A time-varying model for the forward flight dynamics of a flapping-wing micro aerial vehicle is identified from free-flight optical tracking data. The model is validated and used to assess the validity of the widely applied time-scale separation assumption. Based on this assumption, each aerodynamic force and moment is formulated as a linear addition of decoupled time-averaged and time-varying submodels. The resulting aerodynamic models are incorporated in a set of linearized equations of motion, yielding a simulation-capable full dynamic model. The time-averaged component includes both the longitudinal and the lateral aerodynamics and is assumed to be linear. The time-varying component is modeled as a third-order Fourier series, which approximates the flapping dynamics effectively. Combining both components yields a more complete and realistic simulation. Results suggest that while in steady flight the time-scale separation assumption applies well during maneuvers the time-varying dynamics are not fully captured. More accurate modeling of flapping-wing flight during maneuvers may require considering coupling between the time scales.
 , “Design, Aerodynamics, and Vision-Based Control of the DelFly,” International Journal of Micro Air Vehicles, Vol. 1, No. 2, 2009, pp. 71–97. doi:https://doi.org/10.1260/175682909789498288
 , “Development of the Nano Hummingbird: A Tailless Flapping Wing Micro Air Vehicle,” 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2012-0588, Jan. 2012. doi:https://doi.org/10.2514/6.2012-588
 , “The First Takeoff of a Biologically Inspired At-Scale Robotic Insect,” IEEE Transactions on Robotics, Vol. 24, No. 2, 2008, pp. 341–347. doi:https://doi.org/10.1109/TRO.2008.916997 ITREAE 1552-3098
 , Aerodynamics of Low Reynolds Number Flyers,
Cambridge Aerospace Series, Cambridge Univ. Press, New York, 2008, pp. 101–158. doi:https://doi.org/10.1017/CBO9780511551154
 , “Leading-Edge Vortices in Insect Flight,” Nature, Vol. 384, No. 6610, 1996, pp. 626–630. doi:https://doi.org/10.1038/384626a0
 , “Experimental and Numerical Study of Forward Flight Aerodynamics of Insect Flapping Wing,” AIAA Journal, Vol. 47, No. 3, March 2009, pp. 730–742. doi:https://doi.org/10.2514/1.39462 AIAJAH 0001-1452
 , “A Three-Dimensional Computational Study of the Aerodynamic Mechanisms of Insect Flight,” The Journal of Experimental Biology, Vol. 1518, No. 10, 2002, pp. 1507–1518.
 , “Non-Linear Unsteady Aerodynamic Model for Insect-Like Flapping Wings in the Hover. Part 2: Implementation and Validation,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, Vol. 220, No. 3, Jan. 2006, pp. 169–186. doi:https://doi.org/10.1243/09544100JAERO50
 , “Flapping Flight for Biomimetic Robotic Insects: Part I—System Modeling,” IEEE Transactions on Robotics, Vol. 22, No. 4, 2006, pp. 776–788. doi:https://doi.org/10.1109/TRO.2006.875480 ITREAE 1552-3098
 , “An Aerodynamic Model for Flapping-Wing Flight,” The Aeronautical Journal of the Royal Aeronautical Society, Vol. 97, No. 964, 1993, pp. 125–130.
 , “The Aerodynamics of Insect Flight,” Journal of Experimental Biology, Vol. 206, No. 23, 2003, pp. 4191–4208. doi:https://doi.org/10.1242/jeb.00663 JEBIAM 0022-0949
 , “Energy-Minimizing Kinematics in Hovering Insect Flight,” Journal of Fluid Mechanics, Vol. 582, July 2007, pp. 153–168. doi:https://doi.org/10.1017/S0022112007006209 JFLSA7 0022-1120
 , “Flapping Wing Micro Air Vehicle Aerodynamic Modeling Including Flapping and Rigid Body Velocity,” 50th AIAA Aerospace Sciences Meeting, AIAA Paper 2012-0026, Jan. 2012. doi:https://doi.org/10.2514/6.2012-26
 , “Testing and System Identification of an Ornithopter in Longitudinal Flight,” Journal of Aircraft, Vol. 48, No. 2, March 2011, pp. 660–667. doi:https://doi.org/10.2514/1.C031208
 , “Controlled Flight Maneuvers of a Flapping Wing Micro Air Vehicle: A Step Towards the Delfly II Identification,” AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2013-4843, Aug. 2013. doi:https://doi.org/10.2514/6.2013-4843
 , “Approximate Aerodynamic and Aeroelastic Modeling of Flapping Wings in Hover and Forward Flight,” 52nd AIAA/ASME/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA Paper 2011-2008, April 2011.
 , “Modeling and Simulation of the Nonlinear Dynamic Behavior of a Flapping Wings Micro-Aerial-Vehicle,” 42nd AIAA Aerospace Sciences Meeting, AIAA Paper 2004-0541, Jan. 2004. doi:https://doi.org/10.2514/6.2004-541
 , “Modelling and Simulation of Flapping Wing Control for a Micromechanical Flying Insect (Entomopter),” AIAA Modeling and Simulation Technologies Conference, AIAA Paper 2002-4973, 2002. doi:https://doi.org/10.2514/6.2002-4973
 , “Rigid Multi-Body Equations-of-Motion for Flapping Wing MAVs Using Kane’s Equations,” AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009-6158, Aug. 2009. doi:https://doi.org/10.2514/6.2009-6158
 , “Multibody Model of an Ornithopter,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 5, Sept. 2009, pp. 1675–1679. doi:https://doi.org/10.2514/1.43177 JGCODS 0731-5090
 , “Equations of Motion for Flapping Flight,” AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2002-4872, 2002. doi:https://doi.org/10.2514/6.2002-4872
 , “Modeling and Simulation of Nonlinear Dynamics of Flapping Wing Micro Air Vehicles,” AIAA Journal, Vol. 49, No. 5, May 2011, pp. 969–981. doi:https://doi.org/10.2514/1.J050649 AIAJAH 0001-1452
 , “Dynamic Flight Stability in the Desert Locust Schistocerca Gregaria,” Journal of Experimental Biology, Vol. 206, No. 16, Aug. 2003, pp. 2803–2829. doi: https://doi.org/10.1242/jeb.00501 JEBIAM 0022-0949
 , “Rigid vs. Flapping: The Effects of Kinematic Formulations in Force Determination of a Free Flying Flapping Wing Micro Air Vehicle,” International Conference on Unmanned Aircraft Systems, IEEE Publ., Piscataway, NJ, May 2014, pp. 949–959. doi:https://doi.org/10.1109/ICUAS.2014.6842345
 , “Cooperative Control and Modeling for Narrow Passage Traversal with an Ornithopter MAV and Lightweight Ground Station,” Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems, International Foundation for Autonomous Agents and Multiagent Systems, St. Paul, MN, May 2013, pp. 103–110. doi:https://doi.org/10.1.1.309.9545
 , “Dynamic Flight Stability of a Hovering Bumblebee,” The Journal of Experimental Biology, Vol. 208, No. 3, Feb. 2005, pp. 447–459. doi:https://doi.org/10.1242/jeb.01407
 , “Dipteran Insect Flight Dynamics. Part 1: Longitudinal Motion About Hover,” Journal of Theoretical Biology, Vol. 264, No. 2, May 2010, pp. 538–552. doi:https://doi.org/10.1016/j.jtbi.2010.02.018 JTBIAP 0022-5193
 , “Bounded Control of a Flapping Wing Micro Drone in Three Dimensions,” IEEE International Conference on Robotics and Automation, IEEE Publ., Piscataway, NJ, 2008, pp. 164–169. doi:https://doi.org/10.1109/ROBOT.2008.4543203
 , “Control of Longitudinal Flight Dynamics of a Flapping-Wing Micro Air Vehicle Using Time-Averaged Model and Differential Flatness Based Controller,” Proceedings of the 2007 American Control Conference, IEEE Publ., Piscataway, NJ, 2007, pp. 5284–5289. doi:https://doi.org/10.1109/ACC.2007.4283052
 , “Controllability Issues in Flapping Flight for Biomimetic Micro Aerial Vehicles (MAVs),” Proceedings IEEE Conference Decision Control, Vol. 6, IEEE Publ., Piscataway, NJ, Dec. 2003, pp. 6441–6447. doi:https://doi.org/10.1.1.216.7484
 , “A Geometric Control Approach for the Longitudinal Flight Stability of Hovering Insects/FWMAVs,” AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2015-1552, Jan. 2015. doi:https://doi.org/10.2514/6.2015-1552
 , “Effect of the Aerodynamic-Induced Parametric Excitation on the Longitudinal Stability of Hovering MAVs/Insects,” Nonlinear Dynamics, Vol. 78, No. 4, Aug. 2014, pp. 2399–2408. doi:https://doi.org/10.1007/s11071-014-1596-6 NODYES 0924-090X
 , “The Need for Higher-Order Averaging in the Stability Analysis of Hovering, Flapping-Wing Flight,” Bioinspiration & Biomimetics, Vol. 10, No. 1, 2015, Paper 016002. doi:https://doi.org/10.1088/1748-3190/10/1/016002
 , “Dynamic Modeling, Testing, and Stability Analysis of an Ornithoptic Blimp,” Journal of Bionic Engineering, Vol. 8, No. 4, Dec. 2011, pp. 375–386. doi:https://doi.org/10.1016/S1672-6529(11)60043-7 1672-6529
 , “Linear Aerodynamic Model Identification of a Flapping Wing MAV Based on Flight Test Data,” International Journal of Micro Air Vehicles, Vol. 5, No. 4, 2013, pp. 273–286. doi:https://doi.org/10.1260/1756-82184.108.40.2063
 , “Non-Linear Aircraft Flight Path Reconstruction Review and New Advances,” Progress in Aerospace Sciences, Vol. 35, No. 7, 1999, pp. 673–726. doi:https://doi.org/10.1016/S0376-0421(99)00005-6 PAESD6 0376-0421
 , “Black-Box LTI Modelling of Flapping-wing Micro Aerial Vehicle Dynamics,” AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2015-0234, Jan. 2015. doi:https://doi.org/10.2514/6.2015-0234
 , “Behaviour Trees for Evolutionary Robotics: Reducing the Reality Gap,” Master’s Thesis, TU Delft, Delft, The Netherlands, 2014 (unpublished).
 , “Delfly Freeflight, ” Master’s Thesis, TU Delft, Delft, The Netherlands, 2012 (unpublished).
 , “Design, Aerodynamics and Autonomy of the DelFly,” Bioinspiration & Biomimetics, Vol. 7, No. 2, June 2012, Paper 025003. doi:https://doi.org/10.1088/1748-3182/7/2/025003
 , “Modeling a Flapping Wing MAV: Flight Path Reconstruction of the Delfly II,” AIAA Modeling and Simulation Technologies Conference, AIAA Paper 2013-4597, Aug. 2013. doi:https://doi.org/10.2514/6.2013-4597
 , Flight Vehicle System Identification: A Time Domain Methodology,
Progress in Astronautics and Aeronautics, AIAA, Reston, VA, 2006, pp. 79–129.
 , “Application of Parameter Estimation to Aircraft Stability and Control: The Output-Error Approach,” NASA Reference Publ. 1168, 1986.
 , Aircraft System Identification,
AIAA Education Series, AIAA, Reston, VA, 2006, pp. 181–223.
 , “Error Analysis and Assessment of Unsteady Forces Acting on a Flapping Wing Micro Air Vehicle: Free Flight Versus Wind-Tunnel Experimental Methods,” Bioinspiration and Biomimetics, Vol. 10, No. 5, 2015, Paper 056004. doi:https://doi.org/10.1088/1748-3190/10/5/056004 1748-3182