MAGNETIC levitation technology is a promising solution

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1 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE Design and Modeling of a Six Degrees-Of-Freedom Magnetically Levitated Positioner Using Square Coils and 1D Halbach Arrays Haiyue Zhu, Student Member, IEEE, Tat Joo Teo, Member, IEEE, and Chee Khiang Pang, Senior Member, IEEE Abstract This paper presents a novel design of six Degrees-Of-Freedom (DOF) magnetically levitated (maglev) positioner, where its translator and stator are implemented by four groups of 1-Dimensional (1D) Halbach Permanent Magnet (PM) arrays and a set of square coils, respectively. By controlling the eight-phase square coil array underneath the Halbach PM arrays, the translator can achieve six DOF motion. The merits of the proposed design are mainly threefold. First, this design is potential to deliver unlimited-stroke planar motion with high power efficiency if additional coil switching system is equipped. Second, multiple translators are allowed to operate simultaneously above the same square coil stator. Third, the proposed maglev system is less complex in regards to the commutation law and the phase number of coils. Furthermore, in this paper, an analytical modeling approach is established to accurately predict the Lorentz force generated by the square coil with Halbach PM array by considering the corner region effect, and the proposed modeling approach can be extended easily to apply on other coil designs such as circular coil, etc. The proposed force model is evaluated experimentally, and the results show that the approach is accurate in both single and multiple coil cases. Finally, a prototype of the proposed maglev positioner is fabricated to demonstrate its six DOF motion ability. Experimental results show that the Root Mean Square Error (RMSE) of the implemented maglev prototype is around 5 nm in planar motion, and its velocity can achieve up to 1 mm/s. Index Terms Magnetic levitation, planar positioner, 6 DOF motion, Halbach PM array, square coil, modeling of Lorentz force. I. Introduction MAGNETIC levitation technology is a promising solution to achieve great performance for many motion sys- Manuscript received November 6, 15; revised March 6, 16 and June 5, 16; accepted June 3, 16. This work was supported by the Collaborative Research Project under the SIMTech-NUS Joint Laboratory (Precision Motion Systems), Ref: U1-R-4JL. This work was also supported by Singapore MOE AcRF Tier 1 Grant R-63--A H. Zhu is with the SIMTech-NUS Joint Laboratory on Precision Motion Systems, Department of Electrical and Computer Engineering, National University of Singapore, Singapore ( elezhuh@nus.edu.sg). T. J. Teo is a visiting scientist with Precision Engineering Research Group, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 139, USA. He is also with the Mechatronics Group, Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research, Singapore ( dteo@mit.edu). C. K. Pang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore ( justinpang@nus.edu.sg). tems, e.g., precision positioning [1], manipulation [], suspension [3], [4], and haptic interaction [5], due to its non-contact, non-contamination, multi-degrees-of-freedom (DOF), and long-stroke characteristics. Over the past decades, the research on magnetic levitation attracts the attention of many researchers, and various kinds of magnetically levitated (maglev) motion systems are proposed. Generally, these maglev motion systems are realized using either Lorentz force [6] [8] or electromagnetic force [5], [9], [1], and both the movingmagnet design [1], [7] and moving-coil design [11], [1] are proposed for applications with different requirements. Furthermore, the motion range of maglev systems are extended from short-stroke to unlimited-stroke in both translational and rotational axes [9], [1]. In literature, there is one class of maglev positioners [13] [19] that are developed using sets of DOF Moving Magnet Linear Motors (MMLMs) [], [1] as the forcers to provide both the levitation and propulsion force concurrently to deliver 6 DOF motion. This DOF MMLM consists of a 1-Dimensional (1D) Halbach Permanent Magnet (PM) array translator and a three-phase coil stator. Due to the special arrangement of the Halbach PM array, the generated magnetic field is concentrated on one side of the PM array. Correspondingly, the DOF Lorentz force is generated by the three-phase current immersing in the sinusoidal magnetic field. By using the commutation law similar to DQ-decomposition, the Lorentz force for both levitation and propulsion can be decoupled and linearized from the control input. Generally, this kind of the maglev positioner has low system complexity in regards of the commutation law yet restricted to its limited stroke, because large stroke leads to the increasing in size for both the coil and translator, hence degrades the power efficiency. To achieve large-stroke planar motion with improved power efficiency, maglev positioners with -Dimensional (D) Halbach PM array are proposed [] [6]. One design from this class employs the D Halbach PM array with 45 rotated rectangle coils [5], which allows multi-translators to operate simultaneously above the same stator. By actively switching the effective coils underneath the PM array, this design enables the translator to achieve long-stroke planar motion with high power efficiency. With the D Halbach array, circular coils are adopted in [7] so that the maglev positioner can achieve long-stroke planar motion with full rotation ability around the (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE vertical axis. Compared with the 1D Halbach PM array, one demerit of this D Halbach PM array is that its force ripple caused by the harmonic magnetic field is more obvious [13], [5], and many research are also conducted to explore better D Halbach PM array [8] [31] with lower higher-order harmonics to reduce the force ripple. In addition, the phase number of coils needed in this class of maglev positioners is relatively large. Recently, the maglev positioners with 1D Halbach PM array are evolved to deliver long-stroke planar motion by replacing the four coil stators with the multi-layers PCB of orthogonal long coils [3] or the copper strip array [33]. Compared to the D Halbach class, these 1D Halbach-based maglev designs are with lower system complexity and less force ripple. The potential disadvantage of this class could be that only one translator is allowed to operate above the stator and the power efficiency is still relatively lower in large-stroke motion. In this paper, a novel design of 6 DOF maglev positioner is proposed by utilizing 1D Halbach PM arrays and a set of square coils as the translator and stator, respectively. In the proposed design, the major limitations of the existing designs are avoided, that this design inherits the advantages of 1D Halbach PM array maglev design but breaks its shortstroke barrier, so that the system complexity remains low in regards to commutation and current allocation. In theory, it can achieve unlimited-stroke planar motion with high power efficiency if additional coil switching system is equipped. Furthermore, the proposed design allows multi-translators to operate simultaneously above the same stator. In addition, the phase number of coils are reduced from twelve to eight by employing two-phase current configuration for each MMLM, which cuts down the hardware cost and further lowers the system complexity. To accurately predict the Lorentz force generated by the square coil in the proposed design, an analytical force modeling approach is established in this paper, so that the Lorentz force generated by the corner region of the square coil can be modeled effectively. Compared with the model in [34], the proposed approach in this paper is more generic and it can be applied on many coil designs with corner area, e.g., rectangular coil with inner and outer radius and circular ring coil. The rest of the paper are organized as follows. The working principle of the proposed maglev positioner is detailed in Section II and Section III presents the force modeling approach for the proposed maglev design. Results from analytical and numerical analyses are presented on Section IV and Section V shows the implemented prototype of the proposed maglev positioner with the discussions about the experimental results. II. Working Principle of Maglev Positioner The schematic of the proposed 6 DOF maglev positioner is shown in Fig. 1. In this design, the translator, which is formed by four 1D Halbach PM arrays, moves above the square coil stator. In theory, as the number of coils increases, the maglev positioner can achieve unlimited-stroke planar motion with high power efficiency by using the additional coil switching techniques [5], [35] to actively energize the Fig. 1. Schematic of the proposed design of 6 DOF unlimited-stroke maglev positioner. Fig.. Relative position change from (a) to (b) in y-direction with the indicated specifications of the coils and Halbach PM array. effective coils. Each Halbach PM array in the translator forms a forcer delivering both the levitation and propulsion force as indicated in Fig. 1, namely, Magnet Array I and III produce force along the x and z directions while Magnet Array II and IV produce force along the y and z directions. Six coordinate systems are assigned in the maglev positioner, as described in Fig. 1, that the global coordinate system (x c,y c,z c ) is on the coil stator, the local coordinate (x t,y t,z t ) coincides with the center of translator, and (x i,y i,z i ) is for each Halbach PM array i, where i = I, II, III, and IV. The coils in the stator are mapped in global coordinate system, and the sensor system feedback the position information of the translator. Consequently, the corresponding coils underneath each Halbach PM array are energized accordingly to deliver the 6 DOF motion. The design specifications of the Halbach PM arrays and coils are shown in Fig., the width of the square air core in (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

3 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE Fig. 4. (a) Three type of coils, a1 rectangular coil with outer radius, a rectangular coil with inner and outer radius, and a3 circular ring coil, (b) modeling of the corner region in Cartesian and Polar coordinate systems. Fig. 3. Conceptual illustration of relationships between the magnetic field of Halbach PM array and currents in the square coils. the square coil w a = τ, and the length of single square coil l c = 3τ. The number of PMs in each Halbach PM array is 6m, where m = 1,,, and the magnetizations of the Halbach PM array are depicted in Fig. 3. The PMs in the Halbach PM array have the square cross-sections, and their height h m and width w m are both τ. The length of each PM l m is designed as 3nτ where n =, 3,. The gaps l g (shown in Fig. 1) between two Halbach PM arrays should be at least more than the length of two square coils, i.e. l g 6τ, which ensures that the coils below each Halbach PM array will work independently without intersection regions. In this work, one assumption made is that the magnetic field of each Halbach PM array remains constant along the y i direction in the area underneath the array and falls abruptly to zero outside the array. (This assumption will be analyzed in detail in Section IV for practical considerations.) Referring to Fig., consider the situation that coils translate from (a) to (b) in the y i direction while other directions remain unchanged. Under the assumption, the effective parts Region III and IV in Fig. (b) are identical to Region I and II in Fig. (a), respectively. This indicates that the modeling of generated Lorentz force for the Halbach PM arrays can be reduced into a D model, which is independent to their y i axis locations. By energizing the coils along y i direction with same current, there will constantly be n effective coils along the y i direction. To control the maglev positioner, eight-phase current are employed as the driven sources. In Halbach PM array i, two input current I i,1 and I i, are energized in the coils underneath the Halbach PM array. Refer to the Fig. 3, the x and z directions magnetic field of the Halbach PM array are sinusoidal waves with a period of 4τ as indicated. Here, Coil 1 and 3 are energized with the same magnitude current but in the opposite directions, namely I i,1 and I i,1. Similarly, Coil and 4 are energized with I i, and I i,, respectively. By neglecting the corner region behaviors of the square coil, Segment 1 and are separated by τ, which is half of the magnetic field period. This indicates that the magnetic field in these two segments are in opposite directions, and note that the currents in these two segments are of opposite directions, hence the generated force in these two segments are identical in both magnitude and direction. As Coil 1 and 3 are differed by 6τ, Segment 1 and 4 are in a phase difference of 3π in the magnetic field. Therefore, Segment 1 (Coil 1) and Segment 4 (Coil 3) also produce identical force. On the other hand, Coil 1 and are separated by 3π/ (3τ) in the magnetic field, this indicates that when Coil 1 produces zero force, Coil is in the position that produces maximal force, and vice versa, as illustrated in Fig. 3. This naturally avoids the singularity during force commutation. III. Force Modeling for Maglev Positioner In this section, a general force modeling approach to predict the generated Lorentz force by the corners of coil is proposed. Based on this approach, the force model for the square coil utilized in the proposed maglev positioner is established analytically. A. General Force Modeling Method for Coil Corners Rectangular and square coils are commonly used in maglev or other Lorentz-force systems to provide the driving force [4], [5], [36] [38]. In the existing works, the geometry of the coil are typically simplified into a rectangular surface due to its simplicity in force modeling. For example in Fig. 4(a), to predict the Lorentz force of the a1 coil, the force modeling is often concentrated only on the Rectangular Region, while the Corner Region is neglected. This is reasonable in some designs since the Corner Region only accounts a small portion of the whole coil. However, this simplification is not reasonable for some designs. For instance, consider the square coil used in Fig., the corner regions account more than 1/3 of the total coil and as a result, directly neglecting the corner regions in the force modeling is not accurate. In this work, a general force modeling method for predicting the Lorentz force generated by the corner regions of the coil are proposed. With the Halbach PM array being employed in the maglev system, the sinusoidal magnetic fields are considered. Using this method, the force generated by the Corner Regions with inner and outer radius in the rectangular coil (Fig. 4(a), a) or even the circular ring coil (Fig. 4(a), a3) can be modeled in a closed form (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

4 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE The corner region of the coil is considered as a quarter of the round disk as illustrated in Fig. 4(b). The cylindrical coordinate is introduced to facilitate the force modeling, and in D perspective, the modeling can be reduced to the Polar coordinate as indicated in Fig. 4(b). The magnetic field of 1D Halbach PM array is predicted by first order harmonic model [13], [39] in Cartesian coordinate system, given as where B is expressed as B x (x,z) = B e γz sin(γx) B z (x,z) = B e γz cos(γx), (1) B = µ M (1 e γh m ), () π and µ and M represent 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, γ = π/l, where L denotes the pitch of the Halbach PM array, i.e. L = 4w m. Assuming that the center of the corner disk is located at p = (x,y,z ), for any point (p,r,θ) in the corner region, the magnetic field density is expressed in B x (p,r,θ) = B e γz sin(γ(x + r cosθ)) B z (p,r,θ) = B e γz cos(γ(x + r cosθ)), (3) where r [, R], θ [, π/], and R is the radius of the corner disk. Similarly, for a point located (p,r,θ) in corner region, the effective current density J eff is expressed as, J eff (p,r,θ) = J cos(θ), (4) where J is the current density vector in the corner region of the coil. Governed by the Lorentz force law, for any point (p,r,θ) in corner region, the Lorentz force density is expressed as f Corner (p,r,θ) = J(p,r,θ) B(p,r,θ), (5) and the force generated by the corner region in the cylindrical coordinate system is given as F = J Bdv = V z +h c π/ R z f Corner (p,r,θ)r drdθdz. Substitute (3), (4), and (5) into (6), the horizontal force F Corner x is calculated as F Corner x (p ) = z +h c π/ R z J eff (p,r,θ)b z (p,r,θ)rdrdθdh, = JB γ 3 (1 e γh c )e γz ϕ x (x, γ, R), where π/ { ϕ x (x, γ, R) = γrsin(γ(x + Rcosθ)) (6) (7) (8) + cos(γ(x + Rcosθ)) cos(γx )} dθ. cosθ Mag\Pha α(λ): Numerical β(λ): Numerical α(λ): Approximated β(λ): Approximated λ Fig. 5. The numerically obtained α(λ) and β(λ) in λ [, ] with the approximation. Similarly, the vertical force F Corner z (p ) is expressed as where F Corner z (p ) = z +h c π/ R z JB e γz cosθ sin(γ(φ + r cosθ))rdrdθdh, = JB γ 3 (1 e γh c )e γz ϕ z (x, γ, R), π/ { ϕ z (x, γ, R) = γrcos(γ(x + Rcosθ)) (1) sin(γ(x + Rcosθ)) sin(γx )} dθ. cosθ In general, the integral terms ϕ x and ϕ z in (7) and (9) have no simple analytical solutions due to their complexities, and the numerical curving fitting approach is alternatively adopted in this work. Without loss of generality, the radius of the corner disk R in Fig. 4(b) can be defined through the ratio in relative to the width of single PM τ, i.e., R = λτ, where λ denotes a ratio value. Note that γ = π/4τ, as a result, the term γr in (7) and (9) will cancel τ each other. Therefore, the integral terms ϕ x and ϕ z are only related to the radius ratio λ with x and γ, which are (9) ϕ x (x, γ, R) = ϕ x (x, γ, λ) ϕ z (x, γ, R) = ϕ z (x, γ, λ). (11) For a specific design, the ratio λ is fixed as a constant value. Although the analytical solution of ϕ x (x, γ, λ) and ϕ z (x, γ, λ) are unable to obtained, the numerical integral method can be performed to determine the value of the integral at every γx. Furthermore, the obtained results are approximated by a simple analytical expression with the variable γx. Also interpreted by the physical meaning, by considering the corner region of the coil moving along the periodic sinusoidal magnetic field, the resulted force is still periodic. Therefore, this approximation is made as ϕ x (x, γ, λ) α(λ)sin(γx + β(λ)) ϕ z (x, γ, λ) α(λ)cos(γx + β(λ)), (1) where α(λ) and β(λ) represent the magnitude and the phase numerically approximated, respectively. Fig. 5 shows the values of α(λ) and β(λ) from the numerical integral identification (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

5 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE produce equivalent force. With Square II and IV in the same magnetic field but having opposite current directions, the force generated by them always cancel out with each other. By symmetry, the Corner I and II generate identical force, and similar case happens in Corner III and IV. Therefore, the total force generated by one square coil is expressed as, F Coil = (F Square + F Corner I + F Corner IV ), (16) Fig. 6. (a) Representation of the square coil using eight regions with a equivalent rectangular region and (b) uniform model of the square coil in polar coordinate systems. in the range λ [, ] at a step of.5. A 4th order polynomial is employed to approximate the function of α(λ) and β(λ), α(λ) = p 1 λ 4 + p λ 3 + p 3 λ + p 4 λ + p 5 β(λ) = q 1 λ 4 + q λ 3 + q 3 λ + q 4 λ + q 5 λ [, ], (13) where the coefficients are listed as follow p 1 =.55 p =.838 p 3 = p 4 =.37 p 5 =. q 1 =.81 q =.1343 q 3 =.11 q 4 =.974 q 5 = (14) Subsequently, the horizontal force F Corner x (p ) and the vertical force F Corner z (p ) are expressed as F Corner x (p ) = α(λ)jb γ 3 (1 e γh c )e γz sin(γx + β(λ)) F Corner z (p ) = α(λ)jb. γ 3 (1 e γh c )e γz cos(γx + β(λ)) (15) Using this method, the Lorentz force generated by the corner region of different geometry coil in the Halbach PM array s field can be modeled in an analytical form. For example, consider coils a and a3 with inner radius r i and outer radius r o in Fig. 4(a), the corner region force is modeled by quarter disk with radius r o subtracting the quarter disk with radius r i. B. Force Modeling for Single Square Coil To model the magnetic force generated by one square coil in the magnetic field, the coil is represented a combination of eight regions as shown in Fig. 6(a). There are four square regions and four corner regions defined as Square I, Square II, Square III, Square IV, Corner I, Corner II, Corner III, and Corner IV, respectively. As illustrated in Fig. 3, due to the constant phase difference π in the sinusoidal magnetic field, Square I and III will always where F Square represents the force generated by Square I or III, F Corner I, and F Corner IV represent the force generated by Corner I and IV, respectively. For a square coil centered p c = (x c,y c,z c ) in local Halbach PM array coordinate system, F Square can be obtained directly using the Lorentz force law, the horizontal force F Square x (p c ) is expressed as F Square x (p c ) = τ z c +h c z c x c + 3τ x c + τ N t I c B e γz cos(γx)dxdz (17) Nt I c B τ = γ e γz c (e γh c 1)sin(γx), where N t represents the turn number of the coil, I c represents the current of the coil, w c and h c represent the width and height of the coil, respectively. Similarly, the vertical force F Square z (p c ) is given as F Square z (p ) = τ z c +h c z c x c + 3τ x c + τ N ti c B e γz sin(γx)dxdz (18) Nt I c B τ = γ e γz (e γh c 1)cos(γx). The Lorentz force generated by the corner regions of the square coil are modeled based on the proposed method presented previously. Slight differently, the force modeling of two corners in the square coil are treated uniformly. For both Corner I and 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 the x axis. Refer to Corner I and IV in Fig. 6(b), the magnetic field density of a arbitrary point (p c,r,θ) in Corner I or III is B x (p c,r,θ) = B e γz c sin(γ(ν + r cosθ)) B z (p c,r,θ) = B e γz c cos(γ(ν + r cosθ)), (19) where ν and θ are defined as { x + τ/, Corner I ν = x τ/, Corner IV, and { [ ], π/, Corner I θ [ ]. π/, π, Corner IV Based on this definition, the generated magnetic force on both Corner I and IV can be treated together. The horizontal force of Corner i (i = I or IV) F Corneri x (p c ) is given as F Corneri x (p c ) = z c +h c θ τ z c θ 1 J eff (p c )B z (r, θ, z)rdrdθdh, () (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

6 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE TABLE I Value of θ 1 and θ Corner I Corner IV θ 1 π/ θ π/ π Similarly, the vertical force F Corneri z (p c ) is expressed as F Corner i z (p c ) = z c +h c θ τ z c θ 1 J eff (p c )B x (r, θ, z)rdrdθdh, (1) where θ 1 and θ are defined separately for Corner I and IV as in Table. I. Using the results presented previously, the horizontal force F Corner x (p c ) and the vertical force F Corner z (p c ) are expressed as, F Corner i x (p c ) = αn ti c B γ 3 (1 e γh c )e γz c sin(γν + β i ) F Corner i z (p c ) = αn, () ti c B γ 3 (1 e γh c )e γz c cos(γν + β i ) where α 1.144, β I.394, and β IV.394. Therefore, the generated force F Coil x and F Coil z on single square coil are expressed as, F Coil x (p c ) = K x (x c,z c )I c F Coil z (p c ) = K z (x c,z c )I c, (3) where K x (x c,z c ) and K z (x c,z c ) are defined as, K x (x c,z c ) = N tb [ γ (1 e γh c )e γz c τsin(γxc ) + α ( sin(γ(xc + τ γ ) + β I) + sin(γ(x c τ ) + β IV) ) ] K z (x c,z c ) = N tb [ γ (1 e γh c )e γz c τcos(γxc ) + α γ ( cos(γ(xc + τ ) + β I) + cos(γ(x c τ ) + β IV) ) ]. (4) As analyzed in Section II, for each Halbach PM Array i, the generated force is controlled by the two input currents I i,1 and I i,. Therefore, the total force generated by the Halbach PM Array i is the summation of total force generated by two phases of square coils [ ] [ Fi x Kx (x = mn i,z i ) K x (x i + 3τ,z i ) F i z K z (x i,z i ) K z (x i + 3τ,z i ) ][ Ii,1 I i, ], (5) where i denotes each Halbach PM array, i = I, II, III, and IV, and K x and K z are defined in (4), mn is number of coils with current I i, j, j = 1 or for each Halbach PM array, where m and n are the parameters of Halbach PM array defined in Section II. Based on the obtained horizontal and vertical force in each Halbach PM array, the combined force and torque generated Force (N) F Sum z F Square I z F Corner I z F Corner IV z x (mm) Fig. 7. Force generated by Square I, Corner I, Corner IV, and their summation. on the translator of the maglev positioner for 6 DOF motion are calculated as, F I x F x 1 1 F I z F y 1 1 F II x F z F = II z, T x L a L a F III x T y L a L a F III z T z L a L a L a L a F IV x F IV z (6) where L a denotes the arm of force as defined in Fig. 1. Consequently, the 6 DOF motion can be controlled as 6 channels of single-input-single-output (SISO) systems for simplicity, and the control signal vector [F x F y F z T x T y T z ] can be further allocated to four forcers through the inverse relationship of (6), so that the eight-phase current will be injected into the coil array to conduct 6 DOF motion. It is also noted that, although the proposed maglev positioning system is with 6 DOF motion ability, since the Lorentz force is modeled under ideal situation with zero rotational angles of θ x, θ y, and θ z in the above force modeling, the model error will increase when the rotational angles increase, and as a result, this maglev design is not with the ability of full rotation. IV. Simulations and Analysis In this section, several issues are analyzed with some practical considerations, i.e. force effectiveness of the corner regions and the force variations during the operating of maglev positioner. A. Force Effectiveness at the Corner Regions For the square coil utilized in this design, the corner regions account π/(4 + π).43 area of the total coil. In view of this, the effectiveness of the force generated by the corner region should be investigated. From the analytical force model derived in Section III, it is observed that for a corner region and a square region in the magnetic field of Halbach PM array, e.g., Corner I and Square I in Fig. 6(a), the ratio ε f between their peak force magnitudes is given as ε f = max(f Corner) max(f Square ) = α = 51.5%. (7) π (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

7 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE Bx (T) FEA Model Shifted Combined Halbach PM Array Flat Portion P1 P Coils A B C D Position (mm) Fig. 8. Analysis of magnetic field end effect along y-direction for Halbach PM array. Fz (N) n = 4: FEA n = 4: Model n = 3: FEA n = 3: Model n = : FEA n = : Model Position (mm) Fig. 9. Analysis of the force variations in y-direction translation when n =, 3, and 4 in l m = 3nτ. The power efficiency ε p is defined as the ratio between their force per area ε p = max(f Corner/S Corner ) max(f Square /S Square ) = 4 α π = 65.6%, (8) where S Corner and S Square represent the area of the corner region and square region, respectively. Physically, S Corner and S Square are in proportional to the resistance and thermal loss. This indicates the corner region is efficient in force generation. Fig. 7 shows the z-direction force generated by Square I, Corner I, Corner IV, and the summation of these three regions. It is observed there is a slight phase difference between F Square and F Corner, i.e.,.7 in Corner I and.7 in Corner IV. From Fig. 7, it is also noted that if replacing the combination of Square I, Corner I, and Corner II in Fig. 6(a) with a rectangular region Rec I, the force ratio between them is about /3. B. Analysis of End Effect for Halbach PM Array In Section II, an assumption is made for deriving the force model that the magnetic field of Halbach PM array will remain constant inside the PM array and fall abruptly to zero outside the PM array. In other words, the y-direction end effect of the Halbach PM array is neglected in deriving the force model. Since the adopted magnetic field model of Halbach PM array is in D, its y-direction variation of magnetic field is not included, as a result, FEA software (CST Studio) is utilized to investigate the end effect. Fig. 8 plots both the simulated and assumed B x along the y-direction in a case same as Fig., where the air gap is 1 mm, and other detail specifications of the simulation are listed in Table. II. It is noted that the end effect exists evidently in the edges of Halbach PM array, and this will become more obvious as the air gap increases. To analyze the validity of the assumption due to magnetic field end effect, consider a point P1 in Coil C as indicated in Fig. 8, by assumption the magnitude of B x (P1) should be as large as the flat portion of the FEA result, i.e., B a x(p1) =.71 T, but in reality, B x (P1) decreases as it is near the edge of Halbach PM array and B r x(p1) =.63 T, where B a x and B r x denote the B x in assumption and in reality (FEA), respectively. However, the difference of B r x(p1) and B a x(p1) can be compensated by anther point P in Coil A, where the P1 and P are separated by 6 mm, so that they are two points of two coils with exactly same locations. In assumption, P is outside the magnet array so that B a x(p) = T but in reality, B r x(p) =.66 T as the magnetic field do not fall abruptly to zero outside the magnet array due to the effect effect. Furthermore, it is noted that B r x(p1) + B r x(p) B a x(p1). Although P1 is a point chosen as a specific example for illustration, similar trend is valid for all the edge area. By shifting the FEA field among [ 1 ] mm in the Fig. 8 to [5 6] mm, where they are separated by 6 mm as the case of P1 and P, it is observed that the combined field of the shifted field from [ 1 ] mm and the original FEA field among [5 6] mm is almost exactly same with the magnitude of flat portion, which indicates that the end effect acted on Coil C can be compensated internally by Coil A (similar for Coil B and D), so that if the coils near the edges of Halbach PM array, e.g., Coil A, B, C, and D in Fig. 8 are all energized, the assumption made for D field will be always valid, which ensures the accuracy of the derived force model. FEA simulation is also utilized to investigate the force variation in several cases that the coils are moving in y- direction. In the simulation, three different length (l m ) are chosen for the Halbach PM array, that n =, n = 3, and n = 4 in l m = 3nτ, and the coil group in Fig. translate from the initial situation (b) in y-direction with a total distance of one coil length l c. Fig. 9 shows the difference between the FEA result and model are 1.8%, 4.3%, and 3.8% for n =, n = 3, and n = 4, respectively. Furthermore, the maximal force variations in the FEA results are only 1.44%,.63%, and.53% for n =, n = 3, and n = 4, respectively, which indicates that the y-direction translation causes small force variation due to end effect as analyzed above. V. Experiments and Discussions To validate the accuracy of the proposed force model, the prototypes of a Halbach PM array and a stator with TABLE II Detail Specifications of Square Coil and PM Length of the square coil core, τ Height of the square coil, h c 1mm 1mm Turns number of the square coil, N t 3 7 Numbers of PMs in the array 1 Length of PMs in the array, l m 6mm Magnetization magnitude, M 1.3T (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

8 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE Fx (N) Fz (N) 4 (a) (b) Model Measured FEA Position (mm) Fig. 1. Comparison of the recorded force along (a) x and (b) z-directions generated by one square coil with constant 1mm air gap. 4 Model Measured FEA Force (N) Gap (mm) Fig. 11. Comparison of the recorded force generated by one square coil at different z. Fz (N) Fx (N) 5 5 (a) (b) 5 5 Model Measured FEA Position (mm) Fig. 1. Comparison of the recorded force along (a) x and (b) z-directions generated by five square coils with constant 1mm air gap. square coils are fabricated. The detail specifications of the Halbach PM array and square coils are listed as in Table. II. The experimental setup is conducted that the translator of Halbach PM array is mounted on a NSK 3-axes AC-servo gantry system, and a 6-axes Force/Torque sensor (Model: ATI, Mini4) is used to measure the generated force and data is recorded by a LabVIEW program via a National Instruments data acquisition card (model: PCI-635E). For a single square coil, a constant.3 Amp current is used to energize one coil in the stator. Fig. 1 plots the comparison of the recorded force generated by one square coil with constant 1 mm air gap. These results show that the proposed model is accurate except the place where the square coil is near the edge of the Halbach PM array. Without accounting the edge place, the maximal errors of x- and z-directions force are 7.5% and 8.4%, respectively, and the average errors are.4% and 3.3%, respectively. Experiment is also conducted to evaluate the accuracy of proposed model with regards to Fig. 13. Prototype of the proposed maglev positioner, (a) stator of square coil array and (b) whole maglev positioner. x (nm) y (nm) (a) Real (b) Time (s) Fig. 14. Experimental evaluation of the positioning resolution using a series of nm steps in both (a) x and (b) y-axes. different air gaps. Fig. 11 plots the comparison of the recorded force generated by one square coil at different z. This indicates that the relationship between the generated force and air gap z follows an exponential form well. Another experiment is also conducted where five square coils of same phase are energized concurrently with.3 Amp current. Fig. 1 plots the comparison between the prediction of the proposed model, experimental, and FEA results at different x with 1 mm air gap. In this case, the end effect, which is observed in the single square coil case, is eliminated, and the maximal errors of x- and z-directions force are 9.4% and 13.6%, respectively. The average errors are 4.% and 3.9%, respectively. The potential reason for the force error may be that the Halbach PM array and square coil array used in the experiment are manually assembled, so that the unavoidable assembly misalignment will cause certain error, especially the uneven bottom of Halbach PM array and the unexpected rotation angle of square coils. To demonstrate the 6 DOF motion ability of the proposed maglev design, a prototype of maglev positioner is developed, which is shown in Fig. 13. The stator of the maglev positioner, containing four square coil arrays, is shown in Fig. 13(a), Ref (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

9 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE y (mm) 5 Ref f=.1hz f=1hz f=3hz 5 (a) r=5mm 5 5 x (mm) y (µm) 1 1 Ref Real (b) r=1µm f=1hz 1 1 x (µm) Fig. 15. Experimental evaluation of the planar motion using circular references: (a) 5 mm radius with f =.1 Hz, 1 Hz, and 3 Hz; and (b) 1 µm radius with f =1 Hz. y (mm) y (mm) y (mm) 3 (a) v=1mm/s Ref Real x-error y-error (b) v=1mm/s (c) v=1mm/s Time (s) Fig. 16. Experimental evaluation of 3 mm y-axis linear motion using three different velocities: (a) 1 mm/s, (b) 1 mm/s, and (c) 1 mm/s. z (µm) z (µm) 3 (a) Ref Real Error (b) Time (s) Fig. 17. Experimental performance of the vertical motion in z-axis: (a) 5 µm motion with the recorded error signal and (b) 1 µm motion. which ensures the maglev system can perform the motion over a full period of one square coil to evaluate the feasibility. The vertical position of the moving translator is captured by three channels of the Lion Precision CPL9 capacitive sensors with a maximal measurement range of 5 mm, as indicated in Fig. 13(a). Three channels of Renishaw fibre optic laser interferometer with a count resolution of 39.6 nm are used to sense the horizontal positions, so that the 6 DOF position [x y z θ x θ y θ z ] T can be obtained by sensor transformations. Eight Trust TA115 linear current amplifiers are utilized to Error (µm) Error (mm) Error (µm) Error (µm) θx (mrad) θy (mrad) θz (mrad) 3 (a) 3 3 Ref Real Error (b) (c) Time (s) Fig. 18. Experimental performance of the rotational motion with the recorded error signals: (a) 6 mrad θ x motion, (b) 6 mrad θ y motion, and (c) 1 mrad θ z motion. actuate the eight-phase coil array, and a National Instruments (NI) PXI-811 real-time controller is employed as the control hardware for the maglev positioning system to achieve a sampling rate of 5 khz. The x and y-axes positioning resolutions are evaluated in Fig. 14(a) and (b), respectively, where the translator is levitated up by 1 mm constantly. The references containing a series of nm steps are used for tracking, and Fig. 14 indicates that the maximal errors for both axes are kept below nm (5 counts of the sensor resolution), and the Root Mean Square Error (RMSE) for both x and y-axes are 54.5 nm and 49.4 nm, respectively. The combined xy-axes planar motion are evaluated using circular references. Fig. 15(a) shows the case of 5 mm radius circular motion with three frequencies, i.e., f =.1 Hz, 1 Hz, and 3 Hz, and their errors of real radius are 17.5 µm, µm, and 3. µm, respectively. Fig. 15(b) shows the case of 1 µm radius with a 1 Hz frequency motion, and the directional RMSE of Fig. 15(b) is around 74 nm. The experiment is also conducted to evaluate the achievable velocity of the maglev prototype. Fig. 16 shows the motion performance using three velocities during 3 mm motion stroke in y-axis, that (a) 1 mm/s, (b) 1 mm/s, and (c) 1 mm/s, where their maximal errors are 18.5 µm, µm, and 119 µm, respectively. It is noted that the x-axis errors during y-axis motion are also recorded and plotted in Fig. 16, their RMSEs are.14 µm,.86 µm, and 8.6 µm, respectively for Fig. 16(a), (b), and (c), which demonstrates the strong decoupling ability of the implemented maglev prototype. Fig. 17 shows the experimental performance of the vertical motion in z-axis, where Fig. 17(a) shows the levitation height is changed from 5 µm to 3 µm in.5 seconds, and the maximal tracking error is around 5 µm. Fig. 17(b) demonstrates the vertical positioning resolution using a series of 1 µm steps, and its RMSE is calculated as.58 µm. The rotational motions along x, y, and z-axes are also evaluated for the implemented maglev prototype. Since the laser interferometer is sensitive to the rotational angle, another three channels of the Lion Precision CPL9 capacitive sensors are used to sense Error (mrad) Error (mrad) Error (mrad) (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

10 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE the horizontal position in order to demonstrate the rotational motions in relatively large stroke. Fig. 18 plots the rotational motions along x, y, and z-axes using the ramp references, where the levitation height of translator is constantly on 1 mm. Fig. 18(a) shows the θ x motion from -3 mrad to 3 mrad, and it indicates that its maximal tracking error is less than.83 mrad while its RMSE is calculated as.3 mrad. The θ y motion from 3 mrad to -3 mrad is plotted in Fig. 18(b), and its maximal tracking error is less than.8 mrad while its RMSE is calculated as.31 mrad. Finally, Fig. 18(c) shows the θ z motion from -5 mrad to 5 mrad, and its maximal tracking error is less than.18 mrad while its RMSE is calculated as.56 mrad. VI. Conclusion This paper presents a novel design of 6 DOF maglev positioner by using 1D Halbach PM arrays and square coils. The proposed design is potential to deliver unlimited-stroke planar motion with high power efficiency and multi-translators are allowed to operate simultaneously above the same stator, if additional coil switching system is equipped. In addition, the system is less complex in terms of the commutation law and the phase number of coils. An analytical force modeling approach is proposed in this paper to accurately predict the generated Lorentz force between the Halbach PM array and square coil. This includes a general method to model the Lorentz force generated by the corner region of various kinds of coils. Finally, the proposed force model for square coil with Halbach PM array are evaluated experimentally, and a prototype of the proposed maglev positioner is fabricated to demonstrate its 6 DOF motion ability in large stroke, which demonstrates that its RMSE is around 5 nm in planar motion, and the velocity can achieve up to 1 mm/s. References [1] S. Verma, W.-J. Kim, and J. Gu, Six-axis nanopositioning device with precision magnetic levitation technology, IEEE/ASME Trans. Mechatronics, vol. 9, no., pp , Jun. 4. [] J. Boeij, E. Lomonova, and J. Duarte, Contactless planar actuator with manipulator: a motion system without cables and physical contact between the mover and the fixed world, IEEE Trans. Ind. Appl., vol. 45, no. 5, pp , Nov. 9. [3] H. Zhang, B. Kou, Y. Jin, and H. Zhang, Modeling and analysis of a new cylindrical magnetic levitation gravity compensator with low stiffness for the 6-DOF fine stage, IEEE Trans. Ind. Electron., vol. 6, no. 6, pp , Jun. 15. [4] B. Han, S. Zheng, Z. 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11 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/TIE , IEEE [34] H. Zhu, T. J. Teo, and C. K. Pang, Conceptual design and modeling of a six degrees-of-freedom unlimited stroke magnetically levitated positioner, in Proc. of IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics 14, Jul. 14, pp [35] M. Carpita, T. Beltrami, C. Besson, and S. Gavin, Multiphase active way linear motor: proof-of-concept prototype, IEEE Trans. Ind. Electron., vol. 59, no. 5, pp , May. 1. [36] A. J. Hazelton, M. B. Binnard, and J. M. Gery, Electric motors and positioning devices having moving magnet arrays and six degrees of freedom, U.S. Patent, no , Mar. 1. [37] H. Zhang, B. Kou, H. Zhang, and Y. Jin, A three-degree-offreedom short-stroke Lorentz-force-driven planar motor using a Halbach permanent-magnet array with unequal thickness, IEEE Trans. Ind. Electron., vol. 6, no. 6, pp , Jun. 15. [38] T. J. Teo, I.-M. Chen, G. Yang, and W. Lin, Magnetic field modeling of a dual-magnet configuration, Journal of Applied Physics, vol. 1, no. 7, p. 7494, Oct. 7. [39] H. Jiang, X. L. Huang, G. Zhou, Y. B. Wang, and Z. Wang, Analytical force calculations for high-precision planar actuator with Halbach magnet array, IEEE Trans. Magn., vol. 45, no. 1, pp , Oct. 9. Haiyue Zhu (S 13) received the B.Eng. degree in automation from the School of Electrical Engineering and Automation and the B. Mgt. degree in business administration from the College of Management and Economics, Tianjin University, Tianjin, China, in 1, and the M.Sc. degree in electrical engineering from the National University of Singapore (NUS), Singapore, in 13, where he is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering. He joined the Singapore Institute of Manufacturing Technology (SIMTech) NUS Joint Laboratory on Precision Motion Systems in 13, and is an Attached Research Student with the Agency for Science, Technology, and Research (A*STAR), SIMTech. His current research interests include integrated design and control of ultraprecision mechatronics and magnetic levitation technology. Chee Khiang Pang (S 4 M 7 SM 11) received the B.Eng. (Hons.), M.Eng., and Ph.D. degrees from the National University of Singapore (NUS), Singapore, in 1, 3, and 7, respectively, all in electrical and computer engineering. He was a Visiting Fellow in the School of Information Technology and Electrical Engineering (ITEE), University of Queensland (UQ), St. Lucia, QLD, Australia, in 3. From 6 to 8, he was a Researcher (Tenure) with Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo, Japan. In 7, he was a Visiting Academic in the School of ITEE, UQ, St. Lucia, QLD, Australia. From 8 to 9, he was a Visiting Research Professor in the Automation & Robotics Research Institute (ARRI), University of Texas at Arlington (UTA), Fort Worth, TX, USA. Currently, he is an Assistant Professor in Department of Electrical and Computer Engineering (ECE), NUS, Singapore. He is also an A*STAR Singapore Institute of Manufacturing Technology (SIMTech) Associate, Faculty Associate of A*STAR Data Storage Institute (DSI). He is an author/editor of 3 research monographs including Intelligent Diagnosis and Prognosis of Industrial Networked Systems (CRC Press, 11), High-Speed Precision Motion Control (CRC Press, 11), and Advances in High-Performance Motion Control of Mechatronic Systems (CRC Press, 13). His research interests are on ultra-high performance mechatronic systems, with specific focus on advanced motion control for nanopositioning systems, precognitive maintenance using intelligent analytics, and energy-efficient task scheduling considering uncertainties. Dr. Pang is a member of American Society of Mechanical Engineers. He was the recipient of The Best Application Paper Award in The 8th Asian Control Conference (ASCC 11), Kaohsiung, Taiwan, 11, and the Best Paper Award in the IASTED International Conference on Engineering and Applied Science (EAS 1), Colombo, Sri Lanka, 1. He serves as an Associate Editor for Asian Journal of Control, Journal of Defense Modeling & Simulation, Transactions of the Institute of Measurement and Control, and Unmanned Systems, on the Editorial Board for International Journal of Automation and Logistics and International Journal of Computational Intelligence Research and Applications, and on the Conference Editorial Board for IEEE Control Systems Society (CSS). In recent years, he also served as a Guest Editor for International Journal of Automation and Logistics, Asian Journal of Control, International Journal of Systems Science, Journal of Control Theory and Applications, and Transactions of the Institute of Measurement and Control. Tat Joo Teo (M 8) received the B. Eng. degree in mechatronics engineering from Queensland University of Technology, Australia, in 3 and the Ph.D. degree from Nanyang Technological University, Singapore, in 9. He is with Singapore Institute of Manufacturing Technology as a researcher since 1 and is currently a visiting scientist in Massachusetts Institute of Technology. His research interest is to explore the fundamentals of Newtonian mechanics, solid mechanics, kinematics, and electromagnetism to develop high precision mechatronics or robotic systems for micro-/nanoscale manipulation and bio-medical applications. Dr. Teo has published over 4 peer-reviewed articles and has 4 patents granted. In 13, he received the IECON Best Paper Award in the theory and servo design category. In 14, he became the first Singaporean to win the R&D 1 Award, which is the most prestigious international award for technologically-significant products. He currently serves as an Associate Editor for Nanoscience and Nanotechnology Letters (c) 16 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.