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Harmonic Decomposition Models of Flapping-Wing Flight for Stability Analysis and Control Design

Published Online:https://doi.org/10.2514/1.G006447

This paper demonstrates the extension of the harmonic decomposition methodology, originally developed for rotorcraft applications, to the study of the nonlinear time-periodic dynamics of flapping-wing flight. A harmonic balance algorithm based on harmonic decomposition is applied to find the periodic equilibrium and approximate linear time-invariant dynamics about that equilibrium of the vertical and longitudinal dynamics of a hawk moth. These approximate linearized models are validated through simulations against the original nonlinear time-periodic dynamics. Dynamic stability using the linear models is assessed and compared to that predicted using the averaged dynamics. In addition, modal participation factors are computed to quantify the influence of the higher harmonics on the flight dynamic modes of motion. The study shows that higher harmonics play a key role in the dynamics of flapping-wing flight, inducing a vibrational stabilization mechanism that increases the pitch damping and stiffness while reducing the speed stability. This results in stabilization of the pitch oscillation mode and thus of the longitudinal hovering cubic. In addition, the proposed methodology is used to derive open- and closed-loop control laws for attenuating arbitrary state harmonics and to enhance the dynamic response characteristics.

References

  • [1] Adrian L. R. T. and Graham K. T., “Animal Flight Dynamics I. Stability in Gliding Flight,” Journal of Theoretical Biology, Vol. 212, No. 3, 2001, pp. 399–424. https://doi.org/10.1006/jtbi.2001.2387 Google Scholar

  • [2] Khan A. Z. and Agrawal S. K., “Control of Longitudinal Flight Dynamics of a Flapping-Wing Micro Air Vehicle Using Time-Averaged Model and Differential Flatness Based Controller,” 2007 American Control Conference, Inst. of Electrical and Electronics Engineers, New York, July 2007, pp. 5284–5289. https://doi.org/10.1109/ACC.2007.4283052 Google Scholar

  • [3] Sun M. and Xiong Y., “Dynamic Flight Stability of a Hovering Bumblebee,” Journal of Experimental Biology, Vol. 208, No. 3, 2005, pp. 447–459. https://doi.org/10.1242/jeb.01407 CrossrefGoogle Scholar

  • [4] Schenato L., Campolo D. and Sastry S., “Controllability Issues in Flapping Flight for Biomimetic Micro Aerial Vehicles (MAVs),” 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475), Vol. 6, Inst. of Electrical and Electronics Engineers, New York, Dec. 2003, pp. 6441–6447. https://doi.org/10.1109/CDC.2003.1272361 Google Scholar

  • [5] Dietl J. M. and Garcia E., “Stability in Ornithopter Longitudinal Flight Dynamics,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 1157–1152. https://doi.org/10.2514/1.33561 LinkGoogle Scholar

  • [6] Taha H. E., Hajj M. R. and Nafayeh A. H., “Flight Dynamics and Control of Flapping-Wing MAVs: A Review,” Nonlinear Dynamics, Vol. 70, Oct. 2012, p. 907–939. https://doi.org/10.1007/s11071-012-0529-5 CrossrefGoogle Scholar

  • [7] Taha H. E., Woosley C. A. and Hajj M. R., “Geometric Control Approach to Longitudinal Stability of Flapping Flight,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 2, 2015, pp. 214–226. https://doi.org/10.2514/1.G001280 LinkGoogle Scholar

  • [8] Hassan A. M. and Taha H. E., “Differential-Geometric-Control Formulation of Flapping Flight Multi-Body Dynamics,” Journal of Nonlinear Science, Vol. 29, No. 4, 2019, pp. 1379–1417. https://doi.org/10.1007/s00332-018-9520-8 CrossrefGoogle Scholar

  • [9] Sun M., “Insect Flight Dynamics: Stability and Control,” Reviews of Modern Physics, Vol. 86, No. 2, 2014, pp. 615–646. https://doi.org/10.1103/RevModPhys.86.615 CrossrefGoogle Scholar

  • [10] Maggia M., Eisa S. A. and Taha H. E., “On Higher-Order Averaging of Time-Periodic Systems: Reconciliation of Two Averaging Techniques,” Nonlinear Dynamics, Vol. 99, Jan. 2020, p. 813–836. https://doi.org/10.1007/s11071-019-05085-4 CrossrefGoogle Scholar

  • [11] Floquet G., “Sur les Équations Différentielles Linéaires à Coefficients Périodiques,” Annales scientifiques de l’École Normale Supérieure, Vol. 2, No. 12, 1883, pp. 47–88. https://doi.org/10.24033/asens.220 CrossrefGoogle Scholar

  • [12] Bierling T. and Patil M., “Stability and Power Optimality in Time-Periodic Flapping Wing Structures,” Journal of Fluids and Structures, Vol. 38, April 2013, pp. 238–254. https://doi.org/10.1016/j.jfluidstructs.2012.12.006 Google Scholar

  • [13] Su W. and Cesnik C. E. S., “Flight Dynamic Stability of a Flapping Wing Micro Air Vehicle in Hover,” Proceedings of the 52nd AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics, and Materials Conference, AIAA, Reston, VA, April 2011. https://doi.org/10.2514/6.2011-2009 Google Scholar

  • [14] Bolender M. A., “Open-Loop Stability of Flapping Flight in Hover,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2010-7552, 2010. https://doi.org/10.2514/6.2010-7552 LinkGoogle Scholar

  • [15] Sanders J. A., Verhulst F. and Murdock J., “Averaging: The Periodic Case,” Averaging Methods in Nonlinear Dynamical Systems, Part of the Applied Mathematical Sciences Book Series (AMS, Volume 59), Springer, Princeton, NJ, 2007, pp. 21–138, Chaps. 2–6. https://doi.org/10.1007/978-0-387-48918-6 Google Scholar

  • [16] Hassan A. M. and Taha H. E., “Higher-Order Averaging Analysis of the Nonlinear Time-Periodic Dynamics of Hovering Insects/Flapping-Wing Micro-Air-Vehicles,” 2016 IEEE 55th Conference on Decision and Control (CDC), Inst. of Electrical and Electronics Engineers, New York, Dec. 2016. https://doi.org/10.1109/CDC.2016.7799424 Google Scholar

  • [17] Oppenheimer M. W., Weintraub I. E., Sighorsson D. and Doman D. B., “Quarter Cycle Modulation of a Minimally Actuated Biomimetic Vehicle,” AIAA SciTech Forum, AIAA Paper 2014-1467, 2014. https://doi.org/10.2514/1.G000548 Google Scholar

  • [18] Oppenheimer M. W., Weintraub I. E., Sighorsson D. and Doman D. B., “Control of a Minimally Actuated Biometric Vehicle Using Quarter-Cycle Wingbeat Modulation,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 7, 2015, pp. 1187–1196. https://doi.org/10.2514/1.G000548 LinkGoogle Scholar

  • [19] Sigthorsson D. O., Oppenheimer M. W. and Doman D. B., “Flapping-Wing Micro-Air-Vehicle Control Employing Triangular Waves Strokes and Cycle Averaging,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2010-7553, 2010. https://doi.org/10.2514/6.2010-7553 LinkGoogle Scholar

  • [20] Sigthorsson D. O., Oppenheimer M. W. and Doman D. B., “Flapping-Wing Micro-Air-Vehicle 4-DOF Controller Applied to a 6-DOF Model,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2010-7554, 2010. https://doi.org/10.2514/6.2010-7554 LinkGoogle Scholar

  • [21] Finio B. M., Perez-Arancibia N. O. and Wood R. J., “System Identification and Linear Time-Invariant Modeling of an Insect-Sized Flapping-Wing Micro Air Vehicle,” 2011 IEEE/ESJ International Conference on Intelligent Robots and Systems, Inst. of Electrical and Electronics Engineers, New York, 2011, pp. 1107–1114. https://doi.org/10.1109/IROS.2011.6094421 Google Scholar

  • [22] Cheng D. and Deng X., “Translational and Rotational Damping of Flapping Flight and Its Dynamics and Stability at Hovering,” IEEE Transactions on Robotics, Vol. 27, No. 5, 2011, pp. 849–864. https://doi.org/10.1109/TRO.2011.2156170 CrossrefGoogle Scholar

  • [23] Sun M., Wang J. and Xiong Y., “Dynamic Flight Stability of Hovering Insects,” Acta Mechanica Sinica, Vol. 23, June 2007, pp. 231–246. https://doi.org/10.1007/s10409-007-0068-3 CrossrefGoogle Scholar

  • [24] Taha H. E., Tahmasian S., Woosley C. A., Nafayeh A. H. and Hajj M. R., “The Need for Higher-Order Averaging in the Stability Analysis of Hovering, Flapping-Wing Flight,” Bioinspiration & Biomimetics, Vol. 10, No. 1, 2015, pp. 1–15. https://doi.org/10.1088/1748-3190/10/1/016002 CrossrefGoogle Scholar

  • [25] Taha H. E., Kiani M., Hedrick T. L. and Greeter J. S., “Vibrational Control: A Hidden Stabilization Mechanism in Insect Flight,” Science Robotics, Vol. 5, No. 46, 2020, pp. 1–11. CrossrefGoogle Scholar

  • [26] Oppenheimer M. W., Doman D. B. and Sigthorsson D. O., “Dynamics and Control of a Minimally Actuated Biomimetic Vehicle: Part I—Aerodynamic Model,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009-6160, 2009. https://doi.org/10.2514/6.2009-6160 LinkGoogle Scholar

  • [27] Oppenheimer M. W., Doman D. B. and Sigthorsson D. O., “Dynamics and Control of a Minimally Actuated Biomimetic Vehicle: Part II—Control,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009-6161, 2009. https://doi.org/10.2514/6.2009-6161 LinkGoogle Scholar

  • [28] Nogar S. M., Serrani A., Gogulapati A., McNamara J. J., Oppenheimer M. W. and Doman D. B., “Design and Evaluation of a Model-Based Controller for Flapping-Wing Micro Aerial Vehicle,” Journal of Guidance, Control, and Dynamics, Vol. 41, No. 12, 2018, pp. 2513–2528. https://doi.org/10.2514/1.G003293 LinkGoogle Scholar

  • [29] Perez-Arancibia N. O., Whitney J. P. and Wood R. J., “Lift Force Control of a Flapping-Wing Microrobot,” 2011 American Control Conference, Inst. of Electrical and Electronics Engineers, New York, June–July 2011, pp. 4761–4768. https://doi.org/10.1109/ACC.2011.5991157 Google Scholar

  • [30] Saetti U. and Rogers J. D., “Revisited Harmonic Balance Trim Solution Method for Periodically-Forced Flight Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 44, No. 5, 2021, pp. 1008–1017. https://doi.org/10.2514/1.G005553 LinkGoogle Scholar

  • [31] Pandyan R. and Sinha S. C., “Time-Varying Controller Synthesis for Nonlinear Systems Subjected to Periodic Parametric Loading,” Journal of Vibration and Control, Vol. 7, No. 1, 2001, pp. 73–90. https://doi.org/10.1177/107754630100700105 CrossrefGoogle Scholar

  • [32] Colaneri P., Celi R. and Bittanti S., “Constant Coefficient Representation of Discrete Periodic Linear Systems,” Proceedings of the AHS 4th Decennial Specialists’ Conference on Aeromechanics, American Helicopter Soc., Jan. 2004. Google Scholar

  • [33] Lopez M. J. S. and Prasad J. V. R., “Linear Time Invariant Approximations of Linear Time Periodic Systems,” Journal of the American Helicopter Society, Vol. 62, No. 1, 2017, pp. 1–10. https://doi.org/10.4050/jahs.62.012006 CrossrefGoogle Scholar

  • [34] Lopez M. J. S., “Linear Time Invariant Approximations of Linear Time Periodic Systems for Integrated Flight and Vibration Control,” Ph.D. Thesis, Georgia Inst. of Technology, Atlanta, GA, May 2016. Google Scholar

  • [35] Prasad J. V. R., Olcer F. E., Sankar L. N. and He C., “Linear Time Invariant Models for Integrated Flight and Rotor Control,” Proceedings of the 35th European Rotorcraft Forum, Sept. 2009. Google Scholar

  • [36] Lopez M., Prasad J. V. R., Tischler M. B., Takahashi M. D. and Cheung K. K., “Simulating HHC/AFCS Interaction and Optimized Controllers Using Piloted Maneuvers,” Proceedings of the 71st Annual Forum of the American Helicopter Society, May 2015. Google Scholar

  • [37] Padthe A. K., Friedmann P. P., Lopez M. and Prasad J. V. R., “Analysis of High Fidelity Reduced-Order Linearized Time-Invariant Helicopter Models for Integrated Flight and On-Blade Control Applications,” Proceedings of the 41st European Rotorcraft Forum, Sept. 2015. Google Scholar

  • [38] Saetti U. and Horn J. F., “Load Alleviation Flight Control Design Using High-Order Dynamic Models,” Journal of the American Helicopter Society, Vol. 65, No. 3, 2020, pp. 1–15. https://doi.org/10.4050/JAHS.65.032009 Google Scholar

  • [39] Saetti U., Horn J. F., Berger T. and Tischler M. B., “Handling-Qualities Perspective on Rotorcraft Load Alleviation Control,” Journal of Guidance, Control, and Dynamics, Vol. 43, No. 13, Oct. 2020, pp. 1–13. https://doi.org/10.2514/1.G004965 Google Scholar

  • [40] Scaramal M., Horn J. F. and Saetti U., “Load Alleviation Control Using Dynamic Inversion with Direct Load Feedback,” Proceedings of the 77th Annual Forum of the Vertical Flight Society, 2021. Google Scholar

  • [41] Mballo C. E. and Prasad J. V. R., “A Real Time Scheme for Rotating System Component Load Estimation Using Fixed System Measurements,” Proceedings of the 74th Annual Forum of the Vertical Flight Society, 2018. Google Scholar

  • [42] Mballo C. E. and Prasad J. V. R., “Real Time Rotor Component Load Limiting via Model Predictive Control,” Proceedings of the 75th Annual Forum of the Vertical Flight Society, 2019. Google Scholar

  • [43] Peters D. A., Chouchane M. and Fulton M., “Helicopter Trim with Flap-Lag-Torsion and Stall by an Optimized Controller,” Journal of Guidance, Control, and Dynamics, Vol. 13, No. 5, 1990, pp. 824–834. https://doi.org/10.2514/3.25408 LinkGoogle Scholar

  • [44] Peters D. A. and Izadpanah A. P., “Helicopter Trim by Periodic Shooting with Newton-Raphson Iteration,” Proceedings of the 37th Annual Forum of the American Helicopter Society, May 1981. Google Scholar

  • [45] Peters D. A. and Ormiston R. A., “Flapping Response Characteristics of Hingless Rotor Blades by a Generalized Harmonic Balance Method,” NASA TN D-7856, 1975. Google Scholar

  • [46] Ypma T. J., “Historical Development of the Newton–Raphson Method,” SIAM Review, Vol. 37, No. 4, 1995, pp. 531–551. https://doi.org/10.1137/1037125 CrossrefGoogle Scholar

  • [47] Lopez M. J. S. and Prasad J. V. R., “Estimation of Modal Participation Factors of Linear Time Periodic Systems Using Linear Time Invariant Approximations,” Journal of the American Helicopter Society, Vol. 61, No. 4, 2016, pp. 1–4. https://doi.org/10.4050/JAHS.61.045001 CrossrefGoogle Scholar

  • [48] Kokotovic P. V., O’Malley R. E. and Sannuti P., “Singular Perturbations and Order Reduction in Control Theory, An Overview,” Automatica, Vol. 12, No. 2, 1976, pp. 123–132. https://doi.org/10.1016/0005-1098(76)90076-5 CrossrefGoogle Scholar

  • [49] Frazzoli E., Dahlel M. A. and Feron E., “Maneuver-Based Motion Planning for Nonlinear Systems with Symmetries,” IEEE Transactions on Robotics, Vol. 21, No. 6, 2005, pp. 1077–1091. https://doi.org/10.1109/TRO.2005.852260 CrossrefGoogle Scholar

  • [50] Taha H. E., Hajj M. R. and Nayfeh A. H., “Longitudinal Flight Dynamics of Hovering MAVs/Insects,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 3, 2014, pp. 970–978. https://doi.org/10.2514/1.62323 LinkGoogle Scholar

  • [51] Whitney J. P. and Wood R. J., “Aeromechaniics of Passive Rotation in Flapping Flight,” Journal of Fluid Mechanics, Vol. 60, Oct. 2010, pp. 197–220. https://doi.org/10.1017/S002211201000265X Google Scholar

  • [52] Oppenheimer M. W., Doman D. B. and Sigthorsson D. O., “Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions Part I—Aerodynamic Model,” 48th AIAA Aerospace Sciences Meeting, AIAA Paper 2010-1023, 2010. https://doi.org/10.2514/6.2010-1023 LinkGoogle Scholar

  • [53] Doman D. B., Oppenheimer M. W. and Sigthorsson D. O., “Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions Part II—Controller,” 48th AIAA Aerospace Sciences Meeting, AIAA Paper 2010-1024, 2010. https://doi.org/10.2514/6.2010-1024 LinkGoogle Scholar

  • [54] Oppenheimer M. W., Doman D. B. and Sighorsson D. O., “Dynamics and Control of a Biomimetic Vehicle Using Biased Wingbeat Forcing Functions,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 1, 2011, pp. 204–217. https://doi.org/10.2514/1.49735 LinkGoogle Scholar

  • [55] Passaro M. and Lovera M., “LPV Model Identification of a Flapping Wing MAV,” Proceedings of the 4th IFAC Workshop on Linear Parameter Varying Systems, July 2021, pp. 27–32. https://doi.org/10.1016/j.ifacol.2021.08.576 Google Scholar

  • [56] Taha E. H., Woosley C. A. and Hajj M. R., “Geometric Control Approach to Longitudinal Stability of Flapping Flight,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 2, 2016, pp. 214–226. https://doi.org/10.2514/1.G001280 LinkGoogle Scholar

  • [57] Cheng R. P., Tischler M. B. and Celi R., “A High-Order, Time Invariant Linearized Model for Application to HHC/AFCS Interaction Studies,” Proceedings of the 59th Annual Forum of the American Helicopter Society, May 2003. Google Scholar

  • [58] Kautsky J., Nichols N. K. and Van Dooren P., “Robust Pole Assignment in Linear State Feedback,” International Journal of Control, Vol. 41, No. 5, 1985, pp. 1129–1155. https://doi.org/10.1080/0020718508961188 CrossrefGoogle Scholar