Multivariate Prediction Soft Sensor Model for Truck Cranes Based on Graph Convolutional Network and Random Forest

Comments 1: In Section 2, the contents can be shortened since they are well known already, you only need to list suitable references. But the advantages and disadvantages of this method should be pointed out.

Response 1: Thank you for pointing this out. We agree with this comment. I have reduced the content related to GCN and RF, as shown in below.

Input

Variables

Description

Output

Variables

Description

x₁

Rated weight

y₁

Amplitude

x₂

Actual weight

y₂

Boom length

x₃

Multiplier

y₃

Flexiable pump pressure

x₄

Throttle position

y₄

Height

x₅

Engine speed

y₅

Luffing angle

x₆

Luffing handle signal

y₆

Slewing angle

x₇

Winching handle signal

y₇

Winch pump outlet pressure

x₈

Slewing handle signal

y₈

Winch speed

x₉

Telescoping handle signal

/

/

x₁₀

Left slewing current

/

/

x₁₁

Right slewing current

/

/

x₁₂

Flexiable pump control current

/

/

x₁₃

Luffing balance valve control current

/

/

x₁₄

Winch up pump control current

/

/

x₁₅

Winch down pump control current

/

/

x₁₆

Winch motor control current

/

/

 Comments 1: Move the abbreviations from the abstract to the introduction. Mention the contrast methods like GBT, SVR, MLP on the introduction and results section, providing some details regarding its architecture and training process, like how many layers or the amount of decision trees

Response 1: Thank you for pointing this out. We agree with this comment. I have modified the abstract as follows. The structural parameters of GCN-RF are shown in Table 2.

Abstract: Truck cranes, which are crucial construction equipment, need to maintain good operational performance to ensure safe use. However, the complex and ever-changing working conditions they face often make it challenging to test their performance effectively. To address this issue, a Multi-Input and Multi-Output soft sensor technology model is suggested, utilizing a Graph Convolutional Network and Random Forest to predict key performance indicators of crane operations such as luffing, telescoping, winching, and slewing under varying conditions. This method aims to streamline the process of testing and debugging truck cranes, ultimately reducing time and costs. Initially, the Graph Convolutional Network model is employed to extract relevant feature information linked to the target variable. Subsequently, using this feature information and the RF model, multiple decision trees are constructed for regression prediction of the target variables. An operational dataset reflecting the crane's actual working conditions is then generated to assess the Graph Convolutional Network and Random Forest model. The effectiveness of this approach is further confirmed through comparisons with other methods like Gradient Boosting Trees, Support Vector Regression, and Multi-Layer Perceptron.

 Table 2. GCN-RF model parameter settings.

Comments 2: It is hard to understand the big picture, please make a block diagram that explains all the subsistyems on figures 4-7.

Response 2: Thank you for pointing this out. We agree with this comment. I have added the figure of the truck crane and labeled the positions of the four working systems, as shown in Figure 4. I hope this can help you and other readers better understand the movements performed by the truck crane.

Figure 4. Schematic diagram of the crane working systems: Luffing, Slewing, Telescoping, and Winching systems.

Comments 3: In table 1 set input and output variables on different columns.

Response 3: Thank you for pointing this out. We agree with this comment. I have made the corresponding modifications, as shown in Table 1.

Table 1. Variables collected by crane operation system.

¹ Tables may have a footer.

Comments 4: Specify the number of timeseries used, how many for training and how many for validation?

Response 4: Thank you for pointing this out. I have added information about the dataset division, with the training set and test set in a ratio of 8:2.

“We collected operational data from three cranes with lifting capacities of 160t, 200t, and 240t under actual working conditions. Each dataset is divided into a training set and a test set, with a ratio of 8:2. Real-world operations for truck cranes frequently involve both single and compound actions under varying working conditions.”

Comments 5: Can you show the input for the graph neural network? no the timeseries but the actual graph that will be passed to the GNN.

Response 5: Thank you for pointing this out. We agree with this comment. I have modified the GCN-RF schematic diagram to construct graph data as input for the GCN using the adjacency matrix of the sequential data and the weights between neighboring nodes, as shown in Figure 3.

Figure 3. GCN-RF architecture diagram with graph data feature extraction, regression prediction.

Comments 6: In Figs 9 to 16, the left plot is not clear and does not provide a clear visual representation of the predicted variables vs the real dataset. Please replace the plot with a single 2D graph plotting the real data and its predictions. Thus it may be more evident the proposed method advantage. Likewise, right plots can be summarized on a table instead of individual plots. remove all of them. Figs 9 to 16 can be grouped on a single plot with several subplots

Response 6: Thank you for pointing this out. We agree with this comment. I have modified the Figures and add the Tables, as shown in “Response 8”.

Comments 7: How many cranes were evaluated on this study.  What is the cross validation of the model vs real data.

Response 7: Thank you for pointing this out. We agree with this comment. We collected operational data from three cranes with lifting capacities of 160t, 200t, and 240t under actual working conditions. The three datasets were obtained under varying operational environments specific to each crane, which is fundamentally different from experimental data collected under fixed conditions. By comparing different methods and datasets, we can assess the robustness and generalization capability of the models.

Comments 8: Don't itemize the conclusions, write a single paragraph for each one.

Response 8: Thank you for pointing this out. We agree with this comment. For each prediction result, we have provided an analysis as follows.

3.3. Key variables prediction results of the 160t truck crane

The prediction results of the amplitude are shown in Figure 11 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 3.

Figure 11. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 3. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 11 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The prediction curve produced by the GCN-RF demonstrates a close correspondence with the actual curve, indicating a high level of predictive accuracy. Notably, the GBT's prediction curve aligns well with the actual values, exhibiting only minor deviations in specific areas attributed to volatility. In contrast, the prediction curves for the SVR and MLP reveal substantial discrepancies from the actual values, particularly in regions marked by rapid fluctuations. This suggests that the SVR and MLP are less effective in amplitude prediction compared to the GCN-RF and GBT.

Table 3 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8875, indicating its excellent explanatory power and fitting capability. Its MSE and RMSE values are 0.0031 and 0.0556, respectively, which are the lowest among the models, highlighting the GCN-RF's exceptional accuracy in amplitude prediction. The GBT presents an R² value of 0.8680, an MSE of 0.0036, and an RMSE of 0.0603, demonstrating strong predictive performance. In contrast, the SVR exhibits an R² value of 0.6116, an MSE of 0.0108, and an RMSE of 0.1039, indicating average performance in amplitude prediction with relatively lower accuracy. Lastly, the MLP records the lowest R² value of 0.6329, along with an MSE of 0.0101 and an RMSE of 0.1007, reflecting greater prediction errors and inconsistent performance.

The prediction results of the boom length are shown in Figure 12 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 4.

Figure 12. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 4. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 12 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting boom length. It is evident that the prediction curves for the GCN-RF and GBT closely align with the actual curve, indicating a high level of predictive accuracy. Notably, the GCN-RF's curve nearly perfectly matches the actual values. In contrast, the SVR's prediction curve exhibits some deviations from the actual values, particularly in regions with significant data fluctuations. The MLP shows relatively poor performance, as indicated by a larger gap between its prediction curve and the actual values, suggesting a weaker capability in predicting boom length.

Table 4 provides the R², MSE, and RMSE metrics for each model. The GCN-RF exhibits the highest R² value of 0.8946, indicating superior performance in terms of explanatory power and fitting ability. The MSE and RMSE values for this model are 0.0014 and 0.0380, respectively, both of which are the lowest among the models, further confirming the exceptional accuracy of the GCN-RF in predicting arm length. The GBT presents an R² value of 0.8904, an MSE of 0.0014, and an RMSE of 0.0379, also demonstrating strong predictive capability. In contrast, the SVR has an R² value of 0.6372, an MSE of 0.0082, and an RMSE of 0.0906, indicating significantly poorer performance compared to the GCN-RF and GBT, with lower predictive accuracy. The MLP records the lowest R² value of 0.5700, with an MSE of 0.0059 and an RMSE of 0.0768, reflecting greater prediction errors and diminished accuracy.

The prediction results of the flexible pump pressure are shown in Figure 13 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 5.

Figure 13. Prediction results of the flexible pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 5. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 13 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. The results indicate that the prediction curve of the GCN-RF closely aligns with the actual curve, thereby indicating a high level of predictive accuracy. The GBT exhibits a similar trend, maintaining consistency with the actual values overall, although it does present some deviations characterized by fluctuations in specific regions. In contrast, the prediction curves for the SVR and MLP reveal more substantial discrepancies from the actual values, particularly in areas experiencing significant pressure variations, suggesting that these models are less effective than the GCN-RF and GBT in predicting flexible pump pressure.

Table 5 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8689, signifying superior performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0048 and 0.0691, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's high accuracy in predicting flexible pump pressure. The GBT presents an R² value of 0.8124, an MSE of 0.0080, and an RMSE of 0.0897, demonstrating commendable predictive capability. Conversely, the SVR exhibits an R² value of 0.6767, an MSE of 0.0107, and an RMSE of 0.1172, indicating a significantly lower predictive accuracy compared to the GCN-RF and GBT. The MLP records the lowest R² value of 0.7200, with an MSE of 0.0102 and an RMSE of 0.1010, reflecting greater prediction errors and diminished accuracy.

The prediction results of the height are shown in Figure 14 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 6.

Figure 14. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 6. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 14 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. The prediction curves for the GCN-RF and GBT closely align with the actual curve, indicating a high level of predictive accuracy. Notably, the GBT's prediction curve nearly coincides with the actual values. Conversely, the prediction curves for the SVR and MLP display more pronounced deviations from the actual values, particularly in regions characterized by significant height variations, suggesting that these models exhibit inferior performance relative to the GCN-RF and GBT in the height prediction.

Table 6 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.9260, signifying its exceptional explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0014 and 0.0374, respectively, both of which are the lowest among the models, thereby reinforcing the GBT's accuracy in height prediction. The GCN-RF presents an R² value of 0.9248 and still reflects robust predictive capability accompanied by an MSE of 0.0015 and an RMSE of 0.0388, which slightly lower than that of the GBT. In contrast, the SVR exhibits an R² value of 0.8047, with an MSE of 0.0047 and an RMSE of 0.0669, indicating a significantly lower predictive accuracy compared to the GCN-RF and GBT. The MLP records the lowest R² value of 0.8312, with an MSE of 0.0036 and an RMSE of 0.0583. Although it marginally outperforms the SVR, it still demonstrates greater prediction errors.

The prediction results of the luffing angle are shown in Figure 15 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 7.

Figure 15. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 7. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 15 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude angle. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual curve, signifying a high level of predictive accuracy. Notably, the two curves nearly coincide in regions characterized by stable changes. Conversely, the prediction curves for the SVR and MLP display more pronounced deviations from the actual values, particularly in areas with significant fluctuations, indicating that these models exhibit inferior performance relative to the GCN-RF and GBT in amplitude angle prediction.

Table 7 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8952 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0015 and 0.0381, respectively, both of which are the lowest, further corroborating the GCN-RF's high accuracy in amplitude angle prediction. The GBT presents an R² value of 0.8917, an MSE of 0.0015, and an RMSE of 0.0384, demonstrating robust predictive capability. In contrast, the SVR exhibits an R² value of 0.6577, an MSE of 0.0676, and an RMSE of 0.0768, indicating significantly lower predictive accuracy compared to the GCN-RF and GBT models. The MLP records the lowest R² value of 0.6718, with an MSE of 0.0059 and an RMSE of 0.0767, reflecting greater prediction errors and diminished accuracy.

The prediction results of the slewing angle are shown in Figure 16 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 8.

Figure 16. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 8. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 16 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting slewing angle. The prediction curve of the GCN-RF closely aligns with the actual curve, suggesting a high level of predictive accuracy. The GBT still adheres to the overall trend of the observed data while exhibiting some deviation from the actual values. In contrast, the prediction curves for the SVR and MLP display more pronounced discrepancies from the actual values, particularly in regions characterized by significant data fluctuations, thereby indicating inferior performance relative to the GCN-RF and GBT in the task of slewing angle prediction.

Table 8 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8752 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0053 and 0.0728, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high accuracy in slewing angle prediction. The GBT presents an R² value of 0.7975, an MSE of 0.0089, and an RMSE of 0.0944. Although it is slightly less accurate than the GCN-RF, it still demonstrates commendable predictive capability. Conversely, the SVR exhibits an R² value of 0.5463, an MSE of 0.0193, and an RMSE of 0.1390, indicating significantly poorer performance when compared to the GCN-RF and GBT, along with reduced predictive accuracy. The MLP records the lowest R² value of 0.5220, with an MSE of 0.0203 and an RMSE of 0.1423, reflecting greater prediction errors and diminished accuracy.

The prediction results of the winch pump outlet pressure are shown in Figure 17 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 9.

Figure 17. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 9. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 17 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch pump outlet pressure. The prediction curve for the GCN-RF closely aligns with the actual curve, signifying a high level of predictive accuracy. The GBT also demonstrates commendable performance. However, it exhibits noticeable deviations in certain regions characterized by significant fluctuations when compared to the GCN-RF. In contrast, the prediction curves for the SVR and MLP reveal more substantial discrepancies from the actual values, particularly under conditions of heightened data variability, thereby indicating inferior performance relative to the GCN-RF and GBT in the prediction of winch pump outlet pressure.

Table 9 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8948 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for the GCN-RF are 0.0031 and 0.0555, respectively, both of which are the lowest, further substantiating the model's high accuracy in predicting winch pump outlet pressure. The GBT presents an R² value of 0.8089, with an MSE of 0.0068 and an RMSE of 0.0826, demonstrating robust predictive capability. Conversely, the SVR exhibits an R² value of 0.6917, an MSE of 0.0114, and an RMSE of 0.1070, indicating significantly poorer performance in comparison to the GCN-RF and GBT, along with diminished predictive accuracy. The MLP records the lowest R² value of 0.7216, accompanied by an MSE of 0.0094 and an RMSE of 0.0997, reflecting greater prediction errors and reduced accuracy.

The prediction results of the winch speed are shown in Figure 18 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 10.

Figure 18. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 10. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 18 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch speed. The prediction curves for the GCN-RF and GBT closely align with the actual data, suggesting a high level of predictive accuracy. Notably, the prediction curve of the GCN-RF nearly coincides with the actual values. Conversely, the prediction curves for the SVR and MLP models exhibit considerable divergence from the actual values, particularly in areas characterized by significant fluctuations in speed, indicating inferior performance in winch speed prediction relative to the GCN-RF and GBT.

Table 10 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9462 among the four models, signifying superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0010 and 0.0323, respectively, both of which are the lowest, further substantiating the GCN-RF's high accuracy in winch speed prediction. The GBT presents an R² value of 0.9423, an MSE of 0.0013, and an RMSE of 0.0351, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8740, an MSE of 0.0031, and an RMSE of 0.0553, indicating significantly diminished performance compared to the GCN-RF and GBT, with reduced predictive accuracy. The MLP records the lowest R² value of 0.9176, alongside an MSE of 0.0016 and an RMSE of 0.0400. Although it performs slightly better than the SVR model, it still demonstrates greater prediction errors.

3.4. Key variables prediction results of the 200t truck crane

The prediction results of the amplitude are shown in Figure 19 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 11.

Figure 19. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 11. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 19 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The results indicate that the prediction curve of the GCN-RF closely approximates the actual curve, signifying a high level of predictive accuracy. The GBT's prediction curve demonstrates a strong alignment with the actual values, albeit with minor discrepancies in certain fluctuating regions. Conversely, the prediction curves for the SVR and MLP exhibit a significant divergence from the actual values, particularly in areas characterized by substantial data variations, thereby indicating inferior performance in amplitude prediction relative to the GCN-RF and GBT.

Table 11 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8875, signifying superior explanatory power and fitting capability. Its MSE and RMSE values are recorded at 0.0031 and 0.0556, respectively, which are the lowest among the models, further substantiating the GCN-RF's high accuracy in amplitude prediction. The GBT presents an R² value of 0.8680, an MSE of 0.0036, and an RMSE of 0.0603, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.6116, an MSE of 0.0108, and an RMSE of 0.1039, indicating average performance in amplitude prediction with relatively low accuracy. The MLP records the lowest R² value of 0.6329, an MSE of 0.0101, and an RMSE of 0.1007, reflecting greater prediction errors and a lack of stability in its performance.

The prediction results of the boom length are shown in Figure 20 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 12.

Figure 20. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 12. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 20 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting boom length. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual values, suggesting a high degree of predictive accuracy. Notably, the curve for the GCN-RF nearly coincides with the actual data. In contrast, the SVR exhibits some discrepancies from the actual values, particularly in regions characterized by significant fluctuations in the data. The MLP demonstrates comparatively inferior performance, as evidenced by a more pronounced divergence between its prediction curve and the actual values, indicating a reduced efficacy in boom length prediction.

Table 12 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9263, signifying superior fitting performance in the task of boom length prediction. Its MSE and RMSE values are recorded at 0.0032 and 0.0565, respectively, which are the lowest among the evaluated models, thereby reinforcing the GCN-RF's advantage in predictive accuracy. The GBT presents an R² value of 0.9051, an MSE of 0.0042, and an RMSE of 0.0646. While its performance is marginally inferior to that of the GCN-RF, it still demonstrates commendable predictive capability. The SVR exhibits an R² value of 0.9190, which is slightly superior to that of the GBT. However, its MSE and RMSE values are higher, recorded at 0.0074 and 0.1010, respectively, indicating a deficiency in accuracy. The MLP records the lowest R² value of 0.8307, with an MSE of 0.0104 and an RMSE of 0.0860, reflecting that it is comparatively less effective in addressing the boom length prediction challenge.

The prediction results of the flexible pump pressure are shown in Figure 21 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 13.

Figure 21. Prediction results of the flexiable pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 13. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 21 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. The results indicate that the prediction curve produced by the GCN-RF closely approximates the actual curve, suggesting a high level of predictive accuracy. In contrast, the GBT demonstrates slightly inferior performance; while its predictions generally align with the actual values, there are notable discrepancies in certain fluctuating regions. The prediction curves for the SVR and MLP reveal a considerable divergence from the actual values, particularly in areas characterized by abrupt pressure variations, indicating that these models are less effective in predicting flexible pump pressure when compared to the GCN-RF and GBT.

Table 13 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8769, signifying its superior explanatory power and fitting capability compared to the other models. The corresponding MSE and RMSE values for the GCN-RF are 0.0054 and 0.0733, respectively, both of which are the lowest among the models assessed, further substantiating the GCN-RF's high accuracy in accuracy for flexible pump pressure prediction. The GBT presents an R² value of 0.7910, an MSE of 0.0087, and an RMSE of 0.0931, demonstrating a marginally lower performance. The SVR exhibits an R² value of 0.7218, which is inferior to both the GCN-RF and GBT, with MSE and RMSE values of 0.0129 and 0.1135, respectively, indicating suboptimal performance on this dataset. Lastly, the MLP records an R² value of 0.7440, an MSE of 0.0116, and an RMSE of 0.1056, which slightly better than the SVR, reflecting significantly lower than the performance metrics of the GCN-RF and GBT.

The prediction results of the height are shown in Figure 22 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 14.

Figure 22. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 14. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 22 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual values, suggesting a high degree of predictive accuracy. Notably, the GBT exhibits a prediction curve that nearly coincides with the actual measurements. Conversely, the prediction curves for the SVR and MLP demonstrate a considerable deviation from the actual values, particularly in areas characterized by significant height variations. This discrepancy suggests that the performance of the SVR and MLP in height prediction is inferior to that of the GCN-RF and GBT.

Table 14 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves an R² value of 0.9716, which is near 1, signifying exceptional explanatory power and fitting capability. Its MSE and RMSE values are 0.0032 and 0.0564, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's superior accuracy in height prediction. The GBT presents an R² value of 0.9611, with an MSE of 0.0044 and an RMSE of 0.0667, which slightly lower than that of the GCN-RF and still reflects robust predictive performance. The SVR exhibits an R² value of 0.9234, an MSE of 0.0090, and an RMSE of 0.0949, indicating moderate performance in the height prediction task, characterized by relatively low accuracy. The MLP records an R² value of 0.9177, an MSE of 0.0092, and an RMSE of 0.0957, reflecting that both the MLP and SVR models exhibit significant prediction errors.

The prediction results of the luffing angle are shown in Figure 23 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 15.

Figure 23. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 15. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 23 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude angle. The results indicate that the prediction curve generated by the GCN-RF nearly coincides with the actual curve, suggesting an exceptionally high level of predictive accuracy. The GBT also demonstrates commendable performance, as its predictions closely follow the actual values, albeit with minor discrepancies in regions characterized by fluctuations. Conversely, the prediction curves for the SVR and MLP exhibit considerable divergence from the actual values, particularly during intervals of rapid data variation.

Table 15 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves an R² value of 0.9793, which is nearly equal to 1, thereby reflecting its superior explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0027 and 0.0517, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's high accuracy in amplitude angle prediction. The GBT presents an R² value of 0.9660, an MSE of 0.0046, and an RMSE of 0.0678, demonstrating that it retains a robust predictive capacity. The SVR exhibits an R² value of 0.9283, an MSE of 0.0095, and an RMSE of 0.0974, indicating a moderate performance in the amplitude angle prediction task, characterized by relatively lower accuracy. The MLP records an R² value of 0.9150, an MSE of 0.0121, and an RMSE of 0.1046. Consequently, both the MLP and SVR exhibit greater prediction errors and demonstrate less stable performance.

The prediction results of the slewing angle are shown in Figure 24 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 16.

Figure 24. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 16. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 24 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting slewing angle. It is evident that while the prediction curve generated by the GCN-RF exhibits a degree of alignment with the actual curve, its accuracy has diminished relative to prior predictions. The GBT also demonstrates significant deviations from the actual values, particularly in regions characterized by substantial data variability. In contrast, the prediction curves for the SVR and MLP reveal even greater discrepancies from the actual values, especially in areas with frequent fluctuations, thereby indicating a diminished performance in the prediction of slewing angle prediction.

Table 16 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8220 among the four models, signifying a relatively strong explanatory capacity in slewing angle prediction, albeit with a notable decline in performance compared to earlier assessments. The MSE and RMSE values for this model are 0.0118 and 0.1084, respectively, reflecting a certain degree of predictive error. The GBT presents an R² value of 0.7836, with an MSE of 0.0144 and an RMSE of 0.1201, demonstrating that it retains some predictive efficacy for this task. Conversely, the SVR exhibits an R² value of 0.6738, an MSE of 0.0225, and an RMSE of 0.1501, indicating inadequate performance in slewing angle prediction, characterized by low predictive accuracy. The MLP records the lowest R² value of 0.6702, alongside an MSE of 0.0244 and an RMSE of 0.1473, reflecting substantial prediction errors and a lack of stability.

The prediction results of the winch pump outlet pressure are shown in Figure 25 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 17.

Figure 25. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 17. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 25 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch pump outlet pressure. The data presented in the figure indicates that the prediction trajectories of the GCN-RF and GBT closely align with the actual curve, suggesting a superior level of predictive accuracy. Notably, the GBT consistently yields predictions that are in close to the actual values across most scenarios. Conversely, the prediction trajectories of the SVR and MLP demonstrate significant deviations from the actual values, particularly during periods characterized by substantial data fluctuations, indicating that these models exhibit inferior performance relative to the GCN-RF and GBT in the prediction of winch pump outlet pressure.

Table 17 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.8999, signifying its optimal performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0021 and 0.0459, respectively, further substantiating the GBT's high predictive accuracy in this application. The GCN-RF presents an R² value of 0.8932, an MSE of 0.0022 and an RMSE of 0.0474, which slightly lower than that of the GBT model and still indicates robust predictive capability. In contrast, the SVR exhibits an R² value of 0.8119, with an MSE of 0.0067 and an RMSE of 0.0687, indicating average performance in the winch pump outlet pressure prediction and relatively diminished predictive accuracy. The MLP records the lowest R² value of 0.7928, accompanied by an MSE of 0.0074 and an RMSE of 0.0750, reflecting greater prediction errors and a lack of stability in its performance.

The prediction results of the winch speed are shown in Figure 26 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 18.

Figure 26. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 18. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 26 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch speed. The prediction trajectories of the GCN-RF and GBT closely align with the actual curve, suggesting a commendable level of predictive accuracy. Notably, the GBT demonstrates a high degree of correlation with the actual values across most instances. Conversely, the prediction trajectories of the SVR and MLP reveal significant deviations from the actual values, particularly in regions characterized by substantial speed fluctuations, indicating that these models do not perform as effectively as the GCN-RF and GBT in the prediction of winch speed.

Table 18 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.9058, signifying superior performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0042 and 0.0630, respectively, further substantiating the GBT's high predictive accuracy in this context. The GCN-RF presents an R² value of 0.8866, with an MSE of 0.0049 and an RMSE of 0.0699, demonstrating that it also maintains a commendable level of predictive performance. In contrast, the SVR exhibits an R² value of 0.8174, an MSE of 0.0083, and an RMSE of 0.0939, indicating average performance in the winch speed prediction task, with relatively lower predictive accuracy. The MLP records an R² value of 0.8592, an MSE of 0.0061, and an RMSE of 0.0779, reflecting marginally better performance than the SVR. However, it still exhibits considerable prediction errors.

3.5. Key variables prediction results of the 240t truck crane

The prediction results of the amplitude are shown in Figure 27 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 19.

Figure 27. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 19. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 27 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual curve, suggesting a high degree of predictive accuracy. Notably, the prediction curve of the GCN-RF nearly coincides with the actual data. Conversely, the prediction curves for the SVR and MLP exhibit considerable divergence from the actual values, particularly in regions characterized by significant data fluctuations, indicating inferior performance in amplitude prediction relative to the GCN-RF and GBT.

Table 19 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9872 among the four models, signifying superior explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0011 and 0.0330, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high precision in amplitude prediction. The GBT presents an R² value of 0.9770, an MSE of 0.0020, and an RMSE of 0.0448, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8216, an MSE of 0.1234, and an RMSE of 0.1213, indicating markedly inferior to those of the GCN-RF and GBT, thereby reflecting lower prediction accuracy. The MLP records an R² value of 0.9546, an MSE of 0.0039, and an RMSE of 0.0623. While it performs slightly better than the SVR model, it still demonstrates greater prediction errors.

The prediction results of the boom length are shown in Figure 28 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 20.

Figure 28. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 20. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 28 illustrates a comparative analysis of the actual and predicted values generated by the GCN-RF, GBT, SVR, and MLP in the context of boom length prediction. The prediction curves for the GCN-RF and GBT closely approximate the actual curves, indicating a high level of predictive accuracy. Notably, the GCN-RF's prediction curve consistently aligns with the actual values across most instances. Conversely, the prediction curves for the SVR and MLP demonstrate considerable divergence from the actual values, particularly in regions characterized by significant data fluctuations, suggesting inferior performance in boom length prediction relative to the GCN-RF and GBT.

Table 20 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.9223 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0080 and 0.0893, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high precision in boom length prediction. The GBT presents an R² value of 0.8945, an MSE of 0.0108 and an RMSE of 0.1010, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8095, an MSE of 0.0197, and an RMSE of 0.1404, all of which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records the lowest R² value at 0.8573, with an MSE of 0.0144 and an RMSE of 0.1211. While it performs slightly better than the SVR, it still demonstrates greater prediction errors.

The prediction results of the flexible pump pressure are shown in Figure 29 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 21.

Figure 29. Prediction results of the flexible pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 21. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 29 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. It is evident that the prediction curves for the GCN-RF and GBT closely match the actual curves, indicating strong prediction accuracy. Notably, the GCN-RF's prediction curve aligns well with the actual values in most instances. In contrast, the SVR and MLP show a significant discrepancy from the actual values, particularly in areas with large pressure fluctuations, indicating that these models are less effective in predicting flexible pump pressure compared to GCN-RF and GBT.

Table 21 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.8504 among the four models, signifying superior performance in explanatory power and fitting capability. Its MSE and RMSE values are 0.0054 and 0.0736, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in predicting flexible pump pressure. The GBT presents an R² value of 0.8349, an MSE of 0.0065 and an RMSE of 0.0806, demonstrating strong predictive performance. On the other hand, the SVR exhibits an R² value of 0.7716, an MSE of 0.0102, and an RMSE of 0.1009, which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.7983, an MSE of 0.0076, and an RMSE of 0.0872. While it performs slightly better than SVR, it still exhibits greater prediction errors.

The prediction results of the height are shown in Figure 30 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 22.

Figure 30. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 22. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 30 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. It is evident that the prediction curve of the GCN-RF closely aligns with the actual curve, suggesting a high level of accuracy. The GBT also performs well, although it shows some discrepancies from the actual values in areas with significant fluctuations. In contrast, the prediction curves for the SVR and MLP exhibit a greater divergence from the actual values, especially in regions with sharp changes in height, indicating that these models are less effective in height prediction compared to GCN-RF and GBT.

Table 22 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.8804 among the four models, signifying its superior performance in explanatory power and fitting ability. Its MSE and RMSE values are 0.0047 and 0.0685, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in height prediction. The GBT presents an R² value of 0.8228, an MSE of 0.0067, and an RMSE of 0.0821, demonstrating good predictive capability as well. The SVR exhibits an R² value of 0.7097, an MSE of 0.0110, and an RMSE of 0.1054, which are considerably lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.6844, an MSE of 0.0124, and an RMSE of 0.1112, reflecting even greater prediction errors and lower accuracy.

The prediction results of the luffing angle are shown in Figure 31 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 23.

Figure 31. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 23. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 31 illustrates the comparison between actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting the luffing angle. It is evident that the prediction curves for the GCN-RF and GBT models closely align with the actual curves, suggesting a high level of accuracy. Notably, the GCN-RF consistently demonstrates a strong match with the actual values. In contrast, the SVR and MLP exhibit a significant discrepancy from the actual values, particularly in areas with more pronounced angle variations, suggesting that these models are less effective in predicting the luffing angle compared to GCN-RF and GBT.

Table 23 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.9678 among the four models, signifying its superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0011 and 0.0338, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in amplitude angle prediction. The GBT presents an R² value of 0.9261, an MSE of 0.0026, and an RMSE of 0.0511, demonstrating strong predictive performance. The SVR exhibits an R² value of 0.7484, an MSE of 0.0092, and an RMSE of 0.0962, which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.7335, an MSE of 0.0094, and an RMSE of 0.0972, reflecting slightly better performance than SVR but still reflecting greater prediction errors.

The prediction results of the slewing angle are shown in Figure 32 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 24.

Figure 32. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 24. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 32 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting slewing angles. The figure reveals a noticeable gap between the prediction curves and the actual curves across all models, indicating that the prediction accuracy for each model is relatively low in this context. Notably, the SVR and MLP show considerable discrepancies from the actual values, particularly in areas with more significant angle variations, highlighting their limited predictive capabilities. On the other hand, the GCN-RF performs comparatively better than the others, although its results still do not meet expectations when compared to other models.

Table 24 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.5705 among the four models, signifying a relatively better ability to explain and fit the data in slewing angle prediction, though it is still not optimal. Its MSE and RMSE values are 0.0480 and 0.2190, respectively, which are the best among the models but still indicate significant error, reflecting limited prediction accuracy. The GBT presents an R² value of 0.4239, an MSE of 0.0642, and an RMSE of 0.2534, demonstrating average performance. The SVR exhibits an R² value of 0.3255, an MSE of 0.0752, and an RMSE of 0.2752, indicating lower prediction accuracy. Lastly, the MLP records the lowest R² value of 0.2514, with an MSE of 0.0724 and an RMSE of 0.2692, reflecting the weakest performance in slewing angle prediction.

The prediction results of the winch pump outlet pressure are shown in Figure 33 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 25.

Figure 33. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 25. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 33 illustrates the comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting the winch pump outlet pressure. It is evident that the prediction curves for the GCN-RF and GBT closely match the actual curves, suggesting a high level of prediction accuracy. Notably, the GBT consistently aligns well with the actual values. In contrast, the SVR and MLP exhibit a significant disparity between their prediction curves and the actual values, particularly in areas with more data fluctuations, suggesting that these models are less effective than GCN-RF and GBT in the prediction of the winch pump outlet pressure.

Table 25 provides the R², MSE, and RMSE values for each model. The GBT achieves the highest R² value of 0.9195 among the four models, signifying its superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0018 and 0.0430, respectively, which are the lowest, further substantiating the GBT's high precision in predicting winch pump outlet pressure. The GCN-RF presents an R² value of 0.9167, an MSE of 0.0020, and an RMSE of 0.0452, demonstrating strong predictive ability. However, the SVR exhibits an R² value of 0.3736, an MSE of 0.0065, and an RMSE of 0.0804, indicating significantly poorer performance compared to GCN-RF and GBT, with lower prediction accuracy. The MLP records an R² value of 0.8889, an MSE of 0.0028, and an RMSE of 0.0525. While it performs better than SVR, it still does not measure up to GCN-RF and GBT.

The prediction results of the winch speed are shown in Figure 34 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 26.

Figure 34. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 26. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 34 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting the winch speed. It is evident that the prediction curve of the GBT model closely aligns with the actual curve, suggesting a high level of accuracy. The GCN-RF also shows strong consistency with the actual values, following closely behind. In contrast, the SVR and MLP display a greater gap between their prediction curves and the actual values, especially in areas with significant speed variations, indicating that these models are less effective than GBT and GCN-RF in predicting the winch speed.

Table 26 provides the R², MSE, and RMSE values for each model. The GBT achieves the highest R² value of 0.9088 among the four models, signifying the best explanatory power and fitting capability. Its MSE and RMSE values are 0.0030 and 0.0552, respectively, which are the lowest, further substantiating the GBT's high precision in winch speed prediction. The GCN-RF ranks second with an R² value of 0.8690, an MSE of 0.0040, and an RMSE of 0.0631, indicating good predictive performance. The SVR exhibits an R² value of 0.8466, an MSE of 0.0057, and an RMSE of 0.0707, indicating significantly poorer performance compared to GCN-RF and GBT, with lower prediction accuracy. The MLP records an R² value of 0.8701, an MSE of 0.0043, and an RMSE of 0.0657. While it performs better than SVR, it still does not match the effectiveness of GBT and GCN-RF.

Comments 9: How you ensure the generalization of the soft sensor for more than one crane?

Response 9: Thank you for pointing this out. We agree with this comment. We collected operational data from three cranes with different lifting capacities under actual working conditions. The three datasets were obtained under varying operational environments specific to each crane, which is fundamentally different from experimental data collected under fixed conditions. By comparing different methods and datasets, we can assess the robustness and generalization capability of the models. As shown in “Response 8”.

Comments on the Quality of English Language

Check some typos on the manuscript and double check the references and bibtext file, several little fixes required

Response: I have modified the references and the content of the manuscript.

Thank you again for your valuable suggestions.

Comments 1: Please provide some feature maps that the GCN has extracted and make an analysis and comparison between different performance indicators.

Response 1: Thank you for pointing this out. We agree with this comment. We have already added the content on correlation analysis, as follows.

The correlation between variables was analyzed based on data from the actual operating conditions of the crane, identifying input variables that are strongly correlated with the target variables, as shown in Figure 10.

Figure 10. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

The analysis of Figure 10 reveals the correlation between the target variables Amplitude, Boom Length, Flexible Pump Pressure, Height, Luffing Angle, Slewing Angle, Winch Pump Outlet Pressure, and Winch Speed with the input variables, as detailed below.

(1) A significant correlation exists between Amplitude and both the Luffing Handle Signal and the Luffing Balance Valve Control Current. In contrast, the correlation with variables such as Rated Weight, Actual Weight, Throttle Position, and Engine Speed is comparatively weaker or not statistically significant.

(2) A notable positive correlation exists between Boom Length and both the Telescoping Handle Signal and the Luffing Handle Signal. Conversely, the relationships between Boom Length and variables such as Rated Weight, Actual Weight, Throttle Position, and Engine Speed are either weak or not statistically significant.

(3) The Flexible Pump Pressure exhibits a notable positive correlation with several variables, including Flexible Pump Control Current, Telescoping Handle Signal, Luffing Balance Valve Control Current, and Winch Up Pump Control Current. Conversely, variables such as Slewing Handle Signal, Left Slewing Current, and Right Slewing Current demonstrate a weaker or negligible correlation with Flexible Pump Pressure.

(4) The data indicates a positive correlation between Height and the Luffing Handle Signal, Telescoping Handle Signal, and Luffing Balance Valve Control Current. In contrast, the correlation between height and other variables, such as Rated Weight, Engine Speed, and Winch, is comparatively weaker or not statistically significant.

(5) The Luffing Angle exhibits a positive correlation with the Luffing Handle Signal, the control current of the Luffing Balance Valve, and the Telescoping Handle Signal. In contrast, its correlation with variables such as Rated Weight, Engine Speed, and Winch is comparatively weaker or not statistically significant.

(6) The Slewing Angle exhibits a robust positive correlation with the Slewing Handle Signal, Left Slewing Current, and Right Slewing Current. Conversely, variables such as Rated Weight, Engine Speed, and Flexible Pump Control Current demonstrate a weaker correlation or lack significant association with the Slewing Angle.

(7) The analysis reveals a positive correlation between Winch Pump Outlet Pressure and several winch-related control currents, specifically Winch Motor Control Current, Winch Up Pump Control Current, Winch Down Pump Control Current, and Winching Handle Signal. In contrast, the correlation between Winch Pump Outlet Pressure and other variables, including Flexible Pump Control Current, Slewing Handle, Left Slewing Current, and Right Slewing Current, is comparatively weaker.

(8) The analysis reveals a positive correlation between Winch Speed and several variables, including Engine Speed, Winch Motor Control Current, Winch Up Pump Control Current, Winching Handle Signal, and winch down pump control current. In contrast, the correlation between Winch Speed and variables such as Left Slewing Current, Right Slewing Current, and Flexible Pump Control Current is comparatively weaker.

Comments 2: How did you validate the robustness of the proposed method? Please add experiment specifications and results.

Response 2: Thank you for pointing this out. We agree with this comment. We have added operational data from the 160t, 200t, and 240t cranes to validate the generalization capabilities of GCN-RF, as shown “Response 5”.

Comments 3: When predicting the performance indicators, how do we address the coupling effect between different systems, like the slewing, telescopic, winching, and luffing?

Response 3: Thank you for pointing this out. The four handle signals correspond to four actions. For the coupled cases, we used the handle signals to decouple the executed actions, thereby simplifying the operating conditions for subsequent research.

Comments 4: At present, the input variables are sampled with a frequency of 1Hz, which is too low for state monitoring. So, when sampling with high frequency, like 1kHz, how about the modeling performance for the variables in Fig. 9 - 16?

Response 4: Thank you for pointing this out. Due to limitations such as data transmission, data storage, network speed, and working environment, it is challenging to achieve high-frequency data collection in practical operating environments. High-frequency data collection is also part of our ongoing research, aiming to achieve high-frequency data acquisition within the constraints of load capacity. However, based on your suggestion, we attempted to increase the data sampling frequency from 1 Hz to 100 Hz using data interpolation for this study.

The predictions of the eight target variables based on GCN-RF are shown below. As shown, the dataset with high-frequency collection significantly improved the accuracy of the model's predictions, providing valuable insights and references for our ongoing research.

Comments 5: For the predicting results in Fig. 9-16, please provide some local magnification with more details.

 Response 5: Thank you for pointing this out. We agree with this comment. We have added operational data from the 160t, 200t, and 240t cranes to validate the generalization capabilities of GCN-RF, as shown below.

3.3. Key variables prediction results of the 160t truck crane

The prediction results of the amplitude are shown in Figure 11 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 3.

Figure 11. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 3. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 11 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The prediction curve produced by the GCN-RF demonstrates a close correspondence with the actual curve, indicating a high level of predictive accuracy. Notably, the GBT's prediction curve aligns well with the actual values, exhibiting only minor deviations in specific areas attributed to volatility. In contrast, the prediction curves for the SVR and MLP reveal substantial discrepancies from the actual values, particularly in regions marked by rapid fluctuations. This suggests that the SVR and MLP are less effective in amplitude prediction compared to the GCN-RF and GBT.

Table 3 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8875, indicating its excellent explanatory power and fitting capability. Its MSE and RMSE values are 0.0031 and 0.0556, respectively, which are the lowest among the models, highlighting the GCN-RF's exceptional accuracy in amplitude prediction. The GBT presents an R² value of 0.8680, an MSE of 0.0036, and an RMSE of 0.0603, demonstrating strong predictive performance. In contrast, the SVR exhibits an R² value of 0.6116, an MSE of 0.0108, and an RMSE of 0.1039, indicating average performance in amplitude prediction with relatively lower accuracy. Lastly, the MLP records the lowest R² value of 0.6329, along with an MSE of 0.0101 and an RMSE of 0.1007, reflecting greater prediction errors and inconsistent performance.

The prediction results of the boom length are shown in Figure 12 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 4.

Figure 12. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 4. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 12 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting boom length. It is evident that the prediction curves for the GCN-RF and GBT closely align with the actual curve, indicating a high level of predictive accuracy. Notably, the GCN-RF's curve nearly perfectly matches the actual values. In contrast, the SVR's prediction curve exhibits some deviations from the actual values, particularly in regions with significant data fluctuations. The MLP shows relatively poor performance, as indicated by a larger gap between its prediction curve and the actual values, suggesting a weaker capability in predicting boom length.

Table 4 provides the R², MSE, and RMSE metrics for each model. The GCN-RF exhibits the highest R² value of 0.8946, indicating superior performance in terms of explanatory power and fitting ability. The MSE and RMSE values for this model are 0.0014 and 0.0380, respectively, both of which are the lowest among the models, further confirming the exceptional accuracy of the GCN-RF in predicting arm length. The GBT presents an R² value of 0.8904, an MSE of 0.0014, and an RMSE of 0.0379, also demonstrating strong predictive capability. In contrast, the SVR has an R² value of 0.6372, an MSE of 0.0082, and an RMSE of 0.0906, indicating significantly poorer performance compared to the GCN-RF and GBT, with lower predictive accuracy. The MLP records the lowest R² value of 0.5700, with an MSE of 0.0059 and an RMSE of 0.0768, reflecting greater prediction errors and diminished accuracy.

The prediction results of the flexible pump pressure are shown in Figure 13 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 5.

Figure 13. Prediction results of the flexible pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 5. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 13 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. The results indicate that the prediction curve of the GCN-RF closely aligns with the actual curve, thereby indicating a high level of predictive accuracy. The GBT exhibits a similar trend, maintaining consistency with the actual values overall, although it does present some deviations characterized by fluctuations in specific regions. In contrast, the prediction curves for the SVR and MLP reveal more substantial discrepancies from the actual values, particularly in areas experiencing significant pressure variations, suggesting that these models are less effective than the GCN-RF and GBT in predicting flexible pump pressure.

Table 5 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8689, signifying superior performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0048 and 0.0691, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's high accuracy in predicting flexible pump pressure. The GBT presents an R² value of 0.8124, an MSE of 0.0080, and an RMSE of 0.0897, demonstrating commendable predictive capability. Conversely, the SVR exhibits an R² value of 0.6767, an MSE of 0.0107, and an RMSE of 0.1172, indicating a significantly lower predictive accuracy compared to the GCN-RF and GBT. The MLP records the lowest R² value of 0.7200, with an MSE of 0.0102 and an RMSE of 0.1010, reflecting greater prediction errors and diminished accuracy.

The prediction results of the height are shown in Figure 14 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 6.

Figure 14. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 6. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 14 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. The prediction curves for the GCN-RF and GBT closely align with the actual curve, indicating a high level of predictive accuracy. Notably, the GBT's prediction curve nearly coincides with the actual values. Conversely, the prediction curves for the SVR and MLP display more pronounced deviations from the actual values, particularly in regions characterized by significant height variations, suggesting that these models exhibit inferior performance relative to the GCN-RF and GBT in the height prediction.

Table 6 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.9260, signifying its exceptional explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0014 and 0.0374, respectively, both of which are the lowest among the models, thereby reinforcing the GBT's accuracy in height prediction. The GCN-RF presents an R² value of 0.9248 and still reflects robust predictive capability accompanied by an MSE of 0.0015 and an RMSE of 0.0388, which slightly lower than that of the GBT. In contrast, the SVR exhibits an R² value of 0.8047, with an MSE of 0.0047 and an RMSE of 0.0669, indicating a significantly lower predictive accuracy compared to the GCN-RF and GBT. The MLP records the lowest R² value of 0.8312, with an MSE of 0.0036 and an RMSE of 0.0583. Although it marginally outperforms the SVR, it still demonstrates greater prediction errors.

The prediction results of the luffing angle are shown in Figure 15 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 7.

Figure 15. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 7. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 15 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude angle. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual curve, signifying a high level of predictive accuracy. Notably, the two curves nearly coincide in regions characterized by stable changes. Conversely, the prediction curves for the SVR and MLP display more pronounced deviations from the actual values, particularly in areas with significant fluctuations, indicating that these models exhibit inferior performance relative to the GCN-RF and GBT in amplitude angle prediction.

Table 7 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8952 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0015 and 0.0381, respectively, both of which are the lowest, further corroborating the GCN-RF's high accuracy in amplitude angle prediction. The GBT presents an R² value of 0.8917, an MSE of 0.0015, and an RMSE of 0.0384, demonstrating robust predictive capability. In contrast, the SVR exhibits an R² value of 0.6577, an MSE of 0.0676, and an RMSE of 0.0768, indicating significantly lower predictive accuracy compared to the GCN-RF and GBT models. The MLP records the lowest R² value of 0.6718, with an MSE of 0.0059 and an RMSE of 0.0767, reflecting greater prediction errors and diminished accuracy.

The prediction results of the slewing angle are shown in Figure 16 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 8.

Figure 16. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 8. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 16 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting slewing angle. The prediction curve of the GCN-RF closely aligns with the actual curve, suggesting a high level of predictive accuracy. The GBT still adheres to the overall trend of the observed data while exhibiting some deviation from the actual values. In contrast, the prediction curves for the SVR and MLP display more pronounced discrepancies from the actual values, particularly in regions characterized by significant data fluctuations, thereby indicating inferior performance relative to the GCN-RF and GBT in the task of slewing angle prediction.

Table 8 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8752 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0053 and 0.0728, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high accuracy in slewing angle prediction. The GBT presents an R² value of 0.7975, an MSE of 0.0089, and an RMSE of 0.0944. Although it is slightly less accurate than the GCN-RF, it still demonstrates commendable predictive capability. Conversely, the SVR exhibits an R² value of 0.5463, an MSE of 0.0193, and an RMSE of 0.1390, indicating significantly poorer performance when compared to the GCN-RF and GBT, along with reduced predictive accuracy. The MLP records the lowest R² value of 0.5220, with an MSE of 0.0203 and an RMSE of 0.1423, reflecting greater prediction errors and diminished accuracy.

The prediction results of the winch pump outlet pressure are shown in Figure 17 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 9.

Figure 17. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 9. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 17 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch pump outlet pressure. The prediction curve for the GCN-RF closely aligns with the actual curve, signifying a high level of predictive accuracy. The GBT also demonstrates commendable performance. However, it exhibits noticeable deviations in certain regions characterized by significant fluctuations when compared to the GCN-RF. In contrast, the prediction curves for the SVR and MLP reveal more substantial discrepancies from the actual values, particularly under conditions of heightened data variability, thereby indicating inferior performance relative to the GCN-RF and GBT in the prediction of winch pump outlet pressure.

Table 9 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8948 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for the GCN-RF are 0.0031 and 0.0555, respectively, both of which are the lowest, further substantiating the model's high accuracy in predicting winch pump outlet pressure. The GBT presents an R² value of 0.8089, with an MSE of 0.0068 and an RMSE of 0.0826, demonstrating robust predictive capability. Conversely, the SVR exhibits an R² value of 0.6917, an MSE of 0.0114, and an RMSE of 0.1070, indicating significantly poorer performance in comparison to the GCN-RF and GBT, along with diminished predictive accuracy. The MLP records the lowest R² value of 0.7216, accompanied by an MSE of 0.0094 and an RMSE of 0.0997, reflecting greater prediction errors and reduced accuracy.

The prediction results of the winch speed are shown in Figure 18 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 10.

Figure 18. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 10. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 18 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch speed. The prediction curves for the GCN-RF and GBT closely align with the actual data, suggesting a high level of predictive accuracy. Notably, the prediction curve of the GCN-RF nearly coincides with the actual values. Conversely, the prediction curves for the SVR and MLP models exhibit considerable divergence from the actual values, particularly in areas characterized by significant fluctuations in speed, indicating inferior performance in winch speed prediction relative to the GCN-RF and GBT.

Table 10 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9462 among the four models, signifying superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0010 and 0.0323, respectively, both of which are the lowest, further substantiating the GCN-RF's high accuracy in winch speed prediction. The GBT presents an R² value of 0.9423, an MSE of 0.0013, and an RMSE of 0.0351, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8740, an MSE of 0.0031, and an RMSE of 0.0553, indicating significantly diminished performance compared to the GCN-RF and GBT, with reduced predictive accuracy. The MLP records the lowest R² value of 0.9176, alongside an MSE of 0.0016 and an RMSE of 0.0400. Although it performs slightly better than the SVR model, it still demonstrates greater prediction errors.

3.4. Key variables prediction results of the 200t truck crane

The prediction results of the amplitude are shown in Figure 19 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 11.

Figure 19. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 11. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 19 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The results indicate that the prediction curve of the GCN-RF closely approximates the actual curve, signifying a high level of predictive accuracy. The GBT's prediction curve demonstrates a strong alignment with the actual values, albeit with minor discrepancies in certain fluctuating regions. Conversely, the prediction curves for the SVR and MLP exhibit a significant divergence from the actual values, particularly in areas characterized by substantial data variations, thereby indicating inferior performance in amplitude prediction relative to the GCN-RF and GBT.

Table 11 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8875, signifying superior explanatory power and fitting capability. Its MSE and RMSE values are recorded at 0.0031 and 0.0556, respectively, which are the lowest among the models, further substantiating the GCN-RF's high accuracy in amplitude prediction. The GBT presents an R² value of 0.8680, an MSE of 0.0036, and an RMSE of 0.0603, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.6116, an MSE of 0.0108, and an RMSE of 0.1039, indicating average performance in amplitude prediction with relatively low accuracy. The MLP records the lowest R² value of 0.6329, an MSE of 0.0101, and an RMSE of 0.1007, reflecting greater prediction errors and a lack of stability in its performance.

The prediction results of the boom length are shown in Figure 20 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 12.

Figure 20. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 12. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 20 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting boom length. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual values, suggesting a high degree of predictive accuracy. Notably, the curve for the GCN-RF nearly coincides with the actual data. In contrast, the SVR exhibits some discrepancies from the actual values, particularly in regions characterized by significant fluctuations in the data. The MLP demonstrates comparatively inferior performance, as evidenced by a more pronounced divergence between its prediction curve and the actual values, indicating a reduced efficacy in boom length prediction.

Table 12 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9263, signifying superior fitting performance in the task of boom length prediction. Its MSE and RMSE values are recorded at 0.0032 and 0.0565, respectively, which are the lowest among the evaluated models, thereby reinforcing the GCN-RF's advantage in predictive accuracy. The GBT presents an R² value of 0.9051, an MSE of 0.0042, and an RMSE of 0.0646. While its performance is marginally inferior to that of the GCN-RF, it still demonstrates commendable predictive capability. The SVR exhibits an R² value of 0.9190, which is slightly superior to that of the GBT. However, its MSE and RMSE values are higher, recorded at 0.0074 and 0.1010, respectively, indicating a deficiency in accuracy. The MLP records the lowest R² value of 0.8307, with an MSE of 0.0104 and an RMSE of 0.0860, reflecting that it is comparatively less effective in addressing the boom length prediction challenge.

The prediction results of the flexible pump pressure are shown in Figure 21 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 13.

Figure 21. Prediction results of the flexiable pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 13. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 21 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. The results indicate that the prediction curve produced by the GCN-RF closely approximates the actual curve, suggesting a high level of predictive accuracy. In contrast, the GBT demonstrates slightly inferior performance; while its predictions generally align with the actual values, there are notable discrepancies in certain fluctuating regions. The prediction curves for the SVR and MLP reveal a considerable divergence from the actual values, particularly in areas characterized by abrupt pressure variations, indicating that these models are less effective in predicting flexible pump pressure when compared to the GCN-RF and GBT.

Table 13 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8769, signifying its superior explanatory power and fitting capability compared to the other models. The corresponding MSE and RMSE values for the GCN-RF are 0.0054 and 0.0733, respectively, both of which are the lowest among the models assessed, further substantiating the GCN-RF's high accuracy in accuracy for flexible pump pressure prediction. The GBT presents an R² value of 0.7910, an MSE of 0.0087, and an RMSE of 0.0931, demonstrating a marginally lower performance. The SVR exhibits an R² value of 0.7218, which is inferior to both the GCN-RF and GBT, with MSE and RMSE values of 0.0129 and 0.1135, respectively, indicating suboptimal performance on this dataset. Lastly, the MLP records an R² value of 0.7440, an MSE of 0.0116, and an RMSE of 0.1056, which slightly better than the SVR, reflecting significantly lower than the performance metrics of the GCN-RF and GBT.

The prediction results of the height are shown in Figure 22 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 14.

Figure 22. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 14. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 22 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual values, suggesting a high degree of predictive accuracy. Notably, the GBT exhibits a prediction curve that nearly coincides with the actual measurements. Conversely, the prediction curves for the SVR and MLP demonstrate a considerable deviation from the actual values, particularly in areas characterized by significant height variations. This discrepancy suggests that the performance of the SVR and MLP in height prediction is inferior to that of the GCN-RF and GBT.

Table 14 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves an R² value of 0.9716, which is near 1, signifying exceptional explanatory power and fitting capability. Its MSE and RMSE values are 0.0032 and 0.0564, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's superior accuracy in height prediction. The GBT presents an R² value of 0.9611, with an MSE of 0.0044 and an RMSE of 0.0667, which slightly lower than that of the GCN-RF and still reflects robust predictive performance. The SVR exhibits an R² value of 0.9234, an MSE of 0.0090, and an RMSE of 0.0949, indicating moderate performance in the height prediction task, characterized by relatively low accuracy. The MLP records an R² value of 0.9177, an MSE of 0.0092, and an RMSE of 0.0957, reflecting that both the MLP and SVR models exhibit significant prediction errors.

The prediction results of the luffing angle are shown in Figure 23 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 15.

Figure 23. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 15. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 23 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude angle. The results indicate that the prediction curve generated by the GCN-RF nearly coincides with the actual curve, suggesting an exceptionally high level of predictive accuracy. The GBT also demonstrates commendable performance, as its predictions closely follow the actual values, albeit with minor discrepancies in regions characterized by fluctuations. Conversely, the prediction curves for the SVR and MLP exhibit considerable divergence from the actual values, particularly during intervals of rapid data variation.

Table 15 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves an R² value of 0.9793, which is nearly equal to 1, thereby reflecting its superior explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0027 and 0.0517, respectively, both of which are the lowest among the models, further substantiating the GCN-RF's high accuracy in amplitude angle prediction. The GBT presents an R² value of 0.9660, an MSE of 0.0046, and an RMSE of 0.0678, demonstrating that it retains a robust predictive capacity. The SVR exhibits an R² value of 0.9283, an MSE of 0.0095, and an RMSE of 0.0974, indicating a moderate performance in the amplitude angle prediction task, characterized by relatively lower accuracy. The MLP records an R² value of 0.9150, an MSE of 0.0121, and an RMSE of 0.1046. Consequently, both the MLP and SVR exhibit greater prediction errors and demonstrate less stable performance.

The prediction results of the slewing angle are shown in Figure 24 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 16.

Figure 24. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 16. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 24 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting slewing angle. It is evident that while the prediction curve generated by the GCN-RF exhibits a degree of alignment with the actual curve, its accuracy has diminished relative to prior predictions. The GBT also demonstrates significant deviations from the actual values, particularly in regions characterized by substantial data variability. In contrast, the prediction curves for the SVR and MLP reveal even greater discrepancies from the actual values, especially in areas with frequent fluctuations, thereby indicating a diminished performance in the prediction of slewing angle prediction.

Table 16 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.8220 among the four models, signifying a relatively strong explanatory capacity in slewing angle prediction, albeit with a notable decline in performance compared to earlier assessments. The MSE and RMSE values for this model are 0.0118 and 0.1084, respectively, reflecting a certain degree of predictive error. The GBT presents an R² value of 0.7836, with an MSE of 0.0144 and an RMSE of 0.1201, demonstrating that it retains some predictive efficacy for this task. Conversely, the SVR exhibits an R² value of 0.6738, an MSE of 0.0225, and an RMSE of 0.1501, indicating inadequate performance in slewing angle prediction, characterized by low predictive accuracy. The MLP records the lowest R² value of 0.6702, alongside an MSE of 0.0244 and an RMSE of 0.1473, reflecting substantial prediction errors and a lack of stability.

The prediction results of the winch pump outlet pressure are shown in Figure 25 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 17.

Figure 25. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 17. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 25 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch pump outlet pressure. The data presented in the figure indicates that the prediction trajectories of the GCN-RF and GBT closely align with the actual curve, suggesting a superior level of predictive accuracy. Notably, the GBT consistently yields predictions that are in close to the actual values across most scenarios. Conversely, the prediction trajectories of the SVR and MLP demonstrate significant deviations from the actual values, particularly during periods characterized by substantial data fluctuations, indicating that these models exhibit inferior performance relative to the GCN-RF and GBT in the prediction of winch pump outlet pressure.

Table 17 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.8999, signifying its optimal performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0021 and 0.0459, respectively, further substantiating the GBT's high predictive accuracy in this application. The GCN-RF presents an R² value of 0.8932, an MSE of 0.0022 and an RMSE of 0.0474, which slightly lower than that of the GBT model and still indicates robust predictive capability. In contrast, the SVR exhibits an R² value of 0.8119, with an MSE of 0.0067 and an RMSE of 0.0687, indicating average performance in the winch pump outlet pressure prediction and relatively diminished predictive accuracy. The MLP records the lowest R² value of 0.7928, accompanied by an MSE of 0.0074 and an RMSE of 0.0750, reflecting greater prediction errors and a lack of stability in its performance.

The prediction results of the winch speed are shown in Figure 26 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 18.

Figure 26. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 18. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 26 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting winch speed. The prediction trajectories of the GCN-RF and GBT closely align with the actual curve, suggesting a commendable level of predictive accuracy. Notably, the GBT demonstrates a high degree of correlation with the actual values across most instances. Conversely, the prediction trajectories of the SVR and MLP reveal significant deviations from the actual values, particularly in regions characterized by substantial speed fluctuations, indicating that these models do not perform as effectively as the GCN-RF and GBT in the prediction of winch speed.

Table 18 provides the R², MSE, and RMSE metrics for each model. The GBT achieves the highest R² value of 0.9058, signifying superior performance in terms of explanatory power and fitting capability. The corresponding MSE and RMSE values are 0.0042 and 0.0630, respectively, further substantiating the GBT's high predictive accuracy in this context. The GCN-RF presents an R² value of 0.8866, with an MSE of 0.0049 and an RMSE of 0.0699, demonstrating that it also maintains a commendable level of predictive performance. In contrast, the SVR exhibits an R² value of 0.8174, an MSE of 0.0083, and an RMSE of 0.0939, indicating average performance in the winch speed prediction task, with relatively lower predictive accuracy. The MLP records an R² value of 0.8592, an MSE of 0.0061, and an RMSE of 0.0779, reflecting marginally better performance than the SVR. However, it still exhibits considerable prediction errors.

3.5. Key variables prediction results of the 240t truck crane

The prediction results of the amplitude are shown in Figure 27 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 19.

Figure 27. Prediction results of the amplitude based on GCN-RF, GBT, SVR and MLP.

Table 19. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 27 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting amplitude. The results indicate that the prediction curves for the GCN-RF and GBT closely align with the actual curve, suggesting a high degree of predictive accuracy. Notably, the prediction curve of the GCN-RF nearly coincides with the actual data. Conversely, the prediction curves for the SVR and MLP exhibit considerable divergence from the actual values, particularly in regions characterized by significant data fluctuations, indicating inferior performance in amplitude prediction relative to the GCN-RF and GBT.

Table 19 provides the R², MSE, and RMSE metrics for each model. The GCN-RF achieves the highest R² value of 0.9872 among the four models, signifying superior explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0011 and 0.0330, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high precision in amplitude prediction. The GBT presents an R² value of 0.9770, an MSE of 0.0020, and an RMSE of 0.0448, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8216, an MSE of 0.1234, and an RMSE of 0.1213, indicating markedly inferior to those of the GCN-RF and GBT, thereby reflecting lower prediction accuracy. The MLP records an R² value of 0.9546, an MSE of 0.0039, and an RMSE of 0.0623. While it performs slightly better than the SVR model, it still demonstrates greater prediction errors.

The prediction results of the boom length are shown in Figure 28 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 20.

Figure 28. Prediction results of the boom length based on GCN-RF, GBT, SVR and MLP.

Table 20. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 28 illustrates a comparative analysis of the actual and predicted values generated by the GCN-RF, GBT, SVR, and MLP in the context of boom length prediction. The prediction curves for the GCN-RF and GBT closely approximate the actual curves, indicating a high level of predictive accuracy. Notably, the GCN-RF's prediction curve consistently aligns with the actual values across most instances. Conversely, the prediction curves for the SVR and MLP demonstrate considerable divergence from the actual values, particularly in regions characterized by significant data fluctuations, suggesting inferior performance in boom length prediction relative to the GCN-RF and GBT.

Table 20 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.9223 among the four models, signifying superior performance in terms of explanatory power and fitting capability. The MSE and RMSE values for this model are 0.0080 and 0.0893, respectively, both of which are the lowest recorded, further substantiating the GCN-RF's high precision in boom length prediction. The GBT presents an R² value of 0.8945, an MSE of 0.0108 and an RMSE of 0.1010, demonstrating commendable predictive performance. In contrast, the SVR exhibits an R² value of 0.8095, an MSE of 0.0197, and an RMSE of 0.1404, all of which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records the lowest R² value at 0.8573, with an MSE of 0.0144 and an RMSE of 0.1211. While it performs slightly better than the SVR, it still demonstrates greater prediction errors.

The prediction results of the flexible pump pressure are shown in Figure 29 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 21.

Figure 29. Prediction results of the flexible pump pressure based on GCN-RF, GBT, SVR and MLP.

Table 21. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 29 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting flexible pump pressure. It is evident that the prediction curves for the GCN-RF and GBT closely match the actual curves, indicating strong prediction accuracy. Notably, the GCN-RF's prediction curve aligns well with the actual values in most instances. In contrast, the SVR and MLP show a significant discrepancy from the actual values, particularly in areas with large pressure fluctuations, indicating that these models are less effective in predicting flexible pump pressure compared to GCN-RF and GBT.

Table 21 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.8504 among the four models, signifying superior performance in explanatory power and fitting capability. Its MSE and RMSE values are 0.0054 and 0.0736, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in predicting flexible pump pressure. The GBT presents an R² value of 0.8349, an MSE of 0.0065 and an RMSE of 0.0806, demonstrating strong predictive performance. On the other hand, the SVR exhibits an R² value of 0.7716, an MSE of 0.0102, and an RMSE of 0.1009, which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.7983, an MSE of 0.0076, and an RMSE of 0.0872. While it performs slightly better than SVR, it still exhibits greater prediction errors.

The prediction results of the height are shown in Figure 30 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 22.

Figure 30. Prediction results of the height based on GCN-RF, GBT, SVR and MLP.

Table 22. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 30 illustrates a comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting height. It is evident that the prediction curve of the GCN-RF closely aligns with the actual curve, suggesting a high level of accuracy. The GBT also performs well, although it shows some discrepancies from the actual values in areas with significant fluctuations. In contrast, the prediction curves for the SVR and MLP exhibit a greater divergence from the actual values, especially in regions with sharp changes in height, indicating that these models are less effective in height prediction compared to GCN-RF and GBT.

Table 22 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.8804 among the four models, signifying its superior performance in explanatory power and fitting ability. Its MSE and RMSE values are 0.0047 and 0.0685, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in height prediction. The GBT presents an R² value of 0.8228, an MSE of 0.0067, and an RMSE of 0.0821, demonstrating good predictive capability as well. The SVR exhibits an R² value of 0.7097, an MSE of 0.0110, and an RMSE of 0.1054, which are considerably lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.6844, an MSE of 0.0124, and an RMSE of 0.1112, reflecting even greater prediction errors and lower accuracy.

The prediction results of the luffing angle are shown in Figure 31 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 23.

Figure 31. Prediction results of the luffing angle based on GCN-RF, GBT, SVR and MLP.

Table 23. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 31 illustrates the comparison between actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting the luffing angle. It is evident that the prediction curves for the GCN-RF and GBT models closely align with the actual curves, suggesting a high level of accuracy. Notably, the GCN-RF consistently demonstrates a strong match with the actual values. In contrast, the SVR and MLP exhibit a significant discrepancy from the actual values, particularly in areas with more pronounced angle variations, suggesting that these models are less effective in predicting the luffing angle compared to GCN-RF and GBT.

Table 23 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.9678 among the four models, signifying its superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0011 and 0.0338, respectively, both of which are the lowest, further substantiating the GCN-RF's high precision in amplitude angle prediction. The GBT presents an R² value of 0.9261, an MSE of 0.0026, and an RMSE of 0.0511, demonstrating strong predictive performance. The SVR exhibits an R² value of 0.7484, an MSE of 0.0092, and an RMSE of 0.0962, which are significantly lower than those of the GCN-RF and GBT, indicating diminished prediction accuracy. The MLP records an R² value of 0.7335, an MSE of 0.0094, and an RMSE of 0.0972, reflecting slightly better performance than SVR but still reflecting greater prediction errors.

The prediction results of the slewing angle are shown in Figure 32 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 24.

Figure 32. Prediction results of the slewing angle based on GCN-RF, GBT, SVR and MLP.

Table 24. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 32 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting slewing angles. The figure reveals a noticeable gap between the prediction curves and the actual curves across all models, indicating that the prediction accuracy for each model is relatively low in this context. Notably, the SVR and MLP show considerable discrepancies from the actual values, particularly in areas with more significant angle variations, highlighting their limited predictive capabilities. On the other hand, the GCN-RF performs comparatively better than the others, although its results still do not meet expectations when compared to other models.

Table 24 provides the R², MSE, and RMSE values for each model. The GCN-RF achieves the highest R² value of 0.5705 among the four models, signifying a relatively better ability to explain and fit the data in slewing angle prediction, though it is still not optimal. Its MSE and RMSE values are 0.0480 and 0.2190, respectively, which are the best among the models but still indicate significant error, reflecting limited prediction accuracy. The GBT presents an R² value of 0.4239, an MSE of 0.0642, and an RMSE of 0.2534, demonstrating average performance. The SVR exhibits an R² value of 0.3255, an MSE of 0.0752, and an RMSE of 0.2752, indicating lower prediction accuracy. Lastly, the MLP records the lowest R² value of 0.2514, with an MSE of 0.0724 and an RMSE of 0.2692, reflecting the weakest performance in slewing angle prediction.

The prediction results of the winch pump outlet pressure are shown in Figure 33 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 25.

Figure 33. Prediction results of the winch pump outlet pressure based on GCN-RF, GBT, SVR and MLP.

Table 25. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 33 illustrates the comparison between the actual and predicted values of the GCN-RF, GBT, SVR, and MLP in predicting the winch pump outlet pressure. It is evident that the prediction curves for the GCN-RF and GBT closely match the actual curves, suggesting a high level of prediction accuracy. Notably, the GBT consistently aligns well with the actual values. In contrast, the SVR and MLP exhibit a significant disparity between their prediction curves and the actual values, particularly in areas with more data fluctuations, suggesting that these models are less effective than GCN-RF and GBT in the prediction of the winch pump outlet pressure.

Table 25 provides the R², MSE, and RMSE values for each model. The GBT achieves the highest R² value of 0.9195 among the four models, signifying its superior explanatory power and fitting capability. Its MSE and RMSE values are 0.0018 and 0.0430, respectively, which are the lowest, further substantiating the GBT's high precision in predicting winch pump outlet pressure. The GCN-RF presents an R² value of 0.9167, an MSE of 0.0020, and an RMSE of 0.0452, demonstrating strong predictive ability. However, the SVR exhibits an R² value of 0.3736, an MSE of 0.0065, and an RMSE of 0.0804, indicating significantly poorer performance compared to GCN-RF and GBT, with lower prediction accuracy. The MLP records an R² value of 0.8889, an MSE of 0.0028, and an RMSE of 0.0525. While it performs better than SVR, it still does not measure up to GCN-RF and GBT.

The prediction results of the winch speed are shown in Figure 34 and the R², MSE and RMSE of GCN-RF, GBT, SVR and MLP are shown in Table 26.

Figure 34. Prediction results of the winch speed based on GCN-RF, GBT, SVR and MLP.

Table 26. R², MSE and RMSE values of GCN-RF, GBT, SVR and MLP.

Figure 34 illustrates the comparison between the actual and predicted values for the GCN-RF, GBT, SVR, and MLP in predicting the winch speed. It is evident that the prediction curve of the GBT model closely aligns with the actual curve, suggesting a high level of accuracy. The GCN-RF also shows strong consistency with the actual values, following closely behind. In contrast, the SVR and MLP display a greater gap between their prediction curves and the actual values, especially in areas with significant speed variations, indicating that these models are less effective than GBT and GCN-RF in predicting the winch speed.

Table 26 provides the R², MSE, and RMSE values for each model. The GBT achieves the highest R² value of 0.9088 among the four models, signifying the best explanatory power and fitting capability. Its MSE and RMSE values are 0.0030 and 0.0552, respectively, which are the lowest, further substantiating the GBT's high precision in winch speed prediction. The GCN-RF ranks second with an R² value of 0.8690, an MSE of 0.0040, and an RMSE of 0.0631, indicating good predictive performance. The SVR exhibits an R² value of 0.8466, an MSE of 0.0057, and an RMSE of 0.0707, indicating significantly poorer performance compared to GCN-RF and GBT, with lower prediction accuracy. The MLP records an R² value of 0.8701, an MSE of 0.0043, and an RMSE of 0.0657. While it performs better than SVR, it still does not match the effectiveness of GBT and GCN-RF.

Comments 6: Regarding the modeling with a combination of RF, please refer to: 10.1109/ICPHM51084.2021.9486665 and https://doi.org/10.1007/s42417-023-01106-0.

Response 6: Thank you for pointing this out. We agree with this comment. Based on your suggestions, I have modified the GCN-RF model diagram, as shown in Figure 3. Unlike classification problems, we are investigating a regression problem, where the executed actions are determined by the handle signals.

Figure 3. GCN-RF architecture diagram with graph data feature extraction, regression prediction.

Comments: Comments on the Quality of English Language Some spelling and grammar errors have been found, please revise it carefully.

Response 1: Thank you for pointing this out. We agree with this comment. have modified the content of the manuscript.