Indirect magnetic suspension by an actively controlled permanent magnet

Transcription

1 NOLTA, IEICE Paper Indirect magnetic suspension by an actively controlled permanent magnet Akihiro Yamamoto 1, Masayuki Kimura 2a), and Takashi Hikihara 1b) 1 Department of Electric Engineering, Kyoto University Katsura, Nishikyo-ku, Kyoto , Japan 2 School of Engineering, The University of Shiga Prefecture 2500 Hassaka-cho, Hikone, Shiga , Japan a) kimura.m@e.usp.ac.jp b) hikihara@kuee.kyoto-u.ac.jp Received November 5, 2012; Revised February 11, 2013; Published July 1, 2013 Abstract: In this paper, we propose an indirect suspension system that is analogous to a nanosuspension system with a cantilever-like atomic force microscope. The proposed system mainly consists of an electromagnet, a permanent magnet and a target. These magnets are arranged perpendicular to each other. In this system, the current flowing in the electromagnet is the only adjustable parameter, and the magnetic field created by the electromagnet hardly affects the target because of the distance between them. Thus suspension of the target is possible by the dynamically controlled permanent magnet. The indirect suspension system is modeled by a magnetic charge model. The validity of the model is confirmed using a simple system where the permanent magnet is suspended by an electromagnet. A proportional-derivative controller is designed for the indirect suspension using the model. The experimental and numerical results confirm that the target is successfully suspended. Key Words: magnetic suspension, indirect suspension, proportional-derivative control, dynamics 1. Introduction This paper discusses a control method for the manipulation of micro/nanoparticles through numerical simulations and experiments on an analogous macroscopic model. Recently, in the fabrication of nano-devices, multifunctional materials are widely considered as possibilities for application in nanotechnology [1]. Among them, the in situ manipulation of single atoms/molecules is a good candidate for device fabrication. The manipulation has already been achieved to single atoms/molecules on the surface of the material by scanning probe microscopes [2, 3]. Atomic force microscopy (AFM), in particular, has the ability to operate micro/nanoparticles through interaction forces and has been shown to experimentally achieve such a manipulation [4]. Recent studies focus on the theoretical explanation of the mechanism of the manipulation. The approach is necessary to establish assured manipulation using controls [5, 6]. However, the control of nanomanipulation has still been far from 284 Nonlinear Theory and Its Applications, IEICE, vol. 4, no. 3, pp c IEICE 2013 DOI: /nolta.4.284

2 (a) AFM system (b) Magnetic suspension system Fig. 1. (a): Cantilever probe and target in AFM. Target is suspended and manipulated by the cantilever. (b): Electromagnet and permanent magnets. The permanent magnet beneath the electromagnet can easily be suspended by changing the current flowing in the electromagnet. The target is manipulated by the suspended permanent magnet. the realization. One of the reasons is the lack of investigations in terms of the dynamical systems and the control theory. The idea of manipulation of a target through a dynamic cantilever without any contact reveals a novel concept of suspension or levitation technologies. Therefore, the analogous consideration of AFM will be a good technological step to the future consideration of nanomanipulation especially in the case that both an actuator and a manipulation target are in nano-meter scale. Here we consider a target manipulated by the mechanical interaction between the target and a tip of a cantilever in the AFM, as shown in Fig. 1 [1, 7] 1. In the system, a control signal can only be input to the drive system which controls the position of the tip through the dynamics of the cantilever. That is, both the cantilever dynamics and the mechanical interaction between the tip and the target have to be considered for a control scheme. The size of cantilever is of course very huge comparing with the target in the AFM system. When the size of actuator for manipulating nano/micro particle becomes comparable to the target, the dynamics of the actuator will be significant. The main purpose of this paper is to discuss the dynamics of such system through a macro-suspension system which has a similar configuration to the AFM system. For the last several decades, magnetic suspension systems have been studied by many researchers [8, 9]. The possibility of ultra-fine motion control in the nanometer order was confirmed [10 12]. A magnetic suspension system for micromanipulation was also discussed [13]. Therefore, the magnetic suspension system is a good experimental system for investigating the dynamics of suspended bodies and considering a control scheme because of the easiness of measurement and control. In this paper, a magnetic suspension system is produced as an analogous system to the AFM system in terms of the configuration of the driving system, the actuator, and the target. The configuration of them is shown in Fig. 1(b). In the magnetic suspension system, an electromagnet works as the driving system and suspends a permanent magnet with an appropriate control scheme. The suspended permanent magnet exhibits an oscillatory motion around the equilibrium position if it is perturbed. Therefore, the permanent magnet can be thought of as the tip in the AFM system. The target is replaced by another permanent magnet in the magnetic suspension system. To achieve the suspension of the target, the dynamics of the target invoked by the magnetic interaction have to be considered. We refer to the analogous system as the indirect magnetic suspension system. We can expect to achieve the active nanomanipulation control in nano-meter scale through this system. In this paper, we propose an indirect magnetic suspension system that can provide the basis for future study of nanomanipulation systems in nano-meter scale. The organization of this paper is as follows. Section II explains the magnetic forces of an electromagnet, a permanent magnet, and a target. We introduce a model using magnetic charge theory. Section III represents the experimental results obtained from the indirect magnetic suspension system. The behavior of the levitated body is discussed around the equilibrium position determined by setting parameters under proportionalderivative (PD) control. Section IV introduces the dynamic equations of the levitated motions for the 1 The possibility of non-contact suspension and manipulation by a tip of a vibrated cantilever was discussed numerically [5, 6]. 285

3 (a) Electromagnet and two spherical magnets. (b) Configuration of magnetic charges. Fig. 2. Magnetic charge model for indirect suspension system. Magnetic poles of EM, PM and TRGT are modeled by magnetic charges. magnetic body and target, on the basis of the magnetic charge model. In these sections, the dynamics of levitated bodies near the reference positions are analyzed both numerically and experimentally. Section V summarizes the conclusion. 2. Magnetic charge model In the indirect magnetic suspension system, an electromagnet and two spherical permanent magnets are vertically arranged. Figure 2(a) shows the configuration of the suspension system. The electromagnet (EM) mainly suspends the permanent magnet (PM) through the magnetic interaction between them. When the target magnet labeled TRGT is sufficiently far from EM, the magnetic interaction between EM and TRGT becomes weak. In this paper, it is assumed that only PM contributes to the force to suspend TRGT. As shown in Fig. 2(a), there are large gaps between EM, PM, and TRGT. Thus magnetic forces between EM, PM, and TRGT can be approximately described by Coulomb s law for magnetic charges. Figure 2(b) illustrates the configuration of magnetic charges. Each magnetic charge is indicated by a small open circle. For EM, three magnetic charges m 1, m 2,and m 3 (m 1,m 2,m 3 > 0) appear on the surfaces of the E-shaped ferromagnetic core, which is made of E-shaped silicon steel plates. The coil is attached to the center branch of the E-shape, as shown in Fig. 2(a). Because of high magnetic permeability of silicon steel, the surface magnetic flux density reaches a maximum at the end of each branch of the E-shape. Therefore, the magnetic field generated by EM can be approximately represented by the three magnetic charges as shown in Fig. 2(b). The signs of the magnetic charges are determined according to magnetic circuit theory. On the other hand, two magnetic charges appear on the surface of the spherical PM. Here the magnetized orientation corresponds to the z-axis as shown in Fig. 2(b). Thus, the two magnetic charges are located at the top and bottom of each sphere. As shown in Fig. 2(b), it is assumed that the two magnetic charges have the same absolute value but opposite signs. In this paper, the absolute values of the magnetic charges are denoted by m 4 for PM and m 5 for TRGT. 3. Verification of magnetic charge model The magnetic charge model introduced in the previous section is verified both experimentally and by numerical simulations. In this section, for simplicity, we consider a PM-suspension system where only PM is suspended by EM. 3.1 Equation of motion and control method A schematic configuration of the PM-suspension system is shown in Fig. 3. The PM-suspension system is similar to the indirect suspension system but without a target. In this system, we only focused on the z-directional motion. Because of the configuration of the PM-suspension system, any motion in 286

4 Fig. 3. (a) Original configuration. (b) Simplified configuration. Configuration of magnetic charges in PM-suspension system. the x-y plane will not appear if magnetic charges of PM are initially located on the z-axis and there is no disturbance. If PM is disturbed in the x-y plane, a restoring force occurs in any direction of the disturbance, which in turn causes an oscillatory motion in the x-y plane. As long as the disturbance is small, the motion in the x-y plane is negligible in terms of z-directional motion. To derive the equation of motion of PM, the z-directional magnetic field on the z-axis has to be estimated. By experimental measurement of the z-directional magnetic field around the position where PM will be suspended, the dependence of the magnitude with respect to the distance between EM and PM is quite similar to the case that a magnetic charge is located at the origin. Therefore, for simplicity, three magnetic charges on EM, namely, m 1, m 2,andm 3, are represented by only one magnetic charge m 1a, which is located at the origin, as shown in Fig. 3(b). Since the magnitude of the magnetic field is proportional to the current flowing in EM until the magnetic flux saturates inside the core of EM, m 1a can be substituted by p 0 I, where p 0 is a proportional coefficient. In addition, the interaction between m 1a and m 4 is only considered for fitting to experiment. It is still an open problem why the interaction between m 1a and m 4 should be omitted from the equation of motion of PM. One of possible reasons is the effect of image magnetic charge in EM. However, the complex shape of EM makes it difficult to estimate the position and the magnitude of the image magnetic charges. Then, on the basis of experimental result, only m 4 is considered for the equation of motion of PM in this paper. A damping effect caused by air drag is neglected because it is relatively small with respect to that caused by PD control. Consequently, the motion of PM is described by M 1 z 1 = M 1 g m 4p 0 I 4πμ 0 (z 1 r 1 ) 2, (1) where z 1 denotes the center of mass of PM. The other coefficients and constants as well as their values are listed in Table I. The first term on the right-hand side is due to the gravity force and the second therm is due to the interaction between EM and PM. An equilibrium point z 10 > 0 is obtained by setting the left-hand side of Eq. (1) to zero. When the current flowing in EM is fixed at I 0, the equilibrium point is given by m4 p 0 z 10 = r 1 + 4πμ 0 M 1 g I 0. (2) The stability of the equilibrium point z 10 is determined by eigenvalues of a Jacobian matrix obtained by linearizing Eq. (1) around z 10. The Jacobian matrix has two eigenvalues: ( 16πμ0 M 1 g 3 ) 1 4 λ 1 =, λ 2 = λ 1. (3) m 4 p 0 I 0 Since all constants in Eq. (3) are positive, all instances of λ 1 are real numbers and positive, namely, λ 1 > 0. Therefore, z 10 is unstable. To stabilize the equilibrium point, PD control is used in this paper. The current I in Eq. (1) is substituted by 287

5 Table I. List of coefficients, constants, and their values. The magnetic charges m 1a = p 0 I, m 4 and m 5 is determined by curve fitting to the relationship between the magnitude of z-directional magnetic field and the distance which is experimentally measured. Physical Quantity Symbol and Value Gravitational acceleration g Permeability under vacuum μ 0 Mass of PM M 1 =2.66 g Mass of Target M 2 =1.04 g Radius of PM r 1 =5.08 mm Radius of TRGT r 2 =3.80 mm Current flowing in EM I Proportional coefficient of I p 0 = Wb/A Magnetic charge for EM m 1a = p 0 I Magnetic charge for PM m 4 = Wb Magnetic charge for TRGT m 5 = Wb I = I 0 + I ɛ = I 0 + K z1 (z 1 Z 10 )+K vz1 ż 1, (4) where the control gains are denoted by K z1 and K vz1. The variables with superscript ɛ depict the deflection measured from the value at the equilibrium point. The reference position in the PD control scheme is denoted by Z 10. If the reference position is set at z 10, PM is suspended at z 1 = z 10. The state when PM is suspended at z 10 is called the ideal state in this paper. 3.2 Behavior near reference position In this section, the transient behavior of PM is discussed from both a numerical and an experimental perspective. The coefficients in Eq. (1) are estimated experimentally and listed in Table I. Control gains K z1 and K vz1 are given to stabilize z 10 by Eqs. (1) and (4). Here a reference position Z 10 is set at z 10. As an example, the control gains K z1 = 125 A/m and K vz1 = 3 As/m are chosen for the reference current I 0 =0.30 A and the reference position Z 10 =17.64 mm. The transient behavior of PM is simulated by Eqs. (1) and (4) under these conditions. The fourth-order Runge-Kutta method is applied to integrate the differential equations. The numerical result is shown by the dashed curve in Fig. 4. PM is initially located below the reference position. For 0 <t<0.1 s, PM rapidly approaches the equilibrium state. Finally, PM is suspended at z ɛ 1 = z 1 Z 10 =0,i.e., at the reference position. In this case, the reference position is exactly the same as the equilibrium point. Subsequently, the ideal state is finally achieved. Fig. 4. Behavior of PM near z1 ɛ = 0. PM is attracted toward the reference position and is finally suspended at the reference position. The control for the suspension is started at t =0. 288

6 Fig. 5. Schematic diagram of the experimental setup of the PM-suspension system. The position of PM is detected by a CCD laser micrometer, and the detected position is input to a controller installed in the MATLAB/Simulink software. The controller s output is amplified appropriately and applied to EM. Figure 5 shows the schematic of an experimental system for suspending PM. The position of PM is measured by a CCD laser micrometer (VG-035, KEYENCE), and it is input to the controller. The control signal is generated by the controller using MATLAB/Simulink. The amplifier converts the voltage signal to the current signal. To achieve the ideal state in the experiment, the reference position is set at Z 10 =17.46 mm, while the other parameters are set at the same value as in the numerical simulation. The reason for the difference in the reference position is the estimation errors in p 0 and m 4. Their values are individually estimated under the magnetic charge assumption. The experimental result is shown by the solid curve in Fig. 4. PM is attracted toward the reference position, and finally suspended at z1 ɛ = 0. As shown in the figure, the trajectories almost coincide with each other. In the transient state, however, PM in the experiment moves faster than in the numerical simulation. After PM is suspended at the reference position, a small variation is observed. The amplitude of the vibration is about 10 μm. On the other hand, PM does not show such a variation in the numerical simulation. This difference is attributed to the detection error of the laser position sensor because its detection accuracy is within ±50 μm. The experimental result implies that the equation of motion derived by the magnetic charge model is valid around the equilibrium point, although there are small differences from the numerical result in the transient state. That is, the behavior of PM near the reference position can be described by Eq. (1). 3.3 Dependence of equilibrium position on reference position The equilibrium points where PM is suspended under the PD control are given by Z 10 = r 1 + AK z1 ± (AK z1 ) 2 +2A(I 0 K z1 Z10 + K z1 r 1 ), (5) where A =(m 4 p 0 )/(8πμ 0 M 1 g). The equilibrium points obviously depend on the reference position and control gains. In particular, it is necessary to realize the indirect suspension that the equilibrium points can be changed to by the reference position, because TRGT is suspended only by PM. As shown in Eq. (5), there exists two equilibrium points; they are denoted by Z s 10 and Z u 10, where superscripted s and u indicate the stable and unstable equilibrium points, respectively. For the PM-suspension system, an inequality, Z s 10 <Z u 10, is always satisfied, except Z s 10 = Z u 10 = r 1 + AK z1. Figure 6 shows Z s 10 with respect to the reference position Z 10 under the setting I 0 =0.30 A and K z1 = 125 A/m. The experimental result is indicated by the symbol. The solid curve shows the theoretical value of Z s 10. As shown in Fig. 6, they are well in agreement, thus implying that the magnetic charge assumption is applicable and valid in the range 15 mm <Z s 10 < 21 mm. In addition, the dependence of the equilibrium position on the reference position can be correctly described for 16 mm < Z 10 < 19 mm by Eq. (5). If PM is rapidly converged to the equilibrium point by careful 289

7 Fig. 6. Equilibrium position Z s 10 with respect to Z 10 at K z1 = 125 A/m. choice of K vz1, PM can follow the signal of the reference position. That is, it can be controlled at 15 mm <z 1 < 21 mm. The validity of Eq. (5) is also confirmed for other control gains. 3.4 Dependence of equilibrium position on control gain The equilibrium points, Z s 10 and Z u 10, also depend on the control gain of K z1, as shown in Eq. (5). As the control gain increases, the stability reverses from unstable to stable at a certain value of K z1. Thus, the one-parameter diagram of equilibrium points shows how the equilibrium point is stabilized according to the control gain. The diagrams show the range 0 <K z1 < 160 A/m. The current flowing in EM is fixed at I 0 =0.30 A. In this case, the reference position is set at Z 10 = z 10 so that the ideal state is realized. The one-parameter diagram of the equilibrium point is shown in Fig. 7(a) with the experimental data indicated by. If the control gain is enough to suspend the PM, the ideal state is realized, namely, Z ɛ 10 = Z 10 Z 10 = 0. As shown in Fig. 7(a), the two loci of the equilibrium points intersect at K z1 = K b1. The stability of each of the two equilibrium points reverses. This is distinguished as a transcritical bifurcation. When the control gain is less than K b1, the equilibrium point at Z 10 is unstable. However, the diagram shows that the stable equilibrium point exists above the unstable one. Thus, PM approaches EM with the decrease in the control gain from K b1,evenif the reference position is not varied. The bifurcation structure that emerged in Fig. 7(a) disappears when the reference position is varied from z 10. In Figs. 7(b) and 7(c), the bifurcation diagrams are obtained numerically and experimentally when the reference positions are shifted ±0.30 mm from the reference positions in the ideal state. In Fig. 7(b), PM can be suspended at any control gain. This implies that no bifurcation occurs and the suspended position is smoothly changed by the control gain. On the other hand, in Fig. 7(c), PM cannot be suspended in the range of 38.4A/m (= K b2 ) < K z1 < 59.4A/m (= K b3 ). In this case, the increase in the control gain causes the collision of the stable and unstable equilibrium points at K b2.atk b3, the pair of equilibrium points appears. These are classified as saddle-node bifurcations. The experimental result shows characteristics similar to those obtained from the theory derived from the magnetic charge assumptions. We confirm that the theoretical model can represent not only the dynamics around the equilibrium point of the ideal state but also the changes of the equilibrium points for the control gain and reference position. This directly suggests that the magnetic charge model suits the investigation of the indirect suspension system. Hereafter, the magnetic charge model is applied for the design of the indirect suspension system. 4. Indirect suspension of magnetic target In this section, we attempt to suspend a small magnetic ball, TRGT, using a dynamically controlled magnetic ball, PM. The magnetic charge approximation is also used for describing the dynamics of these balls. We numerically estimate the positions where PM and TRGT are suspended and the values 290

8 (a) Z 10 = z 10 (b) Z 10 = z mm (c) Z 10 = z mm Fig. 7. Equilibrium positions Z10 s and Zu 10 with respect to K z1. The solid curves correspond to stable equilibria and the dashed curves to unstable equilibria. Note that the value of z 10 is about mm for theory and mm for experiments. Here, K b1 =47.7A/m, K b2 =38.4A/m, and K b3 =59.4A/m. of control gains needed to achieve indirect suspension on the basis of the equations of motion. By using numerically obtained values, we show that TRGT is suspended by PM. Finally, the efficiency of the equation of motion is compared with the experimental results. 4.1 Equation of motion and control method The indirect suspension system consists of EM, PM, and TRGT, as shown in Fig. 2. For simplicity, the magnetic charge that appeared on each of the three branches of EM is represented by a magnetic 291

9 Fig. 8. Configuration of magnetic charges in the indirect suspension system. charge m 1a in the same manner as in the previous discussion. As a result, the configuration of magnetic charges is obtained for the indirect suspension system, as shown in Fig. 8. Due to the fact that m 4 does not contribute to the interaction between EM and PM, we assume that m 5 is also negligible for the interaction between EM and TRGT. Therefore, the dynamical equations are obtained for PM and TRGT: 4 M 1 z 1 = M 1 g + F 1 (z 1 ) f i (z 1,z 2 ), i=1 (6) 4 M 2 z 2 = M 2 g + F 2 (z 2 )+ f i (z 1,z 2 ), where F 1 (z 1 ) denotes the interaction between EM and PM, and F 2 (z 2 ) denotes the interaction between EM and TRGT. By applying Coulomb s law for the magnetic charge, we have F 1 (z 1 )= m 4p 0 4πμ 0 F 2 (z 2 )= m 5p 0 4πμ 0 i=1 I (z 1 r 1 ) 2, (7) I (z 2 r 2 ) 2. The third terms on the right-hand side of Eq. (6) are derived by the interaction force between PM and TRGT, i.e., f 1 (z 1,z 2 )= ( m 4)( m 5 ) 1 4πμ 0 (z 2 r 2 z 1 + r 1 ) 2, f 2 (z 1,z 2 )= m 4m 5 1 4πμ 0 (z 2 + r 2 z 1 + r 1 ) 2, f 3 (z 1,z 2 )= m (8) 4( m 5 ) 1 4πμ 0 (z 2 r 2 z 1 r 1 ) 2, f 4 (z 1,z 2 )= m 4m 5 1 4πμ 0 (z 2 + r 2 z 1 r 1 ) 2. The parameters in the above equations are listed in Table I along with their corresponding values. The equilibrium positions where PM and TRGT are suspended can be obtained by setting the left-hand side of Eq. (6) equal to zero. The Newton-Raphson method is applied for computing the equilibrium positions. Figure 9 shows the dependency of the equilibrium positions on current. As I increases, z 10 and z 20 tend to depart from EM, whereas the distance between PM and TRGT, as can be observed from z 10 z 20 is almost constant. This implies that the interaction between EM and TRGT is weak. Therefore, TRGT should mainly be suspended by PM. To suspend TRGT, the magnetic force between PM and TRGT has to be controlled because EM barely affects TRGT. According to Eq. (6), the magnetic force toward TRGT can be controlled by the position of PM. In the previous section, we showed that the relationship between the reference position Z 10 and the suspended position Z10 s in Fig. 6 is almost linear. This suggests that the position of PM can be a control parameter for suspending TRGT. That is, TRGT can be suspended by 292

10 Fig. 9. Dependency of equilibrium positions with respect to the current. Table II. Parameter settings for (a): Fig. 10 (b): Fig. 14 (c): Fig. 15. Parameter set (a) (b) (c) I 0 /A Z 10 /mm Z 20 /mm K 1 /A/m K K 3 /s K 4 /As/m changing the reference position for PM. In this study, a PD control method is applied to suspend TRGT. Thus, the reference position Z 10 in Eq. (4) is substituted by Z (z 1,z 2 )=K 2 (z ɛ 2 z ɛ 1)+K 3 (ż 2 ż 1 ), (9) where K 2 and K 3 are the gain constants. The PD control scheme for the indirect suspension is therefore set as I = I 0 + K 1 {z1 ɛ Z (z 1,z 2 )} + K 4 ż 1. (10) In this paper, the offset current I 0 is fixed at 0.4 A for the indirect suspension. The equilibrium positions z 10 and z 20 are set at mm and mm, respectively. The equilibrium point of Eq. (6) is unstable when the control gains are set at zero. The Jacobian matrix, obtained by linearizing Eq. (6) around (z 1, ż 1,z 2, ż 2 )=(z 10, 0,z 20, 0), has two positive real eigenvalues. To achieve the indirect suspension, the control gains for the stabilizing equilibrium point should be combined adequately. The control scheme has four control gains in the parameter space. A domain of the parameter combination is numerically found in the four-dimensional parameter space. All the eigenvalues have a negative real part at the combination of the control gains. The control gains are listed in Table II(a). The numerically simulated transient behavior of the indirect suspension is shown in Fig. 10. The control parameters, including the gains and reference positions, are also listed in Table II(a). The PD control starts at t = 0. PM is initially suspended at an equilibrium position and TRGT is placed on an imaginary stage that prevents TRGT from falling down. After the control starts, PM approaches to TRGT staying at the initial position. When the distance between PM and TRGT becomes sufficiently small, TRGT is attracted to PM. To avoid a collision with each other, PM moves toward EM faster than TRGT. Finally, PM and TRGT approach their respective equilibrium positions. In this case, the reference positions coincide with the equilibrium ones. After the transient behavior, the ideal state is achieved. 4.2 Experimental setup and results To experimentally achieve the indirect suspension, the gap between PM and TRGT and the position of TRGT are measured. For this purpose, another CCD laser micrometer is mounted on the suspension system. The block diagram of the experimental system is shown in Fig. 11. The signal of the two 293

11 (a) PM and TRGT (b) Current I Fig. 10. Numerically simulated behaviors of PM, TRGT, and I in the indirect suspension. (a) Solid and dashed curves correspond to the behavior of PM and TRGT, respectively. The reference position Z 10 =17.85 mm for PM in control, and Z 20 =37.14 mm for TRGT. (b): The current rapidly converges to 0.4 A which is the reference current I 0. sensors is input to the controller. The current flowing in EM is appropriately varied by the controller through the amplifier. The controller is implemented in MATLAB/Simulink; its block diagram is shown in Fig. 12. The reference position is subtracted from the inputs in the first step of the control block diagram. In Block 1, the PD-controlled position z is evaluated with respect to the gap between PM and TRGT. The output of Block 1 z is input to Block 2 as the reference position of PM. Through these blocks, the control current I ɛ is obtained. The offset current I 0 is additionally given to I ɛ at the final step of the control scheme. The output signal is amplified and applied to EM. Although there exists a small difference in the parameter settings between the experiments and numerical simulations, TRGT is successfully suspended by the control scheme mentioned in the previous section. Figure 13 shows the suspended TRGT by the dynamically controlled PM through EM excitation. The parameter settings are listed in Table II(b). The difference of the parameters from the numerical simulation (Table II(a)) is due to disturbance and modeling errors. The transient behavior of PM and TRGT is shown by the solid curves in Fig. 14(a). TRGT is initially supported by a nonmagnetic stage and PM is suspended by a control based on Eq. (4). The control scheme is switched to Eq. (10) at t = 0. The transient behavior at t 0.1 s is similar to that of the numerical simulation obtained in Fig. 10. However, in the experiment, PM and TRGT oscillate together at t>0.1 s. The center of the oscillation seems to converge at t 0. Although small 294

12 Fig. 11. Schematic diagram of experimental system for suspending TRGT indirectly through dynamically controlled PM. Fig. 12. Block diagram of the controller. Fig. 13. Indirect suspension of magnetic target through dynamically controlled PM. fluctuations persist, both TRGT and PM could be suspended around the reference positions until the end of the control. We can conclude that the indirect suspension is achieved. Figure 14(b) represents the current I of EM in the experiment. We observe the presence of both high-frequency and low-frequency currents. The results of Fig. 14(a) suggest that the low-frequency current directly affects the behavior. 4.3 Confirmation of detected equations of motions The difference between the experimental and numerical simulation results is mainly due to the modeling error arising from the estimation of m 4, m 5,andp 0. Figure 15 shows the numerical results with the parameters listed in Table II(c). The oscillations shown in the two graphs at t t 0 coincide with each other qualitatively. However, at t>t 0, they exhibit different features. The fluctuation after t 0 295

13 (a) PM and TRGT (b) Current I Fig. 14. Behaviors of levitated bodies and current in the experiment. Here, the controller s output is applied to EM at t =0. t 0 =0.33 s, the reference position Z 10 =17.85 mm for PM in control, and Z 20 =36.36 mm for TRGT. seems to be caused by the lateral motion in the x-y plane shown in Fig. 2(a) which may be caused by that the initial position of TRGT is slightly away from the z-axis. Since both PM and TRGT are spheres, the lateral motion is wrongly detected as a z-axis motion by the laser position sensors. The lateral motion decays very slowly, because it is not considered in the control scheme. It is difficult to remove the fluctuation after t 0 in this system. To reduce the fluctuation, we believe that more sensors are required to detect the lateral motion. These sensors will enable us to measure the correct position of PM and EM in x-y-z space. 5. Conclusion This paper proposed and examined an indirect suspension system of a magnetic target (TRGT) through a permanent magnet (PM) dynamically controlled by an electromagnet (EM). All magnetic elements EM, PM, and TRGT were modeled by magnetic charges. Then the equation of motion for the z-axis was easily obtained by the Coulomb s law for magnetic charge. For the control scheme, simple PD control was applied. Using this control system, the proposed indirect suspension system is numerically and experimentally achieved. As mentioned in the introduction, the scaling law enables us to apply the result of this paper to particle manipulation in nano/microelectromechanical systems. In these systems, a target is manipulated by multiple actuators. Therefore, the dynamics of these actuators have to be considered simultaneously. In case the size of the actuators becomes comparable to the target object, the effect will be dominant. The authors believe that the results of the indirect suspension system will prove 296

14 (a) PM (b) TRGT Fig. 15. Deflection of PM and TRGT from the reference position. The behaviors within t t 0 =0.33 s and t>t 0 are transient and static ones. beneficial to achieve small particle control in nano/microengineering applications. Acknowledgments This research is partially supported by the Ministry of Education, Culture, Sports, Science and Technology in Japan, The 21st Century COE Program No and the Global COE program. References [1] Y. Sugawara, Y. Sano, N. Suehira, and S. Morita, Atom manipulation and image artifact on Si(111)7 7 surface using a low temperature noncontact atomic force microscope, Appl. Surf. Sci., vol. 188, pp , [2] S.W. Hla, Scanning tunneling microscopy single atom/molecule manipulation and its application to nanoscience and technology, J. Vac. Sci. Technol. B, vol. 23, pp , July/August [3] M. Sitti, Survey of nanomanipulation systems, Proc. the st IEEE Conference on Nanotechnology (IEEE-NANO 2001), Maui, USA, pp , October [4] S. Morita, Y. Sugimoto, N. Oyabu, R. Nishi, Ó. Custance, Y. Sugawara, and M. Abe, Atomselective imaging and mechanical atom manipulation using the non-contact atomic force microscope, J. Electron. Microsc., vol. 53, pp , [5] T. Hikihara and K. Yamasue, A numerical study on suspension of molecules by microcantilever probe, Proc. The Fifth International Symposium on Linear Drives for Industry Applications (LDIA2005), Kobe-Awaji, Japan, pp , September

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