# Estimation of Soil Characteristic Parameters for Electric Mountain Tractor Based on Gauss–Newton Iteration Method

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## Abstract

**:**

## 1. Introduction

## 2. Algorithm for Solving Soil Characteristic Parameters

#### 2.1. Wheel Sinkage

_{0}is obtained from the geometrical relationship.

#### 2.2. Normal Stress of Interface

_{φ}is the friction modulus; and n is the subsidence index.

#### 2.3. Shear Stress of Interface

_{j}of the wheel.

_{j}can be obtained from velocity vector analysis:

_{j}is divided into horizontal movement velocity V and tangential velocity ${V}_{t}$ through vector analysis.

#### 2.4. Mechanics Equation of Wheel–Soil Interaction

#### 2.5. Algorithm for Solving Soil Characteristic Parameters

_{c}, friction modulus k

_{φ}, and subsidence index n. The effects of these parameter changes on vertical load W, hook traction force DP, and driving torque T were experimentally tested. The results show that W, DP, and T are insensitive to changes in the values of the adhesion coefficient c, i.e., the changes in c only slightly change the values of W, DP, and T. Therefore, the solution of c is not considered in this paper, and it is considered as a known value during the solution process. So, these four parameters φ, k

_{c}, k

_{φ}and n are selected here to represent soil characteristics.

## 3. Algorithm Simplification

_{φ}are merged to form joint parameter K

_{L}:

_{L}can be detected through two wheels of different widths during actual operation. Therefore, it can be concluded that:

## 4. Algorithm Solving and Simulation

#### 4.1. Solving Based on Gauss–Newton Iteration Method

^{(k)}input for the solution model. The soil characteristic parameters (K

_{c}, K

_{φ}, φ, and n) constitute the model output x

^{(k+1)}. The process starts from the set initial value and continuously updates until the error is less than the set error threshold ε. Among them, F’(x

^{(k)}) is the Jacobi matrix of the Gauss–Newton iteration method, also known as the sensitivity coefficient.

#### 4.2. Algorithm Simulation and Discussion

## 5. The Use of Identified Soil Parameters

## 6. Conclusions

- (1)
- Due to the complexity and exponential form of the shear stress model, it requires a large amount of computation and high accuracy in subsequent calculations. Therefore, this paper adopted feature extraction to simplify the model from an exponential function to a linear function, and the experimental results prove that the simplification is reasonable.
- (2)
- Owing to the nonlinear characteristics of the algorithm for solving soil characteristic parameters, calculations are difficult. The algorithm was simplified using the Simpson formula and solved using the Newton iteration method. Due to the simplified algorithm not requiring numerical integration for each calculation cycle, the solving speed is improved.
- (3)
- To verify the stability of the solving algorithm, different initial iteration values with an error of 20% or 50% from the true value were selected for simulation calculations of soil characteristic parameters such as internal friction angle, settlement index, and the join parameter of soil cohesion modulus and friction modulus. The results showed that the error was kept within 2%, and the calculation time did not exceed one second, which demonstrates that the algorithm has strong robustness to measurement noise and initial conditions.
- (4)
- In order to further verify the universality of the model, different soil and tractor parameters were used to test the algorithm, and the calculation time did not exceed one second, and the calculation accuracy was basically consistent, which means it can meet the requirements for quickly and accurately identifying soil characteristic parameters.
- (5)
- Through utilizing identified soil parameters to forecast the driving torque needed for a wheeled tractor to traverse terrain and contrasting it with the maximum driving torque that the wheels can generate, one can ascertain whether a tractor is capable of traversing unfamiliar terrain.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Comparison of the distribution curves of the improved shear stress model and the original shear model.

**Figure 6.**Iteration of soil internal friction angle for different soils based on the condition of 20% initial value error from the true value. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

**Figure 7.**Iteration of joint parameter for different soils based on the condition of 20% initial value error from the true value condition. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

**Figure 8.**Iteration of subsidence index for different soils based on the condition of 20% initial value error from the true value. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

**Figure 9.**Iteration of soil internal friction angle for different soils based on the condition of 50% initial value error from the true value. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

**Figure 10.**Iteration of the joint parameter for different soils based on the condition of 50% initial value error from the true value. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

**Figure 11.**Iteration of the subsidence index for different soils based on the condition of 50% initial value error from the true value. (

**a**) Based on wet clay; (

**b**) based on sandy loam.

First Author | Algorithm | Estimate Parameter | Limitations |
---|---|---|---|

Iagnemma [13] | Least square | Cohesion and internal friction angle | Need a lot of trials to verify its applicability in many situations |

Kang [14] | Least square | Cohesion and internal friction angle | Need data conditioning, need an estimate or known value of deformation modulus |

Hutangkabodee [16] | Newton–Raphson method | Internal friction angle, shear deformation modulus, lumped pressure–sinkage coefficient | Improper selection of initial value may lead the wrong solution or no solution |

Ray [18] | Bayesian multiple-model estimation | No approximations of shear and normal stress distributions | The model structure posed for each hypothesis needs to be consistent |

Xue [19] | Trained multiple-output least squares support vector machine | Cohesion, internal friction angle, and shear deformation modulus | Needed to be tested under various conditions |

Li [20] | Adaptive robust extended Kalman filter | Internal friction angle and sinkage exponent | The other terrain parameters are fixed with nominal values |

Soil Parameters | Wet Clay | Sandy Loam |
---|---|---|

Coefficient of adhesion C/kPa | 7.58 | 1.7 |

Internal friction angle φ/° | 14 | 29 |

Shear elasticity of soil K/m | 0.025 | 0.025 |

Soil subsidence index n | 0.85 | 0.7 |

Soil cohesion modulus k_{c} | 43.68 | 5.3 |

Soil friction modulus k_{φ} | 499.3 | 1515 |

Wheel radius r/m | 0.5 | 0.5 |

Wheel width b/m | 0.3 | 0.3 |

Soil Parameters | Ground Type | Iteration Initial Value | Solution Results | Errors/% | Time/s |
---|---|---|---|---|---|

Internal friction angle φ/° | Wet clay | 11.2 | 13.9671 | 0.235 | 0.37 |

Sandy loam | 23.2 | 29.004 | 0.014 | 0.73 | |

Joint parameter K_{L} | Wet clay | 515.92 | 635.522 | 1.8 | 0.37 |

Sandy loam | 1226.16 | 1539.01 | 0.411 | 0.73 | |

Soil subsidence index n | Wet clay | 0.68 | 0.8498 | 0.024 | 0.37 |

Sandy loam | 0.56 | 0.703 | 0.429 | 0.73 |

Soil Parameters | Ground Type | Iteration Initial Value | Solution Results | Errors/% | Time/s |
---|---|---|---|---|---|

Internal friction angle φ/° | Wet clay | 7 | 14.13 | 0.9 | 0.44 |

Sandy loam | 14.5 | 29.004 | 0.014 | 0.84 | |

Joint parameter K_{L} | Wet clay | 322.45 | 635.522 | 1.8 | 0.44 |

Sandy loam | 766.35 | 1533.9 | 0.0783 | 0.84 | |

Soil subsidence index n | Wet clay | 0.425 | 0.8498 | 0.024 | 0.44 |

Sandy loam | 0.35 | 0.705 | 0.714 | 0.84 |

Soil Parameters | Wet Clay |
---|---|

Coefficient of adhesion C/kPa | 7.58 |

Internal friction angle φ/° | 14 |

Shear elasticity of soil K/m | 0.025 |

Soil subsidence index n | 0.85 |

Soil cohesion modulus K_{c} | 43.68 |

Soil friction modulus K_{φ} | 499.3 |

Wheel radius r/m | 1.09 |

Wheel width b/m | 0.45 |

Soil Parameters | Tractors | Iteration Initial Value | Solution Results | Errors/% | Time/s |
---|---|---|---|---|---|

Internal friction angle φ/° | Tractor 1 | 11.2 | 13.9671 | 0.235 | 0.37 |

Tractor 2 | 11.2 | 13.999 | 0.007 | 0.33 | |

Joint parameter K_{L} | Tractor 1 | 515.92 | 635.522 | 1.8 | 0.37 |

Tractor 2 | 476.8 | 596.18 | 0.03 | 0.33 | |

Soil subsidence index n | Tractor 1 | 0.68 | 0.8498 | 0.024 | 0.37 |

Tractor 2 | 0.68 | 0.8503 | 0.035 | 0.33 |

Soil Parameters | Tractors | Iteration Initial Value | Solution Results | Errors/% | Time/s |
---|---|---|---|---|---|

Internal friction angle φ/° | Tractor 1 | 7 | 14.13 | 0.9 | 0.44 |

Tractor 2 | 14.5 | 29.004 | 0.014 | 0.57 | |

Joint parameter K_{L} | Tractor 1 | 322.45 | 635.522 | 1.8 | 0.44 |

Tractor 2 | 766.35 | 1533.9 | 0.0783 | 0.57 | |

Soil subsidence index n | Tractor 1 | 0.425 | 0.8498 | 0.024 | 0.44 |

Tractor 2 | 0.35 | 0.705 | 0.714 | 0.57 |

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**MDPI and ACS Style**

Xi, Z.; Feng, T.; Liu, Z.; Xu, H.; Zheng, J.; Xu, L.
Estimation of Soil Characteristic Parameters for Electric Mountain Tractor Based on Gauss–Newton Iteration Method. *World Electr. Veh. J.* **2024**, *15*, 217.
https://doi.org/10.3390/wevj15050217

**AMA Style**

Xi Z, Feng T, Liu Z, Xu H, Zheng J, Xu L.
Estimation of Soil Characteristic Parameters for Electric Mountain Tractor Based on Gauss–Newton Iteration Method. *World Electric Vehicle Journal*. 2024; 15(5):217.
https://doi.org/10.3390/wevj15050217

**Chicago/Turabian Style**

Xi, Zhiqiang, Tian Feng, Zhijun Liu, Huaijun Xu, Jingyang Zheng, and Liyou Xu.
2024. "Estimation of Soil Characteristic Parameters for Electric Mountain Tractor Based on Gauss–Newton Iteration Method" *World Electric Vehicle Journal* 15, no. 5: 217.
https://doi.org/10.3390/wevj15050217