Optimization of compound performance of two tandem pitching and heaving aerofoils in Martian environment

Muhammad Khan; Li Wang; Fang-Bao Tian; Muhammad Khan; Li Wang; Fang-Bao Tian

doi:10.3934/era.2026059

Electronic Research Archive

2026, Volume 34, Issue 2: 1290-1314. doi: 10.3934/era.2026059

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Optimization of compound performance of two tandem pitching and heaving aerofoils in Martian environment

School of Engineering and Technology, University of New South Wales, Canberra, ACT 2600, Australia

Received: 16 December 2025 Revised: 28 January 2026 Accepted: 02 February 2026 Published: 09 February 2026

Mars exploration has increased the interest in bio-inspired flapping-wing micro aerial vehicles due to their high maneuverability and efficient performance in low Reynolds number environments. This work presents an optimization of the compound performance of two tandem pitching and heaving aerofoils in the Martian environment by using the hybrid method of the reinforcement learning (RL) and the immersed boundary-finite difference method (IB-FDM). The searching parameters include the horizontal and vertical spacing, phase shift, and mean angle of attack (AoA) of the hindwing at a Reynolds number of 1100. It is found that the optimal phase shifts lie close to in-phase and out-of-phase motions, while intermediate phase shifts tend to produce suboptimal results. For the in-phase cases, an increase in AoA and a decrease in horizontal spacing led to an increase in the compound performance. For the out-of-phase cases, a similar trend is observed until the optimal value is reached, after which the performance begins to decline. The effects of vertical spacing vary vastly on a case-to-case basis, depending on other motion and positional parameters.
- pitching and heaving aerofoils,
- bio-inspired flapping wing,
- Martian environment,
- immersed boundary-finite difference method,
- low Reynolds number
Citation: Muhammad Khan, Li Wang, Fang-Bao Tian. Optimization of compound performance of two tandem pitching and heaving aerofoils in Martian environment[J]. Electronic Research Archive, 2026, 34(2): 1290-1314. doi: 10.3934/era.2026059

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Abstract

Mars exploration has increased the interest in bio-inspired flapping-wing micro aerial vehicles due to their high maneuverability and efficient performance in low Reynolds number environments. This work presents an optimization of the compound performance of two tandem pitching and heaving aerofoils in the Martian environment by using the hybrid method of the reinforcement learning (RL) and the immersed boundary-finite difference method (IB-FDM). The searching parameters include the horizontal and vertical spacing, phase shift, and mean angle of attack (AoA) of the hindwing at a Reynolds number of 1100. It is found that the optimal phase shifts lie close to in-phase and out-of-phase motions, while intermediate phase shifts tend to produce suboptimal results. For the in-phase cases, an increase in AoA and a decrease in horizontal spacing led to an increase in the compound performance. For the out-of-phase cases, a similar trend is observed until the optimal value is reached, after which the performance begins to decline. The effects of vertical spacing vary vastly on a case-to-case basis, depending on other motion and positional parameters.

References

[1]	T. Suwa, K. Nose, D. Numata, H. Nagai, K. Asai, Compressibility effects on airfoil aerodynamics at low Reynolds number, in 30th AIAA Applied Aerodynamics Conference, (2012), 3029. https://doi.org/10.2514/6.2012-3029
[2]	W. J. Koning, M. Dominguez, Mars Helicopter Ingenuity Rotor Geometry, Technical report, Ames Research Center, Moffett Field, California, 2024.
[3]	J. D. Eldredge, A. R. Jones, Leading-edge vortices: mechanics and modeling, Annu. Rev. Fluid Mech., 51 (2019), 75–104. https://doi.org/10.1146/annurev-fluid-010518-040334 doi: 10.1146/annurev-fluid-010518-040334
[4]	N. Widdup, L. Wang, J. Young, V. Daria, H. Liu, F. B. Tian, A scaling law for the lift of a bio-inspired wing hovering in low-density compressible flows, J. Fluid Mech., 1003 (2025), A10. https://doi.org/10.1017/jfm.2024.1216 doi: 10.1017/jfm.2024.1216
[5]	U. Pesavento, Z. J. Wang, Flapping wing flight can save aerodynamic power compared to steady flight, Phys. Rev. Lett., 103 (2009), 118102. https://doi.org/10.1103/PhysRevLett.103.118102 doi: 10.1103/PhysRevLett.103.118102
[6]	R. J. Huston, Wind-tunnel measurements of performance, blade motions, and blade air loads for tandem-rotor configurations with and without overlap, Technical report, 1963.
[7]	P. J. Arcidiacono, A method for computation of the induced velocity field of a rotor in forward flight, suitable for application to tandem rotor configurations, J. Am. Helicopter Soc., 9 (1964), 34–45. https://doi.org/10.4050/JAHS.9.34 doi: 10.4050/JAHS.9.34
[8]	D. R. Croom, H. G. Wiley, Investigation at transonic speeds of the hinge-moment and lift-effectiveness characteristics of a single flap and a tandem flap on a 60 degree delta wing, Technical report, 1953.
[9]	V. R. Corsiglia, D. Koenig, Large-scale wind-tunnel tests on an aspect ratio 2.17 delta-wing model equipped with midchord boundary-layer-control flaps, Technical report, 1964.
[10]	T. Broering, Y. Lian, The effect of wing spacing on tandem wing aerodynamics, in 28th AIAA Applied Aerodynamics Conference, (2010), 4385. https://doi.org/10.2514/6.2010-4385
[11]	T. M. Broering, Y. S. Lian, The effect of phase angle and wing spacing on tandem flapping wings, Acta Mech. Sin., 28 (2012), 1557–1571. https://doi.org/10.1007/s10409-012-0210-8 doi: 10.1007/s10409-012-0210-8
[12]	T. M. Broering, Y. Lian, Numerical study of tandem flapping wing aerodynamics in both two and three dimensions, Comput. Fluids, 115 (2015), 124–139. https://doi.org/10.1016/j.compfluid.2015.04.003 doi: 10.1016/j.compfluid.2015.04.003
[13]	A. S. A. Seet, Investigation of Flow Over Oscillating NACA 4421 Airfoils in Tandem Configuration at Low Reynolds Number, PhD thesis, Nanyang Technological University in Singapore, 2019.
[14]	T. Lee, Flow past two in-tandem airfoils undergoing sinusoidal oscillations, Exp. Fluids, 51 (2011), 1605–1621. https://doi.org/10.1007/s00348-011-1173-4 doi: 10.1007/s00348-011-1173-4
[15]	A. Jaroszewicz, J. Sąsiadek, K. Sibilski, Modeling and simulation of flapping wings entomopter in Martian atmosphere, in Aerospace Robotics, Springer, (2013), 143–162. https://doi.org/10.1007/978-3-642-34020-8_12
[16]	N. P. B. Mannam, M. M. Alam, P. Krishnankutty, Review of biomimetic flexible flapping foil propulsion systems on different planetary bodies, Results Eng., 8 (2020), 100183. https://doi.org/10.1016/j.rineng.2020.100183 doi: 10.1016/j.rineng.2020.100183
[17]	J. E. Bluman, J. A. Pohly, M. K. Sridhar, C. K. Kang, D. B. Landrum, F. Fahimi, et al., Achieving bioinspired flapping wing hovering flight solutions on Mars via wing scaling, Bioinspir. Biomim., 13 (2018), 046010. https://doi.org/10.1088/1748-3190/aac876 doi: 10.1088/1748-3190/aac876
[18]	H. Bézard, T. Désert, T. Jardin, J. M. Moschetta, Numerical and experimental aerodynamic investigation of a micro-uav for flying on mars, in 76th Annual Forum & Technology Display, 2020.
[19]	J. L. McCain, Experimental force and wing motion measurements of a bioinspired flapping wing in a martian density condition, 2019. Available from: https://louis.uah.edu/uah-theses/306
[20]	B. Singh, N. Yidris, A. A. Basri, R. Pai, K. A. Ahmad, Study of mosquito aerodynamics for imitation as a small robot and flight in a low-density environment, Micromachines, 12 (2021), 511. https://doi.org/10.3390/mi12050511 doi: 10.3390/mi12050511
[21]	N. Widdup, L. Wang, J. Young, F. B. Tian, A numerical study of flexible flapping wings in highly compressible flows, Phys. Fluids, 37 (2025), 071912. https://doi.org/10.1063/5.0274245 doi: 10.1063/5.0274245
[22]	N. Widdup, L. Wang, J. Young, F. B. Tian, The aerodynamic and acoustic performance of a bumblebee-inspired micro-air vehicle in Martian atmosphere, Phys. Fluids, 37 (2025), 091916. https://doi.org/10.1063/5.0288043 doi: 10.1063/5.0288043
[23]	S. S. Bhat, T. O'Shaughnessy, J. T. Hrynuk, J. Young, F. B. Tian, S. Ravi, Propeller blade performance in low-pressure environments, Phys. Fluids, 38 (2026), In press. https://doi.org/10.1063/5.0311082 doi: 10.1063/5.0311082
[24]	M. Anyoji, M. Okamoto, H. Hidaka, T. Nonomura, A. Oyama, K. Fujii, Planetary atmosphere wind tunnel tests on aerodynamic characteristics of a mars airplane scale model, Trans. JSASS Aerospace Tech. Japan, 12 (2014), 7–12. https://doi.org/10.2322/tastj.12.Pk_7 doi: 10.2322/tastj.12.Pk_7
[25]	J. A. Pohly, C. K. Kang, D. B. Landrum, J. E. Bluman, H. Aono, Data-driven CFD scaling of bioinspired Mars flight vehicles for hover, Acta Astronaut., 180 (2021), 545–559. https://doi.org/10.1016/j.actaastro.2020.12.037 doi: 10.1016/j.actaastro.2020.12.037
[26]	H. Mahjoubi, K. Byl, Steering and horizontal motion control in insect-inspired flapping-wing MAVs: the tunable impedance approach, in 2012 American Control Conference (ACC), (2012), 901–908. https://doi.org/10.1109/ACC.2012.6314655
[27]	H. Mahjoubi, K. Byl, Analysis of a tunable impedance method for practical control of insect-inspired flapping-wing MAVs, in 2011 50th IEEE Conference on Decision and Control and European Control Conference, (2011), 3539–3546. https://doi.org/10.1109/CDC.2011.6160297
[28]	P. A. Oche, G. A. Ewa, N. Ibekwe, Applications and challenges of artificial intelligence in space missions, IEEE Access, 12 (2024), 44481–44509. https://doi.org/10.1109/ACCESS.2021.3132500 doi: 10.1109/ACCESS.2021.3132500
[29]	D. Soler, O. Mariño, D. Huergo, M. de Frutos, E. Ferrer, Reinforcement learning to maximize wind turbine energy generation, Expert Syst. Appl., 249 (2024), 123502. https://doi.org/10.1016/j.eswa.2024.123502 doi: 10.1016/j.eswa.2024.123502
[30]	H. Van Hasselt, A. Guez, D. Silver, Deep reinforcement learning with double q-learning, in Proceedings of the AAAI Conference on Artificial Intelligence, 30 (2016), 2094–2100. https://doi.org/10.1609/aaai.v30i1.10295
[31]	K. Portal-Porras, U. Fernandez-Gamiz, E. Zulueta, R. Garcia-Fernandez, S. E. Berrizbeitia, Active flow control on airfoils by reinforcement learning, Ocean Eng., 287 (2023), 115775. https://doi.org/10.1016/j.oceaneng.2023.115775 doi: 10.1016/j.oceaneng.2023.115775
[32]	Y. Bao, X. Shi, Z. Wang, H. Zhu, N. Srinil, A. Li, et al., Deep reinforcement learning for propulsive performance of a flapping foil, Phys. Fluids, 35 (2023), 103610. https://doi.org/10.1063/5.0169982 doi: 10.1063/5.0169982
[33]	J. Rabault, A. Kuhnle, Accelerating deep reinforcement learning strategies of flow control through a multi-environment approach, Phys. Fluids, 31 (2019), 094105. https://doi.org/10.1063/1.5116415 doi: 10.1063/1.5116415
[34]	G. Lee, Y. Joo, S. U. Lee, T. Kim, Y. Yu, H. G. Kim, Design optimization of heat exchanger using deep reinforcement learning, Int. Commun. Heat Mass Transfer, 159 (2024), 107991. https://doi.org/10.1016/j.icheatmasstransfer.2024.107991 doi: 10.1016/j.icheatmasstransfer.2024.107991
[35]	T. Ji, F. Jin, F. Xie, H. Zheng, X. Zhang, Y. Zheng, Active learning of tandem flapping wings at optimizing propulsion performance, Phys. Fluids, 34 (2022), 047117. https://doi.org/10.1063/5.0084160 doi: 10.1063/5.0084160
[36]	P. I. Frazier, A tutorial on bayesian optimization, preprint, arXiv: 1807.02811.
[37]	N. P. B. Mannam, V. R. Avula, Propulsive performance of tandem flapping wings for autonomous underwater vehicles (AUVs) and surface ships, in 2021 IEEE 4th International Conference on Computing, Power and Communication Technologies (GUCON), (2021), 1–7. https://doi.org/10.1109/GUCON50781.2021.9573893
[38]	H. Nagai, K. Fujita, M. Murozono, Experimental study on forewing–hindwing phasing in hovering and forward flapping flight, AIAA J., 57 (2019), 3779–3790. https://doi.org/10.2514/1.J058335 doi: 10.2514/1.J058335
[39]	W. B. Tay, K. B. Lim, Analysis of non-symmetrical flapping airfoils, Acta Mech. Sin., 25 (2009), 433–450. https://doi.org/10.1007/s10409-009-0259-1 doi: 10.1007/s10409-009-0259-1
[40]	L. Wang, F. B. Tian, Numerical study of sound generation by three-dimensional flexible flapping wings during hovering flight, J. Fluids Struct., 99 (2020), 103165. https://doi.org/10.1016/j.jfluidstructs.2020.103165 doi: 10.1016/j.jfluidstructs.2020.103165
[41]	L. Wang, F. B. Tian, Sound generated by the flow around an airfoil with an attached flap: From passive fluid–structure interaction to active control, J. Fluids Struct., 111 (2022), 103571. https://doi.org/10.1016/j.jfluidstructs.2022.103571 doi: 10.1016/j.jfluidstructs.2022.103571
[42]	K. Isogai, Y. Shinmoto, Y. Watanabe, Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil, AIAAJ, 37 (1999), 1145–1151. https://doi.org/10.2514/2.589 doi: 10.2514/2.589
[43]	L. Wang, F. B. Tian, J. C. Lai, An immersed boundary method for fluid–structure–acoustics interactions involving large deformations and complex geometries, J. Fluids Struct., 95 (2020), 102993. https://doi.org/10.1016/j.jfluidstructs.2020.102993 doi: 10.1016/j.jfluidstructs.2020.102993
[44]	L. Wang, J. Young, F. B. Tian, An immersed boundary method for the thermo–fluid–structure interaction in rarefied gas flows, Phys. Fluids, 36 (2024), 013616. https://doi.org/10.1063/5.0181397 doi: 10.1063/5.0181397
[45]	X. D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212. https://doi.org/10.1006/jcph.1994.1187 doi: 10.1006/jcph.1994.1187
[46]	L. Fu, X. Y. Hu, N. A. Adams, A family of high-order targeted ENO schemes for compressible-fluid simulations, J. Comput. Phys., 305 (2016), 333–359. https://doi.org/10.1016/j.jcp.2015.10.037 doi: 10.1016/j.jcp.2015.10.037
[47]	F. B. Tian, X. Lu, H. Luo, Onset of instability of a flag in uniform flow, Theor. Appl. Mech. Lett., 2 (2012), 022005. https://doi.org/10.1063/2.1202205 doi: 10.1063/2.1202205
[48]	F. B. Tian, W. Wang, J. Wu, Y. Sui, Swimming performance and vorticity structures of a mother-calf pair of fish, Comput. Fluids, 124 (2016), 1–11. https://doi.org/10.1016/j.compfluid.2015.10.006 doi: 10.1016/j.compfluid.2015.10.006
[49]	W. X. Huang, F. B. Tian, Recent trends and progress in the immersed boundary method, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 233 (2019), 7617–7636. https://doi.org/10.1177/0954406219842606 doi: 10.1177/0954406219842606
[50]	L. Wang, F. B. Tian, H. Liu, Numerical study of three-dimensional flapping wings hovering in ultra-low-density atmosphere, Phys. Fluids, 34 (2022), 041903. https://doi.org/10.1063/5.0085021 doi: 10.1063/5.0085021
[51]	J. Schulman, F. Wolski, P. Dhariwal, A. Radford, O. Klimov, Proximal policy optimization algorithms, preprint, arXiv: 1707.06347.
[52]	H. K. Lim, J. B. Kim, J. S. Heo, Y. H. Han, Federated reinforcement learning for training control policies on multiple IoT devices, Sensors, 20 (2020), 1359. https://doi.org/10.3390/s20051359 doi: 10.3390/s20051359
[53]	K. P. Haughn, L. L. Gamble, D. J. Inman, Deep reinforcement learning achieves multifunctional morphing airfoil control, J. Compos. Mater., 57 (2023), 721–736. https://doi.org/10.1177/00219983221137644 doi: 10.1177/00219983221137644
[54]	S. Ravichandiran, Deep Reinforcement Learning with Python: Master Classic RL, Deep RL, Distributional RL, Inverse RL, and more with OpenAI Gym and TensorFlow, Packt Publishing Ltd, 2020.
[55]	V. Nair, G. E. Hinton, Rectified linear units improve restricted Boltzmann machines, in Proceedings of the 27th International Conference on Machine Learning, (2010), 807–814.
[56]	N. Fuengfusin, H. Tamukoh, Nan attacks: Bit-flipping deep neural network parameters to nan or infinity, in 024 1st International Conference on Robotics, Engineering, Science, and Technology (RESTCON), (2024), 33–37. https://doi.org/10.1109/RESTCON60981.2024.10463548
[57]	D. Liu, X. Xiong, Y. Zhang, Action-dependent adaptive critic designs, in IJCNN'01. International Joint Conference on Neural Networks. Proceedings (Cat. No.01CH37222), 2 (2001), 990–995. https://doi.org/10.1109/IJCNN.2001.939495
[58]	G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, et al., Openai gym, preprint, arXiv: 1606.01540.
[59]	M. Andrychowicz, A. Raichuk, P. Stańczyk, M. Orsini, S. Girgin, R. Marinier, et al., What matters in on-policy reinforcement learning? a large-scale empirical study, preprint, arXiv: 2006.05990.
[60]	M. Schilling, Avoid overfitting in deep reinforcement learning: Increasing robustness through decentralized control, in Artificial Neural Networks and Machine Learning – ICANN 2021, (2021), 638–649. https://doi.org/10.1007/978-3-030-86380-7_52
[61]	D. Silver, G. Lever, N. Heess, T. Degris, D. Wierstra, M. Riedmiller, Deterministic policy gradient algorithms, in International Conference on Machine Learning, (2014), 387–395.
[62]	M. Moriche, O. Flores, M. García-Villalba, Generation of thrust and lift with airfoils in plunging and pitching motion, J. Phys. Conf. Ser., 574 (2015), 012163. https://doi.org/10.1088/1742-6596/574/1/012163 doi: 10.1088/1742-6596/574/1/012163
[63]	J. Zhang, X. Y. Lu, Aerodynamic performance due to forewing and hindwing interaction in gliding dragonfly flight, Phys. Rev. E, 80 (2009), 017302. https://doi.org/10.1103/PhysRevE.80.017302 doi: 10.1103/PhysRevE.80.017302
[64]	D. Bie, D. Li, Numerical analysis of the wing–wake interaction of tandem flapping wings in forward flight, Aerosp. Sci. Technol., 121 (2022), 107389. https://doi.org/10.1016/j.ast.2022.107389 doi: 10.1016/j.ast.2022.107389
[65]	Y. Lee, K. B. Lua, Wing–wake interaction: comparison of 2D and 3D flapping wings in hover flight, Bioinspir. Biomim., 13 (2018), 066003. https://doi.org/10.1088/1748-3190/aadc31 doi: 10.1088/1748-3190/aadc31
[66]	O. W. Ata, Aerodynamic performance of advanced ingenuity and dragonfly drones for future space missions to mars and titan, in 2021 4th International Symposium on Advanced Electrical and Communication Technologies (ISAECT), (2021), 01–06. https://doi.org/10.1109/ISAECT53699.2021.9668399
[67]	Z. Terze, V. Pandža, M. Kasalo, D. Zlatar, Discrete mechanics and optimal control optimization of flapping wing dynamics for Mars exploration, Aerosp. Sci. Technol., 106 (2020), 106131. https://doi.org/10.1016/j.ast.2020.106131 doi: 10.1016/j.ast.2020.106131

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