2.1. Analysis of Local Faults in the Centre Gear

Serial-type industrial robot joint servo systems are primarily composed of permanent magnet synchronous motors (PMSM), RV reducers, and mechanical arms, as shown in Figure 1. In the figure, T_em and T_l represent the servo motor output torque and load torque, respectively. The following text will use the RV-20E reducer as an example to establish an electromechanical coupling model under the local fault mode of the centre gear.

In this case, the selected planetary gear transmission has a meshing factor greater than 1, leading to both single-tooth meshing and double-teeth meshing during actual transmission. This results in a periodic variation of meshing stiffness with rotation angle, resembling a square wave. When the localized fault occurs in any gear transmission, it significantly impacts the meshing stiffness, as depicted in Figure 3. Each time the faulty gear engages in meshing, torque fluctuation occurs, causing the transmission error e to become uncertain, resulting in corresponding pulse signals in the input torque.

2.2. Principle of Operation of Servo Motors

The PMSM is utilized as the power source for the test setup in the paper, where the amplitude, frequency, and phase of the torque current are adjustable. By varying the frequency of the stator winding current, the motor rotor can be accelerated or decelerated to balance power fluctuations caused by changes in the load, thereby reducing disturbances to the system. The physical model of the PMSM is illustrated in Figure 4, where θ_m represents the rotor mechanical angular displacement (output angle). In motor control, the electrical displacement angle θ_r is commonly used for calculations, where θ_m and θ_r satisfy θ_m = θ_r/P, with P denoting the number of motor pole pairs.

Where L_d represents the d-axis inductance, L_q represents the q-axis inductance, Φ_f represents the magnetic flux generated by the rotor excitation poles.

The servo motor output torque is related to angular velocity, angular acceleration, and load torque, and it is very challenging to simulate the current signals under each localized fault condition during actual operation. Therefore, when using current signals to monitor the operating conditions of the RV reducer, a deeper analysis of its drive chain system is required.

2.3. Centre Gear Local Fault–Coupling Model

3.1. Testing Platform

Taking IR Joint 3 as an example, the work is similar to a rod rotating around the end point, and the 4–6 axes are viewed as a whole M for force analysis and found that the RV reducer is subjected to time-varying torque. To approximate the torque experienced by the reducer to the actual situation, a load is added to the end effector, and the fluctuation of torque during motion is ignored. Thus, the reducer only experiences the gravitational force G_a = M_g from Axes 4–6, as shown in the force analysis in Figure 5. Here, O represents the centroid of the simplified model, F_c is the centrifugal force exerted on the RV reducer, F_g is its own weight, and the tangential stress generated by the load is F_t = T_l/L_a, where g and L_a represent the gravitational acceleration (9.8 m/s²), the distance from the centroid to the centre of the reducer when in the horizontal position, respectively.

To enable rapid and precise monitoring of the RV reducer with high reliability and the ability to conduct comprehensive performance evaluations even under complex conditions, a magnetic powder brake is utilized to simulate the dynamic torque variations depicted in Figure 6. Corresponding loading experiments are conducted to acquire experimental data under simulated real-world conditions. Firstly, according to the experimental requirements, a servo system with a 23-bit absolute encoder and high-speed control accuracy is selected as the input device. In order to better simulate the load torque, a magnetic powder brake with a controllable loading torque and a maximum torque of 400 N·m is used. Some key parameters of the test platform device are shown in Table 1.

In addition to the measurement device mentioned above, this testing platform also requires some auxiliary equipment, such as a workbench, elastic couplings, and sensor support fixtures. According to research requirements, an open-loop control structure is adopted to build the corresponding testing platform. The installation positions and distribution of each device are shown in Figure 7.

3.2. Fault Setting and Data Acquisition

In this experiment, localized faults only in the centre gear are considered. The different sizes of defects are introduced on one of the teeth, as shown in Figure 8, which simulate common gear faults such as pitting, spalling, and tooth breakage, respectively. Among these, the severity of pitting fault is lower than that of spalling, while tooth breakage represents the most severe fault mode. As the operational state of the reducer is significantly affected by the rotational speed, the theoretical output speeds of the servo motors correspond to the output speeds ω_rv of the RV reducer. The magnetic powder brake is used to simulate the corresponding load conditions, and three loading experiments are carried out in each mode. A total of 12 sets of experimental data are obtained.

After the experimental validation, to avoid excessively high sampling frequencies, which could compromise the quality of low-frequency signals, the sampling frequency is set to 1 kHz following the Nyquist sampling theorem. And get real-time feedback on the current signal of the servo motor by upper computer software. The collected data is regarded as a sample with 4096 data points, and 200 samples are extracted from each group. A total of 819,200 data points are used. The actual current signal (a certain sample sequence) is shown in Figure 9.

Comparing the above time-domain signals, it is found that the trends of the current signals are cyclic with the rotational speed and attitude of the IR joints in Figure 9. The relative currents increase when the load torque becomes larger and do not change significantly for the same operating mode and different rotational speeds. Except for the occurrence of more pulses in the tooth breakage fault mode, the time-domain signals in the pitting and spalling conditions are similar to those in the healthy state, making it difficult to classify the severity of the faults.

3.3. Fault Coupling Model Experimental Verification

Combining with the working principle of the servo motor, the theoretical output speed n₁ = 60f/P is obtained, where f is the pulse frequency output by the servo driver and P is the number of pole pairs of the motor. According to the transmission principle of the RV reducer [33, 34], when the needle gear housing is fixed, the working frequency of the key components of the RV reducer and the meshing frequency of the two-stage transmission can be obtained. The specific calculation formulas and related parameter attributes are shown in Table 2.

Then, the theoretical formula in Table 2 is used to calculate the RV reducer’s corresponding rotation frequency and meshing frequency at three speeds. The specific theoretical calculation values are shown in Table 3.

Finally, we perform the fast Fourier transform analysis on the current signals in Figure 9 to obtain the spectrum plot as shown in Figure 10.

In Figure 10, the corresponding rotation frequency and meshing frequency in Table 2 are marked, where labels (a–d) correspond to the previous text. Combining the theoretical calculation results from Table 3 with the spectrum results in Figure 9, it can be observed that under different operating conditions, the fundamental frequency of the servo motor (f_e), the rotation frequencies of the RV reducer (f₁, f₂, f₃), and the meshing frequencies (f_sp, f_cs) can be effectively extracted. In addition to these frequencies, significant amplitudes are also observed at f_e ± f₁, |f_e ± f_sp|, and |f_e ± f_cs| frequencies (where f₂ and f₃ values are too small, and f_e ± f₂, f_e ± f₃ amplitudes are not significant). From the perspective of experimental analysis, the correctness of the electromechanical coupling model in the case of local failure of the centre gear is verified. At the same time, it is proved that the servo feedback current signal can be used for local fault diagnosis of the RV reducer. Due to the dead zone effect in the servo motor drive system, there will be significant amplitudes near 5 times the motor fundamental frequency. Comparing the spectrum plots of the four states of the centre gear, it is observed that apart from the noticeable continuous f1 sidebands in the case of tooth breakage faults, sidebands are not significant in the case of pitting and spalling fault modes, making it difficult to assess the severity of the faults.

4.1. Based on Time-Domain Statistical Features

To better illustrate the impact of local faults in the RV reducer on the servo feedback current signal, 10 statistical features t1 − t10 are extracted from the current signal, as shown in Table 4.

Using the calculation formulas in Table 4 to determine the characteristics of the data in Figure 9, obtain the time-domain features under three different rotational speed conditions as shown in Figure 11.

Based on the distribution of time-domain features in Figure 12, it can be observed that the trend of feature changes is roughly similar under different rotational speed conditions. The distinct differences in features t1 − t5 are advantageous for classifying local faults in the RV reducer, while features t6 − t10 are less affected by faults. However, due to the interference of the RV reducer’s fluctuation factors, there is an overlapping phenomenon in significantly changing features. Next, the t-SNE algorithm is used to nonlinearly map high-dimensional data to a representational space to achieve feature dimension reduction. This allows for better visualization and analysis of sample features, reducing information redundancy between high-dimensional feature vectors and improving classification efficiency and accuracy. The feature data obtained after dimension reduction are assigned to the same quantified space, resulting in a visualization analysis of time-domain features, as shown in Figure 13.

From the dimensionality reduction results in Figure 13, it can be observed that time-domain features are most sensitive to gear tooth faults, but their effectiveness in recognizing the other three fault modes is poor, especially at a speed of 30°/s where the classification performance is the worst. A comparison of the three classification results indicates that the research on local fault diagnosis of RV reducers using current signal time-domain features poses a challenge.

4.2. Based on Time–Frequency Domain Features

From the spectral analysis results in Figure 10, it can be observed that frequency-domain signals are not conducive to achieving multimode recognition of RV reducers. And it is not introduced repeatedly in this section. Next, the time–frequency domain statistical features of the above signals are extracted for RV reducer fault diagnosis and identification. Wavelet packet decomposition (WPD) can effectively overcome the key difficulty of indistinct representation of current signals and spectral aliasing. It also enhances the accuracy of classification prediction. WPD maps one-dimensional time-domain current signals to two-dimensional time–frequency domain signals. It decomposes the signal into high-frequency and low-frequency components, effectively improving time–frequency resolution while extracting more crucial feature information. The structure of WPD is illustrated in Figure 14.

The experimental results show that optimal results can be obtained when the number of decomposition layers is 4. The energy values of 16 sub-bands from high frequency to low frequency in the fourth layer are extracted as the eigenvalues needed in this paper. Then, the “energy-fault diagnosis” method is established. The time–frequency domain features of the data in Figure 9 are calculated, and the time–frequency domain characteristics under three different speed conditions are obtained, as shown in Figure 15 (disregarding interference from high-frequency components, taking w1 − w14 as 14 feature values).

From the distribution of time–frequency domain features in Figure 15, it can be observed that most features can effectively distinguish between the four operating modes. Particularly, there is significant discrimination between the healthy and tooth fault modes, while the discrimination is less pronounced for the pitting and spalling modes, which hinders fault diagnosis. The visualization analysis of the time–frequency domain features after dimensionality reduction is shown in Figure 16. The dimensionality reduction results correspond to the feature distribution, effectively classifying between the healthy and tooth fault modes but showing poor classification performance for the pitting and spalling faults. Even under a speed condition of 60°/s, all four operating modes can be effectively distinguished, although the discrimination between pitting and spalling faults is relatively close.

4.3. Based on CNN Features

After analyzing the time domain, frequency domain, and time–frequency domain features above, it is evident that employing simple feature extraction methods leads to results that are not sufficiently clear or accurate. With the advancement of computer technology, DL has been widely applied in the extraction of state features and classification prediction of various industrial equipment [35]. Next, CNNs will be utilized to learn and extract deeper-level features from the local relationships within the data.

CNN is a deep feedforward neural network with the characteristics of parameter sharing and local perception. CNN is generally composed of the input layer, convolution layer, activation layer, pooling layer, and fully connected layer. The specific network structure is shown in Table 5.

The initial learning rate of the learning model is set to 0.005, and it decreases by 50% every 10 iterations, for a total of 30 iterations. The data extracted from the fully connected layer is considered as the sample features, and the visualization results obtained after dimensionality reduction using t-SNE are shown in Figure 17.

From the distribution results in Figure 17, it can be observed that after CNNs feature extraction and visualization analysis, the sample data under different fault modes demonstrate good recognition performance for minor faults such as pitting and spalling across various speed conditions. As the severity of faults increases, the feature information of the sample data tends to diverge. Particularly in Figure 17(c), the dispersion is significant for the tooth fault mode. This divergence may affect further classification prediction, indicating that when RV reducers experience severe faults, the collected information becomes disordered and uncontrollable.

4.4. Comparative Analysis of Prediction Results

Generally, classification accuracy is used as an assessment standard of the methods. However, in order to explore better feature extraction methods, a confusion matrix is introduced in the paper. The relationship between the true category of the samples and the classification result is described by the confusion matrix to present the assessment standard of model performance. The confusion matrix is shown in Table 6.

The extracted sample features are then classified using SVM. The features are labeled according to their corresponding failure modes and the feature matrix is divided into a training set and a test set, where the training set is used for model training and the test set is used for model accuracy testing. The training set and test set are 70% and 30%, respectively, and the computation is repeated 10 times, respectively. The resulting confusion matrix under three different features is shown in Figure 18.

In Figure 18, the horizontal axis represents the predicted labels for different modes, while the vertical axis represents the true labels for fault modes. Figures 18(a), 18(b), and 18(c) represent the classification prediction results for three different speeds under time-domain features, and the highest accuracy rate is only 83.75%. Figures 18(d), 18(e), and 18(f) represent the classification prediction results for three different speeds under time–frequency domain features, with an overall accuracy rate higher than 82.5%. In this feature mode, there is potential for local fault diagnosis of RV reducers. Figures 18(g), 18(h), and 18(i) represent the classification prediction results for three different speeds under CNN features, with an average accuracy rate higher than 96%.

The results of the distribution of the confusion matrix in Figure 18 were solved to obtain the mean values of Pre, Rec, and F1 for three different feature extraction methods, as shown in Figure 19.

In Figure 19, these performance metrics Pre, Rec, and F1 perform worst at low rotational speed using time domain statistical features for classification. And it can be seen that rotational speed has a certain influence on the assessment indicators from label (d). Under CNN features, with an average Pre, Rec, and F1 rate higher than 96%. As can be seen from the distribution of results in Figures 18 and 19, using CNN features or other complex feature extraction methods holds the most promising application prospects for local fault diagnosis of RV reducers.