# Development of a Prediction Model for Tractor Axle Torque during Tillage Operation

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## Abstract

**:**

^{2}) of the proposed regression model showed a range of 0.271 to 0.925. Among them, the prediction model E showed an adjusted R

^{2}of 0.925. All of the prediction models were verified using a validation set. All of the axle torque prediction models showed an mean absolute percentage error (MAPE) of less than 2.8%. In particular, Model E, adopting engine torque, engine speed, and travel speed as variables, and Model H, adopting engine torque, tillage depth and travel speed as variables, showed MAPEs of 1.19 and 1.30%, respectively. Therefore, it was found that the proposed prediction models are applicable to actual axle torque prediction.

## 1. Introduction

^{2}) of the prediction model was 0.99, which means that it could replace the existing expensive method. However, such a prediction model using an artificial neural network has difficulty finding the optimal value of a parameter in a learning process, and problems, such as overfitting, may occur. Therefore, artificial intelligence is mainly used for solving nonlinear problems and need not necessarily be used for solving linear problems. Linear regression analysis can be a good alternative to the methods presented above. Linear regression models have been proposed in various research fields and are used to select explanatory variables that are highly related to actual dependent variables and to develop prediction models [20,21,22]. In the field of tractors, there have been many studies using linear regression models to predict tractor traction performance [23,24,25] and fuel efficiency [26]. Kheiralla et al. [25] performed regression analysis on the traction force of a moldboard plow, disk plow, disk harrow, and rotary tiller using the travel speed, tillage depth, and rotor speed as the main variables. Upadhyay and Raheman [27] proposed a specific draft prediction model of a disk harrow using multiple regression analysis based on front gang angle, cone index, tillage depth, and travel speed. The literature review revealed that multiple linear regression-based approaches for predicting axle torque during tillage are rare, and most of them focus on the prediction of traction force.

## 2. Materials and Methods

#### 2.1. Tractor Power Transmission System

#### 2.2. Field Experiment and Data Measurement

#### 2.2.1. Sensor System

^{2}of 0.99 (Tillage depth = −0.2342 × potentiometer value + 32.857). The travel speed was measured using GPS (18 × 5 Hz, Garmin, Olathe, KS, USA) with an accuracy of 5.0%, and the wheel rotational speed was measured using a proximity sensor (Autonics, PRDCMT30-25DO, Seoul, Korea) with hysteresis of the maximum 10% of sensing distance by installing a separate gear jig between the axle case and the wheel. The axle torque was measured using two front torquemeters (Manner Sensortelemetrie GmbH, MW 15 kNm Fu PCM16, Spaichingen, Germany) and two rear torquemeters (Manner Sensortelemetrie GmbH, MW 30 kNm Fu PCM16, Spaichingen, Germany). The linearity deviation of torquemeters is 0.2%. A data acquisition system (IMC, CRONOS compact CRC-400-11, Berlin, Germany) was used to measure field data from each sensor data.

#### 2.2.2. Field Experiment

#### 2.3. Data Processing

#### 2.4. Statistical Descriptions

#### 2.5. Development of Multiple Regression Models

#### 2.5.1. Model Development Procedure

#### 2.5.2. Variable Selection

#### 2.5.3. Regression Model Development

- Linearity: The relationship between the dependent and explanatory variables should be linear. This can be confirmed through correlation analysis.
- Normality: The residuals must have a normal distribution, regardless of the value of the independent variable. The mean of the regression standardized residuals is zero, and a normal distribution with a constant variance is assumed. The normality of the estimation error can be confirmed through the distribution of the regression-standardized residuals and the P–P plots of the regression-normalized residuals of the expected cumulative probability versus actual cumulative probability.
- Independence: The residuals in the dependent variable measurements should not affect each other. A Durbin–Watson test (D.W) is conducted to confirm the independence of the residuals of each model. In general, D.W has a value of 0 to 4, and the closer this value is to 2, the more the residual independence is guaranteed [42].
- Homoscedasticity: Homoscedasticity indicates that the scatter of the explanatory variable should be the same, regardless of the value of the dependent variable. Homoscedasticity can be confirmed by a scatter plot of the standardized predicted values and standardized residuals.
- Multicollinearity: There should be no multicollinearity between the explanatory variables, since the multicollinearity of the explanatory variables means that they are expressed in a linear relationship between the explanatory variables of the prediction model, and this is used to confirm correlations between the prediction variables when developing a prediction model. The variance inflation factor (VIF) was used to detect the multicollinearity of the prediction model [43]. VIFs are calculated for the coefficients of each model with the SPSS 21 software, when regression analysis is conducted. In the selection of the regression model, the problem of collinearity is diagnosed using tolerance and the variance inflation coefficient (VIF). Tolerance is the reciprocal of VIF. Some previous studies have reported that if the VIF is greater than 10, there is a problem of multicollinearity [44,45]. In this study, the upper limit of the VIF was set to 10, and values less than that were adopted.

#### 2.5.4. Verification of Multiple Regression Models

^{2}, mean absolute percentage error (MAPE), root mean square error (RMSE), and relative deviation (RD) were used as an evaluative indicator of the development of each model.

## 3. Results

#### 3.1. Statistical Descriptions

#### 3.2. Correlation Analysis

#### 3.3. Prediction Model

#### 3.3.1. Development of the Regression Model

^{2}, adjusted R

^{2}, and standardized error (S.E) for each regression model. The developed eight regression models showed an adjusted R

^{2}range of 0.271–0.925, indicating that the actual axle torque can be predicted with an accuracy of about 27.1–92.5%. The adjusted R

^{2}of regression model A, which used the engine torque and engine speed as the main variables, was the lowest, at 0.271. This means that the prediction accuracy is somewhat low, because the adjusted R

^{2}is less than 0.3. On the other hand, the adjusted R

^{2}of Model B, with the tillage depth, and Model C, with the slip ratio, were 0.866 and 0.813, respectively. These results showed that they had a high accuracy compared to regression Model A. In addition, Model B showed the highest accuracy among the prediction models using a single input source. In case of using a combination of each source as a variable, adjusted R

^{2}in Models F and E showed lowest (0.867) and highest (0.925), respectively. The S.E was found to be approximately between 12.79 and 39.98 for all the regression models.

#### 3.3.2. Normality Test of Residuals

#### 3.3.3. Independence

#### 3.3.4. Homoscedasticity

#### 3.3.5. Multicollinearity

#### 3.4. Model Verification

^{2}had a range of 0.265 to 0.832, and the MAPE was less than 2.8% for all the regression models. In addition, the RMSE and RD had ranges of 16.50 to 36.65 Nm and 1.53 to 3.39%, respectively. Model A showed the lowest accuracy with a MAPE of 2.73%. Model E showed the best performance, with an R

^{2}of 0.832, MAPE of 1.19%, RMSE of 16.50 Nm, and RD of 1.53%. Thus, it was found that axle torque could best be explained using three explanatory variables (engine torque, engine speed, and travel speed).

## 4. Discussion

^{2}of 0.271, thus it is considered insufficient for use as a model for predicting axle torque. Nevertheless, in the case of Model D, which uses the engine parameter and tillage depth as variables, showed an adjusted R

^{2}of 0.877, and it is higher than that of Model B (R

^{2}adj: 0.866), which uses only the tillage depth as a single variable. Therefore, it is determined that the engine parameter can be used to improve prediction accuracy by combining it with other variables rather than using it in a prediction model as a single variable. Among the single sources, the tillage depth was regarded as the variable that could well explain the axle torque because it has high performance. In all the proposed models except Model A, the range of the adjusted R

^{2}were 0.813–0.925. These results were similar to the results of the prediction model for estimating the major parameters of the tractor such as the engine torque (R

^{2}: 0.835) [18], traction force (R

^{2}: 0.760–0.862) [23], and fuel consumption (R

^{2}: 0.892–0.916) [25]. The basic assumptions of regression analysis such as linearity, normality, independence, homoscedasticity, and multicollinearity were satisfied for all the proposed models except Model A. The D.W value in Model A is 1.34, and it may have difficulty guaranteeing the independence of the residuals. The verification results for the proposed model (MAPE: 1.19–3.61%) showed high prediction performance compared to the results of previous studies (Error: 1.27–3.61%) that predicted the theoretical axle torque according to the tillage depth [15]. Therefore, we believe that the proposed models can be applied to axle torque prediction. In conclusion, in this study, the developed prediction models can well explain the axle torque of the tractor using low-cost sensors or sensor data built into the tractor during tillage operation.

## 5. Conclusions

^{2}of the proposed regression model showed a range of 0.271 to 0.925. Among them, the prediction model E—with engine torque, engine speed, and travel speed as explanatory variables—showed an adjusted R

^{2}of 0.925. All the prediction models were tested for the basic assumptions of the regression analysis: linearity, normality, independence, homoscedasticity, and multicollinearity. All the prediction models proposed using the validation set were verified. As a result, all the prediction models showed an MAPE of less than 2.8%, and, in particular, Models E and H showed MAPEs of 1.19 and 1.30%, respectively, demonstrating a high accuracy. Therefore, it was found that the proposed prediction models are applicable to actual axle torque prediction.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**System installed on the tractor to collect field data for each major part: 1. Engine torque and engine speed, 2. Axle rotational speed, 3. Axle torque, 4. Travel speed, 5. Tillage depth, and 6. Data collection.

**Figure 4.**Procedure for the development and verification of the prediction model using regression analysis.

**Figure 5.**The distribution of the regression standardized residuals for each prediction Model (

**A**–

**H**): Model A = axle torque as a linear function of engine torque and engine speed; Model B = axle torque as a linear function of tillage depth; Model C = axle torque as a linear function of slip ratio and travel speed; Model D = axle torque as a linear function of engine torque, engine speed and tillage depth; Model E = axle torque as a linear function of engine torque, engine speed and travel speed; Model F = axle torque as a linear function of tillage depth and travel speed; Model G = axle torque as a linear function of engine speed, tillage depth and travel speed; and Model H = axle torque as a linear function of engine torque, tillage depth and travel speed.

**Figure 6.**P–P plots of the regression-normalized residuals of the expected cumulative probability versus actual cumulative probability for each regression Model (

**A**–

**H**): Model A = axle torque as a linear function of engine torque and engine speed; Model B = axle torque as a linear function of tillage depth; Model C = axle torque as a linear function of slip ratio and travel speed; Model D = axle torque as a linear function of engine torque, engine speed and tillage depth; Model E = axle torque as a linear function of engine torque, engine speed and travel speed; Model F = axle torque as a linear function of tillage depth and travel speed; Model G = axle torque as a linear function of engine speed, tillage depth and travel speed; and Model H = axle torque as a linear function of engine torque, tillage depth and travel speed.

**Figure 7.**Scatter plot of the standardized predicted values and standardized residuals for each regression Model (

**A**–

**H**): Model A = axle torque as a linear function of engine torque and engine speed; Model B = axle torque as a linear function of tillage depth; Model C = axle torque as a linear function of slip ratio and travel speed; Model D = axle torque as a linear function of engine torque, engine speed and tillage depth; Model E = axle torque as a linear function of engine torque, engine speed and travel speed; Model F = axle torque as a linear function of tillage depth and travel speed; Model G = axle torque as a linear function of engine speed, tillage depth and travel speed; and Model H = axle torque as a linear function of engine torque, tillage depth and travel speed.

**Figure 8.**Relationships between the predicted and measured axle torque for each regression Model (

**A**–

**H**): Model A = axle torque as a linear function of engine torque and engine speed; Model B = axle torque as a linear function of tillage depth; Model C = axle torque as a linear function of slip ratio and travel speed; Model D = axle torque as a linear function of engine torque, engine speed and tillage depth; Model E = axle torque as a linear function of engine torque, engine speed and travel speed; Model F = axle torque as a linear function of tillage depth and travel speed; Model G = axle torque as a linear function of engine speed, tillage depth and travel speed; and Model H = axle torque as a linear function of engine torque, tillage depth and travel speed.

Model | Width × Length × Height | Weight | Applicable Power Range | Furrow |
---|---|---|---|---|

WJSP-8 | 280 × 215 × 125 cm | 790 kg | 60–90 kW | 8 |

Travel Speed | Tillage Depth | Soil Characteristic | ||
---|---|---|---|---|

Texture | Penetration Resistance | Moisture Content | ||

7.09 km/h | 15–20 cm | Silt loam | 1367 kPa | 34.17% |

Parameters | Number of Gear Pairs | Gear Ratio | Efficiency |
---|---|---|---|

Front wheel—gearbox output axle | 6 | 15.8 | 0.941 |

Rear wheel—gearbox output axle | 3 | 22.2 | 0.970 |

**Table 4.**Statistical description of the calibration and validation dataset for each variable used in this study.

Parameter | Mean | Max. | Min. | Range | SD | |
---|---|---|---|---|---|---|

Calibration set (n = 375) | ${T}_{a}$ (Nm) | 1079.60 | 1202.69 | 977.87 | 224.82 | 46.82 |

${T}_{e}$ (Nm) | 316.07 | 392.40 | 252.34 | 140.06 | 23.20 | |

${S}_{e}$ (rpm) | 2265.09 | 2410.23 | 1725.71 | 684.53 | 112.58 | |

${D}_{t}$ (cm) | 16.82 | 22.50 | 13.73 | 8.77 | 1.88 | |

${S}_{t}$ (km/h) | 6.07 | 6.93 | 5.04 | 1.89 | 0.36 | |

$s$ (%) | 13.40 | 18.00 | 9.50 | 8.50 | 1.93 | |

Validation set (n = 125) | ${T}_{a}$ (Nm) | 1080.26 | 1170.26 | 987.65 | 182.61 | 42.38 |

${T}_{e}$ (Nm) | 314.65 | 372.62 | 259.05 | 113.57 | 21.14 | |

${S}_{e}$ (rpm) | 2276.65 | 2407.68 | 1889.99 | 517.69 | 94.25 | |

${D}_{t}$ (cm) | 16.76 | 21.29 | 13.94 | 7.35 | 1.68 | |

${S}_{t}$ (km/h) | 6.08 | 6.90 | 5.24 | 1.66 | 0.34 | |

$s$ (%) | 13.40 | 17.50 | 9.88 | 7.63 | 1.81 |

Pearson Correlation Coefficient (p-Value) | ||||||
---|---|---|---|---|---|---|

Variable | ${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{e}}$ | ${\mathit{S}}_{\mathit{e}}$ | ${\mathit{D}}_{\mathit{t}}$ | ${\mathit{S}}_{\mathit{t}}$ | $\mathit{s}$ |

${\mathit{T}}_{\mathit{a}}$ | 1.000 | 0.505 ** (0.000) | −0.396 ** (0.000) | 0.931 ** (0.000) | −0.688 ** (0.000) | 0.897 ** (0.000) |

${\mathit{T}}_{\mathit{e}}$ | 1.000 | 0.904 ** (0.000) | 0.445 ** (0.000) | 0.132 ** (0.005) | 0.431 ** (0.000) | |

${\mathit{S}}_{\mathit{e}}$ | 1.000 | −0.355 ** (0.000) | −0.033 (0.261) | −0.320 ** (0.000) | ||

${\mathit{D}}_{\mathit{t}}$ | 1.000 | −0.714 ** (0.000) | 0.984 ** (0.000) | |||

${\mathit{S}}_{\mathit{t}}$ | 1.000 | −0.690 ** (0.000) | ||||

$\mathit{s}$ | 1.000 |

Model | Source | Regression Model | R^{2} | R^{2} adj | S.E |
---|---|---|---|---|---|

A | ${f}_{1}$ | ${T}_{a}=1.626{\mathrm{T}}_{\mathrm{e}}+0.138{\mathrm{S}}_{\mathrm{e}}+252.210$ | 0.275 | 0.271 | 39.98 |

B | ${f}_{2}$ | ${T}_{a}=23.189{D}_{t}+689.467$ | 0.866 | 0.866 | 17.16 |

C | ${f}_{3}$ | ${T}_{a}=19.599s-16.860{S}_{t}+919.198$ | 0.814 | 0.813 | 20.26 |

D | ${f}_{1}+{f}_{2}$ | ${T}_{a}=0.419{\mathrm{T}}_{\mathrm{e}}+0.042{\mathrm{S}}_{\mathrm{e}}+21.785{D}_{t}+484.555$ | 0.878 | 0.877 | 16.41 |

E | ${f}_{1}+{f}_{3}$ | ${T}_{a}=2.565{\mathrm{T}}_{\mathrm{e}}+0.302{\mathrm{S}}_{\mathrm{e}}-107.497{S}_{t}+237.209$ | 0.926 | 0.925 | 12.79 |

F | ${f}_{2}+{f}_{3}$ | ${T}_{a}=22.340{D}_{t}-6.175{S}_{t}+741.218$ | 0.867 | 0.867 | 17.11 |

G | ${f}_{1}+{f}_{2}+{f}_{3}$ | ${T}_{a}=-0.047{\mathrm{S}}_{\mathrm{e}}+20.179{D}_{t}-14.662{S}_{t}+935.057$ | 0.876 | 0.875 | 16.54 |

H | ${f}_{1}+{f}_{2}+{f}_{3}$ | ${T}_{a}=0.580{\mathrm{T}}_{\mathrm{e}}+14.461{D}_{t}-40.263{S}_{t}+897.125$ | 0.899 | 0.899 | 14.91 |

Model | Df | SS | MS | F-Value | p-Value | |
---|---|---|---|---|---|---|

A | Regression | 2 | 225,648 | 112,824 | 71 | 0.000 |

Residual | 372 | 594,473 | 1598 | |||

Total | 374 | 820,120 | ||||

B | Regression | 1 | 710,323 | 710,323 | 2413 | 0.000 |

Residual | 373 | 109,797 | 294 | |||

Total | 374 | 820,120 | ||||

C | Regression | 2 | 667,470 | 333,735 | 813 | 0.000 |

Residual | 372 | 152,651 | 410 | |||

Total | 374 | 820,120 | ||||

D | Regression | 3 | 720,217 | 240,072 | 892 | 0.000 |

Residual | 371 | 99,903 | 269 | |||

Total | 374 | 820,120 | ||||

E | Regression | 3 | 759,415 | 253,138 | 1547 | 0.000 |

Residual | 371 | 60,706 | 164 | |||

Total | 374 | 820,120 | ||||

F | Regression | 2 | 711,241 | 355,621 | 1215 | 0.000 |

Residual | 372 | 108,879 | 293 | |||

Total | 374 | 820,120 | ||||

G | Regression | 3 | 718,582 | 239,527 | 875 | 0.000 |

Residual | 371 | 101,538 | 274 | |||

Total | 374 | 820,120 | ||||

H | Regression | 3 | 737,632 | 245,877 | 1106 | 0.000 |

Residual | 371 | 82,488 | 222 | |||

Total | 374 | 820,120 |

Model | Variable | S.E | β | t | p-Value | Tolerance | VIF |
---|---|---|---|---|---|---|---|

A | Constant | 159.556 | 1.581 | 0.115 | |||

${T}_{e}$ | 0.209 | 0.806 | 7.794 | 0.000 | 0.182 | 5.485 | |

${S}_{e}$ | 0.043 | 0.333 | 3.218 | 0.001 | 0.182 | 5.485 | |

B | Constant | 7.992 | 86.274 | 0.000 | |||

${D}_{t}$ | 0.472 | 0.931 | 49.123 | 0.000 | 1.000 | 1.000 | |

C | Constant | 32.052 | 28.678 | 0.000 | |||

$s$ | 0.751 | 0.807 | 26.111 | 0.000 | 0.524 | 1.910 | |

${S}_{t}$ | 3.998 | 0.130 | −4.218 | 0.000 | 0.524 | 1.910 | |

D | Constant | 65.721 | 7.373 | 0.000 | |||

${D}_{t}$ | 0.508 | 0.874 | 42.856 | 0.000 | 0.789 | 1.268 | |

${T}_{e}$ | 0.090 | 0.208 | 4.649 | 0.000 | 0.165 | 6.078 | |

${S}_{e}$ | 0.018 | 0.102 | 2.383 | 0.018 | 0.179 | 5.573 | |

E | Constant | 51.057 | 4.646 | 0.000 | |||

${T}_{e}$ | 0.069 | 1.271 | 37.306 | 0.000 | 0.172 | 5.817 | |

${S}_{t}$ | 1.882 | 0.831 | −57.115 | 0.000 | 0.942 | 1.062 | |

${S}_{e}$ | 0.014 | 0.726 | 21.478 | 0.000 | 0.175 | 5.722 | |

F | Constant | 30.289 | 24.471 | 0.000 | |||

${D}_{t}$ | 0.672 | 0.897 | 33.248 | 0.000 | 0.491 | 2.038 | |

${S}_{t}$ | 3.487 | 0.048 | −1.771 | 0.077 | 0.491 | 2.038 | |

G | Constant | 47.526 | 19.675 | 0.000 | |||

${D}_{t}$ | 0.772 | 0.810 | 26.132 | 0.000 | 0.347 | 2.878 | |

${S}_{e}$ | 0.009 | 0.113 | −5.179 | 0.000 | 0.707 | 1.414 | |

${S}_{t}$ | 3.749 | 0.113 | −3.911 | 0.000 | 0.397 | 2.519 | |

H | Constant | 30.029 | 29.876 | 0.000 | |||

${D}_{t}$ | 0.931 | 0.580 | 15.541 | 0.000 | 0.194 | 5.144 | |

${S}_{t}$ | 4.362 | 0.311 | −9.231 | 0.000 | 0.238 | 4.197 | |

${T}_{e}$ | 0.053 | 0.288 | 10.895 | 0.000 | 0.389 | 2.570 |

Model | A | B | C | D | E | F | G | H |
---|---|---|---|---|---|---|---|---|

D.W | 1.340 | 2.113 | 2.084 | 2.104 | 2.077 | 2.105 | 2.124 | 2.146 |

Model | R^{2} | MAPE (%) | RMSE (Nm) | RD (%) |
---|---|---|---|---|

A | 0.265 | 2.73 | 36.65 | 3.39 |

B | 0.782 | 1.42 | 19.35 | 1.79 |

C | 0.715 | 1.62 | 22.25 | 2.06 |

D | 0.812 | 1.32 | 18.15 | 1.68 |

E | 0.832 | 1.19 | 16.50 | 1.53 |

F | 0.776 | 1.44 | 19.53 | 1.81 |

G | 0.783 | 1.40 | 19.38 | 1.79 |

H | 0.809 | 1.30 | 18.05 | 1.67 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, W.-S.; Kim, Y.-J.; Baek, S.-Y.; Baek, S.-M.; Kim, Y.-S.; Park, S.-U.
Development of a Prediction Model for Tractor Axle Torque during Tillage Operation. *Appl. Sci.* **2020**, *10*, 4195.
https://doi.org/10.3390/app10124195

**AMA Style**

Kim W-S, Kim Y-J, Baek S-Y, Baek S-M, Kim Y-S, Park S-U.
Development of a Prediction Model for Tractor Axle Torque during Tillage Operation. *Applied Sciences*. 2020; 10(12):4195.
https://doi.org/10.3390/app10124195

**Chicago/Turabian Style**

Kim, Wan-Soo, Yong-Joo Kim, Seung-Yun Baek, Seung-Min Baek, Yeon-Soo Kim, and Seong-Un Park.
2020. "Development of a Prediction Model for Tractor Axle Torque during Tillage Operation" *Applied Sciences* 10, no. 12: 4195.
https://doi.org/10.3390/app10124195