Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 1125–1138 | Cite as

Saturation-based actuation for flapping MAVs in hovering and forward flight

  • Haithem E. Taha
  • Ali H. Nayfeh
  • Muhammad R. Hajj
Original Paper


Stringent weight and size constraints on flapping-wing microair-vehicles dictate minimal actuation. Unfortunately, hovering and forward flight require different wing motions and, as such, independent actuators. Therefore, either a hovering or a forward-flight requirement should be included in the mission and design statements of a flapping-wing microair-vehicle. This work proposes a design for an actuation mechanism that would provide the required kinematics in each flight condition using only one actuator. The idea is to exploit the nonlinear dynamics of the flapping wing to induce the saturation phenomenon. One physical spring in the plunging direction is needed along with a feedback of the plunging angle into the control torque of the actuator in the back and forth flapping direction. By detuning the feedback gains away from the saturation requirement, we obtain the flapping kinematics required for hovering. In contrast, tuning the feedback gains to induce the saturation phenomenon transfers the motion into the plunging direction. Moreover, the actuating torque (in the back and forth flapping direction) would then provide a direct control over the amplitude of the plunging motion, while the amplitude of the actuated flapping motion saturates and does not change as the amplitude of the actuating torque increases.


Saturation phenomenon Micro-air vehicles Actuation Nonlinear dynamics 



Mean chord length


Gravitational acceleration

mv, mw

Masses of the vehicle and wing respectively


Radial coordinate along the wing


Wing radius (length)


Area of one wing


Time variable

T, f

Flapping period and frequency

xI, yI, and zI

Inertial fixed frame

xw, yw, and zw

Wing fixed frame


Pitching angle


Back and forth flapping angle


Plunging angle


  1. 1.
    Wood, R.J.: The first takeoff of a biologically inspired at-scale robotic insect. IEEE Trans. Robot. Autom. 24(2), 341–347 (2008) CrossRefGoogle Scholar
  2. 2.
    Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O.: Dynamics and control of a minimally actuated biomimetic vehicle. Part i: Aerodynamic model. 2009-6160. AIAA Guidance, Navigation, and Control Conference Chicago, Illinois Google Scholar
  3. 3.
    Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O.: Wingbeat shape modulation for flapping-wing micro-air-vehicle control during hover. J. Guid. Control Dyn. 33(3), 724–739 (2010) CrossRefGoogle Scholar
  4. 4.
    Schenato, L., Campolo, D., Sastry, S.S.: Controllability issues in flapping flight for biomimetic mavs. ieee Conference on Decision and Control, Maui, HI (2003) Google Scholar
  5. 5.
    Berman, G.J., Wang, Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153,168 (2007) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kurdi, M., Stanford, B., Beran, P.: Kinematic optimization of insect flight for minimum mechanical power. 2010-1420. 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Fl (2010) Google Scholar
  7. 7.
    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) CrossRefGoogle Scholar
  8. 8.
    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. 33, 210–228 (2012) CrossRefGoogle Scholar
  9. 9.
    Taha, H.E., Hajj, M.R., Nayfeh, A.H.: Wing kinematics optimization for hovering micro air vehicles using calculus of variation. J. Aircr. (2013). doi: 10.2514/1.C031969 Google Scholar
  10. 10.
    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) CrossRefGoogle Scholar
  11. 11.
    Sigthorsson, D.O., Oppenheimer, M.W., Doman, D.B.: Flapping wing micro-air-vehicle control employing triangular wave strokes and cycle-averaging. 2010-7553. AIAA, Guidance, Navigation, and Control Conference, Toronto, Ontario, Canada (2010) Google Scholar
  12. 12.
    Chung, S.J., Dorothy, M.: Neurobiologically inspired control of engineered flapping flight. J. Guid. Control Dyn. 33(2), 440–452 (2010) CrossRefGoogle Scholar
  13. 13.
    Stanford, B.K., Beran, P.S., Snyder, R., Patil, M.: Stability and power optimality in time-periodic flapping wing structures. AIAA 2012-1638. 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii (2012) Google Scholar
  14. 14.
    Bhatia, M., Patil, M., Woolsey, C., Stanford, B., Beran, P.: Lqr controller for stabilization of flapping wing mavs in gust environments. AIAA, Atmospheric Flight Mechanics Conference, Minneapolis, Minnesota (2012) Google Scholar
  15. 15.
    Weis-Fogh, T.: Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169–230 (1973) Google Scholar
  16. 16.
    Ellington, C.P.: The aerodynamics of hovering insect flight: iii. Kinematics. Philos. Trans. R. Soc. Lond. B 305, 41–78 (1984) CrossRefGoogle Scholar
  17. 17.
    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 2010-1024 Google Scholar
  18. 18.
    Oppenheimer, M.W., Doman, D.B., Sigthorsson, D.O.: Dynamics and control of a minimally actuated biomimetic vehicle. Part ii: Controller. 2009-6161. AIAA Guidance, Navigation, and Control Conference, Chicago, Illinois Google Scholar
  19. 19.
    Greenwood, D.T.: Advanced Dynamics. Cambridge University Press, Cambridge (2003) CrossRefGoogle Scholar
  20. 20.
    Taha, H.E., Hajj, M.R., Beran, P.S.: Unsteady nonlinear aerodynamics of hovering mavs/insects. AIAA 2013-0504. Aerospace Sciences Meeting, Dallas (2013) Google Scholar
  21. 21.
    Nayfeh, A.H., Mook, D.T., Marshall, L.R.: Nonlinear coupling of pitch and roll modes in ship motion. J. Hydronaut. 7(4), 145–152 (1973) CrossRefGoogle Scholar
  22. 22.
    Haddow, A.G., Barr, A.D.S., Mook, D.T.: Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure. J. Sound Vib. 97(3), 451–473 (1984) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nayfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. Appl. Mech. Rev. 42(11), 175–202 (1989) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973) MATHGoogle Scholar
  25. 25.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) MATHGoogle Scholar
  26. 26.
    Balachandran, B., Nayfeh, A.H.: Nonlinear motions of beam-mass structure. Nonlinear Dyn. 1, 39–61 (1990) CrossRefGoogle Scholar
  27. 27.
    Oueini, S.S., Nayfeh, A.H., Gonaraghi, M.A.: A theoretical and experimental implementation of a control method based on saturation. Nonlinear Dyn. 13, 189–202 (1997) MATHCrossRefGoogle Scholar
  28. 28.
    Oueini, S.S., Nayfeh, A.H., Pratt, J.R.: A nonlinear vibration absorber for flexible structures. Nonlinear Dyn. 15(3), 259–282 (1998) MATHCrossRefGoogle Scholar
  29. 29.
    Hall, B.D., Mook, D.T., Nayfeh, A.H., Preidikman, S.: Novel strategy for suppressing the flutter oscillations of aircraft wings. AIAA J. 39(10), 1843–1850 (2001). doi: 10.2514/2.1190 CrossRefGoogle Scholar
  30. 30.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979) MATHGoogle Scholar
  31. 31.
    Nayfeh, A.H.: Nonlinear Interactions: Analytical, Computational, and Experimental Methods. Wiley, New York (2002) Google Scholar
  32. 32.
    Ellington, C.P.: The aerodynamics of hovering insect flight: ii. Morphological parameters. Philos. Trans. R. Soc. Lond. B 305, 17–40 (1984) CrossRefGoogle Scholar
  33. 33.
    Sun, M., Du, G.: Lift and power requirements of hovering insect flight. Acta Mech. Sin. 19(5), 458–469 (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Haithem E. Taha
    • 1
  • Ali H. Nayfeh
    • 1
  • Muhammad R. Hajj
    • 1
  1. 1.Virginia TechBlacksburgUSA

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