# Development of a Prediction Model for Tractor Axle Torque during Tillage Operation
## 4 Discussion  
In this study, a prediction model was proposed for predicting tractor axle torque. A total of five variables were used to develop axle torque prediction models, and the Pearson correlation coefficients were highest in order of the tillage depth, slip ratio, travel speed, engine torque, and engine speed. The developed Model A using the engine parameter as a variable showed a low adjusted R\({}^{2}\) 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  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **R\({}^{2}\)** & **MAPE (\%)** & **RMSE (Nm)** & **RD (\%)** \\ \hline A & 0.265 & 2.73 & 36.65 & 3.39 \\ \hline B & 0.782 & 1.42 & 19.35 & 1.79 \\ \hline C & 0.715 & 1.62 & 22.25 & 2.06 \\ \hline D & 0.812 & 1.32 & 18.15 & 1.68 \\ \hline E & 0.832 & 1.19 & 16.50 & 1.53 \\ \hline F & 0.776 & 1.44 & 19.53 & 1.81 \\ \hline G & 0.783 & 1.40 & 19.38 & 1.79 \\ \hline H & 0.809 & 1.30 & 18.05 & 1.67 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Actual axle torque prediction performance results for each regression model.  
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.  
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.