SpringWorm: A Soft Crawling Robot with a Large-Range Omnidirectional Deformable Rectangular Spring for Control Rod Drive Mechanism Inspection

Abstract

In this article, a cable-driven elastic backbone worm-like robot (named “SpringWorm”) of decimeter-level size is designed, which has high adaptability in crack inspection of the weld between reactor pressure vessel (RPV) and control rod drive mechanisms. The robot consists of a body that adopts a rectangular helix spring backbone driven by four cables and the flexible claws embedded with distributed electromagnets. Combining the omnidirectional deformation of the backbone and the passive deformation adsorption of the claws, the robot can achieve a variety of gaits. Based on the approaches of geometric analysis and transformation matrices of the coordinate frame, a kinematic model of the cable-driven backbone has been established. Moreover, a mechanical model considering the friction between the cable and the backbone has also been established. The top position and the bending angle of the backbone obtained by the theory, simulation, and experiment are in good agreement. In addition, the errors of the driving force between simulation and experimental results are also small. SpringWorm is 670 g, measures 206 × 65 × 75 mm, has a maximum speed of 8.9 mm/s, and has a maximum payload of 1 kg. The robot can climb over 2-cm-tall steps and 4-cm-deep ditches, and climb and turn on the vertical wall, on the pipe with a radius of 31 cm, and on the spherical surface of RPV.

Introduction

In nature, many biological systems consist mainly of soft tissues and liquids. In fact, soft tissue typically contributes more than 80% of body mass in an adult human. Soft materials make biological organisms flexible, robust, and compliant enough to adapt to their external environment.^1–4 Taking inspiration from biological systems, soft robotics combining materials science, bionics, and robotics has been emerging and growing rapidly.⁴ In contrast to robots made from rigid linkages, soft robots rely on the deformation of soft materials, rather than the displacement of joints and rigid linkages, for motion and other functions.^3,5,6 In general, soft robots can possess more degrees of freedom (DOFs). There are a variety of actuators that have been incorporated into soft crawling robots, including motor cables,⁷ pneumatics,^8,9 shape memory alloys (SMAs),^10–12 piezoelectric actuators,¹³ electroactive polymers,^14–16 and magnetics.^17–19

For example, Vikas et al.⁷ designed a three-dimensional (3D)-printed motor tendon actuated soft robot that can use friction manipulation mechanisms to effect locomotion. Rafsanjani et al.⁸ combined fiber-reinforced soft fluidic actuator and kirigami skins to develop a soft robot. Lin et al.¹¹ developed the soft-bodied GoQBot whose body is made of silicone rubbers with two ventral-axial tunnels to accommodate two SMA coils. Wu et al.¹³ designed a fast-moving and ultrarobust soft robot that is composed of a polyvinylidene difluoride actuated body with a leg-like structure. Gu et al.¹⁴ fabricated a soft wall-climbing robot that was made up of the dielectric elastomer actuators and two electroadhesive feet. Hu et al.¹⁹ designed a soft robot based on a mixture of Ecoflex-10 silicone elastomer and hard magnetic neodymium–iron–boron microparticles, controlled by a time-varying magnetic field to generate different locomotion modes.

Despite the many advantages of a soft robot, its application in engineering has always faced many challenges. Over the past 20 years, there are many types of mobile climbing robots used to inspect marine vessels and large pipelines. The locomotion types include wheels^20,21 and chains, arms and legs,²² and sliding frames.^21,23 The adhesion principles include magnetic adhesion,²⁴ pneumatic adhesion,²⁵ mechanical adhesion,^21,26 and electrostatic adhesion.²⁷ Minor et al.²² constructed a biped robot that is an underactuated five-link biped robot with four joints and three actuators. The size of robot is ∼45 × 45 × 248 mm and weight is 335 g. So the robot has a compact structure, which also increases the complexity of motion planning and joint level control.

Eich and Vögele²⁰ deigned an MINOAS magnetic climbing robot that has two wheels equipped with 50 neodymium magnets each, and the tail has another magnet. The weight of the robot is 670 g, which can realize the inspection of marine vessels. However, there are relatively few inspection robots applied in the reactor pressure vessel (RPV) field. Choi et al.²⁸ developed an inspection robot for the reactor vessel upper head. The robot adopts wheel locomotion and magnetic adsorption with a size of 18 × 13 × 4.5 cm. At present, the wall-climbing robots used in the field of marine vessels and RPV are mainly rigid robots, these robots cannot flexibly deform to adapt the complex environment such as curved surfaces. Therefore, the robot with flexibility of soft robots and robustness of rigid machines would be more suitable to the above cases.

The kinematics of a robot is to study the geometric relation of robot locomotion, and it is the theoretical basis of the position and orientation control when a robot performs an inspection task. A soft robot can also be called a soft continuum robot, and its kinematic model can refer to the modeling approach of a continuum robot. If a constant moment is applied along a beam, the Bernoulli–Euler beam mechanics predicts a constant curvature result.²⁹ The motion of continuum-style robots is generated through the bending of the robot over a given section, which is formulated and explained by a constant curvature.^30–33

The kinematics of the constant curvature continuum robot is decomposed into two submappings.³¹ One is from the actuator space to the configuration space whose parameters describe constant curvature arcs, and the other is from this configuration space to the task space whose parameters describe position and orientation along the backbone. The mapping from the configuration space to task space can be solved by several different approaches, such as the arc geometry approach,^30,33–35 the Denavit–Hartenberg parameter approach,^30,31,35 the Frenet–Serret frame approach,^30,35 and the Integral Representation approach.^35,36 These approaches can arrive at the same results, but the mapping from the actuator space to the configuration space needs to be considered separately for different actuation types.³⁵ For the robots actuated by flexible rods or tendons, some approaches to model the mapping have been proposed by literature.^{30,32,34,35,37}

A cable-driven soft robot (shorted as “SpringWorm”) with an omnidirectional deformable rectangular spring is proposed in this article. This structure can not only bend but also can contract with a large range, but the bending deformation is coupled with the contraction. To address this, a kinematic model was established. Furthermore, a mechanical model of the cable-driven backbone was established based on contact analysis and iterative solutions by using the finite element method (FEM). The theory and simulation results were verified by experiment. At the same time, the effectiveness of a constant curvature model under load is verified by adjusting the initial stiffness of the backbone based on the FEM. Afterward, the motion performance of the robot was tested on some curved surfaces and different terrains, such as cylindrical surfaces, spherical surfaces, steps, and ditches. Finally, SpringWorm was put into use in crack inspection of control rod drive mechanism (CRDM) with a 1:1 model.

Materials and Methods

In this section, the structural composition and characters of the robot are described. Assuming that the bending arc of the cable-driven backbone is with a constant curvature, the forward and inverse kinematic models of the robot can be derived. Furthermore, the FEM model of the backbone was established to obtain the relationship between the displacement and driving force of the cables and the deformation of the backbone.

Design and fabrication

Figure 1b shows that the robot mainly consists of three components: a head claw, an elastic body, and a tail claw. Figure 1c shows that the elastic body includes a driving unit and a deformation unit. As the driving unit is designed, the grooved bearing guides cable onto the reel of the micromotor. The micromotor can implement large torque and precise position control. The rated torque can be determined based on the size and wire diameter of the spring, and the determination of the above parameters needs to iteratively calculate the equivalent spring stiffness with the external load. The deformation unit consists of a spring, two fixed plates, and four driving cables.

FIG. 1.

Structure diagram of SpringWorm. (a) A photograph showing the axonometric view of the robot. (b) A 3D model of the robot. (c) The exploded view of robot structure. (d) A photograph showing passive deformation of the flexible claw. 3D, three dimensional.

The four cables are embedded in the holes at the four corners of the backbone. The backbone has special characteristics and advantages as follows. (1) The backbone has three DOFs, which are the yaw motion (bending around y₀ in Fig. 2), the pitch motion (bending around x₀ in Fig. 2), and the axial translation (contraction along z₀ in Fig. 2). (2) The backbone can adjust its own stiffness by precompression. (3) The rectangular helix spring is more stable than the circular helix spring. (4) Using the four cables to drive the three DOFs backbone is overactuated, which is beneficial to increase load, and the antagonism between the two cables can counteract the torsion deformation in yaw motion and pitch motion. The main deformable structure of the claw is the flexible sheet that is made of stainless steel with a thickness of 0.7 mm and attaches to a connector at the center.

FIG. 2.

An illustration of the kinematic model of a cable-driven spring backbone.

Five cantilever structures are designed in the circumferential direction of the sheet; those structures carry five microelectromagnets of 2-cm-diameter each. The purpose of the above design is mainly to realize the passive contact to curved surface. As shown in Figure 1d, the flexible claw can passively deform to achieve adsorption on the cylindrical surface with a diameter of 300 mm and a wall thickness of 6 mm. Figure 1a shows the SpringWorm prototype that coated the elastic skin whose material is spandex fabric.

Figure 1c shows the fabrication of SpringWorm. (1) The elastic body is fabricated. The rectangular spring is connected to the fixed plate by welding. The fixed mount connects the micromotor to the fixed plate, and the reel and micromotor connected by a fastening screw. The grooved bearing is connected to the bearing seat by a threaded shaft, and the bearing seat is connected to the fixed plate. One end of the cable is clamped by an aluminum sleeve, and the other end is knotted and fixed on the reel. (2) The claws are fabricated. The electromagnet is connected to the flexible sheet through the support frame, and the middle part of the flexible sheet and the connector is tightened by bolts. (3) The elastic body, head claw, and the tail claw are integrated to form the main structure of the robot.

Finally, the connection girder, the cable holder, the camera mount, and the skin were installed, and the fabrication process was completed.

Kinematic model of the cable-driven spring backbone

Based on the deformation analysis of the cable-driven backbone, the centerline of the backbone can be approximately regarded as a curve with constant curvature,³⁵ and the cable can also be assumed to have a constant curvature based on two factors as follows: the cables are embedded into the backbone; and the spring pitch is small. Therefore, the curvature of the cable is approximate to that of the backbone. Moreover, the bending deformation of the backbone is coupled with axial contraction, so the parameters of the configuration space need to be decoupled.

As shown in Figure 2, a Cartesian coordinate frame O₀x₀y₀z₀ is established; the origin O₀ of the base coordinate frame O₀x₀y₀z₀ is located at the center point of the tail fixed plate; the +x₀-axis lies in the plane ABCD toward the midpoint of the edge AB; the +y₀-axis lies in the plane ABCD toward the midpoint of the edge BC; and the +z₀-axis is determined by the right-hand rule. The center point O₁ of the top coordinate frame O₁x₁y₁z₁ is established at the center point of the head fixed plate; the +x₁-axis lies in the plane A₁B₁C₁D₁ toward the midpoint of the edge A₁B₁; the +y₁-axis lies in the plane A₁B₁C₁D₁ toward the midpoint of the edge B₁C₁; and the +z₁-axis is determined by the right-hand rule.

The centerline $\hat{O_{0} O_{1}}$ lies in the δ plane, whose center is O. The initial length of the centerline is l₀, the center angle of the centerline is θ, and the radius of the centerline is r. The angle between the plane δ and the axis +x₀ is φ. l_i is the length of the ith cable, the projection arc of l₁ is $\hat{G G_{1}}$ , and the radius of $\hat{G G_{1}}$ is r₁. The length of O₀A is s. The distance between two holes parallel to the long edge is a, and the distance between two holes parallel to the short edge is b.

By analyzing the geometric relationship between the backbone and the cables in Figure 2, an inverse kinematic model from the actuator space to the configuration space is established; that is, the arc parameters (l, θ, φ) are used to solve the length variations Δl_i of the cables by the following Equations (1)–(6): $s = \frac{\sqrt{a^{2} + b^{2}}}{2}$ (1) $λ = arctan \frac{b}{a}$ (2)

Δ l_{1} = l_{0} - [l - s θ cos (λ + φ)]

(3)

Δ l_{2} = l_{0} - [l - s θ cos (λ - φ)]

(4)

Δ l_{3} = l_{0} - [l + s θ cos (λ + φ)]

(5)

Δ l_{4} = l_{0} - [l + s θ cos (λ - φ)]

(6)

The forward kinematic model from the actuator space to the configuration space is provided as follows. Let the angle φ∈[0, 2π]; to obtain φ, we derive $φ_{1} = arctan (\frac{Δ l_{2} - Δ l_{4} - Δ l_{1} + Δ l_{3}}{Δ l_{2} - Δ l_{4} + Δ l_{1} - Δ l_{3}} cot λ)$ (7)

If Δl₂ + Δl₁ ＜ Δl₃ + Δl₄, then φ∈[π/2, 3π/2] and φ = φ₁ + π; if Δl₂ + Δl₁ ≥ Δl₃ + Δl₄, then φ∈[0, π/2) or (3π/2, 2π] and φ = φ₁ (if φ₁ ＜ 0, φ = φ₁ + 2π).

θ can be expressed as follows: $θ = \frac{Δ l_{2} - Δ l_{4}}{2 s cos (λ - φ)}$ (8)

l can be expressed as follows: $l = l_{0} - \frac{Δ l_{1} + Δ l_{3}}{2}$ (9)

The θ is related to the cross-sectional size and height of the backbone. Combined with the backbone characteristics of SpringWorm, θ∈[0, π] is set. The l is also affected by the cross-sectional size, height, and the number of spring coils, and l∈[n₁d, l₀] is set. Here, n₁ is the number of total coils and d is wire diameter.

Next, the forward kinematic mapping from the configuration space to the task space is analyzed. The forward kinematic analysis is to solve the position (position is represented by the origin of the top coordinate frame O₁x₁y₁z₁) and orientation (orientation is represented by the unit vector of the x₁, y₁, z₁ axes³⁸) of the top of the backbone by using the known arc parameters of the configuration space. The kinematic mapping relationship is represented by the transformation matrix T from the base coordinate frame O₀x₀y₀z₀ to the top coordinate frame O₁x₁y₁z₁. The matrix T consists of one translation transformation and three rotation transformations. $\begin{matrix} T = T_{1} T_{2} T_{3} T_{4} = & T r a n s (\frac{l}{θ} C φ (1 - C θ), \frac{l}{θ} S φ (1 - C θ), \frac{l}{θ} S θ) \\ R o t ({z'}_{1}, φ) R o t ({y''}_{1}, θ) R o t ({z'''}_{1}, - φ) \end{matrix}$ (10)

where Cθ = cosθ, Cφ = cosφ, Sθ = sinθ, Sφ = sinφ.

First, the origin O₀ of the base coordinate frame is translated to O₁, and the translation matrix T ₁ is as follows: $\begin{matrix} T r a n s (\frac{l}{θ} C φ (1 - C θ), \frac{l}{θ} S φ (1 - C θ), \frac{l}{θ} S θ) \\ = [\begin{matrix} 1 & 0 & 0 & \frac{l}{θ} C φ (1 - C θ) \\ 0 & 1 & 0 & \frac{l}{θ} S φ (1 - C θ) \\ 0 & 0 & 1 & \frac{l}{θ} S θ \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}$ (11)

Next, the first rotation matrix T ₂ can be obtained by rotating the angle φ about the new ( ${z'}_{1}$ ) axis: $R o t ({z'}_{1}, φ) = [\begin{matrix} C φ & - S φ & 0 & 0 \\ S φ & C φ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$ (12)

Then, the second rotation matrix T ₃ can be obtained by rotating the angle θ about the new ( ${y''}_{1}$ ) axis: $R o t ({y''}_{1}, θ) = [\begin{matrix} C θ & 0 & S θ & 0 \\ 0 & 1 & 0 & 0 \\ - S θ & 0 & C θ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$ (13)

Finally, the last rotation matrix T ₄ can be obtained by rotating the angle–φ about the new ( ${z'''}_{1}$ ) axis: $R o t ({z'''}_{1}, - φ) = [\begin{matrix} C φ & S φ & 0 & 0 \\ - S φ & C φ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$ (14)

The transformation matrix T can be obtained by multiplying the above four transformation matrices: $T = [\begin{matrix} C^{2} φ C θ + S^{2} φ & C φ C θ S φ - S φ C φ & C φ S θ & \frac{l C φ (1 - C θ)}{θ} \\ C φ C θ S φ - S φ C φ & S^{2} φ C θ + C^{2} φ & S φ S θ & \frac{l S φ (1 - C θ)}{θ} \\ - C φ S θ & - S φ S θ & C θ & \frac{l S θ}{θ} \\ 0 & 0 & 0 & 1 \end{matrix}]$ (15)

Inverse kinematic analysis involves solving the parameters (l, θ, φ) of the configuration space by using the known matrix $_{1}^{0} T$ shown in Equation (16), where p_x, p_y, and p_z denote the coordinates of O₁ in the coordinate frame O₀x₀y₀z₀. α, β, and γ are the unit vectors corresponding to the x₁, y₁, and z₁ axes, respectively. $_{1}^{0} T = [\begin{matrix} α_{x} & β_{x} & γ_{x} & p_{x} \\ α_{y} & β_{y} & γ_{y} & p_{y} \\ α_{z} & β_{z} & γ_{z} & p_{z} \\ 0 & 0 & 0 & 1 \end{matrix}]$ (16)

θ = arccos γ_{z}

(17)

φ_{1} = arctan (\frac{p_{y}}{p_{x}})

(18)

l = \frac{p_{z} θ}{sin θ}

(19)

In Equation (17), θ∈[0, π]. If p_x ≤ 0, then φ∈[π/2, 3π/2] and φ = φ₁ + π. If p_x ≥ 0, then φ∈[0, π/2) or (3π/2, 2π], and φ = φ₁ (if φ₁ ＜ 0, φ = φ₁ + 2π).

Simulation model of the cable-driven spring backbone

During the bending deformation, the spring is subjected to its own gravity, the load from the claw and motors, the friction from ground, and the local support force and friction from the cables. Based on the above factors, modeling by using only kinematics will produce some errors. Therefore, a mechanical model should be established.³⁵ Considering that the contact forces between the backbone and the cables are changing in real time, an analytical model is quite difficult to be built, so a simulation model by the FEM is established as follows.

(1)

Model establishment. The backbone is established as one component, including a spring and two plates on both sides. Moreover, the cable is equivalent to the metal wire with a diameter of 0.2 mm, and the bending stiffness is negligible compared with the spring. The backbone and the cable are defined as 3D solid elements.

(2)

Meshing model. The finite element meshes of the backbone and the cable are set to tetrahedron, the element size of cable is 0.2 mm, the element size of the backbone is 2 mm, and the surface of the hole is set with the refinement operation.

(3)

Defining contact relationships. The bonded contacts between the top surface of the cables and the surface of the head fixed plate are established. The frictional contacts between the cylindrical surface of the cables and the inner wall of the holes are established. The frictional contacts between adjacent spring wires are established.

(4)

Applying loads and setting solution. The specified displacement of the driving cables is applied along the x-axis in Figure 3; the free displacement of the passive cables is applied along the x-axis, and a small axial force is also applied to these cables to avoid relaxation of the cables. The tail fixed plate is fixed. In addition, gravity is applied in the pitch motion. Finally, multiple load steps solution is set; the large deflection should be applied to solve the deformation. The simulation models are presented in Figure 3a–d.

FIG. 3.

The simulation model of the cable-driven spring backbone. (a) Bending angle of 67.068° with displacements of 70 mm of driving cables 1 and 2. (b) Bending angle of 63.717° with displacements of 70 mm of driving cables 1 and 4. (c) Contraction of 59.669 mm with displacements of 60 mm of driving cables 2 and 4. (d) Contraction of 59.848 mm with displacements of 60 mm of driving cables 1, 2, 3, and 4. (e) Bending angle of 58.561° with displacements of 75 mm of driving cables 1 and 4 in the precompressed state of spring. (f) Bending angle of 64.797° with displacements of 75 mm of driving cables 1 and 2 in the spring precompressed state. Where cables 1, 2, 3, and 4 are defined in Figure 2. Where the +x-axis and +y-axis are equivalent to the +z₀-axis and −x₀-axis in Figure 2, respectively.

Due to the existence of the claw and motors, if the stiffness of the backbone is too small, the theory model of constant curvature will have a relatively large error. However, considering the particularity of the spring structure, the initial stiffness can be adjusted by precompression. This compression value can be obtained by means of the FEM. The simulation model uses a same mass that is 283 g to simulate the claw and motors, as shown in Figure 3e. Due to this heavy matter, the head fixed plate moves downward and will no longer be parallel to the tail fixed plate. By driving the cables 1 and 4 shown in Figure 2, the head fixing plate is restored to be parallel to the tail fixed plate, and the displacement of the spring is the precompression value at this time. Figure 3f shows that the backbone yaws on the metal surface under the external load, and the frictional contact relationship between the load and the metal surface is established. Supplementary Video S1 shows the deformation process of the backbone in Figure 3a–d.

Gaits and motion control for SpringWorm

The gaits of the robot adopt the biomimetic crawling mechanism, and the gaits include forward crawling, backward crawling, turning crawling, and 360° turning. The locomotion process of forward crawling is realized by the following steps. First, the head claw of the robot contacts the ground, the tail claw lifts off the ground, and the cables drive the backbone to contract. Next, the tail claw contacts the ground, the head claw lifts off the ground, the cables reset the backbone, and a forward gait period is accomplished. Similarly, other locomotion gaits can also be accomplished. When the tail claw contacts the ground, the backbone can realize omnidirectional rotation and axial translation, and it can also function as a flexible manipulator.

The control of the robot is divided into two modes, automatic gait and remote control. When the robot performs forward gait, backward gait, and turning gait on a plane, it adopts a program-controlled automatic gait. In this mode, configuration parameters such as the specified bending angle, the central arc length, and the deflection angle of the backbone need to be set. Then it is necessary to use inverse kinematics to calculate the actual rotation angle of motor from the specified configuration parameters. While for the complex environment that contains many obstacles and where robot localization is difficult, remote control is required. Based on inverse kinematics, the robot can achieve angle control with a resolution of 2°. When SpringWorm inspects the welds of CRDM, the robot adopts remote control mode.

Results and Discussion

Model validation

An experimental device is shown in Figure 4 to validate the kinematic model and the FEM model. The experimental device consists of the test object, a measurement platform, a fixed device, a displacement platform, and the force-sensing module. Figure 4a shows that the test object consists of a rectangular spring (I) and four cables (II). The measurement platform consists of a polymethyl methacrylate (PMMA) plate (VI) marked Cartesian frame, an upstanding plate (VII) marked Cartesian frame, a smooth polyethylene terephthalate (PET) film (VIII), a ruler (III) fixed at the tail fixed plate, and a slider (IV). The PMMA plate and the upstanding plate can locate the initial position of the backbone.

FIG. 4.

The experimental device for testing the backbone. (a) A 3D model of the experimental device. (b) A photograph of the experimental device.

The PET film can reduce friction and mark the bending angle and top position of the backbone. The passive cables are fixed on the slider, and its displacement can be measured. The fixed device is a plain vice (V), which is utilized to clamp the tail fixed plate; Figure 4a and b shows that the displacement platform is a linear feed device (XI) consisting of a ball screw driven by a stepper motor (XII), a nut, and a guide rail. The force-sensing module consists of power supply, force sensor (IX), sensor fixed plate (X), digital transmitter, and force displayer. The measurement process is illustrated as follows: the cables drive the spring to deform by the displacement platform. The tension force of the driving cables is measured by the force-sensing module, the bending angle and the top position are marked in PET film, and the displacement of the passive cables is recorded by the ruler.

The spring is made of high-carbon steel, and the cables are the braided line (main fabric is 100% polyethylene with high density; DECATHLON). The geometric and mechanical parameters of the backbone and cables are given in Table 1.

Table 1.

The Geometric Parameters and Mechanical Parameters

Term	Description	Value
D₁, D₂	Length and width of the spring cross section	56.8, 46.8 mm
a, b	The distance between of two holes parallel to the long edge and short edge	50.76, 41.36 mm
d	Diameter of spring wires	2.8 mm
t	Pitch	15.8 mm
H ₀	Height of the spring	100.33 mm
n	No. of active coils	6
r ₂	Middle radius of the transition fillet of the spring cross section	5 mm
d ₁	Diameter of holes inside the spring	1.1 mm
s₁, s₂, s₃	Length, width, and thickness of the fixed plate	57, 47, 1 mm
d ₂	Diameter of the cables	0.3 mm
E₁, E₂^a	Young's modulus of the spring and cables	2.0 × 10⁵ Mpa, 1.94 × 10⁵ Mpa
σ₁, σ₂	Poisson's ratio of the spring and cables	0.3
ρ₁, ρ₂	Densities of the spring and cables	7.85 × 10³ kg/m³
μ₁, μ₂, μ₃	Friction coefficients between spring and cables, between adjacent spring wires, between the claw and metal surface	0.1, 0.15, 0.17

In the simulation model, the cable is equivalent to a stainless steel wire with a smaller cross-sectional diameter.

Figure 5a (the yaw motion) and 5b (the pitch motion) shows the variation trend of the driving forces (F₁, F₂) and (F₁, F₄) with the displacement of cable numbers (1, 2) and (1, 4), respectively. The maximum errors of (F₁, F₂) and (F₁, F₄) between the FEM and experiment (Exp) are 10.247% and 5.941%, respectively. The relationship between the driving force and the displacement is approximately linear in the above motions. The number of the driving cables is 2 and 4 in Figure 5c and 1, 2, 3, and 4 in Figure 5d, both of which represent the axial translation, The maximum errors of (F₂, F₄) and (F₁, F₂, F₃, F₄) between the FEM and Exp are 3.572% and 9.423%, respectively. Obviously, there is a linear relationship between the driving force and displacement in the axial translation, and the linear relationship is Hooke's law.

FIG. 5.

Comparison results of driving force for different motion modes between the FEM and experiment (Exp). (a) Comparison results of driving force for the yaw motion, (b) Comparison results of driving force for the pitch motion. (c) Comparison results of driving force (F₂, F₄) for the axial translation. (d) Comparison results of driving force (F₁, F₂, F₃, F₄) for the axial translation. FEM, finite element method.

The main errors are elaborated as follows. The pitch size of the spring used in the experiment has machining irregularity, resulting in error in the spring length. The edge fillet size of the holes used to guide the cables also has machining irregularity. In addition, the fixed plate and the spring were welding, which could not be fully simulated in FEM. In addition, the simplified cable element could also introduce some errors in the FEM. The errors are affected not only by the aforementioned reasons in Figure 5d, but also by the inaccuracy of the force sensor. The force sensor has a measurement tolerance of about ±0.07 N, which may easily introduce some errors in the initial position adjustment of the four cables.

Figure 6a (the yaw motion) and 6b (the pitch motion) shows the variation trend of the bending angle with the displacement of the cables. The blue values represent the errors between the FEM and Exp; and the black values represent the errors between the theory and Exp. The maximum errors of the FEM and theory are 6.320% and 5.369% for the yaw motion, and they are 7.924% and 11.646% for the pitch motion, respectively. The curves show that the bending angle is basically linear with the driving displacement. Due to the simplification of the simulation model, the maximum angle error of the FEM will be slightly larger.

FIG. 6.

Comparison results of bending angle for different motion modes among the theory, FEM, and Exp. (a) Comparison results of bending angle for the yaw motion. (b) Comparison results of bending angle for the pitch motion. (c) Comparison results of bending angle for the pitch motion under the external load. (d) Comparison results of bending angle for the yaw motion under friction.

Figure 6c shows the corresponding relationship between the displacement of the cables and the bending angle of the backbone that is subjected to the load in the pitch motion. The initial straight line in the figure is the precompression of the spring, which is about 17.3 mm. When the antagonism between the cables and the backbone caused by the precompression can offset the excessive deformation caused by the external load, the constant curvature model can obtain a relatively accurate result. The maximum errors of the FEM and theory are 4.212% and 5.093%, respectively. Figure 6d shows the yaw motion of the backbone under external load, which is also subject to the friction from the surface. Under the condition of maintaining the precompression of 17.3 mm, the displacement and the bending angle also maintain an approximately linear relationship, and the maximum errors of the FEM and theory are 8.048% and 6.132%, respectively.

As shown in Figure 7, the motion trajectories of the top position (point O₁ in Fig. 2) with the yaw motion (located in x₀O₀z₀ projection plane) and the pitch motion (located in y₀O₀z₀ projection plane) are shown in a Cartesian frame, and the values on the two sides of the curve are position errors compared with the results of the FEM and theory with the Exp results, respectively. The maximum distance errors of the FEM and theory are 2.109 and 1.597 mm for the yaw motion, and they are 1.693 and 1.899 mm for the pitch motion, respectively. Meanwhile, it is found that the centerline length of the backbone decreases gradually with the bending deformation, which proves that bending deformation is coupled with axial contraction.

FIG. 7.

The top position comparison results of the yaw motion and pitch motion among the theory, FEM, and Exp.

Velocity and load tests

Since the response time of the relays (Omron G2AL-1-E) controlling the electromagnets is less than 12 ms, locomotion speeds mainly depend on two factors: the motor torque characteristics and the maximum spring compression (stride). A higher motor speed and a larger spring compression can lead to a higher locomotion speed. However, the motor speed is limited by the driving torque. When the motor speed is 85 rpm, single gait compression of the spring is about 34 mm, and the speed is 8.9 mm/s.

The load capacity of the robot is not only affected by the performance of the motor, but also by the adsorption force of the claw. When the working voltage of the electromagnet is 32 V, the robot can generate an adsorption force of 16 N on a spherical surface with a diameter of 500 mm and a wall thickness of 3 mm. Furthermore, two groups of the load test are conducted. When the work voltage of the electromagnet is 32 V, SpringWorm can climb on a vertical iron plate with a 2 mm thickness, and carry a weight of 0.5 kg to realize vertical climbing. In the second group, the robot can carry a weight of 1 kg to move on a spherical surface with a radius of 217.8 cm and a thickness of 6 mm. The Supplementary Videos S2 and S3 show the aforementioned test processes, and the robot did not slip or fall throughout the load test processes.

Tests for locomotion performances

Imitating the locomotion of an earthworm in nature, SpringWorm can adapt to some curved surfaces and different terrains. As shown in Figure 8a, the robot can climb over a 2-cm-tall step and a 4-cm-deep ditch by contracting its volume and adjusting its body morphology. Figure 8b demonstrates that the robot can climb on the vertical wall and achieve 90° turning and lateral climbing with only slight slipping. Figure 8c shows that the robot can climbing on a pipe with a radius of 31 cm, and can turn dexterously on this surface. In the climbing process of the pipe with a large curvature, the bending deformation of the elastic backbone and the passive deformation of the flexible claw are combined, which can enable the robot to transform its morphology to match the curvature of surface. These locomotion processes are shown in Supplementary Videos S2, S4, and S5, respectively.

FIG. 8.

Locomotion performances of SpringWorm. (a) Climbing over a step and a ditch. (b) Climbing on the vertical wall. (c) Climbing on a pipe with a radius of 31 cm.

On the spherical surface, the robot can realize a variety of gaits, including forward crawling, backward crawling, lateral climbing, 360° turning, and the pitch motion (Fig. 9b and Supplementary Video S3).

FIG. 9.

Application in CRDM crack inspection of SpringWorm. (a) The model of the RPV closure head and CRDM. (b) SpringWorm is located above the RPV closure head. (c) SpringWorm performs the crack inspection task. CRDM, control rod drive mechanism; RPV, reactor pressure vessel.

Application in CRDM crack inspection

Nuclear RPVs are subjected to extreme loads such as high temperature, high pressure, and nuclear radiation. Cracks are likely to occur at the weld between the RPV closure head and CRDMs, and so, regular inspections are needed. At present, the main challenges are risks of nuclear radiation and inaccessibility of the weld position during inspection.³⁹ In view of the structure of SpringWorm within the decimeter scale, it has the superior capabilities for dexterity, compliance, and high load in a complex environment, and the robot has great potential for the crack inspection of CRDMs. The device consists of an RPV closure head and CRDMs, shown in Figure 9a, which is a 1:1 model with the same physical parameters.

The inspection process mainly relies on remote control and the inspection task is as follows. First, the robot drills through the hole with a 10-cm diameter located on the round wall above the RPV closure head, and arrives at the starting point below the closure head. Next, the robot needs to inspect the welds and go across the CRDMs of diameter 10.2 cm on the spherical closure head of radius 217.8 cm, and the space of adjacent CRDMs is 20 cm. Then, the robot needs to move to the top position of the closure head. Finally, the robot returns to the starting point. Figure 9c and Supplementary Video S6 show that the inside of the RPV is a dark environment, and the robot needs to continuously climb between CRDMs to realize the all-round crack inspection.

Conclusions

In this article, a novel soft robot with structural compliance, motion dexterity, and high load is proposed. These superior capabilities benefit from the ingenious designs of SpringWorm as follows. (1) The backbone can realize three DOFs deformations (yaw motion, pitch motion, and axial translation). (2) The four driving cables are embedded inside the backbone, which can miniaturize the robot and improve the stability and accuracy of motion. (3) The flexible claw adopts the deformable thin plate embedded with distributed electromagnets that ensure the claw contacts a curved surface sufficiently. Therefore, the robot body can transform its morphology to adapt to different surfaces with varying curvatures, such as plane, a cylindrical surface with a curvature radius of 31 cm, and a spherical surface with a curvature radius of 217.8 cm. Furthermore, the robot can adapt to different terrains such as steps, ditches, and even a vertical wall.

Based on the assumption of constant curvature, the forward and inverse kinematic models of the actuator space to the configuration space and the configuration space to the task space of the backbone are derived. Meanwhile, the deformation angle θ and arc length l are decoupled under the conditions that the displacements of the four cables or the top position and orientation have been given. Considering the friction between the cable and the backbone, an FEM model that is more consistent with the real backbone is established. Comparing the results among the theory, FEM, and Exp, we find that the deformation trends of the three results are consistent with acceptable errors lower than 11%. The errors are mainly caused by machining irregularities of the spring and assumption: the curvature of the backbone is regarded as constant.

It is worth noting that the rectangular spring belongs to a highly deformable structure, and its deformation can be approximately solved by the linear elastic deformation theory. In addition, the aforementioned assumption will bring a larger error if the backbone is subjected to an external load. However, the backbone should improve its own initial stiffness through precompressing the backbone, and the assumption is also workable because the initial stiffness can counteract the excessive deformation caused by the external load. The kinematic model and the finite-element model provide guidance for locomotion control, stiffness adjustment of the backbone, motor selection, and representation of the workspace.

Compared with other large-scale rigid robots with multiple DOFs in the fields of marine vessels and RPV, SpringWorm has a compact size, lighter weight, and safe interaction; compared with traditional microlegged robots and wheeled robots, SpringWorm has more DOFs, greater compliance, and load capacity. Therefore, the robot combines the compliance of a soft robot with the large load capacity of a rigid robot. The robot has successfully completed crack inspection for CRDMs with a 1:1 model. In the future, the robot can also work inside other complex equipment, such as aeronautical engines, steam turbine blades, and chemical pipes.

Footnotes

Acknowledgments

The authors would like to appreciate the support and funds.

Authors' Contributions

J.Z. and B.H. conceived the concept. P.Y., J.Z., D.X., and M.L. discussed and designed the structure of the robot. P.Y. built the model of kinematics and simulation. J.Z. and P.Y. carried out the experiments and analyzed the results. D.M.C. and P.Y. wrote the article.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This study was funded by the China Academy of Space Technology (CAST-2021-01-03) and the Shandong Provincial Natural Science Foundation (ZR2021ME088, 2022.01-2024.12).

Supplementary Material

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Supplementary Material

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