A Fast Operation Method for Predicting Stress in Nonlinear Boom Structures Based on RS–XGBoost–RF Model

Abstract

The expeditious and precise prediction of stress variations in nonlinear boom structures is paramount for ensuring the safe, dependable, and effective operation of pump trucks. Nonetheless, balancing prediction accuracy and efficiency by constructing a suitable machine-learning model remains a challenge in engineering practice. To this end, this paper introduces an interpretable fusion model named RS–XGBoost–RF (Random Search–Extreme Gradient Boosting Tree–Random Forest) and develops an intelligent algorithm for the stress prediction of the nonlinear boom structure of concrete pump trucks. Firstly, an information acquisition system is deployed to collect relevant data from the boom systems of ZLJ5440THBBF 56X-6RZ concrete pump trucks during its operational phase. Data pre-processing is conducted on the 2.4 million sets of acquired data. Then, a sample dataset of typical working conditions is obtained. Secondly, the RS algorithm, RF model, and XGBoost model are selected based on their complementary strengths to construct the fusion model. The model fusion condition is established with a focus on prediction efficiency. By leveraging the synergy between search and prediction mechanisms, the RS–XGBoost model is constructed for the prediction of the master hyperparameters of the RF model. This model uses the random search (RS) process to obtain the mapping between the loss function and the hyperparameters. This mapping relationship is then learned using the XGBoost model, and the hyperparameter value with the smallest loss value is predicted. Finally, the RS–XGBoost–RF model with optimized hyperparameters is employed to achieve rapid stress prediction at various detection points of the nonlinear boom structure. The findings demonstrate that, within the acceptable prediction efficiency for engineering practice, the fitting accuracy (R²) of the RS–XGBoost–RF model consistently exceeds 0.955 across all measurement points, with only a few exceptions. Concerning the stress magnitudes themselves, the mean absolute error (MAE) and root mean square error (RMSE) are maintained within the ranges of 2.22% to 3.91% and 4.79% to 7.85%, respectively. In comparison with RS–RF–RF, RS–RF–XGBoost, and RS–XGBoost–XGBoost, the proposed model exhibits the optimal prediction performance. The method delineated in this paper offers valuable insights for expeditious structural stress prediction in the realm of inherent safety within construction machinery.

1. Introduction

The concrete pump truck is a specialized mechanical appliance that employs pressure to convey concrete continuously. Its primary function is utilizing the power generated by the chassis engine to pressurize the concrete within the hopper, subsequently delivering it through the attached pipeline on the boom. The operator exercises control over the movement of the boom to precisely direct the concrete pumped out by the system to the designated pouring point [1]. The boom system, functioning as the material-conveying device of the pump truck, necessitates attributes including high strength, stiffness, stability, work adaptability, safety, and reliability [2]. Stress constitutes a pivotal parameter in assessing the structural performance of pump truck booms. Its significance is irreplaceable in the optimization design and validation of the boom system, quality assurance, and inspection, fault prognosis and prevention, and subsequent maintenance and repair endeavors. Undoubtedly, it exerts a decisive influence on guaranteeing the safe, dependable, and efficient operation of pump trucks [3]. Consequently, conducting stress analysis and application for the nonlinear boom structure of concrete pump trucks holds profound engineering significance and practical value.

Currently, in terms of stress acquisition in the boom structures of concrete pump trucks, conventional approaches encompass the empirical formula method (EFM) [4], analytical method (AM) [5], numerical simulation method (NSM) [6], and experimental measurement method (EMM) [7]. The EFM relies on empirical summarization and practical experience. The stress distribution is estimated through the establishment of empirical formulas or relationships. It is suitable for simple structures and common loading conditions, but the calculation accuracy is low. The AM is grounded in theoretical analysis and mathematical models, employing analytical equations to solve stress distribution. It is suitable for problems with relatively simple geometric shapes and stress conditions. The NSM encompasses finite element analysis (FEA), boundary element analysis (BEA), etc. It is used to discretize complex structures into finite elements or boundary elements. Computers are then used for numerical solutions to obtain structural stress distribution. This method is suitable for various complex structures and stress conditions and can consider issues such as material nonlinearity and contact. But it requires a lot of time consumption. The EMM involves practical measurements, using tools like stress sensors, grating methods, electron holography, etc., to acquire information on stress distribution. This method can provide accurate stress data. However, in practical applications, it may be affected by equipment limitations and cost factors. For example, the unique characteristics of concrete pump trucks, including the numerous working postures of booms, extensive deformation, severe vibration, and limited structural space, present challenges in stress acquisition. Then, it will cause difficulty when arranging strain gauges and susceptibility to damage. This further triggers inaccurate data, signal loss, and system failures. This underscores the manifold drawbacks and limitations inherent in traditional methodologies for acquiring the structural stresses of concrete pump truck booms.

Artificial intelligence technology is advancing, and related industries such as cloud computing, big data, and the Internet of Things (IoT) are moving forward at a fast pace. The use of machine learning regression models to solve practical problems of pump trucks has become a new research trend. Their advantages lie in the following points: (1) Complexity processing [8]: Stress prediction typically involves a multitude of complex data and variables, such as equipment usage conditions, material properties, geometric configurations, and stress-loading conditions. Machine learning can extract mapping patterns and laws from a large amount of data as a way to achieve accurate and efficient predictions. (2) Improvement of efficiency [9]: Machine learning automates the training and prediction of models expeditiously, significantly enhancing predictive efficiency while obviating the need for extensive manual labor and time-consuming trial-and-error iterations and optimizations. (3) Multivariate analysis [10]: Stress prediction typically involves intricate interactions and interdependencies among multiple variables. Machine learning excels in the effective handling of multivariate datasets, unraveling nonlinear relationships, and discerning complex patterns that may exist among these variables.

While the advantages of machine learning regression models in tackling complex engineering challenges are evident, their practical application in predicting stress within pump truck boom structures still confronts three primary challenges. (1) The challenge of balancing prediction accuracy and efficiency in engineering practice. (2) The challenge of solving the issues stemming from poor sample data quality, high-dimensional features, and pronounced nonlinear conditions, which may lead to subpar prediction outcomes. (3) Overcoming instability, inefficiency, and incompleteness are the challenges of intelligent optimization algorithms employed for hyperparameter optimization in machine learning models. To tackle these challenges, this paper introduces the concept of model fusion. From the aspects of model selection, fusion condition setting, and fusion method implementation, a fast method for predicting the stress of nonlinear boom structures based on the RS–XGBoost–RF model is proposed.

The primary contributions of this paper are as follows:

(1)

A criterion for model fusion is proposed to address the challenge of balancing prediction accuracy and efficiency when employing machine learning models for stress prediction in concrete pump truck boom structures. The criterion sets a fitting accuracy threshold r based on the user’s requirements. If the prediction accuracy of a single model achieves the threshold, only the RF model is utilized. Otherwise, model fusion is employed.

(2)

To mitigate the detrimental effects of poor-quality sample data, high feature dimensions, and nonlinear conditions on stress prediction accuracy, an intelligent prediction algorithm based on the RS–XGBoost–RF model is introduced. This algorithm combines the random search algorithm (RS), random forest (RF), and extreme gradient boosting tree (XGBoost), leveraging their complementary strengths.

(3)

Addressing the challenges associated with search instability, inefficiency, and incompleteness in traditional intelligent optimization algorithms used for hyperparameter optimization, the RS–XGBoost prediction model is proposed for the hyperparameter optimization of the RF model. This model capitalizes on the synergy between search and prediction mechanisms.

The structure of the remaining sections of this article is as follows: Section 2 provides a review of relevant literature. Section 3 outlines the methodology. Section 4 presents data sources and analysis. Section 5 focuses on stress prediction for the nonlinear boom structure of pump trucks. Section 6 discusses the results. Section 7 concludes the paper.

2. Background and Related Works

The objective of this paper is to investigate the problem of rapid stress prediction in the nonlinear boom structure of concrete pump trucks within the engineering practice by devising a novel machine-learning model. To tackle this objective, the paper commences by reviewing existing technical specifications and relevant standards for pump trucks, stress analysis, and testing methodologies, as well as machine-learning technologies. Furthermore, it analyzes and discusses this challenge in light of the current research landscape and developmental trends both domestically and internationally.

2.1. Current Technical Specifications and Related Standards for Pump Trucks

The current effective domestic and foreign technical specifications and related standards for concrete pump trucks are GB/T32542-2016 [11], GB/T 39757-2021 [12], GB/T 41495-2022 [13], QC/T 718-2013 [14], ISO 21573-2:2020 [15], BS EN 12001:2012 [16], and so on.

The aforementioned norms and standards offer detailed provisions concerning the structural components and parts of pump trucks, encompassing calculation principles, safe operational procedures, maintenance protocols, repair guidelines, scrapping specifications, and technical parameter inspection procedures. Their overarching objective is to ensure the safety and reliability of pump usage, prolong their operational lifespan, and mitigate the occurrence of failures and accidents. However, during implementation, it becomes apparent that stress emerges as a crucial and indispensable design and evaluation criterion. Specifically, the QC/T 718-2013 standard explicitly mandates stress-related strength and stability calculation and verification methods in accordance with GB/T 3811-2008 [4]. This specification delineates two calculation methodologies: the allowable stress method and the limit state method. Of these, the limit state method is more suitable for structures with large deformations under external loads. The internal forces of the structure are nonlinearly related to the external loads. The core idea is still to add engineering experience and measured data on the basis of the analytical method (AM). Empirical formulas or empirical coefficients are utilized to correct or adjust the analytical results. The stress results obtained are usually conservative.

2.2. Methods of Stress Analysis and Testing

Numerous researchers have delved into stress analysis and testing within the realm of concrete pump trucks. Ren et al. [17] employed the modal reduction method to establish a rigid–flexible coupling model of the pump truck boom system. Their study explored the intricate motion dynamics among the flexible boom, connecting rod, and hinge. They analyzed the dynamic stress variations of the flexible boom frame and validated the correctness and rationality of simulation results through experimental validation. Chen et al. [18] developed a simulation model capable of accurately capturing the dynamic stress characteristics of the boom system. They achieved this by utilizing an approximate mathematical model of concrete flow impact load to address the vibration issues arising from the concrete pumping process. Wu et al. [19] conducted simulations to investigate the extension behavior of cracks in the moving boom using ABAQUS. They provided stress distributions near the crack tip of the corner weld in the boom structure. The work laid the groundwork for fatigue life prediction of the boom structure. Huang et al. [20] established a parametric simulation model of the concrete pump truck boom structure using the APDL language. They conducted a reliability analysis of the boom structure at a specified confidence level of 95%. Then, they obtained the cumulative distribution function of boom displacement and stress. Zhao et al. [21] conducted load tests and finite element simulation analysis on the slewing leg of the pump truck separately. They compared the experimentally obtained actual buckling load and ultimate load with theoretical values derived from simulation. Results indicated the effectiveness of the finite element method in analyzing ultimate load but revealed limitations in buckling load analysis. Pan et al. [22] employed ANSYS for finite element analysis on a three-dimensional model of the boom system to ascertain stresses and strains. Additionally, they utilized ADAMS to establish a simulation model of the hinge point system between each adjacent knuckle boom of the concrete pump truck. They analyzed the change rule of the acceleration and force of the knuckle boom when the boom was in different positions. Their study provided boundary conditions and decision-making insights for determining boom structural parameters.

The aforementioned studies show that in terms of stress acquisition, numerical simulation methods (NSM) do have the ability to simulate complex structures realistically. It can acquire stress distribution of structural components under specific loads or working conditions. However, NSM entails substantial time consumption in model construction and simulation calculations. On the other hand, the experimental/test measurement method (EMM) offers more accurate stress data. Nevertheless, the practical application of EMM is hindered by constraints such as equipment limitations, costs, and environmental factors, making it challenging to conduct experiments or tests.

2.3. Related Technologies for Machine Learning

Currently, there is a relatively limited number of studies focusing on machine learning for nonlinear boom stress prediction in concrete pump trucks. There are several reasons for this. (1) The complexity and specificity of the boom structure. The boom structure is inherently complex, involving the interaction and mutual response of various factors, such as material properties, load characteristics, and usage conditions. This complexity significantly increases the difficulty of developing prediction models. (2) Difficulty in data acquisition. The stress data of the boom structure usually needs to be collected in real-time by sensors and other equipment. However, this acquisition process may be susceptible to environmental factors. This can cause data instability, noise, and even problems such as missing data. Then, it poses challenges in data acquisition. (3) The high cost of data labeling. Machine learning algorithms necessitate a large volume of labeled data for training. Labeling stress data for the boom structure demands specialized knowledge and incurs significant costs. (4) High model interpretability requirements. In the field of engineering, some machine learning algorithms, such as deep neural networks, are usually perceived as black-box models. It is difficult to explain their prediction process.

Nevertheless, the acceptance and utilization of machine learning within the engineering domain are steadily growing, driven by the continuous evolution and diversification of machine learning techniques and their expanding applications. Machine learning methods are increasingly employed to address a wide array of regression problems encountered in engineering practice. Among them, Bernar et al. [23] employed Random Forest (RF), Extreme Gradient Boosting Tree (XGBoost), Multi-Layer Perceptron (MLP), and Support Vector Machines (SVM) to predict the remaining useful life of equipment efficiently. In this way, they addressed the issue of production line downtime caused by equipment failures. Cheng et al. [24] utilized Gaussian regression as the base prediction model. Then, they used the RF model to select optimal features in residual stress sample datasets, ultimately developing a prediction method for surface residual stresses induced by machining. Mudasir et al. [25] introduced a new Digital Twin Driven Framework (DTD) and utilized the RF model to predict the load-carrying capacity of aging tower cranes. The aforementioned studies demonstrate the potential of machine learning for predicting nonlinear boom stress in concrete pump trucks. It holds significant implications. However, relying solely on primitive models often proves insufficient to meet the demands of complex engineering problems in terms of prediction capabilities.

For this reason, most of the studies started with hyperparameter optimization to improve the predictive ability of the model. Among them, Li et al. [26] proposed a novel deep neural network (DNN) model for joint failure load prediction. This model achieves adaptive prediction of failure load under the guidance of the transfer learning (TL) mechanism, significantly reducing computation time. Mahmoodzadeh et al. [27] utilized six heuristic optimization algorithms to optimize hyperparameters of SVM, including particle swarm optimization (PSO), gray wolf optimization (GWO), multivariate universes (MVO), moth optimization (MFO), sine–cosine algorithm (SCA), and spider optimization (SSO). In this way, the effective prediction of working face pressure during tunneling was realized. Mohamed et al. [28] devised an innovative intelligent prediction model grounded in the RF model to evaluate the ultimate bending capacity of circular steel pipes. They compared the effectiveness of three heuristic search algorithms (PSO, Ant Colony Optimization (ACO), and Whale Algorithm (WOA)) in optimizing hyperparameters of the RF model.

The aforementioned studies have notably elevated the predictive capacity of machine learning models through the optimization of hyperparameters. It has already achieved better results in different fields. However, it is not difficult to find that the swarm intelligence optimization algorithm is prone to problems such as unstable, inefficient, and incomplete search in the process of comparative research, especially when facing complex engineering problems. For instance, the BO algorithm [29] searches for the optimal hyperparameters by constructing a probability model of the objective function. It easily falls into the local optimization and is more dependent on the selection of its own initial hyperparameters. The GS algorithm [30] determines the optimal value of hyperparameters by enumerating all the points within the search range. It experiences diminishing computational efficiency as the search space expands. The RS algorithm [31] randomly samples hyperparameter values within the search range. It enhances computational efficiency, but it is constrained by the limited number of searches. This potentially fails to uncover the optimal combination of hyperparameters.

From the existing studies, the machine learning models mentioned provide valuable insights and guidance at a theoretical level for the rapid stress prediction of nonlinear boom structures in concrete pump trucks. These models offer reference value. While, in order to be more effectively applied in engineering practice, it is still necessary to conduct in-depth research by combining the characteristics of the boom structure in concrete pump trucks and the characteristics of the data samples. The stress prediction model that fits the actual needs (balanced prediction accuracy and prediction efficiency) needs to be developed through the research. Hence, this paper proposes an interpretable fusion model RS–XGBoost–RF. The model starts from the aspects of model selection, fusion condition setting, and fusion method implementation, and combines the synergy between search and prediction mechanisms. It is used for fast prediction of nonlinear boom structure stresses in concrete pump trucks.

3. Methodology

3.1. Random Forest (RF)

The random forest (RF) [32], introduced by Breiman in 2001, is a bagging algorithm rooted in decision trees and constitutes an ensemble of multiple decision trees. The decision tree is an algorithm that renders decisions through a tree structure, consisting of root nodes, intermediate nodes, and leaf nodes. Bagging is an ensemble learning algorithm with a parallel structure that generates multiple sets of distinct training samples through bootstrapped sampling. These sets are employed to construct various weak regressor models. Through specific combination strategies, a robust regressor model is then formed. The culmination of decision trees and the bagging algorithms results in the RF model, distinguished by its expeditious training speed, heightened prediction accuracy, robust generalization capability, and not easily overfitting and underfitting. Moreover, it adeptly addresses the nonlinear mapping challenges associated with multiple parameters. The specific construction process of this model is as follows:

(1) Through the application of bootstrap sampling, there is put-back sampling from the sample dataset $T=\{(M_i,N_i),i=1,2,\dots ,n\}$ to obtain k sub-datasets ${\theta }_j(j=1,2,\dots ,k)$. Each of these sub-datasets serves as the foundational root node for a regression tree. After each sampling iteration, the residual sample data are assigned to the out-of-bag (OOB) subset. This OOB data assumes the role of a designated test set. (2) In conjunction with the CART algorithm, the criterion of mean square error minimization is employed to discern the optimal cutoff point for the optimal features. The recursive bifurcation of the regression tree unfolds from top to bottom with the root node as the top. (3) Iterate through step (2) for every regression tree, thereby assembling a ‘forest’. The output of the RF model is determined by averaging the outputs of all the regression trees, as depicted in Equation (1):

$$RF(x)={\displaystyle \sum _{j=1}^{k}F({\theta }_j)}/k$$

(1)

where $RF(x)$ is the output of the RF model, and $F({\theta }_j)$ is the output of a single regression tree.

Various hyperparameters influence distinct facets of the model performance. The RF model is a machine learning model based on the combination of the decision tree and the Bagging algorithm. It encompasses two categories of hyperparameters, namely Bagging hyperparameters and decision tree hyperparameters [33]. See

Table 1. Hyperparameters of the RF model.

Type	Hyperparameter Name	Notation
Hyperparameters of Bagging	n_estimators	number of decision trees
	OBB_score	out-of-bag score
	criterion	characteristic evaluation criteria
Hyperparameters of Decision tree	max_feature	maximum number of features
	max_depth	maximum depth
	min_samples_leaf	minimum number of samples for leaf nodes
	min_samples_split	minimum number of samples required for leaf node splitting
	min_weight_fraction_leaf	minimum sample weights for fraction of leaf nodes
	max_leaf_nodes	maximum number of leaf nodes
	min_impurity_split	minimum impurity split

:

3.2. Extreme Gradient Boosting Tree (XGBoost)

The extreme gradient boosting tree (XGBoost) [34] is a comprehensive learning algorithm rooted in the boosting framework, introduced by Tianqi in 2016. The core lies in building multiple base learners and continuously learning the error value of the previous base learner. The gradual reduction in the deviation in the prediction value is a pivotal aspect. The entire model prediction value is derived through the cumulative accumulation of error values from the base learners. The XGBoost model construction principle is as follows:

Within the sample dataset $(x_i,y_i)$, each set of sample input values $x_i$ comprises t features, corresponding to an output value $y_i$. Subsequently, the predicted value $y_i$ for the ith set of input value samples $x_i$ can be elucidated as follows:

$${\widehat{y}}_i={\displaystyle \sum _{k=1}^m{F}_k(x_i)}$$

(2)

where m is the total number of decision trees. $F_k(x_i)$ is the prediction result of the kth decision tree for the ith set of input value samples $x_i$.

In the XGBoost model, the objective function comprises two components. The first is the model error, denoting the disparity between the predicted value and the true value. The second is the structural error of the model, encompassing the canonical term. The complexity of the model is constrained by regulating the number of leaf nodes and node values through hyperparameters. The objective function can be formulated as follows:

$$\{\begin{array}{l}O={\displaystyle \sum _{i=1}^{n}E(y_i,{\widehat{y}}_i)}+{\displaystyle \sum _{k=1}^K\Omega (F_k)}\\ \Omega (F_k)=\gamma T+2^{-1}\lambda {\displaystyle \sum _{t=1}^T{w}_t^2}\end{array}$$

(3)

where E represents the loss function of the model. It is employed to respond to the fit of the predicted value with the true value. ${\displaystyle \sum _{k=1}^K\Omega (F_k)}$ represents the regularity term. It is introduced to diminish the model complexity and avert overfitting. $\gamma$ serves as the penalty coefficient for the number of leaf nodes. T represents the number of leaf nodes in the kth decision tree. $\lambda$ represents the penalty coefficient for the regular term. $w_t$ represents the value of the tth leaf node in the decision tree.

Assuming that the predicted value obtained after the (m − 1)st iteration of the ith set of input value $x_i$ is ${{\widehat{y}}_i}^{(m-1)}$. The model is incrementally trained using the gradient boosting algorithm, introducing one decision tree at a time. Consequently, the objective function can be reformulated as follows:

$$O^{(m)}={\displaystyle \sum _{i=1}^{n}E(y_i,{{\widehat{y}}_i}^{(m-1)}+F_m(x_i))}+\Omega (F_m)+C$$

(4)

where C is the constant term.

Equation (4) can be derived through a Taylor expansion and subsequent elimination of the constant term:

$$\{\begin{array}{l}{O}^{(m)}\approx {\displaystyle \sum _{i=1}^n\left[p_i{F}_i(x_i)+2^{-1}{q}_i{F}_i{(x_i)}^2\right]}+\Omega (F_m)\\ p_i=\frac{\partial P(y_i,{{\widehat{y}}_i}^{(m-1)})}{\partial {{\widehat{y}}_i}^{(m-1)}}\\ q_i=\frac{{\partial }^{2}P(y_i,{{\widehat{y}}_i}^{(m-1)})}{\partial {({{\widehat{y}}_i}^{(m-1)})}^2}\end{array}$$

(5)

where $p_i$ and $q_i$ are the first- and second-order derivatives of the loss function P, respectively.

To ascertain the minimum value of the loss function, Equation (5) may be reformulated as a quadratic expression concerning variable $w_i$ through the amalgamation of Equation (3):

$$O_{\min }^{(m)}\approx {\displaystyle \sum _{t=1}^T\left[P_i{w}_t+2^{-1}(Q_i+\lambda ){w_t}^2\right]}+\gamma T$$

(6)

where $P_i=\sum p_i$, $Q_i=\sum q_i$.

Solving a quadratic function on $w_t$, the result shows that, for $w_t=-P_i/(Q_i+\lambda )$, the minimum value $O_{\min }$ of $O^{(m)}$ can be expressed as follows:

$$O_{\min }^m=-2^{-1}{\displaystyle \sum _{t=1}^T{P}_i^2/(Q_i+\lambda )}+\gamma T$$

(7)

3.3. Random Search (RS)

Hyperparameters are one of the key factors determining model performance. For a given sample dataset, a set of hyperparameters can invariably be explored to facilitate the comprehensive optimization of the model evaluation indices. Frequently utilized techniques for hyperparameter adjustment encompass the Bayesian optimization algorithm (BO) [35], grid search algorithm (GS) [36], and random search algorithm (RS) [37,38]. In contrast to alternative algorithms, the RS algorithm stands out as a stochastic parameter optimization technique. It randomly samples and combines parameters from diverse search spaces. This will form interleaved non-repetitive combination points. Parameter configurations are subsequently adjusted according to the evaluation results of the combination points. Thus, it finds more beneficial parameter combinations for the model. This renders the RS algorithm computationally efficient.

3.4. Model Evaluations

To assess the predictive performance of the model, commonly employed metrics such as mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²) are chosen as the model evaluation criteria [39]. Specifically, MAE quantifies the average absolute disparity between the predicted values and true values. RMSE signifies the standard deviation of the differences between the predicted and true values. It has a heightened sensitivity to values exhibiting substantial localized variations. Proximity to 0 in both MAE and RMSE implies greater accuracy in the model. R² indicates the goodness of fit between the predicted and true values, and a value of R² closer to 1 denotes the superior predictive performance of the model. The aforementioned evaluation metrics can be mathematically expressed as follows:

$$\{\begin{array}{l}\mathrm{MAE}=n^{-1}{\displaystyle \sum _{i=1}^n|{\widehat{y}}_i-y_i|}\\ \mathrm{RMSE}=\sqrt{n^{-1}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}}\\ {\mathrm{R}}^2=1-{\sum }_{i=1}^n{({\widehat{y}}_i-y_i)}^2/{\sum }_{i=1}^n{({\overline{y}}_i-y_i)}^2\end{array}$$

(8)

where $y_i$ is the true value of the sample data, ${\widehat{y}}_i$ is the predicted value of the regression model, ${\overline{y}}_i$ is the average of the true values of the sample data, and $n$ is the number of samples.

4. Data Sources and Analyses

The data originated from the Concrete Machinery Engineering Technology Research Center affiliated with Zoomlion Heavy Industry Co. The center employed the concrete pump truck information acquisition system to collect pertinent data during the operational phases of the boom of the ZLJ5440THBBF 56X-6RZ concrete pump truck (Changsha, China). For the concrete pump truck, the maximum theoretical conveying capacity stands at 180 m³/h, and the rated working pressure is 42 MPa. The maximum concrete outlet pressure is recorded at 11.3 MPa. The pumping frequency is 0.4333 Hz. The diameter and stroke of the concrete cylinder are Φ260 mm and 2100 mm. The structural configuration of its boom is identified as 56X-6RZ. The maximum height × radius × depth of the boom is 56 × 51 × 40.2 mm. The corresponding minimum spread height of the boom is 15.8 m. The inner and outer diameters of conveying pipes are Φ123 × 133 mm. The rotation angle is ±270°. The lengths of the 1st section boom to the 6th section boom are {11.4 m, 9.02 m, 8.75 m, 11.25 m, 7.08 m, 4.5 m}. The maximum tilt angle of the 1st section boom to the 6th section boom are specified as {90°, 90°, 180°, 180°, 240°, 195°, 120°}.

4.1. Data Acquisition

The concrete pump truck information acquisition system primarily captures the following data: pressure in the boom chamber, overall rotation angle of the boom, tilt angle of each boom, pumping pressure, pumping status, and strain values of detection points at each section boom. Building upon this foundation, the stress rapid prediction method is employed to enable the timely detection and assessment of potential issues in the boom structure, even in cases where strain gauges are compromised. It not only mitigates the risk of boom failure or malfunction but also contributes to enhancing the safety, reliability, and durability of the equipment. Furthermore, it facilitates preemptive maintenance, thereby reducing the cost associated with post-failure repairs.

As shown in Figure 1, the acquisition system comprises the equipment layer, data acquisition layer, data conversion and processing layer, data transmission and storage layer, and data application layer. The equipment layer encompasses six integral components of the concrete pump truck: chassis, boom system, turret, pumping system, hydraulic system, and electrical system. Within the data acquisition layer, rotation angle sensors, tilt angle sensors, pressure sensors, and strain modules play crucial roles. Rotation angle sensors, affixed to the turret, facilitate the measurement of the boom rotation angle. Meanwhile, tilt angle sensors are placed on the boom surface. They maintain parallelism with the boom surface for precise rotation angle measurements. The pressure sensor is installed inside the oil cylinder to measure the pressure during hydraulic oil delivery. Strain modules incorporate strain gauges, strain gauge guards, and strain information converters. Strain gauges are used to obtain the strain information at detection points on the boom structure. The strain gauge guard protects against collisions and environmental factors. Strain information converters perform the conversion of analog signals into digital format. When operational conditions change, the sensors in the acquisition layer promptly detect these alterations. Subsequently, employing the Modus TCP/RTU protocol in the data conversion layer and leveraging the industrial computer in the data processing layer, the prevailing angle, pressure, and strain data are stored locally. Through the GPS/GPRS device in the data transmission layer, the transformed data are transmitted and stored in the cloud using WiFi5/6/6e transmission. The data utilization layer (i.e., terminal) enables users to access, download, analyze, calculate, and mine the data through the cloud interface.

The determination of hazardous sections needs to be based on the structural characteristics and load characteristics of the pump truck boom system. In the plane of boom luffing, it can be simplified as an outwardly extended, simply supported beam. In the plane of boom rotation, it can be simplified as a cantilever beam. From the structural influence line analysis method, it can be seen that the bending moment and shear force near the root of each section of the boom are larger. Combined with the results of the investigation of the actual damage location of the boom structure, the distributions of the hazardous sections and detection points of the boom structure are given [40], which are shown in Section 1-1~Section 5-5 in Figure 1. Considering the practicability of the strain gauge arrangement, the detection points are set in the region near the welds between the main web and the upper and lower flange plates of the cross-section, named M1C1~M4C3.

When collecting information, the sampling frequency for pumping status and strain values of detection points is set at 20 Hz, while the sampling frequency for the remaining characteristic parameters is established at 10 Hz. The data are counted in cycles for every seven normal working days of the equipment. This section gives data for 2.4 million information sets in one of these cycles, as shown in

Table 2. Statistical table of concrete pump truck information acquisition.

Date	Running Dataset Number	Running Dataset Characteristics	Data Volume/Group	Strain Dataset Number	Strain Dataset Characteristics	Data Volume/Group
20211229	Pump-Data [2021122911.db3]	Pressure in the rod chamber of the boom Pressure in the non-rod chamber of the boom Pumping pressure Rotation angle Tilt angles of boom 1, boom 2, boom 3, boom 4, boom 5, and boom 6 Data time	12,601	Strain-Data [2021122911.db3]	Detection points M1C1, M1C2, M1C3, M1C4, M2C1, M2C2, M2C3, M2C4, M4C1, and M4C2 Pumping state Data time	24,694
	Pump-Data [2021122912.db3]		62,635	Strain-Data_ [2021122912.db3]		125,766
20211231	Pump-Data [2021123121.db3]		99,516	Strain-Data_ [2021123121.db3]		199,566
20220101	Pump-Data [2022010100.db3]		36,233	Strain-Data_ [2022010100.db3]		72,905
20220105	Pump-Data [2022010500.db3]		35,692	Strain-Data_ [2022010500.db3]		71,901
20220106	Pump-Data [2022010613.db3]		159,937	Strain-Data_ [2022010613.db3]		320,501
	Pump-Data [2022010618.db3]		40,783	Strain-Data_ [2022010618.db3]		81,762
20220107	Pump-Data [2022010708.db3]		32,622	Strain-Data_ [2022010708.db3]		64,670
	Pump-Data [2022010710.db3]		151,635	Strain-Data_ [2022010710.db3]		304,882
20220118	Pump-Data [2022011803.db3]		36,764	Strain-Data_ [2022011803.db3]		74,092
	Pump-Data [2022011804.db3]		35,248	Strain-Data_ [2022011804.db3]		71,055
	Pump-Data [2022011805.db3]		35,844	Strain-Data_ [2022011805.db3]		71,893
	Pump-Data [2022011815.db3]		144,335	Strain-Data_ [2022011815.db3]		290,221

.

4.2. Data Preprocessing

(1)

Data aggregation

Data aggregation enhances the completeness and uniformity of the collected sample data by eliminating negative ‘impurities’. Nevertheless, due to the potentially disparate sampling frequencies among various types of sensors, different data types may exist independently. Lack of alignment between features can hinder the establishment of a mapping relationship. The time alignment method [41] emerges as the simplest, most direct, and most effective solution. Hence, the least squares method [42] is employed to match and align data acquired by non-homologous sensors based on the criterion of time unification. The specific principle is outlined as follows:

Assume that the sampling periods of the two non-homologous sensors $\alpha$ and $\beta$ are t and T, respectively, and they satisfy the integer ratio relation $n=t/T$. If the moment of the previous common measurement of the two sensors is denoted as p, the moment of the next measurement cycle can be expressed as follows:

$$p^{\prime }=p+t=p+nT$$

(9)

It is evident that between the two measurement moments, the sensor $\beta$ undergoes n measurement processes, yielding n data points (refer to Equation (10)). Here, $q_n$ shares the same measurement moment as sensor $\alpha$:

$$Q_n={\left[q_1,q_2,\dots ,q_n\right]}^{\mathrm{T}}$$

(10)

$U={[q\dot{q}]}^{\mathrm{T}}$ is used to represent the vectors composed of elements and their derivatives in the dataset $Q_n$. Then, the measurement data of sensor $\beta$ can be expressed as:

$$q_i=\widehat{q}+(i-n)\dot{T}\widehat{q}+v_i\hspace{1em}i=1,2,\dots ,n$$

(11)

where $v_i$ is the noise value during the measurement.

Converting Equation (11) to vector format gives the following:

$$Q_n=W_{n}U+V_n=\left[\begin{array}{cc}\begin{array}{l}1\\ 1\\ \dots \\ 1\end{array}& \begin{array}{c}(1-n)T\\ (2-n)T\\ \dots \\ (n-n)T\end{array}\end{array}\right]U+\left[\begin{array}{l}{v}_1\\ v_2\\ \dots \\ v_n\end{array}\right]$$

(12)

Substitute $U={[q\dot{q}]}^{\mathrm{T}}$ into Equation (12) and solve it using the least squares method. The resulting calibrated measurement values for sensor $\beta$ are then as follows:

$$\{\begin{array}{l}\widehat{U}={\left[\begin{array}{cc}\widehat{q}& \widehat{\dot{q}}\end{array}\right]}^{\mathrm{T}}={(W_n^{\mathrm{T}}{W}_n)}^{-1}{W}_n^{\mathrm{T}}{Q}_n\\ q(p^{\prime })=c_1{\displaystyle \sum _{i=1}^n{q}_i}+c_2{\displaystyle \sum _{i=1}^n{q}_i}\end{array}$$

(13)

(2)

Data Cleaning [43]

The integrity, independence, and accuracy of the sample data significantly impact the efficacy of stress prediction algorithms. Data cleaning serves as a standard method for validating sample data. The method mainly includes supplementation of missing data, removal of outliers, and elimination of redundant data. Its core objective lies in the conversion of ‘dirty data’ within the sample dataset into ‘clean data’.

In instances where certain intervals within the sample data exhibit sparsity, the accuracy of data fitting is often compromised due to the absence of key points. To effectively address this issue, the Neighborhood replacement method or Lagrange interpolation is commonly employed to replace and estimate the missing sample data. The former method utilizes the average of surrounding data to replace missing values. While it takes into account data continuity, it exhibits a strong dependency on neighboring values, and it potentially overlooks other attributes of the data. The latter method employs known data points to construct an estimated interpolation across the entire domain in an appropriate basis function. This method is not influenced by local data points. It has robust stability and incremental properties. In this study, the Lagrange interpolation method [44] is employed to fill in the missing features within the stress dataset of nonlinear boom structure detection points in the concrete pump truck. The underlying principle is outlined as follows:

The n − 1 polynomials are employed to approximate the fitting of n known data points in the plane:

$$y=a_0+a_{1}x+a_2{x}^2+\dots +a_{n-1}{x}^{n-1}$$

(14)

where $a_0,a_1,\dots ,a_{n-1}$ are constant terms.

Solve for the constant term by substituting the n known data points $\left\{\left(x_i,y_i\right)|i=1,2,\dots ,n\right\}$ into Equation (15):

$$\{\begin{array}{c}{y}_1=a_0+a_1{x}_1+a_2{x_1}^2+\dots +a_{n-1}{x_1}^{n-1}\hfill \\ y_2=a_0+a_1{x}_2+a_2{x_2}^2+\dots +a_{n-1}{x_2}^{n-1}\hfill \\ \hfill \dots \hfill \\ y_n=a_0+a_1{x}_n+a_2{x_n}^2+\cdots +a_{n-1}{x_n}^{n-1}\hfill \end{array}$$

(15)

The Lagrange interpolation polynomial is obtained by solving the linear algebra determinant as follows:

$$L(x)={\displaystyle {\sum }_{i=0}^n{y}_i{\displaystyle {\prod }_{j=0,j\ne i}^n\left[\left(x-x_j\right)/\left(x_i-x_j\right)\right]}}$$

(16)

In instances where certain intervals within the sample data exhibit excessively dense, redundancy in both data and features may arise. In such cases, the following issues may occur: Firstly, duplicated data may contribute minimally to model fitting while impacting computational efficiency. Secondly, redundant features can elevate the complexity of the model. To address these situations, it is necessary to eliminate duplicate data and redundant features. Duplicate data can be directly removed. For redundant features, the Pearson correlation coefficient [45] is introduced to measure the correlation between features. The larger the absolute value of the correlation coefficient, the stronger the correlation. Two features with absolute values of correlation coefficient between 0.8 and 1.0 are judged as strongly correlated features, and one of them is selected as the redundant feature for deletion. The formula for the Pearson correlation coefficient is as follows:

$$r=\frac{\mathrm{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}=\frac{{\displaystyle \sum _{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(X_i-\overline{X})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{(Y_i-\overline{Y})}^2}}}$$

(17)

where X and Y are the two features to be measured. $\mathrm{cov}(X,Y)$ is the covariance of X and Y. ${\sigma }_X$ and ${\sigma }_Y$ are the standard deviations of X and Y. $X_i$ and $Y_i$ are the ith sample data of features X and Y, respectively. $n$ is the total number of samples. $\overline{X}$ and $\overline{Y}$ are the sample means corresponding to features X and Y, respectively.

In the presence of outliers in the sample data, the representativeness and credibility of the data may be compromised. Outliers include human error, machine malfunctions, irregularities, and so on. They not only diminish the reliability of the data but also pose challenges to data analysis, ultimately affecting model performance. Irregularities in the testing process or instrument malfunctions often lead to the occurrence of outliers. When dealing with outliers, they can be treated as missing values. Equivalent replacement of outliers is performed by the new value replacement method in missing value processing. Several commonly employed methods for identifying outliers include the PauTa Criterion, Grubbs criterion, and Dixon criterion. The selection of a method is often contingent on the volume of sample data available. The nonlinear boom system of concrete pump trucks has many working postures, and there are many factors influencing its structural stresses. A substantial amount of sample data is imperative for accurate predictions. Consequently, the PauTa criterion (also known as the 3$\sigma$ criterion) [46] is adopted for outlier determination. This criterion recognizes the values of sample individual residuals that deviate from the 3$\sigma$ region as outliers. The specific formula is as follows:

$$\left|x_d-\overline{x}\right|\ge 3\sigma$$

(18)

where $x_d$ is the suspected value of the outlier, and $\sigma$ is the standard deviation of the sample data.

(3)

Data after cleaning and aggregation

Raw data often has a series of problems affecting data usability, such as missing values and outliers. The quality of the data will directly affect the performance and computational efficiency of the model. Take some of the data with problems in Table 2 as an example to explain the preprocessing process. Among them, the data related to functional parameters are shown in

Table 3. Some of the functional parameters of the concrete pump truck.

Acquisition Time of Functional Parameters	Pressure in Chamber of Boom/kN		Pumping Pressure/kN	Rotation Angle/°	Tilt Angle of Boom/°
	Rod	Non-Rod			1	2	3	4	5	6
20191229, 12:20:33:24	28.89	12.24	3.7	−145	42	32	−11	−21	−83	−82
20191229, 12:20:33:34	28.89	12.23	3.7	−145	42	32	−11	−21	−83	−82
20191229, 12:20:33:44	28.89	12.23	3.7	−145	NULL	NULL	−11	−21	−83	−82
20191229, 12:20:33:54	28.89	12.22	3.7	−145	NULL	NULL	−11	−21	−83	−82
…
20191229, 13:54:53:39	25.82	2.08	14.11	−142	35	−33	32	−28	−78	−81
20191229, 13:54:53:14	25.79	2.09	14.55	−142	35	−33	31	−28	−78	−81
20191229, 13:54:53:24	25.78	2.08	14.51	−142	35	−33	31	−28	−79	−81
20191229, 13:54:53:44	25.75	2.08	14.44	−143	35	−33	31	−28	−79	−82
…
20191231, 22:56:11:75	0.85	6.62	0	−70	−2	−179	0	−180	−3	−3
20191231, 22:56:11:85	0.85	6.63	0	−70	−2	−179	0	−180	−3	−3
20191231, 22:56:11:95	0.84	6.63	0	−70	−2	−179	0	−180	−3	−3
…
20200118, 03:00:11:13	18.13	3.78	5.03	−204	72	1	−25	−32	−123	−123
20200118, 03:00:11:18	18.12	3.78	5.01	−204	72	4	−25	−32	−123	−123
20200118, 03:00:11:23	18.12	3.79	5	−204	72	6	−25	−32	−123	−123
…
20200-06, 19:11:06:18	32.13	18.00	4.65	−250	38	36	−3	−19	−47	−49
20200106, 19:11:06:23	32.12	17.97	4.62	−250	38	36	−3	−19	−47	−49
20200106, 19:11:06:28	32.11	17.95	4.60	−250	38	36	−3	−19	−47	−49

, and the data related to performance indicators are shown in

Table 4. Some of the performance indicators of the concrete pump truck.

Acquisition Time of Performance Indicators	State	Strain Value/με
		M1C1	M1C2	M1C3	M1C4	M2C1	M2C2	M2C3	M2C4	M4C1	M4C3
20191229, 12:20:33:29	1	3484	−562.3	518.7	−773.0	292.1	−406.3	−452.7	335.8	52	−2.2
20191229, 12:20:33:34	1	3484	−562.3	NULL	−773.0	292.1	−406.4	NULL	335.8	52	−2.2
20191229, 12:20:33:39	1	3484	−562.3	518.6	−773.0	292.1	−406.4	−452.8	335.8	52	−2.1
20191229, 12:20:33:44	1	3484	−562.4	518.6	−773.1	292.1	−406.3	−452.7	335.8	52	−2.2
…
20191229, 13:54:53:41	1	−120.6	−672.6	662.3	−724.4	405.2	−556.0	−593.4	404.8	125.1	480.3
20191229, 13:54:53:14	1	−121.1	−672.4	661.3	−724.1	404.0	−555.8	NULL	404.5	126.8	482.4
20191229, 13:54:53:24	1	−120.4	−671.9	662.7	−722.8	408.0	−557.1	NULL	404.2	129.8	476.9
20191229, 13:54:53:44	1	−120.1	−672.1	661.9	−723.8	407.5	−554.7	NULL	403.9	132.4	469.3
…
20191231, 22:56:11:76	0	3102	−0.9	−303.8	96.9	121.9	−134.9	190.9	−120.1	−4.8	38.2
20191231, 22:56:11:86	0	3102	−0.9	−303.8	96.9	121.9	−135.0	190.9	−120.1	−4.9	38.1
20191231, 22:56:11:96	0	3102	−0.9	−303.9	96.9	121.9	−135.1	190.8	−120.1	−4.9	38.1
…
20200118, 03:00:11:15	1	−331.6	−364.3	361.4	−428.8	233.4	−250.4	−369.6	359.8	−358.6	545.7
20200118, 03:00:11:25	1	−331.9	−364.1	361.2	−428.8	233.3	−251.8	−369.7	359.8	−357.1	545.2
20200118, 03:00:11:35	1	−331.2	−364.5	361.6	−429.7	233.5	−249.8	−368.3	360.1	−358.4	544.2
…
20200106, 19:11:06:27	1	−179.0	−555.2	491.6	−601.5	253.6	−300.7	−409.4	350.3	−162.0	484.1
20200106, 19:11:06:13	1	−179.0	−555.1	491.6	−601.4	253.6	−300.8	−409.4	350.3	−162.0	484.1
20200106, 19:11:06:23	1	−178.9	−555.1	491.6	−601.5	253.6	−300.8	−409.4	350.3	−162.0	484.1

. After data aggregation and data cleaning, the preprocessed data are obtained as shown in Table 5.

As shown in Table 3 and Table 4, there are several problems with the raw data obtained by utilizing the concrete pump truck information acquisition system: (1) The acquisition time of the functional parameters is inconsistent with that of the performance indicators. It is necessary to time-align the raw data. (2) There are missing values in the data, such as “NULL” marked in the table. (3) The sampling frequency is too fast, resulting in roughly the same data in the adjacent time, i.e., data redundancy. At the same time, some features have a strong correlation, such as pumping pressure and state, which can be regarded as redundant features. (4) Some data are abnormal, such as ‘20191231’ in Table 3, where the tilt angle of boom 1 is not within the design range. In view of the above problems, this paper gives the specific result as follows:

Table 5. Some of the preprocessed data.

Acquisition Time		20191229 12:20:33:32	20191229 13:54:53:44	20191231 22:56:11:85	20200118 03:00:11:16	20200118 03:00:11:43	20200106 19:11:06:23
Pressure in chamber of boom/kN	Rod	28.89	25.75	0.85	18.12	18.12	32.12
	Non-rod	12.23	2.08	6.62	3.78	3.79	17.97
Pumping pressure/kN		3.7	14.44	0	5.01	4.99	4.62
Rotation angle/°		−145	−143	−70	−204	−204	−250
The tiltangle of boom/°	1	42	35	2	72	72	38
	2	32	−33	−175	4	6	36
	3	−11	31	4	−25	−25	−3
	4	−21	−28	−176	−32	−32	−19
	5	−83	−79	1	−123	−123	−47
	6	−82	−81	1	−123	−123	−49
Strain value/με	M1C1	3484	−120.1	3101.8	−331.6	−331.9	−178.9
	M1C2	−562.3	−672.1	−0.9	−364.3	−364.1	−555.1
	M1C3	518.6	661.9	−303.8	361.4	361.2	491.6
	M1C4	−773	−723.8	96.9	−428.8	−428.8	−601.5
	M2C1	292.1	407.5	121.9	233.4	233.3	253.6
	M2C2	−406.4	−554.7	135.0	−250.4	−251.8	−300.8
	M2C3	−452.8	−593.1	190.9	−369.6	−369.7	−409.4
	M2C4	335.8	403.9	−120.1	359.8	359.8	350.3
	M4C1	52.0	132.4	−4.9	−358.6	−357.1	−162
	M4C3	−2.2	469.3	38.1	545.7	545.5	484.1

Table 5. Some of the preprocessed data.

Acquisition Time		20191229 12:20:33:32	20191229 13:54:53:44	20191231 22:56:11:85	20200118 03:00:11:16	20200118 03:00:11:43	20200106 19:11:06:23
Pressure in chamber of boom/kN	Rod	28.89	25.75	0.85	18.12	18.12	32.12
	Non-rod	12.23	2.08	6.62	3.78	3.79	17.97
Pumping pressure/kN		3.7	14.44	0	5.01	4.99	4.62
Rotation angle/°		−145	−143	−70	−204	−204	−250
The tiltangle of boom/°	1	42	35	2	72	72	38
	2	32	−33	−175	4	6	36
	3	−11	31	4	−25	−25	−3
	4	−21	−28	−176	−32	−32	−19
	5	−83	−79	1	−123	−123	−47
	6	−82	−81	1	−123	−123	−49
Strain value/με	M1C1	3484	−120.1	3101.8	−331.6	−331.9	−178.9
	M1C2	−562.3	−672.1	−0.9	−364.3	−364.1	−555.1
	M1C3	518.6	661.9	−303.8	361.4	361.2	491.6
	M1C4	−773	−723.8	96.9	−428.8	−428.8	−601.5
	M2C1	292.1	407.5	121.9	233.4	233.3	253.6
	M2C2	−406.4	−554.7	135.0	−250.4	−251.8	−300.8
	M2C3	−452.8	−593.1	190.9	−369.6	−369.7	−409.4
	M2C4	335.8	403.9	−120.1	359.8	359.8	350.3
	M4C1	52.0	132.4	−4.9	−358.6	−357.1	−162
	M4C3	−2.2	469.3	38.1	545.7	545.5	484.1

On this basis, the 2.4 million sets of data from 20211229 to 20220118 in the statistical table of concrete pump truck information acquisition (refer to Table 2) are preprocessed by the aforementioned method. After data aggregation, eight rounds of cleaning are performed on the data. Initial training and testing of the model are carried out using different sample sizes (506, 605, 704, 803, 902, 1001, etc.) and finally, 704 data sets are selected for the study. These typical working condition data, covering a comprehensive range of scenarios, are utilized as the sample dataset. Specific details are illustrated in Figure 2.

(4)

Data conversion

In line with the basic regulation for the verification of the bearing capacity of metal structures in mechanical equipment, the stress calculation is limited to the elastic range of steel materials [41]. Consequently, the strain data can be converted into the stress data for prediction using Equation (19):

$$\sigma =E\epsilon /{10}^6$$

(19)

where E is the elasticity modulus of the material, E = 2.06 × 10⁵ MPa. $\epsilon$ is the microstrain, in με. $\sigma$ is the stress value, in MPa.

As the input variables of the model, each feature possesses distinct dimensions and units. To address the comparability between the indicators and improve the performance of the data in terms of weights, it is imperative to standardize the 704 sets of typical working condition data (as depicted in Figure 2). The results will be more accurate in this way. Therefore, Z-Score standardization [47] is employed, ensuring that the data conforms to a standard normal distribution. The transformation function is expressed as follows:

$$x_{new}=(x-u)/\sigma$$

(20)

where $x$ is the original sample data. $u$ and $\sigma$ are the mean and standard deviation of the sample data.

5. Stress Prediction for Nonlinear Boom Structures of Pump Trucks

The rapid stress prediction method for the nonlinear boom structure of concrete pump trucks aims to address practical engineering problems. It is imperative to not only meet precision requirements in predictions but also enhance prediction efficiency. Currently, a single machine learning regression model exhibits notable efficiency advantages. However, the accuracy of prediction results may not consistently meet the desired goals [48]. Existing studies often utilize optimization algorithms to attain optimal hyperparameters of the regression model [49,50,51]. Although this way can enhance prediction accuracy, it comes with a larger computational load and higher time cost. Furthermore, the optimization process is prone to local optima without finding the global optimal solution due to inherent algorithm characteristics [52]. Taking these considerations into account and utilizing the search–prediction synergy mechanism, this paper constructs the RS–XGBoost model for the hyperparameter prediction of the RF model. The model is constructed considering model selection, fusion condition setting, and fusion method implementation. Consequently, a fusion model (RS–XGBoost–RF) for structural stress prediction is formulated. The specific implementation process for predicting the stress of the nonlinear boom structure in concrete pump trucks is outlined below.

5.1. RS–XGBoost–RF Model

For the construction of fusion models, one needs to consider the fusion conditions, that is, whether fusion is needed. The second aspect is to consider the selection of models, which means fully utilizing the advantages of each model and selecting appropriate models from the perspective of complementary advantages. The third thing to consider is how to integrate.

(1)

Fusion condition

In establishing fusion conditions, both prediction accuracy and efficiency are carefully considered in light of engineering realities. Therefore, the fusion condition is set at the threshold r = R² = 0.95 (in consultation with Zoomlion designers). If the prediction accuracy of a single model surpasses this threshold, only the RF model is utilized. However, if the accuracy falls short of this benchmark, model fusion is implemented.

(2)

Model selection

The characteristic dimension of the stress sample data at the detection points of the boom structure is high and the nonlinear situation is evident. Therefore, the RF model is chosen as the fundamental model for stress prediction. The RF model exhibits exceptional performance in handling nonlinear high-dimensional data [53]. In the process of optimizing the hyperparameters of the RF model, emphasis is placed on achieving a balance between model fit and running speed. Consequently, the two master hyperparameters (n_estimators and max_features [54]) that have a more significant influence on the model are selected for optimization. The remaining hyperparameters retain common or default values. The configurations for each hyperparameter are detailed in

Table 6. Hyperparameter configurations.

Hyperparameter	Settings	Hyperparameter	Settings	Hyperparameter	Settings
n_estimators	To be optimized	max_depth	None	min_weight_ fraction_leaf	0
OBB_score	True	min_samples_leaf	1	max_leaf_nodes	None
criterion	MSE	min_samples_split	2	min_impurity_split	None
max_features	To be optimized

.

The RS algorithm involves the random extraction and combination of hyperparameter values within the search range. It enhances computational efficiency [55]. Consequently, the RS algorithm is chosen as the search algorithm for hyperparameter optimization of the RF model. However, due to the limitation on the number of searches, the RS algorithm does not always identify the optimal hyperparameter solution, and it is prone to unstable optimization results and high time costs. It can be seen that the fundamental concept of a single search is to identify a set of hyperparameters where the objective function (e.g., Gini index or mean square error) attains its optimal value within the range of design variable values (hyperparameters). Due to constraints such as the number of searches or computational accuracy, the RS algorithm often yields a local optimal solution rather than the global optimal solution. To address these limitations and achieve the cooperation of search and prediction, the XGBoost model with strong scalability is selected as the prediction algorithm for hyperparameter optimization, building upon the RS algorithm. This method overcomes issues of instability, inefficiency, and incomplete search.

(3)

Model fusion methods

The RF model serves as the fundamental model for predicting the stress of the nonlinear boom structure. The RS algorithm is employed to search for the two master hyperparameters of the RF model. The loss value MSE for each set of hyperparameters is obtained during the searching process. The MSE is treated as the independent variable, while the hyperparameters constitute the dependent variables. Then, the hyperparameter sample data are formed. The XGBoost model is then employed to train this hyperparameter sample data. Upon completion of training, the model is input with MSE = 0 to predict the hyperparameter value at this point. This method, leveraging the cooperation of search and prediction, achieves hyperparameter optimization for the RF model. The RS–XGBoost–RF model for structural stress prediction is ultimately established.

5.2. Implementation Process of Stress Prediction Based on RS–XGBoost–RF Model

To facilitate application, the implementation process of the RS–XGBoost–RF model for the stress prediction of the nonlinear boom structures is detailed. The specific process is illustrated in Figure 3, and it can be segmented into the following five steps.

Step 1 is sample data construction. The stress sample dataset Str1 of the nonlinear boom structure is randomly divided into a training set Str1_train and a test set Str1_test in a ratio of 7:3. If the fusion condition is satisfied, the RS algorithm is utilized to randomly search for the combination of n_estimators and max_features. Then, the hyperparameter sample data points hpi of the RF model are formed. The iteration step is set to D. Otherwise, the RF model is directly used for the stress prediction. The detailed process is shown in Figure 4.

Step 2 is mapping relationship acquisition. The k-fold cross-validation method [56] is a model performance assessment method that enhances the model generalization capacity. It mitigates the likelihood of overfitting situations. It procedurally partitions the Str_train into k subsets, each containing an equal number of samples. Any single subset is designated as the validation set, while the remaining k − 1 subsets collectively form the new training set. The RF model that has been constructed with hpi undergoes training and validation within this framework. This process iterates k times. The loss function (MSE) is employed to compute the average loss across the k new training sets. This resultant value, denoted as MSEavg, serves as the loss metric for the RF model. Subsequently, the relationship between MSEavg and the hyperparameter sample data point hpi is ascertained. The MSEavg and hpi are stored in the hyperparametric mapping relationship database (Hp_Database). The detailed process is shown in Figure 5.

Step 3 is hyperparametric learning model selection and hyperparameter prediction under inverse control. The better the model fits the sample data, the narrower the disparity between the predicted and true values of the model. This diminished disparity manifests as a reduction in the loss value of the model. Conversely, the optimal hyperparameter values of the model can be ascertained through the minimization of the loss values. The total number of cross-validation iterations is set to S, and it is used as the termination iteration condition of the algorithm. Upon reaching S iterations, the entire set of sample data in the Hp_Database will be trained by the XGBoost model. MSEavg serves as the input feature, while each configuration of hpi is identified as the output result. After the training is completed, MSEavg = 0 is input. Then, the output of this group of hyperparameters hp* is the optimal master hyperparameter hp_best. The detailed process is shown in Figure 6.

Step 4 is the prediction process with directional corrections to the model. Firstly, the sample data in Hp_Database is limited. The MSE values as feature inputs are limited to a certain range. In the process of model learning and prediction, a localized ‘comfort zone’ will emerge. It cannot fit the whole situation. Secondly, because the structural stress value in Str1 is larger, the MSEavg value obtained in Step 2 is also larger. This inevitably affects the accuracy of the optimal solution under the specific objective (MSEavg = 0) in Step 3. Therefore, in the context of the hyperparameter prediction task with a specific objective, this paper proposes an additive strategy with directional corrections. Through the fusion of random search and directed prediction, all sample data in the hyperparameter mapping relation database are trained and tested once for every w search. Simultaneously, the novel hyperparameter sample data point hp_new is predicted under the condition of MSEavg = 0. The hp_new is considered as the hyperparameter sample data point for the next random search. The additional sample data are utilized to continuously correct the learning trend, thus improving the model prediction accuracy. The detailed process is shown in Figure 6.

Step 5, the RS–XGBoost–RF model, configured with the optimal master hyperparameter hp_best, is the final stress prediction model. The RS–XGBoost–RF model is evaluated using the test set from the nonlinear boom structure stress sample data. Ultimately, the findings of this evaluation are explicated. The detailed process is shown in Figure 7.

6. Results and Discussion

The series of models and algorithms in this paper were implemented on a laptop computer with the model number ‘Z7M-KP7GC’. The hardware specifications are as follows: Intel Core i7-8750H processor (Intel Corporation, Santa Clara, CA, USA), NVIDIA GTX1050Ti 4G GDDR5 discrete graphics card (NVIDIA, Santa Clara, CA, USA), 8 GB onboard RAM, and Windows 10 system specification. The results of evaluations of the model, especially the model’s learning time, convergence time, etc., are greatly affected by the hardware specifications. The test results in this paper can be regarded as a reference.

6.1. Stress Prediction Results

In line with the model fusion conditions expounded in Section 5.1, the RF model is deployed for the training and testing of the stress sample data of the nonlinear boom. The fitting accuracies at detection points ‘M1C1’, ‘M1C3’, and ‘M2C3’ are 0.980, 0.963, and 0.957, respectively. They surpass the prescribed threshold of R² = 0.95. The corresponding test outcomes are visually represented in Figure 8.

Except for the detection points ‘M1C1’, ‘M1C3’, and ‘M2C3’, the fitting accuracy for the remaining detection points falls below the prescribed threshold of R² = 0.95. Consequently, adhering to the stress prediction procedure based on the RS–XGBoost–RF model (delineated in Section 5.2), the remaining detection points in Str1 undergo training and testing. In the master hyperparameter optimizing phase of the RF model, the pertinent parameters are configured as follows: the search space for n_estimators and max_features spans [1, 10] and [10, 300], respectively, with integers as the only permissible values. The number of searches is stipulated as S = 200. k = 5 is taken in the k-fold cross-validation. Within the directional model correction, w = 10 is set. The ensuing test results of the RS–XGBoost–RF model, alongside the values of the master hyperparameters, are delineated in Figure 9.

As can be seen from Figure 9, in terms of prediction error, the RMSE and MAE values are followed by larger values due to the larger stress values of the detection points in the sample data. Nevertheless, when evaluated as a percentage of the stress value itself, MAE only accounts for 2.22% to 3.91%. The RMSE is more susceptible to a few outliers, presenting a marginally higher range of approximately 4.79% to 7.85% of the stress value. Consequently, the error associated with the RS–XGBoost–RF model is deemed to reside within a narrow range when considered from a percentage standpoint.

In terms of fitting accuracy, most of the coordinate points consisting of true and predicted values are near the black dashed line ‘y = x’ (horizontal coordinate is x and vertical coordinate is y). This phenomenon signifies the robust fitting capability of the RS–XGBoost–RF model. While a few coordinate points exhibit deviations from the black dashed line, their impact on the overall fitting accuracy is marginal. The prediction accuracies of all stress detection points except ‘M4C1’ are higher than 0.955, and the highest is 0.9805. The results meet the requirements of enterprises for the prediction accuracy of experimental data. Notably, the prediction accuracy for ‘M4C1’ stands at 0.9303. It is not difficult to find that there are many coordinate points deviating from the regression line and residing farther away from it in Figure 5. This signifies the presence of outliers in the data, contributing to the relatively diminished prediction accuracy at this stress detection point. Encouragingly, the 7% accuracy error remains within the acceptable bounds of engineering practice.

In terms of model intricacy, the complexity of the RS–XGBoost–RF model is much higher than that of a single RF model, and the computational workload during model training also increases accordingly. Particularly when the hyperparameters in the stochastic search process assume larger values, the complex model will directly impact computational efficiency.

In terms of the complexity and learning time of the model, the complexity of the RS–XGBoost–RF model is much higher than that of the single RF model. The amount of computation during model training also increases. Therefore, the learning time of the model increases from within 7 s initially to 13–20 min. Especially when the hyperparameters (n_estimators and max_features) in the stochastic search process take larger values, it will greatly increase the model learning time. Therefore, it makes sense to set fusion conditions in the model fusion concept.

In summary, by introducing the concept of model fusion, the RS–XGBoost–RF model proposed in this paper integrates the accuracy and efficiency issues. It can not only ensure the precision of stress prediction but also address efficiency requirements pertinent to the resolution of practical engineering challenges. The RS–XGBoost–RF model can realize the rapid prediction of nonlinear boom stress for concrete pump trucks in a pragmatic sense.

6.2. Sensitivity Analysis

From the preprocessed sample data outlined in Section 4.2, it becomes evident that the primary factors (input features) influencing the stresses in the nonlinear boom structure for concrete pump trucks encompass pressure in the rod chamber of the boom, pressure in the non-rod chamber of the boom, rotation angle, tilt angle of each boom, and pumping pressure. Sensitivity analysis allows for the visualization of the importance of each feature within the prediction model. It is necessary to ensure the consistency between the division of feature weights and the input weights of stress prediction models (including the single RF model and the RS–XGBoost–RF model) when doing the feature sensitivity analysis. The RF feature importance assessment method is used to divide the degree of influence of each feature on the prediction results under different detection points. Higher feature weights signify a more pronounced impact. The importance of a feature within the RF model is derived from the average importance across all decision trees associated with that feature. The calculation of the importance degree of features in the decision tree can be executed by Equation (21). Subsequently, the results of feature weights for each stress detection point are computed, as depicted in Figure 10.

$$\{\begin{array}{l}{I}_f={\displaystyle \sum _{j=1}^m{F}_j}/{\displaystyle \sum _{a=1}^n{F}_a}\\ F_a=p_a{G}_a-p_l{G}_l-p_r{G}_r\end{array}$$

(21)

where, $I_f$ is the importance of the feature f. $F_j$ is the importance of the split node j under the feature f. $F_a$ is the importance of node a. $p_a$, $p_l$ and $p_r$ are the ratios of the number of training samples in node a and the left and right child nodes to the total number of training samples, respectively. $G_a$, $G_l$ and $G_r$ are the impurity of node a and the left and right child nodes, respectively. m is the total number of nodes under the split of the feature f. n is the number of all nodes.

Note: some of the features behave tiny in the distribution of weights for stress detection points. So, labels with less than 4% weight share are omitted.

The importance ratios of various features for each stress detection point are elucidated in Figure 10. Notably, the rotation angle has the largest weight at detection points ‘M1C1’, ‘M2C1’, ‘M2C4’, and ‘M4C3’, garnering weights of 81%, 37%, 26%, and 49%, respectively. The tilt angle of boom 3 emerges as the most influential factor at detection points ‘M1C2’, ‘M1C3’, and ‘M1C4’, commanding weights of 30%, 28%, and 27%, respectively. At detection point ‘M2C2’, the maximum weight is attributed to the tilt angle of boom 1, amounting to 22%. Similarly, the tilt angle of boom 6 attains the maximum weight of 24% at detection point ‘M2C3’. At detection point ‘M4C1’, the maximum weight is assigned to the tilt angle of boom 4, reaching 30%. These characteristics are the predominant influencing factors at their respective detection points.

It can be seen that the weights of the rotation angle and the tilt of booms are more prominent at the stress detection points. However, the weights assigned to pressure in the non-nod chamber of the boom and pumping pressure do not surpass 10%. Despite some features exhibiting lower global weights, their proportions at different detection points vary. Moreover, certain features at the same detection point share a similar proportion to the aforementioned features. This disparity arises from the nonlinear characteristics of the large geometric deformation of the concrete pump truck boom structure, and the numerous and complex working postures. Consequently, across all detection points within the comprehensive target system of stress prediction, existing features cannot be simply deleted in feature engineering processing.

6.3. Convergence Test

This section aims to validate the accuracy of the hyperparameter prediction method based on the RS–XGBoost model in optimizing the hyperparameters of the RF model. From the point of view of computational efficiency and computational accuracy and through convergence analysis, a comparative discussion is conducted on multiple detection points. The discussion essentially contrasts the RS-RF model under the hyperparameter random search algorithm and the RS–XGBoost–RF model under the hyperparameter prediction method. The assessment encompasses considerations of computational efficiency, computational accuracy, and convergence. The iterative convergence curves of both models are depicted in Figure 11.

Figure 11 employs the number of iteration steps as the independent variable and MSE as the dependent variable, illustrating the iteration curves of the RS-RF model and the RS–XGBoost–RF model. The plot distinctly portrays the 200 iterations of both hyperparameter optimization methods. Evidently, throughout the iterative search process, the loss values for both methods consistently decrease and eventually tend toward stability. This unequivocally indicates that both methods adhere to the convergence condition. Thereby, the correctness of the employed methodology is affirmed.

In terms of calculation accuracy, at the convergence point, the RS–XGBoost–RF model exhibits smaller MSE values in comparison to the RS-RF model. This improvement underscores the enhanced calculation accuracy achieved by the RS–XGBoost–RF model. However, in terms of computational efficiency, a comparative analysis reveals that the RS-RF model consistently reaches a stable state earlier than the RS–XGBoost–RF model except for ‘M1C2’ and ‘M1C4’. This observation suggests that the overall efficiency of the hyperparameter prediction method is marginally lower than that of random search. The disparity arises from the fact that, although both methods share an identical number of iteration steps, the RS–XGBoost model engages in additional predictive and directional correction tasks compared to RS in its pursuit of heightened prediction accuracy. Furthermore, the hyperparameter prediction is conducted on the foundation of a random search. The quality of its predictions is heavily reliant on the outcomes of these random searches. Consequently, achieving convergence to a lower loss value necessitates more iterations than the RS algorithm alone.

6.4. Method Comparison and Discussion

The efficacy and precision of the RS–XGBoost–RF model in the stress rapid prediction of nonlinear boom structure necessitate further validation. This section undertakes a detailed analysis and discussion from three distinct perspectives: longitudinal comparison, horizontal comparison, and comparison with other methods. In the context of longitudinal comparison, the RS–XGBoost–RF model is compared with the initial RF model under the progressive enhancement process, as well as with the RS-RF model under random search hyperparameters. The horizontal comparison is to compare the four fusion models (RS–XGBoost–RF model, RS–RF–RF model, RS–RF–XGBoost model, and RS–XGBoost–XGBoost model) consisting of the RS algorithm, RF model, and XGBoost model under the concept of model fusion. The comparison with other methods involves pitting the RS–XGBoost–RF model against the PSO-RF model in terms of hyperparameter optimization. Throughout these comparisons, each model undergoes ten independent runs. The average of the ten model test results is considered as the conclusive evaluation result.

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Longitudinal comparison

The evolution of the RS–XGBoost–RF model unfolds gradually from RF to RS-RF to RS–XGBoost–RF. On the one hand, it is necessary to compare the RS-RF model [57] with the RF model [32]. The impact of the RS algorithm in hyperparameter optimization for the RF model is explored. On the other hand, a comparative analysis is conducted between the RS-RF model and the RS–XGBoost–RF model. The advantages of the cooperation of search and prediction methods in hyperparameter optimization for the RF model are explored. To ensure a fair and reasonable comparison, it is imperative to maintain consistency between the RS-RF model and the RS–XGBoost–RF model in the number of searches. The test results for each model are delineated in Figure 12.

Table 7. Learning time of each model under longitudinal comparison.

Models	M1C2	M1C4	M2C1	M2C2	M2C4	M4C1	M4C3
RF	6.1 s	4.9 s	5.3 s	6.6 s	5.9 s	6.1 s	6.4 s
RS-RF	28.9 s	30.2 s	27.2 s	28.4 s	28.3 s	28.6 s	27.5 s
RS–XGBoost–RF	19 min 48 s	20 min 5 s	16 min 27 s	19 min 33 s	16 min 22 s	13 min 13 s	14 min 8 s

gives the learning time for each model.

Figure 12 illustrates the evaluation results of three models throughout the evolution from RF to RS-RF to RS–XGBoost–RF. A discernible trend of progressively favorable changes is evident. The red line is closer to the black line. This phenomenon signifies an enhancement in the prediction performance of RS-RF compared to the RF model, with the exception of the unclear improvement effects observed in ‘M1C2’ and ‘M2C4’. ‘M2C2’ has a larger improvement. RMSE and MAE are reduced by 23.62% and 22.32% and the improvement in R² is more than 6%. For the remaining detection points, RMSE and MAE exhibit reductions ranging from 6.99% to 9.62% and 13% to 19.75%, and the improvement in R² is 0.06% to 1.81%. The blue line diverging from the red line suggests that the evolution from the RS-RF model to the RS–XGBoost–RF model signifies a qualitative leap. Leveraging the prediction efficacy of the original RS-RF model, the RS–XGBoost–RF model achieves a notable reduction in RMSE and MAE, ranging from 25.39% to 30.88% and 36.98% to 43%, respectively, and the improvement in R² is 1.5% to 5.85%.

From Table 7, the process of hyperparameter optimization consumes quite a lot of time. With the same number of iterations, the learning time of the RS-RF model is about 5–6 times that of the RF model. The RS–XGBoost–RF model, on the other hand, consumes much more time than the RS-RF model due to the need for constant prediction and correction. The good thing is that it has a stable improvement effect on the performance of the model. Compared with the traditional method of stress acquisition, the learning time of the RS–XGBoost–RF model is within a time frame of about 20 min, which is acceptable for engineering practice.

In summary, the RS algorithm avoids solidly adopting default values by constantly trying various hyperparameter pairings. This way has a certain improvement effect on the prediction performance of the RF model, and the model requires less learning time. The cooperation of the search and prediction method, building upon the RS algorithm, excels in predicting hyperparameters with minimized loss values. Although this model has a long learning time, it delves deeper into the predictive capabilities of the model. It enhances the sensitivity of the model to the changing trend in the sample data. The result is a significant improvement in prediction accuracy.

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Horizontal comparison

Under the model fusion concept, variations in fitting results across the data arise from the inherent strengths and weaknesses of each model. Therefore, a comparison is undertaken for the prediction results of four fusion models (RS–XGBoost–RF, RS–RF–RF, RS–RF–XGBoost, and RS–XGBoost–XGBoost). The comparative analysis delves into the impact of the model combination method on the structural stress prediction outcomes under the fusion concept.

For the RS–XGBoost–RF model and RS–RF–RF model, the hyperparameters of the RF model serve as the optimizing target, with the relevant parameter settings detailed in Table 3. In contrast, for the RS–RF–XGBoost model and RS–XGBoost–XGBoost model, the optimizing target shifts to the hyperparameters of the XGBoost model. The optimizing parameters and search space are chosen empirically as follows: the value range of the learning rate [0, 1], the value range of the loss function descent threshold (gamma) [0, +∞], the value range of the L1 regularization parameter (alpha) [0, +∞], the value range of the proportion of randomly selected samples (sub-sample) (0, 1], the value range of the proportion of randomly selected features (Colsample-by tree) (0, 1]. The remaining hyperparameters retain their default values. The test results of each model under horizontal comparison are given in Figure 13.

Table 8. Learning time of each model under horizontal comparison.

Models	M1C2	M1C4	M2C1	M2C2	M2C4	M4C1	M4C3
RS–XGBoost–RF	19 min 48 s	20 min 5 s	16 min 27 s	19 min 33 s	16 min 22 s	13 min 13 s	14 min 8 s
RS–RF–RF	17 min 34 s	17 min 6 s	13 min 27 s	16 min 10 s	14 min 29 s	14 min 46 s	15 min 37 s
RS–RF–XGBoost	2 min 32 s	2 min 48 s	2 min 43 s	2 min 25 s	2 min 33 s	2 miin 39 s	2 min 38 s
RS–XGBoost–XGBoost	2 min 25 s	2 min 34 s	2 min 36 s	2 min 27 s	2 min 18 s	2 min 45 s	2 min 47 s

gives the learning time for each model.

As depicted in Figure 13, in comparison to other fusion models, the RS–XGBoost–RF model exhibits the most exceptional predictive performance. Its goodness-of-fit and error resemble the ‘top’ and ‘bottom’ of a hill, respectively. The former is prominently positioned at the summit and the latter is at the nadir. This configuration signifies that the RS–XGBoost–RF model attains the optimal fit and the lowest error among the considered fusion models.

From the learning time of models shown in Table 8, it is easy to see that the learning time of each model is similar under the same model fusion concept, the same optimization objective, and the same number of algorithm iterations. For different optimization objectives (RF and XGBoost), due to different hyperparameters contained in the models and the different search ranges set, it is difficult to perform a comparative analysis. Therefore, it will not be discussed in detail.

The horizontal comparison elucidates that the adaptability of various fusion models to the sample data diverges due to the inherent strengths and weaknesses inherent in each model. In the training and testing of nonlinear boom structure stress sample data, the primary evaluation criterion lies in the capacity of the model to fit nonlinear high-dimensional data. Here, the RF model outshines the XGBoost model due to its integrated learning approach and robust performance. Commencing from the mapping relationship between Mean Squared Error (MSE) and hyperparameters, the primary examination pertains to the robustness of the model in interpolation and extrapolation when MSE equals zero. This is especially crucial when the prediction target deviates significantly from the actual value of the sample. In this context, the RF-XGBoost model surpasses the RF algorithm due to its potent scalability. The above comparison combined with the principle analysis once again verifies the effectiveness and superiority of the RS–XGBoost–RF model in the rapid stress prediction for the nonlinear boom structure.

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Comparison with other methods

The particle swarm optimization algorithm (PSO) is a type of swarm intelligent optimization algorithm. Each particle continuously explores individual optimal solutions within the search space. Following the computation of fitness for each particle based on the fitness function, the optimal solution for the entire population is extracted as the output value. Numerous studies [13,15] have validated the efficacy of the PSO algorithm as a method for optimizing hyperparameters of the RF model. Consequently, the PSO algorithm is employed to optimize the two master hyperparameters of the RF model. The hyperparameters of the PSO algorithm itself are taken as default values. The relevant test results are presented in

Table 9. Comparison of test results of RS–XGBoost–RF model and PSO-RF model.

Model	Stress Detection Points	RMSE	MAE	R2	Learning Time	Model	Stress Detection Points	RMSE	MAE	R2	Learning Time
PSO-RF	M1C2	9.0813	4.3663	0.9545	62 min 12 s	RS–XGBoost–RF	M1C2	6.5802	3.0359	0.9732	19 min 48 s
	M1C4	11.0836	5.3968	0.9567	49 min 17 s		M1C4	9.2753	4.2724	0.9701	20 min 5 s
	M2C1	7.0128	3.7218	0.9483	35 min 12 s		M2C1	5.2586	2.3532	0.969	16 min 27 s
	M2C2	9.9472	4.4697	0.9332	123 min 46 s		M2C2	6.2874	2.8722	0.9693	19 min 33 s
	M2C4	7.9967	3.9708	0.9496	14 min 8 s		M2C4	5.809	2.3221	0.9729	16 min 22 s
	M4C1	13.6351	7.5705	0.8875	105 min 29 s		M4C1	10.7905	4.3278	0.9303	13 min 13 s
	M4C3	14.1459	7.3841	0.9346	135 min 41 s		M4C3	10.9124	5.1809	0.9632	14 min 8 s

.

As shown in Table 6, in comparison to the PSO-RF model, the RS–XGBoost–RF model exhibits an improvement in R² ranging from 1.34% to 4.28%. The RMSE is reduced by 16.32% to 36.79%, and the MAE is reduced by 20.84% to 42.83%. The learning time of the model is also dramatically reduced. This observation can be attributed to the fact that the hyperparameter search process of the RF model in PSO is significantly influenced by its hyperparameter settings. Additionally, the PSO algorithm has poor population diversity and is prone to premature convergence, thus falling into local optimization. It often leads to long-term costs and the inability to obtain accurate hyperparameters. In turn, it impacts the predictive performance of the PSO-RF model. On the other hand, the RS–XGBoost model fully exploits the robust interpolation and extrapolation capabilities of the XGBoost model by leveraging the RS algorithm as a foundation. The accurate prediction for hyperparameters of the RF model is achieved through further prediction and directional correction. It saves a lot of searching time. Consequently, the RS–XGBoost–RF model demonstrates robust predictive abilities in the stress prediction of the nonlinear boom structure.

7. Conclusions

Metal structure stress acquisition has always been one of the most critical aspects in the field of intrinsic safety of major equipment. It not only plays a decisive role in load-carrying capacity verification but also has a significant impact on the accuracy of fatigue life assessment results. This paper introduces the concept of model fusion. It proposes a fast prediction method of the stress of nonlinear boom structure based on the RS–XGBoost–RF model from the aspects of model selection, fusion condition setting, and fusion method implementation. The main conclusions are as follows:

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Based on the user’s requirements, the decision criterion for model fusion is based on prediction efficiency. If the fitting accuracy of the RF model reaches the threshold r, the prediction results can be directly outputted. Otherwise, model fusion is initiated. Despite the fused model being more complex than the single RF model, it offers improved prediction accuracy within acceptable engineering practice standards. Therefore, establishing model fusion judgment conditions holds practical significance.

(2)

In line with the model fusion concept, the RS algorithm, RF model, and XGBoost model are selected for fusion. It results in four combinations, RS–XGBoost–RF, RS–RF–RF, RS–RF–XGBoost, and RS–XGBoost–XGBoost models. Comparative analysis reveals that all fusion models outperform the original RF model in terms of prediction effectiveness. Additionally, considering the inherent strengths and weaknesses of each model and their compatibility with sample data, it is determined that the RS–XGBoost–RF model exhibits the most superior prediction performance.

(3)

A novel RS–XGBoost prediction model for hyperparameter optimization of RF models is introduced, leveraging the synergy between search and prediction mechanisms. This model predicts optimal hyperparameter combinations through random search. The model achieves accurate and efficient hyperparameter optimization for RF models compared to single RS and PSO algorithms. It overcomes issues related to incomplete, unstable, and inefficient search. Utilizing the optimal RS–XGBoost–RF model, the fit of M4C1 is improved to 0.9303, while the fit of other measurement points remains above 0.955. Furthermore, the errors of mean absolute error (MAE) and root mean square error (RMSE) concerning stress values are constrained within the ranges of 2.22% to 3.91% and 4.79% to 7.85%, respectively. Overall, these errors are minimal, indicating high efficiency.

At present, the fast prediction method of the stress of nonlinear boom structure based on the RS–XGBoost–RF model proposed in this paper still has some limitations. For one, the data in the hyperparameter mapping relationship database are not comprehensively analyzed, i.e., whether every hyperparameter prediction result can play a positive role in the model. For another, the fusion model is not applied to different service scenarios and different types of series of concrete pump truck boom systems from the point of view of model migration. The generalizability of the model needs to be improved.