# Design of Backstepping Sliding Mode Control for a Polishing Robot Pneumatic System Based on the Extended State Observer

^{1}

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## Abstract

**:**

## Featured Application

**Robot Pneumatic System.**

## Abstract

## 1. Introduction

## 2. Construction of System Modeling

_{i}is the amplification factor, and Equations (4) and (5) can be simplified as follows:

## 3. Design of the Backstepping Sliding Mode Pneumatic Cylinder Position Controller

**Assumption**

**1.**

**Assumption**

**2.**

#### 3.1. The Extended State Observer

#### 3.2. Stability Analysis of the ESO

#### 3.3. Backstepping Sliding Mode Control

_{1}:

_{1}, e

_{2}into the equation to obtain the control law:

#### 3.4. Stability Analysis of Backstepping Sliding Mode Control

## 4. Simulation and Experimental Results and Discussion of the Pneumatic System

#### 4.1. Simulation Setup

_{p}, the integral gain K

_{i}, and the differential gain K

_{d}. Due to its few control parameters, a classic trial-and-error method, Ziegler-Nichols, is used to obtain the values of K

_{p}, K

_{i}, and K

_{d}[40]. This method is simple and easy to use, but it is slightly inferior in terms of accuracy and stability. The performance of the backstepping sliding mode controller depends on parameters such as the backstepping transformation function, the sliding mode surface function, and the sliding mode control law function. The determination method of these parameters is more complicated. In Section 3, the control law was derived by an analytical method. Based on this, some parameters were adjusted by referring to some literature [41,42] and combining with our model.

#### 4.2. Simulation Results and Discussions

- (1)
- Mean absolute error (MAE):$${E}_{\mathrm{MAE}}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left|{e}_{1}(i)\right|}$$
- (2)
- Root mean square error (RMSE):$${E}_{\mathrm{RMSE}}=\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left(\left|{e}_{1}(i)\right|-{E}_{\mathrm{MAE}}\right)}^{2}}$$
- (3)
- Integrated time and absolute error (ITAE):$${E}_{\mathrm{ITAE}}=\sum _{i=1}^{N}i{T}_{\mathrm{s}}\left|{e}_{1}(i)\right|$$

#### 4.3. Experimental Setup

- SPEEDAIRE
^{®}Single-acting pneumatic cylinder, model 6CPZ8, which is made from stainless steel, and has a cylinder body with a diameter of 5/16 in, a stroke length 1 in. - Two miniature pressure sensors, type XMLPM60RC23F, for measuring the pressure in the two chambers of the pneumatic cylinder.
- 5/2 solenoid valve B522ADA53C, to control the pneumatic cylinder.
- 24 volts push button to activate the 5/2 proportional solenoid valve.
- A dSPACE DS1104 data acquisition system for receiving sensor signals and triggering control signals.
- A power supply to provide electrical energy.
- A Lenovo Notebook G15, for operation control.

_{s}, is supplied by a gas tank, which is regulated and maintained at a constant pressure by a regulating valve. The input voltage of the proportional solenoid valve is changed by a push button, allowing the pneumatic cylinder rod to move to the expected position. Two pressure sensors, model XMLPM60RC23F, are mounted 10 cm from the cylinder to measure the pressure in the pneumatic cylinder chambers. The dSPACE 1104 data acquisition board, with a real-time data acquisition rate of 0.2 kHz, can receive the analog voltage generated by the sensors and send commands to the push button with the help of a computer. A linear optical encoder is used to measure the displacement of the pneumatic cylinder, connected to the pneumatic cylinder on one end and to the dSPACE DS1104 on the other. The encoder has a resolution of 200 lines per inch and is fixed to the end of the cylinder’s moving rod to ensure correct alignment with the coding strip affixed to the inside of the protective housing. Vibration damping devices such as sponges and brackets are used to protect the encoder, ensuring that the movement of the cylinder rod is accurately captured during sudden strong air pressure shocks. Real-time interaction and recording of data are performed by MATLAB/Simulink (R2017b), which is commonly used in industry.

#### 4.4. Experimental Design

_{d}= 200sin(0.5πt) with F = 40 N and the gas source pressure is set to 4 MPa

_{d}= 200sin(0.5πt) with F = 40 N and the gas source pressure is set to 3 MPa.

#### 4.5. Experimental Results and Discussions

## 5. Conclusions

- (1)
- A mathematical model of the polishing robot pneumatic system is established, considering factors such as cylinder dynamics, valve flow characteristics, and friction.
- (2)
- An extended state observer is designed to estimate unmeasured states such as velocity, acceleration and total disturbance. Stability analysis proves that the estimation error is bounded.
- (3)
- A backstepping sliding mode controller is systematically designed and stability is proved using Lyapunov theory, and the controller achieves robust tracking against disturbances.
- (4)
- Co-simulations in AMESim and MATLAB verify the effectiveness of the proposed control method in tracking various trajectories under friction and load disturbances. It demonstrates superior performance over PID control. Comparative experiments on a test bench further validate the proposed method, showing an over 77% average reduction in tracking error compared to PID control.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 14.**Experimental results under the sinusoidal test condition with gas pressure 4 MPa. (

**a**) Trajectory tracking result. (

**b**) Trajectory tracking error. (

**c**) Result of MEP.

**Figure 15.**Experimental results under the ramp test condition with gas pressure 4 MPa. (

**a**) Trajectory tracking result. (

**b**) Trajectory tracking error. (

**c**) Result of MEP.

**Figure 16.**Experimental results under the step test condition with gas pressure 4 MPa. (

**a**) Trajectory tracking result. (

**b**) Trajectory tracking error. (

**c**) Result of MEP.

**Figure 17.**Experimental results under the sinusoidal test condition with gas pressure 3 MPa. (

**a**) Trajectory tracking result. (

**b**) Trajectory tracking error. (

**c**) Result of MEP.

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

R_{c} | Gas constant | 287 J/kg·K |

T_{d} | Temperature | 293 K |

k | Adiabatic coefficient | 1.4 |

m | Mass | 2.2 kg |

bp | Viscous friction coefficient | 50 Nm·s^{−1} |

D | Piston diameter | 40 mm |

d | Rod diameter | 25 mm |

Name | Symbol | Value |
---|---|---|

PID Controller | K_{p} | 15 |

K_{i} | 1 | |

K_{d} | 3 | |

ESO-Based Backstepping Sliding Mode Controller | k_{1} | 10 |

k_{2} | 10 | |

k_{3} | 1 × 10^{3} | |

c_{1} | 1 × 10^{2} | |

c_{2} | 1 × 10^{3} | |

ESO | q_{1} | 4 × 10^{2} |

q_{2} | 6 × 10^{4} | |

q_{3} | 4 × 10^{6} | |

q_{4} | 1 × 10^{8} |

Control Method | E_{MAE} | E_{RMSE} | E_{ITAE} |
---|---|---|---|

PID | 1.325 × 10^{−3} | 5.213 × 10^{−3} | 18.72 |

ESO-based BSMC | 6.562 × 10^{−5} | 1.519 × 10^{−4} | 1.323 |

Parameter | Value or Range |
---|---|

Gas source pressure | 2–8 MPa |

Displacement of cylinder | 0–500 mm |

The area ratio of cylinder champers | 1:2.5 |

Mass of cylinder | 1.6 kg |

Natural frequency of solenoid valve | 80 Hz |

Relief valve flow rate pressure gradient | 150 L/min/bar |

Name | Symbol | Value |
---|---|---|

PID Controller | k_{p} | 15 |

k_{i} | 10 | |

k_{d} | 0.3 | |

ESO-based BSMC | k_{1} | 50 |

k_{2} | 50 | |

k_{3} | 1 × 10^{3} | |

c_{1} | 5 × 10^{2} | |

c_{2} | 1 × 10^{3} |

Case 1 | Case 2 | Case 3 | Case 4 | Average | |
---|---|---|---|---|---|

PID | 5.9 × 10^{−2} | 7.18 × 10^{−2} | 7.32 × 10^{−2} | 6.4 × 10^{−2} | 6.7 × 10^{−2} |

BSMC | 8.5 × 10^{−3} | 9.73 × 10^{−3} | 3.23 × 10^{−2} | 1.1 × 10^{−2} | 1.54 × 10^{−2} |

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## Share and Cite

**MDPI and ACS Style**

Li, Q.; Ding, B.
Design of Backstepping Sliding Mode Control for a Polishing Robot Pneumatic System Based on the Extended State Observer. *Machines* **2023**, *11*, 904.
https://doi.org/10.3390/machines11090904

**AMA Style**

Li Q, Ding B.
Design of Backstepping Sliding Mode Control for a Polishing Robot Pneumatic System Based on the Extended State Observer. *Machines*. 2023; 11(9):904.
https://doi.org/10.3390/machines11090904

**Chicago/Turabian Style**

Li, Qinsheng, and Birong Ding.
2023. "Design of Backstepping Sliding Mode Control for a Polishing Robot Pneumatic System Based on the Extended State Observer" *Machines* 11, no. 9: 904.
https://doi.org/10.3390/machines11090904