# Integrated Path Tracking Controller of Underground Articulated Vehicle Based on Nonlinear Model Predictive Control

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## Abstract

**:**

## 1. Introduction

- A stepper motor provides torque to drive a hydraulic steering valve and control the articulated angle or angular speed [21];
- A proportional directional control valve (DCV) controls flow into the steering cylinder so that controls the articulated angular speed [22]. However, the proportional DCV usually has a dead zone [20], resulting in small articulated angular speeds not being achieved. Furthermore, when the oil pump is powered by the engine, the engine speed and the opening of the valve port will jointly affect the articulated angular speed, increasing the control difficulty;
- A motor controls the variable displacement pump (VDP) to manage the flow into the steering cylinder, so that controls the articulated angular speed. Compared to the DCV-controlled system, the response time of the pump-controlled actuation system is slower [23];
- Based on solution 3, a variable frequency drive (VFD) is applied to control the speed of an electric motor. Therefore, the flow entering into cylinders can be directly controlled either by the pump’s displacement or by the motor’s variable speed [24].

## 2. Upper-Level Controller

#### 2.1. Nonlinear Model-Predictive Control

#### 2.2. Predictive Model

#### 2.3. Controller Design

## 3. Lower-Level Controller

#### 3.1. Steering Controller

#### Controller Design

#### 3.2. Driving Controller

#### 3.2.1. Controller Design

- Load: no-load ($0\text{}\mathrm{t}$), half-load ($3.5\text{}\mathrm{t}$), full-load ($7\text{}\mathrm{t}$).
- Throttle commands:
- a.
- Fixed: $10\%,\text{}20\%,\text{}30\%,\text{}40\%,\text{}50\%,\text{}60\%,\text{}70\%,\text{}80\%,\text{}90\%,\text{}100\%$;
- b.
- Varying (5 seconds step): $25\%\to 50\%\to 75\%\to 100\%$, $30\%\to 60\%\to 90\%$.

#### 3.2.2. Data Preprocessing

#### 3.2.3. Training

Algorithms 1: The neural network training process. |

Input: training set (control command: engine throttle opening)Input: training set (speed, acceleration)Input: parameters $\lambda $, the learning rate $\eta $, the number of iterations $S$ |

1: for $i=0$ to $S$ do |

2: calculate $E$ for a small batch (number of samples $B\u201dM$) |

3: calculate $\frac{\partial E}{\partial \lambda}$ by back-propagation |

4: $\mathsf{\Delta}\lambda \left(i+1\right)=\mathsf{\Delta}\lambda \left(i\right)-\eta \frac{\partial E}{\partial \lambda}$ |

5: $\lambda \left(i+1\right)=\lambda \left(i\right)+\mathsf{\Delta}\lambda \left(i+1\right)$ update the parameters |

6: end for |

7: Return $\lambda $; return the trained parameters |

## 4. Experimental Vehicle

#### 4.1. Steering-by-Wire System

#### 4.2. Driving-by-Wire System

#### 4.3. Sensors and Controllers

## 5. Field Testing

#### 5.1. Lower-Level Controller Verification

#### 5.1.1. Steering Controller Verification

#### 5.1.2. Driving Controller Verification

#### 5.2. Integrated Path Tracking Controller Testing

#### 5.2.1. Pre-Experiment

#### 5.2.2. Tracking Result

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Relationship between articulated angle, control command, and articulated angular speed: (

**a**) left turn; (

**b**) right turn.

**Figure 6.**Articulated dump truck with no-load condition and full-load condition: (

**a**) no load; (

**b**) full load.

**Figure 20.**Path tracking results at 1 m/s: (

**a**) path comparison results; (

**b**) lateral error and heading error comparison results; (

**c**) computation time comparison results; (

**d**) control inputs comparison results.

**Figure 21.**Path tracking results at 2 m/s: (

**a**) path comparison results; (

**b**) lateral error and heading error comparison results; (

**c**) computation time comparison results; (

**d**) control inputs comparison results.

Major Parameters | Unit | Value |
---|---|---|

Overall vehicle mass | Kg | 7400 |

Maximum load capacity | Kg | 7000 |

Maximum folding angle | deg | ±42 |

Length from front axle to articulated center | mm | 1620 |

Length from rear axle to articulated center | mm | 1923 |

Inside steering radius | mm | 3955 |

Outer steering radius | mm | 5850 |

Tire rolling radius | mm | 519 |

Wheelbase | mm | 1322 |

Engine | DEUTZ-F6L914 | |

Integrated torque converter transmission: | DANA-1201FT20000 |

Parameters | Value |
---|---|

${N}_{\mathrm{p}}$ | 20 |

${N}_{\mathrm{c}}$ | $10$ |

$T$ | $0.1\text{}\mathrm{s}$ |

$P$ | $\mathrm{diag}\left(0.1,0.1,0.5,0\right)\in {\mathbb{R}}^{4\times 4}$ |

$Q$ | $\mathrm{diag}\left(0.01,0.01,0.05,0\right)\in {\mathbb{R}}^{4\times 4}$ |

$R$ | $\mathrm{diag}\left(0.01,0.01\right)\in {\mathbb{R}}^{2\times 2}$ |

${\gamma}_{\mathrm{min}},\text{}{\gamma}_{\mathrm{max}}$ | $-0.73\text{}\mathrm{rad},\text{}0.73\text{}\mathrm{rad}$ |

${v}_{\mathrm{min}},\text{}{v}_{\mathrm{max}}$ | $0\text{}\mathrm{m}/\mathrm{s},\text{}4\text{}\mathrm{m}/\mathrm{s}$ |

${\mathsf{\omega}}_{\mathrm{min}},{\text{}\mathsf{\omega}}_{\mathrm{max}}$ | $-0.17\text{}\mathrm{rad}/s,\text{}0.17\text{}\mathrm{rad}/s$ |

$\mathsf{\Delta}{v}_{\mathrm{min}},\text{}\mathsf{\Delta}{v}_{\mathrm{max}}$ | $-0.3\mathrm{m}/{\mathrm{s}}^{2},\text{}0.3\text{}\mathrm{m}/{\mathrm{s}}^{2}$ |

$\mathsf{\Delta}{\mathsf{\omega}}_{\mathrm{min}},\text{}\mathsf{\Delta}{\mathsf{\omega}}_{\mathrm{max}}$ | $-0.17\text{}\mathrm{rad}/{\mathrm{s}}^{2},\text{}0.17\text{}\mathrm{rad}/{\mathrm{s}}^{2}$ |

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**MDPI and ACS Style**

Sun, N.; Zhang, W.; Yang, J.
Integrated Path Tracking Controller of Underground Articulated Vehicle Based on Nonlinear Model Predictive Control. *Appl. Sci.* **2023**, *13*, 5340.
https://doi.org/10.3390/app13095340

**AMA Style**

Sun N, Zhang W, Yang J.
Integrated Path Tracking Controller of Underground Articulated Vehicle Based on Nonlinear Model Predictive Control. *Applied Sciences*. 2023; 13(9):5340.
https://doi.org/10.3390/app13095340

**Chicago/Turabian Style**

Sun, Nan, Wenming Zhang, and Jue Yang.
2023. "Integrated Path Tracking Controller of Underground Articulated Vehicle Based on Nonlinear Model Predictive Control" *Applied Sciences* 13, no. 9: 5340.
https://doi.org/10.3390/app13095340