1 Introduction

With the growing demand for microscopic manipulation, microrobots have received extensive attention in the past few decades. Due to the advances in material science research and bionic technology, soft bionic robots have evolved siginificantly[1,2,3]. Such robots are considered to be widely applicable for cargo transportation[4], minimally invasive surgery[5, 6] and heavy metal removal[7], making them a popular topic among researchers.

Due to their small size, microrobots cannot accommodate traditional drive modules. Therefore, cableless drives for microrobots have become an important focus for researches in this regard. Currently available researches have proposed the use of the chemical fuel [8, 9], electric field [10, 11], light field [12,13,14], sound field [15, 16], and magnetic field [17,18,19,20] for driving the motion of microrobots. Among them, the magnetic field has been investigated most extensively due to its advantages of high controllability, zero pollution, remote wireless operability, and harmlessness to human body.

Zhao et al. prepared three forms of helical microrobots based on thermoplastic SMP (polylactic acid) materials, and the microrobots have the ability to deform and self-repair after being treated with steel sheets and pre-deformation[21]. Sukho Park and his team members used two-axis Helmholtz coils to generate an alternating magnetic field to drive a tadpole-shaped microrobot tail fin to swing and achieve forward motion[22]. Others in the same research group replaced the two-axis Helmholtz coils with three-axis ones to realize three-dimensional (3D) manipulation of the tadpole-shaped microrobot [23]. Fu et al. created a helical microrobot with a protective cover that can protect organs from damage caused by the helical structure when applied in vivo [24]. When 95 mT was provided by the coil, the microrobot could rotate at 120 rad/s. Pan et al. used a new magnetic field-assisted projection stereolithography (M-PSL) process to fabricate inchworm-shaped microrobots by 3D printing, and drive the microrobot to move by alternately controlling the external magnet [25]. Later, members of the same research group added a gripper structure into the anterior leg of the original microrobot to realize targeted drug delivery. A spiked footpad is designed at the bottom of the microrobot, which improves the adhesion force approximately 30% compared with the microrobot without spiked structure [26]. Ju et al. used the self-made double coil system to realize the local directional magnetization of the microrobot, and used the permanent magnet to control the motion of the inchworm-shaped microrobot, with the motion speed reaching about 22 mm/s [27]. Also for the magnetic structure programming, Qi et al. prepared the original directional magnetic structure in advance, and then encapsulated the original in a certain order in a flexible material to prepare an inchworm-shaped microrobot, which was bent and deformed under the action of a magnetic field [28]. To realize the perception of the microrobot's own motion, Karipoth et al. combined the strain sensor and the soft robot, and realized the feedback control of the soft robot by measuring the strain value during the movement [29]. In addition to controlling the motion with a single magnetic field, Li et al. prepared a soft robot with electrothermal and magnetic responses, and completed a cycle of crawling motion of the soft robot by alternating the two corresponding tasks [30].

In this study, we used NdFeB magnets and polydimethylsiloxane (PDMS) materials to fabricate an inchworm-shaped soft robot, and achieved cableless actuation with Helmholtz coils. The motion characteristics of the soft robot were tested through experiments. Factors affecting the movement speed, including magnet spacing, shape of the microrobot, climbing slope, magnetic field strength, and magnetic field frequency, were examined. The experimental results demonstrated the high flexibility and extensive applicability of the proposed microrobot.

2 Magnetic Drive Systems and Soft Robot Fabrication

2.1 Magnetic Field Construction and Simulation

As shown in Fig. 1a, a 3D Helmholtz coil system was used to control the motion of the soft robot. The structural parameters and discharge parameters of the 3D Helmholtz coil are shown in Table 1. The required signals were generated through a signal generator and amplified by three power amplifiers, so as to change the strength and direction of the magnetic field in the coil and generate a uniform magnetic field.

Fig. 1
figure 1

a Schematic diagram of the magnetically driven soft robot. b Schematic diagram of the soft robot’s motion. c Steps in the soft robot’s motion cycle

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Table 1 The structural parameters and discharge parameters of the 3D Helmholtz coil
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As shown in Fig. 1b, under the action of the magnetic field, the soft robot can realize flexible deformation and forward motion. A motion cycle of the soft robot is shown in Fig. 1c. (1)–(6) are the six steps of the motion cycle, where the magnetic field is used to control the motion posture of the soft robot without cable. As shown in Fig. 2a–e, a 3D Helmholtz coil model was established in COMSOL Multiphysics software, and a rotating magnetic field was set in the X–Y plane. The rotation frequency was set to 1 Hz, and the magnetic field rotated once in a period. As shown in Fig. 2f, the magnetic field can be considered to be uniform in the octagonal region.

Fig. 2
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Finite element analysis of magnetic field. a–e Simulation of rotating magnetic field. f Simulation of uniform magnetic field

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2.2 Soft Robot Fabrication

The soft robot was made of NdFeB magnets and PDMS (Ausbond-194). First, a 3D printer was used to prepare the mold. Then, the cylindrical magnets (3 mm in diameter, 1 mm in height) were placed on both sides of the mold groove in opposite magnetization directions, and their position was fixed by the attraction of the NdFeB magnets attached onto the back of the mold. Next, the PDMS and PDMS prepolymer solutions were mixed at a mass ratio of 10:1. The mixture was let sit for 10 min to eliminate air bubbles, and then poured into the groove of the mold. The soft robot was taken out from the groove. The shape and size parameters are shown in Table 2.

Table 2 Shape and size parameters of the soft robot
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3 Force Analysis of the Soft Robot

When a magnetic field is applied to the soft robot, the NdFeB magnet is subjected to the action of the magnetic moment. Under the action of a uniform magnetic field, as shown in Fig. 3a, the front and rear parts of the soft robot bend due to their opposite magnetization directions. The magnetic moment τ received by the microrobot can be obtained as follows:

$$ {\varvec{\tau}} = {\varvec{M}} \times {\varvec{B}}, $$
(1)

where B is the magnetic flux density and M is the magnetic moment of the NdFeB magnet. As shown in Fig. 3b, the torques generated by the two magnets under the action of the magnetic field are τ1 and τ2.

Under the action of magnetic field, the soft robot has elastic deformation. With the increase of magnetic field strength, the bending degree of the soft robot increases. In the movement process of a cycle of the soft robot, with the increase of the degree of bending, the distance of single step movement is larger. In the case of a certain frequency, the larger the magnetic field strength is, the faster the movement speed of the soft robot will be, as will be explained in Sect. 4.1.

Fig. 3
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a Deformation of the soft robot. b Force analysis of the soft robot. c The relationship between the strength of the magnetic field and the current

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By analyzing the force on the soft robot, the torque balance for point A can be formulated as follows:

$$ \left( {2{\text{G}}L_{1} + \frac{3}{2}{\text{G}}L_{2} - L_{1} N_{2} - L_{2} N_{2} } \right){\text{cos}}\beta_{1} + \left( {\frac{1}{2}{\text{G}}L_{2} - L_{1} N_{2} } \right){\text{cos}}\beta_{2} + \left( {{\text{G}} - N_{2} } \right)P - {\varvec{\tau}}_{1} + {\varvec{\tau}}_{1} = 0, $$
(2)

where N1 and N2 are the supporting forces received at points A and B, respectively; G is the gravity of the magnet; L1 is the distance between the magnet and the front end; L2 is the diameter of the magnet; β1 is the clip between the front of the soft robot and the ground angle; β2 is the angle between the back of the soft robot and the ground; P is the distance between the two magnets; h is the lifting height of the soft robot. By changing the strength and direction of the magnetic field, the attitude of the soft robot can be controlled. The magnetic field generated by the coil is measured by a Tesla meter. The relationship between the magnetic field generated by each axis of the three-axis Helmholtz coil and the current is shown in Fig. 3c. The magnetic field increases linearly with the increase of the current.

We input full-wave signals in the Y-axis direction and sawtooth wave signals in the Z-axis, so that they have the same period and the magnetic field in the y–z plane direction can be changed. τ1 and τ2 vary with the direction of the magnetic field. According to formula (1), under the same magnetic field, when the angle between the magnetic field direction and the magnet polarization direction is different, the magnitude of τ1 and τ2 are different. We call point A the head of the soft robot and point B the tail of the soft robot. Under the magnetic torque, the soft robot bends. The tail overcomes the maximum static friction force at point B, while the head is still in static friction at point A, the head is stuck in place, and the tail moves to the left, as shown in (1)–(3) of Fig. 1c. Similarly, when the head overcomes the maximum static friction force, but the tail does not reach the maximum static friction force, the tail is stuck and the head moves forward, as shown in (4)–(6) of Fig. 1c. Under the combined action of magnetic torque, friction force and elastic force, the soft robot can move forward in the mode shown in Fig. 1c. The speed of the soft robot can be controlled by changing the strength and direction of the magnetic field.

4 Experimental Analysis of the Soft Robot’s Motion Characteristic

4.1 Effects of Magnet Spacings on Speed

To explore the effects of different magnet spacings on the movement speed of the soft robot, we fabricated soft robots with magnet spacings of 25 mm, 20 mm, and 15 mm, respectively, as shown in Fig. 4a. According to the motion state of the soft robot, its motion step length, denoted by s, can be calculated as follows:

$$ s = L - \left( {L_{1} + L_{2} } \right)\left( {{\text{cos}}\beta_{{1{\text{max}}}} + {\text{cos}}\beta_{{2{\text{max}}}} } \right) - P, $$
(3)

where cosβ1max and cosβ2max are the maximum values of the angle between the front and rear ends of the soft robot and the ground during movement. The magnetic induction intensity was set to 120 Gs, and its movement speed was tested by changing the frequency of the magnetic field (3 Hz, 3.5 Hz, 4 Hz, 4.5 Hz, and 5 Hz). Figure 4b shows screenshots of the motion (4 Hz) of the soft robots with a pitch of 25 mm and 15 mm within 0–3 s, respectively. The locomotion experiment videos can be found in Supplementary Video S1. The experimental data of each group are summarized in Fig. 4c. It can be seen that, under a certain frequency, the movement speed of the soft robot increases as the magnet spacing increases. When the magnet spacing remains unchanged, the movement speed increases as the frequency increases. By inference, it can be known that the magnet spacing affects the soft robot’s movement speed primarily by affecting its deformation degree and then its step length. Specifically, as the magnet spacing increases, the middle part of the soft robot is more likely to deform, resulting in increased values of β1max and β2max, increased step length, and finally increased movement speed.

Fig. 4
figure 4

a Soft robots with different magnet spacings: 25 mm, 20 mm, and 15 mm, from left to right. b Images of soft robots with magnetic spacings of 25 mm and 15 mm driven by the same magnetic field. c Movement speed of strip-shaped soft robots with different magnet spacings (25 mm, 20 mm, and 15 mm)

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4.2 Comparing Movement Speeds of Strip- and Dumbbell-Shaped Soft Robots

In addition to strip-shaped soft robots, we also fabricated dumbbell-shaped ones using the same method, as shown in Fig. 5a. With the magnet spacing set to 20 mm and the magnetic induction intensity to 120 Gs, the soft robot’s movement speed was tested by changing the frequency of the magnetic field (3 Hz, 3.5 Hz, 4 Hz, 4.5 Hz, and 5 Hz). The snapshots in Fig. 5b show the motion of the strip- and dumbbell-shaped soft robots with a magnet spacing of 20 mm within 0–3 s (4 Hz). The locomotion experiment videos can be found in Supplementary Video S2. The experimental data are illustrated in Fig. 5c. It can be seen that the dumbbell-shaped soft robot moves at a slightly faster speed than the strip-shaped. However, as the frequency increases, it experiences unstable movement and starts to move slower than the strip-shaped when the frequency exceeds 5 Hz. Due to the shrinkage of the material in the middle part of the soft robot, the dumbbell-shaped soft robot can produce greater deformation than the strip-shaped under the action of the magnetic field, which gives it faster movement speeds than the latter. This lack of material, however, causes the dumbbell-shaped soft robot to move unstably as the frequency goes up. Due to the worse movement stability, the dumbbell-shaped soft robot starts to exhibit a slower movement speed than the strip-shaped when the frequency increases to a certain point. Through the motion performance experiments of the strip- and dumbbell-shaped soft robots, the dumbbell-shaped soft robot can be prepared to reduce the use of materials, and can obtain greater motion speed, which is conducive to explore new structures to improve the motion ability of the soft robots, and lay a foundation for the subsequent design of new structures.

Fig. 5
figure 5

a Image of strip- and dumbbell-shaped soft robots. b Images of strip- and dumbbell-shaped soft robots driven by the same magnetic field. c The relationship between movement speed and frequency of the soft robots

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4.3 Effects of Different Slopes on the Soft Robot’s Climbing Speed

We also explored the effects of different slopes on the climbing speed of the strip-shaped soft robot. The soft robot with L = 41 mm was selected for testing. The magnetic induction intensity was set to 160 Gs and the frequency to 4 Hz. We tested the movement speed of the soft robot by changing the slope. The locomotion experiment videos can be found in Supplementary Video S3. Screenshots of these locomotion videos are shown in Fig. 6. As shown in Fig. 6a–d, the climbing speed of the soft robot was tested at slopes 0°, 5°, 10° and 15° within 0–6 s. The experimental data are illustrated in Fig. 6e. It can be seen that under the same magnetic field, the climbing speed of the soft robot decreases as the slope increases. It reaches 4.875 mm/s at a slope of 15°, about half that at 0°.

Fig. 6
figure 6

Climbing testing at different slopes. a Movement at a slope of 0°. b Movement at a slope of 5°. c Movement at a slope of 10°. d Movement at a slope of 15°. e Climbing speeds at different slopes (0°, 5°, 10°, and 15°)

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4.4 Soft Robot’s Motion Performance

We also conducted a series of experiments to examine the motion performance of the soft robot. The locomotion experiment videos can be found in Supplementary Video S4. Screenshots of these locomotion videos are shown in Fig. 7. One of the applications soft robots is to transport objects. To examine the soft robot’s load carrying capacity, we added a load of 0.3 g, which is equivalent to 0.5 times its own weight, as shown in Fig. 7a. The magnetic induction intensity was set to 120 Gs and the frequency to 5 Hz. Under the action of the magnetic field, the soft robot pulled the load to move forward at a speed of about 4.2 mm/s. In addition, to explore the multi-motion mode of the soft robot, sine and cosine signals were input in the X- and Y-axis coils, the magnetic induction intensity was set to 120 Gs, the frequency to 0.5 Hz, and the X–Y rotating magnetic field was applied to the soft robot. Under the action of the magnetic field, the soft robot can rotate around the Z-axis, while realizing its own periodic motion. As shown in Fig. 7b, a motion cycle involves six steps, and the soft robot can achieve rotational motion after the superposition of the periodic motion. In addition to the forward motion, we also experimented on the steering motion. As shown in Fig. 7c, the positive full-wave signal was input in the X-axis and Y-axis, and then the sawtooth wave signal in the Z-axis, and Y-axis current was reduced and the X-axis one increased as the soft robot kept moving. By changing the magnetic field of the X-axis and Y-axis from time to time, the soft robot realized 90° steering motion in the X–Y plane. In addition, we also explored the ability of the soft robot to push objects. As shown in Fig. 7d, the magnetic induction intensity was set to 120 Gs and the frequency to 5 Hz. Under the action of the magnetic field, the soft robot pushed the load to move forward. We also used the soft robot to splice the two concave and convex objects, and managed to combine the two together, which laid a foundation for the subsequent object splicing process.

Fig. 7
figure 7

a The soft robot pulls the object to move forward. b Circular motion of the soft robot. c Steering motion of the soft robot. d The soft robot pushes the bump blocks together

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5 Conclusion

Bionic flexible soft robots are an important topic of research in the soft robotics field. Bionics can provide inspiring ideas for the design of soft robots. A bionic-inspired design can enable a flexible structure for the soft robot to realize multiple motions. This study provides a rapid method for setting up and fabricating an inchworm-shape soft robot. The fabrication process is simple and requires no additional manual assembly process. By establishing a force model for the soft robot, its motion can be controlled by the magnetic field generated by the Helmholtz coil. The soft robot’s movement speed is affected by different magnet spacings, shapes, step lengths and magnetic fields. By examining its motion modes and applications, we found that the soft robot is capable of pulling/pushing loads and splicing objects. The path planning revealed that in addition to the forward motion, the soft robot can also realize steering motion and rotational motion, which lays a foundation for exploring even more motion modes in the future. However, due to the fabrication accuracy problem, the soft robot still has room to improve for its motion performance, but it can provide some guidance in designing and fabricating bionic soft robots.