# Research on Neural Network Terminal Sliding Mode Control of Robotic Arms Based on Novel Reaching Law and Improved Salp Swarm Algorithm

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## Abstract

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## 1. Introduction

- A novel reaching law is proposed, which utilizes the tanh and sigmoid functions to replace the sign functions in traditional multi-power reaching law. This enables the system state to slide rapidly and accurately onto the sliding surface, suppressing oscillations and enhancing system stability and disturbance rejection capabilities.
- An improved salp swarm algorithm is proposed, incorporating adaptive inertia weight factors and adaptive adjustment strategies to enhance convergence speed, overall performance, and solution accuracy.
- A novel neural network terminal sliding mode controller is proposed and applied to the trajectory tracking control of an ABB robot. The superior control performance of the controller is verified through simulation and experimental validation.

## 2. Controller Design

#### 2.1. Robotic Arm Dynamics Model

#### 2.2. Design of Fast Nonsingular Terminal Sliding Mode Control

#### 2.3. Design of RBF Neural Network

#### 2.4. Control System Stability Analysis

#### 2.4.1. Certificate of Necessity

- $V\left(x\right)>0$ holds for all $x\ne 0$: since $V\left(x\right)$ is asymptotically stable, according to the definition of the Lyapunov function, there exists a positive constant $a$ and a positive constant $b$, such that for all $x$ satisfying $\Vert x\Vert >a$, there is $V\left(x\right)>b$. That is to say, for all non-zero vectors $x$, as long as their paradigm is greater than $a$, the Lyapunov function $V\left(x\right)$ is greater than $b$. Therefore, one can conclude that $V\left(x\right)>0$ holds for all $x\ne 0$.
- $V\left(0\right)=0$: since $V\left(x\right)>0$ holds for all $x\ne 0$, we can deduce that $V\left(0\right)$ must be equal to 0. Otherwise, if $V\left(0\right)$ is greater than 0, then there exists a small neighborhood where $V\left(x\right)>0$. This contradicts the condition that $V\left(x\right)>0$ holds for all $x\ne 0$.
- $\dot{V}\le 0$ holds for all $x\ne 0$: the derivative of the Lyapunov function $V\left(x\right)$ can represent the rate of change of the state of the system. Since $V\left(x\right)$ is asymptotically stable, by the definition of the Lyapunov function, for all $x$ satisfying $\Vert x\Vert >a$, there is $\dot{V}\le 0$. This implies that the Lyapunov function $V\left(x\right)$ is decreasing over the range of these $x$. The derivative of $V\left(x\right)$ holds for all $x\ne 0$: the derivative of $V\left(x\right)$ can represent the rate of change of the state of the system. Also, by the definition of asymptotic stability, $\dot{V}\le 0$ must tend to 0, i.e., $\dot{V}\left(0\right)=0$.

#### 2.4.2. Certificate of Sufficiency

#### 2.5. Improved Salp Swarm Algorithm

## 3. Simulation and Experimental Results

#### 3.1. Simulation Results

#### 3.2. Experiment Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Control torques of different controllers. (

**a**) Joint 1; (

**b**) Joint 2; (

**c**) Joint 3; (

**d**) Joint 4; (

**e**) Joint 5; (

**f**) Joint 6.

**Figure 3.**Position tracking of different controllers. (

**a**) Joint 1; (

**b**) Joint 2; (

**c**) Joint 3; (

**d**) Joint 4; (

**e**) Joint 5; (

**f**) Joint 6.

**Figure 4.**Position tracking errors of different controllers. (

**a**) Joint 1; (

**b**) Joint 2; (

**c**) Joint 3; (

**d**) Joint 4; (

**e**) Joint 5; (

**f**) Joint 6.

**Figure 5.**Velocity tracking of different controllers. (

**a**) Joint 1; (

**b**) Joint 2; (

**c**) Joint 3; (

**d**) Joint 4; (

**e**) Joint 5; (

**f**) Joint 6.

**Figure 6.**Velocity tracking errors of different controllers. (

**a**) Joint 1; (

**b**) Joint 2; (

**c**) Joint 3; (

**d**) Joint 4; (

**e**) Joint 5; (

**f**) Joint 6.

Joint | Angle θ (°) | Offset d (m) | Length a (m) | Twist α (°) |
---|---|---|---|---|

Joint 1 | θ_{1} | 0.290 | 0 | −90 |

Joint 2 | θ_{2} | 0 | 0.270 | 0 |

Joint 3 | θ_{3} | 0 | 0.070 | −90 |

Joint 4 | θ_{4} | 0.302 | 0 | 90 |

Joint 5 | θ_{5} | 0 | 0 | −90 |

Joint 6 | θ_{6} | 0.072 | 0 | 0 |

Joint | Mass (kg) | Position of the Center of Mass (m) |
---|---|---|

Joint 1 | 9.28 | (−0.02819, 0.00002, 0.13210) |

Joint 2 | 3.91 | (−0.00216, 0.00118, 0.39124) |

Joint 3 | 2.94 | (0.00178, −0.01867, 0.61730) |

Joint 4 | 1.33 | (0.00856, −0.22070, 0.62499) |

Joint 5 | 0.55 | (0.01133, −0.29682, 0.62287) |

Joint 6 | 0.01 | (0.01367, −0.36273, 0.61955) |

Joint | GFTSM | RBF-FNTSM | ALSSA-RBFTSM |
---|---|---|---|

Joint 1 | 1.75 × 10^{−3} | 1.31 × 10^{−4} | 4.13 × 10^{−5} |

Joint 2 | 1.83 × 10^{−3} | 4.10 × 10^{−4} | 6.48 × 10^{−5} |

Joint 3 | 2.38 × 10^{−3} | 1.53 × 10^{−4} | 2.19 × 10^{−5} |

Joint 4 | 1.68 × 10^{−3} | 9.02 × 10^{−4} | 1.57 × 10^{−4} |

Joint 5 | 2.79 × 10^{−3} | 6.03 × 10^{−4} | 1.07 × 10^{−4} |

Joint 6 | 2.49 × 10^{−3} | 9.10 × 10^{−4} | 1.16 × 10^{−4} |

Joint | GFTSM | RBF-FNTSM | ALSSA-RBFTSM |
---|---|---|---|

Joint 1 | 3.65 × 10^{−3} | 8.07 × 10^{−4} | 1.66 × 10^{−4} |

Joint 2 | 4.43 × 10^{−3} | 1.50 × 10^{−3} | 2.21 × 10^{−4} |

Joint 3 | 4.84 × 10^{−3} | 9.98 × 10^{−4} | 1.45 × 10^{−4} |

Joint 4 | 3.33 × 10^{−3} | 2.16 × 10^{−3} | 4.49 × 10^{−4} |

Joint 5 | 4.42 × 10^{−3} | 2.12 × 10^{−3} | 3.22 × 10^{−4} |

Joint 6 | 5.65 × 10^{−3} | 2.33 × 10^{−3} | 9.12 × 10^{−4} |

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**MDPI and ACS Style**

Duan, J.; Zhang, H.; Zhang, Q.; Qin, J.
Research on Neural Network Terminal Sliding Mode Control of Robotic Arms Based on Novel Reaching Law and Improved Salp Swarm Algorithm. *Actuators* **2023**, *12*, 464.
https://doi.org/10.3390/act12120464

**AMA Style**

Duan J, Zhang H, Zhang Q, Qin J.
Research on Neural Network Terminal Sliding Mode Control of Robotic Arms Based on Novel Reaching Law and Improved Salp Swarm Algorithm. *Actuators*. 2023; 12(12):464.
https://doi.org/10.3390/act12120464

**Chicago/Turabian Style**

Duan, Jianguo, Hongzhi Zhang, Qinglei Zhang, and Jiyun Qin.
2023. "Research on Neural Network Terminal Sliding Mode Control of Robotic Arms Based on Novel Reaching Law and Improved Salp Swarm Algorithm" *Actuators* 12, no. 12: 464.
https://doi.org/10.3390/act12120464