# Pre-Work for the Birth of Driver-Less Scraper (LHD) in the Underground Mine: The Path Tracking Control Based on an LQR Controller and Algorithms Comparison

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Retrospective: Development of Underground Driver-Less Technology

#### 1.2. Retrospective: Development of Driver-Less Scraper (LHD) in the Underground Mine

## 2. Materials and Methods

#### 2.1. Mathematical Model of Underground Articulated LHD (Scraper)

#### 2.1.1. Kinematics Model of Articulated Scraper

_{1}(X

_{1},Y

_{1}), the rear part of the car body is P

_{2}(X

_{2},Y

_{2}); the center of mass velocity of the front body is v

_{1}, and the center of mass velocity of the rear body is v

_{2}; the length of the front body is L

_{1}, and the length of the rear body is L

_{2}; the slip angle of the front body is ${\mathsf{\alpha}}_{1}$, and the slip angle of the rear body is ${\mathsf{\alpha}}_{2}$; the heading angle of the front body is ${\dot{\theta}}_{1}$ and the heading angle of the rear body is ${\dot{\theta}}_{2}$; the heading angular velocity of the front body is expressed as ${\dot{\theta}}_{1}$, and the heading angular velocity of the rear body is expressed as ${\dot{\theta}}_{2}$.

#### 2.1.2. Location Prediction Model

#### 2.1.3. Deviation Dynamics Model

_{1}+ L

_{2}, it can be obtained:

#### 2.2. Path Tracking of Underground Articulated LHD Based on LQR Controller

#### 2.3. Algorithms

#### 2.3.1. Adaptive GA Algorithm Optimization

- (1)
- Disadvantages of simple genetic algorithms

- (2)
- Improved adaptive genetic algorithm LQR control (LQR–AGA)

#### Encoding

_{1}, q

_{2}, and q

_{3}are genes on the chromosome, while Q is the operation of selection, crossover, and mutation operators after the participation of chromosomes and individuals.

#### Group Value Range

#### Interleaved Mode

#### Variation

#### Parameter Selection

_{1}, q

_{2}, and q

_{3}represent the Q matrix diagonal elements.

- (3)
- Simulation experiment of LQR-AGA control algorithm

#### 2.3.2. Optimization of QPSO Algorithm

- (1)
- Disadvantages of simple PSO algorithm

- (2)
- Quantum Behavior PSO Algorithm (QPSO)

- (3)
- LQR-QPSO control algorithm simulation experiment

#### 2.3.3. ACA Optimization of Ant Colony Algorithm

- (1)
- Ant Colony Algorithm LQR Controller (LQR-ACA)

- (2)
- LQR-ACA control algorithm simulation experiment

## 3. Results

#### 3.1. Comparison of Algorithm Parameter Configuration

#### 3.2. Comparison of Algorithm Results

#### 3.3. Comparison of Simulation Results

## 4. Discussion

- (1)
- The QPSO algorithm has slow operation speed and slow group convergence speed, but it can find the optimal solution;
- (2)
- The AGA algorithm has fast operation speed and fast group convergence, but the optimization result is poor compared with the QPSO algorithm;
- (3)
- The ACA algorithm has slow operation speed and slow population convergence speed, but it can converge to multiple extreme points and has a large space for optimization.

- (1)
- Firstly, the biggest characteristic of the intelligent cluster algorithms is the problem of premature data; in order to solve the problem, the development direction of this kind of algorithm is to improve the intelligent cluster algorithm;
- (2)
- Secondly, in terms of LQR parameter configuration, the fitness function is the main reason for the slow operation speed of the intelligent cluster algorithm; to solve the problem of operation speed, it is necessary to simplify and redefine the fitness function of LQR to reduce the operation time.

- (1)
- Optimize the controller itself.

- (2)
- Algorithm optimization.

## 5. Conclusions

- (1)
- For articulated LHD path tracking control problems, we studied the kinematics model of the articulated LHD body through the analysis of the kinematics modeling, determining the articulated LHD vehicle reference speed of the anchor point, course angular velocity, turning angular velocity, and the mathematical relationship between the scraper speed and steering angular velocity;
- (2)
- For the selection of control scheme, based on the kinematics model of the articulated LHD lateral error identifying the scraper and heading angle error, the error between the steering angle and the curvature of state space, according to the state space, is put forward to the steering angle control to control the amount of articulated LHD vehicle location of LQR controller, linear quadratic linear optimal control;
- (3)
- Aiming at the problem of the difficult parameter selection of the LQR controller, we propose the LQR controller scheme optimized by the intelligent cluster algorithm, compare the advantages and disadvantages of different clustering algorithms, and put forward a feasible implementation scheme for path tracking control of the intelligent scraper.
- (4)
- This paper serves as a guide to starting a conversation; we did not clearly indicate which algorithm (AGA, OPSO, or ACA) is better, because each of them has its own advantages and disadvantages. We will continue research in this direction and hope more and more researchers will be interested in this direction as well.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 9.**(

**A**–

**C**) Population distribution of the 1st, 19th, and 49th generations; (

**D**) Average fitness per generation.

**Figure 14.**Population distribution of the 1st, 31th, and 80th generations; and average fitness per generation.

**Figure 18.**Population distribution of the 1st, 31th, and 80th generations; and average fitness per generation.

Parameter Name | Numerical Value |
---|---|

Distance from front bridge to the articulation point (${\mathrm{L}}_{1}/\mathrm{m}$) | 1.766 |

Rear bridge distance to articulation point distance $({\mathrm{L}}_{2}/\mathrm{m})$ | 1.866 |

Tire diameter (d/m) | 1.32 |

Body width (W/m) | 2.27 |

Articulated steering angle change range $(\mathsf{\gamma}/\mathrm{rad})$ | 0.30π |

Maximum speed ${\mathrm{V}}_{\mathrm{max}}$ (m/s) | 7.2 |

Maximum steering angle speed change range (${\mathsf{\gamma}}_{\mathrm{max}}/\mathrm{rad}\times {\mathrm{s}}^{-1}$) | 0.17 |

Parameter Name | Numerical Value |
---|---|

Population size: N | 30 |

Iterations: G | 50 |

Cross-crossing probability | 0.2 |

Adaptive variation constant | 0.2 |

Variation constant b | 3 |

Population survival range | 0–50 |

Algorithm | Weighted Matrix Q | Linear Feedback Matrix K | Suitability | ||||
---|---|---|---|---|---|---|---|

${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | ||

AGA | 1.4685 | 33.5161 | 33.8515 | 1.0605 | 6.3752 | 6.5008 | 18,220 |

Parameter Name | Numerical Value |
---|---|

Iterations: G | 80 |

Population size: N | 30 |

Termination of the inertia weight ${\mathrm{w}}_{\mathrm{end}}$ | 0.4 |

Initial inertia weight ${\mathrm{w}}_{\mathrm{ini}}$ | 0.9 |

Learning factor ${\mathrm{a}}_{1}$ | 1.5 |

Learning factor ${\mathrm{a}}_{2}$ | 1.5 |

Particle taking value limit ${\mathrm{q}}_{\mathrm{max}}$ | 400 |

Maximum speed ${\mathrm{v}}_{\mathrm{max}}$ | 1.0 |

Initial shrinkage factor of expansion ${\mathsf{\alpha}}_{\mathrm{b}}$ | 1.0 |

Termination shrinkage—expansion factor ${\mathsf{\alpha}}_{\mathrm{e}}$ | 0.5 |

Algorithm | Weighted Matrix Q | Linear Feedback Matrix K | Suitability | ||||
---|---|---|---|---|---|---|---|

${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | ||

QPSO | 40.8844 | 49.0588 | 43.0995 | 5.2700 | 10.6900 | 6.7451 | 17,330 |

Parameter Name | Numerical Value |
---|---|

Ant number: ant | 30 |

Search times: G | 100 |

Hormone play factor ${w}_{ini}$ | 0.4 |

Transfer probability ${P}_{0}$ | 0.2 |

Algorithm | Weighted Matrix Q | Linear Feedback Matrix K | Suitability | ||||
---|---|---|---|---|---|---|---|

${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | ||

ACA | 0.8419 | 7.0752 | 40.6476 | 0.8288 | 4.1523 | 6.5550 | 15,333 |

Algorithm Name | Population Size | Number of Convergence Iterations | Operation Time |
---|---|---|---|

Adaptive genetic algorithm (AGA) | 30 | 50 | 15 min |

Quantum behavior particle swarm algorithm (QPSO) | 30 | 80 | 25 min |

Ant colony algorithm (ACA) | 30 | No convergence | 30 min |

Algorithm | Weighted Matrix Q | Linear Feedback Matrix K | Suitability | ||||
---|---|---|---|---|---|---|---|

${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{k}}_{1}$ | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{3}$ | ||

AGA | 1.4685 | 33.5161 | 33.8515 | 1.0605 | 6.3752 | 6.5008 | 18,220 |

QPSO | 40.8844 | 49.0588 | 43.0995 | 5.2700 | 10.6900 | 6.7451 | 17,330 |

ACA | 0.8419 | 7.0752 | 40.6476 | 0.8288 | 4.1523 | 6.5550 | 15,333 |

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Yu, H.; Zhao, C.; Li, S.; Wang, Z.; Zhang, Y.
Pre-Work for the Birth of Driver-Less Scraper (LHD) in the Underground Mine: The Path Tracking Control Based on an LQR Controller and Algorithms Comparison. *Sensors* **2021**, *21*, 7839.
https://doi.org/10.3390/s21237839

**AMA Style**

Yu H, Zhao C, Li S, Wang Z, Zhang Y.
Pre-Work for the Birth of Driver-Less Scraper (LHD) in the Underground Mine: The Path Tracking Control Based on an LQR Controller and Algorithms Comparison. *Sensors*. 2021; 21(23):7839.
https://doi.org/10.3390/s21237839

**Chicago/Turabian Style**

Yu, Haoxuan, Chenxi Zhao, Shuai Li, Zijian Wang, and Yulin Zhang.
2021. "Pre-Work for the Birth of Driver-Less Scraper (LHD) in the Underground Mine: The Path Tracking Control Based on an LQR Controller and Algorithms Comparison" *Sensors* 21, no. 23: 7839.
https://doi.org/10.3390/s21237839