A Study of the Pneumatic Counterweight of Machine Tools Conventional and Active Pressure Control Method

890 A Study of the Pneumatic Counterweight of Machine Tools Conventional and Active Pressure Control Method Ming-Hung TSAI and Ming-Chang SHIH The pneumatic counterweight method is a suitable counterweight method used for machine tool owing to its high speed and force capacity, combined with low price and clean operation. However, the preset pressure cannot be hold at higher traveling rate if the conventional pneumatic counterweight is applied. In this study, rather than using the conventional pressure compensation method, the active pressure control method was designed as an alternative to the pneumatic counterweight driven by a linear motor. The fuy sliding mode controller was designed and implemented for regulating the pressure using the servo valve. The experimental results demonstrate that the variation of the pressure in the pneumatic cylinder can be hold within the range ±15 kpa by using active pressure control method. Key Words: Active Pressure Control, Pneumatic Counterweight, Fuy Sliding Mode Control 1. Introduction The pneumatic counterweight system enables machine tools to support the weight of a spindle of the machine tool by using air cylinders. The actuator force output is produced by the pressure differential acting across the piston and can be controlled by regulating the pressure in cylinder chambers. The system takes advantage of increasing traveling rate, saving power and reducing the effect of gravity. The pneumatic counterweight system is a suitable method of counterweight method because of its high speed and force capacity, combined with its low price and clean operations. However, if the preset pressure of the cylinder cannot be hold, the actuator force output is not a constant and the feed rate and machining precision is involved. Conventionally, the pneumatic pressure compensation method applied for the pneumatic counterweight system of the servo motor driven machine tool is designed to connect a large air tank with a pneumatic cylinder, so that pressure change in the pneumatic cylinder can be reduced due to the spindle motion. The air tank is connected to the compressed air source, the compressor, and the pressure regulator or the pressure switch is added to achieve the constant pressure. However, if the spindle of the ma- Received 1st March, 2006 No. 06-5029) Department of Mechanical Engineering, National Cheng Kung University, No.1, Ta-Hsueh Rd., Tainan, 701, Taiwan, R.O.C. E-mail: mcshih@mail.ncku.edu.tw chine tool travels at a faster rate, a constant pressure cannot be hold using the conventional method and moreover connecting a large air tank to cylinder would be useless in this situation. Recently, several investigations have developed some different algorithms for controlling pneumatic actuator. Ben-Dov and Salcudean 1) developed a linear force controller to modulate force by voice-coil flapper valves and low-friction cylinders. Shih and Hwang 2) developed modified differential pulse width modulation PWM) method to eliminate the dead one and to improve the nonlinear characteristics between the pressure differences of the cylinder. In Shih and Ma 3), the fuy PWM control method was developed and applied to control the position of a pneumatic cylinder with high speed solenoid valves. Also, Sorli and Pastorelli 4) developed PNM/PWM controller to control chamber pressure of cylinder by two pairs of monostable 2-way valves. Khayati, Bigras and Dessaint 5) developed tractable state-feedback synthesis technique and H controller to control the pneumatic actuator force. Furthermore, Richer and Hurmulu 6) used reduced order sliding mode controller to control pneumatic force with servo valve and obtained a 20 H bandwidth. S. Shibata et al. 7) designed a fuy controller with fuy virtual reference generator to improve the rise time. In this study, an active pressure control method was developed and implemented on the pneumatic counterweight system driven by high speed linear motor. The cylinder pressure was directly regulated with the servovalve. Considering the air compressibility and highly non-

891 Fig. 1 The conventional pneumatic counterweight system of machine tool linear behavior, this study designed the fuy sliding mode controller for the active pressure control method to simplify the system analysis and controller design. The fuy sliding mode controller was designed on the basis of the sliding mode control law 8), 9), 12). The proposed active pressure control method is simple, effective and easy to implement on the pneumatic counterweight of the machine tools. 2. System Description Figure 1 shows the layout of the conventional pneumatic counterweight system of the machine tool. This paper builds the test stand and study the conventional and active pneumatic counterweight system. Figure 2 shows the layout of the test stand. 1 ) Conventional pneumatic counterweight system: Fig. 3 schematically represents the conventional pneumatic counterweight test stand. The cylinder pressure was stabilied by connecting cylinder to the air tank and the air tank pressure was regulated using pressure regulating valve. 2 ) Active pneumatic counterweight control system: Fig. 4 schematically represents the active pneumatic counterweight test stand. An IBM 32 bit compatible microcomputer was applied as a controller. Initially, the computer sent a control signal to trigger an electric control unit ECU) to drive the linear motor. The mass flow rate entering the cylinder is controlled by the servo valve, which is controlled by the computer. The pressure of the pneumatic cylinder or force applied by cylinder can be controlled through controlling mass flow rate using servo valve. Furthermore, the pneumatic force compensates 95 kg of the vertically moved mass driven by the linear motor. The displacement of the cylinder and air pressure are measured by pulse scale and pressure sensor and fed back to the computer. After calculation and comparison, the control sig- Fig. 2 The layout of the test stand nals are sent to control the ECU and linear motor. Finally, the experimental results were compared to those obtained using the conventional method. 3. Basic Equations The model of the pressure control system regulated by servovalve is shown in Fig. 5. This study primarily investigates the dynamics of the pressure in cylinder chamber and is not concerned with the dynamics of the linear motor. The governing equations are derived as follows under the following assumptions: A1) gas is ideal, A2) gas density is uniform in the chamber, A3) flow leakages are neglected, and A4) pressure of air supply is stable in the system. 3. 1 Piston-load dynamics The equilibrium equation for the piston can be expressed as M +βż+ F f = F m +P b A b P a A a + P e A r ) Mg 1) where M is the external load mass, is the piston position, β is viscous friction coefficient, and F f is friction force. The friction force is relation to the velocity of the piston and the characteristic of friction force is shown in Fig. 6. The friction force is described by F fm if ż = 0andF fm2 F fm F fm1 F fm1 F f 1 )e 5ż/v c1 +F f 1 if 0 < ż <v c1 F f = F f 1 +a 1 ż if ż v c1 2) F fm2 F f 2 )e 5ż/v c1 +F f 2 if 0 > ż >v c2 F f 2 +a 2 ż if ż v c2

892 F m is the force applied by linear motor, P a and P b are the absolute pressures in cylinder chambers, P e is the absolute ambient pressure, A a and A b are the piston areas, and A r is the rod area. The weight of mass is balanced by the actuator force output, produced by the pressure differential acting across the piston. 3. 2 Model of the cylinder chambers for the conventional method The conventional pressure control method for pneu- matic counterweight system is to connect the air tank without using servovalve to directly control the pressure in the cylinder chambers. The following equations are obtained: The air in the cylinder chamber A is directly discharged to the atmosphere, so we consider the control volume V b, with densityρ b,massm b, pressure P b,andtemperature T b, the dynamic of the pressure P b can be written as 1), 10) : Fig. 3 Schematic construction of the conventional pneumatic counterweight test stand Fig. 4 Schematic construction of the active pneumatic counterweight test stand a) b) Fig. 5 Schematic representation of the valve controlled pressure control system for a) ż > 0 and b) ż < 0

893 Fig. 6 Friction model of the piston dp b = kp b dt ż+ krt b ṁ b 3) A b where is the displacement of cylinder. Mass rather than volume flow rate is generally used because of the compressibility of air. The equation for the mass flow entering through an effective area A t of an orifice to the air tank can be written as: ) PD1 ṁ t =C 1 A t f 1 signp b P t ) 4) TU1 PD1 ) PD1 if P D1 f 1 = P >C r U1 k 2 R k +1 ) 1/k ) k 1)/k 2k Rk 1) 1 PD1 P U1 if P D1 C r ) k+1)/k 1) where ṁ t is the mass flow entering the air tank, C 1 is discharge coefficient, and C r is the pressure ratio that divides the flow regimes into unchocked and chocked C r = 0.528 for air). P upstream) and P D1 P downstream) are defined as = maxp b,p t ), P D1 = minp b,p t ) where P b is the pressure in cylinder s chamber B, and P t is the pressure in air tank. The mass flow rate of cylinder s chamber B can be presented by ṁ b = ṁ t, while the pressure dynamics of cylinder s chamber B is expressed as: dp b = kp b dt ż krt b ṁ t P b,p t ) 5) A b This equation indicates that the pressure P b is stable if the air tank can provide a corresponding mass flow rate ṁ t, when the mass moves at traveling rate ż. The tank volume is large enough and the pressure variation of the tank can be ignored. Thus we can assume the tank pressure P t is equal to preset pressure. However, the mass flow rate is smaller between the cylinder and the air tank when the cylinder pressure is approximately equal to the tank pressure. Consequently the higher traveling rate produce a larger variation of the cylinder pressure, and connecting a large air tank to cylinder is useless in this condition. 3. 3 Model of the cylinder chamber for the active control method To stabilie the cylinder pressure when the mass moves at a higher traveling rate, this study uses a servovalve to control the cylinder pressure directly. The dynamics of valve spool can be approximated using a first order differential equation because the spool inertial force is much smaller than the force applied by the internal springs of the valve. So, the valve dynamics may approximately be described as: ẋ v = h v x v +e v u 6) where x v is the spool displacement from central position, h v and e v are positive constant, and u is the control signal. The servo valve performance can be obtained from the operating instructions 14) : the critical frequency at the maximum movement stroke of the piston spool is 90 H. Because the movement of valve spool is fast compared with cylinder piston movement, the valve dynamic effects can usually be neglected in this control system. The valve spool stroke can be considered as proportional to the control signal u. Theeffectivearea of servo valveis expressed as: A v = w ev u = k v u 7) h v where w is the area gradient of servovalve. The equation for the mass flow through an the effective area A v of servo valve can be written as: ) PD2 ṁ v =C 2 A v f signu) 8) TU2 f PD2 PD2 ) if P D2 = P >C r U2 k 2 R k +1 ) 1/k ) k 1)/k 2k Rk 1) 1 PD2 P U2 if P D2 C r ) k+1)/k 1) where ṁ v is the mass flow through the valve orifice, C 2 is discharge coefficient, and C r is the pressure ratio that divides the flow regimes into unchocked and chocked C r = 0.528 for air). P upstream) and P D2 P downstream) are defined as { PU2 = P s,p D2 = P b if u 0 = P b,p D2 = P e if u < 0 where P s is the supply pressure, and P e is the discharge pressure.

894 The mass flow rate of the cylinder s chamber B thus can be represented by ṁ b = ṁ v ṁ t, and the pressure dynamics of cylinder is as follows: Table 1 Fuy rule table dp b = kp b dt ż+ krt b [ṁ v P s,p b,p e ) ṁ t P b,p t )]9) A b The pressure in the cylinder s chamber B must be constant in order that the actuator force output is constant. Therefore the mass flow rate though the servovalve must be controlled using active pressure control method, and the pressure change can be reduced when the mass moves at a higher traveling rate. 4. Controller Design When controlling a plant, there is so much mathematical representation used on designing controller. To simplify the system analysis, this study developed the fuy sliding mode controller based on the sliding mode control law 9), 12). This study assumes that the system temperature to be a constant T, and ignore the mass flow rate between the cylinder and tank. The control problem is to get the pressure P b to track the preset pressure P d. The sliding mode index S could be written as: S = P b P d 10) The sliding mode equation becomes Ṡ = Ṗ b 11) Substituting Eqs. 7) and 8) into 9) and setting ṁ t = 0, we can obtain: Ṗ b = kp b ż+ C 2kk v R TP u2 f P D2 / ) u 12) A b Let b = C 2kk v R T, A b the Eq. 12) can be rewritten: Ṗ b = kp b ż+ bp u2 f P D2 / ) u 13) The sliding mode equation can be rewritten: Ṡ = Ṗ b = kp b ż+ bp u2 f P D2 / ) u 14) Assume parameter b may be various in a range: 0 b min b b max Let ˆb = b min b max ) 1/2 and Ṡ = 0, we can get the nominal control law: kp b û = Ż 15) ˆb f P D2 / ) We use the fuy control law u fuy to replace the discontinuous component and the control signal u can be written as: kp b u = û+u fuy = ˆb f P D2 / )ż+u fuy 16) Substituting Eq. 16) into 14), we can get: Ṡ = b/ˆb 1)kP b ż+ b f P D2 / ) u fuy 17) Fig. 7 Fuy membership functions According Lyapunov theory, all the fuy sliding rules should satisfy the requirement as follows: S Ṡ = b/ˆb 1)kP b żs + b f P D2 / ) u fuy S < 0 18) b f P D2 / ) The term in Eq. 18),, is positive. Therefore, in this condition, reducing u fuy leads to decrease S Ṡ as S is positive and increasing u fuy leads to decrease S Ṡ as S is negative. The above statements can be summaried as a rule base for the fuy logic controller and the rule table shown in Table 1. The triangular membership functions for the input S, Ṡ and output u f were shown in Fig. 7 and the center-of-gravity method was used

895 Fig. 8 Scheme of the pneumatic pressure servo control system Fig. 10 Time responses of the system using fuy sliding mode controller with different setting velocity Fig. 9 The time response of the conventional pneumatic counterweight system with different setting velocity for defuification. Figure 8 shows a block diagram of the pneumatic pressure servo control system. 5. Experimental Results The conventional pneumatic counterweight system and the active pneumatic counterweight control system were described above. Their system performance and experimental results are presented as follows. The supply pressure was 600 kpa, and the setting pressure of the lower chamber of cylinder was 320 kpa. The cylinder provides force of 931 N to support the weight of the moved mass. The inner and rod diameter of the cylinder were 0.08 m and 0.03 m, respectively, while the tank volume was 0.08 m 3. The weight of the moved mass driven by linear motor was 95 kg and the stroke was 0.2 m. The different velocity limits are set for the linear motor to clarify the system performance. The sampling frequency was 100 H. Furthermore, the traveling rate was estimated by using velocity estimator based on the least-squares from the discrete position versus time data 13). Figure 9 shows the pressure responses of the counterweight system using the conventional pressure control method. The experimental results show that the higher traveling rate leads to lager pressure variation with the largest pressure variation occurring at the largest traveling rate. The largest pressure variations of the cylinder were 60 kpa in the downward motion and 40 kpa in the upward motion. Moreover, the maximum force change was 259.2 N in the motion process, but the balanced force is just 931 N. Furthermore, the experimental results show that the largest pressure variation in the downward motion exceeds that in the upward motion. The pressure variation can be found to be asymmetrical from experimental results and the asymmetrical conditions of the pressure variation are more obvious for the mass with higher traveling rate. Clearly the motion of the linear motor is involved in these problems, so using the active pressure control method is necessary for the mass with higher traveling rate. Next, this study presents the experimental results using the active pressure control method. The experimental results in Fig. 10 were regulated using the fuy sliding mode control method. The pressure variation e and ė have to be multiplied by the scaling factors, Ge and Gv, sothat the products are normalied in the interval [ 1, 1]. The output signal of the fuy sliding mode controller, u f,is

896 multiplied by the scaling factor, Gu, to correspond with actual operating voltage of the servo valve. The larger scaling factors, Ge and Gv, result in good performance and fast response but may cause oscillatory response. After some trial and error, the scaling factors of the fuy sliding mode control are Ge= 4, Gv = 1.5 andgu= 0.7. In Fig. 10, the largest pressure variation is lower than 15 kpa and the variation of the pneumatic force was lower than 64.8 N in the motion process. These experimental results demonstrate that using the active control method is useful for stabiliing the cylinder pressure and providing more stable actuator force output for the pneumatic counterweight control system. Moreover the pressure variation in the pneumatic cylinder can be controlled to be within the range ±15 kpa in this study. 6. Conclusions This study designed the active pressure control method to modulate the cylinder pressure with a servovalve. The experimental results yield the following conclusions: 1. A lager pressure change would be occurred when the spindle of the machine tools move at a faster speed using the conventional pneumatic counterweight system. 2. The active pressure control method can efficiently reduce the pressure change and provide more stable actuator force output for the pneumatic counterweight system. 3. The pressure variation in the pneumatic cylinder can be controlled to be within ±15 kpa using the active pressure control methods in this study. 4. This study may provide a useful reference for researchers to improve the feed rate and the machining precision for the machine tool. Acknowledgement The support provided by National Science Council of Taiwan NSC 93-2622-E-006-028) is greatly appreciated. References 1 ) Ben-Dov, D. and Dalcudean, S.E., A Force-Controlled Pneumatic Actuator, IEEE Transactions on Robotics and Automation, Vol.11, No.6 1995), pp.906 911. 2 ) Shih, M.C. and Hwang, C.G., Fuy PWM Control the Positions of a Pneumatic Robot Cylinder Using High Speed Solenoid Valves, JSME Int. J., Ser.C, Vol.40, No.3 1997), pp.469 475. 3 ) Shih, M.C. and Ma, M.A., Position Control of a Pneumatic Cylinder Using Fuy PWM Control Method, Mechatronics, Vol.8, No.3 1998), pp.241 253. 4 ) Sorli, M. and Pastorelli, S., Performance of a Pneumatic Force Controlling Servosystem: Influence of Valves Conductance, Robotic and Autonomous Systems, Vol.30, No.3 1995), pp.283 300. 5 ) Khayati, K., Bigras, P. and Dessaint, L.-A., A Robust Feedback Lineariation Force Control of a Pneumatic Actuator, IEEE International Conference on Systems, Man and Cybernetics, 2004), pp.6113 6119. 6 ) Richer, E. and Hurmulu, Y., A High Performance Pneumatic Force Actuator System: Part II Nonlinear Controller Design, Journal of Dynamic Systems, Measurement, and Control, Transactions of the ASME, Vol.122 2000), pp.426 434. 7 ) Shibata, S., Ben-Lamine, M.S., Toyohara, K. and Shimiu, A., Fuy Control of Vertical Pneumatic Servo Systems Using Virtual Reference, JSME Int. J., Ser.C, Vol.42, No.1 1999), pp.79 84. 8 ) Li, T.-H. and Tsai, C.-Y., Parallel Fuy Sliding Mode Control of a Spring Linked Cart-Pole System, IECON Proceedings Industrial Electronics Conference), Vol.1 1998), pp.28 33. 9 ) Lhee, C.-G., Park, J.-S., Ahn, H.-S. and Kim, D.-H., Sliding Mode-Like Fuy Logic Control with Self- Tuning the Dead Zone Parameters, IEEE Transactions on Fuy Systems, Vol.9, No.2 2001), pp.343 348. 10) Richer, E. and Hurmulu, Y., A High Performance Pneumatic Force Actuator System: part I Nonlinear Mathematical Model, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, Vol.122 2000), pp.416 425. 11) Slotine, J.J.E. and Li, W., Applied Nonlinear Control, 1991), Prentice Hall. 12) Renn, J.-C. and Liao, C.-M., A Study on the Speed Control Performance of a Servo-Pneumatic Motor and the Application to Pneumatic Tools, International Journal of Advanced Manufacturing Technology, Vol.23, No.7-8 2004), pp.572 576. 13) Brown, R.H., Schneider, S.C. and Mulligan, M.G., Analysis of Algorithms for Velocity Estimation from Discrete Position Versus Time Data, IEEE Transactions on Industrial Electronics, Vol.39, No.1 1992), pp.11 19. 14) FESTO Inc., Operating Instructions of MPYE-Type Servo Valve, FESTO Inc., 2004).