Homogeneous Parametric Modeling of Airloads


  • Finn Matras Norwegian Univeristy of Science and Technology, Department of Engineering Cybernetics
  • Dirk Peter Reinhardt Norwegian Univeristy of Science and Technology, Department of Engineering Cybernetics
  • Kristoffer Gryte Norwegian Univeristy of Science and Technology, Department of Engineering Cybernetics
  • Morten Dinhoff Pedersen Norwegian Univeristy of Science and Technology, Department of Engineering Cybernetics




Airloads, Homogeneity, Spherical Harmonics, Neural Network, SVD


This work proposes two parametric modeling strategies for steady aerodynamic forces. We point out that airloads are homogeneous and introduce a parametrization based on spherical harmonics and a neural network. The parametrization using spherical harmonics enables an analogue of frequency-based truncation and a variation on the Singular Value Decomposition (SVD), constituting an orthogonal decomposition of the modeled airloads. Since neural networks are universal function approximators, the model based on this allows for more flexible parametrizations, including actuations and model inversions. Both parametrization strategies are showcased for model identification and reduction purposes, highlighting their strengths and weaknesses.


F. Matras, D. P. Reinhardt, and M. D. Pedersen, “Parametrization of airloads using a homogeneity-based orthogonal decomposition,” in 2022 26th International Conference on System Theory, Control and Computing (ICSTCC), November 2022, pp. 553–558.

J. F. Manwell, J. G. McGowan, and A. L. Rogers, Wind energy explained: theory, design and application. John Wiley & Sons, 2010.

T. Burton, N. Jenkins, D. Sharpe, and E. Bossanyi, Wind Energy Handbook. John Wiley & Sons, 2011.

W. Johnson, Helicopter Theory, ser. Dover Books on Aeronautical Engineering Series. Dover Publications, 1994.

R. Beard and T. McLain, Small Unmanned Aircraft: Theory and Practice. Princeton University Press, 2012. [Online]. Available: https://books.google.no/books?id=YqQtjhPUaNEC

M. Hagan, H. Demuth, M. Beale, and O. De Jesus, Neural Network Design 2nd Edition. Martin Hagan, 2014.

Z. Didyk and V. Apostolyuk, “Whole angle approximations of aerodynamic coefficients,” in 2012 2nd International Conference ”Methods and Systems of Navigation and Motion Control” (MSNMC), 2012, pp. 119–121.

K. Gryte, R. Hann, M. Alam, J. Roh´aˇc, T. A. Johansen, and T. I. Fossen, “Aerodynamic modeling of the Skywalker X8 fixed-wing unmanned aerial vehicle,” in 2018 International Conference on Unmanned Aircraft Systems (ICUAS), 2018, pp. 826–835.

J. A. Grauer and E. A. Morelli, “Generic global aerodynamic model for aircraft,” Journal of Aircraft, vol. 52, no. 1, pp. 13–20, February 2015.

B. Simmons and P. Murphy, “Wind tunnel-based aerodynamic model identification for a tilt-wing, distributed electric propulsion aircraft,” 2021.

J. V. Caetano, C. C. de Visser, G. C. H. E. de Croon, B. Remes, C. de Wagter, J. Verboom, and M. Mulder, “Linear aerodynamic model identification of a flapping wing MAV based on flight test data,” International Journal of Micro Air Vehicles, vol. 5, no. 4, December 2013.

J. C. Gibbings, Dimensional analysis. Springer Science & Business Media, 2011.

Encyclopedia of Mathematics. (2021) Homogeneous function. [Online]. Available: http://encyclopediaofmath.org/index.php?title=Homogeneous function&oldid=51769

J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, “Julia: A fresh approach to numerical computing,” SIAM Review, vol. 59, no. 1, pp. 65–98, February 2017.

C. Frye and C. J. Efthimiou, “Spherical harmonics in p dimensions,” 2012.

“NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15, f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. [Online]. Available: http://dlmf.nist.gov/

T. Fukushima, “Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives,” Journal of Geodesy, vol. 87, April 2012.

A. Gil and J. Segura, “A code to evaluate prolate and oblate spheroidal harmonics,” Computer Physics Communications, vol. 108, no. 2, pp. 267–278, February 1998.

X.-G. Wang and T. Carrington, “Using Lebedev grids, sine spherical harmonics, and monomer contracted basis functions to calculate bending energy levels of HF trimer,” Journal of Theoretical and Computational Chemistry, vol. 2, pp. 599–608, 12 2003.

V. I. Lebedev and D. N. Laikov, “A quadrature formulat for the sphere of the 131st algebraic order of accuracy,” Doklady Mathematics, vol. 59, no. 3, pp. 477–481, 1999.

J. Jonkman, S. Butterfield, W. Musial, and G. Scott, “Definition of a 5MW reference wind turbine for offshore system development,” National Renewable Energy Laboratory (NREL), Tech. Rep., January 2009.

M. Innes, E. Saba, K. Fischer, D. Gandhi, M. C. Rudilosso, N. M. Joy, T. Karmali, A. Pal, and V. Shah, “Fashionable modelling with Flux,” CoRR, vol. abs/1811.01457, 2018.

M. Innes, “Flux: Elegant machine learning with Julia,” Journal of Open Source Software, May 2018.

J. Zhuang, T. Tang, Y. Ding, S. Tatikonda, N. Dvornek, X. Papademetris, and J. S. Duncan, “Adabelief optimizer: Adapting stepsizes by the belief in observed gradients,” December 2020.

I. Dunning, J. Huchette, and M. Lubin, “JuMP: A modeling language for mathematical optimization,” SIAM Review, vol. 59, no. 2, pp. 295–320, 2017.

A. W¨achter and L. T. Biegler, “On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming,”Mathematical Programming, vol. 106, pp. 25–57, April 2006.

M. L. Buhl, Jr, “A new empirical relationship between thrust coefficient and induction factor for the turbulent windmill state,” National Renewable Energy Laboratory, Tech. Rep., 8 2005.

F. Golnary, H. Moradi, and K. Tse, “Nonlinear pitch angle control of an onshore wind turbine by considering the aerodynamic nonlinearities and deriving an aeroelastic model,” Energy Systems, vol. 14, pp. 197 – 227, August 2021.

D. Reinhardt, M. D. Pedersen, K. Gryte, and T. A. Johansen, “A symmetry calibration procedure for sensor-to-airframe misalignments in wind tunnel data,” in 2022 IEEE Conference on Control Technology and Applications (CCTA), 2022, pp. 1360–1365.




How to Cite

“Homogeneous Parametric Modeling of Airloads”, Syst. Theor. Control Comput. J., vol. 3, no. 1, pp. 1–11, Jun. 2023, doi: 10.52846/stccj.2023.3.1.44.