3.1. Crawing mechanism analysis

According to the above model and the force analysis of the magnetic film under the magnetic field, the crawling mechanism of the robot is obtained. When there is no magnetic field driving, the magnetic film is not stressed, the two legs are used as support, and the soft robot is stationary. During the first half of the cycle, when the applied magnetic field is upward, the head is pressed down, the belly arches, the front legs are anchored, and the entire body shrinks into an open downward arc, which causes the back legs to slide one step forward. In the second half cycle, the magnetic field is directed downward, the head is raised, the belly is pressed down, the hind legs are imbibed, the robot is extended in an open upward arc, and the front legs slide forward. So the whole robot is crawling forward, cycle by cycle. Of course, when the initial magnetic field is applied downward, the structure stretches and then contracts and can also move forward. Figure 4 shows the magnetic field changing over time and the corresponding motion of the soft robot in one motion period.

Figure 4.

Time-dependent magnetic field and corresponding motion of the soft robot. The magnetic field is sequentially applied on the soft robot in up-and-down manner.

Figure 5.

Schematic diagram of the forces acting on the soft robot.

In order to explain how the deformation of the robot’s body affects the robot’s motion, the worm-like kinematics is studied. Assume that the leg as the fixed end does not slip, the crawling process is shown in Fig. 5. Because the soft-footed worm robot can move in one dimension, it does not need to consider the direction of movement. Taking the forward direction of the soft robot as the positive direction, a global coordinate system is established to represent the position of the two legs of the robot (leg thickness is ignored). The positions of the front and rear legs with respect to the global coordinate system are, respectively, ${G_{xf}}$ and ${G_{xr}}$ , and the length between the legs ${L_x} ={G_{xf}} -{G_{xr}}$ , then $\dfrac{{d{L_x}}}{{dt}} = \dfrac{{d{G_{xf}}}}{{dt}} - \dfrac{{d{G_{xr}}}}{{dt}}$ . $\dfrac{{d{L_x}}}{{dt}}$ represents the two motion stages of a soft robot. If it is less than zero, the front legs are fixed, the back legs are brought forward, and the body is contracted. If greater than zero, the back legs are fixed, the front legs are brought forward, and the body is extended. According to the assumption, there is

(6) \begin{equation} \dfrac{{d{L_x}}}{{dt}}= \left \{{\begin{array}{*{20}{c}}{ - \dfrac{{d{G_{xr}}}}{{dt}},\dfrac{{d{L_x}}}{{dt}} \lt 0}\\[10pt]{\dfrac{{d{G_{xf}}}}{{dt}},\dfrac{{d{L_x}}}{{dt}} \gt 0} \end{array}} \right. \end{equation}

Through integral calculation with the time product, the total displacement of the front foot and the back foot in different periods relative to the global coordinate system can be written as

(7) \begin{equation}{G_{sr}} = \left \{{\begin{array}{*{20}{c}}{\sum \limits _{i = 0}^n{\int _0^t{ - \dfrac{{dGxr}}{{dt}}} } d\tau,T \times n \le \dfrac{T}{2} \times (2n + 1)}\\[10pt]{0,T \times (2n + 1) \le t \le T \times (2n + 2)} \end{array}} \right. \end{equation}

(8) \begin{equation}{G_{sf}} = \left \{{\begin{array}{*{20}{c}}{0,T \times n \le t\dfrac{T}{2} \times (2n + 1)}\\[10pt]{\sum \limits _{i = 0}^n{\int _0^t{ - \dfrac{{dGxr}}{{dt}}} } d\tau,T \times (2n + 1) \le t \le T \times (2n + 2)} \end{array}} \right. \end{equation}

Equation (7) represents the sum of motion displacement values of the first half cycle of the hind leg in different cycles, and Eq. (8) represents the sum of motion displacement values of the second half cycle of the front leg in different cycles. The displacement value of the software intelligent structure in several cycles can be expressed by ${G_{sf}}$ or ${G_{sr}}$ , since the robot’s motion is holistic, then ${G_{sr}} \approx{G_{sf}}$ . Therefore, the average velocity of the software intelligent structure can be defined as

(9) \begin{equation}{G_{\bar vt}} = \frac{1}{{2t}}\int _0^t{\left |{\frac{{d{L_x}}}{{dt}}} \right |} d\tau \end{equation}

According to the above analysis, if the change of $L_x$ is given, the motion speed of the robot can be determined. Therefore, the motion model of the soft crawling robot is established based on its motion attitude.

3.2. Motion model based on static beam bending analysis

The posture of the soft crawling robot is similar to an arch bridge, so it is modeled as an arc. The modeling analysis of the soft crawling robot is shown in Fig. 6. Assume that the torso of the soft robot is bent into a regular arc. During the movement, the fixed support end determined by the force analysis does not slip and move backward. Under these assumptions, the forward distance of the soft robot in a period is calculated, and then the speed of the soft robot is calculated.

Figure 6.

Modeling analysis of soft crawling robot.

When the soft crawling robot is stationary, the distance between the legs is $L$ . When the upward magnetic field is generated, the trunk of the soft crawling robot arches upward. The angle between the front and rear legs and the vertical direction is $\theta _1$ and $\theta _2$ , respectively. The legs extend to form a fan with the trunk, with a radius of $R_1$ and an angle of $\theta$ . The distance $x_1$ between the front and rear legs on the contact surface is shown in Eq. (10).

(10) \begin{equation} x_1=\left(\frac{180\times l}{\theta \times \pi }-H\right)\times (\sin\theta _1+\sin\theta _2) \end{equation}

where $H$ represents the height of soft robot’s leg; $L_1$ represents the distance between the legs of the trunk when the structure arches. $x_1$ represents the distance between the legs of the contact surface when the structure is arched.

When the magnetic field direction is reversed, soft crawling robot from arching state to downward concave. The angle between the front and rear legs and the vertical direction is $\varphi _2$ and $\varphi _1$ , respectively. The legs extend to form a fan with the trunk, with a radius of $R_2$ and an angle of $\varphi$ . The distance $x_2$ between the front and rear legs on the contact surface is shown in Eq. (11).

(11) \begin{equation} x_2=\left(\frac{180\times l}{\varphi \times \pi }\right)\times (\sin\varphi _1+\sin\varphi _2) \end{equation}

where $L_2$ represents the distance between the legs of the trunk when the structure concaves. $x_2$ represents the distance between the legs of the contact surface when the structure is concaved.

The process from (a) to (d) in Fig. 5 is exactly a cycle. Based on the front leg, the forward distance of the front leg is the displacement of the soft crawling robot in one cycle. More specifically, $s$ can be expressed as

(12) \begin{equation} s=x_2-x_1 \end{equation}

The motion time of a period from (a) to (d) is also the change time of an alternating magnetic field, and the alternating frequency of the magnetic field is $f$ . Assuming that the displacement of the soft-climbing robot in each cycle is equal, the moving speed $v$ of the soft robot can be approximated as the product of the moving distance $s$ of each cycle and the $f$ frequency of magnetic field change:

(13) \begin{equation} v=s\times f \end{equation}

According to the formula, when the displacement of each step is determined, the faster the magnetic field change frequency, the greater the motion speed of the software intelligent structure.

In order to determine the bending angle of the dynamic model, the bending condition of the body was analyzed mechanically. The whole soft crawling robot was regarded as a beam structure, and the magnetic field force received by the magnetic film in the uniform magnetic field was regarded as the uniform distribution load. When the bending angle generated by the uniform load on the head is calculated, the robot is regarded as an overstretched simply supported beam structure, as shown in Fig. 7(a); when the bending angle generated by the uniform load on the abdomen is calculated, the robot is regarded as a simply supported beam structure, as shown in Fig. 7(b). The sum of angles produced by the force of the head and abdomen is the change in the angle of the concave or arched legs of the intelligent structure.

Figure 7.

Simply supported beam structure.

From the angle formula of the extended beam, the angles of A and B in Fig. 7(a) are

(14) \begin{equation} \theta _{A1}=\frac{qm^2l}{6EI},\,\,\theta _{b1}=\frac{qm^3l}{12EI} \end{equation}

where $l$ represents the distance between legs; $m$ represents the Length of head magnetic film; $E$ represents the Young’s modulus of trunk material; $I$ represents the moment of inertia of the trunk; and $q$ represents the load force on magnetic film. The length and width of the trunk are, respectively, $b$ and $h$ . The formula of the moment of inertia is

(15) \begin{equation} I=\frac{bh_3}{12} \end{equation}

where $b$ represents the width of the trunk; $h$ represents the thickness of trunk.

(16) \begin{equation} q=\frac{F_{\text{total}}}{m\times b} \end{equation}

where $F_{\text{total}}$ represents total force on the magnetic film; $b$ represents the width of magnetic film.

From the angle formula of the simply supported beam structure, the angles of A and B in Fig. 7(b) are

(17) \begin{align} \begin{aligned} \theta _{A2}=\frac{ql_3}{6EIl}\left(l_2+\frac{l_3}{2}\right)\times \left[\left(l_1+\frac{l_3}{2}\right)\times \left(l+l_2+\frac{l_3}{2}\right)-\frac{1}{4}l_3^2\right] \\[5pt] \theta _{B2}=\frac{ql_3}{6EIl}\left(l_1+\frac{l_3}{2}\right)\times \left[\left(l_2+\frac{l_3}{2}\right)\times \left(l+l_1+\frac{l_3}{2}\right)-\frac{1}{4}l_3^2\right] \end{aligned} \end{align}

where $\theta _{A2}$ represents the angle change of fixed hinge support end; $\theta _{B2}$ represents the angle change of roller bearing end; $l_1$ represents the displacement from uniform load boundary to fixed hinge support; $l_2$ represents the transverse size of the abdominal magnetic film; $l_3$ represents the displacement from uniform load boundary to roller bearing; and $l$ represents the distance between legs, $l=l_1+l_2+l_3$ .

When the body arches upward, the front leg is fixed and the rear leg can move. At this time, the bending angle of the front and rear legs is

(18) \begin{equation} \begin{aligned} \theta _2=\theta _{A1}+\theta _{A2} \\ \theta _1=\theta _{B1}+\theta _{B2} \end{aligned} \end{equation}

When the body concaves upward, the rear leg is fixed and the front leg can move. And the bending angle of the front leg is the angle of the roller shaft support. However, the positions of the two support ends have been interchanged, and the positions of $l_1$ and $l_3$ have also been reversed, so at this time the angles of A and B are changed to:

(19) \begin{equation} \begin{aligned} \theta _{A2}=\frac{ql_1}{6EIl}\left(l_3+\frac{l_1}{2}\right)\times \left[\left(l_2+\frac{l_1}{2}\right)\times \left(l+l_3+\frac{l_1}{2}\right)-\frac{1}{4}l_1^2\right] \\[5pt] \theta _{B2}=\frac{ql_1}{6EIl}\left(l_2+\frac{l_1}{2}\right)\times \left[\left(l_3+\frac{l_1}{2}\right)\times \left(l+l_2+\frac{l_1}{2}\right)-\frac{1}{4}l_1^2\right] \end{aligned} \end{equation}

where $\theta _{A2}$ represents the angle change of roller bearing end. $\theta _{B2}$ represents the angle change of the fixed hinge support end.

The bending angles of the front and rear legs are

(20) \begin{equation} \begin{aligned} \varphi _2=\theta _{A1}+\theta _{A2} \\ \varphi _1=\theta _{B1}+\theta _{B2} \end{aligned} \end{equation}

The simulation results are calculated by MATLAB to determine the relationship between structural size and periodic motion displacement. It can be seen from the above dynamic model of the soft robot that the size of $s$ is determined by many variables, so this paper uses a single variable to calculate.

From the formula, when the other parameters except $H$ are determined, the longer the leg, the greater the displacement of each step. Therefore, the influence of leg length on motion speed can be determined as positive correlation. Except for the leg length, only the parameters $l_1$ and $l$ are not fixed, and the parameters of these fixed values: $m=8.5$ mm, $E=115$ MPa, $b=10$ mm, $h=0.4$ mm, $H=5$ mm, $l_2=10.5$ mm, and $q=5\,\mathrm{N/m^2}$ .

When $l=21$ mm, MATLAB is used to simulate the relationship between the distance $l_1$ from the abdominal magnetic film to the front leg and the displacement $s$ of each step. The results are shown in Fig. 8. The figure shows that with the increase of the distance from the abdominal magnetic film to the front leg, the motion displacement of each step decreases gradually. In fact, the distance between the two magnetic films becomes smaller with the gradual reduction. But the force of the structure will be too concentrated to produce relative sliding.

Figure 8.

The relationship between the distance $l_1$ between the two magnetic films and the displacement $s$ of each step.

When $l_1=6.5$ mm, MATLAB is used to simulate the relationship between the distance $l$ from the abdominal magnetic film to the front leg and the displacement $s$ of each step. The results are shown in Fig. 9. The figure shows that with the larger the distance between the legs, the greater the displacement of each step. However, the distance between the legs cannot be infinitely increased proceed from actual conditions; otherwise, it will lead to the disappearance of structural balance.

Figure 9.

The relationship between the distance $l$ between the two legs and the displacement $s$ of each step.

After simulation, it is concluded that with the increase of the distance from the abdominal magnetic film to the front leg, the movement displacement of each step gradually decreases. The greater the distance between the legs, the greater the movement tendency of each step. Therefore, in practical design, it provides theoretical support for determining the size of each part of the robot. The simulation results of the motion model provide theoretical support for the structural design of the robot. Although the factors of the motion model are not comprehensive, some states of the model are ideal, and there are certain differences with the reality. But it still provides a reference for robot optimization.

4.1. Validation of the model

The actual size of the robot is brought into the dynamic model and compared with the data obtained from the experimental study to verify the correctness of the model. The research object is a soft crawling robot with 4.24 mm leg length and 23 mm spacing. The driving parameters of the magnetic field are 24 V voltage and 10 Hz frequency. Randomly selected from the same motion process of four motion cycles, four periodic structure of the motion posture as shown in Fig. 10, by measuring the movement distance between the legs as shown in Table I.

In Table I, $x_2-x_1$ is the theoretical value derived by the formula, and $s$ is the actual value measured. It can be seen from Table I that the model-predicted value of the distance between the two legs is very close to the experiment result, which proves the correctness of the theoretical modeling. According to Experiment result 4 in Table I, the actual motion distance is far less than the theoretical value. It can be seen from Fig. 10 that the front leg does not remain fixed in the process of arching but slips backward. Especially, the distance of the experiment result 4 slips backward is the largest, about 3 mm, which leads to a small actual step.

From the above analysis, the model is basically consistent with the actual situation. However, the error caused by the slip of the leg makes some theoretical values differ greatly from the actual values. which affects the actual speed of the soft robot.

4.2. Experiment of movement speed

To explore the motion law of soft robot with different structure sizes. Five soft robots of different sizes were selected for velocity measurement experiments under different magnetic fields. Five kinds of size soft robots, as shown in Table II, measure the magnetic field velocity of five kinds of soft robots under different intensity and different frequency.

Table I.

The experiment result and theoretical value of the distance between the two legs of inchworm.

Table II.

Structural dimensions of five soft robots.

Figure 10.

Displacement diagram of robot at different time.

Five types of soft robot are driven under magnetic field, 10 mT magnetic field, and the corresponding velocities under different magnetic field change frequencies are shown in Fig. 11. Figure 11(a) shows the actual curve of motion velocity, and Fig. 11(b) shows the fitting curve of motion velocity.

Figure 11.

The curve of velocity versus frequency of five soft robots with different sizes.

In Fig. 11, we know from curves 2 and 4, when the distance is fixed, the longer the leg, the faster the movement speed. It can be seen from curves 1 and 4 that when the leg length is constant, the greater the distance, the faster the motion speed. The Type 5 software intelligent structure has the longest legs and the spacing is not the largest, but its movement speed is the fastest. This shows that the distance between the two legs cannot be increased indefinitely, otherwise it will affect the speed of movement of the intelligent structure. Of course, the longer the leg is not the bigger the better, too long the leg not only increases the weight of the intelligent structure but also makes the center of gravity too high, reducing the stability of the movement.

As can be seen from Fig. 11, the effects of robot size and voltage frequency on robot crawling speed are obtained through simulation. Because the model does not consider gravity and other factors, the actual speed and the theoretical speed have a certain difference, but the overall trend is consistent, which also provides a reference for the choice of robot drive parameters and robot size. Through theoretical analysis, it is concluded that the robot with Type 5 can get the maximum motion speed under the magnetic field with frequency of 25 Hz, and in the actual experiment, the robot with size 5 can also get the maximum speed of 28.24 mm/s under the magnetic field of 25 Hz (Supplementary video 1).

4.3. Exercise effect under different external environment

The robot was placed in the pipe to crawl, mainly simulating the industrial pipe environment with small aperture. The pipe chosen in this experiment was a transparent rubber tube with an internal space diameter of 12 mm. The pipe was fixed above the moving track and the robot’s moving speed was recorded. As shown in Fig. 12(a), the robot’s maximum motion speed in the pipeline is 13.78 mm/s, which is slower than that in flat ground (supplementary video 2). The main reason is that only the edges of its legs contact the inner wall of the pipeline, and the contact surface is too small.

Figure 12.

Exhibition of robot crawling.

The endoscope lens is mounted on the torso of the robot’s hind leg, with the lens facing upwards. The upper torso of the hind leg is equipped with a camera, which can be used to observe the actual situation on the upper wall of the pipeline (supplementary video 3). The observation effect is shown in Fig. 12(b). In this case, the effect of line arrangement on motion is minimal.