# 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

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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 |

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**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