Real-time milling force monitoring based on a parallel deep learning model with dual-channel vibration fusion

Milling force is one of the most important aspects of milling. Its dynamic excitation effect significantly impacts both product quality and machining productivity. Nevertheless, the force amplitude changes dramatically when the tool and the workpiece begin to contact or separate. Most current research does not consider this phenomenon. This article presents a parallel integration deep learning approach to address the issue. First, this study analyzes the relationship between milling force and vibration signals and sets the dual-channel vibration signals in the same direction as the model’s inputs. Then, this study proposed an encoder-decoder network to realize force monitoring. Considering that the acquired vibration signal contains much noise and needs to be preprocessed, the encoder comprises long-short term memory (LSTM) networks and a fully connected (FC) network to realize adaptive filtering and feature extraction. Multiple-layer FC network forms the decoder part to reconstruct the milling force signal because of the nonlinear relationship between the vibration and force signals. The third is to obtain the parallel monitoring model. The first monitoring model is obtained through the training procedure. The results of the first model are subtracted from the measured cutting force signal to get the residual part. Then, the residual part is set as the output while training the residual monitoring model. Finally, the force monitoring model is derived using the parallel integration method. The experimental results demonstrate that this study’s monitoring model can provide real-time, high-precision, and reliable milling force monitoring under various cutting conditions.


Introduction
The milling operation is an intermittent cutting process using a rotating cutter with multiple teeth [1]. It has been widely used to produce simple to complex components in various materials with high productivity and quality. It has become essential to the automobile, aerospace, and die-mold industries. Cutting force is one of the most critical factors in the milling process. When tool breakage [2], wear [3], chatter [4], or tool-workpiece deflection [5] occurs in the milling process, the cutting force will be changed in amplitude and even frequency. The force amplitude also changes dramatically at the stages when the tool and the workpiece start to contact or move away. Still, most existing research does not consider this phenomenon. Therefore, an accurate real-time milling force monitoring model is essential to monitor the accurate machining status, optimize the cutting parameters, improve the machining productivity and efficiency, and reduce the damage to cutting tools and workpieces due to emergencies.
Since 1941, researchers have researched in-depth to develop accurate and reliable cutting force models for milling operations [6]. With the understanding of the milling process, scholars proceeded from different perspectives and applied various methods to conduct cutting force modeling. At present, these models can be broadly divided into four categories: experimental models [7,8], mechanics-based analytical force models [5,9,10], mechanistic force models [11], and data-driven models [12].
As research goes deeper, more and more variables need to be considered for the experimental models, and more experiments are required. As a result, these models need to consume a lot of time and energy, leading to their unpopularity. Mechanics-based analytical force models require prior knowledge, such as shear, chip flow, and mean friction angles. At the same time, the models also require a lot of experiments with various combinations of work material and cutter geometry. However, these parameters limit its applicability. By applying finite element modeling, mechanistic models predict cutting force in discrete increments. It determines the uncut chip area for each element geometrically and correlates with cutting force components through mechanistic constants. These constants mainly contain workpiece material properties [13], cutting tool material, and geometry [14]. These methods have been the focus of research in recent decades. They make cutting force monitoring more accurate by considering different characteristics, such as size effects [15], cutter run-out [16] and vibration [17], tool posture [18], and flexible workpieces [11]. Recently, a model [19] combining a mechanics-based analytical force model and a mechanism model has also appeared to predict cutting force.
There are broadly two categories in the data-driven model for cutting force monitoring. One is establishing the cutting force model based on adaptive filters, such as Kalman filtering. For example, Postel et al. [12] estimated the cutting force model based on the vibration signal by a Kalman filter. Ref. [20] estimated the Kalman filter model to predict cutting force through a fusion of accelerometer and spindle current signals. Another major category is the neural network models. As mentioned in experimental models, too many parameters must be considered for cutting force monitoring, which requires much time for manual analysis. However, the neural network models can extract these features without much work. Additionally, the neural network model can capture the nonlinear relationship effectively, which is essential for cutting force monitoring.
The neural network models are widely applied in cutting force monitoring and the fields of monitoring tool wear, chatter monitoring, and cutting parameters optimization. In cutting force monitoring, the primary way of using neural networks is to make the cutting parameters, such as spindle rotating speed, feed rate, depth of cut as inputs, and the mean value of cutting force as output. In ref. [21], researchers developed an empirical relationship between the cutting force in end milling operation and the cutting parameters using multiple regression and neural network modeling processes. The results showed that the neural network model is more accurate than the traditional multiple regression model. Ref. [22] compared three machine learning methods in cutting force, surface roughness prediction, and optimization of cutting parameters in the turning process. The results indicated that these models have different advantages and disadvantages. Monitoring cutting force mean values based on cutting parameters still has problems with accuracy and reliability. For example, the mean value does not contain the frequency information, leading to inconvenience for further analysis. To improve the performance of the datadriven models, researchers combined the mechanistic and data-driven models to realize more accurate cutting force monitoring. In ref. [23], the cutting force generated from the mechanistic model is set as an input of the neural network model to improve the force prediction accuracy. In ref. [24], the researchers combined the transfer learning model with the mechanism model to reduce the error of cutting force monitoring.
The models mentioned above also have some limitations in application. Most of these models are used for machining process simulation and cutting parameters optimization but not for real-time cutting force monitoring. Too many factors that appear during the actual milling process that cannot be considered in the simulation can affect the milling process, thus affecting the cutting force. Indirect monitoring methods based on signals generated during the machining process and neural network models are more reasonable for achieving real-time milling force monitoring in the machining process. Teti et al. [25] reviewed sensors and sensing techniques applied for machining process monitoring. The results showed that it might not be a good choice to apply a non-contact displacement sensor [26] or a laser sensor [27] to realize indirect force monitoring. It is desirable to have sensors on the stationary machine tool body. In ref. [28], researchers used the accelerometer to acquire the vibration signal and calculate the force signal by the transfer function. Mostaghimi et al. [20] discussed the vibration and spindle current signals in reconstructing the cutting force signals. According to Liu's research [29], vibration signals acquired at different measurement points also differ.
After the signal acquisition, the most important work is the neural network construction. Various neural networks have been applied to the condition monitoring of the machining process. For example, Thenozhi and Tang [30] implemented the monitoring of cutting forces using SVM and RBF-NN. Among different neural network structures, LSTM [31] can remember the information contained in the signal over a long period and outperform other neural networks in handling temporal information. Thus, it is widely used in time-series tasks such as speech recognition [32], acoustic modeling [33], and sentence embedding [34]. In the field of condition monitoring, LSTM has also been widely applied. Shi et al. [35] realized chatter monitoring during machining based on ON-LSTM, and Sun et al. [36] constructed a realtime chatter monitoring system using the inception network and LSTM network. In terms of cutting force monitoring, Peng et al. [37] combined LSTM with convolution neural networks to accurately predict cutting force under some predefined cutting conditions based on spindle current signal. Denkena et al. [38] used drive signals to predict cutting force in x-, y-, and z-directions through LSTM neural networks and compared the results with a model-based approach.

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The results showed that the LSTM neural network predicted with higher accuracy than the conventional model-based approach. With the development of the machine learning discipline, a structure called encoder-decoder [39] has been proposed and widely used to solve the sequence-to-sequence problem with good results. The encoder part can be seen as the essence of multiple signal filters in series, which perform operations such as dimensionality reduction and filtering on the input signal to extract the useful features from the input signal. The decoder structure then reconstructs the useful features to get the output signal. Compared with the traditional deep learning models, it is more closely matched to the actual process and thus is widely used to solve problems in natural processes. Schlagenhauf et al. [40] applied the encoder-decoder method to anomaly detection in machining processes. Guo et al. [41] combined LSTM with an encoderdecoder structure for tool wear monitoring. Dun et al. [42] realized chatter detection with the encoder-decoder network. Qu et al. [43] implemented the monitoring of cutting force in the cutting process in the frequency domain using the vibration signal through the encoder-decoder.
In summary, studies have been conducted to apply datadriven approaches to cutting force monitoring. These studies have given us a great deal of inspiration and guidance in many aspects, such as the selection of signals, modeling of cutting force monitoring, and the performance of different algorithms. At the same time, these previous works can still be improved to solve more novel problems, such as reducing the input required by the model, improving the monitoring accuracy of the model, and considering more cutting conditions and stages. Therefore, this study uses the vibration signal as the model input, and the encoder-decoder model and the LSTM neural network are combined to establish the force monitoring model. The validation results show that the method proposed in this paper accurately monitors cutting force under variable cutting conditions. This research aims to realize real-time milling force monitoring based on vibration signals. In this study, to ensure monitoring accuracy, two accelerometers are mounted in the x-and y-directions on the spindle to acquire the vibration signals generated during the milling process. Firstly, this study analyzes the relationship between cutting force and vibration signals in different directions. The results show the highest correlation between the cutting force and vibration signals in the same direction. So signals in the same direction will be set as the model's inputs. Secondly is to design the encoder-decoder neural network architecture. As existing research pointed out, the vibration signal acquired by the accelerometers contains the features needed for cutting force monitoring and noise and disturbance. During the cutting process, the cutting force generates vibration at the tooltip, and then the vibration is transmitted to the sensor through the spindle structure. From this process, it can be seen that there is a temporal correlation between the cutting force and the vibration signal so that the LSTM neural network can extract the signal features. However, the vibration transmission process is very complex, and it is difficult to realize feature extraction just by a single layer of LSTM. Therefore, in this study, a two-layer LSTM with different hidden units combined with a one-layer fully connected (FC) neural network constructs the encoder part to perform adaptive filtering and feature extraction of the acquired vibration signal. Usually, the decoder structure is the same as the encoder structure. However, in a stable cutting process, the vibration signal is nonlinear, while the cutting force signal is linear [44]. There is a nonlinear relationship between the two kinds of signals [45]. The input and output signal types differ, so this study's cutting force monitoring problem differs from the traditional series-to-series prediction. The role of the decoder network here is to reconstruct the nonlinear features extracted by the encoder into the linear cutting force signal. To achieve this goal, the decoder structure in this study consists of a multilayer fully connected neural network. Then, researchers divide the force signal into the main and residual parts to realize accurate force monitoring from the extracted features. This study considers the cutting process when the tool starts to contact and separate from the workpiece. When the tool and workpiece start to contact and leave, the cutting force will change significantly, manifested as a sudden increase or decrease of cutting force, which can easily cause the monitoring system to misjudge the machining status, such as tool wear and breakage. Therefore, this part of the changing force needs to be monitored. In fact, in this cutting stage, the cutting process contains both forced and free vibrations due to the absence of constraints. Therefore, two force-vibration systems exist in this phase, so two models are needed to realize cutting force monitoring. The main model outputs the cutting force signal related to forced vibration, and the residual model outputs the cutting force related to free vibration. In this research, we first train the main part model with the data when the tool and workpiece are fully in contact, which is the first model mentioned in the abstract. Then, when training the residual model, we obtain the remainder data by subtracting the cutting force monitoring results obtained from the main part model from the original force signal. The training procedures of the two models are the same. Finally, the two models are connected in parallel to realize the cutting force monitoring, as shown in Fig. 4. The verification results show that the model proposed in this study can realize real-time cutting force monitoring with high accuracy and stability.
The article is organized as follows: "Methodology" describes the theoretical background between cutting force and vibration signals. "Parallel deep learning model" provides the encoder-decoder network's architecture, and the parallel model's structure. "Experiment setup" provides the platform, cutting parameters, and signal acquisition system. "Verification and result discussion" shows the parameters setting process, discusses the relationship between the force and vibration signal, and analyzes the performance of the proposed method. "Conclusion" concludes the paper at last. Figure 1 shows the impact test to get the cross-frequency response function (FRF) between the tooltip and the measurement point on the spindle. The function can be simplified as follows:

The relationship between cutting force and vibration on the spindle housing
where Acc is the response signal acquired by the accelerometer and F is the excitation force added to the tooltip by the hammer. This function shows the causal relationship between the tooltip force and the vibration at the spindle. Based on this relationship, the neural network can predict the cutting force signal with the vibration signal as input. As existing research on modal analysis [29] shows, an accelerometer in one place can hardly acquire all the information about cutting force. So, in this study, two accelerometers will be used to ensure the model's monitoring accuracy, as shown in Fig. 5a. Also, according to ref. [46], the relationship between cutting force and vibration is demonstrated as the following equation: In Eq. (2), X and Y represent the vibration in the x-and y-directions. FRF xx and FRF yy are the cross-frequency response functions (FRF) representing excitation and measurement in the same direction. FRF xy and FRF yx are the cross-FRFs representing the measurement in one direction with excitation in another. FRF xy and FRF yx are equal to zero because the x-and y-directions are orthogonal, and zero crosstalk between the two directions is assumed. From Eq. (2), it can be seen that many research works assume that the cutting forces do not have a causal relationship with vibrations in different directions. The study will do an impact test experiment to verify this theory in the verification procedure.

Parallel deep learning model
Traditional neural networks cannot connect previous information to the present task. However, RNNs (recurrent neural networks) can address this issue. They are networks with loops in them, allowing information to persist. LSTM is a kind of RNN that can deal with the signal which has a relatively long period. Although the cutting force and vibration signal are time-varying in the milling process, LSTM can remember information for long periods to meet the demands of dealing with cutting force signals and vibration signals. This advantage makes it suitable for force monitoring.
The LSTM uses chains to deliver the previous information to the present, from one cell to the next. The cell structure contains four gates: forget gate, input gate i t , output gate o t , and cell state C t . The key to LSTM is the cell state. By adding or deleting information to the cell state, the LSTM can save valuable information from the previous signal, thus realizing the filtering function in the encoder part. Suppose a signal sequence is given as where x t represents the input at the current time t, h t-1 represents the output of the previous cell, and C t-1 is the previous cell state. C t is the current cell state, and h t is the output of the current cell.
The forget gate is composed by a single-layer neural network. It is used to decide what information the current cell needs to forget. The input is h t-1 and x t , and it is calculated as follows: The function is written as follows: where f t is the output of forget gate, W f is the weight, and B f is the bias of the layer.
The input gate has two parts. The process can be written as follows: The function tanh is written by Combined with the calculated results in the forget gate, the C t will be updated as follows: The final step is to calculate the output h t . It is calculated as follows: After h t and C t are calculated, they will be sent to the next cell. The above theory is the mechanism of LSTM.

Encoder-decoder cutting force monitoring model
This study proposed a novel encoder-decoder model to realize force monitoring. The whole encoder-decoder model is shown in Fig. 2.
As shown in Fig. 2, the encoder contains an input layer for inputting the vibration signal, a two-layer LSTM neural network to filter the signal adaptively, and a one-layer FC neural network to extract features for the decoder part. In this study, the sequence size of the input signal is N = 512 . In Fig. 2, D is the number of hidden units, which means that every single cell of LSTM, for instance, the neural network in one cell, also has D hidden layers. In the first LSTM layer, D = N 1 = 200, then, the output size of the first LSTM layer O LSTM1 is [512 × 200] ; in the LSTM layer2, D = N 2 = 120 , the size of the output O LSTM1 is [512 × 120] . C is the number of features. In this study, the input vibration signals are acquired by two accelerometers, so C = 2 .
S represents the size of time steps. It is different from the size of the input signal. Here, the S = 128.
Then, the data will be sent to the FC1 layer. The num fc1 in this layer represents the number of features extracted from the vibration signal. It has an essential impact on the accuracy of the model's output. So, in this study, we will apply RMSE to investigate the effect of num fc1 on the monitoring results.
The number of hidden units of this layer is N 3 . In this study, N 3 = N 2 . The relationship between the LSTM layer2 and FC1 layer is shown as follows: where, In which O f1 is the output of the FC1 layer, its size is [200 × 120] . In this study, W f1 is the weight of the FC1 layer, and the size is [200 × 512] . B f1 is the bias, and the size is To avoid overfitting, the dropout operation will be applied before sending the features to the decoder [47]. This operation will randomly omit the units in the neural network with a certain probability. According to ref.
[47], the FC neural network will achieve the best performance when the probability is 50%. So, in this study, the dropout probability is set as 0.5.
In the decoder part, taking the size of the cutting force signal into consideration, the neuron number of each FC neural network is set as 512. The output of the first layer can be calculated as follows: In which ŷ is the layer's output, its size is [512 × 1] , W f2 is the weight parameter, and its size is [512 × 200] , B f21 is the bias, and the size is [512 × 1] . When the training process is completed, the omitted neurons will be added to the neural network again. To keep the monitoring result the same, W f2 needs to multiply 1∕2 , as follows: The data flow between hidden layers in the FC2 layer is calculated as follows. Suppose the previous layer is the kth layer, the current layer is the lth layer, then (14) W f inish In the decoder part, the number of hidden layers in the FC2 layer represents whether the relationship between vibration and force is linear or nonlinear. When the number of FC2 layers is two or more, it indicates a nonlinear (15) relationship between input and output; when the number of FC2 layers is 1, it means a linear relationship between input and output. In this study, the best number of hidden layers N hl will be evaluated by the monitoring accuracy, and the relationship between vibration and force signal will be found. Finally, the regression layer outputs the predicted cutting force signal and returns the mean square error (MSE), which is used in the backpropagation calculation of the neural network. The formula for calculating the mean square error of the regression layer is where ŷ = ŷ 1ŷ2 ⋯ŷ i ⋯ŷ n , n = 512 is the predicted cutting force and y = y 1 y 2 ⋯ y i ⋯ y n , n = 512 is the measured cutting force. The loss function in this study is the cross-entropy loss, which is shown in Eq. (17). Fig. 3 The workflow of this study Part I Obtain the dataset and hyper parameters Make the impact testing and analyze the correlation between force and vibrations where t = max(y) − y i ∕max(y) , M = 512 , and p i are the predicted probability.
In this study, the method for stochastic optimization is Adam [48]. The theory is not introduced here.

The workflow of the proposed method
The whole workflow of this work is shown in Fig. 3.
As shown in Fig. 3, the work of this study is divided into three parts.
The first part is the obtainment of the dataset and the hyperparameters. In this study, the correlation between the cutting force signal and the vibration signal is analyzed through frequency response analysis experiments to determine the input of the monitoring model, and then experimental data acquisition is performed. The collected signal must be preprocessed before they are processed into data sets for input to the model for training and testing. In the signal acquisition process, due to the vibration transmission process and other factors, the acquired signal often contains disturbance and noise components, such as redundant DC signal and white noise. Therefore, this study normalizes the signal before establishing the data set. The usual normalization processing methods are linear normalization and zeromean normalization methods, and the zero-mean normalization method will lead to distortion of the cutting force signal, so this study uses linear normalization as follows: where x = x 1 x 2 ⋯ x i ⋯ x n is the acquired vibration signal, v is the DC component of the acquired vibration signal, x = x 1x2 ⋯x i ⋯x n is the processed vibration signal for training and testing, y = y 1 y 2 ⋯ y i ⋯ y n (18)  The prediction of f main The prediction of f residual

Parallel intergration
The cutting force prediction f

Fig. 5
The experiment platform is the processed force signal for training and testing, f = f 1 f 2 ⋯ f i ⋯ f n is the acquired force signal, f is the DC component of the acquired force signal. The next step is to get the data set for training and testing. In this study, a signal with 512 points is one sample, and a signal acquired under different cutting conditions will be put together to construct the sample dataset. Then is to set the hyperparameters num fc1 and N hl . The two parameters significantly affect the monitoring accuracy, so they must be set before training and testing. In this study, the performance of models with different hyperparameters is evaluated by two indicators: the RMSE and R-value. They are calculated as follows: where ŷ and y are the same in Eq. (16). ŷ is the mean value of the predicted force; ŷ and y are the variance. After the hyperparameters are obtained, the structure of the model indicates the relationship between the force and vibration. The relationship will be discussed in the verification part.
Secondly, the procedure is to train the main part model. The random sampling method extracts 80% of the sample data set as the training set and the remaining data set as the test set. In the training process, 20% of the training set will be selected as the validation set, and the rest, 80% training set, will be used for model training. Multiple models will be trained simultaneously according to the training set and batch sizes. Weight parameters are shared according to the monitoring accuracy; the validation set is then used to verify the monitoring accuracy. The regression layer returns the MSE value for stochastic  1  2  4000  400  2  3  2  4000  400  3  5  2  4000  400  2  4  1  3  4000  400  5  3  3  4000  400  6  5  3  4000  400  3  7  1  2  4000  600  8  3  2  4000  600  9  5  2  4000  600  4  10  1  2  6000  600  11  3  2  6000  600  12  5  2  6000  600  5  13  1  2  8000  800  14  3  2  8000  800  15  5  2  8000  800  6  16  1  3  8000  800  17  3  3  8000  800  18  5  3  8000  800  7  19  1  2  8000  1200  20  3  2  8000  1200  21  5  2  8000  1200  8  22  1  2  10,000  1000  23  3  2  10,000  1000  24  5  2 10,000 1000 optimization. The training process will stop when the maximum number of iterations is reached. The obtained model will be used to predict the test set and evaluate the model's monitoring accuracy. The total number of iterations can be calculated as follows: where num iteration is the total number of iterations, num batch is the number of batches, and num epoch represents the number of epochs.
Thirdly, train the residual part model and obtain the parallel force monitoring model. After obtaining the main part model, the residual part of the force is obtained by subtracting the monitoring results of the main part model where f main is the monitoring results of the main part model and f residual is the residual part. Then, the residual part model is then trained with the vibration signals as input and the f residual as output. The training process is the same as that of the main part model. After the residual model is obtained, the main part model is concatenated with the residual model to obtain the final cutting force monitoring model, as shown in Fig. 4.
The above process is to obtain the force monitoring model in one direction. It is necessary to predict F x , F y , and F z at the same time, so three different models are required to realize the milling process monitoring.

Experiment setup
The experiment uses the DMU50 five-axis milling machine tool as the platform, and the Kistler milling dynamometer 9265B is placed on the work table. Two accelerometers, PCB 356A66, are mounted on the spindle. The dynamometer is connected to a NI9221 card to acquire the cutting force signal during the milling process, and accelerometers are connected with a NI9234 card to acquire the vibration signal. NI9221 and NI9234 are put together in the NI cDAQ9189 to realize synchronous acquisition, as shown in Fig. 5b. The data sampling frequency is 10240 Hz. The material of the cutting workpiece is aluminum alloy 7075-T651. The tool is a 2-tooth helix milling cutter with a diameter of 16 mm. The total length of the tool is 100 mm, and the hanging length is 35 mm. The experimental platform is Fig. 8 The RMSE of the monitoring model when the num fc1 is different 95% 97% 99% 98% 97% Fig. 9 The monitoring accuracy of the proposed model with different hidden layers shown in Fig. 5a. The feed direction is in the y-direction in the milling process, as shown in Fig. 5c.
According to existing studies in ref. [49], it is known that the cutting force is closely related to feed rate, spindle speed, and cut depth. This study selects a series of cutting parameters provided by the cooperate manufacturers to validate the proposed method, as shown in Table 1.

Verification and result discussion
In this study, since the cutting force model in the three directions of x, y, and z will be established simultaneously, we will use the process of the x-direction cutting force model's establishment as an example to illustrate the model construction steps. The performance of the y-and z-direction force monitoring models will be given at the end of this part.

Obtainment of the dataset
Firstly, the impact test is made to analyze the relationship between force and vibrations in different directions. The accelerometers are placed the same as in Fig. 5a. The results of FRF xx and FRF yx are shown in Fig. 6. Figure 6 shows the results of cross FRFs in the x-and y-directions when given excitation in the x-direction. The results show that although FRF yx is not equal to zero in practice, the magnitude is small, and the coherence curve is bad. The results show that the vibration signal in the y-direction cannot meet the demand of achieving accurate force monitoring in the x-direction. So, in this study, the force monitoring model only uses vibrations in the same direction as the input.
Then, the vibration signals under 24 cutting conditions are acquired by accelerometers A and B, and a dynamometer acquires the cutting force signals. This study takes a signal with 512 points as one sample. The x-direction vibration and cutting force signals under some conditions are shown in Fig. 7.
In Fig. 7, the first column is the x-direction cutting force signal. The second column is the x-direction vibration signal acquired by accelerometer A. The third column is the x-direction vibration signal obtained by accelerometer B.
This study uses Vib A and Vib B as the model's inputs. The cutting force ForceX is the output. The length of a singlesegment signal is 0.05 s, and the total data set for training and testing contains 1748 samples. A total of 80% of the total data set samples are randomly selected to constitute the training set, and the remaining 20% form the test set. Then, choose 20% of samples from the training set randomly to form the validation set. So, there are 1185 samples in the training set, 209 samples in the validation set, and 354 samples in the test set. In the training process, the batch size = 40, the epochs are set as 40, and the total number of iterations is 4640. The computing platform used to train the model is Dell G15. The CPU is i7-11800H, the GPU is RTX3060, the memory is 32 G, and the hard disk capacity is 1 TB.

Parameters obtainment of num fc1 and N hl
The num fc1 in the encoder part reflects the number of features required by the model to reconstruct cutting force. It is an essential factor that affects monitoring accuracy. In this study, RMSE is applied to evaluate the impact of num fc1 . When num fc1 is 1 ~ 900, the monitoring results of RMSE of milling force in the x-direction are shown in Fig. 8.  In Fig. 8, the unit of RMSE is N. It can be seen from Fig. 8 that the RMSE value gradually decreases from 60 to 20 when the num fc1 increases from 1 to 500. The RSME value shows a precipitous decrease when the number of neurons increases to 505. Since then, with the increase of num fc1 , the RMSE value has fluctuated around 10 without any significant change. The results show that the num fc1 significantly influences the monitoring results. When the RMSE is around 10, the monitoring error rate of the x-direction milling force is less than 5%, which has met the requirements of accurate monitoring, so the num fc1 is set as 505. The results also indicate the relationship between the features extracted from the vibration signal and the reconstructed milling force. The deep learning model needs to extract a large number of features from the vibration signal to achieve accurate milling force monitoring.
In Section 3, the authors point out that the number of hidden layers in the decoder part N hl represents the relationship between the vibration and the force signals, which has an important impact on the monitoring accuracy of the proposed model. Therefore, it is necessary to analyze the influence of the N hl on the monitoring results. When the N hl is 1 ~ 5, the monitoring accuracy is shown in Fig. 9.
From Fig. 9, it can be seen that when the N hl is 3, the monitoring accuracy of the neural network is 99%, which indicates that it is the optimal choice. This parameter indicates that the relationship between the vibration signal and the cutting force signal is nonlinear. Meanwhile, when the hidden layer is 1, the monitoring accuracy of the proposed model can reach 95%, which shows that when assuming the relationship between the vibration and the milling force signal is linear, the monitoring model can also achieve high accuracy. By calculating   the contribution to the monitoring accuracy of cutting force, it can be found that the contribution of the linear relationship is 96%, and the nonlinear relationship is 4%. However, when N hl is increased to 4 and above, the monitoring accuracy begins to decrease, which indicates that continuing to increase the nonlinear weight cannot improve the monitoring accuracy but leads to overfitting and reduces the monitoring accuracy. In summary, linear and nonlinear relationships exist between the two signals simultaneously; these relationships significantly impact the model's monitoring accuracy.

Training and testing of the main part model
The training process of single-channel and dual-channel models is shown in Fig. 10. In Fig. 10 and Table 2, model A represents the singlechannel model that uses VibA vibration data as input. Model B represents the dual-channel fusion model. Figure 10 shows that model B has a faster decrease in RMSE and loss in the training process than model A. When the iteration reaches 1000, the B's RMSE drops from 45 to 20. When the training process ends, the convergence RMSE of the validation set is 7.06, and the convergence RMSE of the training set is 6.54. The two results are close, indicating that model B has high monitoring accuracy and no overfitting. Then is model A. The RMSE and loss fall much slower than that of model B. When iterations are about 2500, model A's RMSE is reduced to less than 20. At the end of the training process, model A's convergence RMSE of the training set is 18.56, and the convergence RMSE of the validation set is 18.69. Comparing the convergence speed and RMSE of the two models, the results show that model B's performance is much better than model A's. To fully compare the performance of the two models, this study randomly selects a sample from the training set and analyzes the monitoring results. The performance of the two models is shown in Figs. 11 and 12.   . 17 The monitoring results of the residual model 1 3 The first and second columns in Fig. 11 are the predicted cutting force signals and the measured force signals. The third column is the predicted cutting and measured force signals' spectrum in the frequency domain. By comparing the performance of the two models in Figs. 11 and 12 and Table 3, it can be found that model B's error is much smaller, with an RMSE of 6.5 and an R-value of 0.99, showing that the monitoring force signal is close to the measured signal. The performance of model A is worse than that of model B. The RMSE of model A is 18.59, which is nearly three times that of model B. Its R-value is 0.909, which is much lower than model B's. In this sample data, the performance of the dual-channel fusion model is much better.
Then, compare the performance of the two models on the test set. The results are shown in Figs. 13 and 14. Table 4 shows the RMSE and R-values of the monitoring results.
By comparing the monitoring results of the two models, the performance of the dual-channel model B is still the better one. It can be seen from Fig. 13  This study compares the proposed method with two machine learning methods: basic LSTM and GRU [50]. GRU is another RNN method. The performance of the three models on test set sample data is shown in Fig. 15 and Table 5. All the methods use dual-channel fusion. Figure 15 shows that the method proposed in this study performs the best. Compared with the basic LSTM and GRU methods, the proposed method's monitoring accuracy is the highest, and its monitoring error is also the lowest. While the basic LSTM's monitoring accuracy is lower than 0.7, the GRU's monitoring accuracy is 0.85.

Obtainment of the residual and the parallel force monitoring models
After the main part monitoring model is obtained, the model is used for the milling force monitoring of the whole cutting process. Taking the signal monitoring under the test 1 cutting condition as an example, the results are shown in Fig. 16.
The results in Fig. 16 show that the main part model has a large monitoring error at the beginning and ending stages of the cutting process. There is even more than 100 N error in some periods. Another model that can predict the residual part is required to realize high-accuracy force monitoring of the whole cutting process. The training   Table 6 shows the monitoring accuracy and RMSE at the beginning stage. Table 6 shows that the main part model just reaches an accuracy of 76.2% when the tool and the workpiece begin to come into contact. It cannot meet the demands of force monitoring. The residual part model can reach 94.5% accuracy in monitoring the residual force data. With the help of the residual model, the parallel model can reach 99.3% accuracy in monitoring the cutting force. The residual part model plays a key role in improving the monitoring accuracy. The monitoring results of the residual model and the parallel integration monitoring model are shown in Figs. 17 and 18.
The RMSEs in Table 7 are smaller than those in the above tables due to the calculation's inclusion of the idle portion. By comparing the monitoring results in Figs. 16 and 18, it can be seen that the parallel model can deal with the cutting force monitoring at the starting and ending stages well. The accuracy of the whole cutting force monitoring is also improved compared with the main part model, which is 99.6%. This accuracy has satisfied the cutting force monitoring requirement under most conditions. At the same time, for a single segment of 0.05 s data, the time required for parallel model monitoring is 0.002 s, which fully meets the demand of real-time monitoring, so the method proposed in this study can achieve accurate real-time monitoring of cutting force. Figures 19,20, and 21 are the monitoring results of the models under some working conditions after the parallel integration force monitoring models in three directions, model x , model y , and model z , are obtained. It can be seen from the figure that when the parameters such as cutting speed, feed, and depth of cut are different; the three models can still accurately predict the cutting force in the corresponding direction. The results show that the proposed model can accurately predict cutting force under different conditions.
Ref. [12] proposed a force monitoring method based on the substructure and the Kalman filter, and its error rate is about 7 ~ 9%. In comparison, the proposed method in this study can reach an error rate of less than 2%. Compared with the Kalman filter method, our method reduces the modeling workload and improves the prediction accuracy. The proposed method also has a significant advantage compared with the method using transfer learning [24] with an error rate of 7.38%. Compared with the above models, the model proposed in this paper does not require cutting parameters as inputs, such as spindle speed, feed rate, and depth of cut, and is more suitable for monitoring the cutting forces in the machining process. Also, compared with the basic LSTM neural network [38], the RMSE of the basic LSTM in predicting the cutting force is 35 N in most cases, while the RMSE of our model is less than 10 N. These results show the advantages of the method proposed in this study.

Conclusions
This research proposes a parallel neural network model based on dual-channel vibration fusion for real-time milling force monitoring. The experiment results show that the proposed model can accurately realize real-time cutting force monitoring. The main conclusions are as follows: (1) The parallel integration model is proposed in this article to deal with the problem that the cutting force changes significantly when the tool and the workpiece begin to contact and move away. It realizes accurate and real-time force monitoring of the whole milling process under variable cutting conditions with an accuracy of 99%, such as different rotating speeds, depths of cut, and feed rates. The model needs the vibration signals during the milling process as input, while no cutting parameters are required. (2) This study proposes an encoder-decoder neural network model based on the facts that vibration signals need to be filtered before analyzing and the nonlinear relationship between vibration and cutting force signals. The performance of the single-channel and dual-channel fusion models is also compared. By comparing the monitoring stability, the RMSE, and R-values on the training and testing set, it can be found that the dual-channel model's performance is more stable and accurate than that of the singlechannel model. (3) The hyperparameters significantly influence the monitoring accuracy and reflect the relationship between the force signal and the vibration signal. Firstly, the results show that reasonable num fc1 can achieve a significant decrease of the RMSE in force monitoring from 60 to 10, showing that it is one of the key parameters affecting monitoring accuracy. Secondly, the performance of different N hl shows that both linear and nonlinear relationships exist between the two signals; they significantly impact the model's monitoring accuracy simultaneously. These hyperparameters must first be set to ensure accurate force monitoring.
The model proposed in this article can realize highly accurate force monitoring under various cutting conditions in real time. In the future, to help monitor the milling process and improve product quality and productivity, researchers hope to develop a real-time model to predict cutting force for a wider range of cutting conditions, such as different workpiece materials, cutting tools, and 5-axis cutting conditions. Author contribution Kunhong Chen: methodology, software, experiment, validation, and writing the original draft. Xing Zhang: investigation, review, and editing. Wanhua Zhao: conceptualization, methodology, and supervision.
Funding This work was financially supported by the National Natural Science Foundation of China (nos. 51905410 and 52075426) and the China Postdoctoral Science Foundation (no. BX20180253).

Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Code availability Not applicable.

Declarations
Ethics approval The content studied in this article belongs to the field of ultrasonic processing and does not involve humans and animals. This article strictly follows the accepted principles of ethical and professional conduct.

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Competing interests
The authors declare no competing interests.