Nano positioning control for dual stage using minimum order observer

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Journal of Mechanical Science and Technology 26 (3) (212) 941~947 www.springerlink.co/content/1738-494x DOI 1.

Received Noveber 9 21; Revised June 3 211; Accepted Deceber 6 211)

1 Journal of Mechanical Science and Technology 26 (3) (212) 941~947 DOI 1.17/s Nano positioning control for dual stage using iniu order observer Hong Gun Ki * Departent of Mechanical and Autootive Engineering Jeonju University Jeonju Korea (Manuscript Received Noveber 9 21; Revised June 3 211; Accepted Deceber 6 211) Abstract A nano positioning control is developed using the ultra-precision positioning apparatus such as actuator sensor guide power transission eleent with an appropriate control ethod. Using established procedures a single plane X-Y stage with ultra-precision positioning is anufactured. A global stage for aterialization with robust syste is cobined by using an AC servo otor with a ball screw and rolling guide. An ultra-precision positioning syste is developed using a icro stage with an elastic hinge and piezo eleent. Global and icro servos for positioning with nanoeter accuracy are controlled siultaneously using an increental encoder and a laser interferoeter to easure displaceent. Using established procedures an ultra-precision positioning syste (1 stroke and ±1 n positioning accuracy) with a single plane X-Y stage is fabricated. Its perforance is evaluated through siulation using Matlab. After analyzing previous control algoriths and adapting odern control theory a dual servo algorith is developed for a iniu order observer to secure the stability and priority on the controller. The siulations and experients on the ultra precision positioning and the stability of the ultra-precision positioning syste with single plane X-Y stage and the priority of the control algorith are secured by using Matlab with Siulink and ControlDesk ade in dspace. Keywords: Nano; Positioning; Ultra-precision; Stage; Actuator; Sensor Introduction This paper was recoended for publication in revised for by Associate Editor Moon Ki Ki * Corresponding author. Tel.: Fax.: E-ail address: hki@jj.ac.kr KSME & Springer 212 Nanotechnology can be applied to ost industrial fields including electronics inforation and counication echanics cheistry bioengineering and energy and it exhibits a rearkable potential to change huan civilization [1]. Ultraprecision positioning is a very iportant technology in the developent of nano systes and it encopasses a host of technologies including echanics electronics optics control design and processing. With ultra-precision positioning scanning tunneling icroscopy (STM) or atoic force icroscopy (AFM) easureents and anipulations at the single-ato level and operations in a sall area of few square icrons are possible. Currently in the fields of ultra-icro processing ultra-precision easureent seiconductor wafers optical counication and opto-agnetic eory the deand is for ultra-precision positioning allowing long strokes of hundreds of illieters while aintaining n-level precision. The basic technology to satisfy this deand is urgently needed [1-4]. Recent research into nano positioning concentrated on nano echaniss and control ethods. For exaple the ain ephasis of Lee s research [5] is the developent of a control odel for a dual servo stage with the ai of achieving an overall positioning reproducibility of 1 n for alignent of 2 diaeter wafers. In a previous study [6] the focus was on developing an ultra-precision cutting unit (UPCU) that incorporated 3-axis control using piezo-electric actuators and on enhancing the precision of the current lathe used for ductile ode achining of hardened-brittle aterial where the achining was based on the single crystal diaond. The achining data presented by H. S. Ki et al. show that their proposed approach allows the fabrication of aluinu irrors 62 in diaeter to a for accuracy of.7 μ (peak-to-valley error) [7]. In addition to this uch research has been done on nano positioning echaniss and control ethods to realize nano order control [8-14]. Therefore in response to the deands in the field a ethod is presented to produce an ultra-precision positioning achine that cobines a icro and a global stage. In addition a ethod is proposed for designing an optiu controller that uses a iniu order observer with a dual servo control schee and to which a odern control technique is applied. 2. Dynaic odeling The ultra-precision positioning device proposed in this paper consists of a global stage onto which a icro stage is fixed. The global stage can be constructed in any ways; for this

942 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 (a) (b) Fig. 1. Scheatic of ultra-precision single plane X-Y stage: (a) syste configuration; (b) dynaic odeling.

2 942 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 (a) (b) Fig. 1. Scheatic of ultra-precision single plane X-Y stage: (a) syste configuration; (b) dynaic odeling. research the plane X-Y delta stage schee is used which uses a lead screw tool consisting of a ball screw and a double nut. This stage is driven by an AC servo otor (Mitsubishi MFS- 23) and a drive. Motor revolutions are converted into linear otion by the double-nut type ball screw (2 lead and 19 external diaeter) to convey global tables. In this case the axiu power is 2 W the axiu transfer distance is 1 and the axiu transfer speed is 1 /s. For the icro stage I used the elastic hinge schee and drive the stage using a piezoelectric sensor (Thorlabs AE11D16). The axiu driving range is ±1 μ and the stage is driven within the range of about ±2 μ. Fig. 1 shows an ultra-precision X-Y planar stage. To odel this stage the otion equation of the transfer screw of the global stage is first calculated then the otion equation of the table and the circuit equation of the otor is calculated to find the transfer function by deriving the state equation. The otion equation was derived by analyzing the relation between the icro stage displaceent and the piezoelectric sensor and by easuring input voltage and the voltage across the piezoelectric sensor. Using this process the transfer function ay be found by deriving a state equation for the icro stage. 3. Constructing servo syste For this research the ultra-precision plane X-Y stage is assued to be a double-input single-output syste and an optiu control syste was exained using odern control theory. Because syste design according to classical control theory is based on trial and error it does not generate an optiu control syste. Conversely syste design based on odern control theory using the state-space schee allows designers to design systes having a closed loop polarity (i.e. the desired characteristic equation) or an optiu control for a given perforance index. Furtherore odern control theory allows designers to include initial conditions in their design if required. However using odern control theory to design via the state-space schee requires atheatical representation of the syste s dynaic characteristics. This is different fro the classical schee in which atheatical representations are not required and it is possible to use inaccurate experiental frequency response curves to design [5]. By exploiting the advantages of odern control theory and on the basis of global and icro servo odeling the doubleinput single-output state equation for an ultra-precision plane X-Y stage (described below) was derived and a iniu order observer state-space schee was designed. By so doing it is possible to represent tools with a single state equation and to design global and icro servos as an integral body. Based on the state Eq. (1) I design the type-1 optiu servo syste. The six state variables used in this syste can be easured using a laser interferoeter and the reaining state variables can be easured using the iniu order observer. On this theoretical basis the syste was siulated and a control syste was designed in which neither the icro-servo's extension nor the power current was saturated. 1 a21 a22 a23 a24 a25 b21 a33 b31 Vp x s = xs + 1 V (1) a a a65 a66 b62 x = x x v x x i (2) [ ] T s c c 1 y = x s 1 (3) y = 1 y (4) a [ ] k p 21 C = p a23 = a22 k p 24 C p = a33 kk Ck = RC p p p p 1 C a = a25 = a s 55 2 R = c J + km kk2 a = k R 56 1 a 2 J + 65 = a66 = km Lk L CK p p K G p b21 = G b31 = b62 = RC R L c

3 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~ The left ter in Eq. (7) can be easured and ay be regarded as an output equation. When designing a iniu order observer the left ter in Eq. (7) is the displaceent of the syste and Eq. (7) gives the relationship between a easurable state variable and non-easurable variable. Here xa is a state variable; the reaining state variables are x xb x υ xc x c i. The variables that cannot be easured are related by Fig. 2. Block diagra of the ultra-precise plane X-Y stage with a iniu order observer. In actual systes soe state variables can be directly easured so that it is not necessary to estiate the. Given that n is the order of a state variable the observer for estiating state variables with order less than n is called a iniu order observer. Assue a state vector x that is an n-vector and a power y that is an -vector. Because output variables are represented by a linear cobination of n state variables the iniu order observer has n orders. The state variables in this syste are of order six and the easurable output vector is first order. As a result the state variable x can be easured by laser interferoeter and the reaining state variables are estiated fro the iniu order observer. Fig. 2 shows a iniu order observer applied to this syste. The state equation just derived of the ultra-precision double-input single-output plane X-Y stage can be expressed in the for of a iniu order observer as a atrix as shown below: x = A x + B u. (8) b ba a b Eq. (8) expresses a dynaic characteristic of a state variable that is not easured. The state equation for a iniu order observer is Eq. (8) and the output equation is Eq. (7). Therefore the equation of a iniu order observer is x = ( A ka ) x + A x + Bu+ k( x A x Bu) (9) b bb e ab ba a a e aa a a where k e is the gain atrix for an [(n-1) 1] state observer. To estiate Eq. (9) the derivative of x a is required. However this is not desirable so Eq. (9) is converted to x b k e x a = ( Aaa Ke Aab) x b + ( Aab ke Aaa )y + ( Bb ke Ba ) u (1) using xb key = xb kexa = η x k y = x k x = η (11) b e b e a η = ( Abb ka e ab) η + [( Abb KA e ab) ke + Aba ka e aa]y + ( Ba kb e a) u + [( A KA ) k + A ka ] y+ ( B kb) u. (12) bb e ab e ba e aa a e a A aa = [] A bb = [1 ] a21 a22 a22 a24 a25 a33 A ba = Abb = 1 a55 a56 a65 a 66 b21 b31 B a = [ ] Bb =. b 62 (5) Eqs. (11) and (12) are used to define a iniu order observer. To derive the fifth-order equation of a iniu order observer Eq. (7) is used in Eq. (9) resulting in η = ( a k A ) η + A x + B u+ k A x. (13) aa e ab ba a a e ab b Subtracting Eq. (13) fro Eq. (6) gives x x = ( a k A )( x x ) (14) using b bb e ab b b Here the equation of a easurable state variable is described in Eq. (6): x = A x + A x + B u. (6) a aa a ab b a Also Eq. (6) was translated to derive the following Eq. (7): x A x B = A x. (7) a aa a au ab b e= x x = η η (15) b where e is the equation of order n 1 and Eq. (14) gives e = ( A K A ) e. (16) aa e ab Thus the fifth-order equation for a iniu order observer is given by Eq. (16). A characteristic equation of a iniu order observer is

4 944 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 Fig. 3. SIMULINK for a dual servo for an ultra-precision single plane X-Y stage. SI A + k A bb e ab = ( S μ )( S μ )( S μ )( S μ )( S μ ) = S + as + a S + a S + a (17) (a) where μ 1 μ 2 μ 3 μ 4 μ 5 are the desired characteristic values of a iniu order observer. The gain atrix ke of the observer is obtained after selecting the characteristic values. Using the Ackeoann forula [15] the gain atrix of an observer is 1 Aab Aab A bb 2 ke =Φ ( Abb) Aab A bb 3 Aab A bb 4 Aab A bb Φ ( A ) = A + a A + a A + a I. bb bb 1 BB 3 bb 1 (18) (b) Using this process the gain atrix k e of an observer and the atrix /A /B /H /K of the observer ay be obtained. Each equation atrix is given below: / A = A K A (5 5) / B = B K (5 5) / K = A k A (5 1) bb e ab b eba ab e aa 1 / H = (6 1). ke (19) In testing the ultra-precision positioning of the syste siulation and real-tie control was perfored with MATLAB SIMULINK. The experient consisted of a single servo experient and a dual servo control test. In the single servo control test siulation and real-tie control were perfored using a PID controller for both the global and the icro servo respectively. For the dual servo control test I applied the iniu order observer using odern control theory to copare the results of siulation and real-tie control. For dual servo siulation the iniu order observer under odern control theory was applied to the state equation of a (c) Fig. 4. Sub-SIMULINK of dual servo for ultra-precision single plane X-Y stage: (a) SIMULINK for global servo; (b) SIMULINK for icro servo; (c) SIMULINK iniu order observer. double-input single-output plane stage (derived above) to construct a type-1 optial servo syste and the syste was siulated using MATLAB SIMULINK. Fig. 3 shows a dual servo SIMULINK and Fig. 4 shows

H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 945 Table 1. Step response of the global servo used for siulation. Definition Value Steady-state error Maxiu overshot [%].

Definition Value Steady state error Maxiu overshot [%].1(1%) Rise tie [s].5 Setting tie [s].75 Fig. 7. RTI Siulink of dual servo..6.5 Displaceent( μ ).4.3.2.1 Fig. 5.

5 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~ Table 1. Step response of the global servo used for siulation. Definition Value Steady-state error Maxiu overshot [%].1(1%) Rise tie [s].47 Setting tie [s].62 Fig. 6. Photo of ultra-precision single plane X-Y stage. Table 2. Step response function for icro servo used for siulation. Definition Value Steady state error Maxiu overshot [%].1(1%) Rise tie [s].5 Setting tie [s].75 Fig. 7. RTI Siulink of dual servo..6.5 Displaceent( μ ) Fig. 5. Step response of the dual servo used for siulation. SIMULINK of a iniu order observer. Fig. 5 shows the dual stage displaceents including a global and a icro servo and the siulated displaceent of the global stage according to the SIMULINK dual servo siulation. Fig. 5 shows the displaceent of a icro stage where we see that the syste is driven while the global stage error is corrected by the icro stage indicating that the correct gain control is operating. Tables 1 and 2 show siulated control perforance of a global and a icro servo. A PID controller was installed on a global and a icro servo to perfor positioning tests and satisfactory results were obtained for each tool. These experiental results deonstrate the stability of the global servo and the icro servo. By applying the iniu order observer based on odern control theory for ultra-precision position control real-tie control of a dual servo was carried out. Fig. 6 shows a photograph of this syste and the real-tie control RTI SIMULINK of a dual servo is shown in Fig. 7. Initialization is perfored at progra startup. Fig. 7 shows the RTI SIMULINK of the X-axis (Y-axis has the sae structure). Micro servo step response of Fig. 7 showed the sae result Displaceent( μ ) Tie(sec) (a) Siulation Tie(s) (b) Experient ±1 n Fig. 8. The siulation results and the ultra-precision positioning experient result. as copared siulation of Fig. 5 to experient result of Fig. 8 and resolution of 1 n can be found. Figs. 9 and 1 show a step experient and a resolution ex-

6 946 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 Displaceent() Tie(sec) Fig. 9. Step experient of X-Y axis dual positioning control. x( μ ) y( μ ) θ (arcsec) (a) Fig. 1. Resolution experient of icro positioning control on a vibration-isolation table: (a) with the table on; (b) with the table off. perient of a dual servo respectively. To evaluate the ultraprecision positioning capability of the syste two positioning experients were conducted disregarding the error value easured earlier. In the first experient a 2 n step pulse was used as input to generate a linear displaceent of up to 1 n on a vibration-isolation table (table on) where the external vibration was not notified to syste which was followed by 2-n steps in the opposite direction to get back to the origin. Each displaceent was configured to ove.57 arcsec in steps of.114 arcsec. The data shown in Fig. 1(a) confir that no difficulty occurred during the detailed driving of the 2-n steps and.114 arcsec steps. Furtherore the linear displaceent was1 n and the positioning was.114 arcsec. In the second experient a 2 n step pulse input was used to generate a linear displaceent of up to 1 n on the vibration-isolation table (table off). All the other aspects of the experient were the sae as described for the first experient. The data shown in Fig. 1(b) indicate that the overall perforance of the detailed driving was lower than that obtained using the vibration-isolation table. x(μ) y( μ) θ(arcsec) (b) 4. Conclusions The research on designing an optiu controller for an ultra-precision plane X-Y stage yielded the following results: (1) Using a Matlab siulation the perforance of the control syste for a servo tool that uses an ultra-precision plane X-Y stage positioning syste was evaluated. A dual servo control algorith was devised using a iniu order observer to which odern control theory was applied by coparing and analyzing the conventional control algorith. The controller was verified as having good stability and perforance. (2) The siulation results and the ultra-precision positioning experient result were copared and analyzed through a real-tie ultra-precision positioning experient using Matlab Siulink and ControlDesk fro dspace. The position control algorith and the plane X-Y stage ultra-precision positioning syste were verified as having good perforance and stability. (3) The positioning test of the syste indicates that the resolution for linear translation and rotation is 1 n and.144 arcsec respectively. References [1] N. Lobontiu J. S. N. Paine E. Garcia and M. Goldfarb Corner-filleted flexure hinges ASME J. Mech. Design (21) 123. [2] W. Xu and T. G. King Flexure hinges for piezo-actuator displaceent aplifiers: flexibility accuracy and stress considerations Precision Eng. 19 (1) (1996) 4-1. [3] J. W. Ryu and D. G. Gweon Error analysis of a flexure hinge echanis induced by achining iperfection Precision Eng. 21 (1997) [4] C. W. Lee 3-Axis dual-servo control of an XYTheta-Stage for ultra-precision Positioning Ph.D. Doctoral Thesis of KAIST (1997). [5] L. K. Kwac J. Y. Ki and H. G. Ki Copensation of environent and otion error for accuracy iproveent of ultra-precision lathe International Journal of Modern Physics B 2 (25) (26) [6] H. S. Ki and W. J. Ki Feed-forward control of fast tool servo for real-tie correction of spindle error in diaond turning of flat surfaces International Journal of Machine Tools & Manufacture 43 (23) [7] E. Furukawa M. Mizuno and T. A. Hojo Twin-type piezodriven translation echanis JSPE 28 (1) (1994) [8] P. E. Dupont Avoiding stick-slip through PD control. IEEE Transaction on Autoatic Control 39 (5) (1994) 132. [9] W. Messner and R. Horowitz Identification of a nonlinear function in a dynaical syste Journal of Dynaic Systes Measureent and Control (1993) 115. [1] T. S. Sith V. G. Badai J. S. Dale and Y. Xu Elliptical flexure hinges Rev. Sci. Instru. 68 (3) (1997) [11] M. Holes and D. Truper Magnetic/fluid-bearing stage for atoic-scale otion control JSPE 18 (1) (1996) [12] D. S. Bae R. S. Hwang and E. J. Haug A recursive forulation for real-tie dynaic siulation of echanical sys-

H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~947 947 tes ASME Journal of Mechanical Design 113 (1991) 158-166. [13] T. C. Lin and K. H.

7 H. G. Ki / Journal of Mechanical Science and Technology 26 (3) (212) 941~ tes ASME Journal of Mechanical Design 113 (1991) [13] T. C. Lin and K. H. Yae Linearization of the dynaics of closed-chain echanical systes Mechanics of Sructures and Machines 25 (1) (1997) [14] K. Ogata Moden control engineering Huijoungdang pub. (1993) [15] Using Matlab The MathWorks Inc. Dec. 21. [16] Using SIMULINK The MathWorks Inc. Dec. 21. [17] Real Tie Workshop The MathWorks Inc. Dec. 21. Hong Gun Ki received a B.S. and M.S. degree in Mechanical Engineering fro Hanyang University in 1979 and Dr. Ki then went on to receive his Ph.D degrees fro University of Massachusetts in Dr. Ki is currently the Professor at the Departent of Mechanical & Autootive Engineering at Jeonju University in Jeonju Korea. He is currently serving as an Editor of the J. of KSMTE. Dr. Ki s research interests are in the areas of echanical design production engineering and coposite engineering.

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