Open Access Dynamic Parameters Identification for the Feeding System of Commercial Numerical Control Machine

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1 Send Orders or Reprints to The Open Automation and Control Systems Journal, 13, 5, Open Access Dynamic Parameters Identiication or the Feeding System o Commercial Numerical Control Machine Chen Guangsheng * and Li Haolin School o Mechanical Engineering, University o Shanghai or Science and Technology, Shanghai, China Abstract: The precision o high-speed CNC (Computer Numerical Control) machine are greatly inluenced by the dynamic characteristics o servo system. To establish the servo system model accurately, the internal signals o NC, e.g. motor current and rotate speed were adopt as the input or output signal or the system. The ARMA linear identiication model was also established to identiy the dynamic parameters o the mechanical parts in the servo system, including equivalent inertia, equivalent damping and so on. The riction Stribeck curve was obtained by necessary experiments. The nonlinear riction model was linearized by using higher-order Taylor expansion, and the ive parameters o the Stribeck riction model were identiied. To veriy the eectiveness o the closed-loop identiication, experiments are carried out on the eeding system o a commercial NC machine. Signals o servo motor current and rotating rate which are oered in many modern CNC machine tools are needed or the identiication and experiments results show that the proposed method perorms well with rapid convergence and accurate results and parameters o Stribeck model can be obtained accurately by identiication. The method is suited or industrial condition and has its practicality. Keywords: Computer numerical control machine, eeding system, parameters identiication, dynamic parameters. 1. INTRODUCTION High-speed and high-precision CNC machines are needed in modern manuacture. And the precision o highspeed NC machine is greatly inluenced by the dynamic characteristics o servo system [1]. With regard to the highgrade commercial NC system in closed-loop control, the suitable control parameters o servo system are mainly determined by machine system parameters. However, improper control parameters can introduce higher ollowing error or a servo system. Seriously, it can induce the oscillation o machine system, causing severe degradation in processing quality, even the damage o the machine. Thereore, the dynamic parameters o servo system should irst be obtained to realize the modeling o servo system. Then high-speed machine in high-precision control can be achieved. Model o servo system is a closed-loop and comprised by three loops, with relatively more parameters that are diicult to be identiied. The system identiication o motor or worktable has been investigated by researchers [,3], but in which, experimental platorms o open-loop system are generally required to be established. Whereas, it is diicult to realize open-loop control restricted or commercial CNC system due to its closestructure, especially in the industrial ield. Thereore, the identiication o the commercial CNC system can be only perormed in the condition o closed-loop. In this study, the dynamic parameters such as inertia, damping o servo system, and the model parameters related to nonlinear riction or worktable are identiied using system identiication method. *Address correspondence to this author at the School o Mechanical Engineering, University o Shanghai or Science and Technology, Shanghai, China; Tel: ; Fax: ; cgs-168@163.com. MODELS FOR IDENTIFICATION.1. Model o Servo System Typical CNC machine eeding system is comprised by ampliier, servo motor, ball screw, screw nut, guide rail, worktable. Where, ball screw is driven by motors to exert the movement o screw nut. Thereby, the rotating motion o the motors is transerred to linear motion o working. Worktable is supported by sliding or rolling guide rail. And servo motor and rolling screw are directly connected by couplings. To simpliy the control model, without considering the elasticity o worktable, the mass and damping o all moving parts are all equivalent to the equivalent inertia and equivalent damping o motor shat. The electro-mechanical coupling control system o the whole eeding system is show in (Fig. 1) [4]. And the symbols in (Fig. 1) are deined as ollows: u, the input signal o the controller, I, the current o the AC servo motor, y m, the equivalent displacement o screw nut, l, screw lead, K bs, the transerred coeicient o the nut displacement corresponding to motor rotation angle, : the rotation angle o motor, ù, the rotation velocity o motor K pp, the proportional gain o position loop, K vp, the gain o velocity loop, K cp, the gain o current loop, K t, motor torque constant, L, R, equivalent inductance and equivalent resistance o motor respectively, S, Laplace transorm operator, Je,equivalent inertia, B m, equivalent damping... Stribeck Model O Friction Stribeck model o worktable is an exponential model as well, proposed by Stribeck in 19 [5,6]. Stribeck discovered that riction between two objects was associated with the membrane thickness o the lubricating oil, which varied with the change o the relative movement speed. Thus, riction is expressed by the unction o the relative movement /13 13 Bentham Open

2 16 The Open Automation and Control Systems Journal, 13, Volume 5 Guangsheng and Haolin speed. And this is the known Stribeck curve, as shown in (Fig. ). Seen rom Stribeck curve, three kinds o riction is ound with object s movement. When speed V was little, the static riction reduced to Coulomb riction with the increasing speed, and when speed V increased urther, viscous riction showed a linear relation with speed. In stable-state movement, relations between riction and speed are described as ollows F = F c sgn(v) + v + (F s F c )exp(( v v s ) )sgn(v) (1) Where, F c is coulomb riction orce, F s is static riction orce, V s is the critical Stribeck velocity, is the viscous riction actor, is the Stribeck shape coeicient, usually 1, V s, F c, and Fs are usually obtained by parameters identiication. generate correct identiication result. For a servo system ollowing equation can be obtained [3,7]. Eq. () can be written as: Ater transerred by Laplace transorm, it is: W (s) = b a a s + a [I(s) T (s)/k ] (3) e t Where, W and I are the transerring orms o and i respectively, and suppose a= Be / Je b= Kt / Je. T s is taken as sampling period, discrete expression o Eq. (3) is rewritten as. () u K pp K vp K cp + K t + 1/(Ls+R) T e ω θ y m 1/(J c s+b c ) 1/s K bs (4) Then Te( k)/ K t is equivalent to disturbing current, and it is derived rom riction torque. Assume the disturbance current with is d +, inverse disturbance current is d. Both o them can be considered as the unctions o, and then T e is described as ollows [3]. K e Fig. (1). Simpliied electro-mechanical coupling control system o eeding system. In this equation, and are the unctions illustrating the directions o d + and d respectively, and they can be deined by the ollowing equation: Friction Torque F F s A B C D Where, is the threshold evaluating the direction o velocity, thus Eq. (3) is written as the ollowing dierence equations: F c Y=Fc Viscous Friction o Mixture Friction v s Boundary Friction Velocity v Suppose, And Fig. (). Stribeck curve. 3. PRINCIPLE OF IDENTIFICATION FOR DYNAMIC PARAMETER OF FEEDING SYSTEM 3.1. The Parameter Identiication Using Least Squares Method Based on ARMA Model The least squares method is the most widely used one in system identiication, since it is simple to be applied and can In addition, in the ormulas above, Y and are all known, only remains unknown. Then the least squares method is employed to conduct identiication, and the optimal estimation value o coeicient matrix is:

3 Dynamic Parameters Identiication or the Feeding System The Open Automation and Control Systems Journal, 13, Volume And J e and J e are obtained as Je = Kt / b B e = aj e The input and output signal in the identiication are the current i and rotation velocity o servo motor respectively. 3.. Nonlinear Identiication o Stribeck Model Parameters Eq. (1) shows that the Stribeck model is exponential pattern. Thus the linear identiication is unavailable, and linearization o the nonlinear model must be perormed. Exponential term in Eq.1 was carried out by Taylor expansion, taking 6 orders as example, and the above ormula can be expressed as F = ( a + bv + cv + dv + ev + v + gv )sgn( v) (5) Where Fs Fc Fs Fc Fs Fc a = Fs, b =, c =, d =, 3 vs vs 6vs Fs Fc Fs Fc Fs Fc e=, = g = v 1v 7v s s s In Eq., Friction F can be expressed as the liner power unction o the speed v. When several F i and v i sequences were obtained by experiments, then suppose F = [ F F F F ] T, 1 i N Where, i is current o the motor, J e is the inertia o the moving components and rotating components o the eed mechanism, B e is damp and riction torque is T e.when the machine tool working platorm ed at a uniorm speed and operated with no load, the motor rotated at a uniorm speed, and i was, the above ormula was expressed as: When the damping orce was regarded as part o riction, then T = ikt Where: T is the riction torque when the machine tool ed at a uniorm speed. The relation between the motor riction torque T and rotating speed n can be got on basis o the current i and rotating speed n by the signal collection system, when the machine tool servo axis operated with no load at a uniorm speed under various speeds. This relation can be converted into the relationship between the working platorm eed speed v and riction orce F, and Stribeck curve was then yielded. 4. DESIGN INPUT SIGNAL FOR PARAMETER IDENTIFICATION As or commercial NC system, current i cannot directly be taken as control variable and intput in. Merely programming position signal u can exert eect on the servo system o NC machine. Hence, the so-called ideal excitation signal cannot be obtained directly. it only can be acquired approximately in an indirect way. as y = K bs and = i substitute to Eq. (1) According to Eq. (5), ollowing express can be obtained (6) The optimal estimation value o coeicient matrix was yielded by least square method or the parameter itting, and the riction model parameters can be obtained as. Fs = a c vs = 3d (7) Fc = Fs cvs Fs Fc = b + vs Friction and speed can be expressed as T torque (Nm) and rotating speed (rad/s), and units o the riction model parameters should be adjusted correspondingly as well. Following Equation can be obtained or a servo motor. Assume y is the conic o time, there is y = at /, where, a is a constant, substituting a to the ormula above, i= aje /( KbsKt) + atbe /( KbsKt) + Te / Kt Then k1 = aje /( KbsKt) + Te / Kt, and k = abe /( KbsKt ) are given. When y is the conic o time, current signal reers to the composite o square wave signal and ramp signal. Square wave step signal and ramp signal are the excitation signals usually used in system identiication. Here, the acceleration o displacement curve y is a. Assume that the required excitation signal u is constituted by n dierent accelerating curves. a reers to acceleration o the i curve, u denotes the displacement and v i represents the velocity, then 1 bt 1, < t t1 1 1 bt 11 + b( t t1), t1 < t t u = bt + b ( t t ) + + b ( t t ) 1 + bn( t tn 1), tn 1 < t tn 11 1 n1 n1 n (8)

4 164 The Open Automation and Control Systems Journal, 13, Volume 5 Guangsheng and Haolin In Eq. (8), the position command curve is the conic o time. To realize its G-code programming, the conic is discretized to small straight line segments according to NC interpolation period. Then the small straight line segments are conducted position and eeding velocity programming to generate eective excitation signals. The acceleration command o the excitation signals is assumed as the square wave signals o variable amplitude illustrated as (Fig. 3(a)). And they are transerred to ramp velocity signals by time integral, illustrated as Fig. 3(b). Using time integral again, displacement signal are inally produced, illustrated as (Fig. 3(c)). Corresponding G-code commands are obtained as (Table 1). a v a k t t1 t t3 t4 t5 t6 t7 t8 t9 t (a) Time-Accelerate Curve v k 5. IDENTIFICATION EXPERIMENT FOR SERVO SYSTEM The identiication experiments are perormed in the grinding wheel dressing system o a large NC gear grinding machine, as shown in (Fig. 4). Where, diamond roller is installed on W axis, and high-speed spindle grinding wheel system on Y axis. Then relative synthesize movement between Y axis and W axis can orm tract needed. They are installed separately. The NUM 35 NC system o Schneider Electric Company is applied in the machine. The current i and rotation velocity signal required in identiying J and B can be obtained by the external interace provided by servo driver. In the experiment o equivalent inertia and equivalent damping, the input signal is multiplied by adjusted scale actor c1 to generate excitation signals respectively. Meanwhile, the corresponding system response signals are acquired. One set o input and output signals are illustrated in (Fig. 5). Five dierent coeicients c1 are applied in ive experiments respectively, the identiication results are presented in (Fig. 6). Where, with the gradual increase o coeicient c1, the equivalent rotation inertia and equivalent damping are rapidly convergent ater the second experiment and show stable result. (b) Time-Velocity Curve t y a k t k (c) Time-Displacement Curve t Fig. (3). The generation principle o identiying excitation displacement signal. Table 1. G-Code o the Generated Conic Displacement Curve G Code %3() N L1=.1 L=.7 N3 L1=E91/1+L1 N4 G53 N5 G9 G1 YL1 FL1 N6 G91 G1 Y.1 F1.4 N13 Y. F. N14 Y.3 F.9 N195 Y.1 F1.3 N8 Y.4 F4.1 N9 Y.3 F3.4 N3 Y.3 F.6 N31 Y. F1.9 N33 M Fig. (4). Y axis o servo dressing eeding. i / a Fig. (5). Input and output signals or identiication system o grinding wheel. Identiication experiments o Stribeck model parameters in the servo system also were perormed in the CNC gear grinding machine, and riction o moving components on the Y axis was investigated.the Y axis servo motor torque constant K t =1.1N/A, screw lead L = 5mm. In no-load cases, w i 1 w / rad

5 Dynamic Parameters Identiication or the Feeding System The Open Automation and Control Systems Journal, 13, Volume various speeds were given on the Y axis or repeatable eeding at a uniorm speed. Current i and eeding speed F were collected rom two opposite directions. Due to the same approach, the research adopted the riction in the positive direction or study. The riction torque T and angle speed were shown in (Table ), the discrete points o the angle speed and riction were shown in (Fig. 7). According Eq. and Eq.3, parameters o Stribeck riction model can be got by the least square method as ollows: :1, F s : Nm, F c :1.73Nm, v s :.61 rad/s, :.1985, Nm/rad/s. Optimized parameters can be urther obtained by manual adjustments as ollows: :1, F s : Nm, F c :1.8Nm, v s : rad/s, :.177 Nm/rad/s The itting data are illustration as (Fig. 7) (real line). J/N/m*m..1 Fig. (6). Result o Identiication or inertia and damping. Table. * Data o Current and Velocity rom Servo System when Forth Feeding (rad/s) T (Nm) (rad/s) T (Nm) B J * * * * Coeicient o amplitude B/N*m*s/rad T / NM * Data by experiments Data by itting v /rad/s Fig. (7). Stribeck curve by experiments. 6. CONCLUSIONS In this study, simpliied control model o NC machine servo system is irstly established. The internal signals o NC system, including current and rotation velocity, are employed as the input and output signals to build the linear ARMA model that can identiy the dynamic parameters such as equivalent inertia and equivalent damping. Finally the least squares method is applied to ind solution. High order Taylor expansion method was adopted in this research to achieve the linearization or the nonlinear riction exponential model. Parameters o riction model were also inally obtained by the least square method. To veriy the eectiveness o the method, identiication experiments are conducted on the Y axis o the eeding system o a large NC grinding gear machine. And the results indicate that the method is characterized by rapid identiication convergence and correct result, and can be applicable or the assessment o the systematical lubrication or assembling perormance in engineering applications. CONFLICT OF INTEREST The author(s) conirm that this article content has no conlicts o interest. ACKNOWLEDGEMENT This project was supported by the Natural Science Foundation o Shanghai in China (Grant No. 13ZR1475), the Key Projects in the National Science & Technology Pillar Program o China (Grants No. 1BAF1B) and the National Natural Science Foundation o China (Grant No ). REFERENCES [1] X.S. Mei, M. Tsutsumi and T. Tao, "Study on the compensation o error by stick-slip or high-precision table", International Journal o Machine Tools & Manuacture, vol. 44, pp , 4. [] Q. Liu, L. Er and L.J. Kun, "Overview o characteristics, modeling and compensation o nonlinear riction in servo systems", Systems Engineering and Electronics, vol. 4, no.11, pp ,.

6 166 The Open Automation and Control Systems Journal, 13, Volume 5 Guangsheng and Haolin [3] K. Erkorkmaz and Y. Altintas, "High speed CNC system design. Part II: modeling and identiication o eed driver", International Journal o Machine Tools & Manuacture, vol. 41, pp , 1. [4] C. Guangsheng. "Research on error synthetic compensation and application in large-scale gear grinding machine", Ph.D. thesis, School o mechanical engineering, Xian Jiaotong University, Xian, China, 11. [5] S. Cohn, "Dynamic riction measurement, modeling, and compensation or precise motion control", New Jersey Institute o Technology, New Jersey, [6] Y. Zhou, "Investigation on sensorless evaluation or numeric control machine tool driving system", Ph.D. thesis, School o mechanical engineering, Xian Jiaotong University, Xian, China, 9. [7] L. Yanjun and Z. Ke, Theory o system identiication and application", Beijing, Deend industrial press, 3. Received: August 13, 13 Revised: August 8, 13 Accepted: August 8, 13 Guangsheng and Haolin; Licensee Bentham Open. This is an open access article licensed under the terms o the Creative Commons Attribution Non-Commercial License ( licenses/by-nc/3./) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.