Design and Modeling of a Six Degrees-Of-Freedom ... · non-contamination, multi-Degrees-Of-Freedom...

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Design and Modeling of a Six Degrees-Of-FreedomMagnetically Levitated Positioner Using Square

Coils and 1D Halbach ArraysHaiyue Zhu, Student Member, IEEE, Tat Joo Teo, Member, IEEE,

and Chee Khiang Pang, Senior Member, IEEE

Abstract—This paper presents a novel design of sixDegrees-Of-Freedom (DOF) magnetically levitated (maglev)positioner, where its translator and stator are implemented byfour groups of 1-Dimensional (1D) Halbach Permanent Magnet(PM) arrays and a set of square coils, respectively. By control-ling the eight-phase square coil array underneath the HalbachPM arrays, the translator can achieve six DOF motion. Themerits of the proposed design are mainly threefold. First, thisdesign is potential to deliver unlimited-stroke planar motionwith high power efficiency if additional coil switching system isequipped. Second, multiple translators are allowed to operatesimultaneously above the same square coil stator. Third, theproposed maglev system is less complex in regards to thecommutation law and the phase number of coils. Furthermore,in this paper, an analytical modeling approach is established toaccurately predict the Lorentz force generated by the squarecoil with Halbach PM array by considering the corner regioneffect, and the proposed modeling approach can be extendedeasily to apply on other coil designs such as circular coil, etc.The proposed force model is evaluated experimentally, andthe results show that the approach is accurate in both singleand multiple coil cases. Finally, a prototype of the proposedmaglev positioner is fabricated to demonstrate its six DOFmotion ability. Experimental results show that the Root MeanSquare Error (RMSE) of the implemented maglev prototype isaround 50 nm in planar motion, and its velocity can achieve upto 100 mm/s.

Index Terms—Magnetic levitation, planar positioner, 6 DOFmotion, Halbach PM array, square coil, modeling of Lorentzforce.

I. Introduction

MAGNETIC levitation technology is a promising solu-tion to achieve great performance for many motion sys-

Manuscript received November 6, 2015; revised March 26, 2016 andJune 5, 2016; accepted June 30, 2016. This work was supported by theCollaborative Research Project under the SIMTech-NUS Joint Laboratory(Precision Motion Systems), Ref: U12-R-024JL. This work was also sup-ported by Singapore MOE AcRF Tier 1 Grant R-263-000-A44-112.

H. Zhu is with the SIMTech-NUS Joint Laboratory on Precision MotionSystems, Department of Electrical and Computer Engineering, NationalUniversity of Singapore, Singapore 117583 (e-mail:[email protected]).

T. J. Teo is a visiting scientist with Precision Engineering ResearchGroup, Department of Mechanical Engineering, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA. He is also with the MechatronicsGroup, Singapore Institute of Manufacturing Technology, Agency for Sci-ence, Technology and Research, Singapore 638075 (e-mail:[email protected]).

C. K. Pang is with the Department of Electrical and ComputerEngineering, National University of Singapore, Singapore 117583 (e-mail:[email protected]).

tems, e.g., precision positioning [1], manipulation [2], suspen-sion [3], [4], and haptic interaction [5], due to its non-contact,non-contamination, multi-Degrees-Of-Freedom (DOF), andlong-stroke characteristics. Over the past decades, the researchon magnetic levitation attracts the attention of many re-searchers, and various kinds of magnetically levitated (maglev)motion systems are proposed. Generally, these maglev motionsystems are realized using either Lorentz force [6]–[8] orelectromagnetic force [5], [9], [10], and both the moving-magnet design [1], [7] and moving-coil design [11], [12]are proposed for applications with different requirements.Furthermore, the motion range of maglev systems are extendedfrom short-stroke to unlimited-stroke in both translational androtational axes [9], [10].

In literature, there is one class of maglev positioners [13]–[19] that are developed using sets of 2 DOF Moving Mag-net Linear Motors (MMLMs) [20], [21] as the forcers toprovide both the levitation and propulsion force concurrentlyto deliver 6 DOF motion. This 2 DOF MMLM consistsof a 1-Dimensional (1D) Halbach Permanent Magnet (PM)array translator and a three-phase coil stator. Due to thespecial arrangement of the Halbach PM array, the generatedmagnetic field is concentrated on one side of the PM array.Correspondingly, the 2 DOF Lorentz force is generated by thethree-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 bedecoupled and linearized from the control input. Generally,this kind of the maglev positioner has low system complexityin regards of the commutation law yet restricted to its limitedstroke, because large stroke leads to the increasing in sizefor both the coil and translator, hence degrades the powerefficiency.

To achieve large-stroke planar motion with improved powerefficiency, maglev positioners with 2-Dimensional (2D) Hal-bach PM array are proposed [22]–[26]. One design from thisclass employs the 2D Halbach PM array with 45◦ rotatedrectangle coils [25], which allows multi-translators to operatesimultaneously above the same stator. By actively switchingthe effective coils underneath the PM array, this design enablesthe translator to achieve long-stroke planar motion with highpower efficiency. With the 2D Halbach array, circular coilsare adopted in [27] so that the maglev positioner can achievelong-stroke planar motion with full rotation ability around the




vertical axis. Compared with the 1D Halbach PM array, onedemerit of this 2D Halbach PM array is that its force ripplecaused by the harmonic magnetic field is more obvious [13],[25], and many research are also conducted to explore better2D Halbach PM array [28]–[31] with lower higher-orderharmonics to reduce the force ripple. In addition, the phasenumber of coils needed in this class of maglev positioners isrelatively large. Recently, the maglev positioners with 1D Hal-bach PM array are evolved to deliver long-stroke planar motionby replacing the four coil stators with the multi-layers PCBof orthogonal long coils [32] or the copper strip array [33].Compared to the 2D Halbach class, these 1D Halbach-basedmaglev designs are with lower system complexity and lessforce ripple. The potential disadvantage of this class could bethat only one translator is allowed to operate above the statorand the power efficiency is still relatively lower in large-strokemotion.

In this paper, a novel design of 6 DOF maglev positioneris proposed by utilizing 1D Halbach PM arrays and a setof square coils as the translator and stator, respectively. Inthe proposed design, the major limitations of the existingdesigns are avoided, that this design inherits the advantagesof 1D Halbach PM array maglev design but breaks its short-stroke barrier, so that the system complexity remains low inregards to commutation and current allocation. In theory, itcan achieve unlimited-stroke planar motion with high powerefficiency if additional coil switching system is equipped.Furthermore, the proposed design allows multi-translators tooperate simultaneously above the same stator. In addition, thephase number of coils are reduced from twelve to eight byemploying two-phase current configuration for each MMLM,which cuts down the hardware cost and further lowers thesystem complexity. To accurately predict the Lorentz forcegenerated by the square coil in the proposed design, ananalytical force modeling approach is established in this paper,so that the Lorentz force generated by the corner region ofthe square coil can be modeled effectively. Compared withthe model in [34], the proposed approach in this paper ismore generic and it can be applied on many coil designs withcorner area, e.g., rectangular coil with inner and outer radiusand circular ring coil.

The rest of the paper are organized as follows. The workingprinciple of the proposed maglev positioner is detailed inSection II and Section III presents the force modeling approachfor the proposed maglev design. Results from analytical andnumerical analyses are presented on Section IV and Section Vshows the implemented prototype of the proposed maglevpositioner with the discussions about the experimental results.

II. Working Principle of Maglev Positioner

The schematic of the proposed 6 DOF maglev positioneris shown in Fig. 1. In this design, the translator, which isformed by four 1D Halbach PM arrays, moves above thesquare coil stator. In theory, as the number of coils increases,the maglev positioner can achieve unlimited-stroke planarmotion with high power efficiency by using the additionalcoil switching techniques [25], [35] to actively energize the

Fig. 1. Schematic of the proposed design of 6 DOF unlimited-stroke maglevpositioner.

Fig. 2. Relative position change from (a) to (b) in y-direction with theindicated specifications of the coils and Halbach PM array.

effective coils. Each Halbach PM array in the translator formsa forcer delivering both the levitation and propulsion force asindicated in Fig. 1, namely, Magnet Array I and III produceforce along the x− and z−directions while Magnet Array II andIV produce force along the y− and z−directions. Six coordinatesystems are assigned in the maglev positioner, as described inFig. 1, that the global coordinate system (xc,yc,zc) is on thecoil stator, the local coordinate (xt,yt,zt) coincides with thecenter of translator, and (xi,yi,zi) is for each Halbach PM arrayi, where i = I, II, III, and IV. The coils in the stator are mappedin global coordinate system, and the sensor system feedbackthe position information of the translator. Consequently, thecorresponding coils underneath each Halbach PM array areenergized accordingly to deliver the 6 DOF motion.

The design specifications of the Halbach PM arrays andcoils are shown in Fig. 2, the width of the square air core in




Fig. 3. Conceptual illustration of relationships between the magnetic field ofHalbach PM array and currents in the square coils.

the square coil wa = τ, and the length of single square coillc = 3τ. The number of PMs in each Halbach PM array is 6m,where m = 1, 2, · · · , and the magnetizations of the Halbach PMarray are depicted in Fig. 3. The PMs in the Halbach PM arrayhave the square cross-sections, and their height hm and widthwm are both τ. The length of each PM lm is designed as 3nτwhere n = 2, 3, · · · . The gaps lg (shown in Fig. 1) betweentwo Halbach PM arrays should be at least more than thelength of two square coils, i.e. lg ≥ 6τ, which ensures that thecoils below each Halbach PM array will work independentlywithout intersection regions.

In this work, one assumption made is that the magneticfield of each Halbach PM array remains constant along theyi−direction in the area underneath the array and falls abruptlyto zero outside the array. (This assumption will be analyzed indetail in Section IV for practical considerations.) Referring toFig. 2, consider the situation that coils translate from (a) to (b)in the yi−direction while other directions remain unchanged.Under the assumption, the effective parts Region III and IVin Fig. 2(b) are identical to Region I and II in Fig. 2(a),respectively. This indicates that the modeling of generatedLorentz force for the Halbach PM arrays can be reduced intoa 2D model, which is independent to their yi−axis locations.By energizing the coils along yi−direction with same current,there will constantly be n effective coils along the yi−direction.

To control the maglev positioner, eight-phase current areemployed as the driven sources. In Halbach PM array i,two input current Ii,1 and Ii,2 are energized in the coilsunderneath the Halbach PM array. Refer to the Fig. 3, thex− and z−directions magnetic field of the Halbach PM arrayare sinusoidal waves with a period of 4τ as indicated. Here,Coil 1 and 3 are energized with the same magnitude currentbut in the opposite directions, namely Ii,1 and −Ii,1. Similarly,Coil 2 and 4 are energized with Ii,2 and −Ii,2, respectively.By neglecting the corner region behaviors of the square coil,Segment 1 and 2 are separated by 2τ, which is half of themagnetic field period. This indicates that the magnetic field inthese two segments are in opposite directions, and note thatthe currents in these two segments are of opposite directions,

Fig. 4. (a) Three type of coils, a1 rectangular coil with outer radius, a2rectangular coil with inner and outer radius, and a3 circular ring coil, (b)modeling of the corner region in Cartesian and Polar coordinate systems.

hence the generated force in these two segments are identicalin both magnitude and direction. As Coil 1 and 3 are differedby 6τ, Segment 1 and 4 are in a phase difference of 3π in themagnetic field. Therefore, Segment 1 (Coil 1) and Segment4 (Coil 3) also produce identical force. On the other hand,Coil 1 and 2 are separated by 3π/2 (3τ) in the magnetic field,this indicates that when Coil 1 produces zero force, Coil 2 isin the position that produces maximal force, and vice versa,as illustrated in Fig. 3. This naturally avoids the singularityduring force commutation.

III. Force Modeling for Maglev Positioner

In this section, a general force modeling approach to predictthe generated Lorentz force by the corners of coil is proposed.Based on this approach, the force model for the squarecoil utilized in the proposed maglev positioner is establishedanalytically.

A. General Force Modeling Method for Coil Corners

Rectangular and square coils are commonly used in ma-glev or other Lorentz-force systems to provide the drivingforce [24], [25], [36]–[38]. In the existing works, the geometryof the coil are typically simplified into a rectangular surfacedue to its simplicity in force modeling. For example inFig. 4(a), to predict the Lorentz force of the a1 coil, the forcemodeling is often concentrated only on the Rectangular Re-gion, while the Corner Region is neglected. This is reasonablein some designs since the Corner Region only accounts a smallportion of the whole coil. However, this simplification is notreasonable for some designs. For instance, consider the squarecoil used in Fig. 2, the corner regions account more than 1/3of the total coil and as a result, directly neglecting the cornerregions in the force modeling is not accurate. In this work,a general force modeling method for predicting the Lorentzforce generated by the corner regions of the coil are proposed.With the Halbach PM array being employed in the maglevsystem, the sinusoidal magnetic fields are considered. Usingthis method, the force generated by the Corner Regions withinner and outer radius in the rectangular coil (Fig. 4(a), a2) oreven the circular ring coil (Fig. 4(a), a3) can be modeled in aclosed form.




The corner region of the coil is considered as a quarterof the round disk as illustrated in Fig. 4(b). The cylindricalcoordinate is introduced to facilitate the force modeling, andin 2D perspective, the modeling can be reduced to the Polarcoordinate as indicated in Fig. 4(b). The magnetic field of1D Halbach PM array is predicted by first order harmonicmodel [13], [39] in Cartesian coordinate system, given as

Bx(x,z) = −B0e−γz sin(γx)Bz(x,z) = B0e−γz cos(γx)

, (1)

where B0 is expressed as

B0 =2√

2µ0M0

π(1− e−γhm ), (2)

and µ0 and M0 represent the permeability of the free spaceand the peak magnetization magnitude of PMs, respectively.wm and hm represent the width and height of single PM,respectively. γ is the spatial wave number, γ = 2π/L, whereL denotes the pitch of the Halbach PM array, i.e. L = 4wm.

Assuming that the center of the corner disk is located atp0 = (x0,y0,z0), for any point (p0,r, θ) in the corner region,the magnetic field density is expressed in

Bx(p0,r, θ) = −B0e−γz0 sin(γ(x0 + r cosθ))Bz(p0,r, θ) = B0e−γz0 cos(γ(x0 + r cosθ))

, (3)

where r ∈ [0, R], θ ∈ [0, π/2], and R is the radius of the cornerdisk. Similarly, for a point located (p0,r, θ) in corner region,the effective current density Jeff is expressed as,

Jeff(p0,r, θ) = J cos(θ), (4)

where J is the current density vector in the corner region ofthe coil.

Governed by the Lorentz force law, for any point (p0,r, θ)in corner region, the Lorentz force density is expressed as

fCorner(p0,r, θ) = J (p0,r, θ)×B(p0,r, θ), (5)

and the force generated by the corner region in the cylindricalcoordinate system is given as

F =

∫VJ ×Bdv

=

z0+hc∫z0

π/2∫0

R∫0

fCorner(p,r, θ)r drdθdz.(6)

Substitute (3), (4), and (5) into (6), the horizontal forceFCorner−x is calculated as

FCorner−x(p0) =

z0+hc∫z0

π/2∫0

R∫0

Jeff(p,r, θ)Bz(p,r, θ)rdrdθdh,

=JB0

γ3 (1− e−γhc )e−γz0ϕx(x0, γ, R),

(7)

where

ϕx(x0, γ, R) =

π/2∫0

{γRsin(γ(x0 + Rcosθ))

+cos(γ(x0 + Rcosθ))− cos(γx0)

cosθ

}dθ.

(8)

0 0.5 1 1.5 20

1

2

3

4

λ

Mag\Pha

α(λ): Numerical

β(λ): Numerical

α(λ): Approximated

β(λ): Approximated

Fig. 5. The numerically obtained α(λ) and β(λ) in λ ∈ [0, 2] with theapproximation.

Similarly, the vertical force FCorner−z(p0) is expressed as

FCorner−z(p0) =

z0+hc∫z0

π/2∫0

R∫0

−JB0e−γz cosθ

∗ sin(γ(φ+ r cosθ))rdrdθdh,

=JB0

γ3 (1− e−γhc )e−γz0ϕz(x0, γ, R),

(9)

where

ϕz(x0, γ, R) =

π/2∫0

{γRcos(γ(x0 + Rcosθ))

−sin(γ(x0 + Rcosθ))− sin(γx0)

cosθ

}dθ.

(10)

In general, the integral terms ϕx and ϕz in (7) and (9) haveno simple analytical solutions due to their complexities, andthe numerical curving fitting approach is alternatively adoptedin this work. Without loss of generality, the radius of the cornerdisk R in Fig. 4(b) can be defined through the ratio in relativeto the width of single PM τ, i.e., R = λτ, where λ denotes aratio value. Note that γ = 2π/4τ, as a result, the term γR in(7) and (9) will cancel τ each other. Therefore, the integralterms ϕx and ϕz are only related to the radius ratio λ with x0and γ, which are

ϕx(x0, γ, R) = ϕx(x0, γ, λ)ϕz(x0, γ, R) = ϕz(x0, γ, λ)

. (11)

For a specific design, the ratio λ is fixed as a constant value.Although the analytical solution of ϕx(x0, γ, λ) and ϕz(x0, γ, λ)are unable to obtained, the numerical integral method can beperformed to determine the value of the integral at every γx0.Furthermore, the obtained results are approximated by a simpleanalytical expression with the variable γx0. Also interpreted bythe physical meaning, by considering the corner region of thecoil moving along the periodic sinusoidal magnetic field, theresulted force is still periodic. Therefore, this approximationis made as

ϕx(x0, γ, λ) ≈ α(λ) sin(γx0 +β(λ))ϕz(x0, γ, λ) ≈ α(λ)cos(γx0 +β(λ))

, (12)

where α(λ) and β(λ) represent the magnitude and the phasenumerically approximated, respectively. Fig. 5 shows the val-ues of α(λ) and β(λ) from the numerical integral identification




Fig. 6. (a) Representation of the square coil using eight regions with aequivalent rectangular region and (b) uniform model of the square coil inpolar coordinate systems.

in the range λ ∈ [0, 2] at a step of 0.05. A 4th order polynomialis employed to approximate the function of α(λ) and β(λ),

α(λ) = p1λ4 + p2λ

3 + p3λ2 + p4λ+ p5

β(λ) = q1λ4 + q2λ

3 + q3λ2 + q4λ+ q5

λ ∈ [0, 2], (13)

where the coefficients are listed as follow

p1 = −0.0550 p2 = −0.0838 p3 = 1.3046 p4 = −0.0237p5 = 0.0020 q1 = −0.0281 q2 = 0.1343 q3 = −0.2211q4 = 0.9724 q5 = 1.5342.

(14)Subsequently, the horizontal force FCorner−x(p0) and the

vertical force FCorner−z(p0) are expressed as

FCorner−x(p0) =α(λ)JB0

γ3 (1− e−γhc )e−γz0 sin(γx0 +β(λ))

FCorner−z(p0) =α(λ)JB0

γ3 (1− e−γhc )e−γz0 cos(γx0 +β(λ)).

(15)

Using this method, the Lorentz force generated by the cor-ner region of different geometry coil in the Halbach PM array’sfield can be modeled in an analytical form. For example,consider coils a2 and a3 with inner radius ri and outer radiusro in Fig. 4(a), the corner region force is modeled by quarterdisk with radius ro subtracting the quarter disk with radius ri.

B. Force Modeling for Single Square Coil

To model the magnetic force generated by one square coilin the magnetic field, the coil is represented a combinationof eight regions as shown in Fig. 6(a). There are four squareregions and four corner regions defined as Square I, Square II,Square III, Square IV, Corner I, Corner II, Corner III, andCorner IV, respectively.

As illustrated in Fig. 3, due to the constant phase differenceπ in the sinusoidal magnetic field, Square I and III will always

produce equivalent force. With Square II and IV in the samemagnetic field but having opposite current directions, the forcegenerated by them always cancel out with each other. Bysymmetry, the Corner I and II generate identical force, andsimilar case happens in Corner III and IV. Therefore, the totalforce generated by one square coil is expressed as,

FCoil = 2(FSquare +FCorner I +FCorner IV), (16)

where FSquare represents the force generated by Square Ior III, FCorner I, and FCorner IV represent the force generatedby Corner I and IV, respectively.

For a square coil centered pc = (xc,yc,zc) in local HalbachPM array coordinate system, FSquare can be obtained directlyusing the Lorentz force law, the horizontal force FSquare−x(pc)is expressed as

FSquare−x(pc) = τ

zc+hc∫zc

xc+ 3τ2∫

xc+ τ2

NtIc

wchcB0e−γz cos(γx)dxdz

=

√2NtIcB0τ

γ2wchce−γzc (e−γhc −1)sin(γx),

(17)

where Nt represents the turn number of the coil, Ic representsthe current of the coil, wc and hc represent the width andheight of the coil, respectively. Similarly, the vertical forceFSquare−z(pc) is given as

FSquare−z(p0) = τ

zc+hc∫zc

xc+ 3τ2∫

xc+ τ2

−NtIc

wchcB0e−γz sin(γx)dxdz

=

√2NtIcB0τ

γ2wchce−γz0 (e−γhc −1)cos(γx).

(18)

The Lorentz force generated by the corner regions ofthe square coil are modeled based on the proposed methodpresented previously. Slight differently, the force modeling oftwo corners in the square coil are treated uniformly. For bothCorner I and IV, the poles of Polar Coordinate System I andPolar Coordinate System IV are located in each center of theirround disks, and polar axis are both in the direction of thex−axis. Refer to Corner I and IV in Fig. 6(b), the magneticfield density of a arbitrary point (pc,r, θ) in Corner I or III is

Bx(pc,r, θ) = −B0e−γzc sin(γ(ν+ r cosθ))Bz(pc,r, θ) = B0e−γzc cos(γ(ν+ r cosθ))

, (19)

where ν and θ are defined as

ν =

{x0 +τ/2, Corner Ix0−τ/2, Corner IV ,

andθ ∈

{ [0, π/2

], Corner I[

π/2, π], Corner IV .

Based on this definition, the generated magnetic force on bothCorner I and IV can be treated together. The horizontal forceof Corner i (i = I or IV) FCorner i−x(pc) is given as

FCorner i−x(pc) =

zc+hc∫zc

θ2∫θ1

τ∫0

Jeff(pc)Bz(r, θ, z)rdrdθdh, (20)




TABLE IValue of θ1 and θ2

Corner I Corner IVθ1 0 π/2θ2 π/2 π

Similarly, the vertical force FCorner i−z(pc) is expressed as

FCorner i−z(pc) =

zc+hc∫zc

θ2∫θ1

τ∫0

Jeff(pc)Bx(r, θ, z)rdrdθdh, (21)

where θ1 and θ2 are defined separately for Corner I and IV asin Table. I.

Using the results presented previously, the horizontal forceFCorner−x(pc) and the vertical force FCorner−z(pc) are expressedas,

FCorner i−x(pc) =αNtIcB0

γ3wchc(1− e−γhc )e−γzc sin(γν+βi)

FCorner i−z(pc) =αNtIcB0

γ3wchc(1− e−γhc )e−γzc cos(γν+βi)

, (22)

where α ≈ 1.144, βI ≈ 2.3924, and βIV ≈ −2.3924. Therefore,the generated force FCoil−x and FCoil−z on single square coilare expressed as,

FCoil−x(pc) = Kx(xc,zc)Ic

FCoil−z(pc) = Kz(xc,zc)Ic, (23)

where Kx(xc,zc) and Kz(xc,zc) are defined as,

Kx(xc,zc) =2NtB0

γ2wchc(1− e−γhc )e−γzc

[−√

2τsin(γxc)

+α

γ

(sin(γ(xc +

τ

2) +βI) + sin(γ(xc−

τ

2) +βIV )

)]Kz(xc,zc) =

2NtB0

γ2wchc(1− e−γhc )e−γzc

[−√

2τcos(γxc)

+α

γ

(cos(γ(xc +

τ

2) +βI) + cos(γ(xc−

τ

2) +βIV )

)].

(24)

As analyzed in Section II, for each Halbach PM Array i,the generated force is controlled by the two input currents Ii,1and Ii,2. Therefore, the total force generated by the HalbachPM Array i is the summation of total force generated by twophases of square coils[

Fi−xFi−z

]= mn

[Kx(xi,zi) Kx(xi + 3τ,zi)Kz(xi,zi) Kz(xi + 3τ,zi)

] [Ii,1Ii,2

], (25)

where i denotes each Halbach PM array, i = I, II, III, and IV,and Kx and Kz are defined in (24), mn is number of coilswith current Ii, j, j = 1 or 2 for each Halbach PM array, wherem and n are the parameters of Halbach PM array defined inSection II.

Based on the obtained horizontal and vertical force in eachHalbach PM array, the combined force and torque generated

0 10 20 30 40 50 60 70 80

−1

−0.5

0

0.5

1

x0 (mm)

Force(N

)

FSum−z

FSquare I−z

FCorner I−z

FCorner IV−z

Fig. 7. Force generated by Square I, Corner I, Corner IV, and theirsummation.

on the translator of the maglev positioner for 6 DOF motionare calculated as,

FxFyFzTxTyTz

=

0 0 1 0 0 0 1 01 0 0 0 1 0 0 00 1 0 1 0 1 0 10 0 0 −La 0 0 0 La0 −La 0 0 0 La 0 0La 0 −La 0 −La 0 La 0

FI−xFI−zFII−xFII−zFIII−xFIII−zFIV−xFIV−z

,

(26)

where La denotes the arm of force as defined in Fig. 1. Conse-quently, the 6 DOF motion can be controlled as 6 channels ofsingle-input-single-output (SISO) systems for simplicity, andthe control signal vector [Fx Fy Fz Tx Ty Tz] can be further al-located to four forcers through the inverse relationship of (26),so that the eight-phase current will be injected into the coilarray to conduct 6 DOF motion. It is also noted that, althoughthe proposed maglev positioning system is with 6 DOF motionability, since the Lorentz force is modeled under ideal situationwith zero rotational angles of θx, θy, and θz in the above forcemodeling, the model error will increase when the rotationalangles increase, and as a result, this maglev design is not withthe ability of full rotation.

IV. Simulations and Analysis

In this section, several issues are analyzed with somepractical considerations, i.e. force effectiveness of the cornerregions and the force variations during the operating of maglevpositioner.

A. Force Effectiveness at the Corner Regions

For the square coil utilized in this design, the corner regionsaccount π/(4 + π) ≈ 0.43 area of the total coil. In view ofthis, the effectiveness of the force generated by the cornerregion should be investigated. From the analytical force modelderived in Section III, it is observed that for a corner regionand a square region in the magnetic field of Halbach PM array,e.g., Corner I and Square I in Fig. 6(a), the ratio εf betweentheir peak force magnitudes is given as

εf =max(FCorner)max(FSquare)

=

√2απ

= 51.5%. (27)




−20 0 20 40 60 80

0

0.2

0.4

0.6

0.8

Position (mm)

Bx(T

)

A B C DCoils

Flat Portion

P 2

P 1

Halbach PM Array

FEA Model Shifted Combined

Fig. 8. Analysis of magnetic field end effect along y-direction for HalbachPM array.

0 5 10 15 20 25 30

5

10

15

Position (mm)

Fz(N

)

n = 2: FEA

n = 2: Model

n = 3: FEA

n = 3: Model

n = 4: FEA

n = 4: Model

Fig. 9. Analysis of the force variations in y-direction translation when n = 2,3, and 4 in lm = 3nτ.

The power efficiency εp is defined as the ratio between theirforce per area

εp =max(FCorner/S Corner)max(FSquare/S Square)

=4√

2απ2 = 65.6%, (28)

where S Corner and S Square represent the area of the cornerregion and square region, respectively. Physically, S Corner andS 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 FSquareand FCorner, i.e., 2.07◦ in Corner I and −2.07◦ in Corner IV.From Fig. 7, it is also noted that if replacing the combinationof Square I, Corner I, and Corner II in Fig. 6(a) with arectangular region Rec I, the force ratio between them isabout 2/3.

B. Analysis of End Effect for Halbach PM Array

In Section II, an assumption is made for deriving the forcemodel that the magnetic field of Halbach PM array will remainconstant inside the PM array and fall abruptly to zero outsidethe PM array. In other words, the y-direction end effect of theHalbach PM array is neglected in deriving the force model.Since the adopted magnetic field model of Halbach PM arrayis in 2D, its y-direction variation of magnetic field is notincluded, as a result, FEA software (CST Studio) is utilized toinvestigate the end effect. Fig. 8 plots both the simulated andassumed Bx along the y-direction in a case same as Fig. 2,where the air gap is 1 mm, and other detail specifications ofthe simulation are listed in Table. II. It is noted that the end

effect exists evidently in the edges of Halbach PM array, andthis will become more obvious as the air gap increases.

To analyze the validity of the assumption due to magneticfield end effect, consider a point P1 in Coil C as indicatedin Fig. 8, by assumption the magnitude of Bx(P1) should beas large as the flat portion of the FEA result, i.e., Ba

x(P1) =

0.701 T, but in reality, Bx(P1) decreases as it is near theedge of Halbach PM array and Br

x(P1) = 0.632 T, where Bax

and Brx denote the Bx in assumption and in reality (FEA),

respectively. However, the difference of Brx(P1) and Ba

x(P1)can be compensated by anther point P2 in Coil A, where theP1 and P2 are separated by 60 mm, so that they are two pointsof two coils with exactly same locations. In assumption, P2is outside the magnet array so that Ba

x(P2) = 0 T but in reality,Br

x(P2) = 0.066 T as the magnetic field do not fall abruptlyto zero outside the magnet array due to the effect effect.Furthermore, it is noted that Br

x(P1) + Brx(P2) ≈ Ba

x(P1).Although P1 is a point chosen as a specific example for

illustration, similar trend is valid for all the edge area. Byshifting the FEA field among [−10 0] mm in the Fig. 8 to[50 60] mm, where they are separated by 60 mm as the caseof P1 and P2, it is observed that the combined field of theshifted field from [−10 0] mm and the original FEA fieldamong [50 60] mm is almost exactly same with the magnitudeof flat portion, which indicates that the end effect acted on CoilC can be compensated internally by Coil A (similar for CoilB and D), so that if the coils near the edges of Halbach PMarray, e.g., Coil A, B, C, and D in Fig. 8 are all energized,the assumption made for 2D field will be always valid, whichensures the accuracy of the derived force model.

FEA simulation is also utilized to investigate the forcevariation in several cases that the coils are moving in y-direction. In the simulation, three different length (lm) arechosen for the Halbach PM array, that n = 2, n = 3, and n = 4 inlm = 3nτ, and the coil group in Fig. 2 translate from the initialsituation (b) in y-direction with a total distance of one coillength lc. Fig. 9 shows the difference between the FEA resultand model are 1.8%, 4.3%, and 3.8% for n = 2, n = 3, andn = 4, respectively. Furthermore, the maximal force variationsin the FEA results are only 1.44%, 0.63%, and 0.53% forn = 2, n = 3, and n = 4, respectively, which indicates that they-direction translation causes small force variation due to endeffect 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 IIDetail Specifications of Square Coil and PM

Length of the square coil core, τ 10mm

Height of the square coil, hc 10mm

Turns number of the square coil, Nt 32∗27

Numbers of PMs in the array 12

Length of PMs in the array, lm 60mm

Magnetization magnitude, M0 1.3T




0 20 40 60 80 100 120 140 160−4

−2

0

2

4(a)

Fx(N

)

0 20 40 60 80 100 120 140 160−4

−2

0

2

4(b)

Position (mm)

Fz(N

)

Model Measured FEA

Fig. 10. Comparison of the recorded force along (a) x and (b) z-directionsgenerated by one square coil with constant 1mm air gap.

0 2 4 6 8 10

1

2

3

4

Gap (mm)

Force(N

)

Model Measured FEA

Fig. 11. Comparison of the recorded force generated by one square coil atdifferent z.

0 20 40 60 80 100 120 140 150

−5

0

5(a)

Fz(N

)

0 20 40 60 80 100 120 140 150

−5

0

5

Position (mm)

Fx(N

)

(b)

Model Measured FEA

Fig. 12. Comparison of the recorded force along (a) x and (b) z-directionsgenerated by five square coils with constant 1mm air gap.

square coils are fabricated. The detail specifications of theHalbach PM array and square coils are listed as in Table. II.The experimental setup is conducted that the translator ofHalbach PM array is mounted on a NSK 3-axes AC-servogantry system, and a 6-axes Force/Torque sensor (Model: ATI,Mini40) is used to measure the generated force and data isrecorded by a LabVIEW program via a National Instrumentsdata acquisition card (model: PCI-6035E).

For a single square coil, a constant 0.3 Amp current is usedto energize one coil in the stator. Fig. 10 plots the comparisonof the recorded force generated by one square coil withconstant 1 mm air gap. These results show that the proposedmodel is accurate except the place where the square coil isnear the edge of the Halbach PM array. Without accountingthe edge place, the maximal errors of x- and z-directions forceare 7.5% and 8.4%, respectively, and the average errors are2.4% and 3.3%, respectively. Experiment is also conductedto evaluate the accuracy of proposed model with regards to

Fig. 13. Prototype of the proposed maglev positioner, (a) stator of squarecoil array and (b) whole maglev positioner.

0 2 4 6 8 10

0200400600800

1000x(nm)

(a)

Real Ref

0 2 4 6 8 10

0200400600800

1000

y(nm)

Time (s)

(b)

Fig. 14. Experimental evaluation of the positioning resolution using a seriesof 200 nm steps in both (a) x and (b) y-axes.

different air gaps. Fig. 11 plots the comparison of the recordedforce generated by one square coil at different z. This indicatesthat the relationship between the generated force and air gapz follows an exponential form well.

Another experiment is also conducted where five squarecoils of same phase are energized concurrently with 0.3 Ampcurrent. Fig. 12 plots the comparison between the prediction ofthe proposed model, experimental, and FEA results at differentx with 1 mm air gap. In this case, the end effect, which isobserved in the single square coil case, is eliminated, andthe maximal errors of x- and z-directions force are 9.4% and13.6%, respectively. The average errors are 4.2% and 3.9%,respectively. The potential reason for the force error may bethat the Halbach PM array and square coil array used in theexperiment are manually assembled, so that the unavoidableassembly misalignment will cause certain error, especiallythe uneven bottom of Halbach PM array and the unexpectedrotation angle of square coils.

To demonstrate the 6 DOF motion ability of the proposedmaglev 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),




−5 0 5

−5

0

5

x (mm)

y(m

m)

(a) r=5mm

Reff=0.1Hz

f=1Hz

f=3Hz

−1 0 1

−1

0

1

x (µm)

y(µ

m)

(b) r=1µm f=1Hz

RefReal

Fig. 15. Experimental evaluation of the planar motion using circular refer-ences: (a) 5 mm radius with f =0.1 Hz, 1 Hz, and 3 Hz; and (b) 1µm radiuswith f =1 Hz.

0 10 20 30 40 50

0

30

y(m

m)

(a) v=1mm/s

−20

−10

0

10

20

Error

(µm)

Ref Real x-Error y-Error

0 1 2 3 4 5

0

30

y(m

m) (b) v=10mm/s

−200

−100

0

100

200

Error

(µm)

0 0.1 0.2 0.3 0.4 0.5

0

30

y(m

m) (c) v=100mm/s

Time (s)

−2

−1

0

1

2

Error

(mm)

Fig. 16. Experimental evaluation of 30 mm y-axis linear motion using threedifferent velocities: (a) 1 mm/s, (b) 10 mm/s, and (c) 100 mm/s.

0 2.5 5 7.5 10 12.5 15

500

3000

z(µ

m)

(a)

−80

−40

0

40

80

Error

(µm)

Ref Real Error

0 2 4 6 8 10

998

1000

1002

1004

1006

z(µ

m)

Time (s)

(b)

Fig. 17. Experimental performance of the vertical motion in z-axis: (a)500µm motion with the recorded error signal and (b) 1µm motion.

which ensures the maglev system can perform the motion overa full period of one square coil to evaluate the feasibility. Thevertical position of the moving translator is captured by threechannels of the Lion Precision CPL290 capacitive sensorswith a maximal measurement range of 5 mm, as indicatedin Fig. 13(a). Three channels of Renishaw fibre optic laserinterferometer with a count resolution of 39.6 nm are usedto sense the horizontal positions, so that the 6 DOF position[x y z θx θy θz]T can be obtained by sensor transformations.Eight Trust TA115 linear current amplifiers are utilized to

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

−3

0

3

θx(m

rad)

(a)

−0.1

0

0.1

Error

(mrad)

Ref Real Error

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

−3

0

3

θy(m

rad)

(b)−0.1

0

0.1

Error

(mrad)

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

−5

0

5

θz(m

rad)

Time (s)

(c)

−0.2

0

0.2

Error

(mrad)

Fig. 18. Experimental performance of the rotational motion with therecorded error signals: (a) 6 mrad θx motion, (b) 6 mrad θy motion, and(c) 10 mrad θz motion.

actuate the eight-phase coil array, and a National Instruments(NI) PXI-8110 real-time controller is employed as the controlhardware for the maglev positioning system to achieve asampling rate of 5 kHz.

The x and y-axes positioning resolutions are evaluatedin Fig. 14(a) and (b), respectively, where the translator islevitated up by 1 mm constantly. The references containinga series of 200 nm steps are used for tracking, and Fig. 14indicates that the maximal errors for both axes are kept below200nm (5 counts of the sensor resolution), and the Root MeanSquare Error (RMSE) for both x and y-axes are 54.5 nm and49.4 nm, respectively. The combined xy-axes planar motionare evaluated using circular references. Fig. 15(a) shows thecase of 5 mm radius circular motion with three frequencies,i.e., f = 0.1 Hz, 1 Hz, and 3 Hz, and their errors of realradius are 17.5 µm, 135.6 µm, and 230.2 µm, respectively.Fig. 15(b) shows the case of 1 µm radius with a 1 Hz frequencymotion, and the directional RMSE of Fig. 15(b) is around74 nm. The experiment is also conducted to evaluate theachievable velocity of the maglev prototype. Fig. 16 showsthe motion performance using three velocities during 30 mmmotion stroke in y-axis, that (a) 1 mm/s, (b) 10 mm/s, and (c)100 mm/s, where their maximal errors are 18.5 µm, 131.4 µm,and 1109 µm, respectively. It is noted that the x-axis errorsduring y-axis motion are also recorded and plotted in Fig. 16,their RMSEs are 0.14 µm, 0.86 µm, and 8.6 µm, respectivelyfor Fig. 16(a), (b), and (c), which demonstrates the strongdecoupling ability of the implemented maglev prototype.

Fig. 17 shows the experimental performance of the verticalmotion in z-axis, where Fig. 17(a) shows the levitation heightis changed from 500 µm to 3000 µm in 2.5 seconds, and themaximal tracking error is around 50 µm. Fig. 17(b) demon-strates the vertical positioning resolution using a series of 1 µmsteps, and its RMSE is calculated as 0.58 µm. The rotationalmotions along x, y, and z-axes are also evaluated for theimplemented maglev prototype. Since the laser interferometeris sensitive to the rotational angle, another three channels ofthe Lion Precision CPL290 capacitive sensors are used to sense




the horizontal position in order to demonstrate the rotationalmotions in relatively large stroke. Fig. 18 plots the rotationalmotions 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 θx motion from -3 mrad to 3 mrad,and it indicates that its maximal tracking error is less than0.083 mrad while its RMSE is calculated as 0.032 mrad. Theθy motion from 3 mrad to -3 mrad is plotted in Fig. 18(b), andits maximal tracking error is less than 0.082 mrad while itsRMSE is calculated as 0.031 mrad. Finally, Fig. 18(c) showsthe θz motion from -5 mrad to 5 mrad, and its maximal trackingerror is less than 0.182 mrad while its RMSE is calculated as0.056 mrad.

VI. Conclusion

This paper presents a novel design of 6 DOF maglevpositioner by using 1D Halbach PM arrays and square coils.The proposed design is potential to deliver unlimited-strokeplanar motion with high power efficiency and multi-translatorsare 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 lawand the phase number of coils. An analytical force modelingapproach is proposed in this paper to accurately predict thegenerated Lorentz force between the Halbach PM array andsquare coil. This includes a general method to model theLorentz force generated by the corner region of various kindsof coils. Finally, the proposed force model for square coilwith Halbach PM array are evaluated experimentally, and aprototype of the proposed maglev positioner is fabricated todemonstrate its 6 DOF motion ability in large stroke, whichdemonstrates that its RMSE is around 50 nm in planar motion,and the velocity can achieve up to 100 mm/s.

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Haiyue Zhu (S’13) received the B.Eng. degreein automation from the School of Electrical Engi-neering and Automation and the B. Mgt. degreein business administration from the College ofManagement and Economics, Tianjin University,Tianjin, China, in 2010, and the M.Sc. degree inelectrical engineering from the National Universityof Singapore (NUS), Singapore, in 2013, where heis currently pursuing the Ph.D. degree with the De-partment of Electrical and Computer Engineering.

He joined the Singapore Institute of Manufac-turing Technology (SIMTech)–NUS Joint Laboratory on Precision MotionSystems in 2013, and is an Attached Research Student with the Agencyfor Science, Technology, and Research (A*STAR), SIMTech. His currentresearch interests include integrated design and control of ultraprecisionmechatronics and magnetic levitation technology.

Tat Joo Teo (M’08) received the B. Eng. degreein mechatronics engineering from Queensland Uni-versity of Technology, Australia, in 2003 and thePh.D. degree from Nanyang Technological Univer-sity, Singapore, in 2009.

He is with Singapore Institute of ManufacturingTechnology as a researcher since 2010 and is cur-rently a visiting scientist in Massachusetts Instituteof Technology. His research interest is to explorethe fundamentals of Newtonian mechanics, solidmechanics, kinematics, and electromagnetism to

develop high precision mechatronics or robotic systems for micro-/nano-scale manipulation and bio-medical applications.

Dr. Teo has published over 40 peer-reviewed articles and has 4 patentsgranted. In 2013, he received the IECON Best Paper Award in the theoryand servo design category. In 2014, he became the first Singaporean towin the R&D 100 Award, which is the most prestigious international awardfor technologically-significant products. He currently serves as an AssociateEditor for Nanoscience and Nanotechnology Letters.

Chee Khiang Pang (S’04–M’07–SM’11) receivedthe B.Eng. (Hons.), M.Eng., and Ph.D. degreesfrom the National University of Singapore (NUS),Singapore, in 2001, 2003, and 2007, respectively,all in electrical and computer engineering.

He was a Visiting Fellow in the School of In-formation Technology and Electrical Engineering(ITEE), University of Queensland (UQ), St. Lucia,QLD, Australia, in 2003. From 2006 to 2008, hewas a Researcher (Tenure) with Central ResearchLaboratory, Hitachi Ltd., Kokubunji, Tokyo, Japan.

In 2007, he was a Visiting Academic in the School of ITEE, UQ, St. Lucia,QLD, Australia. From 2008 to 2009, he was a Visiting Research Professorin the Automation & Robotics Research Institute (ARRI), University of Texasat Arlington (UTA), Fort Worth, TX, USA. Currently, he is an Assistant Pro-fessor in Department of Electrical and Computer Engineering (ECE), NUS,Singapore. He is also an A*STAR Singapore Institute of Manufacturing Tech-nology (SIMTech) Associate, Faculty Associate of A*STAR Data StorageInstitute (DSI). He is an author/editor of 3 research monographs includingIntelligent Diagnosis and Prognosis of Industrial Networked Systems (CRCPress, 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 performancemechatronic systems, with specific focus on advanced motion control fornanopositioning systems, precognitive maintenance using intelligent analyt-ics, and energy-efficient task scheduling considering uncertainties.

Dr. Pang is a member of American Society of Mechanical Engineers. Hewas the recipient of The Best Application Paper Award in The 8th AsianControl Conference (ASCC 2011), Kaohsiung, Taiwan, 2011, and the BestPaper Award in the IASTED International Conference on Engineering andApplied Science (EAS 2012), Colombo, Sri Lanka, 2012. He serves as anAssociate 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 Journalof Automation and Logistics and International Journal of ComputationalIntelligence Research and Applications, and on the Conference EditorialBoard for IEEE Control Systems Society (CSS). In recent years, he alsoserved as a Guest Editor for International Journal of Automation and Lo-gistics, Asian Journal of Control, International Journal of Systems Science,Journal of Control Theory and Applications, and Transactions of the Instituteof Measurement and Control.

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