Improved Prediction Model of the Friction Error of CNC Machine Tools Based on the Long Short Term Memory Method

Abstract

Friction is one of important factors that cause contouring errors, and the friction error is difficult to predict because of its nonlinearity. In this paper, a prediction model of the friction error of a servo system is proposed based on the Long Short-Term Memory method (LSTM). Firstly, the transfer function is used to predict the position of the servo system, and then the prediction error of the transfer function is obtained. Secondly, the nonlinear friction error is extracted and predicted by a LSTM network. Finally, the accurate tracking error can be predicted by the proposed combined model. The experimental results show that the proposed model can improve the prediction accuracy of tracking errors dramatically.

1. Introduction

Improving the machining accuracy of five-axis CNC machine tools is of great significance, which has always been one of the hot spots in five-axis CNC machining [1]. In order to improve the contour accuracy, Song et al. [2] added a certain compensation value to the command reference voltage source of each interpolation cycle to compensate the tracking error. Zhang et al. [3] proposed an iterative tracking error precompensation method to improve the tracking accuracy. Friction is an inevitable nonlinear phenomenon in five-axis CNC machine tools [4]. Because of the adverse effects of nonlinear friction, two strategies are commonly used to improve tracking accuracy. One is to build accurate friction models and another is to design advanced observers and controllers, by which the friction can be compensated. Because friction modeling and prediction play a great role in reducing the tracking error of five-axis CNC machine tools. Now many friction prediction methods have been proposed, such as iterative learning [5], parameter tuning [6], and so on.

The typical friction characteristics of lubricated metal surfaces in contact are traditionally described as Stribeck curves. With the increase of speed under partial fluid lubrication, the friction between lubricated metal surfaces decreases, which is called the Stribeck effect. Scientists have proposed many models for this nonlinear effect, including “LuGre model” [7], “Coulomb model” [8], “Tustin model” [9], and so on. Although these three classical models can compensate for the Stribeck effect to a certain extent, they cannot effectively compensate for the adverse effects of viscous friction because they do not contain viscous friction terms. At the same time, based on the LuGre model, the controller proposed by Dai et al. [10] can effectively suppress the force error spike of the low velocity region via the deduced adaptive law and bi-observer, and Xi et al. [11] can estimate the nonlinear friction behavior of machine shafts with a long stroke distance.

Although the dynamic friction model established without considering the static friction term has achieved many successful applications, a good static friction model can reflect about 90% of the real friction [12]. For the friction model, Xu et al. [13] proved that nonlinear friction had a great influence on tracking turntables by studying the Column-Viscous model. Klaus et al. [14] used the controlled magnetic field to overcome the viscous friction. Shi and Zhang [15] proposed a friction correction method to realize the Stribeck effect in dynamic simulations. Pennestrì [16] compared the performance of eight friction models for simulating viscous friction and pre-sliding displacement and concluded that the LuGre model is more suitable for reproducing friction effects.

Because of the characteristics of nonlinear static friction, the modeling process is more complex, and the data-driven modeling method can simplify the modeling process. At present, there are many nonlinear modeling algorithms, such as the genetic algorithm [17], neural network [18], and other nonlinear algorithms [19,20,21]. The neural network has the ability to predict the complex nonlinear relationship between the input and output of the system [22].

The above research fruits model the friction, but the nonlinear friction error, which is defined as the tracking error resulting from the nonlinear friction, cannot be obtained directly.

This paper uses the model-driven method to model the servo system. According to Yin et al. [23], a mathematical theoretical model was established for the motor, transmission mechanism, and position sensor in the servo feed system, and the parameters of the model were obtained by system identification. The transfer function residual, which is the prediction error of the transfer function, can be obtained by the difference between the actual trajectory and the trajectory predicted by the model simulation. The transfer function residual mainly comes from the influence of nonlinear friction, which only occurs in the pre-slip state. Xi et al. [24] regard the separation displacement as the boundary between the pre-slip state and the slip state. However, a scientific method of identifying the separation displacement has not been proposed and M. Yang et al. [25] proposed a two-stage friction model only related to speed, which can easily predict the position of the pre-slip state during the movement of CNC machine tools. Therefore, this paper regards the break velocity as the boundary between the pre slip state and the slip state.

This paper proposes an improved prediction model of friction error based on the Long Short-Term Memory method. Firstly, according to the friction break velocity proposed in [25], the boundary of the pre-sliding state and sliding state is obtained. Secondly, according to the separation boundary obtained by the break speed and the residual error predicted by the servo system model, we separate the nonlinear friction error. Thirdly, the LSTM is used to model the nonlinear friction error, and a prediction model of nonlinear friction errors is obtained. Finally, the combined model of the transfer function and LSTM is used to predict the tracking error accurately.

The rest of this paper is organized as follows. Section 2 deduces the transfer function of the servo system after analyzing the dynamics of the servo system. Section 3 deals with the friction error after analyzing the characteristics of friction, and it builds a LSTM model according to the properties of the nonlinear friction error. Finally, the tracking error is accurately predicted based on the combined model. Section 4 describes the experimental results, and Section 5 gives the conclusion.

2. Model-Based Tracking Error Prediction

In the position control mode, the servo system is composed of three loops, which are the current loop, speed loop, and position loop. When the transfer function of the servo system is derived theoretically, not only the three-loop control, but also the influence of the mechanical transmission structure should be considered. In theory, the more factors considered in the modeling, the more accurate the model is, which also increases the complexity of the modeling. The servo system three-loop control and mechanical transmission structure are analyzed below, and the transfer function is derived.

(1) The current loop

The current loop is the innermost loop of the three-loop control of the servo system. The simplified current loop of a servo system is shown in Figure 1. In the figure the current input to the servo motor is $i_q$, while $i_q$ also determines the output torque of the motor $T_e$. In this paper, the electrical characteristics of the current link are ignored, and the current loop is represented by the proportional link with the proportional coefficient of $K_t$. The transfer function of the current loop is:

(2) The speed loop

The speed loop is the middle loop of the three-loop control of the servo system. The control principle of the speed loop is shown in Figure 2. In the figure, $T_l$ and $T_d$ are the load torque and torque disturbance of a servo system, respectively. $J$ is the effective inertia of a servo system, and $B$ is the equivalent damping coefficient of a servo system.

From Figure 2, the speed loop transfer function can be expressed as:

$$G_s\left(s\right)=\frac{G_v\left(s\right)·K_t}{J\cdot s+B+G_v\left(s\right)·K_t}$$

(1)

Here, $G_v\left(s\right)$ is:

$$G_v\left(s\right)=K_v\left(1+\frac{1}{{\tau }_s}\right)$$

(2)

where $K_v$ and ${\tau }_s$ are the proportional factor and time constant in speed, respectively.

(3) The position loop

The position loop is the outermost loop of the three-loop control of the servo system. The input of the position loop is the reference position $x_r,$ and the output is the actual position $x_a$.

From Figure 3, the transfer function of the position loop can be expressed as:

$$G\left(s\right)=\frac{x_a}{x_r}=\frac{G_p\left(s\right)G_s\left(s\right)}{1+G_p\left(s\right)G_s\left(s\right)}$$

(3)

In order to improve the tracking control performance, the feedforward control is usually added to the position loop:

From Figure 4, the position loop transfer function with feedforward can be written as:

$$G\left(s\right)=\frac{x_a}{x_r}=\frac{F\left(s\right)G_s\left(s\right)+G_p\left(s\right)G_s\left(s\right)}{1+G_p\left(s\right)G_s\left(s\right)}$$

(4)

(4) Mechanical transmission structure

If the motor, coupling, ball screw, and workbench are regarded as rigid bodies and the nonlinear friction torque is ignored, the motor torque balance formula can be expressed as:

$$T_e=J\frac{d\omega }{dt}+B\omega +T_d+T_l=J\frac{d\omega }{dt}+B\omega +T_f+T_n+T_l$$

(5)

where $\omega$ is the angular speed of a servo system, and $T_e$, $T_l$, $T_f,$ and $T_n$ are the output torque load torque, nonlinear friction torque disturbance, and other torque disturbances of a servo system. The torque disturbance is the sum of $T_f$ and $T_n$. $J$ is the effective inertia of a servo system, and $B$ is the equivalent damping coefficient of a servo system.

Then the control principle of the mechanical transmission structure can be expressed as Figure 5:

By analyzing the three-loop control principle and the mechanical transmission structure of the servo system, the simplified control block diagram of the servo system is obtained.

According to Figure 6, when ignoring $T_d$ and T_i, the simplified transfer function for the servo system reference position and the actual position can be expressed as:

$$G\left(s\right)=\frac{x_a\left(s\right)}{x_r\left(s\right)}=\frac{F\left(s\right)G_v\left(s\right)K_t+G_p\left(s\right)G_v\left(s\right)K_t}{J{s}^2+\left(B+K_t\cdot G_v\left(s\right)\right)\cdot s+G_p\left(s\right)G_v\left(s\right)K_t}$$

(6)

The friction torque disturbance includes the linear portion $B\omega$ and nonlinear portion $T_f$. $T_f$ can be obtained by the nonlinear friction force described by:

$$f_f\left(\dot{x}\right)=\left[f_c+\left(f_s-f_c\right)e^{-{\left(\frac{\dot{x}}{{\dot{x}}_s}\right)}^2}\right]·\mathrm{sgn}\left(\dot{x}\right)$$

(7)

where $f_f\left(\dot{x}\right)$ is the frictional force, which is the function of the velocity $\dot{x}$. $f_c$ and $f_s$ are the minimum levels of column friction and nonlinear friction, respectively. ${\dot{x}}_s$ is the lubricant parameters, which is determined by the empirical experiments.

$T_f$ is obtained by:

$$T_f=f_f\left(\dot{x}\right)r$$

(8)

The friction error caused by the linear portion can be predicted as one part of the tracking error by the transfer function [26,27]. Yet, the nonlinear friction error is difficult to predict because of the strong nonlinearity of the friction torque. This paper proposes to use the long short term memory method is model the friction error, which is show in the next section.

3. Prediction of the Nonlinear Friction Error

3.1. Achieving the Nonlinear Friction Error Based on the Transfer Function

Nonlinear friction only occurs when the speed increases from zero or decelerates to zero. As shown in Figure 7, the nonlinear friction modeling should consider three situations:

$$\left\{\begin{array}{c}{v}_1=0\\ v_2=0\\ v_3\ne 0\end{array}\right. \left\{\begin{array}{c}{v}_1\ne 0\\ v_2=0\\ v_3=0\end{array}\right.\left\{\begin{array}{c}{v}_1\ne 0\\ v_2=0\\ v_3\ne 0\end{array}\right.$$

where $v_1,v_2$, and $v_3$ are the instantaneous velocity at three consecutive moments during tool movement.

By the method in [25], the boundary line between the pre-slip state and slip state can be determined by the break velocity $v_{break}$:

$$v_{break}=\sqrt{2a_0{x}_{break}}$$

(9)

$a_0$ is the acceleration when $v=0$. $x_{break}\approx \frac{v_{break}^2}{2a0}$, by identifying $v_{break}^2$ and $a_0$ to obtain the separation displacement. After the separation displacement is calculated, $v_{break}$ can be calculated. Using $v_{break}$ and the pre-slip region, the nonlinear friction error can be extracted from the transfer function residual.

3.2. Nonlinear Friction Error Prediction Model Based on the LSTM

3.2.1. Selection of Neural Network Models

The input and output data of the servo system have a strong time correlation, that is, the response at the current time is closely related to the previous input data. The input and output data of the servo system are typical time series data.

Recurrent Neural Network (RNN) is a kind of neural network that studies the input–output relationship of sequence data. Different from other neural networks, RNN not only considers the input at the current time, but also considers the impact of information from the past period on the output at the current time. Specifically, RNN will save the past information and apply it to the calculation output at the current time. Although simple RNN has achieved good results in dealing with the problem of time series, RNN is unable to grasp the nonlinear relationship with a large time span. Therefore, when learning long series, the phenomenon of gradient disappearance—that is, the “long-term dependencies problem” of RNN—appears easily. In order to solve this problem, many scholars have made different improvements based on RNN. The LSTM is a commonly used algorithm that has achieved great success in sequence data prediction.

The network principle of LSTM is shown in Figure 8, and the three different boxes represent the application of the same functions at different time steps. The LSTM introduces three thresholds to control data flow—the forget gate, input gate, and output gate—and records historical information by cell state.

The task of the forget gate is to accept the cell state at the last moment $C_{t-1}$, and decide to retain and forget which part of $C_{t-1}$, namely, $C_{t-1}$ multiplied by a forgetting factor $f_t$. The forgetting factor $f_t$ is decided by the output from the previous time $h_{t-1}$ and current input data $x_t$ through the $\sigma$ activation function.

$$f_t=\sigma \left(W_f·\left[h_{t-1},x_t\right]+b_f\right)$$

(10)

The function of the input gate is to determine which new information is stored in the current cell state. The input gate consists of two parts. One part is to use $h_{t-1}$ and $x_t$ through the $\sigma$ activation function that determines what values need to be updated to get updated information $i_t$. $W_f$ is the weight matrix for $f_t$. The other part is to obtain the temporary state at the current time $\widetilde{C_t}$ through the tanh activation function. i_t and $\widetilde{C_t}$ can be calculated according to (11) and (12):

$$i_t=\sigma \left(W_i·\left[h_{t-1},x_t\right]+b_i\right)$$

(11)

$$\widetilde{C_t}=\text{tanh}\left(W_C·\left[h_{t-1},x_t\right]+b_C\right)$$

(12)

Figure 8 shows the process of obtaining $C_t$ by using $C_{t-1}$ at the last moment, $f_t$, updated information $i_t,$ and the temporary state at the current time $\widetilde{C_t}$. $W_i$ and $W_C$ are the weight matrix for $i_t$ and $\widetilde{C_t}$.

$C_t$ can be calculated by:

$$C_t=f_t·C_{t-1}+i_t·\widetilde{C_t}$$

(13)

The function of the output gate is to determine the output value according to the cell state: apply the output at the last time $h_{t-1}$ and the current input data $x_t$ through the $\sigma$ activation function to get $o_t$. Combined with the current cell state C_t to get the final output $h_t$ through the tanh activation function, $o_t$ and $h_t$ can be calculated according to (14) and (15).

$$o_t=\sigma \left(W_o·\left[h_{t-1},x_t\right]+b_o\right)$$

(14)

$$h_t=o_t·\text{tanh}\left(C_t\right)$$

(15)

By introducing cell state, forget gate, input gate, and output gate, the LSTM effectively solves the long-term dependencies problem, and it is a very suitable model for long sequence problems. Considering that the input and output of the servo system are typical time series data, the LSTM is selected to predict the servo system transfer function residual.

3.2.2. Design of the LSTM

The input and output data of the LSTM greatly affect the performance of the neural network. If the correlation between the input and output is very low or the input is basically constant, the learning effect of the LSTM will be affected, and the performance of the LSTM will be greatly deteriorated. Therefore, it is necessary to analyze the servo system to select the appropriate input and output.

Because the nonlinear friction error is one of the main factors of the tracking error of the servo system, the nonlinear friction error is taken as the output of the LSTM.

In order to determine the input characteristics of the LSTM, the following analysis is given.

The tracking error of servo system $e_s$ is the deviation between the reference position $x_r\left(t\right)$ and the actual position $x_a\left(t\right)$. $e_s$ is:

$$e_s=x_r\left(t\right)-x_a\left(t\right)$$

(16)

The transfer function between the tracking error $e_s$ and the reference position $x_r\left(t\right)$ can be expressed as:

$$\frac{e_s}{x_r}=1-\frac{x_a}{x_r}$$

(17)

According to the control theory, the Laplace transform is performed. The above transfer function is denoted by ${\mathsf{\Phi }}_e\left(s\right)$, and we expand it at $s=0$:

$${\mathsf{\Phi }}_e\left(s\right)={\displaystyle \sum }_{i=0}^{\infty }\left[\frac{1}{i!}{\mathsf{\Phi }}_e{}^{\left(i\right)}\left(0\right)·s^i\right]$$

(18)

Similarly, set the Laplace transform of the reference position input $x_r\left(t\right)$ as $R\left(s\right)$ and the Laplace transform tracking error $e_s$ as $E\left(s\right)$, so the relationship between and $E\left(s\right)$ is:

$$E\left(s\right)={\mathsf{\Phi }}_e\left(s\right)·R\left(s\right)={{\displaystyle \sum }}_{i=0}^{\infty }\left[\frac{1}{i!}{\mathsf{\Phi }}_e{}^{\left(i\right)}\left(0\right)·R\left(s\right)s^i\right]$$

(19)

Therefore, the actual output $x_a\left(t\right)$ of the servo system can be expressed as:

$$x_a\left(t\right)=x_r\left(t\right)+e_s=x_r\left(t\right)+{{\displaystyle \sum }}_{i=0}^{\infty }\left[\frac{1}{i!}{\mathsf{\Phi }}_e{}^{\left(i\right)}\left(0\right)·R^{\left(i\right)}\left(t\right)\right]$$

(20)

A simplified expression of the reference input and the actual output of the servo system can be obtained by expanding (20) and ignoring the high-order infinitesimal:

$$x_a\left(t\right)=x_r\left(t\right)+{\xi }_v·x_i^{\prime }\left(t\right)+{\xi }_a·x_i^{″}\left(t\right)$$

(21)

Among them, ${\xi }_v=-{\mathsf{\Phi }}_e^{\left(1\right)}\left(0\right),{\xi }_a=-{\mathsf{\Phi }}_e^{\left(2\right)}\left(0\right)/2$, correspond to the coefficients of velocity and acceleration, respectively.

According to (21), the tracking error of the servo system is mainly affected by velocity and acceleration. The velocity can be obtained by the first derivative of displacement versus time, and the acceleration can be obtained by the second derivative of displacement versus time. Of course, other higher-order derivatives of displacement versus time, such as jerk, will also have an impact on the tracking error of the servo system, but the extent of its impact can be ignored.

Therefore, the velocity and acceleration are taken as the input kinematics characteristics to predict the residual of the transfer function of the servo system.

3.3. Accurate Prediction of the Tracking Error Based on the Combined Model of the Transfer Function and LSTM

Combining the transfer function established in Section 3.1 with the LSTM-based nonlinear friction error prediction model established in Section 3.2, the combined model of the servo system can be obtained and the actual position can be accurately predicted, which is shown in Figure 9.

First, the actual position of the servo system output is acquired through the actual operation reference command. The predicted actual position is obtained through the transfer function, and the difference between them is the transfer function residual. Then, the nonlinear friction error is extracted from the transfer function residual. Then, the reference command is differentiated to obtain the velocity and acceleration motion characteristics, which are inputs of the LSTM for training together with the nonlinear friction error. For the new reference command, the actual position is also predicted by the transfer function, the velocity v and acceleration a are obtained by differentiation, then the nonlinear friction error is predicted by the LSTM, and the position predicted by the transfer function is added to the predicted nonlinear friction error to obtain the accurate position prediction.

4. Experimental Results

Figure 10 shows the experimental setup, which is a five-axis machine tool. In the setup, the X-axis servo system is used to verify the proposed prediction method, and it can be verified that other axes have the similar performance. The control system of the machine is driven by a six-axis motion control card by Googol Tech. The software DriverStudio is used to sample the position, velocity, and current information with a sampling period of 1ms. The absolution of its encoder is 10,000 pulse/rev, and the prediction accuracy is expected to be within 10 μm. Figure 11 shows the input reference position and tracking error of a M-serial linear trajectory.

After filtering the sampled data by a low-pass filter, the X-axis transfer function is identified:

$$Gs=\frac{1351s+4.952\times {10}^5}{s^2+1109s+4.952\times {10}^5}$$

(22)

By using the transfer function, the tracking error can be predicted, then the transfer function residual is obtained by the method in Section 3, which is shown in Figure 12. Compared with the actual tracking error of M-serial linear trajectory shown in Figure 11, the transfer function method has reduced the maximum prediction error from 40 $\mathsf{\mu }\mathrm{m}$ to 20 $\mathsf{\mu }\mathrm{m}$.

According to (9), $v_{break}$ can be obtained as $v_{break}\approx 1.98 \mathrm{mm}/\mathrm{s}$. According to Section 3.1, the nonlinear friction error can be calculated by the transfer function residual, which is the prediction error by the transfer function method. The results are shown in Figure 13.

By using the LSTM, the nonlinear friction error can be predicted. When the predicted nonlinear friction error is added to the tracking error predicted by the transfer function, the tracking error becomes more accurate. By the proposed combination of the transfer function and the LSTM methods, the prediction error of the tracking error is shown in Figure 14. Compared with the prediction error by the transfer function method shown in Figure 12, the proposed method has reduced the maximum prediction error from 20 μm to 3.5 μm. It can be seen that the prediction error of the tracking error can be reduced because the nonlinear friction error can be estimated by the LSTM methods.

In order to verify the proposed method, a curve is used instead of the M-serial linear trajectory. Figure 15 shows the reference position and tracking error of a curve.

The transfer function in (22) is used to predict the tracking error and the transfer function residual by the method in Section 3. The transfer function residual is shown in Figure 16, and the nonlinear friction error is shown in Figure 17. Compared with the actual tracking error of curve trajectory shown in Figure 15, the transfer function method has reduced the maximum prediction error from 44 $\mathsf{\mu }\mathrm{m}$ to 27 $\mathsf{\mu }\mathrm{m}$.

Similarly, the transfer function residual and nonlinear friction error are the output of the LSTM. According to the predicted results, the former prediction accuracy can reach $20 \mathsf{\mu }\mathrm{m},$ and the latter prediction accuracy can reach $4 \mathsf{\mu }\mathrm{m}$. The predicted results are shown in Figure 18.

By the proposed combination of the transfer function and the LSTM methods, the prediction error of the tracking error is shown in Figure 18. Compared with the prediction error by the transfer function method shown in Figure 16, the proposed method has reduced the maximum prediction error from 27 μm to 8 μm. It can be seen that the proposed method is still applicable to curve motion, which can not only greatly reduce the tracking error but also has universality.

According to the above experimental results, the nonlinear friction error can be predicted accurately by the LSTM method. After the LSTM deep-learning network predicts the nonlinear friction error, the prediction error of the tracking error is much reduced.

5. Conclusions

Improving the machining accuracy of five-axis CNC machine tools is of great significance, and friction is one of important factors that cause contouring error. Yet, the friction error is difficult to predict because of its nonlinearity. To improve the prediction accuracy of tracking errors, a prediction model of the friction error based on the LSTM is proposed. First, in the position control mode, the transfer function of the servo system is derived and used to predict the tracking error, and then the residual of the transfer function can be calculated. Second, according to the characteristics of nonlinear friction, the nonlinear friction error is separated from the transfer function residual. Finally, the LSTM is used to train the nonlinear friction error to obtain the prediction model of the nonlinear friction error. By combining the nonlinear friction error prediction model and the transfer function, the tracking error of the servo system can be predicted accurately. The experimental results show that the transfer function method will generate a prediction error of about 20$\mathsf{\mu }\mathrm{m}$ to predict the M-serial linear trajectory, and the combined model of the LSTM and the transfer function can reduce the prediction error to 3.5$\mathsf{\mu }\mathrm{m}$. The same result can be obtained by predicting the tracking error of a curve.