# Iterative Learning Sliding Mode Control for UAV Trajectory Tracking

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## Abstract

**:**

## 1. Introduction

## 2. Iterative Learning Sliding Mode Control

#### 2.1. ILSMC Design

**Definition**

**1.**

**Assumption**

**1.**

#### 2.1.1. Equivalent Control

#### 2.1.2. Learning Control

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. System Description and Modeling

#### 3.1. Kinematics

#### 3.2. Quadcopter Dynamics

#### 3.3. Discrete-Time Model

## 4. Integrated ILSMC for UAV Attitude Control

#### 4.1. Inner-Loop PID Controller

#### 4.2. Outer-Loop ILSMC

#### 4.3. Implementation Procedure

- Step 1: Declare ${I}_{xx},{I}_{yy},{I}_{zz}$, ${K}_{p},{K}_{i},{K}_{d},c,\mu ,\lambda $.
- Step 2: Set ${x}_{d}\left(k\right)$, $j=0$, and ${\widehat{u}}_{il{c}_{\left(j\right)}}\left(k\right)=0$.
- Step 3: Compute the ILSMC ${\widehat{u}}_{\left(j\right)}\left(k\right)$ from (61) as a reference to the inner loop.
- Step 4: Compute, from the measured states ${x}_{\left(j\right)}\left(k\right)$, ${e}_{\left(j\right)}\left(k\right)$, ${\sigma}_{\left(j\right)}\left(k\right)$, ${S}_{\left(j\right)}\left(k\right)$, and the selected TPI.
- Step 5: Check if the tracking performance requirement is met to terminate the learning process. Otherwise, proceed to Step 6.
- Step 6: Set $j=j+1$, update ${\widehat{u}}_{il{c}_{\left(j\right)}}\left(k\right)$ from (60), then return to Step 3.

## 5. Simulation Results

#### 5.1. Step Response in Nominal Conditions

#### 5.2. Trajectory Tracking Performance under Disturbances and Uncertainties

## 6. Experimental Validation

#### 6.1. Experimental Setup

#### 6.2. Real-Time Data Validation Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMC | Sliding mode control |

ILC | Iterative learning control |

ILSMC | Iterative learning sliding mode control |

UAV | Unmanned aerial vehicle |

PID | Proportional–integral–derivative |

FL | Feedback linearization |

CoG | Center of gravity |

TPI | Tracking performance index |

ITAE | Integral time absolute error |

ATSMC | Adaptive twisting sliding mode control |

AFTC | Finite-time control scheme. |

## References

- Grzonka, S.; Grisetti, G.; Burgard, W. A fully autonomous indoor quadrotor. IEEE Trans. Robot.
**2012**, 28, 90–100. [Google Scholar] [CrossRef][Green Version] - Grisetti, G.; Stachniss, C.; Burgard, W. Non-linear constraint network optimization for efficient map learning. IEEE Trans. Intell. Transp. Syst.
**2009**, 10, 428–439. [Google Scholar] [CrossRef][Green Version] - Flint, M.; Polycarpou, M.; Fernandez-Gaucherand, E. Cooperative control for multiple autonomous UAV’s searching for targets. In Proceedings of the 41st lEEE Conference on Decision and Control, Las Vegas, NV, USA, 10–13 December 2002. [Google Scholar]
- Brown, A.; Anderson, D. Trajectory optimization for high-altitude long endurance UAV maritime radar surveillance. IEEE Trans. Aerosp. Electron. Syst.
**2020**, 56, 2406–2421. [Google Scholar] [CrossRef][Green Version] - Metni, N.; Hamel, T. A UAV for bridge inspection: Visual servoing control law with orientation limits. Autom. Constr.
**2007**, 17, 3–10. [Google Scholar] [CrossRef] - Herwitz, S.R.; Johnson, L.F.; Dunagan, S.E.; Higgins, R.G.; Sullivan, D.V.; Zheng, J.; Lobitz, B.M.; Leung, J.G.; Gallmeyer, B.A.; Aoyagi, M.; et al. Imaging from an unmanned aerial vehicle: Agricultural surveillance and decision support. Comput. Electron. Agric.
**2004**, 44, 49–61. [Google Scholar] [CrossRef] - Choi, Y.C.; Ahn, H.S. Nonlinear control of quadrotor for point tracking: Actual implementation and experimental tests. IEEE/ASME Trans. Mechatron.
**2015**, 20, 1179–1192. [Google Scholar] [CrossRef] - Park, J.; Kim, Y.; Kim, S. Landing site searching and selection algorithm development using vision system and its application to quadrotor. IEEE Trans. Control Syst. Technol.
**2015**, 23, 488–503. [Google Scholar] [CrossRef] - Prasenjit, M.; Steven, W. Direct adaptive feedback linearization for quadrotor control. In Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, MN, USA, 13–16 August 2012; American Institute of Aeronautics and Astronautics: Minneapolis, MN, USA, 2012. [Google Scholar]
- L’afflitto, A.; Anderson, R.B.; Mohammadi, K. An introduction to nonlinear robust control for unmanned quadrotor aircraft. IEEE Control Syst. Mag.
**2018**, 38, 102–121. [Google Scholar] [CrossRef] - Qingtong, W.; Honglin, W.; Qingxian, W.; Mou, C. Backstepping-based attitude control for a quadrotor UAV using nonlinear disturbance observer. In Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China, 28–30 July 2015. [Google Scholar]
- Chincholkar, S.H.; Jiang, W.; Chan, C.-Y. Continuous nonsingular terminal sliding mode control of DC–DC boost converters subject to time-varying disturbances. IEEE Trans. Circuits Syst. II Exp. Briefs
**2020**, 67, 92–96. [Google Scholar] [CrossRef] - Ma, H.; Xiong, Z. Sliding mode control for uncertain discrete-time systems using an adaptive reaching law. IEEE Trans. Circuits Syst. II Exp. Briefs
**2021**, 68, 722–726. [Google Scholar] [CrossRef] - Ali, S.U.; Samar, R.; Shah, M.Z.; Bhatti, A.I.; Munawar, K.; Al-Sggaf, U.M. Lateral guidance and control of UAVs using second-order sliding modes. Aerosp. Sci. Technol.
**2016**, 49, 88–100. [Google Scholar] [CrossRef] - Sankaranarayanan, V.; Mahindrakar, A.D. Control of a class of underactuated mechanical systems using sliding modes. IEEE Trans. Robot.
**2009**, 25, 459–467. [Google Scholar] [CrossRef] - Tian, B.; Yin, L.; Wang, H. Finite-time reentry attitude control based on adaptive multivariable disturbance compensation. IEEE Trans. Ind. Electron.
**2015**, 62, 5889–5898. [Google Scholar] [CrossRef] - Qi, Y.; Zhang, S.; Jiang, F.; Zhou, H.; Tao, D.; Li, X. Siamese local and global networks for robust face tracking. IEEE Trans. Image Process.
**2020**, 29, 9152–9164. [Google Scholar] [CrossRef] [PubMed] - Zhou, M.; Feng, Y.; Xue, C.; Han, F. Deep convolutional neural network based fractional-order terminal sliding-mode control for robotic manipulators. Neurocomputing
**2020**, 416, 143–151. [Google Scholar] [CrossRef] - Bristow, D.A.; Tharayil, M.; Alleyne, A.G. A survey of iterative learning control. IEEE Control Syst. Mag.
**2006**, 26, 96–114. [Google Scholar] - Yu, M.; Li, C. Robust Adaptive Iterative Learning Control for Discrete-Time Nonlinear Systems with Time-Iteration-Varying Parameters. IEEE Trans. Syst. Man Cybern. Syst.
**2017**, 47, 1737–1745. [Google Scholar] [CrossRef] - Uchiyama, M. Formation of high speed motion pattern of mechanical arm by trial. Trans. Soc. Instrum. Control. Eng.
**1978**, 14, 706–712. [Google Scholar] [CrossRef][Green Version] - Arimoto, S.; Kawamura, S.; Miyazaki, F. Iterative learning control for robot systems. In Proceedings of the IECON, Tokyo, Japan, 22–26 October 1984; pp. 393–398. [Google Scholar]
- Moore, K.L. Iterative learning control for deterministic systems. In Advances in Industial Control; Springer: London, UK, 1993. [Google Scholar]
- Tayebi, A.; Abdul, S.; Zaremba, M.B.; Ye, Y. Robust iterative learning control design: Application to a robot manipulator. IEEE/ASME Trans. Mechatron.
**2008**, 13, 608–613. [Google Scholar] [CrossRef][Green Version] - Mezghani, M.; Roux, G.; Cabassud, M.; Lann, M.V.L.; Dahhou, B.; Casamatta, G. Application of iterative learning control to an exothermic semibatch chemical reactor. IEEE Trans. Control Syst. Technol.
**2002**, 10, 822–834. [Google Scholar] [CrossRef] - Zhu, Q.; Song, F.; Xu, J.; Liu, Y. An internal model based iterative learning control for wafer scanner systems. IEEE/ASME Trans. Mechatron.
**2019**, 24, 2073–2084. [Google Scholar] [CrossRef] - Nikooienejad, N.; Maroufi, M.; Moheimani, R. Iterative learning control for video-rate atomic force microscopy. IEEE/ASME Trans. Mechatron.
**2021**, 26, 2127–2138. [Google Scholar] [CrossRef] - Madani, T.; Benallegue, A. Adaptive control via backstepping technique and neural networks of a quadrotor helicopter. IFAC Proc. Vol.
**2008**, 41, 6513–6518. [Google Scholar] [CrossRef][Green Version] - Hoang, V.T.; Phung, M.D.; Ha, Q.P. Adaptive twisting sliding mode control for quadrotor unmanned aerial vehicles. In Proceedings of the 2017 Asian Control Conference (ASCC 2017), Gold Coast, Australia, 17–20 December 2017; pp. 671–676. [Google Scholar]
- Tian, B.; Lu, H.; Zuo, Z.; Zong, Q. Adaptive finite-time attitude tracking of quadrotors with experiments and comparisons. IEEE Trans. Ind. Electron.
**2019**, 66, 9428–9438. [Google Scholar] [CrossRef] - He, X.; Guo, D.; Leang, K.K. Repetitive control design and implementation for periodic motion tracking in aerial robots. In Proceedings of the 2017 American Control Conference (ACC), Seattle, WA, USA, 24–26 May 2017; pp. 5101–5108. [Google Scholar]
- Dong, J.; He, B. Novel fuzzy PID-type iterative learning control for quadrotor UAV. Sensors
**2019**, 19, 24. [Google Scholar] [CrossRef] [PubMed][Green Version] - Adlakha, R.; Zheng, M. An Optimization-Based Iterative Learning Control Design Method for UAV’s Trajectory Tracking. In Proceedings of the 2020 American Control Conference (ACC), Denver, CO, USA, 1–3 July 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 1353–1359. [Google Scholar]
- Chen, Y.; Moore, K.L. Harnessing the nonrepetitiveness in iterative learning control. In Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, 10–13 December 2002; IEEE: Piscataway, NJ, USA, 2002; Volume 3, pp. 3350–3355. [Google Scholar]
- Sarpturk, S.Z.; Istefanopulos, Y.; Kaynak, O. On the stability of discrete-time sliding mode control systems. IEEE Trans. Autom. Control
**1987**, 32, 930–932. [Google Scholar] [CrossRef] - Norrlöf, M.; Gunnarsson, S. Time and frequency domain convergence properties in iterative learning control. Int. J. Control
**2002**, 75, 1114–1126. [Google Scholar] [CrossRef] - Guilherme, V.R.; Manuel, G.O.; Francisco, R.R. An integral predictive/nonlinear H
_{∞}control structure for a quadrotor helicopter. Automatica**2010**, 46, 29–39. [Google Scholar] - Craig, J.J. Introduction to Robotics—Mechanics and Control, 2nd ed.; Addison-Wesley Publishing Company, Inc.: Reading, MA, USA, 1989. [Google Scholar]
- Hoang, V.T.; Phung, M.D.; Dinh, T.H.; Ha, Q.P. System architecture for real-time surface inspection using multiple UAVs. IEEE Sens. J.
**2020**, 40, 4430–4441. [Google Scholar] [CrossRef][Green Version]

Parameter | Value | Unit |
---|---|---|

m | $1.5$ | kg |

l | $0.205$ | m |

g | $9.81$ | m/s${}^{2}$ |

${I}_{xx}$ | $9.1\times {10}^{-3}$ | kg m${}^{2}$ |

${I}_{yy}$ | $16.4\times {10}^{-3}$ | kg m${}^{2}$ |

${I}_{zz}$ | $24.1\times {10}^{-3}$ | kg m${}^{2}$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

${c}_{\varphi}$ | 50 | $\mu $ | 10 |

${c}_{\theta}$ | 50 | $\lambda $ | $0.9$ |

${c}_{\psi}$ | 20 | - | - |

UAV Angle (degrees) | ATSMC | AFTC | PD | PD-ILC | ILSMC |
---|---|---|---|---|---|

Roll | $5.03$ | $2.03$ | $3344.2$ | $39.91$ | $0.255$ |

Pitch | $3.87$ | $1.40$ | $3468.4$ | $55.60$ | $0.261$ |

Yaw | $3.47$ | $2.42$ | $3323.0$ | $149.89$ | $0.444$ |

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**MDPI and ACS Style**

Nguyen, L.V.; Phung, M.D.; Ha, Q.P. Iterative Learning Sliding Mode Control for UAV Trajectory Tracking. *Electronics* **2021**, *10*, 2474.
https://doi.org/10.3390/electronics10202474

**AMA Style**

Nguyen LV, Phung MD, Ha QP. Iterative Learning Sliding Mode Control for UAV Trajectory Tracking. *Electronics*. 2021; 10(20):2474.
https://doi.org/10.3390/electronics10202474

**Chicago/Turabian Style**

Nguyen, Lanh Van, Manh Duong Phung, and Quang Phuc Ha. 2021. "Iterative Learning Sliding Mode Control for UAV Trajectory Tracking" *Electronics* 10, no. 20: 2474.
https://doi.org/10.3390/electronics10202474