1 Introduction

Microrobots have several unique advantages such as highly accurate maneuverability and the ability to access small spaces. Possible biomedical applications of microrobots include minimally invasive diagnosis, localized treatment, and micromanipulation [1, 2]. Various actuation methods for microrobots are currently being investigated, for example, chemical catalytic reaction [3], biological actuation [4], and magnetic actuation [5–8]. The use of external magnetic fields, in particular, has strong advantages such as wireless controllability and safe use in clinical applications. Helical propulsion for microrobots, inspired by the movements of bacterial flagella, is a well-studied and advantageous method especially in a low-Reynolds-number regime [9–11]. Such microrobots are composed of a ferromagnetic head and a helical tail. An applied rotating external magnetic field induces torque on the ferromagnetic head, and the resultant rotational movement of the head is converted into linear motion by the helical tail. Honda et al. studied helical propulsion of microrobots with hard magnetic heads [5]. They fabricated several types of helical microrobots, each with a hard magnetic head and a corkscrew- or woodscrew-like tail. Recently, smaller helical microrobots were realized: Zhang et al. developed helical microrobots approximately 30 μm long [6]. They can be batch fabricated by means of a self-scrolling technique. A nickel thin plate is used as the ferromagnetic head because of its compatibility with batch fabrication. The same research group demonstrated a coordinated swarm-like behavior with three artificial bacterial flagella [7]. Ghosh et al. developed approximately 1-μm-long swimmers using a shadow-growth method and surface coating with cobalt [8].

A few reports exist on the selective control of magnetic microrobots. For example, a segmented surface that generates an electrostatic force [12] and microrobots with different resonant frequencies [13] are used in order to allow selective actuation when using multiple robots. These methods are currently limited to actuation on flat surfaces. In addition, selective control of multiple spiral-shaped microrobots has also been studied [14]. In this study, the researchers developed a method for selective actuation of millimeter-sized microrobots by changing the shapes of the microrobots and demonstrated selective control of two microrobots placed in an agar-gel phantom; the characteristic “step-out” frequency unique to each microrobot was used for selective actuation. The step-out frequency is the frequency at which the microrobot becomes unsynchronized with the rotational magnetic field when the induced torque becomes smaller than that of the fluid and frictional drag. The step-out frequency of a microrobot is determined by the magnitude of the magnetic field and the robot design, i.e., the shape of its tail, strength of magnetization, and volume of its magnetic head [7, 15]. By changing these design parameters, each microrobot can have a unique step-out frequency, therefore making selective actuation of microrobots feasible. For instance, when a rotating magnetic field of a specified frequency is applied, microrobots with higher step-out frequencies are actuated, whereas those with lower step-out frequencies do not move. Although this method has shown good results in agar gel, it will not work in water or Newtonian fluids because robots still move even at frequencies higher than the step-out frequency and maneuverability suffers. Moreover, the step-out frequency and velocity of each microrobot need to be measured in advance as this method relies on the difference in step-out frequencies among the microrobots. The step-out frequency varies according to the viscosity of the liquid and the boundary conditions.

This paper proposes a selective control method for magnetic helical swimming microrobots. The proposed method allows selective control over the motion of each microrobot (Fig. 1) regardless of changes in fluid viscosity and boundary conditions (e.g., the wall effect). In addition, it is not necessary to measure the step-out frequency in advance because it does not utilize differences in step-out frequency. These features are greatly advantageous because it is difficult to predict the local viscosity and the wall effect around moving microrobots. In the proposed method, the heads of the robots have different shapes and are made of a soft-magnetic material such as nickel for easy batch fabrication. Two different rotational magnetic fields are used to drive the movement of the individual microrobots.

Fig. 1
figure 1

Concept of selective control for two types of helical microrobots. Each robot can move independently within the same magnetic field

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2 Selective control of microrobots

2.1 Theory: magnetic anisotropic torque

An external magnetic field can induce torque on a ferromagnetic body and tends to align the longer side of the body in the direction of the magnetic field. In this paper, a soft-magnetic body with a small coercivity and large permeability is used for the microrobot heads. A soft-magnetic body is susceptible to an external magnetic field, and its degree of magnetization can be easily varied by changing the direction and strength of the applied field [16, 17].

A column-shaped soft-magnetic body is illustrated in Fig. 2. The figure shows the direction of the external magnetic field, Hext, magnetization, M, and axis of symmetry, X. Both Hext and M are vectors with ampere per meter as their units of measurement. The demagnetizing factors in the x-, y-, and z-axes are defined as N x , N y , and N z . These factors are constrained by the following relation:

$$ \mathop{N}\nolimits_x + \mathop{N}\nolimits_y + \mathop{N}\nolimits_z = 1 $$
(1)
Fig. 2
figure 2

Column-shaped soft-magnetic body placed in an external magnetic field, H ext . Line X represents the axis of geometric symmetry. Angle θ is the angle between the axis of symmetry and the external magnetic field, whereas angle φ is the angle between the axis of symmetry and the magnetization axis, M

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The demagnetizing factor can be calculated only for special shapes such as a sphere or prolate spheroid. The design of the column used in our microrobotic heads is 0.6 mm in diameter and 5 mm in length, and its ratio of length to diameter is 8.33. Assuming that the column-shaped robot is prolate spheroidal, the demagnetizing factors can be calculated as N x  = 0.0267, N y  = 0.487, and N z  = 0.487[18].

Magnetization is proportional to the internal magnetic field of a soft-magnetic body, H i .

$$ {\mathbf{M}} = \chi \mathop{{\mathbf{H}}}\nolimits_{{\mathbf{i}}}, $$
(2)

where χ is a variable parameter that represents the magnetic susceptibility, and H i is described as

$$ \mathop{{\mathbf{H}}}\nolimits_{{\mathbf{i}}} = \mathop{{\mathbf{H}}}\nolimits_{{{\mathbf{ext}}}} + \mathop{{\mathbf{H}}}\nolimits_{{\mathbf{d}}}, $$
(3)

where H ext and H d are the external and demagnetizing magnetic fields, respectively. The magnetic susceptibility χ varies depending on the magnetization. The demagnetizing field H d is described as

$$ \mathop{{\mathbf{H}}}\nolimits_{{\mathbf{d}}} = - {\mathbf{NM}}, $$
(4)

where N is a diagonal matrix described as

$$ {\mathbf{N}} = diag(\begin{array}{*{20}{c}} {\mathop{N}\nolimits_x } & {\mathop{N}\nolimits_x } & {\mathop{N}\nolimits_x } \\ \end{array} ). $$
(5)

By substituting Eqs. 2 and 4 in Eq. 3, the following relation is obtained:

$$ {\mathbf{M}} = \mathop{{\mathbf{\chi }}}\nolimits_{{\mathbf{a}}} \mathop{{\mathbf{H}}}\nolimits_{{{\mathbf{ext}}}}, $$
(6)

where χ a is a diagonal matrix of the extrinsic susceptibility components, which is described as follows:

$$ \mathop{{\mathbf{\chi }}}\nolimits_{{\mathbf{a}}} = diag\left( {\begin{array}{*{20}{c}} {\frac{\chi }{{1 + \chi \mathop{N}\nolimits_x }}} & {\frac{\chi }{{1 + \chi \mathop{N}\nolimits_y }}} & {\frac{\chi }{{1 + \chi \mathop{N}\nolimits_z }}} \\ \end{array} } \right). $$
(7)

The magnetic susceptibility of nickel is of the order of 102, whereas the demagnetizing factor is of the order of 10-2; therefore, a change in magnetic susceptibility is not negligible. In general, in a weak magnetic field below the level of saturation, the magnetic susceptibility increases as the magnetization increases, up to a certain point [16]. The value of χ a varies as magnetic susceptibility χ varies. The torque induced on the microrobotic head, T m , is

$$ \mathop{{\mathbf{T}}}\nolimits_{{\mathbf{m}}} = \mathop{\mu }\nolimits_0 \mathop{V}\nolimits_m {\mathbf{M}} \times \mathop{{\mathbf{H}}}\nolimits_{{{\mathbf{ext}}}}, $$
(8)

where μ 0 and V m are the permeability of free space and the volume of the component. When the magnetization is sufficiently low, as described by \( |M{| < }{M_s} \), where M s is the saturation magnetization, the magnitude of the torque can be described as

$$ \left| {\mathop{{\mathbf{T}}}\nolimits_{{\mathbf{m}}} } \right| = \frac{{\mathop{\mu }\nolimits_0 \mathop{V}\nolimits_m \left| {\mathop{N}\nolimits_x - \mathop{N}\nolimits_y } \right|}}{{2\left( {\frac{1}{\chi } + \mathop{N}\nolimits_x } \right)\left( {\frac{1}{\chi } + \mathop{N}\nolimits_y } \right)}}\mathop{{\left| {\mathop{H}\nolimits_{{ext}} } \right|}}\nolimits^2 \sin \left( {2\theta } \right). $$
(9)

Magnetic susceptibility χ is smallest when angle θ between magnetic field H ext and axis of symmetry X is 90°. The degree of susceptibility increases as angle θ decreases and peaks at 0°. Utilizing this feature, we have developed the ability to selectively and individually control the movement of microrobots, as described in subsection 2.2.

2.2 Design and control of soft-magnetic heads

We propose two microrobotic head designs that employ a soft-magnetic body. By rotating the direction of an external magnetic field, we can control the swimming motion of these robots. The first head design involves the use of a bar-shaped soft-magnetic body with an easily magnetized axis, while the other uses a cross-shaped soft-magnetic body with two easily magnetized orthogonal axes. The cross-shaped soft-magnetic body was designed by orthogonally attaching two bar-shaped magnetic bodies. The heads are attached to helical tails, as shown in Fig. 3, where the long axis of the ferromagnetic head is perpendicular to the axis of the helical tail. The heads of the microrobots can be rotated by rotating the external magnetic field, and this rotational motion is then converted into translational motion by the robot’s helical tail.

Fig. 3
figure 3

Microrobots with bar- (top) and cross-shaped heads (bottom). Torque T1 and T2 induced by an external magnetic field are converted into translational motion of velocity V

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Figure 4 shows the torque induced on each soft-magnetic head. The parameters θ, M 1 , and M 2A , and M 2B in the figure are the angle between the axis of symmetry and the direction of the external magnetic field, the degree of magnetization of the bar-shaped soft-magnetic body, and the degrees of magnetization of the cross-shaped body in axes A and B, respectively. The difference between the direction of an external magnetic field and the magnetization of a soft-magnetic body generates torque on a ferromagnetic body. The torque induced on each magnetic body results in the following:

$$ \left| {\mathop{{\mathbf{T}}}\nolimits_1 } \right| = \left| {\mathop{{\mathbf{T}}}\nolimits_{{2A}} } \right| = \frac{{\mathop{\mu }\nolimits_0 \mathop{V}\nolimits_m \left| {\mathop{N}\nolimits_x - \mathop{N}\nolimits_y } \right|}}{{2\left( {\frac{1}{{\mathop{\chi }\nolimits_A }} + \mathop{N}\nolimits_x } \right)\left( {\frac{1}{{\mathop{\chi }\nolimits_A }} + \mathop{N}\nolimits_y } \right)}}\mathop{{\left| {\mathop{{\mathbf{H}}}\nolimits_{{{\mathbf{ext}}}} } \right|}}\nolimits^2 \sin \left( {2\theta } \right)\,{\text{and}} $$
(10)
$$ \left| {\mathop{{\mathbf{T}}}\nolimits_{{2B}} } \right| = \frac{{\mathop{\mu }\nolimits_0 \mathop{V}\nolimits_m \left| {\mathop{N}\nolimits_x - \mathop{N}\nolimits_y } \right|}}{{2\left( {\frac{1}{{\mathop{\chi }\nolimits_B }} + \mathop{N}\nolimits_x } \right)\left( {\frac{1}{{\mathop{\chi }\nolimits_B }} + \mathop{N}\nolimits_y } \right)}}\mathop{{\left| {\mathop{{\mathbf{H}}}\nolimits_{{{\mathbf{ext}}}} } \right|}}\nolimits^2 \sin \left( {2\theta } \right), $$
(11)

where \( |{T_1}| \), \( |{T_{{{2}A}}}| \), and \( |{T_{{{2}B}}}| \)denote the torque induced by an external magnetic field. χ A and χ B denote the magnetic susceptibilities of bar A and bar B, respectively. Magnetic susceptibility χ A is higher than χ B when angle θ is smaller than π/4 due to the non-linearity of the magnetic susceptibility. By superimposing the two torques \( |{T_{{2B}}}| \) and \( |{T_{{2B}}}| \), the torque induced on a cross-shaped magnetic body, \( |{T_2}| \), is expressed as follows:

$$ \left| {\mathop{{\mathbf{T}}}\nolimits_2 } \right| = \left| {\mathop{{\mathbf{T}}}\nolimits_{{{\mathbf{2A}}}} + \mathop{{\mathbf{T}}}\nolimits_{{{\mathbf{2B}}}} } \right|. $$
(12)
Fig. 4
figure 4

Bar- and cross-shaped heads in a uniform external magnetic field. The end of axis A of the cross-shaped head is marked with a dot for illustrating its orientation. Angle θ is the angle between the axis of symmetry and the external magnetic field. The angle between axis B and the magnetic field is described as \( \frac{\pi }{2} - \theta \)

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From Eq. 3, the degree of magnetization of a cross-shaped body in axis A \( \left( {|{M_{{2A}}}|} \right) \) is larger than that in axis B \( \left( {|{M_{{2B}}}|} \right) \) under the condition of θ < π/4, because χ A is larger than χ B . Thus, the following inequality is obtained:

$$ \left| {\mathop{{\mathbf{T}}}\nolimits_2 } \right| = \left| {\mathop{{\mathbf{T}}}\nolimits_{{{\mathbf{2A}}}} + \mathop{{\mathbf{T}}}\nolimits_{{{\mathbf{2B}}}} } \right| > 0. $$
(13)

Therefore, both the bar- and cross-shaped heads rotate in correspondence with the rotation of the external magnetic field. Torque \( |{{\text{T}}_2}| \) is smaller than torque \( |{T_1}| \) because the opposite directional torque, \( |{T_2}_B| \), is larger than zero.

To drive each microrobot selectively, we propose two methods for rotating an external magnetic field: a continuous rotation of the field (Fig. 5) and a repeated rotation that takes place in 90° steps (Fig. 6). Figure 5 shows that the two heads rotate 180° in correspondence with a 180° rotation of the external magnetic field. This rotational method was used to actuate both types of microrobotic heads. Figure 6 shows the rotation angle of the microrobotic heads during repeated rotations in 90° steps. With the first 90° rotation of the field, both heads rotate 90°. The bar-shaped head, however, rotates back to its initial orientation when the rotational angle is set back to 0° to initiate a second rotation. Meanwhile, the cross-shaped head maintains its orientation because the magnetized axis changes from axis A to axis B. In this way, the net rotational angle of the bar-shaped head is 0°, but the cross-shaped head actually rotates, thus giving us selective control of its movement. For a 180° rotation of the bar-shaped head, a continuous magnetic field rotation is used. Then, two 90° rotations in the reverse direction are used to return the head to its initial orientation. It therefore takes twice as long to rotate the bar-shaped head than to rotate the cross-shaped head. Table 1 illustrates examples of the net rotational angle of the microrobotic heads when these two methods are applied. These examples demonstrate that selective control of microrobotic heads is feasible.

Fig. 5
figure 5

Motions of the bar- and cross-shaped heads within the continuous rotation of the external magnetic field. Both the heads rotate 180° in accordance with the magnetic field rotation

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Fig. 6
figure 6

Motions of the bar- and cross-shaped heads within a repeated 90° rotation. Only the cross-shaped head rotates in accordance with the magnetic field rotation. The net rotational angle of the bar-shaped head is 0°

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Table 1 Net rotational angles of the microrobotic heads
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3 Experiment

In order to validate the developed control methods, scaled-up models were fabricated for both head types. An experimental setup was also developed to measure the rotational frequency of the models.

The orthogonal three-axis coil setup is shown in Fig. 7. The videos were taken from the window of the coil at the side. The experiments were performed in silicone oil with a viscosity of 100 cSt (KF-96-1-100, Shin-Etsu Silicones, Japan) (Fig. 8). The container holding the silicone oil was 42 mm in diameter and 81 mm deep. The container was placed at the center of the coil setup, which consisted of three pairs of coils; each pair generated a uniform magnetic field of up to 3.5 mT at around the center. The strength and direction of the magnetic field were tuned by controlling the current in the coils. The current values were controlled using a PC with a digital analog converter (PCI-3343A, Interface, Japan) and amplifiers (4-Q-DC Servo Control LSC 30/2, Maxon Motor, Switzerland). Function generator software (BPC-0600, Interface, Japan) was used to generate periodic signals. The scaled-up prototype is approximately 15 mm long and 5 mm in diameter (Fig. 9). The tail was fabricated using rapid prototyping (VisiJet®EX2000, HD3000, 3D Systems). Cylindrically shaped Styrofoam (diameter, 5 mm; height, 3 mm) was glued to the top of the tail of the microrobot models in order to avoid rapid sinking or rising in the silicone oil. The nickel wires used in the heads of the models were 0.60 mm in diameter and 5 mm in length (purity 99 + %, Nilaco, Japan). The bar-shaped head was made of a single nickel wire, whereas the cross-shaped head was made of two orthogonally placed nickel wires.

Fig. 7
figure 7

Electromagnetic coils and a container filled with silicone oil

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Fig. 8
figure 8

Scaled-up prototypes swimming in silicone oil

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Fig. 9
figure 9

Scaled-up prototype of a microrobot. a Nickel head (0.6 mm in diameter and 5 mm in length), b Styrofoam, and c a microrobotic tail made using rapid prototyping

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4 Results and discussion

The rotational frequencies of the scaled-up models in silicone oil were measured in continuous and repeated 90° rotations of the magnetic field. We define 360° of rotation as a complete rotation, which corresponds to the frequency 1 Hz. A frequency of 1 Hz thus indicates 360° rotation of the continuous rotational magnetic field or four repeated rotations in 90° steps.

Figure 10 shows the experimental results for the bar- and cross-shaped heads with continuous rotation of the external magnetic field. The bar-shaped and cross-shaped heads were synchronized at frequencies of up to 0.9 and 0.3 Hz, respectively. Figure 11 shows the results for the repeated rotations in 90° steps. Only the cross-shaped head rotated as intended, with a step-out frequency of 0.25 Hz. In the experiment, the rotation steps were set to slightly greater than 90° to ensure that the magnetization of each long axis of the cross-shaped head was induced in alternation.

Fig. 10
figure 10

Rotational frequencies of the prototypes during the continuous rotation of the external magnetic field

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Fig. 11
figure 11

Rotational frequencies of the prototypes during the repeated rotations of the external magnetic field with 90° steps

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The step-out frequency of the cross-shaped head was lower than that of the bar-shaped head when the continuous rotational magnetic field was used. As described in Eq. 13, the driving torque induced on the cross-shaped head was smaller than that on the bar-shaped head. This can be explained as follows. The cross-shaped head has two long axes. One long axis is highly magnetized, while the other long axis (which is perpendicular to the first long axis) is weakly magnetized. The torque induced on the weakly magnetized axis decreases the total magnitude of the induced torque. The torque induced on the bar-shaped head reaches a maximum when the angle between its long axis and the magnetic field is 45°, as shown in Eq. 10. On the other hand, the torque induced on the cross-shaped head is nullified when the angle between its long axis and the magnetic field is 45°, due to the symmetry of the head, as shown in Eq. 12. If the angle exceeds 45°, the cross-shaped head begins to rotate in the opposite direction, because one of the long axes, which shares a smaller angle with the direction of the magnetic field, tends to be aligned in that direction. As shown in Fig. 10, the step-out frequency of the bar-shaped head model was approximately three times greater than that of the cross-shaped model under a continuous rotational magnetic field.

Theoretically, the step-out frequencies of the cross-shaped head model in the two applied magnetic fields should be equal; however, the experimental results differed slightly. The step-out frequency was 0.25 Hz in the magnetic field with 90° rotation steps, as shown in Fig. 11, a value that is slightly lower than that in the continuous rotational magnetic field, as shown in Fig. 10. There are some possible reasons for this. First, the steps used were not exactly 90°, but were slightly larger, so the angular velocity was also slightly larger than that of the continuous magnetic field. Thus, there was an increase in the drag torque, and as a result, a decrease in the step-out frequency. Second, because the fluid flow surrounding the microrobots was not a complete Stokes flow, the effect of inertia, which was previously neglected, still remained. The cross-shaped head did not rotate in a completely continuous manner under a magnetic field with 90° steps because of the slight time gap between each 90° rotation. Based on the assumption that the inertia effect still remained, the torque required to rotate the initially stopped object was larger than the torque required to rotate the rotating object. Thus, the step-out frequency of the cross-shaped head was lower for repeated rotation of the external magnetic field (0.25 Hz, in Fig. 11) than it was for continuous rotation (0.3 Hz, in Fig. 10).

We demonstrated microrobotic swimming in silicone oil with low frequency, but the same technique is applicable for microrobotic actuation in water with high frequency. In the experiments, silicone oil with a viscosity of 100 cSt was used for scaling. The torque loaded by the surrounding fluid, Tdrag, is described as

$$ \mathop{T}\nolimits_{{drag}} \propto \mu \mathop{L}\nolimits^3 \omega, $$
(14)

where μ is the viscosity of the fluid and L and ω are the characteristic length and angular velocity of the microrobot, respectively. As can be seen in Eq. 3, the torque induced by the external magnetic field, T m , is proportional to the volume of the ferromagnetic body. At smaller dimensions, both T drag and T m decrease at the same rate because both the torques are proportional to the third power of the length. At a lower viscosity, the step-out frequency increases. The viscosity of the silicone oil was 100 cSt, which is 100 times larger than that of water. Therefore, theoretically, the step-out frequency in water would be 100 times larger than that in the silicone oil.

In theory, this selective control method is applicable to a three-dimensional environment because one of the robots remains in the initial position while the other one is moving. This principle is not limited to just one- or two-dimensional control. The problem that may arise, however, is that the orientation of the robot may be changed by a number of factors, such as gravity, obstacles (e.g., walls), or the external flow in a three-dimensional environment. Moreover, the control of more than two robots may become possible by adding more and different designs and steps of rotation. For example, the four bars crossing with 45° steps might be able to be selectively actuated by a 45°-step rotational field. In this case, however, the total torque would become less than that of the two bars crossing with 90° steps because the shape becomes close to the circular disc upon which magnetic torque cannot be induced along the central axis. These factors pose a challenge that will be investigated in a future study.

5 Conclusion

In this paper, novel selective control methods for magnetic microrobots were proposed, and scaled-up prototypes were fabricated and tested in silicone oil. The experimental results are in agreement with those of a theoretical analysis, and selective control was demonstrated using the prototypes. The prototype with a bar-shaped head was synchronized at a higher frequency compared to the prototype with a cross-shaped head, just as the theoretical model predicted. The step-out frequencies of the models were very low owing to the high viscosity of silicone oil. The developed methods are applicable for microrobotic actuation in high-frequency rotational magnetic fields when the fluid used has lower viscosity (e.g., water). Nickel was used as the microrobotic head material, thus allowing easy batch fabrication of the microrobots. Our future work in this area includes developing accurate models to simulate nonlinearity of magnetic susceptibility and to fabricate microrobots at the microscale.