Actively Controlled Hexapole Electromagnetic ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2015.2503274, IEEE/ASME Transactions on Mechatronics

Actively Controlled Hexapole Electromagnetic Actuating System Enabling 3-D Force Manipulation in Aqueous Solutions Fei Long, Student Member, IEEE, Daisuke Matsuura, and Chia-Hsiang Menq, Senior Member, IEEE

Abstract This paper presents the design, modeling and calibration of an over-actuated and actively controlled hexapole electromagnetic actuating system, which is implemented to stabilize and propel a microscopic magnetic particle in aqueous solutions. Compared to the 2-D quadruple magnetic tweezers in [1] and the 3-D magnetic actuator in [2], the new electromagnetic actuating system achieves significantly greater force generation capability and much improved controllability of 3-D force in the workspace. These improvements are accomplished through enhanced design and synthesis of the six electromagnetic poles and improved current allocation of the over-actuated system. A lumped parameter hexapole force model of the system is derived and validated through experimental calibration. It is then used to devise inverse modeling, which is implemented to realize real-time current allocation to enable the most effective manipulation of the 3-D magnetic force exerting on the particle at the center of the workspace. The system is designed for use with live cell experiments. Fei Long is with Mechanical and Aerospace Engineering Department of the Ohio State University, Columbus, OH, 43210 USA. (E-mail: [email protected]; address: W387, 201 W.19th Ave., Columbus, OH, 43210; Tel: 01614-772-5743). Chia-Hsiang Menq is with Mechanical and Aerospace Engineering Department of the Ohio State University, Columbus, OH, 43210 USA. (E-mail: [email protected]; address: W394, 201 W. 19th Ave., Columbus, OH, 43210; Tel: 01-614-292-4232). Daisuke Matsuura is with Department of Mechanical Sciences and Engineering, TAKEDA Lab, Tokyo Institute of Technology, Tokyo 152-8552, JAPAN (e-mail: [email protected]; address: 2-12-1-I6-16 Ookayama, Meguro-ku, Tokyo 152-8552, JAPAN; Tel&Fax: 03-5734-2177). He was a postdoctoral researcher in Mechanical and Aerospace Engineering Department of the Ohio State University from 2008 to 2011.

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Actively Controlled Hexapole Electromagnetic Actuating System Enabling 3-D Force Manipulation in Aqueous Solutions Fei Long, Student Member, IEEE, Daisuke Matsuura, and Chia-Hsiang Menq, Senior Member, IEEE Abstract – This paper presents the design, modeling and calibration of an over-actuated and actively controlled hexapole electromagnetic actuating system, which is implemented to stabilize and propel a microscopic magnetic particle in aqueous solutions. Compared to the 2D quadruple magnetic tweezers in [1] and the 3-D magnetic actuator in [2], the new electromagnetic actuating system achieves significantly greater force generation capability and much improved controllability of 3-D force in the workspace. These improvements are accomplished through enhanced design and synthesis of the six electromagnetic poles and improved current allocation of the over-actuated system. A lumped parameter hexapole force model of the system is derived and validated through experimental calibration. It is then used to devise inverse modeling, which is implemented to realize real-time current allocation to enable the most effective manipulation of the 3-D magnetic force exerting on the particle at the center of the workspace. The system is designed for use with live cell experiments.

of mechanical connections employed [7]. It has been applied to study biomolecules from isolated model systems [8, 9] rather than from their native and fully active environment. Optical tweezers is another modern technique that is useful for studying biological systems under physiological conditions [10-16]. It is well suited for quasi-static force measurement [17] and dynamic force sensing [18]. Although it is inherently stable, heating is a main issue that needs to be resolved [19], especially when large trapping force is desired. One feasible approach is to implement real-time Brownian motion control to reduce heating [20, 21]. Another problem is that unwanted trapping of debris due to the lack of specificity in optical trapping limited the application in probing biological samples [15]. Magnetic tweezers use magnets and/or electromagnets to propel microscopic magnetic particles, which serve as measurement probes and exert force on biological samples. They have several advantages over optical tweezers. Magnetic field is intangible and safe to most biological materials. It is specific to magnetic particles and there is no heat generated in the process. It is, however, necessary to employ feedback control to achieve stable magnetic trapping since the magnetic force field is inherently unstable [22]. Therefore, without active control, the magnetic particles/probes are often functionalized and anchored to target bio samples to avoid instability. Many magnetic actuators are, thus, actually simple force appliers, wherein the force is adjusted according to a precalibrated force function [36]. They were used in a wide range of applications, ranging from manipulating biological macromolecules [23-27], probing cell membranes [28-30], to characterizing intracellular properties [31-34]. Among various magnetic tweezers, some could only apply forces in a single direction, using only one coil-actuated pole [35,36] or two poles facing each other [23,32], some were 2-D systems with multiple poles [33,37], and a few were able to generate forces in 3-D space [27]. These magnetic tweezers were employed as simple force appliers, without motion control, wherein magnetic forces were applied and the induced motions of magnetic particles were recorded by appropriate measurement systems. Accurate force modeling and inverse modeling of electromagnetic actuation are essential for effective force manipulation and for enabling stable magnetic trapping via feedback control. The development of a 2-D quadrupole magnetic tweezers was reported in [1], wherein four tip-shaped electromagnetic poles were employed to control 2-D magnetic force to propel a microscopic magnetic particle, serving as a

Index Terms- Electromagnetic actuation, over-actuated system, current allocation, magnetic particle stabilization, magnetic force generation. I. INTRODUCTION Probing biological samples and manipulating biological processes have become important techniques in the study of cell mechanics and mechanobiology [3]. Especially, a dream of using modern instruments to probe biomolecules in live cells has been shared by many researchers. Technologies such as Atomic force microscopy (AFM) [4], optical tweezers and magnetic tweezers are most widely used for biology applications [5]. AFM offers high spatial resolution and enables force probing and scanning [6]. AFM, however, remains as a 2-D surface tool due to its kinematic constraints and the restriction This paper is submitted on 3/31/2015 for review. This work was supported in part by CMMI of NSF under Grant 1067962 and 1200017. Chia-Hsiang Menq and Fei Long are with Mechanical and Aerospace Engineering Department of the Ohio State University, Columbus, OH, 43210 USA (e-mail: [email protected] (Chia-Hsiang Menq) and [email protected] (Fei Long)). Daisuke Matsuura is with Department of Mechanical Sciences and Engineering, TAKEDA Lab, in Tokyo Institute of Technology, Tokyo 1528552, JAPAN (e-mail: [email protected]). He was a postdoctoral researcher in Mechanical and Aerospace Engineering Department of the Ohio State University from 2008 to 2011.



force-sensing probe. It was applied to characterize the mechanical property of live cells using a functionalized probe [41]. A lumped-parameter analytical magnetic force model was derived to characterize the nonlinearity of the magnetic force with respect to the input currents and its position dependency in the workspace. Its extension to the realization of a 3-D hexapole magnetic actuator was detailed in [2]. Employing a visual particle tracking system [43], a visual servo control system was developed to enable precise motion control of the magnetically propelled particle in the 3-D workspace [40], wherein a simple inverse model was derived and used to implement a nonlinear feedback control law to realize stable magnetic trapping. These two multi-pole electromagnetic actuating systems had electromagnetic poles made of thin (approximately 100μm thick) high-permeability nickel-iron magnetic alloy (permalloy) film. Whereas the design of these systems had advantages, such as easy in fabrication and assembly, it had several limitations. First, it was necessary to use specially made sample chambers, therefore, standard culture dishes commonly used for live cell experiments could not be employed. Second, small cross-sectional area of the thin film resulted in large magnetic reluctance, and thus yielded small magnetic flux and saturation flux density. Other than hardware design, inverse modeling is also important for multi-pole electromagnetic actuating systems, which are usually over-actuated systems. The issue raised by redundancy was solved through applying constant constraint in [40, 42]. But the use of constant constraints resulted in excessive actuation effort, severely limiting the force generation capability. A micro-robot system [39] implemented a suboptimal inverse model from pseudo inverse, which cannot be applied in our system due to the nonlinear nature of the force model. A multi-pole magnetic actuator [44, 45] could steer a ferrofluid drop by optimal control effort. But the sampling rate was limited to 15Hz due to the cumbersome calculation. Moreover, 3-D realization has not been achieved. In this paper, we present the design, modeling, and calibration of an over-actuated and actively controlled hexapole electromagnetic actuating system, which is implemented to stabilize and propel a microscopic magnetic particle in aqueous solutions. The system is designed for use with live cell experiments. The design and synthesis of the six electromagnetic poles, including geometric design and material selection, result in higher magnetic saturation limit, improving the force generation capability of the actuating system. An optimal inverse model is derived and implemented to realize real-time current allocation to enable the most effective manipulation of the 3-D magnetic force exerting on the particle at the center of the workspace. Compared to the 2-D quadrupole magnetic tweezers in [1] and the 3-D magnetic actuator in [2], the hexapole electromagnetic actuating system achieves significantly greater force generation capability and much improved controllability of 3-D force in the workspace. The content of the paper is organized as follows. Section II presents the design and synthesis of hexapole electromagnetic

actuation, and analysis of the resulting magnetic field. Section III introduces a hexapole magnetic force model and presents the development of inverse modeling for optimal current allocation at the center of the workspace. Active control, including feedback linearization, stabilization, and experimental results, is presented in Section IV. Calibration and validation of the force model are described in Section V. Section VI presents the force generation capability. Finally, conclusion is made in Section VII. II. DESIGN OF HEXAPOLE ELECTROMAGNETIC ACTUATION A. Design, synthesis, and fabrication Since a single electromagnetic pole can only generate attractive force exerting on the magnetic particle, three pairs of electromagnetic poles are employed and their pole tips are placed symmetrically on three orthogonal axes to enable generation of 3-D forces. In order to significantly increase the force generation capability, these sharp-tipped poles are made of cone-shaped 1018 steel rods, and are assembled to concentrate the magnetic flux into the workspace. Each electromagnetic pole is actuated through an individual coil. All the coils and poles are then magnetically connected through a magnetic yoke, which helps increase the magnetic flux density and the flux gradient in the workspace. The six pole tips enclose the workspace, wherein the specimen and the magnetic particle are placed. The hexapole electromagnetic actuator is assembled with two overlaid motion stages to form a unique experimental apparatus. The apparatus is integrated with an inverted microscope equipped with a visual particle tracking system [43]. The visual sensing can achieve sub-nm resolution in x and y axis and the resolution in z axis is under 5nm. The tracking error in x, y is under 5nm in large range (60um x 60um) and under 10nm in z axis within 20um range. This error is negligible considering the motion range and the size of the bead (4.5um in diameter). In order to have the optical path free of blockage, a rigid body rotation is applied to the three pairs of electromagnetic poles along with their three orthogonal axes such that their tips are on two parallel horizontal planes, i.e., one upper plane and the other lower plane. Fig. 1(a) shows the design of a manual x-y stage and the lower three electromagnetic poles, assembled on an x-y-z piezo motion stage. The coarse x-y stage achieves an 8mm×8mm working range, and can be used to locate a specimen cell. The lower three poles are fixed under the culture dish, which holds live sample cells so that the bottom cover glass of the culture dish, whose thickness is approximately 100 m , can be placed inbetween upper three and lower three poles; the gap between them is as small as possible to generate strong magnetic force. The upper three magnetic poles are assembled on the yoke ring and their pole tips sunk into medium filled in the dish, as



shown in Fig. 1(b). They can be easily disassembled to replace culture dish to perform live cell experiments in practical use. The arrangement also makes cleaning easy, necessary to avoid cell contamination. The magnetic particle is to be stabilized and steered within the 3-D workspace, whereas the position of the sample dish with respect to the 3-D workspace is controlled by the 3-axis piezo stage.

whole setup is integrated with an inverted microscope (Olympus IX 81) and a dry 60x objective lens is used for visual measurement. A vision-based measurement method with sub-nanometer resolution [43] is employed to provide position feedback, wherein a speed CMOS camera (Mikrotron MC3010) and Image grabbing/processing card (Matrix Odyssey XCL) are used. A 4.5um bead (Dynabead M-450 Epoxy) is used in the experiment. The coils are driven by six linear power amplifiers (Micro Dynamics, BTA-28V-6A) with 10K bandwidth and are set to current mode. A piezo positioner (PI P-721 PIFOC) is used to drive the lens for image calibration. The whole setup is put on a Smart Table (Newport) and a vibration isolation table (Herzan TS-150) to remove vibrations. B. Finite element analysis of the magnetic field

Fig.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and three upper poles.

Finite Element Method (FEM) is employed to analyze quantitatively the magnetic field produced by the actuating system and to visualize its spatial distribution, particularly within the workspace. A CAD model imitating the real setup is built and meshed (Fig. 2(b)) using the ANSYS environment for FEM calculation. When applying current to the coil, the magnetic field produced by the hexapole actuator can be computed using FEM analysis.

Fig.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5) forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for flux generation) and a measurement coil (for flux measurement). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate system. (b) CAD model and meshing of the hexapole actuator

The fabricated prototype is shown in Fig.2 (a). The electromagnetic poles and the magnetic yoke are fabricated with 1018 steel, low-carbon (0.18% carbon) steel with high saturation limit (over 2T). However, one drawback of using such material is the hysteresis effect, which will be addressed in the future through hysteresis modeling or real-time sensing and control. At the present time, superparamagnetism is assumed to characterize force generation capability of the actuating system. The diameter of the poles is about 6mm. Three upper poles are about 45mm long, whereas the lower poles are 42mm long and are milled to form a flat platform to support the culture dish. Whereas sharper tips can be produced using advanced machining processes, the radius of the pole tips of the prototype is 40mm , which is included in the finite element analysis (Fig. 2(b)). The distance from the workspace center to the tips of all six poles is adjustable due to the flexibility achieved in the new design. Its nominal value of 500µm is used in all analyses and experiments reported in this paper. The actuation coils, realized for experiments and winded around the protrusions of the yoke, have 70 turns. The

Fig.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic flux density distribution (unit: Tesla). (b) The magnetic flux density vectors near the workspace center (actuation coordinate). (c) The magnetic field vectors near the tip of pole 1.

Fig.3 shows the ANSYS analysis result by actuating coil 1 with 1A current. It can be seen from Fig.3 (a) (top view of measurement coordinate) that the magnetic field forms a closed loop under the guidance of magnetic poles and the yoke, which makes the magnetic flux density and the flux gradient in the workspace significantly higher. Fig.3 (b) shows the magnetic flux density vectors, associated with the actuation coordinate system, within a 100 m cube centered at the origin of the workspace. Due to the direction of the input current, which is counter clockwise, these vectors all point to the tip of



pole 1. The magnetic field vectors near the tip of the electromagnetic pole 1 are shown in Fig.3 (c). It can be seen that they strongly converge to the tip of the electromagnetic pole. Fig.3 (b) and Fig.3 (c) validate the assumption that sharp tip behaves like a point charge. The magnetic flux density is measured in Tesla (104 Gauss) in Fig. 3. Fig.4 shows the contour plot of the magnitude of the magnetic flux density (measured in Gauss) when applying 1A current to coil 1. A similar analysis for the thin-foil design was given in Fig. 6 of [2]. Specifically, the contour plots of two sections, i.e. the horizontal plane and the vertical plane, are displayed and the gradient of the magnetic field is clearly visualized. The magnetic flux density in the vicinity of the workspace center of this hexapole actuator increases by twofold from that of the actuator with the thin-foil design. Moreover, the saturation limit of the magnetic pole is over 2T, which is much higher than that of permalloy pole with 0.9T saturation limit in the previous design. This improvement allows each 70-turn coil to be actuated up to 3 Amperes, the force generation capability is greatly increased. The detailed force generation calculation will be addressed in section VI.

where B(p) is the magnetic induction vector at the specified position p  [ x, y,z ]T , km  0 4 ,  0 is the permeability of vacuum, qi is the magnetic charge defined by qi  i 0 , wherein  i is the magnetic flux, ri (p) is the norm of the displacement vector ri (p) pointing from the ith magnetic charge to the specified location, and u i (p ) is the unit directional vector, i.e., ui (p)  ri (p) ri (p) . The magnetic flux is determined by the magnetomotive force and the reluctance of the air according to Hopkinson’s law: Φ  [1, 2 , 3 , 4 , 5 , 6 ]T = K I [F1, F2 , F3 , F4 , F5 , F6 ]T a , wherein the magnetomotive force is proportional to the input current, i.e., Fi  N c I i , N c is the turns of the coil, a is the lumped magnetic reluctance from the pole tip to the workspace center in the air, and K I is the 6×6 flux distribution matrix describing the magnetic coupling among 6 poles since they are connected by a magnetic yoke (Fig.2 (b)). The magnetic charges Q =

 q1 q2 q3 q4 q5 q6 

T

can thus be related to the input currents,

I  [ I1 I 2 I3 I 4 I5 I6 ]T , as described by the following equation, Q

Φ

0



Nc

 0 a

KII

(2)

Fig.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the measurement coordinate system. (a) |B| in the horizontal plane (top view). (b) |B| in the vertical plane (side view).

C. Hexapole magnetic field model It can be seen from Fig. 3(c) that the magnetic field produced by a sharp tipped pole looks to the magnetic particle in the workspace of the actuator as though it is generated by a point source [46]. Therefore, a hexapole magnetic field model, consisting of six magnetic poles/charges, is used to characterize the magnetic field within the workspace of the hexapole electromagnetic actuator. The resulting magnetic field produced by the hexapole actuating system can be obtained by summing the field associated with each of the six electromagnetic poles. In other words, the resulting magnetic field in the workspace can be modeled as [2], 6

B(p) 

k i 1

m

qi ui (p) 2 ri (p)

Fig.5. Validation of the hexapole magnetic field model: (a) comparison of magnetic induction vectors, and (b) normalized norms of error vectors. (c) the definition of the fitting error

The spatial distribution of the magnetic field within the workspace of the actuating system and its dependence on the input currents can be determined using the above equations once the reluctance and workspace radius, the distance from the magnetic charge to the workspace center, of the actuating system are known. They are determined by best fitting the magnetic inductance vector based on Eq. (1) with that

(1)



calculated using FEM analysis (Fig. 3(b)). The sum of the squared norm of error vectors is selected as the objective function, which is minimized to determine the two optimal values, i.e., workspace radius  900m , which is the distance variable defining the location of qi , and the air reluctance a =6.3×108A/Wb. The fitted result is shown in Fig.5, wherein vector plots are compared in (a) and normalized norms of error vectors in (b). It can be seen that norms of error vectors are mostly smaller than 1% of the norm of the associated flux density. This validates the hexapole magnetic field model. The , is shown as calibrate flux distribution matrix, denoting as K FEM I follows in Eq. (3). The first entry is normalized to 5/6, which is the same value as theoretical K I from magnetic circuit analysis [1].

K FEM I

 0.8333  -0.4042 -0.1935  -0.2481 -0.2481  -0.1935

-0.2312 -0.1935 -0.1375 -0.1375 -0.1935  1.0388 -0.2481 -0.1012 -0.1012 -0.2481  -0.1375 0.8333 -0.2312 -0.1375 -0.1935   -0.1012 -0.4042 1.0388 -0.1012 -0.2481  -0.1012 -0.2481 -0.1012 1.0388 -0.4042   -0.1375 -0.1935 -0.1375 -0.2312 0.8333 

which has 5/6 as diagonal entries and -1/6 as off-diagonal entries. g I is the force gain determined by the size and magnetic property of the magnetic particle and that of the magnetic circuit. An effective maximum force associated with the maximum input current, I max , is derived,   N c   0 2 Fmax  g I I max  3Vkm2   5   20 (   20 )  0a

(5)

to characterize the force generation capability of a regular hexapole electromagnetic actuating system. It is used to normalized the magnetic force and form a dimensionless expression, F (pˆ , I ) ˆ T T Fˆi (pˆ , Iˆ )  i  I K I Li (pˆ )K I Iˆ , Fmax

(3)

(6)

where Iˆ  I I max is the normalized input current. It is evident that this dimensionless force field characterizes the spatial distribution of the force field produced by the hexapole electromagnetic actuating system.

It is worth mentioning that the theoretical K I matrix is used to derive the force model in the next section while the purpose of this FEM analysis is to verify the multi-charge model. Moreover, the K I matrix will also be experimentally verified after active control being established.

B. Inverse model based on constant constraints Whereas the force model described above establishes the relationship between the input currents and the resulting magnetic force exerting on the magnetic particle, inverse modeling is necessary for the practical use of the implemented electromagnetic actuating system. Since the actuator is an over-actuated system, direct inverse of Eq. (6) is impossible. It was shown in [40] that the redundancy could be removed by imposing three constant constraints. This approach will be briefly summarized and the associated limitations will be discussed. Since the three pairs of electromagnetic poles are placed symmetrically on the three orthogonal axes of the actuation coordinate system, in the following, all analysis refer to this coordinate system. Imposing three constant constraints, i.e.,

III. FORCE MODEL AND INVERSE MODELING A. Hexapole magnetic force model Without being magnetized to saturation, the magnetic force exerted on a superparamagnetic microscopic particle placed in the field is F  (1 2)(m  B) , where F is the gradient force, m  (3V 0 )  [(   0 ) (   2 0 )]B is the effective magnetization of the magnetic particle,  is the permeability of the particle, and V is the volume of the particle. An analysis beyond the linear magnetization of the particle can be found in [38]. By substituting Eq. (1) and Eq. (2) into the gradient force, each component of the gradient force can be cast into a quadratic form,

Fi (pˆ , I)  g I IT KTI Li (pˆ )K I I, i  x, y, z ,

2

 2  I max , 

Iˆ1  Iˆ2  cx , Iˆ3  Iˆ4  cy , and Iˆ5  Iˆ6  cz , and denoting three

effective

input

currents,

i.e.,

 Iˆ

= [ Iˆx ,  Iˆy ,  Iˆz ]T =

[ Iˆ1  Iˆ2 , Iˆ3  Iˆ4 , Iˆ5  Iˆ6 ]T , an exact linear relationship between the

effective input currents and the dimensionless force at the

(4)

center, i.e., Fˆ c  Fˆ (pˆ  0, Iˆ ) , is derived,

ˆ is the position vector normalized with respect to where p workspace’s radius , which is the effective working radius, Li (pˆ ) is the matrix characterizing the position-dependence of

Fˆ c  2 A Iˆ ,

(7)

where A is a constant actuation matrix,

the magnetic gradient. The 6×6 flux distribution matrix K I is adopted from the same magnetic circuit analysis reported in [2],



A  diag[(2cx  cy  cz ),(cx  2cy  cz ),(cx  cy  2cz )]

input current vector to generate the desired force will reduce the heat generation from the coils. On the other hand, it will enhance the force generation capability of the actuating system. The squared norm of the input current vector is related to the effective input current vector and three constraints as,

(8)

The inverse model at the center can then be derived from Eq. (7) and Eq. (8), 1 1 [ Iˆ1 , Iˆ3 , Iˆ5 ]  b  A 1Fˆ c , 2 4

(9) ||Iˆ ( , )||2 

and [ Iˆ2 , Iˆ4 , Iˆ6 ] 

1 1 b  A 1Fˆ c , 2 4





1 || Iˆ ( , )||2  cx2  c 2y  c z2 , 2

(12)

(10) 1 2

where  Iˆ ( , )  A 1rˆ ( , ) according to Eq. (7). The objective

where b  [c x ,c y ,c z ]T is a constant bias introduced by the three

function can then be cast as

constant constraints. It is worth noting that whereas the use of three constant constraints leads to exact inverse relationship at the center, the constant bias also results in unnecessarily large input currents as evident in Eq. (9) and Eq. (10). It can also be seen from Eq. (7) that due to the imposed constant constraints the magnetic force increases linearly, instead of increasing quadratically as in Eq. (6), with the input currents, whereby the force generation capability is severely degraded.

1 1 J (cx , cx , cx ; , )  {cx2  c y2  cz2 }  ||A1rˆ ( ,  )||2 . 2 8

(13)

Minimizing this objective function yields the three optimal constraints, cx ( , ) , cy (, ) and cz ( , ) , which are orientation dependent. They are compared with constant constraints used in [2, 40] and displayed in Fig. 7.

C. Optimal inverse model at the center of the workspace It can be seen from Eq. (6) that the hexapole actuating system is capable of generating 3-D force in arbitrary direction at any spatial position in the 3-D workspace, and the resulting force is scalable, i.e., ||Fˆ (pˆ , Iˆ )||  ||Iˆ||2 . When the desired force is expressed in magnitude and orientation in the spherical coordinate system (Fig. 6), i.e., Fˆ d  ||Fˆ d ||rˆ ( , ) , where

Fig.7. Three orientation-dependent optimal constraints compared with their corresponding constant constraints.

rˆ  [cos  cos ,cos  sin  ,sin  ]T is the unit vector in the radial

direction, the optimal inverse solution can, therefore, be cast as, ˆ , pˆ )  ||F ˆ ||1/ 2 I  rˆ ( , ), pˆ  . (11) Iˆ opt (F d d opt

Let Fˆ c  rˆ ( , ) and substitute cx ( , ) , cy (, ) and cz ( , ) into b and A in Eq. (9) and Eq. (10), the optimal current allocation, Iˆ uoptint ( , ) , associated with the unit force can be

(13)

determined. The result is compared with that obtained using constant constraints and displayed in Fig. 8. It can be seen that each of the six input currents resulted from optimal current allocation is significantly smaller than that associated with constant constraints.

Fig.6. Illustration of the desired force in the spherical coordinate system.

At the center, the optimal current Iˆ uoptint ( , ) associated with the unit force, i.e., Fˆ d  rˆ ( , ) , can be obtained by finding three optimal constraints, namely cx ( , ) , cy (, ) , and cz ( , ) , and minimizing the squared norm of the input current vector, i.e., ||Iˆ ( , )||2 . This solution will improve current allocation of the over-actuated system. On one hand, minimizing the norm of

Fig.8. Optimal current allocation compared with that obtained using constant constraints.



IV. STABILIZATION

optimal current allocation, the Brownian motion standard deviation reduced to (52.36nm, 64.51nm, 28.62nm) compared with (107.32nm, 69.95nm, 28.62nm) while using constant constraints. As shown in the block diagram of the feedback control system (Fig. 9), the positioning fluctuation of the particle attributed to at least two disturbances, i.e., the random thermal force FT and the force error F due to imperfect inverse modeling. Whereas the two methods are based on the same force model and both yield theoretically exact inverse solutions, the force error F is likely different as the realization of the two methods operate with different actuation currents (Fig. 10). The results shown in Fig.11 imply that the modeling error likely becomes greater when increasing the actuation currents.

A feedback control system was implemented and used to stabilize the magnetic particle placed in the workspace of the hexapole electromagnetic actuating system. A block diagram of the control system is shown in Fig. 9, wherein p d is the desired position, p m the measured position, Fd the control effort determined by the feedback controller, ∆𝐅 the force error due to imperfect inverse modeling, FMT the resulting magnetic force, and FT the thermal force. The position of the particle was measured using a 3-D vision-based particle tracking system [43], wherein a CMOS camera was employed to acquire the image of the particle at 200 frames per second. The image grabbing board is used to process the visual measurement algorithm and 200 fps is close to the limit of calculation capability. Current allocation derived from inverse modeling was implemented to overcome the issue raised by over-actuation and to achieve feedback linearization, part (1) in Fig.9. Together with the constant-gain feedback controller, part (2) in Fig.9, it stabilized the magnetic particle and suppressed the disturbances introduced by the random thermal force.

Fig.10. Six input currents of the hexapole actuator

Fig.9. Block diagram of the feedback control system achieving stabilization

The bead dynamics can be described by the Langevin equation [47], ,

(14)

where m is the mass of the particle,  the drag coefficient of the particle in aqueous solution. As the inertia term is very small when compared to the damping force , the bead dynamics is often reduced to a 1st order system. Both methods for current allocation, i.e. the one based on optimal inverse modeling and the other using constant constraints, were implemented. The feedback controller was a constant-gain PI controller. Experiments were conducted to stabilize the particle in water and the results were shown in Fig. 10 and Fig. 11 to compare the performance of the two methods. It can be clearly seen from Fig. 10 that when using optimal current allocation each of the six input currents absolute value is significantly smaller than its counterpart. This result validates the theoretical analysis developed for optimal inverse modeling. The Brownian motion displayed in Fig. 11 also shows improved performance of stabilization when using

Fig.11. Stabilization of the magnetic particle at the center of the workspace

V. CALIBRATION AND VALIDATION OF THE FORCE MODEL Whereas the hexapole magnetic force model has a sound theoretical basis, numerical values of two model parameters, i.e., the force gain and the flux distribution matrix, are not exactly known without experimental calibration. Any discrepancies introduce errors in inverse modeling and degrade the effectiveness of current allocation for force generation and control.



This is due to the fact that significant amount of the three lower poles’ material was removed to form a flat platform to support the culture dish. It is worth noting that whereas the calibrated flux distribution matrix in Eq. (15) precisely quantifies the effect of the geometric difference between the upper and lower poles on the flux distribution in the six electromagnetic poles, it does not fully quantify the effect in the workspace. The geometric difference also changes the way that the magnetic field converges to the pole tip and thus the field distribution in the workspace. Another experiment was performed to calibrate the force gain of the actuating system and to improve the result given by Eq. (15). In the experiment, the magnetic particle was steered along a linear trajectory and propelled at a constant speed in viscous liquid. Moreover, the setup is degaussed after each trajectory tracking. When traveling at constant velocity, v p ,

The nominal flux distribution matrix K I was obtained through the magnetic circuit analysis presented in [2]. Due to magnetic leakage, which is not negligible as can be seen from the results in Fig. 3(a), the exact values of K I differ from the nominal values. An experimental setup along with an experimental procedure was devised to calibrate the flux distribution matrix by utilizing the electromagnetic induction, which is widely used to extract information from the magnetic field [48, 49]. In the experiment, each individual actuation coils were excited one by one sequentially and the six induction voltages were measured simultaneously using the six measurement coils (Fig.2 (a)). The current applied to the ith actuation coil together with the voltage readings from the six measurement coils was used to determine the ith column of the flux distribution matrix according to the Faraday’s Law, i.e., E (t )   N m d  (t) dt , where N m is the number of turns of the

the magnetic force exerting on the particle is balanced by the viscous force, i.e. Fv    v p , where  is the drag coefficient,

measurement coil,  (t) is the magnetic flux, and E (t ) is the induction voltage. A typical experimental result is shown in Fig. 12, wherein a sinusoidal actuation current applied to coil 1 and six measured induction voltages are displayed.

which can be calibrated by using the Power-Spectral-Density (PSD) curves as described in [42]. The viscous fluid used in the experiment was glycerol, which was much denser than water. Therefore, it required greater magnetic force to balance the viscous force. The particle employed was a 4.5 m magnetic spherical bead and the drag coefficient was calibrated to be about 8.5×10-6N.s/m.

Fig.12. Actuation current applied to coil 1 and voltage readings from the six measurement coils.

Fig.13. Twelve linear trajectories on the horizontal plane passing the center of the workspace (in measurement coordinate)

The calibrated flux distribution matrix is denoted as Kˆ I . It was determined after completing the calibration procedure,  0.6022  0.0103   0.0294 ˆ KI    0.1540  0.1805   0.0235

0.0124 0.0285 0.1507 0.1668 0.0229 0.9322 0.1740 0.0787 0.0680 0.1780 0.1655 0.6291 0.0121 0.1458 0.0319  0.0712 0.0112 0.9040 0.0746 0.1501 0.0712 0.1521 0.0769 0.9026 0.0095  0.1726 0.0331 0.1506 0.0123 0.6122

.

With reference to the measurement coordinate, twelve linear trajectories, as shown in Fig. 13, having distinct directions on the horizontal plane passing the center of the workspace were planned. In the experiment, the particle was controlled to travel along each trajectory one by one at a speed of 6.5  m s . The viscous force was, therefore, 55pN along the travel direction. The particle’s position in the workspace was measured using the vision-based particle tracking system. With reference to the actuation coordinate, the three components of the particle’s position, measured and target, are displayed as time sequences in the left column of Fig. 14. It can be seen that the particle followed the target trajectories accurately. Three components of the calculated viscous force in the actuation coordinate are displayed in the right column of Fig. 14 as time

(15)

Whereas the off-diagonal elements of the matrix quantify the degree of couplings among the six electromagnetic poles, diagonal elements are actuation gains of the six poles. It is seen that the actuation gains associated with the three lower poles are significantly smaller than those with the three upper poles.



sequences synchronized with the particle’s motion. They were used to serve as measurements of the magnetic force.

The results of three options are presented in this paper. First, the nominal flux distribution matrix is used in Eq. (16), i.e., K I (c)  K I , and the force gain vector 𝐠 𝐼 is determined while minimizing the objective function. The calibrated force gain vector is g I  [5.37,6.82,6.93]T , measured in pN , and the bestfitted results are shown in Fig. 16.

Fig.14. Particle motion and viscous force displayed with reference to the actuation coordinate, wherein black dashed lines are target motion and color solid lines are measured motion (plotted in time sequences). Fig. 16. The best-fitted results when using the nominal flux distribution matrix to calibrate the force gain vector.

Fig.15. Six actuation currents associated with motion control

Associated with the real-time motion control in the experiment, six actuation currents were known and displayed in Fig. 15. Since the magnetic force is related to the actuation currents through the hexapole magnetic force model, calibration can be accomplished through best fitting by minimizing the following objective function using MATLAB Optimization Toolbox, ,

Fig. 17. The best-fitted results when using the measured flux distribution matrix to calibrate the force gain vector

Second, the measured flux distribution matrix is used, i.e., as in Eq. (14). The force gain vector is g I  [9.08,9.64,8.39]T , and the fitted results are shown in Fig.

(16)

17. It can be seen that model calibration using the second option yields better results. Third, in order to further improve model calibration, a weighting coefficient, c , is employed, i.e., row 1, 3, and 6 of

where 𝑁 denotes the number of total time instants, i.e., 𝑁 = 200𝐻𝑧 × 72.06𝑠𝑒𝑐𝑜𝑛𝑑 = 14412. The magnetic force model Eq.

ˆ is scaled by c , to account for the difference between the K I

(4) is cast as from the point view of calibration. It differs from Eq. (4) in two ways. First, it adopts a force gain vector with three components instead of a scalar gain. This allows the force model to have different force gains in different directions. Second, the flux distribution matrix is denoted as to provide options for model calibration.

lower poles and the upper poles in the distribution of the magnetic flux density in the workspace, and the modified flux distribution matrix is used in Eq. (16). The improved results are shown in Fig. 18. The value of c and the force gain vector



gI

are

determined

simultaneously,

i.e.,

c =1.2

The force generation capability of a hexapole electromagnetic actuating system is dictated by the design and synthesis of the electromagnetic poles and by the inverse modeling derived for current allocation. According to Eq. (5), a design yielding a higher force gain and allowing larger maximum input current will lead to greater effective force possibly generated by the actuating system, i.e., 𝐹𝑚𝑎𝑥 = 2 𝑘𝐼 𝐼𝑚𝑎𝑥 . Compared with the design of the 3-D actuator in [40], the new design of the hexapole electromagnetic actuating system yields a fourfold increase in force gain. While the maximum input current allowed by the 3-D actuator in [40] was 1.5 A, the new design was expected to have larger maximum input current.

and

g I  [7.56,8.55,7.62] . T

Fig. 18. The best-fitted results when using the modified flux distribution matrix to calibrate the scaling factor and the force gain vector simultaneously.

A quantitative measure, i.e., e  J (g I , c) (3N ) , is defined to evaluate and compare the three calibration results. The average error reduces from 12.42pN to 10.04pN, and further to 8.25pN. It is worth noting that a significant component of the average error is attributed to the thermal force, which is about 4pN in our calibration experiments. The actual modeling error is, therefore, significantly smaller than the calculated average error and it is very small compared to the calibration force range, which is -55pN~55pN, i.e. 110pN. After this force model being calibrated, the matrix that best describes the magnetic field is obtained by multiplying the

Fig.19. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right)

An experiment was conducted to determine the maximum input current allowed by the newly developed actuating system. In the experiment, an actuation current was applied to an individual electromagnetic pole and its strength was increased monotonously while a hall-effect sensor was used to measure the magnetic flux density near the pole tip. The test was applied sequentially to two poles, namely P2 (an upper pole) and P3 (a lower pole). As shown in Fig. 19, the flux density increased linearly with respect to the input current and neither test reached magnetic saturation while the actuation current was increased to 3A. It is worth noting that the placement of the sensor tip strongly affects the absolute value of the readings of the hall-effect sensor. The objective of the test was to examine the linear range of the electromagnetic actuation. The hexapole magnetic force model of the newly developed actuating system and that of the 3-D actuator in [40] are used to calculate three force envelopes when a 4.5µm magnetic bead is placed at the center of the workspace. Force generation capabilities are compared and shown in Fig. 20, wherein the three force envelopes are calculated using nominal force models and they are spatially symmetric. It can be seen that the force generation capability of the newly developed actuating system is significantly increased through design and synthesis of the actuating system and the realization

1st, 3rd and 6th row of Kˆ I (Eq. (15)) by c, i.e. 1.2. To compare in Section II.C, the 1st entry of with K FEM I 5/6, denote as

is normalized to

in Eq. (17).

(17)

accord It can be seen that diagonal terms in and K FEM I well, whereas the off-diagonal terms have discrepancies, which are likely due to 1) K FEM is obtained directly from flux density I while is calibrated from force, and 2) nominal value of the material property is used in FEM analysis. VI. FORCE GENERATION CAPABILITY



property characterization. A hexapole magnetic force model was employed to investigate the force generation capability of the actuating system and to lay the foundation for inverse modeling. Optimal inverse modeling was derived. It solved the redundancy problem of the over-actuated system and led to the realization of optimal current allocation to enable the most effective manipulation of the 3-D magnetic force exerting on the particle at the center of the workspace. Several aspects to improving the performance of the hexapole electromagnetic actuating system are identified. First, optimal inverse modeling needs to be extended to the entire workspace to enable superior motion control of the particle away from the center and to enhance its applications that require larger workspace. Second, the delay in the feedback control loop needs to be shortened to improve the control performance; which can be realized through developing highspeed vision-based particle tracking techniques. Third, it is necessary to investigate and reduce the effect of hysteresis on force generation and control through modeling [50] or realtime estimation [18].

of optimal current allocation. Fig. 21 shows the comparison of three force envelopes, which are calculated using calibrated force models. Whereas reduction of the force generation capability of the newly developed system is noticeable due to the removing of significant amount of the material from the three lower poles, significant improvement of force generation capability remains evident.

REFERENCES Fig. 20. Comparing three force envelopes calculated using nominal force models: 3-D actuator using current allocation based on constraint constraints (blue), newly developed actuating system using current allocation based on constraint constraints (red), and newly developed actuating system using optimal current allocation (grey).

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Fig. 21. Comparing three force envelopes calculated using calibrated force models: 3-D actuator using current allocation based on constraint constraints (blue), newly developed actuating system using current allocation based on constraint constraints (red), and newly developed actuating system using optimal current allocation (grey).

VII.

CONCLUSION

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Fei Long received the B.S degree from Department of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China, in 2009. He is currently a Ph.D. candidate in the Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH. His research interests include modeling, control, sensing and estimation of precision electromechanical systems.



Daisuke Matsuura obtained his PhD degree in mechanical engineering at Tokyo Institute of Technology, Japan, in 2008 (Motion Control of Hyper Redundant Robot). After three years of post-doctoral research experience in Precision Measurement and Control Laboratory at the Ohio State University (2008-2011), he was appointed assistant professor at Tokyo Institute of Technology in 2011, with his main activities as design and analysis of elastic redundant mechanisms and development of magnetically actuated noncontact precise manipulation systems. Chia-Hsiang Menq (M’03-SM’08) received the B.S. degree from National Tsing-Hua University, Taiwan, in 1978 and the M.S. and Ph.D. degrees from Carnegie-Mellon University in 1982 and 1985, respectively, all in mechanical engineering. Since 1985, he has been with the Ohio State University, where he is currently Professor and Ralph W. Kurtz Chair in Mechanical Engineering. His research has four areas of focus: 1) control, sensing, and instrumentation; 2) metrology, precision engineering, and manufacturing; 3) imaging and mechanical characterization of live cells; and 4) visual sensing and visual servo control. Prof. Menq is a Fellow of the American Society of Mechanical Engineers (ASME), the Association for the Advancement of Science (AAAS), and the Society of Manufacturing Engineers (SME).



List of figures and table captions Fig.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and three upper poles. Fig.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5) forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for flux generation) and a measurement coil (for flux measurement). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate system. (b) CAD model and meshing of the hexapole actuator Fig.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic flux density distribution (unit: Tesla). (b) The magnetic flux density vectors near the workspace center (actuation coordinate). (c) The magnetic field vectors near the tip of pole 1. Fig.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the measurement coordinate system. (a) |B| in the horizontal plane (top view). (b) |B| in the vertical plane (side view). Fig.5. Validation of the hexapole magnetic field model: (a) comparison of magnetic induction vectors, and (b) normalized norms of error vectors. (c) the definition of the fitting error Fig.6. Illustration of the desired force in the spherical coordinate system. Fig.7. Three orientation-dependent optimal constraints compared with their corresponding constant constraints. Fig.8. Optimal current allocation compared with that obtained using constant constraints. Fig.9. Block diagram of the feedback control system achieving stabilization Fig.10. Six input currents of the hexapole actuator Fig.11. Stabilization of the magnetic particle at the center of the workspace Fig.12. Actuation current applied to coil 1 and voltage readings from the six measurement coils. Fig.13. Twelve linear trajectories on the horizontal plane passing the center of the workspace (in measurement coordinate) Fig.14. Particle motion and viscous force displayed with reference to the actuation coordinate, wherein black dashed lines are target motion and color solid lines are measured motion (plotted in time sequences). Fig.15. Six actuation currents associated with motion control Fig.16. The best-fitted results when using the nominal flux distribution matrix to calibrate the force gain vector. Fig.17. The best-fitted results when using the measured flux distribution matrix to calibrate the force gain vector Fig.18. The best-fitted results when using the modified flux distribution matrix to calibrate the scaling factor and the force gain vector simultaneously. Fig.19. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right) Fig.20. Comparing three force envelopes calculated using nominal force models: 3-D actuator using current allocation based on constraint constraints (blue), newly developed actuating system using current allocation based on constraint constraints (red), and newly developed actuating system using optimal current allocation (grey). Fig. 21. Comparing three force envelopes calculated using calibrated force models: 3-D actuator using current allocation based on constraint constraints (blue), newly developed actuating system using current allocation based on constraint constraints (red), and newly developed actuating system using optimal current allocation (grey).
