# Backstepping Fuzzy Sliding Mode Control for the Antiskid Braking System of Unmanned Aerial Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of Antiskid Braking System for UAV

#### 2.1. UAV Mathematical Model

#### 2.2. Slip Ratio Model

#### 2.3. EMA Mathematical Model

#### 2.4. Control Objectives

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

## 3. Control Strategy

#### 3.1. ABS Controller

#### 3.2. EMA Controller

#### 3.3. Fuzzy Corrector

#### 3.4. Stability Analysis

## 4. Results and Analysis

#### 4.1. Experimental Results for the EMA

#### 4.2. HIL Experimental Results for the Slip Ratio (Dry Runway Condition)

#### 4.3. HIL Experimental Results for the Slip Ratio (Icy Runway Condition)

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Tian, R.N.; Jiao, Z.X.; Huang, K.J.; Liu, X.C.; Jing, G.H. Aircraft anti-skid braking control based on pressure servo control using high-speed on/off valve. In Proceedings of the 2016 IEEE Chinese Guidance, Navigation and Control Conference, Nanjing, China, 12–14 August 2016. [Google Scholar] [CrossRef]
- Lin, C.M.; Li, H.M. Intelligent hybrid control system design for antilock braking systems using self-organizing function-link fuzzy cerebellar model articulation controller. IEEE Trans. Fuzzy Syst.
**2013**, 21, 1044–1055. [Google Scholar] [CrossRef] - Sharkawy, A.B. Genetic fuzzy self-tuning PID controller for antilock braking systems. Eng. Appl. Artif. Intell.
**2010**, 23, 1041–1052. [Google Scholar] [CrossRef] - Mi, C.T.; Lin, H.; Zhang, Y. Iterative learning control of antilock braking of electric and hybrid vehicles. IEEE Trans. Veh. Technol.
**2005**, 54, 486–494. [Google Scholar] [CrossRef] - Yuan, D.L.; Wei, J.; Qu, Y.H.; Wu, J.Z. Simulation of Hydraulic Brake Built-in Test System for a Certain UAV. In Proceedings of the 32nd Chinese Control Conference, Xi’an, China, 26–28 July 2013. [Google Scholar]
- Sun, H.; Yan, J.G.; Qu, Y.H.; Ren, J. Sensor fault-tolerant observer applied in UAV anti-skid braking control under control input constraint. J. Syst. Eng. Electron.
**2017**, 28, 126–136. [Google Scholar] [CrossRef] - Tang, Y.G.; Zhang, X.Y.; Zhang, D.L.; Zhao, G.; Guan, X.P. Fractional order sliding mode controller design for antilock braking systems. Neurocomputing
**2013**, 111, 122–130. [Google Scholar] [CrossRef] - Cao, W.P.; Mecrow, B.C.; Atkinson, G.J.; Bennett, J.W.; Atkinson, D.J. Over view of electric motor technologies used for more electric aircraft. IEEE Trans. Ind. Electron.
**2012**, 59, 3523–3531. [Google Scholar] [CrossRef] - Li, B.Q.; Chen, X.L.; Lin, H.; Dai, Z.Y. Enhanced stability dynamic surface control for aircraft antiskid braking system using electromechanical actuator. Syst. Eng. Electron.
**2016**, 38, 1139–1145. [Google Scholar] [CrossRef] - Dincmen, E.; Guvenc, B.A.; Acarman, T. Extremum-seeking control of ABS braking in road vehicles with lateral force improvement. IEEE Trans. Control Syst. Technol.
**2014**, 22, 230–237. [Google Scholar] [CrossRef] - Mehdi, F.; Asghar, M.; Michael, H.; Kate, S.M. A cross-entropy method for optimising robotic automated storage and retrieval systems. Int. J. Prod. Res.
**2018**, 56, 6450–6472. [Google Scholar] [CrossRef] - Mehdi, F.; Reza, T.M. A scalarization-based method for multiple part-type scheduling of two-machine robotic systems with non-destructive testing technologies. Iran. J. Oper. Res.
**2019**, 10, 1–17. [Google Scholar] [CrossRef] - Peng, X.Y.; Jia, M.F.; He, L.; Yu, X.; Lv, Y.B. Fuzzy sliding mode control based on longitudinal force estimation for electro-mechanical braking systems using BLDC motor. CES Trans. Electr. Mach. Syst.
**2018**, 2, 142–151. [Google Scholar] [CrossRef] - Kayacan, E.; Oniz, Y.; Kaynak, O. A grey system modeling approach for sliding-mode control of antilock braking system. IEEE Trans. Ind. Electron.
**2009**, 19, 767–773. [Google Scholar] [CrossRef] - Velimir, C.; Dragan, A. Adaptive neuro-fuzzy wheel slip control. Expert Syst. Appl.
**2013**, 40, 5197–5209. [Google Scholar] [CrossRef] - Corno, M.; Gerard, M.; Verhaegen, M.; Holweg, M. Hybrid ABS Control Using Force Measurement. IEEE Trans. Control Syst. Technol.
**2012**, 20, 1223–1235. [Google Scholar] [CrossRef] - Choi, S.B. Antilock Brake System with a Continuous Wheel Slip Control to Maximize the Braking Performance and the Ride Quality. IEEE Trans. Control Syst. Technol.
**2008**, 16, 996–1003. [Google Scholar] [CrossRef] - Rajesh, R.; Gridsada, P.; Damrongrit, P.; Jae, Y.L. Algorithms for real-time estimation of individual wheel tire-road friction coefficients. IEEE/ASME Trans. Mechatron.
**2012**, 17, 1183–1195. [Google Scholar] [CrossRef] - Wei, Z.; Xu, J.; Halim, D. Braking force control strategy for electric vehicles with load variation and wheel slip considerations. IET Electr. Syst. Transp.
**2017**, 7, 41–47. [Google Scholar] [CrossRef] - Zhang, X.; Lin, H. UAV Anti-Skid Braking System Simulation. In Proceedings of the 2018 37th Chinese Control Conference, Wuhan, China, 25–27 July 2018. [Google Scholar] [CrossRef]
- Zhang, M.; Nie, H.; Zhu, R.P. Design and Dynamics Analysis of Anti-skid Braking System for Aircraft with Four-wheel Bogie Landing Gears. Chin. J. Mech. Eng.
**2011**, 24, 77–284. [Google Scholar] [CrossRef] - Huang, C.; Jiao, Z.X.; Shang, Y.X. Antiskid braking control with on/off valves for aircraft applications. J. Aircr.
**2013**, 50, 1869–1879. [Google Scholar] [CrossRef] - Mercorelli, P. A Two-Stage Sliding-Mode High-Gain Observer to Reduce Uncertainties and Disturbances Effects for Sensorless Control in Automotive Applications. IEEE Trans. Ind. Electron.
**2015**, 62, 5929–5940. [Google Scholar] [CrossRef] - Mercorelli, P. An adaptive and optimized switching observer for sensorless control of an electromagnetic valve actuator in camless internal combustion engines. Asian J. Control
**2014**, 16, 959–973. [Google Scholar] [CrossRef] - Wu, Y.Q.; Yu, X.H.; Man, Z.H. Terminal Sliding Mode Control Design for Uncertain Dynamic Systems. Syst. Control Lett.
**1998**, 34, 281–287. [Google Scholar] [CrossRef] - Dong, S.; Jiao, Z.X.; Sun, X.H.; Liu, X.C. Dynamic allocation algorithm for the gain of UAV nose wheel steering and differential braking based on decomposition control. In Proceedings of the 2016 IEEE International Conference on Aircraft Utility Systems, Beijing, China, 10–12 October 2016. [Google Scholar] [CrossRef]
- Jung, H.; Choi, S.B. Real–time Individual Tire Force Estimation for an All-wheel Drive Vehicle. IEEE Trans. Veh. Technol.
**2018**, 67, 2934–2944. [Google Scholar] [CrossRef] - Lin, W.C.; Lin, C.L.; Hsu, P.M.; Wu, M.T. Realization of Anti-Lock braking Strategy for Electric Scooters. IEEE Trans. Ind. Electron.
**2014**, 61, 2826–2833. [Google Scholar] [CrossRef] - Sun, W.C.; Zhang, J.H.; Liu, Z.Y. Two-Time-Scale Redesign for Antilock Braking Systems of Ground Vehicles. IEEE Trans. Ind. Electron.
**2019**, 66, 4577–4586. [Google Scholar] [CrossRef] - Naseri, F.; Farjah, E.; Ghanbari, T. An Efficient Regenerative Braking System Based on Battery/Supercapacitor for Electric, Hybrid, and Plug-In Hybrid Electric Vehicles With BLDC Motor. IEEE Trans. Veh. Technol.
**2017**, 66, 3724–3738. [Google Scholar] [CrossRef] - Woodburn, D.; Wu, T.; Zhou, L.; Hu, Y. High-Performance Electromechanical Actuator Dynamic Heat Generation Modeling. IEEE Trans. Aerosp. Electron. Syst.
**2014**, 50, 530–541. [Google Scholar] [CrossRef] - Tee, K.P.; Ge, S.S.; Tay, E.H. Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica
**2009**, 45, 918–927. [Google Scholar] [CrossRef] - Yu, S.H.; Yu, X.H.; Shirinzadeh, B.J.; Man, Z.H. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica
**2005**, 41, 1957–1964. [Google Scholar] [CrossRef] - Ju, J.Y.; Zhao, Y.R.; Zhang, C.R.; Liu, Y.F. Vibration suppression of a flexible-joint robot based on parameter identification and fuzzy PID control. Algorithms
**2018**, 11, 189. [Google Scholar] [CrossRef] [Green Version] - Liang, B.; Zhu, Y.Q.; Li, Y.R.; He, P.J.; Li, W.L. Adaptive Nonsingular Fast Terminal Sliding Mode Control for Braking Systems with Electro-Mechanical Actuators Based on Radial Basis Function. Energies
**2017**, 10, 1637. [Google Scholar] [CrossRef] [Green Version] - Chen, X.L.; Lin, H.; Ma, D.Q. Sliding-mode extremum-seeking control for all-electric active braking system in unmanned aerial vehicle. Control Theory Appl.
**2015**, 32, 1439–1448. [Google Scholar] [CrossRef]

**Figure 8.**Comparison of EMA controller performance. (

**a**) PID control; (

**b**) proposed method; (

**c**) error for PID control; (

**d**) error for proposed method.

**Figure 9.**Comparison of EMA controller performance. (

**a**) Pressure without the fuzzy corrector; (

**b**) pressure with the fuzzy corrector; (

**c**) pressure error without the fuzzy corrector; (

**d**) pressure error with the fuzzy corrector;.

**Figure 10.**Slip ratio control performance of the ABS controller under dry runway conditions. (

**a**) Aircraft velocity and wheel velocity in PID control; (

**b**) aircraft velocity and wheel velocity in the proposed control method; (

**c**) slip ratio in PID control; (

**d**) slip ratio in the proposed control method; (

**e**) adhesive coefficient between wheel and runway in PID control; (

**f**) adhesive coefficient between wheel and runway in the proposed control method.

**Figure 11.**Slip ratio control performance of the ABS controller in icy runway conditions. (

**a**) Aircraft velocity and wheel velocity in PID control; (

**b**) aircraft velocity and wheel velocity in the proposed control method; (

**c**) slip ratio in PID control; (

**d**) slip ratio in the proposed control method; (

**e**) adhesive coefficient between wheel and runway in PID control; (

**f**) adhesive coefficient between wheel and runway in the proposed control method.

$\mathbf{\Delta}{\mathit{k}}_{\mathbf{2}}$$\mathbf{\Delta}\mathit{\tau}$ | ${\mathit{z}}_{\mathbf{2}}$ | |||||
---|---|---|---|---|---|---|

$\mathit{N}\mathit{B}$ | $\mathit{N}\mathit{S}$ | $\mathit{Z}\mathit{E}$ | $\mathit{P}\mathit{S}$ | $\mathit{P}\mathit{B}$ | ||

${\dot{z}}_{2}$ | $NB$ | $B$$B$ | $S$$M$ | $M$$B$ | $S$$M$ | $B$$S$ |

$NS$ | $B$$B$ | $S$$M$ | $M$$B$ | $S$$M$ | $B$$S$ | |

$ZE$ | $B$$M$ | $S$$M$ | $M$$B$ | $S$$M$ | $B$$M$ | |

$PS$ | $B$$S$ | $S$$M$ | $M$$B$ | $S$$M$ | $B$$B$ | |

$PB$ | $B$$S$ | $S$$M$ | $M$$B$ | $S$$M$ | $B$$B$ |

Name | Description | Value | Name | Description | Value |
---|---|---|---|---|---|

$a$ | Distance between front wheel and center of gravity of UAV | 2.1 m | $b$ | Distance between back wheel and center of gravity of UAV | 1.6 m |

$h$ | Distance between ground and center of gravity of UAV | 1.5 m | ${I}_{w}$ | Moment of inertia of the back wheel | 0.3 kg·m^{2} |

$g$ | Gravity acceleration | 9.8 m/s^{2} | $n$ | Number of the back wheels | 2 |

${C}_{L}$ | Aerodynamic lift coefficient | 0.66 | ${\rho}_{a}$ | Air density | 1.2 g/L |

${S}_{L}$ | Aerodynamic lift area | 5.1 m^{2} | ${C}_{P}$ | Aerodynamic drag coefficient | 0.15 |

$m$ | Weight of UAV | 650 kg | ${S}_{P}$ | Aerodynamic drag area | 3.7 m^{2} |

$r$ | The radius of the back wheel | 0.18 m | ${k}_{T}$ | Torque constant | 0.3 N·m/A |

${B}_{v}$ | Viscous damping coefficient | 0.02 N·m/rad | $J$ | Inertia moment of BLDCM | 0.0002 Kg·m^{2} |

${c}_{b}$ | Stiffness coefficient of brake disc | 5 × 10^{6} N/m | ${k}_{emf}$ | Back electromotive force constant | 0.5 V·rad/s |

$R$ | Stator winding resistance | 2.1 Ω | $\eta $ | The transmission ratio of EMA | 21 |

$L$ | Stator winding inductance | 1.2 mH | ${L}_{0}$ | The lead of the ball screw | 0.003 m |

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**MDPI and ACS Style**

Zhang, X.; Lin, H.
Backstepping Fuzzy Sliding Mode Control for the Antiskid Braking System of Unmanned Aerial Vehicles. *Electronics* **2020**, *9*, 1731.
https://doi.org/10.3390/electronics9101731

**AMA Style**

Zhang X, Lin H.
Backstepping Fuzzy Sliding Mode Control for the Antiskid Braking System of Unmanned Aerial Vehicles. *Electronics*. 2020; 9(10):1731.
https://doi.org/10.3390/electronics9101731

**Chicago/Turabian Style**

Zhang, Xi, and Hui Lin.
2020. "Backstepping Fuzzy Sliding Mode Control for the Antiskid Braking System of Unmanned Aerial Vehicles" *Electronics* 9, no. 10: 1731.
https://doi.org/10.3390/electronics9101731