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Switching Wireless Control for Longitudinal Quadrotor Maneuvers

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Abstract

To improve the maneuverability of remotely guided quadrotors in longitudinal motion, the design goal of independent control of the horizontal and the vertical velocity, is satisfied. For the sake of the communication delays, a synchronization / signal reconstruction algorithm, imposing constant delays, is applied and the design goal is satisfied by introducing a stepwise safe switching procedure for time delay dynamic controllers. The proposed controllers are designed to satisfy I/O decoupling, stability and atmospheric disturbance attenuation. The controllers are determined via a mixed analytic and metaheuristic design procedure. To increase the range of the external commands, the stepwise safe switching algorithm is based on the approximate minimization of a composite criterion including the infinity norm and the H2 norm of the variations around trim points under hard constraints for the steady state performance and the overshoot of the performance variables. The satisfactory performance of the proposed control scheme is illustrated through simulations for a climbing maneuver passing through different target operating areas.

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Availability of data and materials

All authors state that all data and materials as well as software application or custom code support their published claims and comply with field standards. The present paper is an enhanced version the paper entitled “Wireless Longitudinal Motion Controller Design for Quadrotors” that has been presented in ICUAS 2020 – International Conference on Unmanned Aircraft Systems that was held in Athens (Greece) from 1 September 2020 until 4 September 2020.

Abbreviations

Symbol:

Description

𝜃, q :

Pitch angle and pitch rate of the quadrotor

v x, v z :

Horizontal and vertical velocities of the quadrotor

ω f, ω r :

Front and rear motor velocities

V f, V r :

Front and rear motor voltage supplies (actuatable inputs)

v a, x, v a, z :

Horizontal and vertical velocities of the ambient air (non-measurable disturbances)

m q, J q, y :

Quadrotor mass and moment of inertia

k dt, x, k dt, z :

Translation drag coefficients in the x and z directions of the earth reference frame

k a, y :

Rotational aerodynamic friction coefficient around the y axis

k l, g, d :

Lift coefficient of the propellers, gravitational acceleration and distance between the quadrotor center of mass and the rotation axis of the propellers

a 0, a 1, a 2, b :

Electric motor parameters

x, u, ξ, y :

State, input, disturbance and performance output vectors

x i, u j, ξ j, y j :

Elements of x, u, ξ, y (i = 1,...,6; j = 1,2)

\(\bar {x}, \bar {u}, \bar {\xi }, \bar {y}\) :

Nominal points of x, u, ξ, y

\(\bar {x}_{i}, \bar {u}_{j}, \bar {\xi }_{j}, \bar {y}_{j}\) :

Elements of \(\bar {x}\), \(\bar {u}\), \(\bar {\xi }\) and \(\bar {y}\) (i = 1,...,6, j = 1,2)

\(\bar {x}_{2}^{*}, \bar {x}_{3}^{*}\) :

Horizontal and vertical quadrotor nominal velocities

Δ x, Δ u, Δ ξ, Δ y :

Deviations around \(\bar {x}\), \(\bar {u}\), \(\bar {\xi }\), \(\bar {y}\)

δ x, δ y :

State and performance output vectors of the linear approximant

δ ω, δ ω j :

Vector of external inputs and its elements (j = 1,2)

A, B, Q;C :

Linear approximant system matrices; Performance output matrix

τ 1, τ 2 :

Transmission / reconstruction delays of the control commands at the actuatable inputs and the measurable state variables at the feedback term of the controller

K, G;k 1, k 2 :

Feedback and precompensator matrices; 1st and 2nd rows of K

H Ω, H Ξ :

Closed loop transfer matrices mapping the external inputs and the disturbances to the performance variables

h j :

Diagonal elements of the decoupled closed loop transfer matrix (j = 1,2)

T i, j, σ, δ σ :

Positive real parameters of the closed loop system (i = 1,...,5; j = 1,2)

J d, e d, max :

Disturbance attenuation cost and threshold

\({\left (\cdot \right )_{c}}, {\left (\cdot \right )_{w}}\) :

Center value and half width of the argument parameter

n loop, n rep, λ :

Metaheuristic algorithm parameters

\(\bar {\chi }\) :

Nominal point vector including nominal inputs and outputs

\({\mathscr{S}}_{L}(\bar {\chi })\) :

Open loop linear approximant evaluated at \(\bar {\chi }\)

\({{\mathscr{C}}}(\bar {\chi })\) :

Controller evaluated at \(\bar {\chi }\)

\({{\mathscr{S}}}_{L,CL}(\bar {\chi })\) :

Closed loop system resulting after application of \({{\mathscr{C}}}(\bar {\chi })\) to \({{\mathscr{S}}_{L}}(\bar {\chi })\)

\({{\mathscr{S}}}_{NL,CL}(\bar {\chi })\) :

Closed loop system resulting after application of \({{\mathscr{C}}}(\bar {\chi })\) to the nonlinear model

\({J_{\infty } }, {J_{2}}, {J_{1}}, {J_{o}},\tilde J\) :

Performance metrics

J T, γ :

External control input area metric and threshold

ε 1, ε o, ε, μ :

Bounds of the performance metrics

\(\varepsilon _{\max \limits }, \varepsilon _{s}\) :

Positive numbers to define the settling zone and convergence around steady state value

\({\mathbb X}\) :

Finite set of nominal operating point vectors

\({\mathbb T}(\bar {\chi }_{j} )\) :

Target operating area around the nominal point vector \(\bar {\chi }_{j} \in {\mathbb {X}}\)

\(\tilde e, e_{1}, e_{o}\) :

Thresholds of precision in the target operating area

Λ j :

Set of adjacent nominal trim point vectors

ρ, ρ s, ρ f, ρ ψ :

Trim point vectors (ψ = 1,...,σ)

n T, N j :

Total number of transitions for estimation the target area through uniform gridded search and number of samples per coordinate

T a, r h, p a :

Air temperature, relative humidity and atmospheric pressure

ρ a, ρ a,0 :

Air density and its nominal value

\({\hat k}_{dt,x}, {\hat k}_{dt,z}, {\hat k}_{a,y}, {\hat k}_{l}\) :

Uncertain aerodynamic parameters

References

  1. Zhang, D., Qi, H., Wu, X., Xie, Y., Xu, J.: The quadrotor dynamic modeling and indoor target tracking control method. Mathematical problems in engineering (2014)

  2. Magalhaes, G.M., Santos, E.G., Borges, L.M., Cunha, A.E.C.: Comparison and Implementation of Control Strategies for a Quadrotor. In: XIII Simpósio Brasileiro de Automação Inteligente, pp. 133–138 (2017)

  3. Garcia, R.A., Rubio, F.R., Ortega, M.G.: Robust PID control of the quadrotor helicopter. IFAC Proceedings Volumes 45(3), 229–234 (2012)

    Article  Google Scholar 

  4. Moreno-Valenzuela, J., Perez-Alcocer, R., Guerrero-Medina, M., Dzul, A.: Nonlinear PID-type controller for quadrotor trajectory tracking. IEEE/ASME Trans. Mechatron. 23(5), 2436–2447 (2018)

    Article  Google Scholar 

  5. Chehadeh, M.S., Boiko, I.: Design of rules for in-flight non-parametric tuning of PID controllers for unmanned aerial vehicles. Journal of the Franklin Institute 356(474–491) (2019)

  6. Outeiro, P., Cardeira, C., Oliveira, P.: Lqr/mmae height control system of a quadrotor for constant unknown load transportation. In: Proceedings of the 13th APCA International Conference on Control and Soft Computing (CONTROLO), pp 389–394 (2018)

  7. Safaei, A., Mahyuddin, M.N.: Optimal model-free control for a generic MIMO nonlinear system with application to autonomous mobile robots. Int. J. Adapt. Control 3, 792–815 (2018)

    Article  MathSciNet  Google Scholar 

  8. Ozturk, O., Ozkan, H.M.: Optimal control of quadrotor unmanned aerial vehicles on time scales. Int. J. Diff Eq 13, 41–54 (2018)

    MathSciNet  Google Scholar 

  9. Sabatino, F.: Quadrotor control modeling, nonlinear control design, and simulation. Master’s Degree Project, KTH Electrical Engineering (2015)

  10. Arrosida, H., Effendi, R., Agustinah, T., Pramudijanto, J.: Design of decoupling and nonlinear PD Controller for Cruise Control of a Quadrotor. In: International Seminar on Intelligent Technology and its Applications (ISITIA) (57–61) (2015)

  11. Leonardo, S., Zaira, P., Duque, M.: Nonlinear control of the airship cruise flight phase with dynamical decoupling. In: Electronics, Robotics and Autonomous Mechanics Conference, pp. 473–477 (2008)

  12. Neto, G.G., dos Santos Barbosa, F., de Oliveira, Jr, J.G., Angelico, B. A.: State feedback decoupling control of a 2DoF Helicopter. In: XIII Simpósio Brasileiro de Automação Inteligente, pp. 398–403 (2017)

  13. Kouvakas, N.D., Koumboulis, F.N.: Independent velocity control of the longitudinal motion of quadrotors. In: Proceedings of the 7th International Conference on Systems and Control, pp. 194–200 (2018)

  14. Cai, W., She, J., Wu, M., Ohyama, Y.: Disturbance suppression for quadrotor UAV using sliding-mode-observer-based equivalent-input-disturbance approach. ISA Transactions (2019)

  15. Wang, X., Sun, S., van Kampen, E.-J., Chu, Q.: Quadrotor fault tolerant incremental sliding mode control driven by sliding mode disturbance observers. Aerosp. Sci. Technol. 87, 417–430 (2019)

    Article  Google Scholar 

  16. Wang, B., Yu, X., Mu, L., Zhang, Y.: Disturbance observer-based adaptive fault-tolerant control for a quadrotor helicopter subject to parametric uncertainties and external disturbances. Mech. Syst. Signal Process. 120, 727–743 (2019)

    Article  Google Scholar 

  17. Rios, H., Falcon, R., Gonzalez, O.A., Dzul, A.: Continuous sliding-mode control strategies for quadrotor robust tracking: real-time application. IEEE Trans. Ind. Electron. 66(2), 1264–1272 (2019)

    Article  Google Scholar 

  18. Chen, M., Xiong, S., Wu, Q.: Tracking flight control of quadrotor based on disturbance observer. IEEE Transactions on Systems, Man, and Cybernetics, Systems (2019)

  19. Qian, L., Liu, H.H.T.: Path following control of a quadrotor UAV with a cable suspended payload under wind disturbances. IEEE Trans. Ind. Electron. 67(3), 2021–2029 (2020)

    Article  Google Scholar 

  20. Yuan, Y., Cheng, L., Wang, Z., Sun, C.: Position tracking and attitude control for quadrotors via active disturbance rejection control method. Science China: Information Sciences, pp. 62 (2018)

  21. Baldini, A., Felicetti, R., Freddi, A., Longhi, S., Monteriu, A.: Direct position control of an octarotor unmanned vehicle under wind gust disturbance. In: Proceedings of the 2019 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 684–691 (2019)

  22. Gośliński, J., Kasiński, A., Giernacki, W., Owczarek, P., Gardecki, S.: A study on coaxial quadrotor model parameter estimation: an application of the improved square root unscented Kalman filter. J. Intell. Robot. Syst. 95, 491–510 (2019)

    Article  Google Scholar 

  23. Shraim, H., Awada, A., Youness, R.: Survey on quadrotors: configurations, modeling and identification, control, collision avoidance, fault diagnosis and tolerant control. IEEE Aerosp. Electron. Syst. Mag. 33 (7), 14–33 (2018)

    Article  Google Scholar 

  24. Xiong, S., Chen, M., Wu, Q.: Predictive control for networked switch flight system with packet dropout. Appl. Math. Comput. 354, 444–459 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Gonzaleza, A., Cuencab, A., Balaguerb, V., Garcia, P.: Event-triggered predictor-based control with gain-Scheduling and extended state observer for networked control systems. Inform. Sci. 491, 90–108 (2019)

    Article  MathSciNet  Google Scholar 

  26. He, L., Zhang, J., Hou, Y., Liang, X., Bai, P.: Time-varying formation control for second-order discrete-time multi-agent systems with directed topology and communication delay. IEEE Access 7, 33517–3352 (2019)

    Article  Google Scholar 

  27. Guerrero, J.A., Garcia, P.C., Challal, Y.: Quadrotors formation control: a wireless medium access aware approach. J. Intell. Robot. Syst. 70, 221–231 (2013)

    Article  Google Scholar 

  28. Koumboulis, F.N., Tzierakis, K.G.: Meeting transfer function requirements via static measurement output feedback, vol. 335B, pp 661–667 (1998)

  29. Kouvakas, N.D., Koumboulis, F.N., Giannaris, G.L., Vouyioukas, D.: Wireless longitudinal motion controller design for quadrotors. In: Proceedings of the 2020 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 1350–1358 (2020)

  30. Koumboulis, F.N., King, R.E., Stathaki, A.: Logic-based switching controllers – a stepwise safe switching approach. Inform. Sci. 177(13), 2736–2755 (2007)

    Article  Google Scholar 

  31. Koumboulis, F.N., Tzamtzi, M.P.: Multivariable step-wise safe switching controllers. In: Proceedings of the International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC’05) (2005)

  32. Koumboulis, F.N., Kouvakas, N.D., Giannaris, G.L., Vouyioukas, D.: Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Trans. 60, 312–320 (2016)

    Article  Google Scholar 

  33. Rosario-Gabriel, I., Cortes, H.R.: Aircraft longitudinal control based on the Lanchester’s Phugoid dynamics model. In: Proceedings of the 2018 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 924–929 (2018)

  34. Dhadekar, D.D., Talole, S.E.: Robust fault tolerant longitudinal aircraft control. IFAC-PapersOnLine 51(1), 604–609 (2018)

    Article  Google Scholar 

  35. Gossmann, F., Svaricek, F., Gabrys, A.: Control of longitudinal aircraft motion with loadcase robustness using LPV-control with partly-measurable parameters. In: Proceedings of the 2018 AIAA Guidance, Navigation, and Control Conference, pp. 1–13 (2018)

  36. Koumboulis, F.N., Kouvakas, N.D.: I/O Decoupling with simultaneous disturbance rejection of general neutral time delay systems via a measurement output feedback dynamic controller. In: Proceedings of the 21st Mediterranean Conference on Control and Automation (MED), pp. 890–895 (2013)

  37. Giannaris, G.L., Kouvakas, N.D., Koumboulis, F.N., Vouyioukas, D.: Towards remote control of planar redundant robotic manipulators. In: Proceedings of the IEEE 21st International Conference on Intelligent Engineering Systems (INES 2017), pp. 231–236 (2017)

  38. Koumboulis, F.N., Kouvakas, N.D.: A three term controller for ride comfort improvement. In: Proceedings of the 19th Mediterranean Conference on Control & Automation (MED), pp. 114–119 (2011)

  39. Derafa, L., Madani, T., Benallegue, A.: Dynamic modelling and experimental identification of four rotors helicopter parameters. In: Proceedings of the IEEE International Conference on Industrial Technology (ICIT) (2006)

  40. Hakim, T.M.I., Arifianto, O.: Implementation of dryden continuous turbulence model into simulink for LSA-02 flight test simulation proceedings of the 5th international seminar of aerospace science and technology (2017)

  41. Cervin, A., Henriksson, D., Lincoln, B., Eker, J., Årzén, K.-E.: How does control timing affect performance? analysis and simulation of timing using Jitterbug and TrueTime. IEEE Control. Syst. Mag. 23(3), 16–30 (2003)

    Article  Google Scholar 

  42. Kaya, D., Kutay, A.T.: Aerodynamic modeling and parameter estimation of a quadrotor helicopter. In: Proceedings of the AIAA Atmospheric Flight Mechanics Conference (2014)

  43. Davis, E.D.: Aerodynamic force interactions and measurements for micro quadrotors, PhD Thesis. School of Information Technology and Electrical Engineering The University of Queensland (2018)

  44. Picard, A., Davis, R.S., Gäaser, M., Fujii, K.: Revised formula for the density of moist air (CIPM-2007). Metrologia 45, 149–155 (2008)

    Article  Google Scholar 

  45. Koumboulis, F.N., Kouvakas, N.D.: Block decoupling of general neutral multi delay systems. Proceedings of the International Conference on Emerging Technologies and Factory Automation (ETFA) (2011)

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Funding

No funding was received to assist with the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Information and Communication Systems Engineering, University of the Aegean, Karlovassi, Samos, 83200, Greece

    George L. Giannaris & Demosthenes Vouyioukas

  2. Robotics and Automation Laboratory, National and Kapodistrian University of Athens, Euripus Campus, Psahna Evias, 34400, Greece

    Nikolaos D. Kouvakas & Fotis N. Koumboulis

Authors
  1. George L. Giannaris
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  2. Nikolaos D. Kouvakas
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  3. Fotis N. Koumboulis
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  4. Demosthenes Vouyioukas
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Contributions

All authors (G. L. Giannaris, N. D. Kouvakas, F. N. Koumboulis and D. Vouyioukas) contributed to the study conception and design as well as material preparation, data collection, analysis and manuscript preparation (first draft or final). All authors read and approved the final manuscript.

Corresponding author

Correspondence to Fotis N. Koumboulis.

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Appendix

Appendix

$$ \begin{array}{@{}rcl@{}}{\tilde{a}_{1,1}} &=& {a_{4,5}}{b_{5,1}}\left( {{a_{2,2}}\left\{ {{a_{3,3}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right] - 2{a_{3,5}}{b_{5,1}}{k_{2,3}}} \right\} - 2{a_{2,5}}{a_{3,3}}{b_{5,1}}{k_{2,2}}} \right)\\{{\tilde{a}}_{1,2}} &=& - {a_{4,5}}{b_{5,1}}\left[ {2{a_{2,5}}{b_{5,1}}\left( {{a_{3,1}}{k_{2,3}} - {a_{3,3}}{k_{2,1}}} \right) + {a_{2,1}}\left\{ {{a_{3,3}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right] } \right.} \right.\\ &&\left. {\left. { - 2{a_{3,5}}{b_{5,1}}{k_{2,3}}} \right\}} \right]\\{{\tilde{a}}_{1,3}} &=& - {a_{4,5}}{b_{5,1}}\left[ {2\left( {{a_{2,1}}{a_{3,5}} - {a_{2,5}}{a_{3,1}}} \right){b_{5,1}}{k_{2,2}} + {a_{2,2}}\left\{ {{a_{3,1}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]} \right.} \right.\\&&\left. {\left. { - 2{a_{3,5}}{b_{5,1}}{k_{2,1}}} \right\}} \right], {\tilde{a}_{1,5}} = {a_{4,5}}b_{5,1}^2\left[ {{a_{2,1}}{a_{3,3}}{k_{2,2}} + {a_{2,2}}\left( {{a_{3,1}}{k_{2,3}} - {a_{3,3}}{k_{2,1}}} \right)} \right]\\{\tilde{a}_{1,6}} &=& {a_{4,5}}b_{5,1}^2\left[ {{a_{2,1}}{a_{3,3}}{k_{2,2}} + {a_{2,2}}\left( {{a_{3,1}}{k_{2,3}} - {a_{3,3}}{k_{2,1}}} \right)} \right]\\{{\tilde{a}}_{2,1}} &=& - {a_{4,5}}{b_{5,1}}\left\{ { - 2{b_{5,1}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}}} \right) + {a_{3,3}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]} \right.\\ &&\left. { + {a_{2,2}}\left[ {{a_{3,3}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]} \right\}\\{{\tilde{a}}_{2,2}} &=& {b_{5,1}}\left[ {{a_{2,1}}{a_{4,5}}\left[ {{a_{3,3}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right] + } \right.{a_{2,5}}\left( {{a_{3,3}}\left\{ {2{a_{4,5}}{b_{5,1}}{k_{2,4}} } \right.} \right.\\&&\left. {\left. {\left. { + {a_{4,4}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right\} - 2{a_{4,5}}{b_{5,1}}{k_{2,1}}} \right)} \right]\\{{\tilde{a}}_{2,3}} &=& {b_{5,1}}\left[ {{a_{4,5}}\left\{ {{a_{3,1}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right] - 2{a_{3,5}}{b_{5,1}}{k_{2,1}}} \right\}} \right.\\&&\left. { + {a_{2,2}}\left( {{a_{3,5}}\left\{ {2{a_{4,5}}{b_{5,1}}{k_{2,4}} + {a_{4,4}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right\} + {a_{3,1}}{a_{4,5}}} \right)} \right]\\{\tilde{a}_{2,4}} &=& {a_{4,5}}{b_{5,1}}\left( {{a_{2,2}}\left\{ {{a_{3,3}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right] - 2{a_{3,5}}{b_{5,1}}{k_{2,3}}} \right\} - 2{a_{2,5}}{a_{3,3}}{b_{5,1}}{k_{2,2}}} \right)\\{{\tilde{a}}_{2,5}} &=& - {b_{5,1}}\left( {{b_{5,1}}\left[ {{a_{4,5}}\left( {{a_{2,1}}{k_{2,2}} + {a_{3,1}}{k_{2,3}}} \right) - {a_{3,3}}\left( {{a_{4,5}}{k_{2,1}} + {a_{2,5}}{a_{4,4}}{k_{2,2}}} \right)} \right]} \right.\\&&\left. { + {a_{2,2}}\left\{ {{a_{3,3}}\left[ {{a_{4,5}}{b_{5,1}}{k_{2,4}} + {a_{4,4}}\left( {{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right] - {b_{5,1}}\left( {{a_{4,5}}{k_{2,1}} + {a_{3,5}}{a_{4,4}}{k_{2,3}}} \right)} \right\}} \right)\\{{\tilde{a}}_{2,6}} &=& b_{5,1}^2\left\{ {{a_{3,3}}\left( {{a_{4,5}}{k_{2,1}} - {a_{2,5}}{a_{4,4}}{k_{2,2}}} \right) - {a_{4,5}}\left( {{a_{2,1}}{k_{2,2}} + {a_{3,1}}{k_{2,3}}} \right)} \right.\\&&\left. { + {a_{2,2}}\left[ {{a_{4,5}}\left( {{k_{2,1}} - {a_{3,3}}{k_{2,4}}} \right) + {a_{4,4}}\left( {{a_{3,3}}{k_{2,5}} - {a_{3,5}}{k_{2,3}}} \right)} \right]} \right\}\\{\tilde{a}_{3,1}} &=& {a_{4,5}}{b_{5,1}}\left[ {{a_{2,2}} + {a_{3,3}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]\\{{\tilde{a}}_{3,2}} &=& {b_{5,1}}\left( { - {a_{2,5}}\left\{ {2{a_{4,5}}{b_{5,1}}{k_{2,4}} + {a_{4,4}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right.} \right.\\&&\left. {\left. { + {a_{3,3}}\left[ {{a_{4,4}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right\} - {a_{2,1}}{a_{4,5}}} \right)\\{{\tilde{a}}_{3,3}} &=& {b_{5,1}}\left( { - {a_{3,5}}\left\{ {2{a_{4,5}}{b_{5,1}}{k_{2,4}} + {a_{4,4}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right.} \right.\\&&\left. {\left. { + {a_{2,2}}\left[ {{a_{4,4}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]} \right\} - {a_{3,1}}{a_{4,5}}} \right)\\{{\tilde{a}}_{3,4}} &=& - {a_{4,5}}{b_{5,1}}\left\{ { - 2{b_{5,1}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}}} \right) + {a_{3,3}}\left[ {{a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]} \right.\\&&{ + {a_{2,2}}\left[ {{a_{3,3}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]}\\{\tilde{a}}_{3,5} &=& {b_{5,1}}\left\{ { - {b_{5,1}}\left[ {{a_{4,5}}{k_{2,1}} + {a_{4,4}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}}} \right)} \right] + {a_{3,3}}\left[ {{b_{5,1}}\left( {{a_{4,5}}{k_{2,4}}} \right.} \right.} \right.\\&&\left. {\left. { - {a_{2,5}}{k_{2,2}}} \right) + {a_{4,4}}\left( {{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right] + {a_{2,2}}\left[ {{b_{5,1}}\left( {{a_{4,5}}{k_{2,4}} - {a_{3,5}}{k_{2,3}}} \right)} \right.\\&&\left. {\left. { + {a_{4,4}}\left( {{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right) + {a_{3,3}}\left( {{a_{4,4}} + {a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right]} \right\}\\{{\tilde{a}}_{3,6}} &=& b_{5,1}^2\left\{ {{a_{2,5}}\left( {{a_{3,3}} + {a_{4,4}}} \right){k_{2,2}} + {a_{3,5}}\left( {{a_{2,2}} + {a_{4,4}}} \right){k_{2,3}}} \right.\\&&\left. { + {a_{4,5}}\left[ {\left( {{a_{2,2}} + {a_{3,3}}} \right){k_{2,4}} - {k_{2,1}}} \right] - \left[ {{a_{2,2}}{a_{3,3}} + \left( {{a_{2,2}} + {a_{3,3}}} \right){a_{4,4}}} \right]{k_{2,5}}} \right\}\\{\tilde{a}_{4,1}} &=& - {a_{4,5}}{b_{5,1}}, {\tilde{a}_{4,2}} = {a_{2,5}}{b_{5,1}}\left[ {{a_{3,3}} + {a_{4,4}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]\\{\tilde{a}_{4,3}} &=& {a_{3,5}}{b_{5,1}}\left[ {{a_{2,2}} + {a_{4,4}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,6}} - {k_{2,5}}} \right)} \right]\\{\tilde{a}_{4,4}} &=& {a_{4,5}}{b_{5,1}}\left[ {{a_{2,2}} + {a_{3,3}} + {a_{5,5}} + {b_{5,1}}\left( {{k_{2,5}} + {k_{2,6}}} \right)} \right]\\{{\tilde{a}}_{4,5}} &=& - {b_{5,1}}\left[ { - {b_{5,1}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}} - {a_{4,5}}{k_{2,4}}} \right) + {a_{4,4}}\left( {{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right.\\&&\left. { + {a_{3,3}}\left( {{a_{4,4}} + {a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right) + {a_{2,2}}\left( {{a_{3,3}} + {a_{4,4}} + {a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right]\\{\tilde{a}_{4,6}} &=& b_{5,1}^2\left[ {\left( {{a_{2,2}} + {a_{3,3}} + {a_{4,4}}} \right){k_{2,5}} - {a_{2,5}}{k_{2,2}} - {a_{3,5}}{k_{2,3}} - {a_{4,5}}{k_{2,4}}} \right]\\{\tilde{a}_{5,2}} &=&- {a_{2,5}}{b_{5,1}}, {\tilde{a}_{5,3}} = - {a_{3,5}}{b_{5,1}}, {\tilde{a}_{5,4}} = - {a_{4,5}}{b_{5,1}}\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} {\tilde{a}_{5,5}} &=& {b_{5,1}}\left( {{a_{2,2}} + {a_{3,3}} + {a_{4,4}} + {a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right), {\tilde{a}_{5,6}} = - b_{5,1}^2{k_{2,5}}, {\tilde{a}_{6,5}} = - {b_{5,1}}\\{\tilde b_{1,1}} &=& {a_{4,5}}{a_{5,5}}{b_{5,1}}\left[ {{a_{2,2}}\left( {{a_{3,3}}{k_{2,1}} - {a_{3,1}}{k_{2,3}}} \right) - {a_{2,1}}{a_{3,3}}{k_{2,2}}} \right]\\{{\tilde b}_{2,1}} &=& {b_{5,1}}\left\{ {{a_{4,5}}{a_{5,5}}\left( {{a_{2,1}}{k_{2,2}} + {a_{3,1}}{k_{2,3}}} \right) - {a_{3,3}}\left[ {{a_{2,5}}{a_{4,4}}{a_{5,5}}{k_{2,2}} + {a_{4,5}}\left( {{a_{5,5}}{k_{2,1}}} \right.} \right.} \right.\\&& \left. {\left. {\left. { - {a_{2,1}}{k_{2,2}}} \right)} \right]} \right\} + {a_{2,2}}\left\{ {{a_{3,3}}\left[ {{a_{4,5}}{b_{5,1}}\left( {{a_{5,5}}{k_{2,4}} - {k_{2,1}}} \right) + {a_{4,4}}{a_{5,5}}\left( {{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)} \right] } \right.\\&&\left. { - {b_{5,1}}\left[ {{a_{3,5}}{a_{4,4}}{a_{5,5}}{k_{2,3}} + {a_{4,5}}\left( {{a_{5,5}}{k_{2,1}} - {a_{3,1}}{k_{2,3}}} \right)} \right]} \right\}\\{{\tilde b}_{3,1}} &=& {b_{5,1}}\left[ {{a_{4,4}}{a_{5,5}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}}} \right) + {a_{4,5}}\left( {{a_{5,5}}{k_{2,1}} - {a_{2,1}}{k_{2,2}} - {a_{3,1}}{k_{2,3}}} \right)} \right] \\&& - {a_{3,3}}\left\{ {{a_{4,4}}\left( {{a_{5,5}}{b_{5,1}}{k_{2,6}} - {a_{2,5}}{b_{5,1}}{k_{2,2}} + a_{5,5}^2} \right) - {b_{5,1}}\left[ {{a_{2,5}}{a_{5,5}}{k_{2,2}} } \right.} \right.\\&&\left. {\left. { + {a_{4,5}}\left( {{k_{2,1}} - {a_{5,5}}{k_{2,4}}} \right)} \right]} \right\} + {a_{2,2}}\left\{ { - {b_{5,1}}\left[ {{a_{3,5}}{a_{5,5}}{k_{2,3}} + {a_{4,5}}\left( {{k_{2,1}} - {a_{5,5}}{k_{2,4}}} \right)} \right] } \right.\\&& + {a_{4,4}}\left( {{a_{5,5}}{b_{5,1}}{k_{2,6}} - {a_{3,5}}{b_{5,1}}{k_{2,3}} + a_{5,5}^2} \right) + \\&&+ \left. {{a_{3,3}}\left[ {{a_{4,5}}{b_{5,1}}{k_{2,4}} + \left( {{a_{4,4}} + {a_{5,5}}} \right){b_{5,1}}{k_{2,6}} + a_{5,5}^2 + 2{a_{4,4}}{a_{5,5}}} \right]} \right\}\\{{\tilde b}_{4,1}} &=& {a_{4,4}}a_{5,5}^2 - {a_{4,5}}{b_{5,1}}{k_{2,1}} - {a_{2,5}}{a_{4,4}}{b_{5,1}}{k_{2,2}} - {a_{2,5}}{a_{5,5}}{b_{5,1}}{k_{2,2}} \\&&- {a_{3,5}}{a_{4,4}}{b_{5,1}}{k_{2,3}} - {a_{3,5}}{a_{5,5}}{b_{5,1}}{k_{2,3}} + {a_{4,5}}{a_{5,5}}{b_{5,1}}{k_{2,4}} + {a_{4,4}}{a_{5,5}}{b_{5,1}}{k_{2,6}} \\&&+ {a_{3,3}}\left[ {{a_{5,5}}\left( {2{a_{4,4}} + {a_{5,5}}} \right) + {b_{5,1}}\left( {{a_{4,5}}{k_{2,4}} - {a_{2,5}}{k_{2,2}}} \right) + \left( {{a_{4,4}} + {a_{5,5}}} \right){b_{5,1}}{k_{2,6}}} \right] \\&&+{a_{2,2}}\left[ {{a_{3,3}}{a_{4,4}} + 2{a_{3,3}}{a_{5,5}} + 2{a_{4,4}}{a_{5,5}} + a_{5,5}^2 } \right.\\&&\left. { + {b_{5,1}}\left( {{a_{4,5}}{k_{2,4}} - {a_{3,5}}{k_{2,3}}} \right) + \left( {{a_{3,3}} + {a_{4,4}} + {a_{5,5}}} \right){b_{5,1}}{k_{2,6}}} \right]\\{{\tilde b}_{5,1}} &=& {b_{5,1}}\left( {{a_{2,5}}{k_{2,2}} + {a_{3,5}}{k_{2,3}} - {a_{4,5}}{k_{2,4}}} \right) - {a_{5,5}}\left( {2{a_{4,4}} + {a_{5,5}}} \right) - \left( {{a_{4,4}} + {a_{5,5}}} \right){b_{5,1}}{k_{2,6}}\\&& - {a_{3,3}}\left( {{a_{4,4}} + 2{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right) - {a_{2,2}}\left( {{a_{3,3}} + {a_{4,4}} + 2{a_{5,5}} + {b_{5,1}}{k_{2,6}}} \right)\\{\tilde b_{6,1}} &=& {a_{2,2}} + {a_{3,3}} + {a_{4,4}} + 2{a_{5,5}} + {b_{5,1}}{k_{2,6}} \end{array} $$

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Giannaris, G.L., Kouvakas, N.D., Koumboulis, F.N. et al. Switching Wireless Control for Longitudinal Quadrotor Maneuvers. J Intell Robot Syst 102, 42 (2021). https://doi.org/10.1007/s10846-021-01405-2

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