Integrated Design of a Planar Maglev System for Micro Positioning

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005 American Control Conference Jne 8-0, 005. Portland, OR, USA hc06.6 Integrated Design of a Planar Maglev System for Micro Positioning Mei-Yng Chen, Chia-Feng sai, Hsan-Han Hang and Li-Chen F.

nt.ed.tw) Abstract According to what the 6-DOF magnetic levitation system has proposed at last year. In the continos work, this research desire to pgrade positioning precision.

1 005 American Control Conference Jne 8-0, 005. Portland, OR, USA hc06.6 Integrated Design of a Planar Maglev System for Micro Positioning Mei-Yng Chen, Chia-Feng sai, Hsan-Han Hang and Li-Chen F.Department of Electrical Engineering, China Institte of echnology, aipei, aiwan, R.O.C.(mychen@cc.chit.ed.tw). Department of Electrical Engineering, National aiwan University, aipei, aiwan, R.O.C.(lichen@ccms.nt.ed.tw) Abstract According to what the 6-DOF magnetic levitation system has proposed at last year. In the continos work, this research desire to pgrade positioning precision. he redesign concept attempts to keep the good performance in the whole jorney of moving rather than the point-to-point positioning precision. Based on these concepts, the force model was modified in considering the variation from the displacement to the magnetic forces first, and avoids the raint of the attractive levitation in replacing the replsive levitation. Finally, the concept of relative place to bild the measring system was adopted. All of the performance of the improved framework will be demonstrated in the experimental reslts. Keywords: Maglev, Precision positioning, Adaptive sliding-mode controller. Introdction Electrical machines have existed since electromagnetic phenomena were first systematically researched in the 9 th Centry. raditionally, the electromagnetic drives are rotational motors, and the rotational movement is converted by means of the transmissions when the linear movement is reqired for application. Mostly, the piezoelectric actators only handle the small moving range[][], the ball-screws case distrbances and backlash de to roghness of the bearing elements[3][4], and the linear motors have end effect in a motion stroke. herefore, other proper resoltions was needed that can extend their application fields and satisfy these reqirements as well no contamination, no friction, high speed, and can be operated in a high vacm environment and also a clean room. Hereto magnetic levitation (abbreviated as Maglev) mechanism was gener- Active Coils PMs Carrier Figre. Photograph of the improved framework ally believed that it is the optimal soltions[5][6]. Based on the design theories and experience of the planar Maglev system, which was proposed at last year[7], in development, the carrier in adding a one-shaped strctre was redesign and replacing the cylindrical coils with larger one to generate the niform magnetic field near the srface of the coils. Likewise, the relative circlar magnets are replaced with larger one to decrease the air gap and generate the powerfl magnetic forces, too. Additionally, in order to have the expectable larger moving range(mm), the replsive levitation instead of the attractive one was employed. he photograph of the improved framework is shown in Fig... Magnetic Forces and System Modeling. Magnetic forces formlation Gap Sensors Based on the previos research[7][8], from the theoretical analysis, the magnetic forces with the mltiplication of displacement fnction and fed crrent can be presented and simplified as: /05/$ AACC 3066

2 F g ( x, y, z), () rec / cyl, rec / cyl, rec / cyl where is the fed crrent, sbscript is denoted as the forces act perpendiclar to the srface of the permanent magnet, and rec and cyl are denoted as the rectanglar and cylindrical coil, respectively. y F x F 3 L l L 3 L =l z x F 4 F 5 F 6 Actally, the height in levitation is always stationary in this system. herefore, for the bottom sbsystem, what most concern will be the inflence of variation of the displacement on the magnetic forces in the horizontal direction rather than in the vertical direction. When a aylor series expansion of the analytical magnetic force eqation is trncated after the first order term, within a small range arond the nominal position, the forces formlation Eq.() can be expressed as: F( d, ) F0 F F pp0, 0 dd0, p 0 dd d,() 0 F0 kd d k where k d and k represents the force-displacement factor and force-crrent factor respectively. Neglecting the hysteresis, F 0 is zero, and removing in the final form, the force eqation with a linear expression becomes k d k, cyl d, cyl o, cyl, cyl cyl k d k, (3) rec d, rec o, rec, rec rec where the d ocyl, means the absolte distance form the measred point to the center of the srface of cylindrical coil in the horizontal direction and the d orec, means the distance form the measred point to the center of the srface of rectanglar coil in the direction perpendiclar to this srface. Finally, there is a fairly important point that needs to be considered. As mentioned in the force model simplification, we jst se the half-part crve of the fll-range distribtion for representation in virte of the symmetry. herefore, to represent the fll-range distribtion well in the linearized form, there shold be some modifications. According to the symmetry of the force distribtion, we can apply the distribtion behavior on one side to describe that on the other side and hence Eq.(3) has to be rewritten as follows: k d k, cyl d, cyl o, cyl, cyl cyl k d k. (4) rec d, rec o, rec, rec rec F (a) Figre. Carrier strctre and force relation. (a) op view of the carrier. (b) Front view of the carrier. System modeling he component of the planar Maglev system inclde: six PMs attached on the moving part so-called carrier, while six coils monted on a fixed part. hree of these six coils are arranged in the bottom and mainly generate levitating forces as well as torqes arond x- and y- axis. And the three side-coils with rectangle shaped mainly prodce lateral forces and torqe arond z-axis as well. Figre shows the design concepts combining with the forces relations. hen, some assmptions are made. First, it is assmed that the mtal effects in this system are very small and can be ignored. he second, in or operations, the performance of carrier s translation is mainly concerned and its rotation is jst reglated. hat means the anglar displacement of the carrier arond each axis is very small. De to these small anglar displacements, it is assmed that the attitde arond z-axis will not be copled with that arond the other two axes. De to the system s strctre, by the Newton s and Eler s law, the total forces and torqes eqations can be expressed as: F x F F mx 3, F y F F my 3, F z F F F mg mz 4 5 6, x Fl 5 Fl 6 I xx, y F l 4 F l I 6 yy, z F3L I zz, (5) where m=0.65kg, l =0.6m, L =0.m, l =L =0.m, and l 3 =0.06m. From Eq.(5), the governing eqations of the rigid body can be expressed in the vector form, and after sbstitting the position fnction with the linearized reslt from Eq.(5), the motion eqation can be express as: (b) 3067

3 MX K U K D G, (6) d o each variable is defined as follows: X x y z, M diag[ m, m, m, I, I, I ], xx yy zz , U G [0 0 mg 0 0 0], D [ d d d d d d ], o o, rec, x o, rec, y o, rec, o, cyl, x o, cyl, y o, cyl, K d C diag k d, rec, x, k d, rec, y, k d, rec,, k d, cyl, x, k d, cyl, y, k d, cyl,, K Cdiag k, k, k, k, k, k C, rec, x, rec, y, rec,, cyl, x, cyl, y, cyl, l l L l l,, and where the sbscript d/ denotes the force-displacement or force-crrent factor, rec/cyl denotes the rectanglar or cylindrical coil, and the final one x/y/ denotes the location of the sbsystem in the x, y direction or -shaped bearing. In this dynamic model, in addition to the dimensions of the configration and the distribtion and strength of the magnetic force can be measred in advance, so the system s six states: x, y, z,,, and can be on-line measred. 3. Controller Design Refer to the linearized plant model Eq.(6) with an additional ncertainty term: MX K U K D Gw, (7) d o where the additional term w is a 6 vector considered as the noise and inpt distrbances de to imperfect system model and states transformation, and it is assmed its valves are bonded. According to Eq.(7), the bias command inpt is decided with Bias U K G. (8) If the control inpt was designed as the sm of one specific controller U ctrl and the bias U Bias as: U U U, (9) ctrl Bias then Eq.(7) cab be rewrote as MX K U K D w. (0) ctrl d o 3. Controller Design Before the design of adaptive controller, the error state E was defined as the difference between crrent state X and reference signal X f, i.e. E X X f. herefore, Eq.(0) can be rewritten as: E X AU B D W, () r ctrl d o where A M K, Bd M Kd, and W M w. Frthermore, divide the last term W into a ant ncertainty W and a varying ncertainty W var, and represent the error-state eqation with E X AU B D W W. () r ctrl d o var De to the assmption of the additional term w, it leads the varying ncertainty to be bonded, W var W MAX. Next, design a sliding srface S as follow: S E E, (3) [,,,,, ], and 0. where diag In this application, the state error E and the derivative of E will be reglated to zero, simltaneosly. De to this reason, if the sliding srface tends to zero within finite time can be proved, then E and E are also forced to zero exponentially. o relate the sliding srface to the dynamics of motion, the time derivative of the sliding srface is: S E E. (4) In this research, an adaptive controller is applied in this research, which is capable of estimating parameters of the system online while controlling the system simltaneosly. After the system parameters have estimated, Eq.() with these estimates in the control command can be sed. So, the estimates acqired from the online estimator can be sbstitted and derive as: ˆ ˆ ˆ d o r Uctrl A ( vb D W X Nsat( S)), (5) where N is a positive vale and sat(.) is the satration fnc- i 3068

4 tion defined as: sat( S) sat( s ), sat( s ), sat( s ), sat( s ), sat( s ), sat( s ),(6) where si i s, for i=~6. i sat( si) if i sii i si i herefore, in Eq.(5), the symbols was sed with head to denote the estimation terms of their real vales, and afterward, applying the adaptive law to design the estimation terms. Additionally, this controller was modified in the designing bondary vale of the satration fnction to satisfy N WMAX Wvar. hen, the adding PID controller, v, can be expressed as v G() t, de Gt () KP EKI EKD, (7) dt where K D, K P, and K I, are parameters to be tned. Sbseqently, from designing the realizable liner controller G t, which can merge E term into the controller. hs, sbstitting Eq.(5) into Eq.(4), we can obtain: S A U v B D W N sat( S), (8) Ctrl d o where A, B d, and W are denoted as the estimation errors with A A Aˆ, B ˆ d Bd B, and d D ˆ D D, respectively. If the estimation errors can converge to zero asymptotically, the S will become S v N sat( S). (9) According to the theory of sliding mode control[9], Eq.(9) reveals that time derivative of the sliding srface S will decrease when S is positive and will increase when S is negative ntil the sliding srface converges to zero. 3. Stability Analysis As mentioned above, in order to eliminate the estimation errors, the adaptive laws have to be fond ot. Applying direct Lyapnov method and defining a Lyapnov fnction candidate V, it is a positive definite fnction: d d W V S S tr A A tr B B, (0) tr W 3 where,, and 3 are all positive diagonal matrixes. hen the differential of the Lyapnov fnction candidate with time variance can be express as: d d 3 W V S S tr A A tr B B tr W. () Sbstitte Eq.(9) into Eq.(). hen, it can be rearranged as following: V = S ( vnsat( S)) tr A ( SUCtrl A ). () tr B ( SD B ) tr W ( S W ) d o d 3 Now, adaptive laws were derived as: A Aˆ SU, Ctrl B Bˆ SD, d d o W Wˆ S. (3) 3 In order to maintain stability and to prevent satration of the control effort, A ˆ mst be bonded away from zero in Eq.(3). One method to avoid A ˆ going throgh zero is to modify the adaptive law sing a projection method. Sch a modification does not affect the convergence of A and is achieved by sing a prior knowledge of A, sch that A A( t0 ), where A ( t 0 ) is defined as the lower bond. Applying the projection method to the adaptive law, we obtain ˆ ˆ SU () ( 0 ) () AS if A t A t A t. (4) 0 otherwise At the initial time, Aˆ ( t 0 ) is chosen so that Aˆ ( t0 ) A ( t0 ). If these eqations hold, Eq.(8) will become V S ( vnsat( S)) S S 0. (5) By sing the Barbalat s Lemma[9], we can establish that S L, S L and so that: St (). De to S convergence to zero, it can be readily verified that E coverage to zero asymptotically. In other words, the all state variables and time derivatives all converge to zero eventally, which is the goal of designing the controller for this system. 4. Experimental Reslts 3069

5 he experimental hardware, inclding the main body, sensor system, driver system and controller hardware have bilt. A nmber of experimental reslts, inclding the transient and the steady-state responses in different sitations, will also be provided in this section to demonstrate the performance of this system with the controller presented in section 3. o test the largest moving range, mm-scale motion, we try to execte the positioning motion in y direction and reglate the other five states to zero position. Figre 3 shows the transient responses to the sitation with the largest initial transitional displacement range, i.e., 7mm in this system. From the trajectories of state, one can see that the all error signals converge to the set-points. In addition to the m-scale motion, the system response for a contining stepping command with each step eqal to 0.mm are shown in Fig 4. he step-train test reslt demonstrates the satisfactory performance in y-axis motion and stiffness in other 5 axes. he final precision is made p 5m in translation which is estimated to reach a limit of the sensor device. o explain the meaning of the reslt, it is necessary to consider the resolving capability of the measring device firstly. Namely, the repeatable accracy is abot 5m. Additionally, the other way to show the tracking capability is to profile a desired circle. Figre 5 present the experimental reslts of this circling motion which radis is eqal to.0mm at freqency 0.4Hz. According to what we have revealed in above experimental reslts, here the positioning precision will be compared with previos work. From able., it is evident to see that the positioning precision of the improved configration is better than that before. 5. Conclsion As a continos work, in this paper, the pgraded positioning precision has sccessflly improved throgh some redesign concepts. hese redesign on the configration inclding the adoption of replsive levitation, increase of the actating forces, and consolidation of the strctre of carrier. By experimental reslts, excellent performance indices have been observed which ascertain that the objectives were sccessflly achieved; the pgrade of the whole performance when moving, precision positioning, and expectable larger moving range. able : he root-mean-sqare-error (RMSE) performance of the previos work and of crrent work Largest Step-train Circle RMSE motion response motion Previos work mm 00-3 mm 0.mm Crrent work mm mm 40-3 mm 6. Acknowledgement his research is sponsored by National Science Concil, R.O.C., nder the grant NSC E PAE. 7. Bibliography [] Sitti, M. and Hashimoto, H., wo-dimensional fine particle positioning sing a piezoresistive cantilever as a micro/nano-maniplator, 999 IEEE International Conference on Robotics and Atomation, Vol. 4, pp , 999. [] Hector M. Gtierrez, Pal I. Ro. Sliding-Mode Control of a Nonlinear-Inpt System: Application to a Magnetically Levitated Fast-ool Servo, IEEE ran. on Indstrial Electronics,Vol. 45, no. 6, pp. 9 97, 998. [3] Li X and Bin Yao, Adaptive robst precision motion control of linear motors with negligible electrical dynamics: theory and experiments, IEEE/ASME ran. on Mechatronics, Vol. 6 Isse: 4, pp , 00. [4] Pai-Yi Hang, Yng-Yew Chen and Min-Shin Chen Position-dependent friction compensation for ballscrew tables, Proceedings of the 998 IEEE International Conference on Control Applications, Vol., pp , 998. [5] Kwang Sk Jng and Yoon S Baek, Stdy on a novel contact-free planar system sing direct drive DC coils and permanent magnets, IEEE/ASME rans. Mechatron., vol. 7, no., March 00. [6] Lex Molenaar, A novel planar magnetic bearing and motor configration applied in a positioning stage, Ph.D thesis, Delft University of echnology, 000. [7] Mei-Yng Chen, zo-bo Lin, Chia-Feng sai, and Li-Chen F, Novel Design of a 6-DOF Planar Maglev Positioning System, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp,.90-95,

6 Figre 3: ransient responses of the carrier with largest translation displacement. Figre 4: 0.mm step inpt control. [8] J. G. David, Introdction to Electrodynamics. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 98. [9] Jean-Jacqes E. Slotine, Weiping Li., Applied nonlinear control, Prentice Hall, 990. Figre 5:.0mm, 0.4 Hz circling motion in XY plot. 307

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