# Friction Feedforward Compensation Composite Control of Continuous Rotary Motor with Sliding Mode Variable Structure Based on an Improved Power Reaching Law

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Continuous Rotary Motor State Space Modeling

^{3}/s); ${K}_{q}$ is the flow gain (m

^{2}/s); ${X}_{v}$ is the spool displacement (m); ${K}_{c}$ is the flow pressure coefficient (m

^{3}/(s·Pa)); and ${P}_{L}$ is the external load pressure (MPa).

^{3}/rad); $\theta $ is the displacement (rad); ${C}_{tm}$ is the total leakage coefficient (m

^{3}/(s·Pa)); ${V}_{t}$ is the total volume of the connecting pipe, motor and servo valve chamber (m

^{3}); and ${\beta}_{e}$ is the effective volume modulus of elasticity (Pa).

^{2}); ${B}_{m}$ is the viscous damping factor (N·m/(rad/s)); $G$ is the spring stiffness of the load (N·m/rad); and ${T}_{L}$ is the load moment acting on the motor (N·m).

_{L}, P

_{L}, the output angular displacement of continuous rotary motor $\theta \left(s\right)$ is related to the spool displacement of the electro-hydraulic servo valve ${X}_{v}\left(s\right)$, and the sine disturbance torque ${T}_{L}\left(s\right)$ is as follows:

_{h}= $\frac{{K}_{ce}}{{D}_{m}}\sqrt{\frac{{J}_{t}{\beta}_{e}}{{V}_{t}}}+\frac{{B}_{m}}{4{D}_{m}}$, ${K}_{ce}$ is the total flow-pressure coefficient of the valve-controlled motor(m

^{3}/(s·Pa)), ${K}_{ce}={K}_{c}+{C}_{tm}$.

^{3}/s), ${Q}_{0}={K}_{q}\xb7{X}_{v}$; ${K}_{sv}$ is the flow gain (m

^{3}/(s·A)); ${\omega}_{sv}$ is the intrinsic frequency of the servo valve (rad/s); and ${\xi}_{sv}$ is the damping ratio of the servo valve.

## 3. Continuous Rotary Motor Friction Torque Modeling and Compensation

#### 3.1. Continuous Friction Model

_{f}is the value of the motor friction torque (N·m); ${k}_{i}\in Ri=1,2,\dots ,6$; $\mathrm{tanh}$ is the hyperbolic tangent function; and $\dot{q}$ is the motor angular velocity (degree/sec).

- (1)
- The model is symmetrical about the origin and applies to the motor’s bi-directional rotational state;
- (2)
- The static friction factor can be described when k
_{6}= 0, where $\mathrm{tanh}\left({k}_{2}\dot{q}\right)-\mathrm{tanh}\left({k}_{3}\dot{q}\right)$ captures the Stribeck phenomenon, where the friction factor decreases as the speed of the motor system continues to increase; - (3)
- ${k}_{6}\dot{q}$ is viscous friction, capturing the viscosity resistance between the relative moving parts of the motor due to the viscosity of the lubricant;
- (4)
- ${k}_{4}\mathrm{tanh}\left({k}_{5}\dot{q}\right)$ indicates Coulomb friction and exists in a motor system without viscous friction.

#### 3.2. Identification of Friction Model Parameters

_{1}, k

_{2}, k

_{3}, k

_{4}, k

_{5}, k

_{6}, and the model is non-linear. In order to improve the recognition accuracy of the friction model, a heuristic algorithm, genetic algorithm, is used to recognize the model parameters.

_{F}is the model-identified friction moment value.

_{c}= 0.9, variation probability P

_{m}= 0.1, and iteration counter t = 0;

#### 3.3. Friction Compensation

## 4. Sliding Mode Variable Structure Controller Based on Improved Power Reaching Law

#### 4.1. Reaching Laws

#### 4.2. Application of Reaching Laws

_{i}is as follows:

#### 4.3. Improving the Power Reaching Law

_{1}, c

_{2}is the sliding mode gain, ${c}_{1}>0,{c}_{2}0$; e is the motor tracking error, defined as follows:

## 5. Simulations

#### 5.1. Determination of System Modal Parameters

_{1}and c

_{2}are determined empirically. In order to ensure that the system movement point can quickly converge to the sliding mode surface without large steady-state errors, the value of c is not too large; the values of k

_{1}and k

_{2}are slightly smaller to reduce the steady-state errors and to meet the tracking performance of the system; the value of $\alpha $ is fine-tuned from the initial value to avoid overshooting. After several simulations, the values of the parameter variables of the sliding mode variable structure controller in this paper are as follows:

#### 5.2. Simulink Simulation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wang, X.J. Continuous Rotary Motor Electro-hydraulic Servo System Based on the Improved Repetitive Controller. J. Harbin Inst. Technol.
**2010**, 5, 731–734. [Google Scholar] - Li, C.; Yan, Y.; Yang, Y. The Coordinate System Design and Implementation of the Spacecraft’s Attitude Simulation based on Five Axis Turntable. In Proceedings of the 2018 IEEE 3rd Advanced Information Technology, Electronic and Automation Control Conference, IAEAC, Chongqing, China, 12–14 October 2018; pp. 388–393. [Google Scholar]
- Feng, W. Research on the Control Performance of Large Displacement Continuous Rotary Electro-Hydraulic Servo Motor. Master’s Thesis, Harbin Institute of Technology, Harbin, China, 2011. Volume 3. pp. 1–26. [Google Scholar]
- Ma, Y. Application of QFT in the Control System of Continuous Rotary Electro-Hydraulic Servo Motor. Master’s Thesis, Harbin Institute of Technology, Harbin, China, 2008. Volume 6. pp. 7–11. [Google Scholar]
- Wensel, R.G.; Metcalfe, R.; Pothier, N.E.; Russell, B.G. O-ring Seal Studies for Space Shuttle Solid Rocket Booster Joints. Can. Aeronaut. Space J.
**1988**, 34, 204–212. [Google Scholar] - Wei, L.J.; Han, S.X.; Xiong, Q.H.; Lv, L.; Duan, J. Effect of O-ring seal groove chamfer radius on sealing performance. Hydraul. Pneum. Seals
**2016**, 36, 72–75. [Google Scholar] - Nikas, G.K.; Burridge, G.; Sayles, R.S. Modelling and Optimization of Rotary Vane Seals. ARCHIVE Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2007**, 221, 699–715. [Google Scholar] [CrossRef] - Li, G.; Zhao, Q.; Li, Y.; Guo, B. Research on the sealing structure of electro-hydraulic servo oscillating motor. Lubr. Seal.
**2015**, 40, 9–13. [Google Scholar] - Peng, Y.; Yu, X.; Tan, L. Research on friction characteristics and compensation of feed servo system based on improved Dahl model. Mod. Manuf. Eng.
**2014**, 114, 117–121. [Google Scholar] - Simoni, L.; Beschi, M.; Visioli, A.; Åström, K.J. Inclusion of the Dwell Time Effect in the LuGre Friction Model. Mechatronics
**2020**, 66, 102345–102352. [Google Scholar] [CrossRef] - Ni, F.; Liu, H.; Kai, D. Identification and compensation of GMS friction model based on speed observer. J. Electr. Mach. Control
**2012**, 16, 70–75. [Google Scholar] - Li, Y.; Zeng, Y.; Pan, Q.; Jiang, X. Friction model of hydraulic cylinder considering pressure effect. J. Agric. Mach.
**2020**, 51, 418–426. [Google Scholar] - Jiang, W.D.; Wang, H.T.; Zhang, S.H.; Ge, H.X.; Zhou, Z.D. Study of Buck converter sliding mode control method based on improved power reaching law. Electr. Drives
**2021**, 51, 58–63. [Google Scholar] - Huo, A.; Zhang, S.; Wu, S. Sliding Mode Variable Structure Control of the Steerable Drilling Stabilized Platform Based on Disturbance Observer. J. Phys. Conf. Ser.
**2021**, 1894, 012033–012041. [Google Scholar] [CrossRef] - Huang, X.P.; Ma, D.; Chen, X. Sliding Mode Variable Structure Control on Vienna Rectifier. In Proceedings of the 2020 Chinese Control and Decision Conference (CCDC), Hefei, China, 22–24 August 2020; IEEE: Piscataway, NJ, USA, 2020. [Google Scholar]
- Makkar, C.; Dixon, W.E.; Sawyer, W.G.; Hu, G. A New Continuously Differentiable Friction Model for Control Systems Design. In Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Monterey, CA, USA, 24–28 July 2005; pp. 600–605. [Google Scholar]
- Stotsky, A. Adaptive Estimation of the Engine Friction Torque. In Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 15 December 2005; IEEE: Piscataway, NJ, USA, 2005. [Google Scholar]
- Stotsky, A.A. Data-driven algorithms for engine friction estimation. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2006**, 221, 901–909. [Google Scholar] [CrossRef] - Slotinej, J.; Sastrys, S. Tracking Control of Non-Linear Systems Using Sliding Surfaces, with Application to Robot Manipulators. Int. J. Control
**1983**, 38, 465–492. [Google Scholar] [CrossRef] [Green Version] - Qin, T.; Lu, D.L.; Zheng, G.J.; Lei, X.; Wang, T. Study on the sliding mode variable structure control of a stable aiming system based on PSO. Mod. Manuf. Technol. Equip.
**2020**, 56, 49–53. [Google Scholar] - Rakhtala, S.M.; Ahmadi, M. Twisting control algorithm for the yaw and pitch tracking of a twin rotor UAV. In Proceedings of the 2015 2nd International Conference on Knowledge-Based Engineering and Innovation (KBEI), Tehran, Iran, 5–6 November 2015; pp. 276–284. [Google Scholar]
- Hou, H.; Yu, X.; Xu, L.; Rsetam, K.; Cao, Z. Finite-Time Continuous Terminal Sliding Mode Control of Servo Motor Systems. IEEE Trans. Ind. Electron.
**2020**, 67, 5647–5656. [Google Scholar] [CrossRef] - Li, G.; Ding, Y.; Feng, Y.; Li, Y. AMESim simulation and energy control of hydraulic control system for direct drive electro-hydraulic servo die forging hammer. Int. J. Hydromechatronics
**2019**, 2, 203–225. [Google Scholar] [CrossRef] - Kato, T.; Xu, Y.; Tanaka, T.; Shimazaki, K. Force control for ultraprecision hybrid electric-pneumatic vertical-positioning device. Int. J. Hydromechatronics
**2021**, 4, 185–201. [Google Scholar] [CrossRef]

Model Parameters | k_{1} | k_{2} | k_{3} | k_{4} | k_{5} | k_{6} |
---|---|---|---|---|---|---|

Parameter values | 24.26 | −12.77 | −468.3 | 24.55 | 12.34 | 0.3057 |

X | 0 | 0.027 | 0.058 | 0.092 | 0.11 | 0.2 | 0.415 | 0.562 | 0.854 | 1 |

Y | 24.17 | 24.11 | 24.08 | 24.18 | 24.34 | 24.49 | 24.71 | 24.75 | 24.82 | 24.83 |

Reaching Law (Math.) | Mathematical Expressions |
---|---|

Isokinetic reaching law | $\dot{s}=-\epsilon \mathrm{sgn}(s),\epsilon 0$ |

Exponential reaching law | $\dot{s}=-ks-\epsilon \mathrm{sgn}(s),k0,\epsilon 0$ |

Power reaching law | $\dot{s}=-k|s{|}^{\alpha}\mathrm{sgn}(s),k0,1\alpha 0$ |

Variable speed reaching law | $\dot{s}=-\epsilon \left|x\right|\mathrm{sgn}(s),\epsilon 0$ |

Frequency/Hz | Magnitude Error/% | Phase Error/° | ||
---|---|---|---|---|

PID | SMC | PID | SMC | |

13 | 7.1 | 3.5 | 5.381 | 1.872 |

14 | 11.8 | 5.9 | 5.542 | 3.526 |

15 | 16.1 | 8.6 | 5.758 | 4.858 |

16 | 20 | 11.3 | 5.937 | 6.333 |

PID | SMC | |
---|---|---|

Scheme 5 | 5.0805 × 10^{−7} | 8.9490 × 10^{−8} |

MSE | 6.8143 × 10^{−7} | 2.9915 × 10^{−7} |

MAE | 2.1814 × 10^{−5} | 8.0668 × 10^{−6} |

RMSE | 2.2028 × 10^{−5} | 9.4552 × 10^{−6} |

MAPE | 0.0074 | 0.0078 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, X.; Bai, B.; Feng, Y.
Friction Feedforward Compensation Composite Control of Continuous Rotary Motor with Sliding Mode Variable Structure Based on an Improved Power Reaching Law. *Electronics* **2023**, *12*, 1447.
https://doi.org/10.3390/electronics12061447

**AMA Style**

Wang X, Bai B, Feng Y.
Friction Feedforward Compensation Composite Control of Continuous Rotary Motor with Sliding Mode Variable Structure Based on an Improved Power Reaching Law. *Electronics*. 2023; 12(6):1447.
https://doi.org/10.3390/electronics12061447

**Chicago/Turabian Style**

Wang, Xiaojing, Bocheng Bai, and Yaming Feng.
2023. "Friction Feedforward Compensation Composite Control of Continuous Rotary Motor with Sliding Mode Variable Structure Based on an Improved Power Reaching Law" *Electronics* 12, no. 6: 1447.
https://doi.org/10.3390/electronics12061447