Conceptual Design and Modeling of a Six Degrees-of-Freedom Unlimited Stroke Magnetically Levitated Positioner*

Transcription

1 4 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM) July 8-, 4. Besançon, France Conceptual Design and Modeling of a Six Degrees-of-Freedom Unlimited Stroke Magnetically Levitated Positioner* Haiyue Zhu Student Member; IEEE, Tat Joo Teo Member; IEEE and Chee Khiang Pang Senior Member; IEEE Abstract Magnetic levitation technology is a promising solution to achieve ultra-precision motion. This paper presents a novel conceptual design of a 6 degrees-of-freedom (DOF) magnetically levitated (maglev) planar positioner. The advantages of the proposed maglev positioner includes that, it is able to deliver unlimited planar motion stroke with good power efficiency, allow multi-translators simultaneously above the same stator and also with low system complexity. The proposed design employs four groups of D Halbach PM arrays and a set of square coils as the translator and stator, respectively. Furthermore, an analytical modeling approach is proposed to model the Lorenz force of the square coil accurately, which considers the corner area effect of the coil model. By controlling the currents energized in the coils underneath the Halbach PM array, the translator delivers the desired 6-DOF motions. Finally, FEA simulation is conducted to validate the accuracy of the proposed force model, and limited variance is observed. I. INTRODUCTION Magnetic levitation technology is a promising solution to achieve ultra-precision motion in vacuum environment due to its non-contact, friction-less, and unlimited stroke characteristics. Two degrees-of-freedom (DOF) moving magnet linear motor (MMLM) is proposed in [] as the actuator to provide forces for magnetically levitated (maglev) positioner. This MMLM uses a Halbach permanent magnet (PM) array to deliver magnetic field, which is stronger than a conventional PM array. By energizing the coils under the Halbach PM array, a coupled levitation and propulsion force in the z- and x-axis respectively can be generated based on the Lorentz-force law. These kind of maglev planar motion positioners [] [4] utilizes four sets of MMLMs to generate 6-DOF motion with low system complexity, while its motion stroke is limited. To achieve unlimited stroke planar motion, maglev positioners using dimensional (D) Halbach PM array are proposed [5] [8], that only the corresponding coils under the PM array are energized efficiently instead of the all coils. One design of maglev positioner using D Halbach *This work is funded and supported by Singapore Institute of Manufacturing Technology (SIMTech) Haiyue Zhu and Chee Khiang Pang are with Department of Electrical and Computer Engineering, National University of Singapore, Singapore 7583, and also members of the SIMTech-NUS Joint Laboratory (Precision Motion Systems), c/o Department of Electrical and Computer Engineering, National University of Singapore, Singapore 7576 elezhuh@nus.edu.sg, justinpang@nus.edu.sg Tat Joo Teo is with Mechatronics Group, SIMTech, 7 Nanyang Drive, Singapore 63875, and also members of the SIMTech-NUS Joint Laboratory (Precision Motion Systems), c/o Department of Electrical and Computer Engineering, National University of Singapore, Singapore 7576 tjteo@simtech.a-star.edu.sg Fig.. Schematic of the proposed design of 6-DOF unlimited stroke maglev planar positioner. PM array with 45 rotated rectangle coils is presented in [8], which enables the translator to achieve 6-DOF long stroke motion and allow multi-translators simultaneously. Although researches are conducted to design better D Halbach PM array [9] [] with lower high-order harmonics to reduce the force ripple, ideal D Halbach array as D case is not practical to fabricate []. Recently, -DOF MMLM based maglev positioner [] is modified and extended in [3] to deliver long stroke 6-DOF motion. This design utilizes multilayers PCB of orthogonal long coils and D Halbach PM array as the stator and translator, respectively. Hence the system complexity and force ripple are reduced. The potential demerits are that only one translator is allowed above the stator, and lower power efficiency. Control, switching and commutation strategies of these Halbach PM array based maglev planar motor are described in [4] [6]. In this paper, a novel design of 6-DOF unlimited stroke maglev planar positioner is proposed by utilizing four groups of D Halbach PM arrays and a sets of square coils as the translator and stator, respectively. Although the translator used in this design is adopt similarly as in [], the proposed design avoids the stroke limitation constrained, and in theory it can achieve unlimited stroke planar motion as long as /4/$3. 4 IEEE 569

2 Fig.. (a) Relative positions change in y-direction between coils and PM array (b) Decompose the square coil into 8 regions. the coils expends in the stator. Besides, the number of amplifiers used in the positioner is reduced from to 8, which lower the hardware cost of the system and simplify the commutation complexity. Compared with [3], which is also a unlimited stroke design, the proposed design allows multi-translators concurrently above the same stator, and the power efficiency is also improved due to the fact that only the coils underneath the translator is energized with current to generate force, while other coils are switched off. To control the motion of the maglev positioner, analytical force model is derived to precisely model the generated Lorentz force between the translator and the stator. In this paper, we presents a modelling approach to analytically derive the Lorenz force generated by the coil corners, through the introduced cylindrical coordinate system. And finally, FEA simulation is conducted to validate the accuracy of the proposed force model. II. DESIGN TOPOLOGY In this proposed design, the translator carrying four Halbach PM arrays moves above the stator, which is consisted of square coil sets. As the coils expand, the motor can achieve infinite stroke planar motion. Each Halbach PM array in the translator forms a forcer delivering force in two directions, namely, Magnet Array I and Magnet Array III produce force in x and z directions, while Magnet Array II and Magnet Array IV produce force in y and z directions, which is indicated in Fig.. The specification of coil sets is shown in Fig. a, the length of single square coil l c is 3τ, and the length of square air core in the coil is τ. Each Halbach PM array consists 6m PMs, where m =,,..., and the magnetizations of Halbach PM array are depicted in Fig. 3. Each PM in the Halbach PM array is of square cross-section, with both τ in height h m and width w m, and the length of PMs l m is designed to be 3nτ, where n =,,... The gaps l g between two Halbach PM arrays should cover the length of two square coils, i.e. l g = 6τ, ensuring that the coils below each Halbach PM array will work independently without intersection regions. Assumption is made that the magnetic field of each Halbach PM array remains constant along their y i direction Fig. 3. Conceptual illustration of relationships between the magnetic field of Halbach PM array and currents in the square coils. in the area underneath the array and falls abruptly to zero outside the array. Referring to Fig. a, consider Coil Group and Coil Group, which Coil Group is translated from Coil Group only in y i direction, the effective coil Region III and Region IV (Fig. a) in Coil Group will always be identical to Region I and Region II in Coil Group, respectively. This indicates that the modeling of generated force of each Halbach PM array can be reduced to a D problem, which is uncorrelated of y i axis location. And if the coils along y i direction are energized with same current, there will be always constant n effective coils along the y i direction in Coil Group, which is identical with the situation in Coil Group. Eight input currents are employed to control the maglev planar motor, denoted as I i, and I i,, where i = I, II, III and IV corresponds to each Halbach PM array. And in each forcer, there are two input currents energized in the coils underneath the corresponding Halbach PM array. Take Fig. 3 for example, Coil and Coil 3 are injected with same magnitude currents but with opposite direction, as I i, and I i,, and similarly, Coil and Coil 4 are injected with I i, and I i,, respectively. Neglecting the corner behavior of the square coil (detail analysis on corner behavior will be presented in next section), Segment and Segment are separated by τ, which is half the pitch of the Halbach PM array, indicating that their magnetic fields are in opposite directions, and the currents in these two segments are of opposite directions, hence the force generated in these two segments are identical in both magnitudes and directions. Since Coil and Coil 3 are separated by 6τ, Segment and Segment 4 are in phase difference of 3π, therefore similarly Segment (Coil ) and Segment 4 (Coil 3) produce identical force. In the other hand, Coil and Coil are separated by 3τ (3π/), this indicates that when Coil produce zero force, Coil is in the situation that produce maximal force, and vice versa, which is illustrated in Fig. 3. Such design will naturally avoid the singularity of force matrix in the force modeling. For each Halbach PM array, the coils in same y i axis location (coil groups in Fig. a) are energized 57

3 with same current. For case of Coil Group, three coils underneath the array are energized while in case of Coil Group, four coils are all energized to ensure smoothness during the coil switching. Six coordinate systems are assigned in this maglev planar motor, as described in Fig., that the global coordinate (x c,y c,z c ) is mounting on the stator, local coordinate (x t,y t,z t ) is coincident with the center of the translator, and (x i,y i,z i ) is for each Halbach PM array. Each coil in stator are mapped in global coordinate, and the sensor systems feedback the position information of the translator, and then the corresponding coils underneath each Halbach PM array are energized accordingly as described. III. ELECTROMAGNETIC FORCE ANALYSIS A. Analytical Force Model Analytical force modeling for the proposed maglev planar motor design is presented in this section. The magnetic field density of D Halbach PM array is predicted by first order harmonic model [7], which is obtained by using Maxwell equation combined with scalar potential and boundary equation [8], given as where B is defined as B x (x,z) = B e γz sin(γx) B z (x,z) = B e γz cos(γx), () B = µ M ( e γh m ), () π and µ and M denote the permeability of the free space and the peak magnetization magnitude of PMs, respectively. w m and h m represent the width and height of single PM, respectively. γ is the spatial wave number equal to π/l, where L denotes the pitch of the Halbach PM array, i.e. L = 4w m. To analyse the magnetic force generated in the field of Halbach PM array, the square coil is modeled as shown in Fig. b, that the coil is viewed as the combination of 8 regions. There are four square regions and four corner regions in one coil, which we define as Square I, Square II, Square III, Square IV, Corner I, Corner II, Corner III and Corner IV, as shown in Fig. b, respectively. As illustrated in Fig. 3, due to the constant phase difference π, Square I and Square III will always produce equivalent force. And since Square II and Square IV are in same magnetic field but with opposite current directions, the force generated by them always cancel with each other. By symmetry, the Corner I and Corner II generate identical force, and similar case happens in Corner III and Corner IV. Therefore, the total force generated by single square coil can be modeled as, F Coil = (F Square + F Corner I + F Corner IV ), (3) where F Square denotes the force generated by Square I or Square III, F Corner I and F Corner IV represent the force generated by Corner I and Corner IV, respectively. Fig. 4. Force model of the corner regions and the introduced polar coordinate systems (cylindrical coordinate systems in 3 dimension). Governed by Lorentz-force law, the force generated by an energized coil of rectangle cross-section in the magnetic field is expressed as F = N I Bdv, (4) w c h c where N denotes the number of turns in the coil, V represents the volume of coil, I is the current vector, w c and h c are the width and height of the coil, respectively. For square coil centered p = (x,y,z ) in local Halbach PM array coordinate system, F Square can be obtained directly by substituting () into (4), the horizontal force F Square x (p ) is expressed as F Square x (p ) = τ = z +h c z x + 3τ x + τ V JB e γz cos(γx) JB τ e γz (e γh c )sin(γx )dxdz γ Similarly, the vertical force F Square z (p ) is given as F Square z (p ) = τ = z +h c z x + 3τ x + τ JB e γz sin(γx) JB τ e γz (e γh c )cos(γx )dxdz γ, (5), (6) To model the force in the corner regions of the square coil, the corner is viewed as quarter of the round disk, as illustrated in Fig. 4. Cylindrical coordinate system is introduced to facilitate the force modeling (Polar coordinate system in D views as shown in Fig. 4). For both Corner I and Corner IV, the poles of Polar Coordinate System I and Polar Coordinate System IV are located in each center of their round disks, and polar axis are both in the direction of x axis. Through these two introduced polar coordinate systems, the generated force F Corner I and F Corner IV can be modeled unified. Consider Corner I and Corner IV in Fig. 4, for a point that located in (r, θ) in their polar coordinate systems, the effective current density J e f f can be expressed as, J e f f = J cos(θ), (7) 57

4 And the magnetic field density of a point (r,θ) in Corner I or Corner III is B x (p,r,θ) = B e γz sin(γ(φ + r cosθ)) B z (p,r,θ) = B e γz cos(γ(φ + r cosθ)), (8) where φ and θ are defined as { x + π/, Corner I φ = x π/, Corner IV, and { [ ], π/, Corner I θ [ ]. π/, π, Corner IV By this definition, the generated magnetic force on both Corner I and Corner IV can be treated together. For a point (p,r,θ) in corners, the force density vector can be calculated as f Corner i (p,r,θ) = J(p ) B(p,r,θ), (9) where i = I or IV. Consider a surface layer that cover infinitesimal along the coil s height, the generated force df Corner i on this surface can be obtained by the integral over its volume in cylindrical coordinate system, therefore the force on the whole corner is modeled as z +h c θ τ F Corner i (p ) = f Corner i (p,r,θ)r drdθdh, () z where, θ are defined separately in Corner I and Corner IV as, Corner I Corner IV π/. θ π/ π The horizontal force F Corner x (p ) is F Corner i x (p ) = z +h c θ τ z J e f f (p )B z (p,r,θ)rdrdθdh, Substitute (7) (8) to (), F Corner x can be calculated as F Corner i x (p ) = JB γ 3 ( e γh c )e γz + θ cos(γ(φ + τcosθ)) cos(γφ) cosθ Similarly, the vertical force F Corner z (p ) is F Corner i z (p ) = z +h c θ τ z JB e γz cosθ sin(γ(φ + r cosθ))rdrdθdh, = JB γ 3 ( e γh c )e γz θ sin(γ(φ + τcosθ)) sin(γφ) cosθ () { γτsin(γ(φ + τcosθ)) } dθ () { γτcos(γ(φ + τcosθ)) } dθ (3) Magnitude Fig Numerical: Integral () Numerical: Integral (3) Approximated: Integral () Approximated: Integral (3) φ (m) Numerical integral results and the approximated analytical models. Generally, the integral terms of θ in () and (3) have no simple solutions due to their complexities. However, an alternative approach can be adopted that, the numerical integral can be performed off-line, then curve fitting technique can be employed to fit the numerical solution to a simple function. And from the physical meaning, it is noted that () and (3) are same periodic functions with π/ phase difference, due the property of the Halbach PM array s vertical and horizontal fields. Since γ = π/4τ, γτ will cancel τ in each other, therefore, the integral terms in () and (3) are actually independent of τ, which indicates that the obtained numerical results are generic to general motor specifications with different τ. Fig. 5 shows the numerical solutions of the integral terms in () and (3) for Corner I, trigonometric functions are used to approximate the numerical solutions as indicated. Therefore, the horizontal force F Corner x (p ) and the vertical force F Corner z (p ) can be finally expressed by, F Corner i x (p ) = αjb γ 3 ( e γh c )e γz sin(γφ + ϕ i ) F Corner i z (p ) = αjb, (4) γ 3 ( e γh c )e γz cos(γφ + ϕ i ) where α and ϕ i are the parameters determining from the numerical integral solutions, that α.44, ϕ I.394 and ϕ IV.394. Note although numerical technique is utilized to approximate a simple expression of analytical formula, this modeling approach is generic to simply obtain all specification round corner area of coils in sinusoidal magnetic field. Therefore, the force generated by single square coil F Coil x and vertical force F Coil z can be expressed as, F Coil x (p ) = K x (x,z )J F Coil Z (p ) = K z (x,z )J, (5) 57

5 Magnitude F Sum z F Square I z.8 F Corner I z F Corner IV z x (m) Force F x f.5 F z f F x p F z p F y f x (m) Fig. 6. Force generated by Square I, Corner I, Corner IV and their summation. where K x (x,z ) and K z (x,z ) are defined as, K x (x,z ) = JB γ ( [ e γh c )e γz τsin(γx ) + α ( sin(γ(x + τ γ ) + ϕ I) + sin(γ(x τ ) + ϕ IV) ) ] K z (x,z ) = JB γ ( [ e γh c )e γz. τcos(γx ) + α ( cos(γ(x + τ γ ) + ϕ I) + cos(γ(x τ ) + ϕ IV) ) ] (6) As analyzed in Section II, for each Hlabach PM Array i, the forcer is controlled by two input currents, i.e. I i, and I i,. These two currents correspond to two groups of coils underneath the PM array, which coils in each group generate identical force. Therefore, the total force generated by Hlabach PM Array i is the summation of force generated by two groups of coils, [ ] Fi x = mn F i z [ Kx (x i,z i ) K x (x i + 3τ,z i ) K z (x i,z i ) K z (x i + 3τ,z i ) ][ Ii, I i, ], (7) where i denotes each Halbach PM array, i =I, II, III and IV, and K x and K z are defined in (6), m and n are the specification parameters of Halbach PM array defined in Section II, namely, mn is number of coils with current I i, j, j = or for each Halbach PM array. Since the force generated on each forcer are modeled as above, the total force and torque generated on the whole translator can be obtained easily by matrix transformation as presented in [4], [6]. IV. SIMULATIONS AND DISCUSSIONS A. Force Effectiveness From (5) and (6), it is observed that for a corner region and a square region, e.g. Corner I and Square I of Fig. b, in the same sinusoidal magnetic field (Halbach PM array), the ratio between their peak force magnitudes is given as, λ = max F Corner max F Square = α = 5.5%, (8) π Fig. 7. Force of one square coil underneath Halbach PM array predicted by the analytical model and recorded from the FEA simulations with mm gap. with a slight phase difference, i.e..7 in Corner I and.7 in Corner IV. Fig. 6 shows the z-direction force generated by Square I, Corner I, Corner IV and the summation of these three regions, the specifications are detailed in Table. From Fig. 6, it is noted that if replace the Segment in Fig. 3 by a segment of straight line coil with same length, the ratio of Segment s peak force magnitude to the straight line coil s peak force magnitude is about /3. B. FEA Verification To verify the accuracy of the proposed force model, FEA simulation via CST STUDIO SUITE platform is conducted to predict the force of one square coil underneath the Halbach PM array. The detail specifications of the square coil and Halbach PM array are listed in Table. I. TABLE I Detail Specifications of FEA Simulation Length of the square coil core, τ Height of the square coil, h c mm mm Turns number of the square coil, N 5 Current magnitude in single turn, I 3 Amp Numbers of PMs in the array 5 Length of PMs in the array l m, 6 mm Fig. 7 plots the force predicted by the proposed analytical model against the force recorded from the FEA simulations based on a single square coil underneath the Halbach PM array with mm gap. It is observed that, when comparing the predicted horizontal force F x p from the proposed model with the FEA simulated force F x f, the maximum and average variations are.6% and.97%, respectively. For the vertical force F z p predicted from the proposed model and F z f from FEA simulation, the maximum and average variations are.5% and.79%, respectively. Besides, the generated y-direction force F y f is also obtained from the 573

6 Force F x f F z f F x p.5 F z p F y f x (m) Fig. 8. Force of one square coil underneath Halbach PM array predicted by the analytical model and recorded from the FEA simulations with mm gap. FEA simulation, which is not modeled in the analytical force model. The maximal and average F y f are.83 N and.9 N, which is reasonable to omit in the analytical force model. Fig. 8 plots the analytical model predicted force against the recorded force from the FEA simulations with mm gap. Similarly, when comparing the predicted horizontal force F x p from the proposed model with the FEA simulation F x f, the maximum and average variations are.8% and.7%, respectively. For the vertical force F z p predicted from the proposed model and F z f from FEA simulation, the maximum and average variations are.34% and.86%, respectively. And the generated maximal and average y- direction force F y f in FEA simulation are.35 N and.9 N, which is reasonable to omit in the analytical force model. V. CONCLUSION This paper presents a novel design of 6-DOF maglev planar positioner which can achieve unlimited stroke planar motion. This proposed design employs four groups of D Halbach PM arrays and a set of square coils as the translator and stator, respectively. Therefore, it has both the advantages of low system complexity and allowing multitranslators above the same stator concurrently. Analytical model is derived to precisely model the generated Lorentz force between the translator and the stator by decomposing the square coil into eight regions, and a modeling approach is proposed to drive the force generated in the corner area of the coils. Finally, FEA simulation is conducted to validate the accuracy of the proposed force model, which shows the limited variance in comparison to the proposed model. Acknowledgment The authors acknowledge the support from the SIMTech- NUS Joint Laboratory (Precision Motion Systems); Ref: U-R-4JL, and Singapore MOE AcRF Tier Grant R- 63--A44- References [] D. L Trumper., W. J. Kim, and Williams M. E., Design and analysis framework for linear permanent-magnet machines, IEEE Transactions on Industry Applications, Vol. 3, No., pp , March 996. [] W. J. Kim, High-precision planar magnetic levitation, Ph. D. Thesis, Massachusetts Institute of Technology, 997. [3] R. J. Hocken, D. L. Trumper, C. Wang, Dynamics and control of the UNCC/MIT sub-atomic measuring machine, CIRP Annals - Manufacturing Technology, Vol. 5, No., pp ,. [4] R. Fespermana, O. Ozturka, R. Hockena, S. D. Rubenb, T. C. Tsaob, J. Phippsa, T. Lemmonsa, J. Briena, and G. Caskeya, Multi-scale alignment and positioning system MAPS, Precision Engineering, Vol. 36, No. 4, pp , October. [5] Ir. J.C. Compter, Electro-dynamic planar motor, Precision Engineering, Vol. 8, No., pp. 7 8, April 4. 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