex wake of a hovering animal. J. Fluid Mech. 91, 697-730 (1979)
101. Rayner, J.M.V.: A vortex theory of animal flight. Part 2. The forward flight of birds. J. Fluid Mech. 91, 731-763 (1979)
102. Reissner, E.: Effect of finite span on the air load distributions for oscillating wings, I - Aerodynamic theory of oscillating wings of finite span. Tech. Rep. 1194, NACA T.N. (1947)
103. Reissner, E., Stevens, J.E.: Effect of finite span on the air load distributions for oscillating wings, II - Methods of calculation and examples of application. Tech. Rep. 1195, NACA T.N. (1947)
104. Sane, S.P.: The aerodynamics of insect flight. J. Exp. Biol. 206, 4191-4208 (2003)
105. Sane, S.P., Dickinson, M.H.: The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. J. Exp. Biol. 205, 1087-1096 (2002)
106. Schenato, L., Campolo, D., Sastry, S.S.: Controllability issues in flapping flight for biomemetic MAVs. In: IEEE Conference on Decision and Control (2003)
107. Schlichting, H., Truckenbrodt, E.: Aerodynamics of the Airplane. McGraw-Hill, New York (1979)
108. Seo, K., Chung, S.J., Slotine, J.J.E.: Cpg-based control of a turtle-like underwater vehicle. Auton. Robots 28(3), 247-269 (2010)
109. Shkarayev, S., Silin, D.: Applications of actuator disk theory to membrane flapping wings. AIAA J. 48(10), 2227- 2234 (2010)
110. Shyy, W., Lian, Y., Tang, J., Viiero, D., Liu, H.: Aerodynamics of Low Reynolds Number Flyers. Cambridge University Press, Cambridge (2008)
111. Sigthorsson, D.O., Oppenheimer, M.W., Doman, D.B.: Flapping wing micro-air-vehicle control employing triangular wave strokes and cycle-averaging. AIAA, Washington. 2010-7553
112. Sira-Ramirez, H., Agrawal, S.K.: Differentially Flat Systems. Dekker, New York (2004)
113. Stanford, B.K., Beran, P.S.: Analytical sensitivity analysis of an unsteady vortex-lattice method for flapping-wing optimization. J. Aircr. 47(2), 647-662 (2010)
114. Stanford, B.K., Beran, P.S., Snyder, R., Patil, M.: Stability and power optimality in time-periodic flapping wing structures. In: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii. AIAA, Washington (2012). 2012-1638
115. Stolpe,M., Zimmer, K.: Der Vogelflug. Akademische Verlagsgesellschaft, Leipzig (1939)
116. Strogatz, S.: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (1994). Perseus Books Group
117. Su,W., Cesnik, C.E.S.: Coupled nonlinear aeroelastic and flight dynamic simulation of a flapping wing micro airvehicle. In: International Forum on Aeroelasticity and Structural Dynamics (2009)
118. Su, W., Cesnik, C.E.S.: Nonlinear aeroelastic simulations of a flapping wing micro air vehicle using two unsteady aerodynamic formulations. In: 51st AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Orlando, Florida (2010)
119. Su, W., Cesnik, C.E.S.: Flight dynamic stability of a flapping wing MAV in hover. In: 52nd AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado (2011)
120. Sun, M., Du, G.: Lift and power requirements of hovering insect flight. Acta Mech. Sin. 19(5), 458-469 (2003)
121. Sun, M., Tang, J.: Lift and power requirements of hovering flight in drosophila virilis. J. Exp. Biol. 205, 2413-2427 (2002)
122. Sun, M., Tang, J.: Unsteady aerodynamic force generation by a model fruitfly wing in flapping motion. J. Exp. Biol. 205, 55-70 (2002)
123. Sun, M., Wang, J., Xiong, Y.: Dynamic flight stability of hovering insects. Acta Mech. Sin. 23(3), 231-246 (2007)
124. Sun, M., Wu, J.H.: Lift generation and power requirements of fruit fly in forward flight with modeled wing motion. J. Exp. Biol. 206, 3065-3083 (2003)
125. Sun, M., Xiong, Y.: Dynamic flight stability of a hovering bumblebee. J. Exp. Biol. 208, 407 (2005)
126. Sussmann, H.J., Liu, W.: Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories. In: 30th IEEE Conf. Decision Control, New York, pp. 425-437 (2001)
127. Tabaddor, M., Nayfeh, A.H.: An experimental investigation of multimode responses in a cantilever beam. J. Vib. Acoust. 119, 532-538 (1997)
128. Taha, H.E., Hajj, M.R., Nayfeh, A.H.: Optimization of wing kinematics for hovering MAVs using calculus of variation. In: 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN (2012)
129. Taylor, C.: Contribution of compound eyes and ocelli to steering of locusts in flight: I-II. J. Exp. Biol. 93, 1-31 (1981)
130. Taylor, G.K., Bomphrey, R.J., Hoen, J.: Insect flight dynamics and control. In: AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada (2006)
131. Taylor, G.K., Thomas, A.L.R.: Animal flight dynamics II. Longitudinal stability in flapping flight. J. Theor. Biol. 214, 351-370 (2002)
132. Taylor, G.K., Thomas, A.L.R.: Dynamic flight stability in the desert locust. J. Theor. Biol. 206, 2803-2829 (2003)
133. Taylor, G.K., Zbikowski, R.: Nonlinear time periodic models of the longitudinal flight dynamics of desert locusts. J. R. Soc. Interface 2, 197-221 (2005)
134. Thomas, A.L.R., Taylor, G.K.: Animal flightdynamics I. Stability in gliding flight. J. Theor. Biol. 212, 399-424 (2001)
135. Vela, P.A.: Averaging and control of nonlinear systems (with application to biomimetic locomotion). Ph.D. thesis, California Institute of Technology, Pasadena, CA (2003)
136. Von Karman, T., Sears, W.R.: Airfoil theory for nonuniform motion. J. Aeronaut. Sci. 5(10), 379-390 (1938)
137. Wagner, H.: Über die Entstehung des dynamischen Auftriebs von Tragflugeln. Z. Angew. Math. Mech. 5, 17- 35 (1925)
138. Walker, G.T.: The flapping fligh of birds. J. R. Aero. Soc. 29, 590-594 (1925)
139. Walker, G.T.: The flapping fligh of birds II. J. R. Aero. Soc. 31, 337-342 (1927)
140. Wang, Z.J.: Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183-210 (2005)
141. Wang, Z.J., Birch, J.M., Dickinson, M.H.: Unsteady forces in hovering flight: computation vs experiments. J. Exp. Biol. 207, 449-460 (2004)
142. Weis-Fogh, T.: Energetics of hovering flight in hummingbirds and drosophila. J. Exp. Biol. 56, 79-104 (1972)
143. Weis-Fogh, T.: Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169-230 (1973)
144. Weis-Fogh, T., Jensen, M.: Biology and physics of locust flight. I. Basic principles of insect flight. A critical review. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 239, 415-458 (1956)
145. Willmott, A.P., Ellington, C.P.: The mechanisms of flight in the hawkmoth manduca sexta, I: Kinematics of hovering and forward flight. J. Exp. Biol. 200(21), 2705-2722 (1997)
146. Willmott, A.P., Ellington, C.P., Thomas, A.L.R.: Flow visualization and unsteady aerodynamics in the flight of the hawkmoth manduca sexta. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 352, 303-316 (1997)
147. Wood, R.J.: The first takeoff of a biologically inspired atscale robotic insect. IEEE Trans. Robot. Autom. 24(2), 341-347 (2008)
148. Xiong, Y., Sun, M.: Dynamic flight stability of a bumble bee in forward flight. Acta Mech. Sin. 24, 25-36 (2008)