# Simulation Research on the Grouser Effect of a Reconfigurable Wheel-Crawler Integrated Walking Mechanism Based on the Surface Response Method

^{1}

^{2}

^{*}

## Abstract

**:**

## Featured Application

**The reconfigurable wheel-crawler walking mechanism involved in this study is a type of walking mechanism which can manage a high moving speed and good terrain adaptability given the unstructured terrain existing in the Qinghai–Tibet Plateau’s scientific research station. The mechanism can achieve the purpose of switching between circular tire mode and triangular crawler mode without changing the working parts through the integration of wheeled and tracked components. In this study, the grouser effect of the walking mechanism was analyzed, and the parameters with the largest traction force were obtained, which can guide the machining of a prototype of the reconfigurable wheel-shoe integrated walking mechanism.**

## Abstract

## 1. Introduction

## 2. Structural Analysis of the Reconfigurable Wheel-Crawler Integrated Walking Mechanism

## 3. Theoretical Analysis of the Grouser Effect

#### 3.1. Relationship between the Vertical Load and Subsidence of the Grouser

_{1}is the unit of pressure of the ground acting on the track shoe; p

_{2}is the unit of pressure of the soil acting on the unit of pressure of the track shoe’s thickness; and F

_{ma}

_{x}is the maximum tractive force produced by the soil.

_{c}is the coefficient of soil cohesion, k

_{φ}is the coefficient of the soil’s internal friction angle, z is the amount of soil subsidence and n is the soil deformation index.

_{1}of the ground acting on the track shoe and the unit of pressure p

_{2}of the soil acting on the grouser are the following, respectively.

_{2}is the equivalent width of the grouser and B

_{1}is the width of the track shoe minus the width of the grouser.

#### 3.2. Calculation and Analysis of the Traction Force under the Grouser Effect

_{1}acting on the surface of the track shoe, the force F

_{2}acting on the bottom of the grouser, the force F

_{3}acting on both sides of the track shoe and the grouser, and the force F

_{4}acting on the grouser. The shear stress of a single-track shoe on the ground is

_{1 =}0,

_{3}acting on both sides of the track shoe and the grouser is

_{4}acting on the spine was analyzed using the principle of a retaining wall, as shown in Figure 7. According to Rankine’s earth pressure theory, under the action of a uniformly distributed load p on the track shoe, the passive ground pressure on the horizontal grouser is

_{φ}is the flow value.

_{4}acting on the grouser can be obtained as follows

_{1}is the number of horizontal grousers, n

_{1}can be 0 or 1; and n

_{2}is the number of inclined grousers, either 0 or 2. When n

_{1}= 1 and n

_{2}= 0, the force on the grouser of a T-shaped track, a π-shaped track, and a V-shaped track can be solved; when n

_{1}= 1 and n

_{2}= 2, the force on the grouser of a K-shaped track can be solved; and when n

_{1}= 0 and n

_{2}= 2, the force on the grouser of a figure-eight track can be solved.

## 4. Simulation Test of the Grouser Effect

#### 4.1. Determination of the Significant Factors of the Grouser Effect

_{1}acting on the surface of the track shoe, the force F

_{2}acting on the bottom of the grouser, the force F

_{3}acting on both sides of the track shoe and the grouser, and the force F

_{4}acting on the grouser and the bulldozing resistance R collectively affects the hitch pull. Through an in-depth analysis, it was found that the parameters related to the crawler such as b, l, W, h, and λ, and soil parameters such as k

_{c}, k

_{φ}, γ, and N

_{φ}have a major impact on the traction force. Among these, the moving speed of the robot and the track’s parameters are controllable factors, but the soil parameters are uncontrollable factors. The traction process model in Figure 9 was obtained.

#### 4.2. Construction of the Dynamic Simulation Model

#### 4.3. Design of the Simulation Experiment

#### 4.3.1. The Full Factorial Experimental Design

#### 4.3.2. Response Surface Design Experiments

## 5. Simulation Test of the Grouser Effect

#### 5.1. Establishment of the Regression Equation of the Grouser Effect

^{2}was 0.982 and the adjusted R

^{2}was about 0.966, indicating that the output data were good. Among these statistics, the p-values of the response values of the grouser’s height h, the grouser’s pitch l, h

^{2}, h×l and λ×h were all less than 0.05, indicating that these parameters were significant. The p-values of λ×l, l

^{2}, and λ

^{2}were all greater than 0.05, indicating that these parameters were not significant and needed to be optimized. For the ratio λ of the grouser’s thickness to the length of the track shoe, the p-value was 0.0524, which is greater than 0.05, but since this response exists in the corresponding interaction λ*h, λ must be retained, marked with ^. In summary, the factors with significant correlation in ANOVA were marked with * in Table 7.

^{2}decreased because of the reduction in the number of items in the model, but the difference between R

^{2}and the adjusted R

^{2}was smaller. At the same time, the measured value and predicted value of the traction force were compared, as shown in Figure 13; this also shows that the simplified regression model is better.

#### 5.2. Coupling Analysis of Multiple Factors

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bai, Y.; Sun, L.; Zhang, M.; Liu, X.; Li, Z. Progress of research on terramechanics for tracked mobile robots. Mech. Des.
**2020**, 10, 1–13. [Google Scholar] [CrossRef] - Bekker, M.G. Theory of Land Locomotion; The Univ. of Michigan Press: Ann Arbor, MI, USA, 1956. [Google Scholar]
- Bekker, M.G. Off-Road Locomotion: Research and Development in Terramechanics; The Univ. of Michigan Press: Ann Arbor, MI, USA, 1960. [Google Scholar]
- Xiao, W.; Zhang, Y. Design of Wheel of Manned Lunar Rover and Research on Terramechanics Model for Wheel-Terrain Interaction Based on Elastic Wheel. J. Mech. Eng.
**2016**, 10, 119–125. [Google Scholar] [CrossRef] - Wismer, R.D.; Luth, H.J. Off-road traction prediction for wheeled vehicles. J. Terramech.
**1974**, 10, 49–61. [Google Scholar] [CrossRef] - Wong, J.-Y.; Reece, A.R. Prediction of rigid wheel performance based on the analysis of soil-wheel stresses part I. Performance of driven rigid wheels. J. Terramech.
**1967**, 4, 81–98. [Google Scholar] [CrossRef] - Bekker, M.G. Mechanics of off-the-road locomotion. Proc. Inst. Mech. Eng. Automob. Div.
**1962**, 16, 25–44. [Google Scholar] [CrossRef] - Liang, D.; Deng, Z.; Gao, H.; Tao, J.; Iagnemma, K.D.; Liu, G. Interaction mechanics model for rigid driving wheels of planetary rovers moving on sandy terrain with consideration of multiple physical effects. J. Field Robot.
**2015**, 32, 827–859. [Google Scholar] [CrossRef] - Guo, J.; Li, W.; Gao, H.; Ding, L.; Deng, Z. In-situ wheel sinkage estimation under high slip conditions for grouser-wheeled planetary rovers: Another immobility index. Mech. Mach. Theory
**2021**, 158, 104243. [Google Scholar] [CrossRef] - Yang, C.B.; Gu, L.; Lv, W.W. Study of Factors with Effects on Tracked Vehicle Driving Resistance Basis of Bekker Theory. Appl. Mech. Mater.
**2013**, 288, 80–83. [Google Scholar] [CrossRef] - Yang, C.B.; Gu, L.; Li, Q. Finite element simulation of Track shoe and Ground adhesion. Appl. Mech. Mater.
**2014**, 644–650, 402–405. [Google Scholar] [CrossRef] - Yang, C.; Cai, L.; Liu, Z.; Tian, Y.; Zhang, C. A calculation method of track shoe thrust on soft ground for splayed grouser. J. Terramech.
**2016**, 65, 38–48. [Google Scholar] [CrossRef] - Yokoyama, A.; Nakashima, H.; Shimizu, H.; Miyasaka, J.; Ohdoi, K. Effect of open spaces between grousers on the gross traction of a track shoe for lightweight vehicles analyzed using 2d dem. J. Terramech.
**2019**, 90, 31–40. [Google Scholar] [CrossRef] - Li, J.; Bie, E.; Wu, Z.; Jin, S.; He, J. Research on Maximum Traction Characteristics of 3D Model of Track Shoe-Soil. Agric. Equip. Veh. Eng.
**2016**, 57, 6. [Google Scholar] [CrossRef] - Li, J.; Li, Q.; Zhou, J.; Zhang, Y.; Xie, C. Analysis of Track-Terrain Interaction on Soft Soil. Acta Armament.
**2012**, 12, 1423–1429. [Google Scholar] - Gan, L.; Huang, Q.; Chen, S. Study on Design and Traction Performance of Ostrich-foot Sandy Track Shoe. Mech. Sci. Technol. Aerosp. Eng.
**2022**, 41, 673–680. [Google Scholar] - Xu, Z.; Liu, Y.; Yang, G.; Xia, J.; Dou, Z.; Meng, Q.; Xu, X. Research on contact model of track-soft sediment and traction performance of four-tracked seabed mining vehicle. Ocean Eng.
**2022**, 259, 111902. [Google Scholar] [CrossRef] - Yang, C.B.; Dong, M.M.; Gu, L.; Li, Q.; Gao, X.D. Research on soil shear strength considering the shape of grouser. Trans. Beijing Inst. Technol.
**2015**, 35, 1118–1121. [Google Scholar] [CrossRef] - Li, J.; Sun, S.; Peng, J. Experimental Study on Effect of Structural Parameters of Track Shoe on Traction Force. Agric. Equip. Veh. Eng.
**2015**, 55, 15–18. [Google Scholar] - Chen, Z.; Xue, D.; Wang, G.; Cui, D.; Fang, Y.; Wang, S. Simulation and optimization of the tracked chassis performance of electric shovel based on DEM-MBD. Powder Technol.
**2021**, 390, 428–441. [Google Scholar] [CrossRef] - Zhou, L.; Gao, J.; Hu, C.; Li, Q. Numerical simulation and testing verification of the interaction between track and sandy ground based on discrete element method. J. Terramech.
**2021**, 95, 73–88. [Google Scholar] [CrossRef] - Yang, K.; Gan, L.; Chen, S.; Xu, X.; Sun, X.; Zhou, Q.; Jiang, X.; Hu, D.; Liu, D.; Wang, Z. Research on the design and traction characteristics of a vehicle track shoe for sandy land. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2022**, 236, 5826–5835. [Google Scholar] [CrossRef] - Taghinezhad, J.; Alimardani, R.; Masdari, M.; Mahmoodi, E. Performance optimization of a dual-rotor ducted wind turbine by using response surface method. Energy Convers. Manag. X
**2021**, 12, 100120. [Google Scholar] [CrossRef] - Ahmed, M.; Ayub, A.; Sheikh, N.A.; Shahzad, M.W.; Haroon, M.; Imran, M. Thermodynamic optimization and performance study of supercritical CO
_{2}thermodynamic power cycles with dry cooling using response surface method. Int. Commun. Heat Mass Transf.**2023**, 142, 106675. [Google Scholar] [CrossRef] - Kostić, S.; Vasović, N.; Sunarić, D. Slope stability analysis based on experimental design. Int. J. Geomech.
**2016**, 16, 3–36. [Google Scholar] [CrossRef] - Kostić, S.; Vasović, N.; Marinković, B. Robust optimization of concrete strength estimation using response surface methodology and Monte Carlo simulation. Eng. Optim.
**2017**, 49, 864–877. [Google Scholar] [CrossRef]

**Figure 2.**The process of switching to the walking mode of the reconfigurable wheel-crawler integrated mechanism. (

**a**) The circular wheel mode; (

**b**) the swing arm is recovered, and the traveling mechanism is switched from the circular wheel mode to the triangular crawler mode; (

**c**) the triangular crawler mode; (

**d**) the triangular crawler mode with the bracket is below; (

**e**) the process of changing from the triangular crawler mode to the circular wheel mode, with the bracket is retracting, the swing arm is expanding; (

**f**) the circular wheel mode after the mode switched.

**Figure 3.**The more common track shoe shapes are as follows: (

**a**) T-shape track shoe; (

**b**) π-shape track shoe; (

**c**) V-shape track shoe; (

**d**) K-shape track shoe.

**Figure 4.**Schematic diagram of the interaction force between the track shoe and the soil. (

**a**) Schematic diagram of the interaction force between an L-shaped track shoe and the soil. (

**b**) Schematic diagram of the interaction force between a π-shaped track shoe and the soil. (

**c**) Schematic diagram of the interaction force between a V-shaped track shoe and the soil.

**Figure 5.**Schematic diagram of the interaction force between a K-shaped track shoe and the soil. (

**a**) Side view of the force diagram of a K-shaped track shoe. (

**b**) Dimension diagram of a K-shaped track shoe.

**Figure 14.**Analysis of residual error. (

**a**) Scatterplot of the predicted values versus the residuals; (

**b**) normal probability plot.

**Figure 15.**Scatterplot of the residuals vs. responses. (

**a**) Scatterplot of the residuals vs. h; (

**b**) scatterplot of the residuals vs. l; (

**c**) scatterplot of the residuals vs. λ.

Component | Quantity | Parameters | |
---|---|---|---|

Driving wheel | 1 | Number of Teeth | 15 |

Base Circle Radius/mm | 21.3844 | ||

Pitch Circle Radius/mm | 22.3844 | ||

Induction wheel | 2 | Inner Flange Radius/mm | 26 |

Wheel Radius/mm | 23.5 | ||

Inner Flange Width/mm | 35 | ||

Total Width/mm | 45 | ||

Road wheel | 12 | Wheel and Hub Width/mm | 50 |

Wheel Radius/mm | 13 | ||

Hub Width/mm | 30 | ||

Track shoes | Determined by width | Pin Radius/mm | 1.65 |

Track Link Left Length/mm | 13 | ||

Left Pin Height/mm | −10.2, 3 | ||

Track Link Height/mm | 6 | ||

Grouser Width/mm | 129 |

Definitions | Parameters |
---|---|

Terrain Stiffness (k_{c})/kN·m^{−(n+1)} | 3.7389 × 10^{−2} |

Terrain Stiffness (k_{φ})/kN·m^{−(n+1)} | 1.0425 × 10^{−2} |

Exponential Number (n) | 1.1 |

Cohesion (c)/kPa | 3.3 × 10^{−3} |

Shearing Resistance Angle/° | 33.7 |

Shearing Deformation Modulus/MPa | 25 |

Sinkage Ratio | 1.9 × 10^{−2} |

Response | Parameters | ||
---|---|---|---|

−1 | 0 | 1 | |

λ | 0.1 | 0.2 | 0.3 |

h | 5 | 15.3 | 25.6 |

l | 12 | 16 | 20 |

v | 0.18 | 0.36 | 0.54 |

No. | Model | λ | h | l | v | FX |
---|---|---|---|---|---|---|

1 | −−−− | 0.1 | 5 | 12 | 0.18 | 47.04 |

2 | −−−+ | 0.1 | 5 | 12 | 0.54 | 24.25 |

3 | −−+− | 0.1 | 5 | 20 | 0.18 | 32.9 |

4 | −−++ | 0.1 | 5 | 20 | 0.54 | 23.55 |

5 | −+−− | 0.1 | 25.6 | 12 | 0.18 | 183.92 |

6 | −+−+ | 0.1 | 25.6 | 12 | 0.54 | 180.94 |

7 | −++− | 0.1 | 25.6 | 20 | 0.18 | 151.34 |

8 | −+++ | 0.1 | 25.6 | 20 | 0.54 | 149.58 |

9 | 0000 | 0.2 | 15.3 | 16 | 0.36 | 72.94 |

10 | 0000 | 0.2 | 15.3 | 16 | 0.36 | 72.94 |

11 | 0000 | 0.2 | 15.3 | 16 | 0.36 | 72.94 |

12 | +−−− | 0.3 | 5 | 12 | 0.18 | 44.23 |

13 | +−−+ | 0.3 | 5 | 12 | 0.54 | 64.07 |

14 | +−+− | 0.3 | 5 | 20 | 0.18 | 29.92 |

15 | +−++ | 0.3 | 5 | 20 | 0.54 | 23.94 |

16 | ++−− | 0.3 | 25.6 | 12 | 0.18 | 168.33 |

17 | ++−+ | 0.3 | 25.6 | 12 | 0.54 | 113.91 |

18 | +++− | 0.3 | 25.6 | 20 | 0.18 | 107.56 |

19 | ++++ | 0.3 | 25.6 | 20 | 0.54 | 107 |

No. | Model | λ | h | l | FX |
---|---|---|---|---|---|

1 | −−− | 0.1 | 5 | 12 | 47.04 |

2 | −−+ | 0.1 | 5 | 20 | 32.9 |

3 | a00 | 0.1 | 15.3 | 16 | 83.31 |

4 | −+− | 0.1 | 25.6 | 12 | 183.92 |

5 | −++ | 0.1 | 25.6 | 20 | 151.34 |

6 | 0a0 | 0.2 | 5 | 16 | 48.56 |

7 | 00a | 0.2 | 15.3 | 12 | 86.52 |

8 | 000 | 0.2 | 15.3 | 16 | 84.36 |

9 | 000 | 0.2 | 15.3 | 16 | 84.36 |

10 | 000 | 0.2 | 15.3 | 16 | 84.36 |

11 | 000 | 0.2 | 15.3 | 16 | 84.36 |

12 | 000 | 0.2 | 15.3 | 16 | 84.36 |

13 | 000 | 0.2 | 15.3 | 16 | 84.36 |

14 | 00A | 0.2 | 15.3 | 20 | 80.05 |

15 | 0A0 | 0.2 | 25.6 | 16 | 154.4 |

16 | +−− | 0.3 | 5 | 12 | 44.23 |

17 | +−+ | 0.3 | 5 | 20 | 29.92 |

18 | A00 | 0.3 | 15.3 | 16 | 92.52 |

19 | ++− | 0.3 | 25.6 | 12 | 168.33 |

Definitions | Parameters |
---|---|

Average | 90.838 |

Standard deviation | 45.553742 |

Mean standard error | 9.7389129 |

95% upper limit of the mean | 111.22178 |

95% lower limit of the mean | 70.454211 |

Number | 20 |

Number of missing values | 0 |

Definitions | Estimated Value | Standard Error | T | p | Sig. |
---|---|---|---|---|---|

Intercept | 85.568 | 2.764663 | 20.95 | <0.0001 | * |

h | 56.29 | 2.54312 | 22.13 | 0.00000 | * |

l | −12.827 | 2.54312 | −5.04 | 0.00050 | * |

h×h | −3.545 | 2.843294 | −1.25 | 0.01563 | * |

h×l | −4.095 | 4.849539 | −0.84 | 0.01716 | * |

λ×h | 0.535 | 4.849539 | 0.11 | 0.04026 | * |

λ | −5.595 | 2.54312 | −2.20 | 0.05244 ^ | * |

λ×l | 14.1 | 4.849539 | 2.91 | 0.24089 | |

l×l | −8.1125 | 2.843294 | −2.85 | 0.41818 | |

λ×λ | −6.6975 | 2.843294 | −2.36 | 0.91434 | |

S = 8.0421 | R^{2} = 0.982056 | Adjustment R^{2} = 0.965906 | R-sq. (predict) = 80.28% |

Definitions | Estimated Value | Standard Error | T | p |
---|---|---|---|---|

Intercept | 84.856 | 2.475419 | 34.28 | <0.0001 |

h | 56.29 | 2.475419 | 22.74 | 0.00000 |

l | −12.827 | 2.475419 | −5.18 | 0.00018 |

h×h | 11.964 | 3.500771 | 3.42 | 0.00459 |

h×l | −8.1125 | 2.767602 | −2.93 | 0.01169 |

λ×h | −6.6975 | 2.767602 | −2.42 | 0.03091 |

λ | 84.856 | 2.475419 | −2.26 | 0.04161 |

S = 7.828 | R^{2} = 0.97898 | Adjustment R^{2} = 0.967697 | R-sq. (predict) = 90.84% |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, P.; Tang, S.; Sun, Z.
Simulation Research on the Grouser Effect of a Reconfigurable Wheel-Crawler Integrated Walking Mechanism Based on the Surface Response Method. *Appl. Sci.* **2023**, *13*, 4202.
https://doi.org/10.3390/app13074202

**AMA Style**

Zhou P, Tang S, Sun Z.
Simulation Research on the Grouser Effect of a Reconfigurable Wheel-Crawler Integrated Walking Mechanism Based on the Surface Response Method. *Applied Sciences*. 2023; 13(7):4202.
https://doi.org/10.3390/app13074202

**Chicago/Turabian Style**

Zhou, Pengfei, Shufeng Tang, and Zaiyong Sun.
2023. "Simulation Research on the Grouser Effect of a Reconfigurable Wheel-Crawler Integrated Walking Mechanism Based on the Surface Response Method" *Applied Sciences* 13, no. 7: 4202.
https://doi.org/10.3390/app13074202