 Original Article
 Open Access
 Published:
Modeling and Configuration Design of Electromagnetic Actuation Coil for a Magnetically Controlled Microrobot
Chinese Journal of Mechanical Engineering volume 32, Article number: 63 (2019)
Abstract
Noncontact actuated microbeads have attracted a lot of attention in recent years because of its enormous potential in medical, biological, and industrial applications. Researchers have proposed a multitude of electromagnetic actuation (EMA) systems consisting of a variety of coil pairs. However, a unified method to design and optimize a coil pair according to technical specifications still does not exist. Initially, this paper presented the modeling of an untethered ferromagnetic particle actuated by externally applied magnetic field. Based on the models, a simple method of designing and optimizing the EMA coil pair according to technical specifications, was proposed. A loopshaped coil pair generating uniform magnetic and gradient fields was chosen to demonstrate this method clearly and practically. The results of the optimization showed that the best distance to radius ratio of a loopshaped coil pair is 1.02 for a uniform magnetic field and 1.75 for a uniform gradient field. The applicability of the method to other shapes of coil configuration was also illustrated. The best width to distance ratio for a squareshaped coil pair is 0.558 and 0.958 for uniform magnetic and gradient fields, respectively. The best height to width ratio and distance to width ratio for a rectangleshaped coil pair is h/w = [0.9,1.1], d/w = [0.5,0.6] for uniform magnetic field and h/w = [1.0,1.2], d/w = [0.9,1.1] for uniform gradient field. Furthermore, simulations of a microparticle tracking the targeted trajectory were conducted to analyze the performance of the newly designed coils. The simulations suggested the ability of manipulating microparticles via the coils designed by our proposed method. The research mainly proposed a unified design and optimization method for a coil pair, which can support researchers while designing a specific coil pair according to the technical requirements. This study is aimed at researchers who are interested in EMA system and microrobots.
Introduction
Microrobots hold great promise for a large amount of applications, including minimally invasive surgery, targeted drug delivery, and micromanipulation [1,2,3]. However, as the size of the microrobots decreases, traditional mechanical structures, based on the use of links and joints, suffer from lack of direct contact during the delivering of motion and energy supply. One feasible solution is electromagnetic actuation (EMA), specifically, magnetic microrobots. Magnetic microrobots [4,5,6] are simple devices actuated by an externally applied magnetic field, which exerts combinations of magnetic forces and torques upon the microrobots without any direct contact. Three main driving fields have been proposed: alternating field, rotating field, and gradient field.
An alternating field is a field whose amplitude and direction vary over time according to certain rules. Two main alternatingfieldbased actuation mechanisms have been proposed in recent years: oscillatinguniformfieldbased and pulsedfieldbased actuation. An oscillating uniform field can oscillate or rotate within a certain angle, which produces thrust force to propel the microrobot to swim forward similar to the tail fin of a fish [7,8,9]. A pulsed magnetic field can induce stickslip behaviors in the microrobot [10, 11], causing it to translate on arbitrary surfaces.
A rotating field is a field that has moving polarities in which its opposite poles rotate around a central axis, which drives the microrobot to rotate together with the field and produces propulsive motion like the helicalshaped bacteria. The rotatingfieldbased actuation can be grouped into three main mechanisms: rolling, propelling, and swimming [12]. This method of actuation is favored by numerous researchers because of its simplicity. The fabrication, control, and application of an artificial bacterial flagella actuated by a rotating magnetic field was introduced in [13,14,15] and optimization of the robot shape and rotating field frequency were discussed in [14, 16].
Generally, owing to the manufacturing difficulties, poor turning performance, and low motion speed of the microrobots actuated by alternating or rotating fields, the gradient field is a preferable alternative. The gradient field can exert magnetic forces and torques on any ferromagnetic particle. A host of gradientfieldbased driving mechanisms were reported in the past several years. Different configurations of gradientfieldbased EMA systems [17,18,19,20,21,22] were proposed for various purposes. Further, to improve controllability and mane maneuverability of the microrobots, a variety of novel methods were also proposed. An enhanced EMA system was proposed to enable the effective manipulation of microparticles in a considerably larger workspace with higher effectiveness [23,24,25]. A steering system with coaxial Maxwell and Helmholtz coil pairs rotating around the workspace to control the orientation and direction of magnetic thrust vector was proposed to achieve an inplane threedegreesoffreedom motion with the minimum number of coils [26]. An EMA system consisting of a pair each of stationary and rotational Helmholtz–Maxwell coils can manipulate a microrobot in 3dimensional space and in the phantom of a blood vessel [27]. A novel system that was composed of one conventional pair of Maxwell and Helmholtz coils and one newly developed pair of gradient and uniform saddle coils demonstrated a more geometrically compact and magnetically efficient system than the conventional systems [28, 29]. In addition, special applications of magnetic robots were also developed. A system called OctoMag was designed to control intraocular microrobots for delicate retinal procedures [30]. A novel method of controllable disassembly of paramagnetic nanoparticle chains, using a predefined dynamic field generated by a threeaxis Helmholtz electromagnetic coils system, was proposed for microrobotic cargo delivery [31].
However, even though various gradientfieldbased systems and methods have been proposed, a unified method to design and optimize a specific gradientfieldbased coil pair configuration according to technical requirements is yet to be developed. This paper presented the modeling of the behaviors of an object actuated by a gradient field, including the current, magnetics, and dynamic behaviors. Furthermore, based on the models, a general and simple method was proposed for EMA coil configuration design and optimization according to certain technical specifications. In addition, the design of uniform magnetic and gradient fields with a loopshaped coil, acting as the basic shape of the coil, is used to illustrate the designing and optimizing process. Applicability of the optimization method to other kinds of coil configuration is also illustrated. Finally, through simulations, the validity of the models and the simple method was evaluated.
This paper is organized as follows: Section 2 presents the modeling equations of microrobot controlled by a gradient field. Section 3 discusses the EMA coil configuration design and optimization method and an example is used to demonstrate it clearly. Applicability of the method to other coil configurations is also discussed in this section. Section 4 provides the simulation results using the optimized coil pair and certain discussions are presented. The conclusion is summarized in Section 5.
Modeling of Electromagnetic Actuation
Any ferromagnetic object can be actuated by a gradientfieldbased EMA system through three stepbystep processes: (1) The currents flowing in the coils induce nonuniform magnetic field; (2) This externally applied field exerts a combination of forces and torques on the object; (3) The exerted forces and torques propel the object to move. Three models were constructed to describe the three processes, respectively.
Current Induces Magnetic Field
As is shown in Figure 1, we initially consider the magnetic field distribution induced by a single loopshaped finite long currentcarrying coil. The central axis of the coil is aligned with zaxis of the cylindrical coordinate. The length and radius of the coil are l and a, respectively. The magnitude of current flowing in the coil is I, with number of turns N.
The magnetic field [32, 33] at the position of \( (\rho ,z) \) can be expressed as follows:
where
However, because of the elliptic integrals, E, K, Π, fast calculation of the magnetic field is difficult. Besides, using these formulae to design a magnetic field quantitatively is an impossible task. Therefore, optimization of the equations was required. According to Ref. [34], elliptic integrals can be expanded as a sum of infinite series. We used the first few items of the elliptic integral expansions to approximate the accurate value, thus, the elliptic integrals can be approximated by:
The truncation errors were less than 0.2% and were minor enough to be neglected for the calculation of magnetic field. The truncation error of elliptic integrals can occur at any point in space; specifically, it affects the whole scale. Further, in the process of optimizing the coil configuration described in Section 3, normalized values of optimization operators are used for convenience of comparison, which eliminated the wholescale truncation errors and did not affect the design and optimization result described in Section 3.
Substituting Eqs. (6)‒(8) into Eqs. (1)‒(5) yields:
Quantitative and fast computation of the magnetic field of a loop coil through Eqs. (9)‒(10) was realizable. Therefore, it was used to design and optimize an EMA coil pair configuration, which will be introduced in Section 3.
Further, squareshaped or rectangleshaped coils are also widely used. We present below the magnetic field distribution expressions of these coils for the convenience of optimization. As shown in Figure 2, the field distribution in the xyplane can be expressed as:
where
Magnetic Field Exerts Forces and Torques on Object
We considered an object created using ferromagnetic material owing to the high susceptibility and magnetization saturation of the material. Any ferromagnetic object in the magnetic field is influenced by the magnetic force and torque induced from the gradient and intensity of the externally applied nonuniform magnetic field [35]. The magnetic force and torque exerted on the object can be expressed by:
where B is the value of intensity—measured in units of Tesla (T)—of the magnetic field at the body’s center of mass. B can also be expressed as H, which is simply related to B as \( {\varvec{B}} = \mu_{0} {\varvec{H}}, \) where \( \mu_{0} = 4\uppi\, \times \,10^{  7} {\text{T}} \cdot {\text{m/A}} \) is the permeability of free space. M is the magnetization at the body’s center of mass. v is the volume of the ferromagnetic object. \( \nabla \) is the gradient operator:
Magnetization M can be calculated as:
\( \chi_{a} \) is the equivalent susceptibility of the object expressed as:
where \( \chi \) is the susceptibility of the object’s material, n_{x}, n_{y}, n_{z} are the demagnetizing factors [36] along (x, y, z)axes, respectively, of the ellipsoidal object or its magnetically equivalent ellipsoid [37] with semiminor and semimajor axes aligned with respect to the body frame system’s axes.
The components of the magnetization M in (x, y, z)axes directions can be represented as M_{x}, M_{y}, and M_{z}, respectively. To express the magnetic force and torque in a more intuitive form, Eqs. (14) and (15) can be rewritten, after some manipulation, as:
In Eqs. (14)‒(20), M, H, and \( \chi_{a} \) are all written with respect to the global coordinate system Oxyz, which has the same orientation as the body frame system \( O_{1}  x_{1} y_{1} z_{1} \), shown in Figure 3(1).
If the body frame system and global system do not have the same orientation, as shown in Figure 3(2), then Eq. (17) can be updated as:
where R is the rotation matrix relating the body frame system with respect to the global system, reflecting the impact of the pose of the ellipsoid on the magnetization M.
Forces and Torques Propel the Object to Move
Dynamic behaviors of the object can be modeled by Newton’s equation for translational motion and Euler’s equation for rotational motion, respectively:
where g is gravitational acceleration, \( \sum {{\varvec{F}}_{mag} } , \, \sum {{\varvec{M}}_{mag} } \) are magnetic forces and torques, respectively, and \( \sum {{\varvec{F}}_{ext} } , \, \sum {{\varvec{M}}_{ext} } \) are external forces and torques, respectively. \( \ddot{x} \) is the translational acceleration of the body frame attached to the body’s center of mass with respect to the global frame. \( {\varvec{I}}\dot{\omega }, \, \omega \times {\varvec{I}}\omega \) reflect Coriolis and centrifugal effect, respectively.
EMA Coil Configuration Design and Optimization Method
As stated before, although various EMA systems consisting of different types of coil pair configurations were proposed in recent years, no one has reported a universal method to design and optimize a gradientfieldbased coil pair. This section will introduce a simple method on this issue. We considered a 2dimensional EMA system as an example to introduce the method step by step.
Characteristics of the Field
The first step was to determine the most remarkable and suitable characteristic of the field according to technical specifications.
From Eqs. (19) and (20), it is easy to observe that the magnetic force and torque are functions of position and current. If the magnitude and gradient of the magnetic field in the region of interest were assumed to be constants or pseudoconstants, i.e., \( \partial B = 0{\text{ or const}}, \) \( B = 0{\text{ or const,}} \)we could obtain singlevalued mappings from currents to magnetic force and torque, which would significantly simplify the design and optimization process. Therefore, the most remarkable characteristic was uniformity of the uniform magnetic and gradient fields. Furthermore, this is also the most common characteristic in most of the gradientfield based EMA systems as is described in Section 1.
Choosing Appropriate Types of Coil Pairs
The next step was to determine the fundamental shape of the finite long currentcarrying coil, which can be a loopshaped coil [38], a rectangleshaped coil [39], or even a saddleshaped coil, and the parameters to be optimized. In this process, certain major factors including compactness of the EMA system, space utilization, and manufacturing difficulties and expenses was taken into consideration.
Generally, it is acknowledged that the conventional loopshaped Helmholtz coil pair induces uniform magnetic field and the conventional loopshaped Maxwell coil pair induces uniform gradient field. They are the most commonly used coil pairs in gradientfieldbased EMA systems. However, there are no reports to prove that they are the best choices to induce such a field and the reasons for their current coil parameters. To overcome this limitation, we considered a loopshaped coil pair as an example to illustrate the method to optimize the coil configuration clearly and practically. Section 3.6 will present the applicability of the optimization method on other coil configurations.
For a loopshaped coil pair as shown in Figure 4, r, d, and 2l are the distance, radius, and length of the coils, respectively. I_{1}, I_{2} are the magnitude of currents flowing in the coil pair. N is the number of turns of the wire. 2l varies with the number of turns N of the current and the size of the wire, which does not have a direct impact on the field distribution. Conversely, I, r, and d have a big impact on the magnetic field distribution, magnitude, and even the type of magnetic field. Therefore, the main optimization task was to optimize them.
Optimization of the Coil Pair
Optimization Operators
To generalize the optimization process, certain universal optimization operators were defined to meet various requirements of the field. Here, we present a list of these operators:

1)
\( \partial B = 0, \, B = const \) ensures a uniform field.

2)
\( \partial B = const, \, B = 0 \) ensures a uniform gradient field.

3)
\( \partial B = 0, \, B = 0 \) ensures no magnetic field existing in a specific direction.

4)
\( \partial^{2} B = 0 \) ensures absolute uniformity of the uniform magnetic field or the uniform gradient field.

5)
\( S\left( {args} \right) \) calculates the maximum usable area (MUA) of the magnetic field that conforms to specific uniformity requirements, where args are parameters that need to be optimized.

6)
\( M\left( {args} \right) \) calculates the maximum magnitude of the uniform magnetic field or the maximum gradient of the uniform gradient field, where args are parameters that need to be optimized.
Optimization of the LoopShaped Coil Pair That Induces Uniform Magnetic Field
If the currents in the two coils flow in the same direction, i.e., \( I_{1} = I_{2} = I \), the coil pair is likely to induce uniform magnetic field along the zaxis in the central area as the field component along the ρaxis generated by the two coils cancel each other, and thereby remaining the component along the zaxis. The magnetic field of the coil pair is the superposition of the two coils and can be obtained from Eqs. (9) and (10).
The magnetic field along the ρaxis and its partial derivative with respect to the ρaxis are expressed as:
where
\( i_{\rho } \) represents a comprehensive influence of parameters that are not required to be optimized, such as currents, number of turns, permeability of free space, and diameter of the wire.
The magnetic field along the zaxis and its partial derivative with respect to the zaxis are expressed as:
where
From Eqs. (24)‒(32), it is observed that the following optimization operators are naturally satisfied in the central area of the loopshaped coil pair:
Eq. (33) ensures no magnetic field exists along the ρaxis. Eq. (34) ensures a uniform field along the zaxis. Therefore, any loopshaped coil pair with currents flowing in the same direction in each coil induces uniform magnetic field near the central area, the magnitude of which can be expressed in cylindrical coordinate system \( O  \rho z \) as:
From Eq. (35), it can be observed that the magnitude of the uniform field is a singlevalued function of current I in the central area. However, the geometry parameter of the coil pair, i.e., d/r is still unknown. Besides, this parameter has a great impact on the magnitude and MUA of the uniform field. Figure 5 shows the uniform field distribution in the ρzplane and its MUA for two different values of d/r. The colormap shows the normalized magnitude \( B(x,y)/B(0,0) \) of the field. The region enclosed by the red line represents the MUA, which is defined as the area with perfect uniformity of the field that never deviates from the field at the origin by more than a specific amount of deviation.
The next step was to identify an optimal d/r. Optimization operators M(d/r) and S(d/r) were used. For the convenience of calculation, we defined S(d/r) = MUA, M(d/r) = B_{0}(d/r), where B_{0}(d/r) is the magnitude of the field at the origin for a specific d/r. The results are shown in Figure 6.
Figure 6(1) shows M(d/r) and S(d/r) of the field. For different uniformity deviations of 2%, 1%, and 0.5%, the S(d/r) reaches the maximum when d/r = 1.05, 1.04, and 1.02, respectively. Meanwhile, the magnitude of the field M(d/r) is also large. Figure 6(2) shows the magnetic field distribution for d/r = 1.02. Figure 6(3) shows its MUA for different uniformity deviations of 2%, 1%, 0.05%.
Thus, to obtain the best uniform magnetic field, we use of a coil pair with currents flowing in the same direction in each coil and with d/r = 1.02 was a perfect choice. In a special case where d/r = 1.00, it represents a conventional Helmholtz coil pair. The result proved that the conventional Helmholtz coil pair is one of the best loopshaped coil pairs to induce uniform magnetic field.
Optimization of the LoopShaped Coil Pair That Induces Uniform Gradient Field
Similarly, if the currents flow in opposite directions in each coil, i.e., \( I_{1} =  I_{2} = I \), the coil pair is likely to induce uniform gradient field as the magnitude of the field generated by the two coils cancel each other in the central area. The field distribution and its partial derivative can be similarly obtained from Eqs. (9) and (10) expressed as follows:
where
From Eqs. (36)‒(40), we observed that in the central area of the loopshaped coil pair the following operators were naturally satisfied:
Eqs. (41) and (42) ensure uniform gradient field both along the ρ and zaxis. Therefore, any loopshaped coil pair with currents flowing in opposite directions in each coil induces uniform gradient field near the central area, the gradient magnitude of which can be expressed in cylindrical coordinate system \( O  \rho z \) as follows:
where
From Eqs. (43) and (44), it can be observed that the magnitude of the uniform gradient field is a singlevalued function of current I in the central area. Besides, d/r also has a great impact on the amplitude and MUA of the gradient field. Figure 7 shows the gradient field distribution and its gradient along the zaxis in the ρzplane.
Figure 7(1) shows the magnetic field distribution of the uniform gradient field. Figure 7(2) shows its gradient along the zaxis. The region enclosed by red circles in Figure 7(3) and 7(4) represent the MUA for d/r = 1.1 and 1.6, respectively.
We defined S(d/r) = MUA, M(d/r) = ∂B_{z0}(d/r), where B_{z0}(d/r) is the magnitude of the gradient at the origin along the zaxis for a specific d/r. The results are shown in Figure 8.
Figure 8(1) shows M(d/r) and S(d/r) of the field. For different uniformity deviations of 2%, 1%, and 0.5%, the S(d/r) reaches the maximum when d/r = 1.78, 1.77, and 1.75, respectively. Meanwhile, the magnitude of the gradient is also large. Figure 8(2) shows the field distribution for d/r = 1.75. Figure 8(3) shows the gradient distribution along zaxis and its MUA for different uniformity deviations of 2%, 1%, and 0.5%.
Therefore, to obtain the best uniform gradient field, the use of a coil pair with currents flowing in opposite directions in each coil and with d/r = 1.75 was a perfect choice. In a special case where d/r = 1.73, it represents a conventional Maxwell coil pair. The result proved that the conventional Maxwell coil pair is one of the best loopshaped coil pairs to induce uniform gradient field.
Implementing the 2dimensional EMA System
After optimizing the coil parameters, we implemented the 2dimensional EMA system by combining the two kinds of coils. We chose the superposition of uniform magnetic and gradient fields as the most remarkable characteristic of the field. Thus, we can choose two pairs of orthogonal uniform field coil with parameter d/r = 1.02 to induce a uniform field. A single pair of uniform gradient field coils with parameter d/r = 1.75 is sufficient to induce uniform gradient field because it can induce gradients in both the radial and axial directions. Figure 9 shows the schematic of the EMA system.
Discussion of the Optimization Method
In this section, we proposed a simple method to design and optimize a coil pair. Initially, this method was implemented by determining characteristics of the magnetic field, which is assumed as a superposition of the uniform magnetic and gradient fields in this case. Then, we chose different shapes of the coil pair and its parameters to be optimized. Subsequently, appropriate optimization operators were chosen to optimize the coil pair. Finally, the EMA system was implemented by combining the optimized coil pair. A large variety of EMA systems [24, 27, 28, 38] were proposed and optimized through this method, although they were not strictly implemented in a stepbystep manner.
This method achieves several advantages in design and optimization of EMA systems composed of Xshaped coil pair, including but not limited to, loopshaped, ovalshaped, squareshaped, rectangleshaped, capsuleshaped, and even saddleshaped coil pairs. Theoretically speaking, the optimization process is applicable for coil pair configurations of any shape, to quantitatively analyze and optimize it, provided the magnetic field distribution expressions and its partial derivative with respect to position can be obtained. Moreover, because the most significant characteristics of the field were determined in Section 3.1, the main difficulties of the process lie in the optimization of the parameters of the coil. Determining the ideal methods for choosing the best parameters of the coil pair and appropriate optimization operators are major concerns.
Derivation of the magnetic field distribution of the Xshaped coil required a great deal of patience, as it was the fundamentals of the subsequent studies, and it was prone to errors. This paper also presents the magnetic field distribution of a single rectangleshaped coil in Section 2, which can be directly used in the design and optimization process.
However, this method also has its own limitations. It mainly concentrates on coil pairs generating a uniform magnetic or gradient field or the superposition of these fields. Therefore, this method mainly focuses on the design and optimization of a gradientbasedfield EMA system. It was not appropriate to apply this to the other two kinds of driving fields, i.e., alternating and rotating field, because of the essential change in the driving mechanism.
Applicability of the Optimization Method
As discussed in Section 3.5, the newly proposed design and optimization method is applicable to coil pairs in any shape. Here, we briefly present the optimization results of applying it to the squaredshaped and rectangleshaped coil pairs. The EMA system composed of other shapes of coil pair configurations could be optimized through the same process discussed in Sections 3.1‒3.4.
For a squareshaped coil pair, the parameter to be optimized was d/w, where w is width of the square coil, d is distance between the two coils.
The optimization result of the squaredshaped coil pair inducing uniform magnetic field was shown in Figure 10. Figure 10(1) shows M(d/w) and S(d/w) of the field. For different uniformity deviations of 2%, 1%, and 0.5%, S(d/w) reaches the maximum when d/w = 0.590, 0.570, and 0.558, respectively. Meanwhile, the magnitude of the field is also large. Figure 10(2) shows the distribution of the field for d/w = 0.558. Figure 10(3) shows the MUA of the uniform field for different uniformity deviations of 2%, 1%, and 0.5%.
The optimization result of the squaredshaped coil pair inducing uniform gradient field was shown in Figure 11. Figure 11(1) shows M(d/w) and S(d/w) of the field. For different uniformity deviations of 2%, 1%, and 0.5%, S(d/w) reaches the maximum when d/w = 0.982, 0.966, and 0.958, respectively. Meanwhile, the magnitude of gradient of the field is also large. Figure 11(2) shows the distribution of the field for d/w = 0.958. Figure 11(3) shows the gradient distribution along zaxis the MUA of the uniform gradient for different uniformity deviations of 2%, 1%, and 0.5%.
For a rectangleshaped coil pair, the parameters to be optimized were h/w and d/w, where w and h are the width and height of the rectangle coil, respectively, d is distance between the two coils.
The optimization result for the rectangleshaped coil pair inducing uniform magnetic field is shown in Figure 12. The best parameters are h/w = [0.9, 1.1], d/w = [0.5, 0.6], of which the best values are h/w = 1.015, d/w = 0.56.
The optimization result for the rectangleshaped coil pair inducing uniform gradient field is shown in Figure 13. The best parameters are h/w = [1.0, 1.2], d/w = [0.9, 1.1], of which the best values are h/w = 1.10, d/w = 1.00.
Other coil configurations were basically a combination of loop/arc coil and line coil. The optimization process was the same as the loop and rectangleshaped coil pairs.
Simulation Results of Motion Control
Numerical simulations of target trajectory tracking were performed to test the controllability of the coils designed in Section 3. Thus, we verified the validity of the models and coil pair configuration design and optimization method. The trajectory tracking procedure is described as follows: Initially, a desired 2dimensional (2D) targeted trajectory X_{0} of the microparticle was specified. Secondly, the position of the microparticles were detected by the position measure technique, and the velocity of the particle was determined using the traditional position differentiation method. Thirdly, proper currents flowing in the Helmholtz coils were calculated through feedback control algorithms and the currents flowing in the Maxwell coils were maintained constant. Finally, magnetic forces were generated by the currentcarrying coils to drive the microparticle. Thus, the microparticle was controlled automatically to track the targeted trajectory.
Trajectory tracking of the microparticle was implemented through the control flow chart shown in Figure 14. The control feedback block is an inputtostate stability controller [25], which computes the currents to be applied to the coils, by the following equation:
where k_{0}, k_{1}, k_{2}, and \( \delta \) are parameters of the controller. \( \mu \) is the damping coefficient of the medium. m is mass of the microparticle. E_{1} and E_{2} are position and velocity tracking errors, respectively, which are calculated based on targeted trajectory and the measured trajectory provided by the position measurement block, given by:
where \( {\varvec{X}}_{0} , \, {\dot{\varvec{X}}}_{0} \) are the desired position and velocity, respectively. \( {\tilde{\varvec{X}}},{\dot{\tilde{\varvec{X}}}} \) are the measured position and velocity, respectively. The control feedback block computes currents flowing in the two orthogonal Helmholtz coils and allows the system to pull a microparticle towards the targeted position.
Further, measurement errors and environmental noises were both taken into consideration to simulate the real environment. Therefore, the measured trajectory and dynamic force were expressed as:
where N denotes the measurement errors generated by the random function in MATLAB, with an amplitude of [−0.01, 0.01]*X. F_{noise} denotes the environmental noises generated by MATLAB random function, with an amplitude of [−0.1, 0.1]*F_{mag}.
The microparticle was manipulated to track an inplane 2D spiral curve expressed as
Figure 15 shows the simulation results for controller parameter of \( k_{0} = 10^{5} , \, k_{1} = k_{2} = 40, \, \delta = 0.5 \). Figure 15(1)‒15(3) show the target trajectory (in blue) of the particle and the tracking trajectory (in red) at various time instants, t = 1.0 s, 7.0 s, 12.0 s, respectively. Figure 15(4) shows the position tracking errors E_{1}. Figure 15(5) shows the velocity tracking errors E_{2}. Figure 15(6) shows the feedback current I_{helm}. The simulation results demonstrated good controllability of the EMA system.
Conclusions

(1)
This paper proposed the modeling of an EMA system, including the three processes: 1) Different shapes of single currentcarrying coil induced magnetic field; 2) The externally applied field exerted magnetic forces and torques on a ferromagnetic object; 3) The exerted forces and torques drove the ferromagnetic object to move. The three models fully described the behaviors of a ferromagnetic object controlled by the EMA system.

(2)
A simple and efficient method for coil configuration design and optimization was proposed, which mainly focused on gradientfieldbased driving EMA system. The method included four steps: determining the most critical characteristics of the magnetic field according to the specific requirements; choosing appropriate types of coil, including shapes of the coil and coil parameters to be optimized; optimizing the coil configuration by choosing appropriate optimization operators as listed in Section 3.3.1; and finally, implementing the EMA system using the optimized coil pair.

(3)
A loopshaped coil pair was chosen to demonstrate the method clearly and practically. The optimization result demonstrated that the best distance to radius ratio of a loopshaped coil pair was 1.02 for uniform magnetic field and 1.75 for uniform gradient field, respectively. We observed that these are the values demonstrated by the conventional Helmholtz and Maxwell coil pairs. The newly proposed method is applicable to anyshaped coil pair. Applicability of the method on other coil configurations, such as squareshaped and rectangleshaped coil pair, was discussed, the results of which were demonstrated.

(4)
Simulations of a microparticle tracking the target trajectory were conducted to show the performance of the coil pair designed by the newly proposed method. The results demonstrated good controllability.
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Authors’ Contributions
XJ took on most of the research work, including the theoretical research and modeling, proposal and establishment of the new method and simulation work, and paper writing of the manuscript. WG put forward a great variety of valuable suggestions on some key theory points, assisted with the theory researching and method validity, so that the research work can be carried out smoothly. All authors read and approved the final manuscript.
Authors’ Information
Xiaolong Jing, born in 1993, is currently a master candidate at Institute of Design and Control Engineering for Heavy Equipment, School of Mechanical Engineering, Shanghai Jiao Tong University, China. He received his bachelor’s degree from Shanghai Jiao Tong University, China, in 2017. His research interests include robotics, magnetically controlled microrobot and machine vision.
Weizhong Guo, is currently a professor at School of Mechanical Engineering, Shanghai Jiao Tong University, China. He received his PhD degree from Shanghai Jiao Tong University, China, in 1999. He was an Assistant Professor at SJTU (19992001) and Associate Professor at SJTU (20012009) and became a Full Professor at SJTU in 2009. His research mainly focuses on parallel robots and modern mechanisms.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by Aerospace Research Project (Grant No. 040102).
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Jing, X., Guo, W. Modeling and Configuration Design of Electromagnetic Actuation Coil for a Magnetically Controlled Microrobot. Chin. J. Mech. Eng. 32, 63 (2019). https://doi.org/10.1186/s100330190378y
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Keywords
 Microrobot/particle
 Coil configuration design and optimization
 Electromagnetic actuation
 Trajectory tracking