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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 39 FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH CONTROL OF CNC MACHINE TOOLS KOTARO NAGAOKA and TOMONORI SATO Advanced Technology R&D Center, Mitsubishi Electric Corporation 8-1-1 Tsukaguchi-honmachi, Amagasaki, Hyogo 661-8661, Japan Abstract: A practical feedforward controller for the continuous path control of a CNC machine tool is presented. A servo control system for continuous path co ntrol requires the following behaviors: the prevention of mechanical vibrations, the reduction of path tracking errors, and the making of a symmetric path to reduce errors between the forward and backward paths. To overcome these problems, a method that has a two-degrees-of-freedom structure with a reference model and that is also equipped with a mechanical vibration compensator to suppress vibrations is proposed. The effectiveness of the proposed method is shown in the simulation results. Keywords: Servo Control, Feedforward Control, Continuous Path Control, Vibration, and Symmetry. 1. INTRODUCTION High speed and high accuracy machining is required because of the need for short lead times, high productivity, and complicated workpieces. To meet these requirements, servo control systems for computerized numerical control (CNC) machine tools are demanded to minimize tracking errors even in high speed and high acceleration conditions. In order to minimize errors, the improvements to the servo responses by means of high feedback gains and feedforward controls are attempted. But when a high speed and high acceleration command is given, the high servo response causes a problem wherein mechanical vibration occurs and the path becomes wavy, spoiling the tracking accuracy. In spite of the attempts of the position signal, which is acquired from the detector, to track the commanded position accurately, errors between the path of the machine load and the commanded path are increased because of the low frequency vibration characteristic of the detector position to the load position. Considering that reducing machine weight prevents high stiffness, a sophisticated controller that can attain a high response without causing machine vibration is needed. Although several servo controllers for reducing vibration have been proposed, most of them are applied to point-to-point controls, and few to continuous path controls [Itoh et al, 2004; Pao and Singhose, 1995]. Some methods that use a disturbance observer in a closed loop are proposed [Hori, 1996], but there are difficulties in applying these for industrial uses such as in CNC machine tools because stability might have been lost. The Zero Phase Error Tracking Controller (ZPETC) [Tomizuka, 1994; Torfs et al, 1998] is hard to implement because the complete characteristics of the closed loop system must be known and a high order compensator is necessary. For a servo controller for industrial use such as in CNC machine tools, the vibration suppressing filter (the notch filter), which attenuates the vibration of the load position, is often inserted at the position command signal to suppress mechanical vibrations. This method has the advantage of being easy to implement and easy to tune. Nevertheless, there is the problem of the tracking error becoming larger. Moreover, asymmetric errors, which are errors between the actual forward path and the actual backward path of the same shape, should be reduced to avoid a scratch that is made by a bi-directional scanning feed motion for a die and mold. However, almost all servo control methods hardly consider this problem of asymmetry and end with a poor rate of machining accuracy. This paper presents a feedforward controller that is based on a tw o-degrees-of-freedom controller [Iwasaki et al, 1996]. The controller has a reference model and a mechanical vibration compensator to make the machine load position follow the output of the reference model. The reference model is also designed to consider path symmetry. By using this controller, a reduction in path tracking errors, a highly symmetric path, and vibration suppression are

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 40 achieved. In the next section, the required control characteristics for continuous path control are discussed. In section 3, the structures of the proposed and the conventional methods are described, and the control characteristics are investigated. Furthermore, in section 4, numerical simulations are carried out to compare both control methods. Conclusions are given in section 5. 2. REQUIRED CONTROL CHARACTERISTICS 2.1 Trajectory Error Parameters In this section, the required control characteristics are presented. Trajectory error parameters which show tracking errors are defined as follows to show the performance of a continuous path control of a servo system because tracking errors appear most frequently when the corner shape and the circular shape are commanded. (a) Radial reduction When a circular shape is commanded, the radius of the actual path becomes smaller than the commanded one because of a delay in the servo response of each axis. The radial reduction is defined as this reduction. A smaller radial reduction represents a higher tracking ability. (b) Edge unsharpness When a corner shape is commanded, as a result of a delay in the servo response of each axis, the actual path does not reach the vertex of the commanded corner and becomes an unsharpened edge. In this paper, edge unsharpness is defined as the length from the vertex of a commanded corner to the crossed point of the actual path and the bisector of an angle of the commanded corner. Smaller edge unsharpness represents a higher tracking ability. (c) Asymmetric error If the actual path for a corner shape command is not symmetric with the bisector of an angle of a commanded corner, the actual forward path does not correspond to the actual backward path in a bi-directional movement along the same path. In this paper, the asymmetric error is defined as the maximum value of the error between the forward path and the backward path in the direction of the bisector of an angle of a commanded corner. A description of radial reduction is shown in Fig. 1. Descriptions of edge unsharpness and asymmetric error are shown in Fig. 2. To enhance the performance of a continuous path control, a servo controller for a CNC machine tool is required to possess the following characteristics: a high tracking ability to increase the dimensional accuracy of the workpiece and high symmetry to improve the roughness of the surface in bi-directional cutting. Therefore, all three trajectory error parameters (radial reduction, edge unsharpness, and asymmetric error) are required to be small. Commanded path Actual path Radial reduction Velocity Fig. 1 Radial reduction Angle Commanded path Actual path (forward) Actual path (backward) Asymmetric error Edge unsharpness Velocity Fig. 2 Edge unsharpness and asymmetric error 2.2 Calculation of Error Parameters Equations that can derive error parameters from the transfer function of a servo control system are introduced in this section. It is assumed that the transfer function of the servo control system from the position command to the load position of each axis is the same transfer function () Gs , where is the Laplace operator. To represent the response of the servo control system, the time delay is introduced as follows: 1() lim Gs . (1) (a) Radial reduction Radial reduction is given by ' rGjr , (2) where is the velocity of the circular command, is the commanded radius, and is an imaginary unit. (b) Edge unsharpness Firstly, the time function of the actual path (i.e. the path of the load position) when a corner shaped command is added to the servo control system is derived. The y axis is set to the bisector of an angle of the commanded corner and the x axis is set perpendicular to y axis.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 41 Assuming the position command reaches to the vertex of the corner at , the coordinate value of the position command () ct and () ct is written as follows: () cos ct v t , () sin ct v t , where is the velocity of the corner command along the path and is the exterior angle of the corner. Then, the coordinate value of the actual path () and () becomes () cos ( ) sin ( ) ( 0) () () sin 2 ( 0) t xd pt v t t vtt t pt Gs vttt , where [()] stands for the inverse Laplace transform of () . Consequently, edge unsharpness is given by () ' eyd . (3) (c) Asymmetric error Setting the time function ( ) et to the difference between the forward path and the backward path for the corner command along the y axis, it is said that () () (2 ) yyyd et pt p t t . Thus, the asymmetric error is given by max[ ( )] ' ay et . (4) 3. DESIGN OF SERVO CONTROLLER 3.1 Modeling of the Mechanical System In order to design a controller, a model of the mechanical system is required. There are, however, many vibration modes. In general, the most influential mode is the one that has the lowest eigenfrequency, because the vibration of a higher frequency is attenuated faster. A controller will then be designed to suppress the lowest mode of vibration and the model of the mechanical system is simplified to a two-mass system as shown in Fig. 3 . This model consists of the first mass (mass ), the second mass (mass ), a spring element (stiffness ), and a damper element (viscous constant ). The first mass receives force by an actuator such as a motor. The position of the first mass is detected as the detector position to send to the controller as feedback. The position of the second mass represents the load position which determines the actual path. The transfer function from the actuator force to the detector position is written as 21 () 21 fd Gs Ms , and the one from the detector position to the load position is written as () 21 dl Gs , where is the resonant frequency, is the anti- resonant frequency, is the resonant damping ratio, is the anti-resonant damping ratio, and is the total mass, which is given by 12 11 rf MM , , 12 11 MM , , 12 MM . Actuator force Detector position Load position Mass 1 Mass 2 Fig. 3 Model of mechanical system 3.2 Structure of the Conventional Method The conventional controller for the servo control of a CNC machine tool is shown in Fig. 4 . In this figure, is the commanded position, is the detector position, is the load position, dis is the disturbance force, () Cs is the velocity loop compensator, () Cs is the position loop compensator, () Gs is the notch filter, and and are the coefficients that change the efficiency of the feedforward control. The notch filter removes those vibrations with frequencies that are resonant frequencies of the closed loop system and suppresses the vibration of the load position. Ordinarily, proportional (P) and proportional- integrate (PI) compensators are used as position and velocity loop compensators, respectively. Although there are various ways to design notch filters, a bi-quad filter is generally used [Dumetz et al, 2001; Ellis, 1991]. This filter is explained as follows:

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 42 12 () 21 Gs , where is the notch frequency and is the notch damping ratio. Feedforward controller Feedback controller Mechanical system Conventional controller () Gs () Cs () Cs () dl Gs () fd Gs dis Fig. 4 Conventional Servo Control Method An FIR filter, such as a moving average filter can be also used as a notch filter, but it is not adopted in this paper because the radial reduction becomes large and it requires a lot of memory to implement. 3.3 Analysis of the Conventional Method The transfer function of the whole control system (i.e. the transfer function from the position command to the load position) is necessary to compute the trajectory error parameters and to estimate its control abilities. The transfer function is given by 22 22 21 () () 21 2 1 () 11 ff na fb nv nv ss Gs Gs Gs sss ZZ ZZ ZZ , where and are the resonant frequency and the damping ratio of the closed loop system, respectively, and () fb Gs and () ff Gs are polynomials which are determined by the parameters of the feedback system. Assuming that and are set equally to and , respectively, and ignoring the higher order terms to observe the behavior of the lower frequency mode, it is approximated that () 21 Gs . (5) The analytic trajectory error parameters are then calculated as follows: The time delay in Eq. (1) is obtained by . The radial reduction in Eq. (2) is obtained by 24 24 21 ZZ ' nn vv rr . If the angular velocity of the command ( vr ) is much smaller than , the following approximation is carried out: 22 11 ' rd vv rr . The edge unsharpness in Eq. (3) is obtained by sin 0.54 sin 22 ' ed vtv . The asymmetric error in Eq. (4) is obtained by 0.309 sin 0.155 sin 22 ' ad vtv . 3.4 Structure of the Proposed Method In the conventional method, the response of the whole control system is fixed by the notch frequency. If the resonant frequency of the closed loop system is lower, then the response speed must be lower and the asymmetric error must be larger to prevent vibration. In the proposed method, a two-degrees-of-free dom control with the reference model [Koyama and Yano, 1991] is used so that the feedforward controller and the feedback controller can be designed independently. The feedforward controller is designed to achieve vibration suppression and a symmetric path. A proposed servo controller is shown in Fig. 5 . In this figure, () Gs is the reference model, () Gs is the position feedforward compensator, () Gs is the velocity feedforward compensator, and () Gs is the force feedforward compensator. The feedback controller is the same as that of a conventional controller.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 43 Feedforward controller Feedback controller Mechanical system Proposed controller () Cs () Cs () dl Gs () fd Gs () Gs () Gs () Gs () Gs dis Fig. 5 Proposed Servo Control Method Position, velocity, and force feedforward compensators are designed from a model of the mechanical system. Introducing a model transfer function from the actuator force to the detector position () fdm Gs and from the detector position to the load position () dlm Gs , each feedforward compensator is designed as follows: () () cdlm Gs G s , () () cdlm Gs sG s , 11 () () () cfdmdlm Gs G sG s . Assuming that dis () () fdm fd GsGs , and () () dlm dl GsGs , a transfer function from the position command to the load position is obtained by () () Gs G s . Namely, the response characteristics of the whole system are determined only by the reference model and not influenced by the feedback controller. To design a reference model, the following things must be considered: (a) Maintaining symmetry from input to output to improve the symmetry of the actual path. (b) Having a high frequency cutoff characteristic for the rejection of noise and high order vibration from the commanded signal. A linear phase characteristic across the passband is necessary to keep symm etry. Bessel filters are well-known filters that possess this characteristic [Carlitz, 1957; Franklin et al, 1994]. Then, a reference model is designed into a lowpass filter that has the pole placement of a Bessel filter. A higher order Bessel filter achieves a higher symmetry and a higher cutoff characteristic, but is more difficult to implement. On the other hand, considering the order of the numerators of a feedforward compensator can be up to four if the model of the mechanical system is a two-mass system, the order of the reference model must be larger than four to make the outputs of the feedforward compensators smooth. 3.5 Analysis of the Proposed Method A lowpass filter, which has the pole placement of a fifth order Bessel filter, is used as a reference model. Then, the transfer function of the reference model is written as 23 4 5 23 4 5 () 41 1 1 9963945 rr r r Gs ss s s KK K K , where is the gain of the reference model. Thus, the analytic trajectory error parameters are calculated as follows: The time delay in Eq. (1) is obtained by . The radial reduction in Eq. (2) is obtained by ' Kr , where 24 6 8 10 () 1 9 567 2835 59535 893025 ZZZZZ . If the angular velocity of the command ( vr ) is much smaller than , the following approximation is carried out: 22 11 18 18 ' rd vv rr . The edge unsharpness in Eq. (3) is obtained by 0.276655 sin 0.276655 sin 22 ' ed vtv . The asymmetric error in Eq. (4) is obtained by 0.00884626 sin 0.00884626 sin tv ' . 4. COMPARISON BETWEEN TWO METHODS 4.1 Analytic Comparison In this section, the tracking control ability of the proposed method is compared to the conventional

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 44 method. Each of the trajectory error parameters calculated in section 3.3 and section 3.5 are summarized in Table 1. The path of the corner shaped command for the conventional method and the proposed method is shown in Fig. 6 and Fig. 7 , respectively. The coordinate value of x axis and y axis are normalized by the values cos( ) tv and sin( ) tv , respectively. According to these table and figures, it is clear that the proposed method has a higher tracking ability and greater symmetry than the conventional method. Table 1 Comparison of trajectory error parameters Convention- al method (A) Proposed method (B) Ratio (B/A) (/) vr 0.25 0.056 0.22 sin( / 2) 0.54 0.28 0.51 sin( / 2) 0.15 0.0088 0.06 V XEQU V XUK P EQOOCPFGFRCVJ CEVWCN RCVJ Fig. 6 Corner path in conventional method V XEQU V XUK P EQOOCPFGFRCVJ CEVWCN RCVJ Fig. 7 Corner path in proposed method 4.2 Case Study Numerical simulations are carried out to compare the two methods. However, while the mechanical model for designing the controller is a two-mass system, the actual mechanical system for the simulation is set as a three- mass system, in order to estimate the influence of modeling errors. The parameters of the actual mechanical system are shown in Table 2. In addition, the Bode diagram of the mechanical system is shown in Fig. 8 . The commanded path for the simulations is defined as follows: Corner shaped, a feedrate of 5mm/s, and a corner angle of /2 rad. In the conventional method, the frequency and the damping ratio of the notch filter is set to be the same as those of the first vibration mode, and the feedforward ratio and varies from 0.7 to 1.05, where . In the proposed method, the parameters of position, velocity, and force feedforward compensators are determined by the resonant frequency, the anti-resonant frequency, and the damping ratio of the first vibration mode of the mechanical system. The gain of the reference model varies from 30 rad/s to 55 rad/s. The simulations are executed using MATLAB/Simulink. Table 2 Parameters of mechanical system 1st mode 2nd mode Resonant frequency 200rad/s 400rad/s Anti-resonant frequency 150rad/s 300rad/s Damping ratio 0.15 0.05 /CIPK VWFGF$ 2JCUGFGI HTQO VQZ HTQO VQZ (TG WGPE T CFUGE Fig. 8 Bode diagram of mechanical system The relationship between vibration amplitude and edge unsharpness, which is obtained by the simulation, is shown in Fig. 9 . In this paper, vibration amplitude is defined as the maximum amplitude of the actual path in the perpendicular direction of the commanded path after the

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 45 commanded path passes the vertex of the corner. In both methods, vibration amplitude becomes larger when edge unsharpness becomes smaller, but when comparing the same vibration amplitude, the values of edge unsharpness in the proposed method are half of those in the conventional method. The relationship between vibration amplitude and asymmetric error, which is obtained by the simulation, is shown in Fig. 10 . The asymmetric error becomes larger when the vibration amplitude becomes smaller in the conventional method, but both the asymmetric error and the vibration amplitude are smaller in the proposed method. The actual path in each method (in the case in which the vibration amplitude is 1.5 m) is depicted in Fig. 11. Edge unsharpness and asymmetric error become smaller in the proposed method. XK DTCVK QP O EQTPGTGTTQT O RTQRQUGFOGVJQF EQPXGPVK QPCN OGVJQF Fig. 9 Simulation results of relationship between vibration and corner error XK DTCVK QP O CU[OOGVTK EGTTQT O RTQRQUGFOGVJQF EQPXGPVK QPCN OGVJQF Fig. 10 Simulation results of relationship between vibration and asymmetric error ZOO [OO EQOOCPFGFRCVJ CEVWCN RCVJ (a) Conventional method ( 0.75 ) ZOO [OO EQOOCPFGFRCVJ CEVWCN RCVJ (b) Proposed method ( 45 rad/s) Fig. 11 Simulated paths in both methods 5. CONCLUSIONS A practical control method for the continuous path control of a CNC machine tool was proposed. The method featured a two-de grees-of-freedom control system and a compensator for the suppression of mechanical vibration. The results of the analysis and the numerical simulation show that the trajectory error parameters such as radial reduction, edge unsharpness, asymmetric error, and vibration amplitude are totally reduced by the proposed method. REFERENCES Carlitz L. 1957, “A Note On The Bessel Polynomials”. In Duke Math Journal , vol. 24, Pp151-162. Dumetz E., Hende F. V., and Barre P. J. 2001, “Resonant Load Control Methods Application to High-Speed Machine Tool with Linear Motor”. In Proceedings of 8 th IEEE International Conference on Emerging Technology and Factory Automation , vol. 2, Pp 23-31. Ellis G. 1991, In Control System Design Guide , 2 nd ed. San Diego, CA: Academic Press.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 46 Franklin G. F., Powell J. D., and Emami-Naeini A. 1994, Feedback Control of Dynamic Systems , 3rd ed. Reading, MA: Addison-Wesley. Hori Y. 1996, “A Review of Torsional Vibration Control Methods and a Proposal of Disturbance Observer- based New Techniques”. In Proceedings of 13 th IFAC World Congress , Pp7-13. Itoh K., Iwasaki M., and Matsui N. 2004, “Optimal Design of Robust Vibration Suppression Controller Using Genetic Algorithms”. In IEEE Transactions on Industrial Electronics , vol. 51, no. 5, Pp947-953. Iwasaki T., Sato T., Morita A., and Maruyama H. 1996, “Auto-Tuning of Two-Degree-of-Freedom Motor Control for High-Accuracy Trajectory Motion”. In Control Engineering Practice , vol. 4, no. 4, Pp537-544. Koyama M. and Yano M. 1991, “Two Degrees of Freedom Speed Controller using Reference System Model for Motor Drives”. In Proceedings of 4 th European Conference on Power Electronics and Applications , Pp60-65. Pao L. Y. and Singhose W. E. 1995, “On the Equivalence of Minimum Time Input Shaping with Traditional Time-Optical Control”. In Proceedings of 4 th IEEE Conference on Control Applications , Pp1120-1125. Tomizuka M. 1994, “Zero Phase Error Tracking Algorithm for Digital Control”. ASME Journal of Dynamic Systems, Measurement, and Control , vol. 109, no. 1, Pp583-592. Torfs D. E., Vuerinckx R., Swevers J., and Schoukens J. 1998, “Comparison of Two Feedforward Design Methods Aiming at Accurate Trajectory Tracking of the End Point of a Flexible Robot Arm”. In IEEE Transactions on Control Systems Technology , vol. 6, no. 1, Pp2-14. BIOGRAPHY Kotaro Nagaoka received his Master of Engineering degree from Kyoto University in 2001. He is currently a researcher in Advanced Technology R&D Center, Mitsubishi Electric Corporation and working on development of motion control systems for factory automation. Tomonori SATO received the B.E. and M.E in Engineering from Kyoto University, Japan, in 1990 and 1992 respectively. He joined the R&D Lab. in Mitsubishi Electric Co. in 1992. He received the Ph.D. degree in Engineering from Kyoto University, Japan in 2003. His research interests include motion control, machining process control and servo control for machine tool.

NAGAOKA et al FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH IJ of SIMULATION Vol 7 No 8 ISSN 1473804x online 14738031 print 39 FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH CONTROL OF CNC MACHINE TOOLS KOTARO NAGA ID: 23181

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 39 FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH CONTROL OF CNC MACHINE TOOLS KOTARO NAGAOKA and TOMONORI SATO Advanced Technology R&D Center, Mitsubishi Electric Corporation 8-1-1 Tsukaguchi-honmachi, Amagasaki, Hyogo 661-8661, Japan Abstract: A practical feedforward controller for the continuous path control of a CNC machine tool is presented. A servo control system for continuous path co ntrol requires the following behaviors: the prevention of mechanical vibrations, the reduction of path tracking errors, and the making of a symmetric path to reduce errors between the forward and backward paths. To overcome these problems, a method that has a two-degrees-of-freedom structure with a reference model and that is also equipped with a mechanical vibration compensator to suppress vibrations is proposed. The effectiveness of the proposed method is shown in the simulation results. Keywords: Servo Control, Feedforward Control, Continuous Path Control, Vibration, and Symmetry. 1. INTRODUCTION High speed and high accuracy machining is required because of the need for short lead times, high productivity, and complicated workpieces. To meet these requirements, servo control systems for computerized numerical control (CNC) machine tools are demanded to minimize tracking errors even in high speed and high acceleration conditions. In order to minimize errors, the improvements to the servo responses by means of high feedback gains and feedforward controls are attempted. But when a high speed and high acceleration command is given, the high servo response causes a problem wherein mechanical vibration occurs and the path becomes wavy, spoiling the tracking accuracy. In spite of the attempts of the position signal, which is acquired from the detector, to track the commanded position accurately, errors between the path of the machine load and the commanded path are increased because of the low frequency vibration characteristic of the detector position to the load position. Considering that reducing machine weight prevents high stiffness, a sophisticated controller that can attain a high response without causing machine vibration is needed. Although several servo controllers for reducing vibration have been proposed, most of them are applied to point-to-point controls, and few to continuous path controls [Itoh et al, 2004; Pao and Singhose, 1995]. Some methods that use a disturbance observer in a closed loop are proposed [Hori, 1996], but there are difficulties in applying these for industrial uses such as in CNC machine tools because stability might have been lost. The Zero Phase Error Tracking Controller (ZPETC) [Tomizuka, 1994; Torfs et al, 1998] is hard to implement because the complete characteristics of the closed loop system must be known and a high order compensator is necessary. For a servo controller for industrial use such as in CNC machine tools, the vibration suppressing filter (the notch filter), which attenuates the vibration of the load position, is often inserted at the position command signal to suppress mechanical vibrations. This method has the advantage of being easy to implement and easy to tune. Nevertheless, there is the problem of the tracking error becoming larger. Moreover, asymmetric errors, which are errors between the actual forward path and the actual backward path of the same shape, should be reduced to avoid a scratch that is made by a bi-directional scanning feed motion for a die and mold. However, almost all servo control methods hardly consider this problem of asymmetry and end with a poor rate of machining accuracy. This paper presents a feedforward controller that is based on a tw o-degrees-of-freedom controller [Iwasaki et al, 1996]. The controller has a reference model and a mechanical vibration compensator to make the machine load position follow the output of the reference model. The reference model is also designed to consider path symmetry. By using this controller, a reduction in path tracking errors, a highly symmetric path, and vibration suppression are

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 40 achieved. In the next section, the required control characteristics for continuous path control are discussed. In section 3, the structures of the proposed and the conventional methods are described, and the control characteristics are investigated. Furthermore, in section 4, numerical simulations are carried out to compare both control methods. Conclusions are given in section 5. 2. REQUIRED CONTROL CHARACTERISTICS 2.1 Trajectory Error Parameters In this section, the required control characteristics are presented. Trajectory error parameters which show tracking errors are defined as follows to show the performance of a continuous path control of a servo system because tracking errors appear most frequently when the corner shape and the circular shape are commanded. (a) Radial reduction When a circular shape is commanded, the radius of the actual path becomes smaller than the commanded one because of a delay in the servo response of each axis. The radial reduction is defined as this reduction. A smaller radial reduction represents a higher tracking ability. (b) Edge unsharpness When a corner shape is commanded, as a result of a delay in the servo response of each axis, the actual path does not reach the vertex of the commanded corner and becomes an unsharpened edge. In this paper, edge unsharpness is defined as the length from the vertex of a commanded corner to the crossed point of the actual path and the bisector of an angle of the commanded corner. Smaller edge unsharpness represents a higher tracking ability. (c) Asymmetric error If the actual path for a corner shape command is not symmetric with the bisector of an angle of a commanded corner, the actual forward path does not correspond to the actual backward path in a bi-directional movement along the same path. In this paper, the asymmetric error is defined as the maximum value of the error between the forward path and the backward path in the direction of the bisector of an angle of a commanded corner. A description of radial reduction is shown in Fig. 1. Descriptions of edge unsharpness and asymmetric error are shown in Fig. 2. To enhance the performance of a continuous path control, a servo controller for a CNC machine tool is required to possess the following characteristics: a high tracking ability to increase the dimensional accuracy of the workpiece and high symmetry to improve the roughness of the surface in bi-directional cutting. Therefore, all three trajectory error parameters (radial reduction, edge unsharpness, and asymmetric error) are required to be small. Commanded path Actual path Radial reduction Velocity Fig. 1 Radial reduction Angle Commanded path Actual path (forward) Actual path (backward) Asymmetric error Edge unsharpness Velocity Fig. 2 Edge unsharpness and asymmetric error 2.2 Calculation of Error Parameters Equations that can derive error parameters from the transfer function of a servo control system are introduced in this section. It is assumed that the transfer function of the servo control system from the position command to the load position of each axis is the same transfer function () Gs , where is the Laplace operator. To represent the response of the servo control system, the time delay is introduced as follows: 1() lim Gs . (1) (a) Radial reduction Radial reduction is given by ' rGjr , (2) where is the velocity of the circular command, is the commanded radius, and is an imaginary unit. (b) Edge unsharpness Firstly, the time function of the actual path (i.e. the path of the load position) when a corner shaped command is added to the servo control system is derived. The y axis is set to the bisector of an angle of the commanded corner and the x axis is set perpendicular to y axis.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 41 Assuming the position command reaches to the vertex of the corner at , the coordinate value of the position command () ct and () ct is written as follows: () cos ct v t , () sin ct v t , where is the velocity of the corner command along the path and is the exterior angle of the corner. Then, the coordinate value of the actual path () and () becomes () cos ( ) sin ( ) ( 0) () () sin 2 ( 0) t xd pt v t t vtt t pt Gs vttt , where [()] stands for the inverse Laplace transform of () . Consequently, edge unsharpness is given by () ' eyd . (3) (c) Asymmetric error Setting the time function ( ) et to the difference between the forward path and the backward path for the corner command along the y axis, it is said that () () (2 ) yyyd et pt p t t . Thus, the asymmetric error is given by max[ ( )] ' ay et . (4) 3. DESIGN OF SERVO CONTROLLER 3.1 Modeling of the Mechanical System In order to design a controller, a model of the mechanical system is required. There are, however, many vibration modes. In general, the most influential mode is the one that has the lowest eigenfrequency, because the vibration of a higher frequency is attenuated faster. A controller will then be designed to suppress the lowest mode of vibration and the model of the mechanical system is simplified to a two-mass system as shown in Fig. 3 . This model consists of the first mass (mass ), the second mass (mass ), a spring element (stiffness ), and a damper element (viscous constant ). The first mass receives force by an actuator such as a motor. The position of the first mass is detected as the detector position to send to the controller as feedback. The position of the second mass represents the load position which determines the actual path. The transfer function from the actuator force to the detector position is written as 21 () 21 fd Gs Ms , and the one from the detector position to the load position is written as () 21 dl Gs , where is the resonant frequency, is the anti- resonant frequency, is the resonant damping ratio, is the anti-resonant damping ratio, and is the total mass, which is given by 12 11 rf MM , , 12 11 MM , , 12 MM . Actuator force Detector position Load position Mass 1 Mass 2 Fig. 3 Model of mechanical system 3.2 Structure of the Conventional Method The conventional controller for the servo control of a CNC machine tool is shown in Fig. 4 . In this figure, is the commanded position, is the detector position, is the load position, dis is the disturbance force, () Cs is the velocity loop compensator, () Cs is the position loop compensator, () Gs is the notch filter, and and are the coefficients that change the efficiency of the feedforward control. The notch filter removes those vibrations with frequencies that are resonant frequencies of the closed loop system and suppresses the vibration of the load position. Ordinarily, proportional (P) and proportional- integrate (PI) compensators are used as position and velocity loop compensators, respectively. Although there are various ways to design notch filters, a bi-quad filter is generally used [Dumetz et al, 2001; Ellis, 1991]. This filter is explained as follows:

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 42 12 () 21 Gs , where is the notch frequency and is the notch damping ratio. Feedforward controller Feedback controller Mechanical system Conventional controller () Gs () Cs () Cs () dl Gs () fd Gs dis Fig. 4 Conventional Servo Control Method An FIR filter, such as a moving average filter can be also used as a notch filter, but it is not adopted in this paper because the radial reduction becomes large and it requires a lot of memory to implement. 3.3 Analysis of the Conventional Method The transfer function of the whole control system (i.e. the transfer function from the position command to the load position) is necessary to compute the trajectory error parameters and to estimate its control abilities. The transfer function is given by 22 22 21 () () 21 2 1 () 11 ff na fb nv nv ss Gs Gs Gs sss ZZ ZZ ZZ , where and are the resonant frequency and the damping ratio of the closed loop system, respectively, and () fb Gs and () ff Gs are polynomials which are determined by the parameters of the feedback system. Assuming that and are set equally to and , respectively, and ignoring the higher order terms to observe the behavior of the lower frequency mode, it is approximated that () 21 Gs . (5) The analytic trajectory error parameters are then calculated as follows: The time delay in Eq. (1) is obtained by . The radial reduction in Eq. (2) is obtained by 24 24 21 ZZ ' nn vv rr . If the angular velocity of the command ( vr ) is much smaller than , the following approximation is carried out: 22 11 ' rd vv rr . The edge unsharpness in Eq. (3) is obtained by sin 0.54 sin 22 ' ed vtv . The asymmetric error in Eq. (4) is obtained by 0.309 sin 0.155 sin 22 ' ad vtv . 3.4 Structure of the Proposed Method In the conventional method, the response of the whole control system is fixed by the notch frequency. If the resonant frequency of the closed loop system is lower, then the response speed must be lower and the asymmetric error must be larger to prevent vibration. In the proposed method, a two-degrees-of-free dom control with the reference model [Koyama and Yano, 1991] is used so that the feedforward controller and the feedback controller can be designed independently. The feedforward controller is designed to achieve vibration suppression and a symmetric path. A proposed servo controller is shown in Fig. 5 . In this figure, () Gs is the reference model, () Gs is the position feedforward compensator, () Gs is the velocity feedforward compensator, and () Gs is the force feedforward compensator. The feedback controller is the same as that of a conventional controller.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 43 Feedforward controller Feedback controller Mechanical system Proposed controller () Cs () Cs () dl Gs () fd Gs () Gs () Gs () Gs () Gs dis Fig. 5 Proposed Servo Control Method Position, velocity, and force feedforward compensators are designed from a model of the mechanical system. Introducing a model transfer function from the actuator force to the detector position () fdm Gs and from the detector position to the load position () dlm Gs , each feedforward compensator is designed as follows: () () cdlm Gs G s , () () cdlm Gs sG s , 11 () () () cfdmdlm Gs G sG s . Assuming that dis () () fdm fd GsGs , and () () dlm dl GsGs , a transfer function from the position command to the load position is obtained by () () Gs G s . Namely, the response characteristics of the whole system are determined only by the reference model and not influenced by the feedback controller. To design a reference model, the following things must be considered: (a) Maintaining symmetry from input to output to improve the symmetry of the actual path. (b) Having a high frequency cutoff characteristic for the rejection of noise and high order vibration from the commanded signal. A linear phase characteristic across the passband is necessary to keep symm etry. Bessel filters are well-known filters that possess this characteristic [Carlitz, 1957; Franklin et al, 1994]. Then, a reference model is designed into a lowpass filter that has the pole placement of a Bessel filter. A higher order Bessel filter achieves a higher symmetry and a higher cutoff characteristic, but is more difficult to implement. On the other hand, considering the order of the numerators of a feedforward compensator can be up to four if the model of the mechanical system is a two-mass system, the order of the reference model must be larger than four to make the outputs of the feedforward compensators smooth. 3.5 Analysis of the Proposed Method A lowpass filter, which has the pole placement of a fifth order Bessel filter, is used as a reference model. Then, the transfer function of the reference model is written as 23 4 5 23 4 5 () 41 1 1 9963945 rr r r Gs ss s s KK K K , where is the gain of the reference model. Thus, the analytic trajectory error parameters are calculated as follows: The time delay in Eq. (1) is obtained by . The radial reduction in Eq. (2) is obtained by ' Kr , where 24 6 8 10 () 1 9 567 2835 59535 893025 ZZZZZ . If the angular velocity of the command ( vr ) is much smaller than , the following approximation is carried out: 22 11 18 18 ' rd vv rr . The edge unsharpness in Eq. (3) is obtained by 0.276655 sin 0.276655 sin 22 ' ed vtv . The asymmetric error in Eq. (4) is obtained by 0.00884626 sin 0.00884626 sin tv ' . 4. COMPARISON BETWEEN TWO METHODS 4.1 Analytic Comparison In this section, the tracking control ability of the proposed method is compared to the conventional

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 44 method. Each of the trajectory error parameters calculated in section 3.3 and section 3.5 are summarized in Table 1. The path of the corner shaped command for the conventional method and the proposed method is shown in Fig. 6 and Fig. 7 , respectively. The coordinate value of x axis and y axis are normalized by the values cos( ) tv and sin( ) tv , respectively. According to these table and figures, it is clear that the proposed method has a higher tracking ability and greater symmetry than the conventional method. Table 1 Comparison of trajectory error parameters Convention- al method (A) Proposed method (B) Ratio (B/A) (/) vr 0.25 0.056 0.22 sin( / 2) 0.54 0.28 0.51 sin( / 2) 0.15 0.0088 0.06 V XEQU V XUK P EQOOCPFGFRCVJ CEVWCN RCVJ Fig. 6 Corner path in conventional method V XEQU V XUK P EQOOCPFGFRCVJ CEVWCN RCVJ Fig. 7 Corner path in proposed method 4.2 Case Study Numerical simulations are carried out to compare the two methods. However, while the mechanical model for designing the controller is a two-mass system, the actual mechanical system for the simulation is set as a three- mass system, in order to estimate the influence of modeling errors. The parameters of the actual mechanical system are shown in Table 2. In addition, the Bode diagram of the mechanical system is shown in Fig. 8 . The commanded path for the simulations is defined as follows: Corner shaped, a feedrate of 5mm/s, and a corner angle of /2 rad. In the conventional method, the frequency and the damping ratio of the notch filter is set to be the same as those of the first vibration mode, and the feedforward ratio and varies from 0.7 to 1.05, where . In the proposed method, the parameters of position, velocity, and force feedforward compensators are determined by the resonant frequency, the anti-resonant frequency, and the damping ratio of the first vibration mode of the mechanical system. The gain of the reference model varies from 30 rad/s to 55 rad/s. The simulations are executed using MATLAB/Simulink. Table 2 Parameters of mechanical system 1st mode 2nd mode Resonant frequency 200rad/s 400rad/s Anti-resonant frequency 150rad/s 300rad/s Damping ratio 0.15 0.05 /CIPK VWFGF$ 2JCUGFGI HTQO VQZ HTQO VQZ (TG WGPE T CFUGE Fig. 8 Bode diagram of mechanical system The relationship between vibration amplitude and edge unsharpness, which is obtained by the simulation, is shown in Fig. 9 . In this paper, vibration amplitude is defined as the maximum amplitude of the actual path in the perpendicular direction of the commanded path after the

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 45 commanded path passes the vertex of the corner. In both methods, vibration amplitude becomes larger when edge unsharpness becomes smaller, but when comparing the same vibration amplitude, the values of edge unsharpness in the proposed method are half of those in the conventional method. The relationship between vibration amplitude and asymmetric error, which is obtained by the simulation, is shown in Fig. 10 . The asymmetric error becomes larger when the vibration amplitude becomes smaller in the conventional method, but both the asymmetric error and the vibration amplitude are smaller in the proposed method. The actual path in each method (in the case in which the vibration amplitude is 1.5 m) is depicted in Fig. 11. Edge unsharpness and asymmetric error become smaller in the proposed method. XK DTCVK QP O EQTPGTGTTQT O RTQRQUGFOGVJQF EQPXGPVK QPCN OGVJQF Fig. 9 Simulation results of relationship between vibration and corner error XK DTCVK QP O CU[OOGVTK EGTTQT O RTQRQUGFOGVJQF EQPXGPVK QPCN OGVJQF Fig. 10 Simulation results of relationship between vibration and asymmetric error ZOO [OO EQOOCPFGFRCVJ CEVWCN RCVJ (a) Conventional method ( 0.75 ) ZOO [OO EQOOCPFGFRCVJ CEVWCN RCVJ (b) Proposed method ( 45 rad/s) Fig. 11 Simulated paths in both methods 5. CONCLUSIONS A practical control method for the continuous path control of a CNC machine tool was proposed. The method featured a two-de grees-of-freedom control system and a compensator for the suppression of mechanical vibration. The results of the analysis and the numerical simulation show that the trajectory error parameters such as radial reduction, edge unsharpness, asymmetric error, and vibration amplitude are totally reduced by the proposed method. REFERENCES Carlitz L. 1957, “A Note On The Bessel Polynomials”. In Duke Math Journal , vol. 24, Pp151-162. Dumetz E., Hende F. V., and Barre P. J. 2001, “Resonant Load Control Methods Application to High-Speed Machine Tool with Linear Motor”. In Proceedings of 8 th IEEE International Conference on Emerging Technology and Factory Automation , vol. 2, Pp 23-31. Ellis G. 1991, In Control System Design Guide , 2 nd ed. San Diego, CA: Academic Press.

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K. NAGAOKA et al: FEEDFORWARD CONTROLLER FOR CONTINUOUS PATH I.J of SIMULATION, Vol. 7, No. 8 ISSN 1473-804x online, 1473-8031 print 46 Franklin G. F., Powell J. D., and Emami-Naeini A. 1994, Feedback Control of Dynamic Systems , 3rd ed. Reading, MA: Addison-Wesley. Hori Y. 1996, “A Review of Torsional Vibration Control Methods and a Proposal of Disturbance Observer- based New Techniques”. In Proceedings of 13 th IFAC World Congress , Pp7-13. Itoh K., Iwasaki M., and Matsui N. 2004, “Optimal Design of Robust Vibration Suppression Controller Using Genetic Algorithms”. In IEEE Transactions on Industrial Electronics , vol. 51, no. 5, Pp947-953. Iwasaki T., Sato T., Morita A., and Maruyama H. 1996, “Auto-Tuning of Two-Degree-of-Freedom Motor Control for High-Accuracy Trajectory Motion”. In Control Engineering Practice , vol. 4, no. 4, Pp537-544. Koyama M. and Yano M. 1991, “Two Degrees of Freedom Speed Controller using Reference System Model for Motor Drives”. In Proceedings of 4 th European Conference on Power Electronics and Applications , Pp60-65. Pao L. Y. and Singhose W. E. 1995, “On the Equivalence of Minimum Time Input Shaping with Traditional Time-Optical Control”. In Proceedings of 4 th IEEE Conference on Control Applications , Pp1120-1125. Tomizuka M. 1994, “Zero Phase Error Tracking Algorithm for Digital Control”. ASME Journal of Dynamic Systems, Measurement, and Control , vol. 109, no. 1, Pp583-592. Torfs D. E., Vuerinckx R., Swevers J., and Schoukens J. 1998, “Comparison of Two Feedforward Design Methods Aiming at Accurate Trajectory Tracking of the End Point of a Flexible Robot Arm”. In IEEE Transactions on Control Systems Technology , vol. 6, no. 1, Pp2-14. BIOGRAPHY Kotaro Nagaoka received his Master of Engineering degree from Kyoto University in 2001. He is currently a researcher in Advanced Technology R&D Center, Mitsubishi Electric Corporation and working on development of motion control systems for factory automation. Tomonori SATO received the B.E. and M.E in Engineering from Kyoto University, Japan, in 1990 and 1992 respectively. He joined the R&D Lab. in Mitsubishi Electric Co. in 1992. He received the Ph.D. degree in Engineering from Kyoto University, Japan in 2003. His research interests include motion control, machining process control and servo control for machine tool.

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