Research on Fault Diagnosis Method with Adaptive Artificial Gorilla Troops Optimization Optimized Variational Mode Decomposition and Support Vector Machine Parameters

Abstract

To address the issue of intelligent optimization algorithms being prone to local optima, resulting in insufficient feature extraction and low fault-type recognition rates when optimizing Variational Mode Decomposition and Support Vector Machine parameters, this paper proposes a fault diagnosis method based on an improved Artificial Gorilla Troops Optimization algorithm. The Artificial Gorilla Troops Optimization algorithm was enhanced using Logistic chaotic mapping, a linear decreasing weight factor, the global exploration strategy of the Osprey Optimization Algorithm, and the Levy flight strategy, improving its ability to escape local optima, adaptability, and convergence accuracy. This algorithm was used to optimize the parameters of Variational Mode Decomposition and Support Vector Machine for fault diagnosis. Experiments on fault diagnosis with two datasets of different sample sizes showed that the proposed method achieved a diagnostic accuracy of no less than 98% for samples of varying sizes, with stable and reliable results.

1. Introduction

Motors and bearings, as important components of modern industrial devices and machinery, are prone to various faults under the combined influence of long-term continuous operation and internal and external environmental factors. These faults can affect normal production, cause economic losses, and pose safety hazards. Therefore, diagnosing faults in motors and bearings is of great significance [1]. Fault diagnosis in industrial equipment such as motors and bearings is primarily achieved through signal analysis and the development of diagnostic models [2]. This process is typically divided into three key stages: vibration signal processing, fault feature extraction, and fault identification and classification. Among these, the fault feature extraction stage is critical, as the adequacy of the extracted features directly determines the accuracy of subsequent diagnostics. Therefore, feature extraction is considered the core of the entire fault diagnosis process.

For vibration signal feature extraction, methods such as Empirical Mode Decomposition (EMD) [3], Local Mean Decomposition (LMD) [4], Empirical Wavelet Decomposition (EWT) [5,6], and Variational Mode Decomposition (VMD) [7,8] are widely used. However, EMD suffers from mode mixing and endpoint effects, and it is sensitive to noise and subtle signal changes, resulting in poor decomposition stability [9,10]. LMD is prone to pseudo-fluctuations when processing signal endpoints, and its decomposition process is complex with high computational demands, limiting its applicability [11,12]. EWT’s decomposition performance depends on the selection and design of wavelet filters and has limited frequency resolution, making it ineffective for handling non-stationary signals [13,14]. By contrast, VMD addresses the issues of mode mixing, endpoint effects, decomposition stability, and applicability found in EMD, LMD, and EWT by using variational optimization and mode bandwidth constraints. VMD demonstrates more stable performance when decomposing nonlinear and non-stationary signals [15].

When using VMD to extract vibration signal features, it is necessary to manually select the total number of mode components, $K$, and the quadratic penalty factor, $\alpha$. These two parameters determine the performance of VMD [16]. Since $\alpha$ and $K$ typically take values within a certain range, selecting values based solely on experience or comparing a few values and then drawing conclusions lacks systematic analysis and rationality. In recent years, swarm intelligence optimization algorithms have gained widespread attention from researchers for their application in parameter optimization for Variational Mode Decomposition (VMD). These algorithms are favored for their strong global search capabilities, adaptability, and ability to overcome the limitations of experience-based parameter selection, where the chosen parameter values may not be optimal. Reference [17] proposed using the kernel estimate for mutual information (KEMI) as a fitness function and employing a genetic algorithm (GA) to optimize VMD parameters, achieving the identification of single and multiple defects in constant and variable speed bearings. However, a GA relies on the evolution of the population, requiring numerous iterations to find the optimal solution, significantly increasing computational cost in parameter optimization problems. Reference [18] proposed a hybrid diagnostic method based on vibration signals, using the Gray Wolf Optimization algorithm (GWO) to optimize VMD parameter combinations. However, GWO lacks an adaptive mechanism and cannot dynamically adjust algorithm parameters based on feedback during the search process, leading to insufficient search accuracy. Reference [19] employed the Sparrow Search Algorithm (SSA) to optimize the parameters to be determined in VMD and Maximum Correlated Kurtosis Deconvolution (MCKD), but the SSA also suffers from issues such as initial population random creation, leading to uneven population distribution and a lack of diversity. Additionally, when the warning value is less than the set safety threshold, the search ability of the discoverer gradually decreases as the algorithm iterations increase, resulting in insufficient spatial search area and low convergence efficiency. Therefore, this paper selects the Artificial Gorilla Troops Optimizer (GTO), which has diversified search strategies, and addresses its shortcomings by making improvements. The improved GTO is used to optimize the parameters $\alpha$ and $K$ of VMD, and to extract vibration signal features.

Using vibration signal features for fault diagnosis requires classifying and identifying the processed signals. Choosing an appropriate diagnostic method is crucial. In the field of fault diagnosis using vibration signals, machine learning methods such as k-Nearest Neighbors (k-NN), Random Forest (RF), and Support Vector Machines (SVMs) are widely employed for diagnosing faults in motors and other equipment [20,21,22,23]. Moreover, deep learning approaches, including Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM) networks, are also commonly used to achieve fault diagnosis [24]. Reference [25] utilized genetic algorithms to select important features from statistical features extracted from motor signals, employing three machine learning methods—K-Nearest Neighbors, decision trees, and Random Forest—for training and testing to achieve precise fault identification. Reference [26] proposed a multi-signal fault diagnosis method based on a Deep Convolutional Neural Network (DCNN), capable of learning from various types of sensor signals. By extracting features from vibration and current signals, the model achieved accurate induction motor fault diagnosis. Reference [27] applied the MPE threshold method to select appropriate VMD parameters for decomposing motor vibration signals and employed the cuckoo search algorithm to optimize the global optimal solution of the Support Vector Machine, resulting in high fault identification rates. These methods each have their respective advantages and disadvantages. In this paper, considering the background of motor and motor bearing vibration signals, the availability of fault samples, and the nonlinear and non-stationary characteristics of vibration signals, we choose the SVM classifier, which supports nonlinear classification and has good generalization ability, for fault diagnosis.

In summary, to address the issue of insufficient fault vibration signal feature extraction leading to low subsequent diagnostic accuracy, this paper presents an improvement of the Artificial Gorilla Troops Optimization algorithm using four strategies to enhance its search performance and adaptability, resulting in a robust intelligent optimization algorithm. This improved algorithm is employed to optimize the parameters of Variational Mode Decomposition and Support Vector Machine, thereby enhancing the generalization capability of the fault diagnosis model. As a result, the model achieves high diagnostic accuracy across different diagnostic targets, including bearings and motors, and different sample sizes of faults. This work provides a reliable diagnostic method for the field of fault diagnosis.

2. Improved Artificial Gorilla Force Algorithm Fusing Osprey and Levy Flight (OLGTO)

2.1. The Original Artificial Gorilla Force Algorithm

Artificial gorilla troop optimization algorithm [28] is a new type of intelligent optimization algorithm proposed by mathematizing the collective social life habits of gorillas. The algorithm mainly simulates the population migration and courtship behavior of gorillas and other life behaviors to find the optimal behavior, and it has the advantages of strong ability to find the optimal and fast convergence speed, etc. The GTO algorithm mainly consists of an exploration phase and a development phase, and the development phase contains two kinds of behaviors, namely, following the silverback gorillas and competing for the adult females.

First, the GTO is initialized by setting parameters such as $N$, $MaxIt$, $\beta$, $p$, and $W$. Here, $N$ represents the number of gorillas in the population; $MaxIt$ is the maximum number of iterations; $p$ controls the migration strategy of the gorillas to unknown locations, with values ranging between 0 and 1; $\beta$ is set to 0.3 to calculate the intensity of the gorillas’ violent behavior; $W$ determines the two mechanisms in the exploitation phase, with a value of 0.5. Subsequently, the gorilla population is randomly initialized in the search space.

Second, during the exploration phase, the silverback gorilla leads the other gorillas to live in groups in the natural environment. This phase primarily involves three mechanisms for global search: migrating to unknown locations, migrating to known locations, and migrating to other gorillas’ locations. Mechanism 1 allows gorillas to randomly explore the space, with the execution condition $ran{d}_1<p$; Mechanism 2 enhances the algorithm’s exploration of the space, with the execution condition $ran{d}_2\ge 0.5$; Mechanism 3 strengthens the algorithm’s ability to escape local optima, with the execution condition $ran{d}_2<0.5$. The specific formulas for the exploration phase are as follows in Equations (1)–(6):

$$GX(t+1)=\left\{\begin{array}{ll}(UB-LB)\times r_1+LB& ,ran{d}_1<p\\ (r_2-C)\times X_r(t)+L\times H& ,ran{d}_2\ge 0.5andran{d}_1\ge p\\ X(t)-L\times (L\times (X(t)-GX(t)))+r_3\times (X(t)-G{X}_r(t))& ,ran{d}_2<0.5andran{d}_1\ge p\end{array}\right.$$

(1)

$$C=F\times \left(1-{\displaystyle \frac{t}{MaxIt}}\right)$$

(2)

$$F=\cos (2\times r_4)+1$$

(3)

$$L=C\times l$$

(4)

$$H=Z\times X(t)$$

(5)

$$Z=[-C,C]$$

(6)

In the formulas, $X(t)$ and $GX(t+1)$ represent the current position and the position in the next iteration of a gorilla, respectively, while $X_r(t)$ and $G{X}_r(t)$ are positions of randomly selected gorillas. $t$ denotes the current iteration number. $r_1$, $r_2$, $r_3$, $r_4$, $l$ and $rand$ are all random numbers; $r_1,r_2,r_3,l,rand\in [0,1]$, while $r_4\in [-1,1]$. $UB$ and $LB$ represent the upper and lower bounds of the variables, respectively. $Z$ is a random number within the range $[-C,C]$, where the parameter $C$ has significant variation in the initial stages and gradually decreases later on. $L$ indicates the leadership ability of the silverback gorilla, which might make incorrect decisions due to lack of experience in finding food or managing the group. At the end of the exploration phase, the fitness values of $GX(t)$ and $X(t)$ are calculated and compared. If the fitness value of $GX(t)$ is smaller, the position $X(t)$ is replaced by $GX(t)$.

Finally, during the exploitation phase, the algorithm employs two behavioral mechanisms: following the silverback gorilla and competing for mature females. The parameter $C$ is used to switch between these mechanisms. If $C\ge W$, the gorillas follow the silverback gorilla. The silverback, as the leader of the group, guides the gorillas to food sources and ensures the safety of the group. All gorillas in the group adhere to the decisions made by the silverback. If $C<W$, competition occurs. The silverback gorilla may age and die, allowing a blackback gorilla to potentially become the leader, or other male gorillas may challenge the silverback in combat to dominate the group.

When $C\ge W$, the mechanism for following the silverback gorilla is described by the following equations:

$$GX(t+1)=L\times M\times (X(t)-X_{silverback})+X(t)$$

(7)

$$M={\left({\left|{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^{N}G{X}_i(t)}\right|}^g\right)}^{{\displaystyle \frac{1}{g}}}$$

(8)

$$g=2^L$$

(9)

In the equations, $X_{silverback}$ represents the position of the silverback gorilla (the optimal position). $G{X}_i(t)$ denotes the position of each candidate gorilla during the iteration. $N$ indicates the total number of gorillas.

When $C<W$, the calculation formula for the competitive adult female mechanism is shown in Equation (10).

$$\left\{\begin{array}{l}GX(t)=X_{silverback}-(X_{silverback}\times Q-X(t)\times Q)\times A\\ Q=2\times r_5-1\\ A=\beta \times E\\ E=\left\{\begin{array}{l}{N}_1,rand\ge 0.5\\ N_2,rand<0.5\end{array}\right.\end{array}\right.$$

(10)

where $Q$ represents the intensity of gorilla competition; $A$ denotes the coefficient of the competition degree; and $E$ represents the impact of violence on the dimensions of the solution. $r_5$ and $rand$ are random numbers within the interval $[0,1]$. When $rand\ge 0.5$, $E$ is a random value from a normal distribution and within the problem dimensions; otherwise, $E$ is a random value from a normal distribution.

At the end of the development phase, a population operation is conducted, which involves estimating the fitness values of all $GX$ individuals. If $GX(t)<X(t)$, the $GX(t)$ individuals replace the $X(t)$ individuals. The best solution (the minimum fitness value) obtained from the entire population is considered the silverback gorilla.

2.2. Improved Artificial Gorilla Force Algorithm (OLGTO)

Based on the iterative optimization process of the GTO algorithm, it is known that the GTO algorithm has a good capability to avoid local optima, achieving high convergence accuracy and speed. However, in the early stages of iteration, the large number of formula parameters prevents timely information exchange among gorillas, adversely affecting the algorithm’s ability to escape local optima and impacting convergence speed and accuracy in later stages. To address these issues, the algorithm is improved in four aspects.

(1) We have adopted the Logistic chaotic map to initialize the gorilla population, enhancing the diversity of the population. This modification allows the algorithm to more uniformly cover the search space during the initial phase, resulting in higher convergence precision and reducing the risk of the population falling into local optima.

(2) An improvement in the weight factor $W$ has been made to balance the algorithm’s global search and the ability to escape local optima. This adjustment allows the algorithm to more effectively avoid local optima during the convergence process, thereby enhancing the overall optimization performance, with improvements in both convergence speed and solution accuracy.

(3) The Fish Hawk Optimization Algorithm has been used to replace the second formula in the original GTO algorithm’s global exploration strategy in the first phase. This change aims to improve the global search capability of the algorithm. The modified algorithm demonstrates stronger global search capabilities in complex search spaces, enabling faster convergence to the optimal solution and avoiding the reduction in global search performance due to delayed information exchange among populations caused by excessive parameters.

(4) After the development phase, Levy flight has been applied to update the positions of the gorillas, which enhances the robustness of the algorithm. This application makes the algorithm more effective at escaping local optima in the later stages of iteration.

2.2.1. Logistic Chaotic Mapping Initializes Gorilla Population

Logistic chaotic mapping is a simple yet nonlinear mapping with complex dynamic behavior used to generate chaotic sequences. Compared to other chaotic mappings, it has the advantages of a relatively simple expression, strong adaptability, and the generated sequences tend to stabilize without infinitely increasing or decreasing [29].

Using Logistic chaotic mapping to initialize the population can enhance population diversity, thereby improving the optimization performance and global search capability of the algorithm. The Logistic chaotic mapping is mapped to the search space through Equations (11) and (12).

$$Y_{n+1}=\alpha \cdot Y_n\cdot (1-Y_n)$$

(11)

$$Z_n=a+(b-a)\cdot Y_n$$

(12)

where $\alpha \in [0,4]$ is the control parameter of the system. When $\alpha$ is within the range $[3.57,4]$, the system enters a chaotic state [30], and in this study, $\alpha$ is set to 4. $Y_n$ is a random number within the interval $[0,1]$; $Y_n$ represents the chaotic sequence generated by Equation (11); $Z_n$ denotes the chaotic sequence mapped to the search space; a and b are the lower and upper bounds of the search space, respectively.

Firstly, a chaotic sequence of length 30 (where the population size is 15 and the search space dimension is 2) is generated using Equation (11) and iterated accordingly. Next, the generated chaotic sequence is mapped to the search space with two-dimensional ranges of $[100,2500]$ and $[3,10]$ using Equation (12). Finally, the initial population is generated as described by Equation (13).

$$Y=\left[\begin{array}{cc}{y}_{1,1}& y_{1,2}\\ y_{2,1}& y_{2,2}\\ ⋮& ⋮\\ y_{15,1}& y_{15,2}\end{array}\right]$$

(13)

2.2.2. Linearly Decreasing Weight Factors

Due to the GTO algorithm’s inability to effectively control the iteration step size during the iterative process, once an optimal solution is identified, other individuals quickly converge towards this optimal solution, leading to premature convergence to a local optimum and losing the opportunity for further exploration of the global optimum. However, by employing the linear decreasing strategy for the convergence factor as shown in Equation (14), the balance between the exploration and exploitation behaviors of the GTO algorithm can be improved [31]. This approach enables the algorithm to comprehensively explore the search space while also performing fine-tuned exploitation near high-quality solutions, thereby enhancing the algorithm’s adaptability and robustness across different problems.

$$W(t)=W_{\max }-{\displaystyle \frac{W_{\max }-W_{\min }}{T_{\max }}}\times t$$

(14)

where $W_{\max }$ and $W_{\min }$ represent the initial and final weight factors, with values of 1.5 and 0.4, respectively. $t$ and $T_{\max }$ denote the current iteration number and the maximum number of iterations. The convergence factor starts with a higher weight in the early stages to maintain strong global search capabilities and linearly decreases as the iteration progresses, thereby enhancing the algorithm’s local exploitation ability.

2.2.3. Integrated Osprey Algorithm Global Exploration Strategy

In the original GTO algorithm, the second formula in Equation (1) of the first-stage exploration strategy contains numerous parameters, which leads to a slower information exchange among gorillas and subsequently affects the global search speed and convergence performance of the algorithm. To address this issue, the more efficient Osprey Optimization Algorithm is used to replace the second formula in Equation (1) during the first-stage global exploration strategy [32]. This introduces randomness into the global search process, thereby enhancing the convergence performance of the GTO algorithm. The global exploration strategy of the Osprey Optimization Algorithm is described by Equations (15)–(17).

$$x_{i,j}^{P1}=x_{i,j}+r_{i,j}\cdot (S{F}_{i,j}-I_{i,j}\cdot x_{i,j})$$

(15)

$$x_{i,j}^{P1}=\left\{\begin{array}{cc}{x}_{i,j}^{P1},& l{b}_j\le x_{i,j}^{P1}\le u{b}_j\\ l{b}_j,& x_{i,j}^{P1}<l{b}_j\\ u{b}_j,& x_{i,j}^{P1}>u{b}_j\end{array}\right.$$

(16)

$$X_i=\left\{\begin{array}{cc}{X}_i^{P1},& F_i^{P1}<F_i\\ X_i,& else\end{array}\right.$$

(17)

where $x_{i,j}^{P1}$ represents the new position information of the i-th osprey in the j-th dimension during the first stage; $S{F}_{i,j}$ denotes the selected fish for the i-th osprey in the j-th dimension; $r_{i,j}$ is a random number within the interval $\left[0,1\right]$; $I_{i,j}$ is a random number from the set $\left\{1,2\right\}$; and $u{b}_j$ and $l{b}_j$ denote the upper and lower bounds of the optimization search space, respectively. The updated formula for the gorilla position is as follows:

$$GX(t+1)=\left\{\begin{array}{ll}(UB-LB)\times r_1+LB& ,ran{d}_1<p\\ X_r(t)+r_2\times (X(t)-I\times X_r(t))& ,ran{d}_2\ge 0.5andran{d}_1\ge p\\ X(t)-L\times (L\times (X(t)-GX(t)))+r_3\times (X(t)-G{X}_r(t))& ,ran{d}_2<0.5andran{d}_1\ge p\end{array}\right.$$

(18)

2.2.4. Levy Flight Strategy

In the exploitation phase of the GTO algorithm, the population leader (the silverback gorilla) is responsible for guiding the population towards food sources. At this stage, the accuracy of the silverback gorilla’s exploration is particularly crucial. Therefore, incorporating the Levy flight strategy [33] into the gorilla position update formula (Equation (7)) during the GTO algorithm’s exploitation phase helps the algorithm utilize known optimal solutions while exploring new potential solutions. The Levy flight strategy employs step lengths with heavy-tailed distributions, as described by Equation (19). This strategy allows for both small and large jumps, where large jumps enable the algorithm to explore new areas of the search space and escape local optima, while small jumps facilitate detailed search in known high-quality regions. This multi-scale search capability enhances the algorithm’s flexibility in handling complex problems.

$$RL={\displaystyle \frac{u}{{\left|v\right|}^{\frac{1}{\lambda }}}}$$

(19)

where $\lambda$ is the power-law exponent of the Levy distribution, typically in a range of $[1,2]$ and is set to 1.5 in this study. $v~N(0,1)$, $u~N(0,{\sigma }_u^2)$, and the standard deviation of $u$ is calculated using Equation (20).

$${\sigma }_u={({\displaystyle \frac{\mathsf{\Gamma }(1+\beta )\cdot \sin (\frac{\pi \beta }{2})}{\mathsf{\Gamma }(\frac{1+\beta }{2})\cdot \beta \cdot 2^{\frac{\beta -1}{2}}}})}^{{\displaystyle \frac{1}{\beta }}}$$

(20)

where $\mathsf{\Gamma }$ is the gamma function.

The improved gorilla position update formula is given by Equation (21), as follows:

$$GX(t+1)=RL\times (X_{silverback}-X(t))+X_{silverback}$$

(21)

This paper improves the GTO algorithm based on the above strategies, balancing the algorithm’s exploration and exploitation capabilities and enhancing the convergence speed. The improved algorithm is used to optimize the parameters of VMD and SVM. The specific process of the improved GTO algorithm is shown in Figure 1.

2.3. Improved Algorithm Testing

To validate the effectiveness and superiority of the improved GTO algorithm, it is tested on the single-peaked function F6 and the multi-peaked function F12 from the CEC2005 benchmark function set. The results are compared with the Whale Optimization Algorithm (WOA), Dung beetle optimization algorithm (DBO), Grey Wolf Optimizer (GWO), Northern Goshawk Optimization (NGO), Harris Hawks Optimization (HHO), the original GTO algorithm, and the improved GTO algorithm (OLGTO). Functions F6 and F12 are defined by Equations (22) and (23), respectively.

$$F_6={\Sigma }_{i=1}^n{(|x_i+0.5|)}^2$$

(22)

$$\begin{array}{c}\begin{array}{ll}{F}_{12}(x)=& {\displaystyle \frac{\pi }{n}}\{10{\sin }^2(\pi y_1)+{\displaystyle \sum _{i=1}^{n-1}{(y_i-1)}^2}\\ & [1+10{\sin }^2(\pi y_{i+1})]+{(y_n-1)}^2\}\\ & +{\displaystyle \sum _{i=1}^{n}u(x_i,10,100,4)}\end{array}\\ y_i=1+{\displaystyle \frac{x_i+1}{4}}\\ u(x_i,a,k,m)=\left\{\begin{array}{cc}k(x_i-a),& x_i>a\\ 0,& -a<x_i<a\\ k{(-x_i-a)}^m,& x_i<a\end{array}\right.\end{array}$$

(23)

where the dimensions and optimal minimum values for the F6 and F12 functions are 30 and 0, respectively. The search ranges for these functions are [−100, 100] and [−50, 50], respectively.

The population size and maximum number of iterations for the seven algorithms are uniformly set to 100 and 1000, respectively. In the OLGTO and GTO algorithms, the exploration probability p and control parameter Beta are set to 0.03 and 3, respectively. The weight factor W for GTO is set to 0.5. OLGTO employs a linear decreasing strategy and Levy flight strategy, with parameters $W_{\max }=1.5$, $W_{\min }=0.4$, and $\lambda =1.5$. The convergence results of the tested algorithms are shown in Figure 2.

In the entire search space, unimodal functions have only one global optimum and are primarily used to evaluate the algorithm’s exploitation capability, i.e., its ability to converge to the optimal solution. Unimodal functions are relatively simple, allowing for clear measurement of convergence speed and accuracy. On the other hand, multimodal functions contain multiple local optima and are highly suitable for assessing the algorithm’s exploration capability. Complex multimodal functions can test the algorithm’s ability to avoid local optima and find the global optimum while evaluating the balance between exploration and exploitation. From the comparative results in Figure 2, it can be observed that for unimodal functions, the OLGTO algorithm consistently converges at a faster rate and ultimately achieves higher convergence accuracy than the other six algorithms. For multimodal functions, except for DBO and OLGTO, the WOA, GTO, GWO, NGO, and HHO algorithms all experience a significant decline in convergence speed after 200 iterations, falling into local optima and resulting in lower final convergence accuracy. Although the OLGTO algorithm shows a noticeable decrease in convergence speed around the 400th iteration, indicating a tendency to fall into local optima, it quickly resumes its convergence and reaches the optimal solution by the 579th iteration. This demonstrates the algorithm’s strong balance between exploration and exploitation and its ability to escape local optima. In summary, the OLGTO algorithm exhibits strong adaptability compared to other algorithms, effectively avoiding local optima. This proves the effectiveness and superiority of the improved algorithm.

3. Fault Diagnosis Model Construction

3.1. VMD Fault Feature Extraction

Variational Mode Decomposition (VMD) is an improved method of the traditional Empirical Mode Decomposition (EMD), suitable for processing nonlinear and non-stationary signals. The principle of VMD is to decompose the original signal into a set of $K$ intrinsic mode functions (IMFs), each with a central frequency. This decomposition is achieved through variational optimization. Additionally, VMD introduces Lagrange multipliers and a quadratic penalty factor $\alpha$, and uses the Alternating Direction Method of Multipliers (ADMM) to solve the variational optimization problem to obtain the various modes and their corresponding central frequencies. By appropriately setting the total number of modes $K$ and the value of the quadratic penalty factor $\alpha$, VMD can overcome issues encountered by EMD when processing complex signals, such as mode mixing and endpoint effects, thereby enhancing VMD’s adaptability to different signal decompositions. VMD is used for fault feature extraction in vibration signals, employing a composite index: permutation entropy/mutual information entropy as the fitness function. Permutation entropy reflects the complexity and irregularity of time series, with lower permutation entropy indicating more regular and periodic decomposed sequences [34]. Mutual information entropy measures the correlation between different information contained in different modes of VMD, with higher mutual information entropy indicating stronger correlation [35]. Therefore, the composite index of permutation entropy/mutual information entropy can balance the regularity and correlation of the decomposition modes. The calculation formulas for permutation entropy, mutual information entropy, permutation entropy/mutual information entropy, and the fitness value are shown in Equations (24)–(27), respectively.

$$PE=-{\displaystyle \sum \left(p(i)\ast \log \left(p\left(i\right)\right)\right)}$$

(24)

$$MI={\displaystyle \sum {\displaystyle \sum \left[P\left(X,Y\right)\ast \log \left(P\left(X,Y\right)/\left(P\left(X\right)\ast P\left(Y\right)\right)\right)\right]}}$$

(25)

$$CI={\displaystyle \frac{PE}{MI}}$$

(26)

$$fitness=\underset{1~K}{\min }\left\{CI\right\}$$

(27)

where $p(i)$ denotes the probability of the i-th permutation; $X$ and $Y$ are two random variables; $P(X,Y)$ represents the joint probability distribution of $X$ and $Y$; and $P(X)$ and $P(Y)$ denote the marginal probability distributions of $X$ and $Y$, respectively. When decomposing with VMD, the IMF component with the minimum fitness value is considered the current optimal IMF component. The mean, variance, peak value, kurtosis, root mean square, peak factor, impulse factor, waveform factor, and margin factor of the optimal IMF are then calculated, providing a 9-dimensional time-domain feature vector as the characteristic vector of the optimal IMF.

3.2. SVM Classification Model

Support Vector Machine is a flexible supervised learning model widely applied in fault diagnosis due to its high classification performance and ability to handle high-dimensional data. The core concept of SVM is to find an optimal separating hyperplane that divides samples of different classes. To achieve this, SVM solves the optimization problem as shown in Equation (28).

$$\begin{array}{ll}\underset{\omega ,b,\xi }{\min }& {\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\displaystyle \sum _{i=1}^N{\xi }_i},\\ s.t.& y_i(\omega \cdot x_i+b)\ge 1-{\xi }_i,i=1,2,\dots ,N,\\ & {\xi }_i\ge 0,i=1,2,\dots ,N\end{array}$$

(28)

where $\omega$ is the normal vector of the hyperplane, $b$ is the bias term, and ${\xi }_i$ are slack variables used to handle non-separable data. The regularization parameter $C$ controls the trade-off between training error and margin. A larger $C$ imposes a higher penalty on misclassification, potentially leading to overfitting, while a smaller $C$ allows for more misclassification, which may improve generalization ability [36].

Furthermore, SVM utilizes a kernel function to map data from a low-dimensional space to a high-dimensional space, making the sample data linearly separable in the higher-dimensional space. A commonly used kernel function is the Radial Basis Function (RBF) kernel, as shown in Equation (29).

$$K(x_i,y_i)=\exp \left(-\sigma {‖x_i-y_j‖}^2\right)$$

(29)

where $\sigma$ is the kernel parameter that determines the influence range of a single sample. Setting appropriate values for $C$ and $\sigma$ can enhance the model’s generalization ability, reduce the complexity of high-dimensional calculations, and thus improve the accuracy of fault diagnosis [37].

3.3. OLGTO-VMD-OLGTO-SVM Fault Diagnosis Model

To obtain suitable values for the number of modes $K$ in the Variational Mode Decomposition (VMD), the quadratic penalty factor $\alpha$, the regularization parameter $C$, and the kernel parameter $\sigma$, and to effectively extract vibration signal features to improve the accuracy of fault diagnosis models, this paper proposes the use of the OLGTO algorithm. The OLGTO algorithm iteratively selects the optimal values for the critical parameters of the VMD and SVM models, establishing the OLGTO-VMD-OLGTO-SVM fault diagnosis model. The overall design process of the model is shown in Figure 3, with the specific steps as follows:

Step 1: Initialize the settings for the gorilla population size, the maximum number of iterations, and other relevant parameters.

Step 2: Distribute the gorilla population randomly and uniformly in the search space using Logistic chaotic mapping. At this stage, the position vectors of the gorilla individuals represent the initial values of $\alpha$, $K$, $C$, and $\sigma$.

Step 3: Use the composite metric of permutation entropy/mutual information entropy as the fitness function to calculate the initial fitness values of the gorilla individuals and sort them in ascending order to determine the optimal fitness value and position.

Step 4: Update the optimal fitness value and position according to Formulas (18), (21), and (10) during the exploration and exploitation phases.

Step 5: Check if the current number of iterations exceeds the maximum number of iterations. If it does, the optimization process ends, and the optimal fitness value and position of the gorilla individuals, representing $\alpha$, $K$, $C$, and $\sigma$, are outputted. Otherwise, return to Step 2 and continue the optimization process.

Step 6: Substitute the optimal parameters $\alpha$ and $K$ into the VMD to extract features from the vibration signals. Divide the dataset into training and testing sets.

Step 7: Substitute the optimal parameters $C$ and $\sigma$ into the SVM to construct the fault diagnosis model and input the dataset to achieve fault diagnosis.

4. Fault Diagnosis Experiment Verification and Analysis

4.1. Data Preparation

In practical applications, it is necessary to consider the potential insufficiency of fault sample data. Therefore, fault diagnosis experiments utilize both the small-sample vibration data from the drive end of motor bearings from Case Western Reserve University and the large-sample vibration data from three-phase asynchronous motors to, respectively, verify the feasibility and superiority of the OLGTO-VMD-OLGTO-SVM fault diagnosis model.

The motor bearing fault diagnosis experiment at Case Western Reserve University (CWRU) involves a 2-horsepower (1.5 kW) motor, a torque sensor, a power meter, and an electronic controller. The test bearings, SKF6205, were subjected to single-point faults using electrical discharge machining, creating defect diameters of 0.1778 mm, 0.3556 mm, and 0.5334 mm. An accelerometer placed above the bearing housing on the drive end collected vibration acceleration signals at a sampling rate of 12 kHz. The experiment analyzed ten states: one normal operating condition (labeled as 1) at 1797 r/min and nine fault conditions (three each for inner race defects, rolling element defects, and outer race defects) corresponding to the different defect diameters (labeled as 2, 5, and 8 for inner race; 3, 6, and 9 for rolling element; and 4, 7, and 10 for outer race). Time series vibration data were segmented using a sliding window of 1000, with each sample containing 2048 data points. Each state type included 120 samples, resulting in a dataset of 1200 samples. Due to space limitations, only the first 1 s samples for the normal state and each defect type with a 0.1778 mm defect diameter are shown in Figure 4.

The data collection for the three-phase induction motor is illustrated in Figure 5. The experiment involves a 1-horsepower (750 W) three-phase induction motor, a piezoelectric accelerometer, and a single-stage gearbox. The motor model is YXVF0M-2, with an adjustable speed range of 0–5000 r/min. The accelerometer model is CK-EU-50B, with a measurement range of 50 g and a sampling frequency of 5 kHz. The gearbox is a single-stage gearbox with a module of 1.5 and a gear ratio of 75:25. This study examines five fault conditions: normal operating state (labeled as 1) at a speed of 1500 r/min, motor rotor imbalance fault (labeled as 2), motor rotor misalignment fault (labeled as 3), motor bearing fault (labeled as 4), motor warped rotor fault (labeled as 5), and motor rotor broken bar fault (labeled as 6). Data for these six states were processed using the same sliding window method, with each sample containing 2048 data points. A total of 2000 sample sets were collected for each state, resulting in a dataset of 12,000 samples. The sampled signals for each state during the first second are shown in Figure 6.

4.2. Motor Bearing Fault Diagnosis Experiment

To validate the feasibility of the OLGTO-VMD-OLGTO-SVM method, four sets of experiments were conducted. (1) The motor bearing vibration signals were directly input into the SVM model for diagnosis. (2) The motor bearing vibration signals were directly input into the OLGTO-SVM model for diagnosis. (3) The motor bearing vibration signals were input into the SVM model for diagnosis after feature extraction using OLGTO-VMD. (4) The motor bearing vibration signals were input into the OLGTO-SVM model for diagnosis after feature extraction using OLGTO-VMD.

For experiments (1) and (2), the raw data were processed into a multi-dimensional sequence matrix of $1200\times 2048$ and input as $1\times 2048$ one-dimensional sequences into the SVM and OLGTO-SVM models, respectively. In experiment (1), the parameters $C$ and $\sigma$ were set to 3 and 1.5, respectively.

For experiments (3) and (4), the OLGTO algorithm was first used to optimize the parameters $\alpha$ and $K$ in the VMD for feature extraction from the bearing vibration signals. The initial parameter settings are shown in

Table 1. OLGTO-VMD parameter settings.

Gorilla Population Size	Maximum Iterations	Optimization Parameters	Search Range
15	20	$\alpha$	$[100,2500]$
		$K$	$[3,10]$

. For each bearing state, 120 feature vectors were extracted (10 states in total, resulting in 1200 sets). The $1200\times 2048$ data samples were reduced to $1200\times 9$ feature vector samples after feature extraction. The OLGTO-VMD results are shown in

Table 2. OLGTO-VMD result (bearing).

Bearing Status	$\mathit{\alpha }$	$\mathit{K}$	Fitness Value	Optimal IMF Component
1	110	10	0.53519	IMF1
2	2104	10	0.72607	IMF2
3	110	9	0.70307	IMF1
4	2491	5	0.72564	IMF2
5	1989	7	0.75217	IMF1
6	1980	10	0.65067	IMF1
7	2271	10	0.66239	IMF1
8	2082	9	0.68290	IMF1
9	2372	10	0.68985	IMF1
10	2491	10	0.79858	IMF1

.

The iterative convergence curves are shown in Figure 7. Due to space limitations, only the convergence curves for the first four bearing states are displayed.

As shown in Table 2 and Figure 7, the total number of mode decompositions for each vibration signal is not less than 5, but the optimal IMF components are consistently IMF1 and IMF2. This indicates that the main characteristic components are fully extracted in the first few mode components after OLGTO-VMD. Additionally, for all 10 states, the optimal fitness values are achieved before the end of the iterations, demonstrating the algorithm’s effective optimization performance.

Secondly, for each bearing state, the first 80 sets were used as the training set, and the last 40 sets were used as the testing set. The parameters $C$, $\sigma$, and data input method for experiment (3) were the same as those in experiment (1). For experiment (4), the training set samples were first input into the OLGTO-SVM model for training. The initial parameter settings for the OLGTO-SVM model are shown in

Table 3. OLGTO-SVM parameter settings.

Gorilla Population Size	Maximum Iterations	Optimization Parameters	Search Range
10	30	$C$	$[{10}^{-3},{10}^3]$
		$\sigma$	$[{10}^{-8},{10}^3]$

. Through iteration, the optimal positions of the gorillas, representing the optimal parameters $C$ and $\sigma$ of the SVM, were obtained. The ultimate goal of fault diagnosis is to achieve accurate classification of different fault types. Therefore, the diagnostic accuracy as shown in Equation (30) was used as the fitness function. The results of OLGTO optimizing the SVM parameters are shown in

Table 4. OLGTO-SVM parameter optimization results (bearing).

Algorithm Model	Regularization Parameter $\mathit{C}$	Kernel Parameter $\mathit{\sigma }$	Experiment
OLGTO-SVM	471.6035	747.5808	(2)
	20.2627	42.4093	(4)

, and the fitness curves are shown in Figure 8.

$$Accuracy={\displaystyle \frac{N_{correct}}{N_{total}}}\times 100\%$$

(30)

As shown in Figure 8, the fitness values for experiments (2) and (4) reach their optimum at the 16th and 3rd iterations, respectively, and remain stable thereafter. This indicates that the algorithm performs well in optimization. The optimal parameters corresponding to the best fitness obtained through the iterative optimization of the OLGTO algorithm were used as the optimal parameters for the SVM model to test the sample data. The results of optimizing the SVM parameters are shown in Table 4.

In summary, the fault diagnosis results for experiments (1) through (4) are shown in

Table 5. Comparison of experimental results of motor bearing fault diagnosis.

Methods	Accuracy
SVM	29.5%
OLGTO-SVM	29.75%
OLGTO-VMD-SVM	94.5%
OLGTO-VMD-OLGTO-SVM	99.5%

.

The confusion matrices for the fault diagnosis results are shown in Figure 9.

From the experimental results above, it can be seen that the SVM model performs poorly in identifying vibration signals without feature extraction. Even when optimized using OLGTO, there is no significant improvement in the final results. However, after feature extraction using the OLGTO-VMD method, the SVM achieves high diagnostic accuracy. Specifically, the OLGTO-VMD-OLGTO-SVM model achieves a diagnostic accuracy of 99.5%, which is a five-percentage-point improvement over the 94.5% accuracy of the OLGTO-VMD-SVM model. This validates the feasibility of applying the OLGTO-VMD-OLGTO-SVM method in fault diagnosis.

4.3. Fault Diagnosis Experiment of Three-Phase Asynchronous Motor

To validate the superiority of the OLGTO-VMD-OLGTO-SVM method, this method was compared with five other methods: OLGTO-VMD-CNN-BiLSTM (Convolutional Neural Network-Bidirectional Long Short-Term Memory), OLGTO-VMD-KNN (K-Nearest Neighbors), OLGTO-VMD-RF (Random Forest), OLGTO-VMD-BP (Back Propagation Neural Network), and OLGTO-VMD-ELM (Extreme Learning Machine). These six methods were applied to the fault diagnosis of an actual three-phase asynchronous motor, and their performance was compared. The experimental process was as follows: First, the vibration signals were processed using OLGTO-VMD to extract signal features. Then, the first 1200 sets of each motor state were used as the training set, and the last 800 sets were used as the testing set. These sets were input into OLGTO-SVM, CNN-BiLSTM, KNN, RF, BP, and ELM for training and diagnosis, respectively. The parameter settings for the OLGTO-VMD feature extraction were the same as those in Table 1. For each state of the three-phase asynchronous motor, 2000 feature vectors were extracted (six states in total, resulting in 12,000 sets). The $\mathrm{12,000}\times 2048$ sample data were reduced to $\mathrm{12,000}\times 9$ feature vector samples after feature extraction. The convergence curves of the parameter optimization iterations are shown in Figure 10, and the decomposition results of the motor vibration signals are shown in

Table 6. OLGTO-VMD result (Electric motors).

Motor States	$\mathit{\alpha }$	$\mathit{K}$	Fitness Value	Optimal IMF Component
1	2054	10	0.65074	IMF1
2	1176	7	0.67622	IMF1
3	2319	10	0.65835	IMF1
4	144	8	0.93037	IMF1
5	955	10	0.61136	IMF1
6	1545	7	0.661504	IMF1

.

As shown by the decomposition results in Table 6, the total number of mode decompositions $K$ for the three-phase asynchronous motor vibration signals is not less than seven, but the optimal IMF components are consistently IMF1 and IMF2. This indicates that the primary characteristic information of the vibration signals is concentrated in the first few IMF components after OLGTO-VMD. Furthermore, as observed from the convergence curves in Figure 10, the optimal fitness value for each state is achieved before the iterations are completed, demonstrating the algorithm’s effective optimization performance.

After feature extraction, multi-dimensional feature vector samples were obtained. These samples were input as one-dimensional sequences into OLGTO-SVM, CNN-BiLSTM, KNN, RF, BP, and ELM for training and diagnosis. The initial parameter settings for the OLGTO-SVM model were the same as those in Table 3. Using diagnostic accuracy as the fitness function, the parameter optimization results are shown in Figure 10.

As shown in Figure 11, it can be observed that the fitness value reaches its optimum for the first time during the fifth iteration and remains stable thereafter with no fluctuations, indicating an overall stable trend. The optimal parameters $C$ and $\sigma$ obtained from the iterative optimization process of the OLGTO algorithm are then input into the SVM model to test the samples. The optimization results of the SVM model with the optimal parameters are shown in

Table 7. OLGTO-SVM parameter optimization results (electric motors).

Algorithm Model	Algorithm Model $\mathit{C}$	Algorithm Model $\mathit{\sigma }$
OLGTO-SVM	81.4547	588.5758

.

Additionally, the CNN-BiLSTM model was trained using the Adam optimizer with a maximum of 100 training epochs, an initial learning rate of 0.01, and a learning rate adjustment factor of 0.01 starting after 70 epochs. The regularization parameter was set to 0.001. The KNN model was constructed using the fitcknn function, with the number of nearest neighbors set to 1 and the distance metric type set to Euclidean. The RF model was constructed using the TreeBagger function with a minimum leaf size of 1, no out-of-bag predictions, no surrogate splits, and the number of decision trees set to 10. For the BP model, the optimal number of hidden layer nodes was set to 15. The hidden layer used the tansig activation function, the output layer used the purelin activation function, and the Levenberg–Marquardt algorithm was used as the training function. The maximum number of training iterations was set to 1000, with a learning rate of 0.1, a training goal error of 0.00001, a momentum factor of 0.01, a minimum gradient of 1 × 10⁻⁶, and a maximum validation failure count of 6. The ELM model was used in regression mode with a sine activation function and 100 hidden layer nodes. The diagnostic results of the six methods are shown in

Table 8. Comparison of experimental results of motor fault diagnosis.

Methods	Accuracy
OLGTO-VMD-KNN	92.6667%
OLGTO-VMD-CNN-BiLSTM	92.8958%
OLGTO-VMD-ELM	95.0833%
OLGTO-VMD-BP	98.125%
OLGTO-VMD-RF	98.4375%
OLGTO-VMD-OLGTO-SVM	98.6458%

.

The confusion matrices for the diagnostic results of the six methods are shown in Figure 12.

From the comparison results above, it can be seen that despite the large data sample size, the OLGTO-VMD-OLGTO-SVM method still achieves a diagnostic accuracy of 98.6458%. This represents a significant improvement compared to the accuracies of 92.6667%, 92.8958%, and 95.0833% achieved by the OLGTO-VMD-KNN, OLGTO-VMD-CNN-BiLSTM, and OLGTO-VMD-ELM methods, respectively. While the improvements over OLGTO-VMD-BP and OLGTO-VMD-RF, which achieved 98.125% and 98.4375% accuracy, respectively, are smaller, it is important to note that a 0.3~0.5% increase in accuracy still represents a considerable number of data samples given the large sample size. Therefore, the method proposed in this paper is more reliable for practical applications, demonstrating the superiority of the OLGTO-VMD-OLGTO-SVM method.

5. Conclusions

(1) To address the deficiency of the Artificial Gorilla Troops Optimization algorithm in the early iteration stages, where the numerous formula parameters hinder timely information exchange between gorillas, reducing the algorithm’s global search capability and ability to escape local optima, and thus degrading convergence accuracy, improvements were made using Logistic chaotic mapping, a linear decreasing weight factor strategy, the global exploration strategy of the Osprey Optimization Algorithm, and the Levy flight strategy. Tests on unimodal and multimodal functions showed that the improved GTO algorithm possesses excellent global search capabilities and effectively avoids local optima.

(2) The selection of parameters $\alpha$ and $K$ in VMD determines the effectiveness of vibration signal decomposition and feature extraction, while the parameters $C$ and $\sigma$ in SVM determine the final diagnostic accuracy. The improved GTO algorithm was used to optimize the parameters of both VMD and SVM, overcoming the difficulty of manually selecting optimal parameters and enhancing the generalization ability and adaptability of the fault diagnosis model.

(3) Fault diagnosis experiments on motor bearings and three-phase asynchronous motors validated the feasibility and superiority of the OLGTO-VMD-OLGTO-SVM method. This method can accurately identify fault types for both large and small sample data.