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IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 12, DECEMBER 2014

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Modeling of a Two Degrees-of-Freedom Moving Magnet Linear Motor for Magnetically Levitated Positioners Tat Joo Teo1,3 , Haiyue Zhu1,2,3 , and Chee Khiang Pang1,2 1 SIMTech-NUS

Joint Laboratory on Precision Motion Systems, Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 2 Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 3 Agency for Science, Technology, and Research, Singapore Institute of Manufacturing Technology, Singapore 638075 This paper presents a novel analytical model that accurately predicts the current–force characteristic of a 2 degrees-of-freedom moving magnet linear motor (MMLM), where its translator is formed by a Halbach permanent magnet (PM) array. Unlike existing theoretical models, the uniqueness of this proposed model is based on a derived magnetic field model that accounts for the magnetic flux leakage at the edges of the Halbach PM array. Hence, it can be used to model an MMLM that employs a low-order Halbach PM array effectively. To implement the proposed model in high sampling rate control system, a model-based approximation approach is proposed to simplify the model. The simplified model minimizes the computation complexity while guarantees the accuracy of the current–force prediction. MMLM prototype with two separate translators, i.e., one with a single magnetic pole Halbach PM array and the other with six magnetic poles Halbach PM array, were developed to evaluate the accuracy of the proposed models. Index Terms— Current–force characteristic, low-order Halbach permanent magnet (PM) array, magnetically levitated positioner, model-based approximation approach, moving magnet linear motor (MMLM).

I. I NTRODUCTION

M

AGNETIC levitation technology is a promising solution to achieve ultraprecision motion in vacuum environment due to its noncontact, friction-less, and unlimited stroke characteristics. Among various forms of the magnetically levitated (Maglev) motion stages presented in recent literatures, one group of Maglev planar motion stages [1]–[5] were developed using sets of 2 degrees-of-freedom (DoF) moving magnet linear motors (MMLMs), as shown in Fig. 1(a). The schematic diagram of the 2 DoF MMLM is shown in Fig. 1(b), a Halbach permanent magnet √(PM) array is employed to deliver magnetic field, which is 2 stronger than conventional N–S 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 Lorentzforce law. According to Fig. 1(a), these kind of Maglev planar motion stages mainly used four sets of MMLMs to generate 6 DoF motion. Subsequently, similar designs are demonstrated in a multiscale alignment and positioning system to reach subnanoscale resolution [5], where the Maglev stage delivers the lateral motion. Recent advancements also demonstrated the Maglev stages have the potentials to deliver planar motions with unlimited displacement strokes [6]–[14]. In many applications, minimizing the size and mass of the motion system is preferred in consideration of space and energy limitations. For motion stages, reducing the moving mass also increases its natural frequency, and thus enhances control bandwidth. Besides, in a coarse–fine motion system, the fine motion stage with smaller mass also decreases the Manuscript received November 26, 2013; revised March 13, 2014, April 8, 2014, and May 24, 2014; accepted July 4, 2014. Date of publication July 22, 2014; date of current version December 12, 2014. Corresponding author: T. J. Teo (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2341646

Fig. 1. Schematic diagram. (a) Maglev planar positioner constructed via four. (b) Conventional 2 DoF MMLM. (c) Proposed MMLM with a single magnetic pole Halbach PM array.

coupling effect and interference between coarse and fine motions [15], [16]. For Maglev stages that are formed by these MMLM, having a low-order Halbach PM array can be an attracting solution if minimizing of the size and mass is required. Here, low order is defined with small number of magnetic poles. For example, the Halbach PM array with one magnetic pole shown in Fig. 1(c) is considered as a low-order Halbach PM array. From past literatures, several methods were proposed to model the current–force characteristic of the MMLM, e.g., the Maxwell stress tensor method [1], the coupledflux method [17], and the co-energy method [18]. All these methods assumed that the magnetic field behavior of a Halbach PM array to be infinite sinusoidal harmonics. Based on this assumption, these models are effective for predicting the current–force characteristic of the MMLM with a Halbach PM array that consists of a large number of magnetic poles. Examples include the Maglev stages that used MMLM with eight [2], ten [3], and six magnetic poles [5]. However, the

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assumption made by these models has neglected the magnetic flux linkages at the edges of the PMs and such leakages have significant contribution to the magnetic field behavior. Hence, existing models are not suitable for predicting the current– force characteristic of an MMLM with low-order Halbach PM array since they are unable to predict the forces generated by those energized coils underneath these edges accurately. This paper presents a novel force modeling approach that accurately predicts the current–force characteristic of a 2 DoF MMLM with low-order Halbach PM array by accounting the magnetic flux leakages at the edges of PMs. To implement the proposed model in a high sampling rate control, a model-based approximation approach was proposed to obtain a simplified representation of the proposed model. This simplified model minimizes the computation complexity while guarantees the accuracy of the current–force prediction. Finally, MMLM prototype with two separate translators, i.e., one with a single magnetic pole Halbach PM array and the other with six magnetic poles Halbach PM array was developed to evaluate the accuracy of the proposed force model and experimental results are discussed in detail. II. A NALYTICAL F ORCE M ODELING OF H ALBACH PM A RRAY This section presents the magnetic field modeling of a Halbach PM array via the magnetic current model and subsequently on the derivation of the analytical force model. A. Magnetic Field Modeling From past literatures, the magnetic field of the Halbach PM array is predicted by either the harmonic model [9] or the transfer relationship [1], which can be eventually solved as boundary-value problems using the Maxwell equation and the scalar potential [19]. Using this approach, the general expressions of the magnetic flux density components in the x-axis, Bx , and the z-axis, Bz , are given as √ 2 2μ0 M0 (1−e−2γ1h m )e−γ1 (z+h m ) sin(γ1 x) Bx (x, z) = − π √ 2 2μ0 M0 Bz (x, z) = (1−e−2γ1h m )e−γ1 (z+h m ) cos(γ1 x) (1) π where μ0 and M0 denote the permeability of the free space and the peak magnetization magnitude of PMs, respectively. Referring to Fig. 2, wm and h m represent half the width and height of a single PM respectively. γ1 represents the first spatial wave number, i.e., 2π/L, where L represents the pitch of the Halbach PM array. From (1), the magnetic flux linkage at the edges of the Halbach PM array was not considered and hence any force model derived using this magnetic field solution is ineffective in predicting the forces generated by the energized coils underneath the edge of the PMs. In this paper, the magnetic current model [20] of a rectangular PM was used to model the magnetic field and flux linkage of the Halbach PM array. Based on the Maxwell equations, the differential form of the field equations for magnetostatic

Fig. 2. (a) Schematic diagram of a rectangular PM and (b) its crosssectional view with equivalent surface charge. (c) Halbach PM array with global coordinate system.

Fig. 3.

PMs 1, 2, and 2R with respective local coordinate systems.

analysis are expressed as [20] ∇ ·B = 0 ∇ ×H = J

(2)

where B, H, and J represent the magnetic flux density, the magnetic field intensity, and the current density, respectively. The magnetic flux density can be expressed as the curl of the magnetic vector potential, i.e., B = ∇ × A. Using a constitutive relation, B = μ0 H, and the Coulomb Gauge, ∇ · A = 0, (2) was rewritten as ∇ 2 A = −μ0 J.

(3)

The solution of (3) can be expressed in an integral form using the free-space Green’s function [20]. Assuming that the length of each rectangular PM has an infinite length, i.e., lm → ∞, solving the Green’s function can be simplified as a 2-D field problem. Therefore, the x- and z-component of the magnetic flux density of a rectangular PM is given as μ0 M0 (x + wm )2 + (z − h m )2 Bx (x, z) = ln 4π (x + wm )2 + (z + h m )2 (x − wm )2 + (z − h m )2 − ln (x − wm )2 + (z + h m )2 2h m (x + wm ) μ0 M0 Bz (x, z) = arctan 2π (x + wm )2 + z 2 − h 2m 2h m (x − wm ) − arctan . (4) (x − wm )2 + z 2 − h 2m To model the magnetic field of the Halbach PM array, the different magnetization directions of PMs [Fig. 2(c)] need to be considered. In this paper, PM 2R, which rotates 90° anticlockwise with respect to PM 2, was introduced to predict the magnetic field of PM 2 using (4). Based on Fig. 3, for an arbitrary point located below the PM 2, i.e., p0 = (α, β)T

TEO et al.: MODELING OF A 2 DoF MMLM

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with β < −h m , the magnetic flux density can be obtained from a corresponding point located in (−β, α) in PM 2R through simple coordinate transformation 2

B(p0 ) = 2 R2R 2R B(2R R2 p0 )

reexpressed as μ0 M0 (α + wm )2 + (β − h m )2 ln 4π (α + wm )2 + (β + h m )2 (α − wm )2 + (β − h m )2 − ln (α − wm )2 + (β + h m )2 μ0 M0 (−β + h m )2 + (α − wm )2 =− ln 4π (−β + h m )2 + (α + wm )2 (−β −h m )2 + (α−wm )2 − ln . (10) (−β −h m )2 + (α+wm )2

1

Bx (α, β) =

(5)

where i B denotes the magnetic flux density of PM i and j Ri represents the rotation matrix from the coordinate system of PM i to the coordinate system of PM j given as cos(θ ) − sin(θ ) R(θ ) = (6) sin(θ ) cos(θ ) where θ represents the rotation angle about the y-axis. Thus, the magnetic field of PM 2 can be predicted from PM 2R using Bx (α, β) =

2

Bz (α, β) =

2

Bz (−β, α) 2R − Bx (−β, α). 2R

(7)

In theory, the magnetic flux density of the Halbach PM array can be predicted from (4) and (7) directly with some simple coordinate transformations. In practice, it is a challenge in solving arctan(tan(x)) = x + kπ because it is a multisolutions function with a set of k values, i.e., k = {. . . , −2, −1, 0, 1, 2, . . .}. Referring to Fig. 3, this multisolutions function causes a limitation if (4) and (7) are used to predict 2RBz along Line I for g < wm because it is essential but time consuming to identify the correct solution. In this paper, the difference of arctan function was used to avoid the multiple solutions and to simplify the magnetic field model of the Halbach PM array. The difference of arctan function is given as γ1 − γ2 . (8) arctan(γ1 ) − arctan(γ2 ) = arctan 1 + γ1 γ2

Based on Fig. 3, using (4) to determine the x- and z-component of the magnetic field density from PM 2R at (−β, α) will result to solutions similar as (9) and (10), respectively. Through this observation with the derivations of (7), (9), and (10), the interrelation of the x- and z-component of the magnetic flux density between PMs 1 and 2 at (α, β) with β < −h m can be written as

Bz (α, β) = 1Bz (α, β)

1

Bz (α, β) μ0 M0 2h m (α + wm ) = arctan 2π (α + wm )2 + β 2 − h 2m 2h m (α − wm ) − arctan (α − wm )2 + β 2 − h 2m μ0 M0 arctan =− 2π 2 + α2 − β 2 ) 4h m wm (h 2m − wm × 4 2 +w4 +2(h 2 −w2 )(α 2 −β 2 )+(α 2 +β 2 )2 h m −6h 2m wm m m m 2wm (−β + h m ) μ0 M0 arctan =− 2 2π (−β + h m )2 + α 2 − wm 2wm (−β − h m ) − arctan . (9) 2 (−β − h m )2 + α 2 − wm

Similarly, the x-component of the magnetic field density from PM 1, 1Bx , at an arbitrary point (α, β) can be

(11)

and

Bx (α, β) = −1Bz (α, β)

2

2

Bz (α, β) = −2RBx (−β, α).

(12)

Refers to (12), the multiple solutions problem when solving 2RB (−β, α) via (7) will be avoided by solving an equivalent z term, −1Bz (α, β). Similar derivations were conducted for another two PMs, i.e., PMs 3 and 4, and their interrelation of the x- and z-component of the magnetic flux density can be written as

Using (4) and (8), the z-component of the magnetic field density from PM 1, 1Bz , at an arbitrary point (α, β) can be reexpressed as 1

Bx (α, β) = −2RBx (−β, α)

1

Bx (α, β) =

3

Bx (−β, α)

2R

Bz (α, β) = − Bz (α, β)

3

1

(13)

and

4

Bx (α, β) = 1Bz (α, β) Bz (α, β) =

4

Bx (−β, α)

2R

(14)

where β < −h m . Subsequently, the x- and z-component of the magnetic flux density are decomposed into two basic components, Bmain (α, β) = 1Bz (α, β) and Bside (α, β) = 2RBx (−β, α), and the x- and z-component of magnetic flux density of the Halbach PM array, H B, are expressed as Bx (p) =

H

ρmain (i )Bmain (p − pi ) + ρside (i )Bside (p − pi )

Nm

Bz (p) =

H

σmain (i )Bmain (p − pi ) + σside (i )Bside(p − pi )

Nm

(15)

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Fig. 4.


Schematic diagram of a Halbach PM array with single coil.

respectively. Vc and Sc represent the volume of effective coil and the area of the coil cross section, respectively. The effective coil length is assumed to be same as lm and the rectangular cross section of the coil has a width of 2wc and height of 2h c . The overall force generated by an MMLM is determined by the summation of forces generated by all the PMs that form the Halbach PM array. Therefore, the force generated along the x- and z-axis are expressed as Fx (pc ) = K x (pc )Ic Fz (pc ) = K z (pc )Ic

with

μ0 M0 2h m (x + wm ) arctan 2π (x + wm )2 + z 2 − h 2m 2h m (x − wm ) − arctan (x − wm )2 + z 2 − h 2m μ0 M0 (−z + h m )2 + (x − wm )2 Bside (p) = ln 4π (−z + h m )2 + (x + wm )2 (−z − h m )2 + (x − wm )2 − ln (16) (−z − h m )2 + (x + wm )2

Bmain (p) =

where p represents the a coordinate (x, z) with respect to the global frame with z < −h m , pi denotes the center position of PM i with i = 1, 2, . . . , Nm , and Nm represents the number of PMs in the array. σmain (i ), σside (i ), ρmain (i ), and ρside (i ) are the coefficients determined by the configuration of the Halbach PM array expressed as ⎧ ⎧ i = 4n + 1 ⎨ 1, ⎨ −1, i = 4n +2 σmain (i ) = −1, i = 4n + 3 , σside (i ) = 1, i = 4n +4 ⎩ ⎩ 0, i = 2n + 2 0, i = 2n +1 ⎧ ⎧ ⎨ −1, i = 4n + 2 ⎨ −1, i = 4n +1 i = 4n + 4 , ρside (i ) = 1, i = 4n +3 ρmain (i ) = 1, ⎩ ⎩ 0, i = 2n + 1 0, i = 2n +2 (17) where n ∈ Z∗ with Z∗ denotes the set of positive integers. Here, the assumption was made that the widths of vertically magnetized PMs 1 and 3 are equal to the widths of horizontally magnetized PMs 2 and 4, as shown in Fig. 3. However, this decomposition approach is also generic for general problems where the widths of PMs are nonidentical by having four components to describe the flux density of PMs instead. Therefore, this provides more flexibility in designing the MMLM especially when unequal levitation and propulsion forces are desired.

(19)

where pc represents the center of the coil’s cross-sectional area as shown in Fig. 4 and Ic represents the magnitude of current flow in one single turn of the coil with K x and K z written as K x (pc ) = τ [σmain (i )φmain (pc −pi )+σside (i )ϕside(pc −pi )] Nm

K z (pc ) = τ [ρmain (i )φmain (pc −pi )+ρside (i )ϕside(pc −pi )] Nm

(20) where

φmain (x, z) =

Bmain (x, z) d x dz

μ0 M0 [φ (x −wc , z −h c )+φ (x + wc , z + h c ) 2π

−φ

(x + wc , z − h c ) − φ (x − wc , z + h c )] ϕside (x, z) = Bside (x, z) d x dz =

=

μ0 M0 [ϕ (x −wc , z −h c )+ϕ (x + wc , z + h c ) 4π −ϕ (x + wc , z − h c ) − ϕ (x − wc , z + h c )] .

Here, φ (x, z) = φ+ (x, z) − φ− (x, z) and ϕ (x, z) = ϕ+ (x, z) − ϕ− (x, z). Using φ± to define either φ+ or φ− , φ± (x, z) is expressed as 2h m −z 2h m +z R2 + z R3 φ± (x, z) = z(wm ± x)R1 +z 4 4 (x ± wm )2 − h 2m − h m (wm ± x) R4 + i 2 (x ± wm )2 − h 2m +h m (wm ± x) R5 −h m z + i 2 (21)

B. Analytical Force Modeling Governed by Lorentz-force law, the force generated by an energized rectangular coil under the Halbach PM array (Fig. 4) is expressed as

Jc × H B dv = τ Ic × H B ds (18) F= Vc

Sc

where Jc represents the volume current distribution in the coil, τ = Nt lm /4wc h c , Ic represents the current in the wire, Nt is the number of wire turns in the coil with wc , and h c representing half of the width and height of the coil,

where

2h m (wm ± x) (x ± wm )2 + z 2 − h 2m R2 = ln[(z − h m )2 + (x ± wm )2 ] R1 = arctan

R3 = ln[(z + h m )2 + (x ± wm )2 ] z(i h m − wm ∓ x) R4 = arctan h 2 + (x ± wm )2 m z(i h m + wm ± x) R5 = arctan . h 2m + (x ± wm )2


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Using ϕ± to define either ϕ+ or ϕ− , ϕ± (x, z) is expressed as ϕ± (x, z) = z(2h m ∓ z)T1 + h 2m − (x − wm )2 T2 + h 2m − (x + wm )2 T3 + (wm − x)(±h m − z)T4 + (wm + x)(±h m − z)T5 + 2wm z where

2 (z ∓ h m )2 + x 2 − wm T1 = arctan 2wm (h m ∓ z) z ∓ hm T2 = arctan wm − x z ∓ hm T3 = arctan wm + x

(22)

T4 = ln[(x − wm )2 + (z ∓ h m )2 ] T5 = ln[(x + wm )2 + (z ∓ h m )2 ]. The proposed force model is applicable for the MMLM with low- and high-order Halbach PM array. Note that φmain and ϕside decrease close to zero once the distance between the coil and the PM is larger than a certain length. Therefore, only those adjacent PMs are required to be calculated in (20). Similar with the other force models proposed in [1], [17], and [18], the magnetic field underneath the middle of the Halbach PM array can be assumed in periodical form. Hence, the force modeling of those coils underneath the middle of the Halbach PM array can be simplified by only calculating coils under one period of Halbach PM array and subsequently multiply by its equivalent number of periods instead of calculating separately. III. M ODEL -BASED A PPROXIMATION A PPROACH For applications that require high bandwidth closed-loop control, the prediction of force is performed on every sampling. Hence, the proposed force model needs to be approximated by a simplified model, which is more computational efficient. From [21], the curve fitting method is a common approach to approximate a mathematic representation based on given experimental results. However, using this method directly to approximate a simplified model, based on the proposed 2-D force model, will lead to a complex and inaccurate solution. In this section, a model-based approximation approach is proposed to obtain the simplified model, which largely reduces the computational time and yet guarantees the accuracy. This approach is generic to obtain the approximation results of different magnitude of air gaps, as well as different dimensions of PMs and coils. From (19) and (20), it is observed that the computation effort is largely concentrated on the two components, i.e., φmain and ϕside , which are the integral of the magnetic field within the cross section of the coil. Therefore, one direct simplification is to treat the magnetic field as a uniform field with its magnitude being chosen from the center of the coil’s cross section. Using this simplification approach, φmain and ϕside are

expressed as

2h m (x + wm ) 2M0 μ0 wc h c arctan π (x + wm )2 + z 2 − h 2m 2h m (x −wm ) − arctan (x −wm )2 + z 2 − h 2m M0 μ0 wc h c (−z + h m )2 + (x − wm )2 d ϕside (x, z) = ln π (−z + h m )2 + (x + wm )2 (−z −h m )2 +(x − wm )2 − ln . (−z − h m )2 +(x +wm )2 (23)

d (x, z) = φmain

Termed as the direct model, (23) ignores the distribution of magnetic field within the cross section of the coil and should produce certain inaccuracy. Therefore, an approximad (x, z) and tion approach is proposed to approximate φmain d ϕside (x, z) to be closer to φmain (x, z) and ϕside (x, z), where the value of z in (23) can be tuned as z op instead. The relationship between the tuned value z op and z is expressed as z op = z f (z). To obtain the function z f (z) in φmain , an optimization is created with the following objective function: φ d (x, z op ) − φmain (x, z). min (24) z op

main

x

For each value of z within the range as specified, a numerical search can be performed to find an optimal value for z op d that obtains the least absolute error between φmain (x, z op ) and φmain (x, z) on the summation of x. To prevent inaccuracy, the range of x during the optimization process can be chosen as (−4wm and 4wm ) because the magnetic field decreases close to zero when out of this range. The summation is performed on x at a discrete step of wm /100. During the optimization process, z can be chosen discretely and the range of z is dependent on the design specification of vertical stroke. As the specifications of the Halbach PM array and the coil are known, the optimal value of z op for each z can be obtained by a numerical linear search. Therefore, z f (z) was solved as a simple 1-D fitting problem between the real position z and the obtained z op via the optimization process. Similar approach can be applied to obtain z f (z) in ϕside . Subsequently, the approximated models, which are obtained from the approximation approach, are expressed as f

φmain (x, z) 2M0 μ0 wc h c 2h m (x +wm ) = arctan π (x +wm )2 +z f (z)2 −h 2m 2h m (x −wm ) − arctan (x −wm )2 +z f (z)2 −h 2m f

ϕside (x, z) (−z f (z) + h m )2 + (x −wm )2 M0 μ0 wc h c = ln π (−z f (z)+h m )2 + (x +wm )2 (−z f (z)−h m )2 + (x −wm )2 − ln . (−z f (z)−h m )2 +(x +wm )2 (25) f φmain

f ϕside

and in (25) can be substituted in (20) to These replace the terms of φmain and ϕside , which is denoted as the simplified model.

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TABLE I D IFFERENT T YPES OF C ONFIGURATIONS

Fig. 6. MER results obtained from all six types of configurations with different L m values.

Fig. 5. Numerical search results for the optimal z op corresponding to the different z positions for all three types of configurations.

Fig. 5 shows the results of the numerical search for the optimal z op corresponding to different z positions for three different types of configurations with L m = 7.5 mm. Each type of configuration is listed in Table I. Fig. 5 shows that z f (z) between z and z op can be solved as a 1-D curve fitting problem using simple polynomial functions. It was also observed that the optimal z op for same position z are tuned differently in three different types to approximate the φmain and ϕside closer. In this paper, the maximal error rates (MERs) was defined to investigate the accuracy of the approximation model for certain air gap between the coil and PM. The MER, which accounts for the maximum errors between the derived {φmain , f f ϕside } and the approximated {φmain , ϕside }, is expressed as f f maxx φmain − φmain maxx ϕside − ϕside 2. MER = + maxx |φmain | maxx |ϕside | (26) The results of the MER based on different types of configurations with a fixed air gap of 1 mm between the coil and PM array are shown in Fig. 6. The MER results are the highest for Types 1 and 4, i.e., not exceeding 9%. For both configuration types, wc is equal to wm and can be considered as the extreme case since it is inefficient for wc to be larger than wm . For other configuration types, the MERs are less than 4%. Although this

Fig. 7. Comparison between the derived {φmain , ϕside }, the approximated f f d d } from the direct model. , ϕside {φmain , ϕside }, and {φmain

evaluation was conducted based on a fixed air gap of 1 mm, the MER results and trend can also be applied to different air gaps. By maintaining a constant ratio between L m and air gap, changing the L m can lead to various air gap values. Thus, the obtained MER results suggest that the approximated model is a generic solution for different dimensions of coils and PMs, and air gaps between the coil and PM array. To evaluate the accuracy of the approximated model, the f f values of {φmain , ϕside } were compared against the values of d d {φmain , ϕside } obtained via the direct method and the values of {φmain , ϕside } obtained from (23). The evaluation was conducted using Type 1 with L m = 75 mm and results are plotted in Fig. 7. It is guaranteed that approximation model should not be worse than the direct method in any situation, since the direct method is one special case that included in the approximation model, where z op = z. Based on the comparison results, it indicates that the approximation


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Fig. 8. (a) Experimental setup of force measurement of MMLM with either a translator prototype made up of (b) single magnetic pole Halbach PM array or (c) six magnetic poles Halbach PM array. Fig. 10. B x measured from the single magnetic pole Halbach PM array prototype verse the predicted B x obtained from the harmonics and proposed models with 3.5 mm air gap.

Fig. 9. Bz measured from the single magnetic pole Halbach PM array prototype verse the predicted Bz obtained from the harmonics and proposed models with 2 mm air gap.

model (25) is more accurate than the direct method (23) in predicting the values of {φmain , ϕside }. IV. E XPERIMENTAL I NVESTIGATIONS AND D ISCUSSION In this section, experimental investigations are conducted based on the MMLMs with both low-order and typical Halbach PM arrays. A. MMLM With Single Magnetic Pole Halbach PM Array An MMLM prototype was developed based on a single magnetic pole Halbach PM array, as shown in Fig. 8(b). First, an experimental investigation was conducted to evaluate the accuracy of the proposed magnetic field model, using this PM array. The magnetic flux density was measured by a Lake-Shore 3-axes Gaussmeter (Model 460) and data were recorded by a LabVIEW program via a National Instruments data acquisition card (Model PCI-6035E). Figs. 9 and 10 plot the Bz and Bx of the PM array obtained experimentally and

analytically via the harmonic and the proposed models at air gaps of 2 and 3.5 mm, respectively. Comparing between the Bz predicted by the proposed magnetic field model and the measured values, a maximum variation of 4.70% and an average variation of 1.69% are obtained. Between the Bx predicted by the proposed magnetic field model and the measured values, the maximum and average variations are 4.56% and 2.76%, respectively. Clearly from Figs. 9 and 10, results show that the traditional harmonic field model is less effective in predicting the magnetic field of single magnetic pole Halbach PM array. This investigation shows that the proposed magnetic field model is effective in predicting the magnetic field of the low-order magnetic pole Halbach PM array as it predicts the flux leakages at the edges accurately. Second, experimental investigation was also conducted to evaluate the accuracy of the proposed force model. The experimental setup is shown in Fig. 8(a) where the translator with Halbach PM array was mounted on a NSK three axes ac-servo gantry system. A force/torque sensor (Model ATI, Mini40) was used to measure the generated force. Similar to the previous recording method, the measured force were recorded by a LabVIEW program via a National Instruments data acquisition card (Model PCI-6035E). During the measurement, only one coil within the stator was energized with a current of 3 A while the gantry machine carried the translator to move in a step of 0.2 mm along the x-direction at a certain air gap while the generated forces were measured at each step concurrently. To reduce the noise disturbances, 200 measurements were conducted at each step and the average value of these data were recorded. Before going to the Halbach PM array, an experiment is conducted to investigate the force generated by single magnet, as it is noted that the accuracy of the proposed force model largely depends on accuracy of the two force components, i.e., φmain and ϕside . Fig. 11 plots the generated forces recorded from the prototype against the forces predicted via the approximated model in Section III and analytical model in Section II

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Fig. 11. Forces measured from one single magnet, i.e., PM 1 (Fig. 3), with 1 mm air gap between the coil and PM verse the predicted forces obtained from the proposed approximated and analytical models.

Fig. 12. Forces measured from one single magnet, i.e., PM 2 (Fig. 3), with 2 mm air gap between the coil and PM verse the predicted forces obtained from the proposed approximated and analytical models.

based on a single magnet (PM 1 in Fig. 3) with an air gap of 1 mm between coils and PM, and 3 A input current. For the translation force along the x-axis, the predicted Fx between the approximated and analytical models are almost identical. When comparing the predicted Fx with the actual Fx generated from the prototype, the maximum and average variations are 4.5% and 1.45%, respectively. As for the levitation force along the z-axis, the predicted Fz obtained from the approximated model are similar to the predicted Fz obtained from the analytical model. When comparing the predicted Fz with the actual Fz generated from the prototype, a maximum variation of 7.17% and an average variations of 1.28% are obtained. Similarly, Fig. 12 plots the generated force recorded from the prototype against the forces predicted via the approximated and analytical models based on a single magnet PM 2 (Fig. 3) with an air gap of 2 mm between coils and PM. From Fig. 12, the maximum variation between the actual and predicted


Fx is 8.66% while the average variation is 2.31%. When comparing the predicted and actual Fz , a maximum variation of 10.49% and an average variations of 5.39% are obtained. At different air gaps, the predicted Fx and Fz obtained from the approximated and analytical models are also identical, while both agree with the measured force. The observed error is considered to be caused by two reasons: 1) manufacturing ununiformity of PMs. Ideally, the magnetization in PM is uniform in only one direction, however, due to the manufacturing ability, the fabricated PMs magnetization is not absolute ideal and 2) limitation of force sensor. The force sensor may not work very well in small force magnitude region. It is also noted that the computation time here using the original analytical and the approximated models for one point’s force are 6.4 × 10−5 and 2.9 × 10−6 s (Intel i7-3770 CPU 3.40 GHz), respectively. This shows that the approximated model is a simplified yet accurate model that can be used to replace the proposed analytical model when high bandwidth closed-loop control is required. The schematic details of the MMLM are shown in Fig. 15(a). During actual operations, only the six adjacent coils underneath the Halbach PM array are energized with currents. The six coils with current Ik , which represents the magnitude of current in respective coil and k denotes the coil number, i.e., k = 1, 2, . . . , 6, are shown in Fig. 15(a). Based on the developed MMLM, the first, second, and third coils are connected together with the fourth, fifth, and sixth coils in the opposite polarity. According to the wiring sequence and (19), the force model for this MMLM prototype is expressed as K x (pc1 ) − K x (pc4 ) K x (pc2 ) Fx =− K z (pc1 ) − K z (pc4 ) K z (pc2 ) Fz ⎡ ⎤ I −K x (pc5 ) K x (pc3 ) − K x (pc6 ) ⎣ 1 ⎦ I2 (27) −K z (pc5 ) K z (pc3 ) − K z (pc6 ) I3 where pci , i = 1, 2, . . . , 6, denotes the center position of Coil i . The negative symbol in (27) suggests that the forces in coils and the Halbach PM array are action and reaction forces in opposite directions. In addition, the force generated from the Halbach PM array was computed from the forces generated from the coils. Fig. 15(b) shows the flowchart to implement the modelbased approximation approach to simplify the force model in real-time control. The analytical model in Section II can be determined based on the design specification directly. Next, the optimization problem created in Section III can be solved off-line. Based on the results of the optimization problem, the simplified model can be obtained. Finally, the simplified model, which is to replace the terms of φmain and ϕside in (20), will be implemented in real-time control on-line algorithm. For this design, the z f (z) can be obtained separately for both f f simplified models of φmain and ϕside (25) as f

φmain : z f (z) = 0.9968z − 0.0001 f

ϕside : z f (z) = 0.9910z − 0.0002.

(28)

Based on the MMLM prototype, the forces obtained from the proposed approximated model, the traditional harmonics


Fig. 13. Forces measured from the MMLM prototype with 1 mm air gap between the coil and PM verse the predicted forces obtained from the proposed and harmonic models.

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Fig. 15. (a) Schematic details of the MMLM prototype with a moving translator. (b) Flowchart of model implementation in real-time control applications.

predicted Fx obtained from the proposed approximated model is 7.21%, and the average variation is 2.44%. When compared against the actual Fz , the predicted Fz obtained from the proposed approximated model has a maximum and average variations of 12.17% and 4.51%, respectively. As for the predicted Fx obtained from the traditional harmonics model, a maximum variation of 78.86% and the average variation of 32.45% are obtained when compared against the actual Fx . In addition, the maximum and average variations between the actual Fz and predicted Fz obtained from the traditional harmonics model are 131.68%, and 49.59%, respectively. The results obtained from these investigations suggest that the accuracy of the approximated model in predicting the current– force characteristic of the MMLM prototype with a single magnetic pole Halbach PM array is consistent even at different air gaps. Fig. 14. Forces measured from the MMLM prototype with 2 mm air gap between the coil and PM verse the predicted forces obtained from the proposed and harmonic models.

model, and the experimental measurements are plotted in Figs. 13 and 14, with air gaps of 1 and 2 mm between the PM and coil arrays, respectively. From Fig. 13, the maximum variation between the actual Fx and predicted Fx obtained from the proposed approximated model is 6.48% while the average variation is 2.30%. When comparing the actual Fz against the predicted Fz obtained via the proposed approximated model, a maximum variation of 10.22% and an average variations of 3.54% are obtained. On the other hand, the maximum variation between the actual Fx and predicted Fx obtained from the traditional harmonics model is 79.93% while the average variation is 32%. In addition, the maximum and average variations between the actual Fz and predicted Fz obtained from the traditional harmonics model are 128.59% and 49.49%, respectively. From Fig. 14, the maximum variation between the actual Fx obtained from the prototype with 2 mm air gap and

B. MMLM With Six Magnetic Poles Halbach PM Array To demonstrate the generality of the proposed models, a separate translator formed by six magnetic poles Halbach PM array [Fig. 8(c)] was also fabricated to evaluate the accuracy of the proposed magnetic field and force models. First, the magnetic flux density along the x- and z-axis was measured, and Figs. 16 and 17 plot the Bz and Bx of the six magnetic poles PM array obtained experimentally and analytically via the harmonic and proposed models at air gaps of 5 and 3.5 mm, respectively. When comparing the Bz predicted by the traditional harmonic model and the measured values, the maximum and average variations are 29.89% and 5.04%, respectively. Between the Bx predicted by the traditional harmonic model and the measured values, a maximum variation of 32.13% and an average variation of 6.35% are obtained. Results show that the harmonic field model is effective in predicting the magnetic field of multiple magnetic poles Halbach PM array. When comparing between the Bz predicted by the proposed magnetic field model and the measured values, a maximum variation of 4.82% and an average variation of

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Fig. 16. Bz measured from a six magnetic pole Halbach PM array prototype verse the predicted Bz obtained from the harmonics and proposed models with 5 mm air gap.

Fig. 18. Forces along the x-axis measured from the MMLM prototype with a six magnetic poles Halbach PM array at 0.5 mm air gap verse the predicted forces obtained from the proposed and harmonic models.

Fig. 17. B x measured from a six magnetic pole Halbach PM array prototype verse the predicted Bz obtained from the harmonics and proposed models with 3.5 mm air gap.

Fig. 19. Forces along the z-axis measured from the MMLM prototype with a six magnetic poles Halbach PM array at 0.5 mm air gap verse the predicted forces obtained from the proposed and harmonic models.

2.03% are obtained. Between the Bx predicted by the proposed magnetic field model and the measured values, the maximum and average variations are 8.28% and 4.01%, respectively. These comparisons show that the proposed magnetic field model is also able to predict the magnetic field accurately even for multiple magnetic poles Halbach PM array. Therefore, this investigation shows that the proposed magnetic field model is an effective and generic solution for analyzing a Halbach PM array regardless of the numbers of magnetic poles. Finally, the forces along the x- and z-axis generated from the MMLM prototype with a six magnetic poles Halbach PM array at 0.5 mm air gap were measured and plotted against the analytical models in Figs. 18 and 19, respectively. From Fig. 18, the maximum variation between the actual

and predicted Fx obtained from the proposed approximated model is 6.27% while the average variation is 3.19%. On the other hand, the maximum and average variations between the actual and predicted Fx obtained from the traditional harmonics model are 30.99% and 5.63%, respectively. Based on Fig. 19, the maximum variation between the actual and predicted Fz obtained from the proposed approximated model is 12.01% while the average variation is 5.23%. The maximum variation between the actual and predicted Fz obtained from the traditional harmonics model is 45.87% while the average variation is 7.42%. Therefore, this investigation shows that the proposed approximated model is more accurate in predicting the current–force relationship of a MMLM even for translator with multiple magnetic poles Halbach PM array.


V. C ONCLUSION This paper presents an analytical force modeling approach that accurately predicts the current–force characteristic of the 2 DoF MMLM. Unlike existing theoretical force models, the uniqueness of this model is that it accounts for the flux leakage at the edges of the Halbach PM array, due to a proposed magnetic field model. Therefore, the proposed model is suitable for both low- and high-order Halbach PM array. Especially for the MMLM with low-order Halbach PM array, the resulted Maglev positioners become smaller and lighter, comparing with existing designs. Furthermore, the proposed model is also applicable for Halbach PM arrays with nonidentical width and height in single PM, this additionally increase the design freedom for some specific applications. The proposed magnetic field model for Halbach PM array is formulated based on the magnetic current model. Subsequently, an analytical force model was derived from the magnetic field model via the Lorentz-force principle. To facilitate the high sampling rate control applications, an approximation approach was used to obtain an approximated model, which is a simplified mathematical representation of the proposed analytical force model. This approximated model reduces the computational time and complexity while guarantees the accuracy of the predicted current–force characteristic of an MMLM. MMLM prototype was developed with translators with both single and six magnetic poles Halbach PM arrays. Experimental investigations have shown that the proposed magnetic field and force models are accurate in predicting the magnetic field and force for both the Halbach PM arrays. ACKNOWLEDGMENT This work was supported in part by the Collaborative Research Project under the SIMTech-NUS Joint Laboratory (Precision Motion Systems), Ref: U12-R-024JL, and in part by SingaporeMOE AcRF Tier 1 under Grant R-263-000-A44-112. R EFERENCES [1] D. L. Trumper, W.-J. Kim, and M. E. Williams, “Design and analysis framework for linear permanent-magnet machines,” IEEE Trans. Ind. Appl., vol. 32, no. 2, pp. 371–379, Mar. 1996. [2] W. J. Kim, “High-precision planar magnetic levitation,” Ph.D. dissertation, Dept. Mech. Eng., Massachusetts Inst. Technology, Cambridge, MA, USA, 1997. [3] M. E. Williams, “Precision six degree of freedom magnetically-levitated photolithography stage,” Ph.D. dissertation, Dept. Mech. Eng., Massachusetts Inst. Technology, Cambridge, MA, USA, 1997. [4] R. J. Hocken, D. L. Trumper, and C. Wang, “Dynamics and control of the UNCC/MIT sub-atomic measuring machine,” CIRP Ann., Manuf. Technol., vol. 50, no. 1, pp. 373–376, 2001. [5] R. Fesperman et al., “Multi-scale alignment and positioning system—MAPS,” Precision Eng., vol. 36, no. 4, pp. 517–537, Oct. 2012. [6] X. Lu and I.-U.-R. Usman, “6D direct-drive technology for planar motion stages,” CIRP Ann., Manuf. Technol., vol. 61, no. 1, pp. 359–362, 2012. [7] H. Ohsaki and Y. Ueda, “Numerical simulation of mover motion of a surface motor using Halbach permanent magnets,” in Proc. Int. Symp. Power Electron., Elect. Drives, Autom. Motion, Taormina, Italy, May 2006, pp. 364–367. [8] J. de Boeij, E. Lomonova, and A. Vandenput, “Modeling ironless permanent-magnet planar actuator structures,” IEEE Trans. Magn., vol. 42, no. 8, pp. 2009–2016, Aug. 2006.

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[9] J. W. Jansen, C. M. M. van Lierop, E. A. Lomonova, and A. J. A. Vandenput, “Modeling of magnetically levitated planar actuators with moving magnets,” IEEE Trans. Magn., vol. 43, no. 1, pp. 15–25, Jan. 2007. [10] C. M. M. van Lierop, J. W. Jansen, A. A. H. Damen, E. A. Lomonova, P. P. J. van den Bosch, and A. J. A. Vandenput, “Model-based commutation of a long-stroke magnetically levitated linear actuator,” IEEE Trans. Ind. Appl., vol. 45, no. 6, pp. 1982–1990, Nov./Dec. 2009. [11] H.-S. Cho, C.-H. Im, and H.-K. Jung, “Magnetic field analysis of 2-D permanent magnet array for planar motor,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3762–3766, Sep. 2001. [12] W. Min et al., “Analysis and optimization of a new 2-D magnet array for planar motor,” IEEE Trans. Magn., vol. 46, no. 5, pp. 1167–1171, May 2010. [13] J. Peng and Y. Zhou, “Modeling and analysis of a new 2-D Halbach array for magnetically levitated planar motor,” IEEE Trans. Magn., vol. 49, no. 1, pp. 618–627, Jan. 2013. [14] P. Berkelman and M. Dzadovsky, “Magnetic levitation over large translation and rotation ranges in all directions,” IEEE Trans. Mechatronics, vol. 18, no. 1, pp. 44–52, Feb. 2013. [15] Q. Xu, “Design and development of a flexure-based dual-stage nanopositioning system with minimum interference behavior,” IEEE Trans. Autom. Sci. Eng., vol. 9, no. 3, pp. 554–563, Jul. 2012. [16] L. Wang, J. Zheng, and M. Fu, “Optimal preview control of dual-stage actuators system for triangular reference tracking,” in Proc. 10th IEEE Int. Conf. Control Autom. (ICCA), Jun. 2013, pp. 164–169. [17] I. J. C. Compter, “Electro-dynamic planar motor,” Precision Eng., vol. 28, no. 2, pp. 171–180, Apr. 2004. [18] H. Jiang, X. Huang, G. Zhou, Y. Wang, and Z. Wang, “Analytical force calculations for high-precision planar actuator with Halbach magnet array,” IEEE Trans. Magn., vol. 45, no. 10, pp. 4543–4546, Oct. 2009. [19] T. J. Teo, I.-M. Chen, G. Yang, and W. Lin, “Magnetic field modeling of a dual-magnet configuration,” J. Appl. Phys., vol. 102, no. 7, pp. 074924-1–074924-12, 2007. [20] E. P. Furlani, Permanent Magnet and Electromechanical Devices. New York, NY, USA: Academic, 2001. [21] M.-Y. Chen, T.-B. Lin, S.-K. Hung, and L.-C. Fu, “Design and experiment of a macro–micro planar maglev positioning system,” IEEE Trans. Ind. Electron., vol. 59, no. 11, pp. 4128–4139, Nov. 2012.

Tat Joo Teo (M’09) received the B.Eng. (Hons.) degree in mechatronics engineering from the Queensland University of Technology, Brisbane, QLD, Australia, in 2003, and the Ph.D. degree in mechanical and aerospace engineering from Nanyang Technological University, Singapore, in 2009. He joined the Singapore Institute of Manufacturing Technology, Singapore, in 2009, as a Researcher with the Mechatronics Group. He holds four patents granted and two provisional patents filed. His current research interests include ultraprecision system, compliant mechanism theory, parallel kinematics, electromagnetism, electromechanical system, thermal modeling and analysis, energy-efficient machine, and topological optimization. Dr. Teo serves as a Technical Reviewer of the IEEE T RANSACTIONS ON M ECHATRONICS ; the IEEE T RANSACTIONS ON ROBOTICS , M ECHANISM , AND M ACHINE T HEORY ; and IFAC Mechatronics Journal. He was a recipient of the Best Session Paper Award in the 39th Annual Conference of the IEEE Industrial Electronics Society in 2013. One of his patents was selected as the recipient of 2014 R&D 100 Award.

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 2010, and the M.Sc. degree in electrical engineering from the National University of Singapore (NUS), Singapore, in 2013, 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 2013, and is an Attached Research Student with the Agency for Science, Technology, and Research, SIMTech. His current research interests include integrated servo-mechanical design of precision mechatronics and magnetic levitation technology.

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Chee Khiang Pang (Justin) (S’04–M’07–SM’11) received the B.Eng. (Hons.), M.Eng., and Ph.D. degrees in electrical and computer engineering from the National University of Singapore (NUS), Singapore, in 2001, 2003, and 2007, respectively. He was a Visiting Fellow with the School of Information Technology and Electrical Engineering (ITEE), University of Queensland (UQ), Brisbane, QLD, Australia, in 2003. From 2006 to 2008, he was a Researcher (Tenure) with Central Research Laboratory, Hitachi Ltd., Tokyo, Japan. In 2007, he was a Visiting Academic with the School of ITEE. From 2008 to 2009, he was a Visiting Research Professor with the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX, USA. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering at NUS. He is also an Associate with the Agency for Science, Technology, and Research (A*STAR), Singapore Institute of Manufacturing Technology, and a Faculty Associate with the A*STAR Data Storage Institute. He has authored and edited three research monographs, Intelligent Diagnosis and Prognosis of Industrial Networked Systems (CRC Press, 2011), High-Speed Precision Motion Control (CRC Press, 2011), and Advances in High-Performance Motion Control of Mechatronic Systems


(CRC Press, 2013). 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 the American Society of Mechanical Engineers. He is currently serving as an Associate Editor of the Journal of Defense Modeling and Simulation and Transactions of the Institute of Measurement and Control; on the editorial board of the International Journal of Advanced Robotic Systems, the International Journal of Automation and Logistics, and the International Journal of Computational Intelligence Research and Applications; and on the conference editorial board of the IEEE Control Systems Society. In recent years, he also served as a Guest Editor of the Asian Journal of Control, the International Journal of Systems Science, the Journal of Control Theory and Applications, and the Transactions of the Institute of Measurement and Control. He was a recipient of the Best Application Paper Award at the 8th Asian Control Conference, Kaohsiung, Taiwan, in 2011, and the Best Paper Award at the International Association of Science and Technology for Development, International Conference on Engineering and Applied Science, Colombo, Sri Lanka, in 2012.