Dynamic parameters’ identification for the feeding system of computer numerical control machine tools stimulated by G-code

Abstract

This study proposed a dynamic parameters’ identification method for the feeding system of computer numerical control machine tools based on internal sensor. A simplified control model and linear identification model of the feeding system were established, in which the input and output signals are from sensors embedded in computer numerical control machine tools, and the dynamic parameters of the feeding system, including the equivalent inertia, equivalent damping, worktable damping, and the overall stiffness of the mechanical system, were solved by the least square method. Using the high-order Taylor expansion, the nonlinear Stribeck friction model was linearized and the parameters of the Stribeck friction model were obtained by the same way. To verify the validity and effectiveness of the identification method, identification experiments, circular motion testing, and simulations were conducted. The results obtained were stable and suggested that inertia and damping identification experiments converged fast. Stiffness identification experiments showed some deviation from simulation due to the influences of geometric error and nonlinear of stiffness. However, the identification results were still of reference significance and the method is convenient, effective, and suited for industrial condition.

Keywords

High speed and high precision feeding system parameter identification modeling friction

Introduction

High speed and high precision are the important development directions of computer numerical control (CNC) machines. However, accuracy and stability problems arise because of the increase in the rotating speed of the spindle and the feeding speed of servo system. High feed rate means the enlargement of the system tracking error. To reduce the tracking error and improve the machining accuracy of the machine tools, researchers have taken some measures on the controller of the machine tool, such as improving the position and speed gains of the machine controller and using the feedforwards of speed or acceleration.^1,2 However, the high-gain parameters affect the stability of the control system. In addition, the stability problem of high-speed cutting, that is, the cutting instability (such as chatter) in the processing process in the case of large cut depth, high rotation speed of the spindle, and high feeding speed condition, often plagues the researchers and users. To solve the problems, it is needed to conduct system parameter identification and accurate dynamics modeling for the feed system of the CNC machine tools.

The parameter identification and modeling of the servo system have been repeatedly researched. Under the open-loop control condition of the motor, Kaan Erkorkmaz and Altintas³ identified the dynamic parameters of the worktable, including inertia, damping, and Coulomb friction, using unbiased least square algorithm. Chyun-Chau Fuh and Tsai⁴ proposed an identification method for the dynamic parameter of the servo system using the continuous and disordered signals with fixed band width as drive signals. The test proved that the method effectively avoided the disturbance of the uncertain high-frequency signals and presented convergent identification results. Varanasi and Nayfeh⁵ drove the system using the sinusoidal signal generated by the signal generator. On the basis of analyzing the amplitude–frequency characteristics of the input and output signals, the frequency-response function of the system was obtained. Pislaru et al.⁶ from the University of Huddersfield in the United Kingdom employed the Morle wavelet to identify the natural frequency and damping ratio of the feed system of the NC machine tool. The results obtained were further compared with that obtained by Bode diagram method. The comparison result revealed that the wavelet identification method showed higher identification accuracy than Bode diagram method. Lee et al.⁷ realized the modal parameter identification of ball screw using the profile error measurement and iterative method of planar grating.

Although there are many methods^2,3 for precisely identifying the parameters of dynamic equations as well as the effects of friction, it is described in the published lectures such as Söderström and Stoica⁸ and Armstrong-Helouvry.⁹ But in these methods, special experimental platforms generally required to be established or special controller are needed in order to generate stimulating signal for system identification, some of identification are under open-loop control condition. Whereas these methods are difficult to realize for commercial CNC system due to its close structure, especially in the industrial field.

The servo system of the machine tool contains position, velocity, and current sensors. Meanwhile, commercial NC system also provides users with increasing open resources and allows users to access the sensor information conveniently. Obviously, the method of parameter identification using the information from internal sensor is more applicable to the in-machine or online detection in the processing and production field. However, the parameter identification of the servo system based on internal sensor has rarely been reported in previous researches.

In this study, a simplified control model and linear identification model of the feeding system were established, and the dynamic parameters for the feeding system of commercial CNC machine tools are identified using system identification method under closed-loop condition; to stimulate dynamic of the feeding system, special programmed G-code is used, and the method is suitable for commercial CNC system in industrial field.

The feed system model for CNC machine tools

Typical CNC machine feeding system generally comprises servo driver, alternating current (AC) servo motor, and the mechanical system. The mechanical system is composed of connector, ball screw, screw nut, guide rail, worktable, and so on, as shown in Figure 1. The motor drives the rotation of the ball screw and the movement of screw nut in turn to realize the transformation from the rotational motion of motor to the linear motion of worktable. Worktable is supported by sliding or rolling guide rail. The servo motor and the ball screws were connected by the coupler directly or the reducer. In this figure, $θ$ refers to the rotational angle of the motor; K_s, J_s, and B_s are the torsional stiffness, inertia, and damping of the screw, respectively; K_j and B_j represent the stiffness and damping from the worktable to the screw nut, respectively.

Figure 1.

Schematic diagram of a feeding system in CNC machine tools.

The torsional stiffness K_s and translational stiffness K_j can be equated to linear equivalent stiffness K of screw nut pair, while the inertia and damping of the screw and motor shaft are to be equated to the equivalent inertia J_e and equivalent damping B_e of the motor shaft. The full closed-loop servo system model was obtained thereby,¹⁰ as shown in Figure 2.

Figure 2.

The closed-loop servo system model with the friction.

As shown in Figure 2, the system is nested of current loop, speed loop, and position loop. The feedback of the speed loop is from the motor encoder, while that of the position loop is sourced from the highly accurate grating scale. u is the input signal of the controller; y_m is the equivalent displacement of the screw nut transformed by motor rotation angle; K_bs is the transformation coefficient of the nut displacement corresponding to the motor rotation angle; y is the displacement of the worktable; $θ$ is the rotation angle of the motor; i is the motor current; $ω$ is the rotational speed of the motor; K_pp is the proportional gain of the position loop; K_vp is the proportional gain of the speed gain; K_cp is the gain of the current loop; K_t is the torque constant of the motor; L and R are the equivalent inductance and resistance of the motor, respectively; J is the equivalent inertia of the motor; F_f is the frictional force; and s is the Laplace transform operator. In the figure, K_t, L, R, K_pp, K_vp, and K_cp are all directly available from relevant manuals or the parameter setting of the CNC system. The parameters of worktable such as J_e, B_e, M, K, and B should be identified. K is the connection stiffness and F_f is the frictional force of the screw nut and guide rail in case of neglecting the viscous effect.

The principal for the identification on the dynamic parameter of the feed system

As for the double mass system in Figure 2, the differential equation of the motor shaft can be established as

{iK}_{t} - K_{bs} (d_{y} + sgn (ω) g_{b}) K = J_{e} \overset{\cdot}{ω} + B_{e} ω

(1)

where $d_{y} = y_{θ} - y$ , the position difference of the encoder and grating scale; g_b is the backlash which emerges while feeding direction is changed; sgn(ω) is the direction function defined as follows

{\begin{matrix} sgn (ω) = 1, ω \geq 0 \\ sgn (ω) = 0, ω < 0 \end{matrix}

By Laplace transformation, the equation is written as

W (s) = {\frac{K_{t}}{J_{e} s + B}}_{e} {I (s) - k_{bs} K [D_{y} (s) + sgn (ω) g_{b}] / K_{t}}

where W, I, and $D_{y}$ are the Laplace transformation forms of ω, i, and d_y separately. The equation above is converted into

W (s) = \frac{K_{t} / J_{e}}{s + B_{e} / J_{e}} {I (s) - k_{bs} K [D_{y} (s) + sgn (ω) g_{b}] / K_{t}}

Let $a = B_{e} / J_{e}$ , $b = K_{t} / J_{e}$ , the equation above is rewritten as

W (s) = \frac{b}{a} \times \frac{a}{s + a} {I (s) - k_{bs} K [D_{y} (s) + sgn (ω) g_{b}] / K_{t}}

After discretization, there is

ω (k) = \frac{b}{a} \times \frac{1 - e^{- {aT}_{s}}}{z - e^{- {aT}_{s}}} {i (k) - k_{bs} K [d_{y} (k) + sgn (ω) g_{b}] / K_{t}}

where T_s is the sampling period; the difference equation is expressed as

\begin{array}{l} ω (k + 1) = e^{- a T_{s}} ω (k) + \frac{b}{a} (1 - e^{- a T_{s}}) \\ {i (k) - k_{b s} K [d_{y} (k) + sgn (ω) g_{b}] / K_{t}} \end{array}

(2)

Let

\begin{array}{l} Y^{0} = [\begin{array}{l} ω_{1} \\ ω_{2} \\ \cdot \cdot \cdot \\ ω_{k - 1} \end{array}] Y^{0} = [\begin{array}{l} ω_{0} \\ ω_{1} \\ ⋮ \\ ω_{k - 1} \end{array}], \\ Φ^{0} = [\begin{array}{l} \begin{matrix} ω_{0} & i_{0} & d y_{0} & sgn (ω_{0}) \end{matrix} \\ \begin{matrix} ω_{1} & i_{1} & d y_{1} & sgn (ω_{1}) \end{matrix} \\ ⋮ \\ \begin{matrix} ω_{k - 1} & i_{k - 1} & d y_{k - 1} & sgn (ω_{k - 1}) \end{matrix} \end{array}] \end{array}

and

θ^{0} = {[e^{- {aT}_{s}} \frac{b}{a} (1 - e^{- {aT}_{s}}) i (k) - \frac{b}{a} (1 - e^{- {aT}_{s}}) {Kk}_{bs} / K_{t} - \frac{b}{a} (1 - e^{- {aT}_{s}}) {Kk}_{bs} / K_{t}]}^{T}

where $ω_{k}$ , $i_{k}$ , and $d_{k}$ refer to the poison sequences of the rotation speed of motor, current, and the position difference of encoder and grating scale sampled by the data acquired system at the kth time.

According to equation (2), the matrix vector equation is obtained

Y^{0} = Φ^{0} • θ^{0}

In the equation above, $Y^{0}$ and $Φ^{0}$ are known except $θ^{0}$ . According to the least square method,¹¹ we get

{\hat{θ}}^{0} = {({(Φ}^{0})^{T} Φ^{0})}^{- 1} {(Φ}^{0})^{T} Y^{0}

(3)

and

a = \frac{- \ln (θ^{0} (1))}{T_{s}}, b = \frac{a θ^{0} (2)}{(1 - e^{- {aT}_{s}})}

Dynamic parameters can be obtained as follows

{\hat{J}}_{e} = \frac{K_{t}}{b}, {\hat{B}}_{e} = {aJ}_{e}, \hat{K} = \frac{- θ^{0} (3) K_{t}}{θ^{0} (2)}

As for the worktable, the differential equation below is supported

(y_{θ} - y) K = M \overset{\cdot\cdot}{y} + B \overset{\cdot}{y} + F_{f}

(4)

where $F_{f}$ is the friction force to worktable and can be obtained in the next section. In a similar way, the difference equation is obtained as

\begin{array}{l} y (k + 2) = (1 + e^{- c T_{s}}) y (k + 1) - e^{- c T_{s}} y (k) \\ + \frac{d T_{s}}{c} (1 - e^{- c T_{s}}) [d_{y} (k + 1) - F_{f} (k + 1) / K] \end{array}

(5)

where c = B/M and d = K/M.

Using the same identification methods, [ $\hat{M} \hat{B}$ ] can be recognized.

Followed by the parameters’ identification, four signals were used, including the current i of the motor, the rotation speed $ω$ of the motor, the position signal y from the grating scale, and the position signal y_m from the encoder. The friction force sequence $F_{f}$ is accessible through the friction force identification test described in the following section. In a popular CNC systems currently, such as Siemens, Fanuc, and Num, the four signals above are all available through proper interfaces. In addition, the rotation speed of the motor can also be obtained through the difference in the position signals of the encoder.¹²

The parameter identification for the Stribeck friction model

The frictional force modeling is necessary for accurate control of the servo system of the NC machine tool. Stribeck friction model¹³ can accurately describe the characteristics of the friction at low speed and is most widely used at present. When the worktable of the machine tool moves along the guide rail at uniform speed under no-load condition, since $\overset{\cdot}{ω} = 0$ , and the viscous damping force is treated as part of the friction, then

T_{f} = {iK}_{t}

(6)

where $T_{f}$ is the frictional torque of the machine tool at uniform feed speed condition.

If the servo axis of the machine tool was run at uniform speed while different speeds under no-load condition, the relationship of the current i and rotational speed n of the motor can be obtained through the signal collection system. This relationship can be further converted into the correspondence relationship of the feed speed v and friction F_ft. The Stribeck curve is acquired finally.

Stribeck model is described as follows¹³

F_{ft} = F_{c} sgn (v) + (F_{s} - F_{c}) \exp (- {(\frac{v}{v_{s}})}^{δ}) sgn (v) + σ v

(7)

F_c, F_s, and σ represent Coulomb, static, and viscous friction coefficients, respectively. V_s and δ are the Stribeck velocity and the Stribeck shape factor, respectively. According to Jamaludin et al.,¹⁴δ = 1 or 2; to simplify the calculation, δ = 1 in this article.

Let

y = \exp (- \frac{v}{v_{s}})

According to the Taylor equation, the equation above can be expanded by sixth order as

y = \exp (- \frac{v}{v_{s}}) = 1 - \frac{v}{v_{s}} + \frac{v^{2}}{2! v_{s}^{2}} - \frac{v^{3}}{3! v_{s}^{3}} + \frac{v^{4}}{4! v_{s}^{4}} - \frac{v^{5}}{5! v_{s}^{5}} + \frac{v^{6}}{6! v_{s}^{6}}

Equation (7) is rewritten as

F_{ft} = (a + bv + {cv}^{2} + {dv}^{3} + {ev}^{4} + {fv}^{5} + {gv}^{6}) sgn (v)

(8)

where

\begin{matrix} a = F_{s}, b = σ - \frac{F_{s} - F_{c}}{v_{s}}, c = \frac{F_{s} - F_{c}}{2 v_{s}^{2}}, d = - \frac{F_{s} - F_{c}}{6 v_{s}^{3}}, \\ e = \frac{F_{s} - F_{c}}{24 v_{s}^{4}}, f = - \frac{F_{s} - F_{c}}{120 v_{s}^{5}}, g = \frac{F_{s} - F_{c}}{720 v_{s}^{6}} \end{matrix}

In case of obtaining multiple sets of $F_{fti}$ and $v_{i}$ sequences from the test, let

F = [\begin{matrix} F_{ft 1} & F_{ft 2} & \dots & F_{fti} & \dots & F_{ftN} \end{matrix}]^{T}

θ = [\begin{matrix} a & b & c & d & e & f & g \end{matrix}]

Φ = [\begin{array}{l} \begin{matrix} sgn (v_{1}) & v_{i} sgn (v_{1}) & v_{i}^{2} sgn (v_{1}) & \dots & v_{i}^{6} sgn (v_{1}) \end{matrix} \\ ⋮ \\ \begin{matrix} sgn (v_{i}) & v_{i} sgn (v_{i}) & v_{i}^{2} sgn (v_{i}) & \dots & v_{i}^{6} sgn (v_{i}) \end{matrix} \\ ⋮ \\ \begin{matrix} sgn (v_{N}) & v_{N} sgn (v_{N}) & v_{N}^{2} sgn (v_{N}) & \dots & v_{N}^{6} sgn (v_{N}) \end{matrix} \end{array}]

Thus, equation (6) is altered into

F = Φ θ

(9)

Using least square method, $\hat{θ}$ is obtained. Furthermore, the parameters of the friction model are acquirable in turn as

F_{s} = a, v_{s} = - \frac{c}{3 d}, F_{c} = F_{s} - 2 {cv}_{s}^{2}, σ = b + \frac{F_{s} - F_{c}}{v_{s}}

In the model above, the viscous damping force is treated as part of the friction. However, the viscous damping force is excluded from the fictional force in the identification model above.

Thus

F_{f} = F_{t} - σ v

(10)

Identification experiment for dynamic parameters of the servo system of a large-scale CNC gear grinding machine

The identification experiment is conducted on the dressing system of the grinding wheel of a large-scale CNC gear grinding machine, as shown in Figure 3(a). The diamond dressing wheel is installed on feeding axis W, while the high-speeding grinding wheel is mounted on axis Y. The axes Y and W are independent from each other by the involute trajectory integrated through linkage interpolation. The machine employs the NUM 3050 CNC system produced by Schneider Electric. The current i and rotation speed signal ω needed in the identification for J and B are accessible through the external interfaces provided by servo drivers. Since the position loop of the NUM system can provide 1 VPP simulation signal, the position signals of encoder and grating scale (y_m and y) needed in the K identification are directly available from the hardware circuit interface of the grating scale and encoder, as shown in Figure 3(b).

Figure 3.

Identification testing setup: (a) the dressing servo feed system of the large-scale NC gear grinding machine and (b) data collection setup.

To ensure the feed system stimulated continuously, initial input signal of the identification for dynamic parameter beyond the friction is generated from specially designed CNC G-code. Under the driving effect of G-code, the position signal of the machine tool was exhibited as the secondary segmented curve with constant acceleration. Figure 4 displays the curves of the position, speed, and acceleration of the machine tool trajectory.

Figure 4.

The input signals used in the identification: (a) time–accelerate curve, (b) time–velocity curve and (c) time–position curve.

The identification experiments are repeated for five times for the feed axes Y and W with input signals in the same shape while amplitudes are 1, 1.5, 2, 2.5, and 3, respectively. Figure 5(a) demonstrates a group of identification signals including the motor rotation speed w and current i of axis Y. The difference signal dy and position signal y from grating scale of axis Y are shown in Figure 5(b).

Figure 5.

The input and output signals for the identification: (a) identification signals for inertia and damping and (b) identification signals for the stiffness.

Figure 6 shows the identification results. The identification results suggest that the equivalent rotational inertia J_e and equivalent damping B_e converge rapidly at the second test and present constant results. By contrast, B and K fluctuate violently as shown in Figure 7. M is directly available through the design data of the machine.

Figure 6.

The identification results of the inertia J_e and damping B_e of axis Y: (a) results of J_e and (b) results of B_e.

Figure 7.

The identification result on the stiffness K of axis Y: (a) results of B and (b) results of K.

The identified dynamic parameters of axes Y and W for the large-scale CNC gear grinding machine except the friction are listed in Table 1.

Table 1.

The dynamic parameters of the axes Y and W.

Axis	J_e (N/m²)	B_e (N m s/rad)	M (kg)	B (N/(m/s))	K (N/m)
Y	0.00738	0.02744	968	20,455.3	3.2254×10⁸
W	0.00132	0.00819	200	3833.4	9.0922×10⁷

The parameter identification experiments for Stribeck model of the servo system were performed on the dressing system of the grinding wheel of a large-scale CNC gear grinding machine. With the data acquisition system, the current i in the drive motor was recorded, while jogging the axes back and forth at various speeds ranging from 0.1 to 42 rad/s. More tests were performed in the lower velocity region to better identify the boundary and partial fluid lubrication characteristics. Typical test results for the axis Y are shown in Figure 8. According to equations (6) and (8), the parameters of Stribeck fiction model are identified as shown in Table 2. By substituting the parameters obtained into equation (7), the Stribeck curve was obtained, as shown in Figure 8 with solid line. It can be seen that the curve drawn is highly consistent with the experimental data.

Figure 8.

The forward friction test data on axis Y and the Stribeck curve obtained.

Table 2.

Results of the parameters’ identification for Stribeck model.

δ	F_s (N m)	F_c (N m)	V_s (rad/s)	σ (N m/rad/s)
1	1.4873	1.0080	3.6543	0.01707

As one of the most common motions in CNC machining process, circular motion can directly reflect the dynamic characteristics of the servo system at low and high speeds.¹⁰ Therefore, the parameter identification method is validated by comparing the simulation and experimental results of the circular motion in the dressing system for the grinding wheel of the large-scale CNC grinding machine. Control model of the servo system is small different with the model described in Figure 2 with P controller; this controller introduces the speed feedforward coefficient K forward into the position loop and a proportional–integral (PI) controller into the speed loop. The servo system simulation model of axes Y and W established corresponded to the controller. The parameters of simulation model of axis Y involve K_forward = 72%, K_ppx = 8202 mm/min/mm, K_vpx = 130 mA/r/min, T_ix = 20 ms, K_t = 1.1, K_bsx =7.9577 × 10⁻⁴, J = 1.34 × 10⁻³ kg m², L = 12.8 mH, R = 1.37 Ω, M = 968 kg, K_x = 5.5 × 10⁸ N/m, and B = 27,742 N s/m. The friction model uses the Stribeck model. The parameters in Table 2 are transformed into the values of worktable, namely, F_s = 1.8690 × 10³ N, F_c = 1.2666 × 10³ N, and V_s = 2.9080 × 10⁻³ m/s.

The simulation conditions are as follows: F = 2 m/min and the radius R of the circular motion is 25 mm. The simulation results are displayed as the dotted line in Figure 9.The experimental conditions of the machine tool are consistent with the simulation conditions. The NC controller parameters are identical with the controller parameters of the simulation model. The test results are shown as the solid line in this figure.

Figure 9.

Comparison of experimental results and simulation results of the circular motion of servo system.

The results of the simulation and experiment are shown in the figure. To ensure that the results are displayed normally in the polar co-ordinate, all the values displayed in Figure 9 are adjusted by adding a constant (here 2 × 10⁻⁵) to the contour errors. The curves by the simulation and the experiments show their similar profiles and behaviors at different places especially at four quadrants of the circular motion. As shown in the figure, there appears to be a good correspondence between the two sets of the results for the protrusions on the horizontal axis. Although the correspondence is not good enough for the protrusions on the vertical axis, the test results suggested that the errors between the curves by the simulation and experiment are no more than 3 μm, and dynamic behaviors shown by the simulation and experiment curves including emerge time, lasting-time, and the maximum value of height of the protrusions are in good consistency. This outcome proves that the dynamic characteristics of the simulation system constructed are consistent with those of the real system. Meanwhile, the dynamic parameters used in the simulation are basically consistent with those in real system. However, there are differences between the values of simulation and identification parameters.

To index the performance of the presented methods, relative error between identification and simulation parameters is obtained as in Table 3.

Table 3.

Relative error for identification parameters to simulation for Y axis.

J_e	B_e	B	K	F_s	F_c	V_s	σ
10.2%	18.2%	26.3%	−70%	20.6%	−20.4%	25.6%	10.5%

From the table, all relative errors are no more than 27% except K. The evident differences are attributed to the stiffness difference among the joint faces of the worktable. Especially, when the joint faces move downward at different speeds, the contact stiffness presents large variations.¹⁵ Even though the identification result can also provide references for the evaluation on the overall stiffness of the worktable. They are applicable to evaluate the variations in the overall mechanical stiffness of the machine tool in different working environments or different life phases of a machine tool and thus to realize the rapid performance evaluation and fault diagnosis of the servo system of the CNC machine tools.

Conclusion

This article presented an identification method for the dynamic parameters of the servo system using the internal signals of the CNC system. The identification model for the dynamic parameters of the servo system was first established. By solving this model using the least square method, the dynamic parameters including equivalent inertia, equivalent damping, worktable damping, and the overall stiffness of the mechanical system were available. Using the experimental method, the friction curve of the servo system of the NC machine tool was obtained. By high-order Taylor expansion on the Stribeck curve, the nonlinear model was linearized, and the five parameters of the Stribeck friction model were solved using the least square fitting method. Using the dynamic parameters obtained, the simulation model for the servo system of the large-scale CNC gear grinding machine was established. To verify the validity of the method, identification experiment was conducted to verify the effectiveness of identification method. The stable experiment results suggested that inertia and damping identification experiment converged fast. Stiffness identification experiments showed some error under the influences of nonlinear error and stiffness. However, the identification results were still of reference significance. Using the identification parameters, the circular motion of the CNC gear grinding machine system was simulated using the model and tested by the experiments; the results of comparison suggested that the identification method proposed in this article is valid.

Footnotes

Academic Editor: Yaguo Lei

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This project was supported by the Natural Science Foundation of Shanghai of China (Grant No. 13ZR1427500), the Key Projects in the National Science & Technology Pillar Program of China (Grant No. 2012BAF01B02), and the National Natural Science Foundation of China (Grant No. 51005158).

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