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Backstepping-based supertwisting sliding mode attitude control for a quadrotor aircraft subjected to wind disturbances: experimental validation

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Abstract

Quadrotor unmanned aerial vehicles are increasingly explored due to their potential applications in many daily activities, including surveillance, cinematography, product delivery, and so on. Designing a flight control system for such a vehicle is very challenging because of its nonlinear dynamics and its sensitivity to the influence of disturbances. To meet part of this challenge, here a robust backstepping supertwisting sliding mode control (BSTSMC) has been developed to achieve a good tracking of the targeted trajectories. The proposed flight control system can drive the quadrotor's attitude to their reference values rapidly and, at the same time, reduce the impact of external perturbations and model inaccuracies. Experimental tests on an X450 quadrotor were performed to prove the practicality and disturbance rejection capability of the suggested control scheme.

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Data availability

Data will be available on reasonable request after the publication of this paper.

Abbreviations

\(\mathfrak{I}=\left(\boldsymbol{\Phi },\boldsymbol{\Theta },\boldsymbol{\Psi }\right)\) :

Quadrotor attitude

\(\dot{\mathfrak{I}}=(\dot{{\varvec{\Phi}}},\dot{{\varvec{\Theta}}},\dot{{\varvec{\Psi}}})\) :

Attitude rate

\(\dot{{\varvec{\xi}}}=(\dot{{\varvec{x}}},\dot{{\varvec{y}}},\dot{{\varvec{z}}})\) :

Linear velocities

\({\varvec{E}}=({{\varvec{O}}}_{{\varvec{E}}},{{\varvec{X}}}_{{\varvec{E}}},{{\varvec{Y}}}_{{\varvec{E}}},{{\varvec{Z}}}_{{\varvec{E}}})\) :

Earth-frame

\({{\varvec{J}}}_{{\varvec{x}}},{{\varvec{J}}}_{{\varvec{y}}},{{\varvec{J}}}_{{\varvec{z}}}\) :

Moments of inertia

\({\varvec{m}}\) :

Quadrotor mass

\({\varvec{g}}\) :

Gravity acceleration

\({{\varvec{J}}}_{{\varvec{r}}}\) :

Rotor inertia

\(\boldsymbol{\mho }={{\varvec{\omega}}}_{1}-{{\varvec{\omega}}}_{2}+{{\varvec{\omega}}}_{3}-{{\varvec{\omega}}}_{4}\) :

Residual angular velocity of motor

\({{\varvec{\omega}}}_{{\varvec{i}}}\) :

Angular velocity of rotor \({\varvec{i}}{\varvec{\epsilon}}\left\{1,2,3,4\right\}\).

\({{\varvec{U}}}_{1}\) :

Lift force

\({{\varvec{\tau}}}_{\boldsymbol{\Phi }}\) :

Roll torque

\({{\varvec{\tau}}}_{\boldsymbol{\Theta }}\) :

Pitch torque

\({{\varvec{\tau}}}_{\boldsymbol{\Psi }}\) :

Yaw torque

\({{\varvec{K}}}_{\mathbf{\hslash }}\) :

Drag coefficients

\(\ddot{\mathfrak{I}}=(\ddot{{\varvec{\Phi}}},\ddot{{\varvec{\Theta}}},\ddot{{\varvec{\Psi}}})\) :

Angular accelerations

\({\varvec{\xi}}=({\varvec{x}},{\varvec{y}},{\varvec{z}})\) :

Cartesian positions

\(\ddot{{\varvec{\xi}}}=(\ddot{{\varvec{x}}},\ddot{{\varvec{y}}},\ddot{{\varvec{z}}})\) :

Linear accelerations

\({\varvec{B}}=({{\varvec{O}}}_{{\varvec{B}}},{{\varvec{X}}}_{{\varvec{B}}},{{\varvec{Y}}}_{{\varvec{B}}},{{\varvec{Z}}}_{{\varvec{B}}})\) :

Body-frame

\({{\varvec{d}}}_{\boldsymbol{\Phi }}^{{\varvec{I}}{\varvec{u}}{\varvec{n}}},{{\varvec{d}}}_{\boldsymbol{\Theta }}^{{\varvec{I}}{\varvec{u}}{\varvec{n}}},{{\varvec{d}}}_{\boldsymbol{\Psi }}^{{\varvec{I}}{\varvec{u}}{\varvec{n}}}\) :

Internal unmodeled dynamics

\({{\varvec{d}}}_{\boldsymbol{\Phi }},{{\varvec{d}}}_{\boldsymbol{\Theta }},{{\varvec{d}}}_{\boldsymbol{\Psi }}\) :

Exogenous perturbations

\({\Delta }_{\boldsymbol{\Phi }},\boldsymbol{ }{\Delta }_{\boldsymbol{\Theta }},\boldsymbol{ }{\Delta }_{{\varvec{\Psi}}}\) :

Lumped disturbances

\({\mathbf{X}}_{1\mathbf{d}}=\left({\boldsymbol{\Phi }}_{{\varvec{d}}},{\boldsymbol{\Theta }}_{{\varvec{d}}},{\boldsymbol{\Psi }}_{{\varvec{d}}}\right)\) :

Desired attitude angles

\({\mathbf{X}}_{1}=\left(\boldsymbol{\Phi },\boldsymbol{\Theta },\boldsymbol{\Psi }\right)\) :

Current attitude angles

\({{\varvec{e}}}_{1}\) :

Attitude tracking error

\({{\varvec{V}}}_{1},{{\varvec{V}}}_{2}\) :

Lyapunov functions

\({{\varvec{k}}}_{1},{{\varvec{k}}}_{2}\) :

Positive parameters for the backstepping control design

\({{\varvec{\Upsilon}}}_{1},{{\varvec{\Upsilon}}}_{2}\) :

Positive parameters for the supertwisting algorithm design

\({\varvec{s}}\) :

Sliding surface

\({\varvec{s}}{\varvec{i}}{\varvec{g}}{\varvec{n}}\left(\boldsymbol{*}\right)\) :

Standard signum function

References

  1. Islam S, Liu PX, El Saddik A (2014) Robust control of four-rotor unmanned aerial vehicle with disturbance uncertainty. IEEE Trans Industr Electron 62:1563–1571

    Article  Google Scholar 

  2. Mofid O, Mobayen S (2018) Adaptive sliding mode control for finite-time stability of quad-rotor UAVs with parametric uncertainties. ISA Trans 72:1–14

    Article  Google Scholar 

  3. Liu H, Xi J, Zhong Y (2017) Robust attitude stabilization for nonlinear quadrotor systems with uncertainties and delays. IEEE Trans Industr Electron 64:5585–5594

    Article  Google Scholar 

  4. Hassani H, Mansouri A, Ahaitouf A (2020) Mechanical modeling, control and simulation of a quadrotor UAV. In: International conference on electronic engineering and renewable energy. Springer, pp 441–449

  5. Tilki U, Erüst AC (2021) Robust adaptive backstepping global fast dynamic terminal sliding mode controller design for quadrotors. J Intell Rob Syst 103:1–12

    Article  Google Scholar 

  6. Garcia PC, Lozano R, Dzul AE (2006) Modelling and control of mini-flying machines, Springer

  7. Naidoo Y, Stopforth R, Bright G (2011) Quad-rotor unmanned aerial vehicle helicopter modelling and control. Int J Adv Rob Syst 8:45

    Article  Google Scholar 

  8. Carrillo LRG, López AED, Lozano R, Pégard C (2013) Modeling the quad-rotor mini-rotorcraft. In: Quad rotorcraft control, Springer, pp 23–34

  9. Zhang X, Li X, Wang K, Lu Y (2014) A survey of modelling and identification of quadrotor robot. In: Abstract and applied analysis, Hindawi

  10. Bolandi H, Rezaei M, Mohsenipour R, Nemati H, Smailzadeh SM (2013) Attitude control of a quadrotor with optimized PID controller

  11. Hassani H, Mansouri A, Ahaitouf A (2022) Modeling and trajectory tracking of an unmanned quadrotor using optimal PID controller. WITS 2020, Springer, pp 457–467

  12. Liu C, Pan J, Chang Y (2016) PID and LQR trajectory tracking control for an unmanned quadrotor helicopter: experimental studies. In: 2016 35th Chinese control conference (CCC). pp 10845–10850, IEEE

  13. Das A, Lewis F, Subbarao K (2009) Backstepping approach for controlling a quadrotor using lagrange form dynamics. J Intell Rob Syst 56:127–151

    Article  MATH  Google Scholar 

  14. Zhou L, Zhang J, She H, Jin H (2019) Quadrotor UAV flight control via a novel saturation integral backstepping controller. Automatika 60:193–206

    Article  Google Scholar 

  15. Madani T, Benallegue A (2006) Backstepping control for a quadrotor helicopter. In: 2006 IEEE/RSJ international conference on intelligent robots and systems, pp 3255–3260, IEEE

  16. Hassani H, Mansouri A, Ahaitouf A (2019) Control system of a quadrotor UAV with an optimized backstepping controller. In: 2019 international conference on intelligent systems and advanced computing sciences (ISACS), pp 1–7

  17. Mohd Basri MA, Danapalasingam KA, Husain AR (2014) Design and optimization of backstepping controller for an underactuated autonomous quadrotor unmanned aerial vehicle. Trans FAMENA 38:27–44

    Google Scholar 

  18. Basri MAM, Husain AR, Danapalasingam KA (2015) Enhanced backstepping controller design with application to autonomous quadrotor unmanned aerial vehicle. J Intell Rob Syst 79:295–321

    Article  Google Scholar 

  19. Basri MAM (2018) Design and application of an adaptive backstepping sliding mode controller for a six-DOF quadrotor aerial robot. Robotica 36:1701–1727

    Article  Google Scholar 

  20. Xuan-Mung N, Hong SK (2019) Robust backstepping trajectory tracking control of a quadrotor with input saturation via extended state observer. Appl Sci 9:5184

    Article  Google Scholar 

  21. Jia Z, Yu J, Mei Y, Chen Y, Shen Y, Ai X (2017) Integral backstepping sliding mode control for quadrotor helicopter under external uncertain disturbances. Aerosp Sci Technol 68:299–307

    Article  Google Scholar 

  22. Li Z, Ma X, Li Y (2020) Robust trajectory tracking control for a quadrotor subject to disturbances and model uncertainties. Int J Syst Sci 51:839–851

    Article  MathSciNet  MATH  Google Scholar 

  23. Xiong J-J, Zheng E-H (2014) Position and attitude tracking control for a quadrotor UAV. ISA Trans 53:725–731

    Article  Google Scholar 

  24. Al-Dujaili AQ, Falah A, Humaidi AJ, Pereira DA, Ibraheem IK (2020) Optimal super-twisting sliding mode control design of robot manipulator: design and comparison study. Int J Adv Rob Syst 17:1729881420981524

    Google Scholar 

  25. Adil HMM, Ahmed S, Ahmad I (2020) Control of MagLev system using supertwisting and integral backstepping sliding mode algorithm. IEEE Access 8:51352–51362

    Article  Google Scholar 

  26. Jayakrishnan HJ (2016) Position and attitude control of a quadrotor UAV using super twisting sliding mode. IFAC-PapersOnLine 49:284–289

    Article  MathSciNet  Google Scholar 

  27. Labbadi M, Cherkaoui M (2020) Novel robust super twisting integral sliding mode controller for a quadrotor under external disturbances. Int J Dyn Control 8:805–815

    Article  MathSciNet  Google Scholar 

  28. Kahouadji M, Mokhtari MR, Choukchou-Braham A, Cherki B (2020) Real-time attitude control of 3 DOF quadrotor UAV using modified super twisting algorithm. J Franklin Inst 357:2681–2695

    Article  MathSciNet  MATH  Google Scholar 

  29. Muñoz F, González-Hernández I, Salazar S, Espinoza ES, Lozano R (2017) Second order sliding mode controllers for altitude control of a quadrotor UAS: real-time implementation in outdoor environments. Neurocomputing 233:61–71

    Article  Google Scholar 

  30. Hassani H, Mansouri A, Ahaitouf A (2022) Processor in the loop experiments of an adaptive trajectory tracking control for quadrotor UAVs

  31. Noordin A, Mohd Basri MA, Mohamed Z, Mat Lazim I (2021) Adaptive PID controller using sliding mode control approaches for quadrotor UAV attitude and position stabilization. Arab J Sci Eng 46:963–981

    Article  Google Scholar 

  32. Mofid O, Mobayen S, Wong W-K (2020) Adaptive terminal sliding mode control for attitude and position tracking control of quadrotor UAVs in the existence of external disturbance. IEEE Access 9:3428–3440

    Article  Google Scholar 

  33. Hassani H, Mansouri A, Ahaitouf A (2020) A new robust adaptive sliding mode controller for quadrotor UAV flight. In: 2020 IEEE 2nd international conference on electronics, control, optimization and computer science (ICECOCS), pp 1–6

  34. Hassani H, Mansouri A, Ahaitouf A (2022) Adaptive fast terminal sliding mode control for uncertain quadrotor based on butterfly optimization algorithm (BOA). WITS 2020, Springer, pp 353–364

  35. Mechali O, Xu L, Huang Y, Shi M, Xie X (2021) Observer-based fixed-time continuous nonsingular terminal sliding mode control of quadrotor aircraft under uncertainties and disturbances for robust trajectory tracking: theory and experiment. Control Eng Pract 111:104806

    Article  Google Scholar 

  36. Eliker K, Grouni S, Tadjine M, Zhang W (2021) Quadcopter nonsingular finite-time adaptive robust saturated command-filtered control system under the presence of uncertainties and input saturation. Nonlinear Dyn 104:1363–1387

    Article  Google Scholar 

  37. Hassani H, Mansouri A, Ahaitouf A (2021) Robust autonomous flight for quadrotor UAV based on adaptive nonsingular fast terminal sliding mode control. Int J Dyn Control 9:619–635

  38. Moreno JA, Osorio M (2012) Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans Autom Control 57:1035–1040

    Article  MathSciNet  MATH  Google Scholar 

  39. Almakhles DJ (2019) Robust backstepping sliding mode control for a quadrotor trajectory tracking application. IEEE Access 8:5515–5525

    Article  Google Scholar 

  40. Hassani H, Mansouri A, Ahaitouf A (2022) Robust finite-time tracking control based on disturbance observer for an uncertain quadrotor under external disturbances. J Robotics

  41. Hassani H, Mansouri A, Ahaitouf A (2022) Robust hybrid controller for quadrotor UAV under disturbances. Int J Model Ident Control 40:195–203

  42. Quan Q (2017) Modeling and evaluation of propulsion system. In: Introduction to multicopter design and control. Springer, pp 73–95

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Funding

The authors did not receive support from any organization for the submitted work.

Author information

Authors and Affiliations

  1. Laboratory of Intelligent Systems, Geo-Resources and Renewable Energies Faculty of Sciences and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco

    Hamid Hassani & Ali Ahaitouf

  2. Laboratory of Intelligent Systems, Geo-Resources and Renewable Energies, School of Applied Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco

    Anass Mansouri

Authors
  1. Hamid Hassani
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  2. Anass Mansouri
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  3. Ali Ahaitouf
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Contributions

HH contributed to the conceptualization, analysis and interpretation of data, software, validation, writing—original draft, and approval of the version of the manuscript to be published. AM was involved in the supervision, analysis and interpretation of data, review and editing, and approval of the version of the manuscript to be published. AA helped in the supervision, analysis and interpretation of data, review and editing, and approval of the version of the manuscript to be published.

Corresponding author

Correspondence to Hamid Hassani.

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The authors declare that they have no conflict of interest.

Appendix

Appendix

The control torque based on the backstepping sliding mode control [19]:

$$\tau = \frac{1}{g\left( X \right)}\left( {e_{1} + k_{1} \dot{e}_{1} + \ddot{X}_{1d} - f\left( X \right) - k_{2} s - \varepsilon sign\left( s \right)} \right)$$
(24)

with \(s = X_{2} - \dot{X}_{{1{\rm{d}}}} - k_{1} e_{1}\) being the sliding surface.

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Hassani, H., Mansouri, A. & Ahaitouf, A. Backstepping-based supertwisting sliding mode attitude control for a quadrotor aircraft subjected to wind disturbances: experimental validation. Int. J. Dynam. Control 11, 1285–1296 (2023). https://doi.org/10.1007/s40435-022-01004-5

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