Design, Analysis and Experiment of a Modular Deployable Continuum Robot

Abstract

Continuum robots, possessing great flexibility, can accomplish tasks in complex work scenes, regarded as an important direction in robotics. However, the current continuum robots are not satisfying enough in terms of fabrication and maintenance, and their workspace is limited by structure and other aspects. In this paper, to address the above problems, a modular deployable robot, which adopts an origami structure instead of a flexible hinge, is proposed. A fabrication method is innovated, the Spherical Linkage Parallel Mechanism (SLPM) unit is optimized, and the installation and fabrication process of the robot is simplified through modularization. The forward kinematics and inverse kinematics of the robot and its workspace are analyzed by using the screw theory. The prototype of the robot is constructed, and its folding performance, bending performance, and motion accuracy are tested, and the error analysis and compensation optimization are carried out. After the optimization, the position error of the robot is reduced by about 65%, and the standard deviation is greatly lowered, which effectively improves the motion accuracy and stability of the robot.

1. Introduction

Continuum robots are typical bionic robots inspired by many organisms that have flexible structures similar to “invertebrates”, such as elephant trunks, octopus tentacles, and the tongues of some animals, which do not have rigid skeletons but have excellent bending performance and great motion capabilities. By mimicking these flexible structures, continuum robots can realize winding and grasping operations in a variety of complex environments, and maintain a certain end-loading capacity. Therefore, the research of continuum robots has become one of the most active directions in robotics research in recent years [1].

Previous research has proposed a variety of structural solutions suitable for continuum robots. Walker proposed a continuum robot with a backbone and systematically studied the structural design [2,3], kinematics [4,5], and intelligent control [6], which made an outstanding contribution to the promotion of the spine-type continuum robot. Xu [7,8] proposed a continuum robot with a super-elastic NiTi alloy skeleton, which fully utilizes the material’s properties of both flexibility and rigidity to enhance the end-loading capacity. Sun [9] proposed a method based on 3D topology to optimize the torsional stiffness and rotational flexibility of different axes of flexible joints, which allowed the robot to successfully achieve high torsional stiffness while maintaining bending flexibility. Similarly, Geng [10] proposed a continuum robot with tendons and a spine made of a NiTi alloy, and enabled the robot to maintain a high enough accuracy to avoid collisions during spatial manipulation. Ranzani [11] proposed a liquid-driven continuum robot, which consists of two modules that can be independently controlled. Greer [12] proposed a pneumatic artificial muscle-driven continuum robot, which can be fully flexible for safer human–robot interaction. Brandon [13] applied pressure-driven topology optimization to a pneumatic continuum robot to explore the optimal robot structure. Chautems [14] proposed a magnetically driven variable-stiffness continuum robot, which utilizes a magnetic catheter with variable stiffness to achieve robot shape locking and reduce operation difficulty. In recent years, concentric tube continuum robots (CTCRs) have received a lot of attention, which utilize relative rotation and extension between tubes to achieve localization and orientation of the robot end as well as control over the robot’s attitude [15,16,17]. Concentric tube robots are very slender, and the actuators are integrated and attached to the tubes so that they can access the organs and vascular system of the human body to assist in minimally invasive surgical treatments [18], which has a promising future in the medical field. There are also some research attempts to combine continuum robots with parallel mechanisms. Rucker [19] proposed the concept of the parallel continuum robot (PCR), developed an efficient computational method for robot equivalent modeling suitable for real-time simulation and control [20], investigated the effects of design parameters on robot flexibility and mechanical properties [21], and explored its application in teleoperated surgery [22,23]. Wu [24] proposed a multi-constraint parallel continuum robot on this basis to realize the precise calibration of the end position. Lilge [25] used a continuum structure instead of the rigid linkage of the parallel mechanism to achieve light weight and miniaturization, enabling safe interaction with the environment. Similarly, Mauze [26] designed a PCR with three flexible struts that can achieve very high accuracy in a large workspace. The parallel continuum robot synthesizes the flexibility of the continuum robot with the high accuracy and flexibility of the parallel robot.

However, current continuum robots are not designed conveniently enough to fabricate and maintain them, and their structure limits their workspace. More and more modular continuum robots (MCRs) have appeared in recent works. Castledine [27] proposed a continuum robot with modular kinematic segments, each of which allows for bending in two degrees of freedom while minimizing torsional motions, and the robot can be connected in series with multiple segments. Yang [28] proposed a modular continuum robot with hybrid rigid–flexible motion segments, where the modular design simplifies the control and sensing system and improves the end-loading capacity. Gomez [29] proposed a modular scalable continuum robot, which was designed to be more applicable to the mining operation, with mechanical interface and variable length. Wu [30] systematically investigated modular continuum robot dynamic modeling, considering the robot’s stretching, retracting, bending, and twisting, which provided a reference for modular continuum robot dynamic modeling. Overall, modular continuum robots possess great expandability and flexibility and are relatively easy to maintain and store. However, they are still complicated in terms of fabrication. In recent years, the art of origami has received the attention of more scholars who have been studying origami mechanisms and utilizing the structural rationality and shape variability of origami-inspired works, thus helping engineers design creative structures in architecture, medicine, and aerospace [31,32,33,34]. Origami structures have been shown to solve the problem of difficult modular fabrication [35,36], but there is still a relatively complex process involved in their fabrication, and the longevity of the robot cannot be guaranteed.

In this paper, we synthesize the advantages of continuum robots, parallel mechanisms, and origami structures, and propose a modular deployable continuum robot, which is modeled on the structural design of continuum robots and consists of multiple modules. As shown in Figure 1, each module has multiple cells, and the cells are made with origami structures, the equivalent substitution of parallel mechanisms [37]. This paper also establishes the modular kinematic model and analyzes the workspace of the robot. Finite element simulations on the modular unit are conducted, analyzing its stiffness and its ability to resist lateral force and torsion, formulating the control strategy of the robot, and building the control system of the robot. Finally, an experimental study of the robot’s kinematic performance was carried out to verify the validity and accuracy of the robot’s kinematic model and control system.

2. Design of SLPM Unit Based on Origami Structure

Flexible hinge joints have the shortcomings of easily falling off and a short lifespan. In this paper, based on the SLPM unit proposed by Guan [34] et al., a Polyethylene Terephthalate (PET) membrane is adopted for fabricating flexible hinges and connecting parts of the moving and static platforms. Resin rigid plates, hereinafter referred to as rigid plates, are fabricated by 3D printing. The above parts constitute the folding structure and, together with the connecting plate and spring, constitute the SLPM unit. To avoid the interference of the thickness of the rigid plate on the folding structure, this paper adopts a hole–shaft fit, instead of the common fastener. The upper and lower boards are set to connect each other with short shafts and holes, and the latter ones are cut out on the PET membrane. The size of the PET membrane is 60 mm × 70 mm, and the size of the two rigid plates is 60 mm × 50 mm with 1 mm thickness.

As the thickness of the PET film increases, the folding resistance and stiffness are improved, but it also leads to difficulty in folding, which affects the robot’s folding performance and bending performance. Therefore, we take the thickness of the PET membrane as 0.1 mm and 0.2 mm, which requires less force for folding and is easy for the robot to stretch and retract; also, the membrane will not be damaged because of the rigid plate deformation.

A gap is required between the rigid plates for unit folding, and the size of the gap will directly affect the folding effect and life of the deployable structure. As shown in Figure 2, the gap value should be larger than the sum of the thicknesses of the two rigid plates; however, the gap being set too large would cause an offset of the original structure, leading to excessive fatigue or even damage to the membrane. Therefore, a suitable gap value needs to be selected. Lee et al. [38] used the Euler–Bernoulli beam theory to estimate the energy accumulated during the membrane folding process, and based on Equation (1), it was calculated that when the thickness of the membrane is 0.1 mm, the least energy is accumulated when the gap is 3.30 mm, and when the thickness of the membrane is 0.2 mm, the least energy is accumulated when the gap is 3.46 mm.

$${\displaystyle \frac{l}{t_b+0.5t_m}}\approx \pi$$

(1)

where $l$ represents the gap value, $t_b$ represents the rigid plate thickness, and $t_m$ represents the PET membrane thickness.

The robot body adopts a modular design method. To realize the control of the robot’s end position and develop additional functions such as path planning, obstacle avoidance, etc., the prototype of the continuum robot fabricated in this paper consists of three modules connected in series, and each module consists of three units, increasing the robot’s degree of freedom and satisfying the requirements of both the workspace and the control precision. The cell fabrication process is shown in Figure 3.

To reduce the influence of the robot’s self-weight and improve the control accuracy, stiffness, and bending strength, the top module of the robot uses the 0.1 mm PET membrane for the cell, and the other two modules use the 0.2 mm PET film. Each module is controlled by three drive wires; the wiring is shown in Figure 4. The assembled robot prototype is shown in Figure 5.

3. Kinematic Analysis

3.1. Forward Kinematics

The forward kinematics of SLPM-based continuum robots is analyzed. First, the forward kinematics model of an SLPM cell is established based on its equivalent configuration. Then, based on the assumption of the constant curvature model among different cells of the same module, the conclusion of one cell is generalized to one module. In turn, the generalization is made to the continuum robot.

The moving platform and static platform of the SLPM unit are the same as those of the 3-RSR, and the origami flexible hinge is used instead of the kinematic chain, which improves its defects, is prone to wear and tear, and has poor loading capacity. Therefore, any SLPM unit can be equivalent to a 3-RSR parallel mechanism. The mechanism consists of three identical and uniformly distributed spatial kinematic chains, each consisting of two rotating pairs and one spherical pair [39]. The schematic diagram of the equivalent parallel mechanism of the SLPM unit is shown in Figure 6. The drive wires are set to positions $A_0{B}_0{C}_0$ and $A_e{B}_e{C}_e$, where they form equilateral triangles. The equivalent spatial ball hinge is located at the intersection of six pieces of folded paper, denoted by $S_i(i=1,2,3)$, and the equivalent rotating vice is located at the midpoint of the two platforms, denoted by $R_{0i}$, $R_{ei}(i=1,2,3)$.

Zhuang et al. [40] analyzed the degrees of freedom of the SLPM units to be three by the method based on screw theory. Without considering the driving mode, the degree of freedom of the module composed of three SLPM units is nine. Due to the synchronous motion between the SLPM units, the analysis can be simplified by adopting the assumption of segmental constant curvature, which means the deformation form of the units in the same module is the same so that the overall number of degrees of freedom of the robot is nine, regardless of how many units there are.

The kinematic characteristics of the SLPM equivalent mechanism are shown in Figure 6. The motion of the SLPM cell is shown in Figure 7. In this section, the kinematic modeling and analysis of the SLPM cell and the robot are carried out using the Product of Exponentials (PoE) method [41], which is based on screw theory.

The origin coordinate system of the body is the O₀ coordinate system, whose origin is located in the center of the triangle ${\mathit{A}}_{\mathbf{0}}{\mathit{B}}_{\mathbf{0}}{\mathit{C}}_{\mathbf{0}}$. The positive direction of the $x_0$ axis points to the point ${\mathit{A}}_{\mathbf{0}}$. The $z_0$ axis is perpendicular to the plane ${\mathit{A}}_{\mathbf{0}}{\mathit{B}}_{\mathbf{0}}{\mathit{C}}_{\mathbf{0}}$. The $y_0$ axis meets the right-handed rule. At the same time, we establish the proposed body coordinate system ${\mathit{O}}_{\mathit{e}}$, the origin of which is located in the center of the triangle ${\mathit{A}}_{\mathit{e}}{\mathit{B}}_{\mathit{e}}{\mathit{C}}_{\mathit{e}}$. The positive direction of the $x_e$ axis points to the point ${\mathit{A}}_{\mathit{e}}$. The $z_e$ axis is perpendicular to the plane ${\mathit{A}}_{\mathit{e}}{\mathit{B}}_{\mathit{e}}{\mathit{C}}_{\mathit{e}}$. The $y_e$ axis meets the right-handed rule. We set the intersection line between plane ${\mathit{A}}_{\mathbf{0}}{\mathit{B}}_{\mathbf{0}}{\mathit{C}}_{\mathbf{0}}$ and plane ${\mathit{A}}_{\mathit{e}}{\mathit{B}}_{\mathit{e}}{\mathit{C}}_{\mathit{e}}$ to be $\mathit{L}$ and make the vertical line of straight line $\mathit{L}$ through ${\mathit{O}}_{\mathbf{0}}$ intersect straight line $\mathit{L}$ at point $\mathit{D}$. Take $\alpha$ as the angle between ${\mathit{O}}_{\mathbf{0}}\mathit{D}$ and the $x_0$ axis, and $\theta$ as the angle between the end moving platform and the origin static platform. The end moving platform can be regarded as obtained by rotating the origin static platform around a certain rotation axis by an angle $\theta$ [35].

The three-dimensional motion of a rigid body can be expressed as a rotation around an axis and a translation along that axis, similar to the motion of a nut on a bolt [42]. According to the definition of rotation, we can split the transformation from the origin static platform to the end moving platform into a rotational transformation around the origin axis and a translation transformation, where the rotational transformation is the rotation of the unit vector g parallel to the straight line $\mathit{L}$ around the origin by an angle $\theta$, which is denoted as Motion I, and the translation transformation is the translation of the length r along the ${\mathit{O}}_{\mathbf{0}}{\mathit{O}}_{\mathit{e}}$ direction, which is denoted as Motion II.

From the screw theory, the screw can be expressed as a six-dimensional vector formed by a pair of three-dimensional vectors, written as

$$\mathit{S}=\left[\begin{array}{c}\mathit{s}\\ {\mathit{s}}_0\end{array}\right]=\left[\begin{array}{c}\mathit{s}\\ r\times \mathit{s}+h\mathit{s}\end{array}\right]$$

(2)

where $\mathit{s}$ is the spin axis vector and ${\mathit{s}}_0$ is the vector used to determine the axis position and spin distance.

In this paper, the motion is described by $\left(\mathrm{M},\mathit{s},h\right)$, where $\mathrm{M}$ is a fixed point on the helix axis, $\mathit{s}$ is the spin axis vector, and $h$ is the pitch. The velocity spin can be defined as

$$\mathbf{T}=\left[\begin{array}{c}\omega \\ v\end{array}\right]=\left[\begin{array}{c}\omega \\ r\times \omega +h\omega \end{array}\right]=\omega \left[\begin{array}{c}\mathit{s}\\ {\mathit{s}}_0\end{array}\right]=\omega \mathit{S}$$

(3)

$$r=\left(r_O-r_M\right)$$

(4)

where $\omega$ is the angular velocity of the rigid body about the axis of the spin, $v$ is the velocity of point $\mathrm{M}$ on the rigid body, $r$ is the position vector pointing from point $\mathrm{M}$ to the axis, $r_O$ is the position vector perpendicular to the axis of the spin, $r_M$ is the position vector from $O$ to $\mathrm{M}$, and $\mathit{S}$ is the six-dimensional vector form of the spin.

From this, one can derive the spin distance of the velocity spins:

$$h={\displaystyle \frac{\omega ·v}{\omega ·\omega }}={\displaystyle \frac{\mathit{s}·{\mathit{s}}_{\mathbf{0}}}{\mathit{s}·\mathit{s}}}$$

(5)

This confirms that the translational component $h\omega$ of the velocity spin is equivalent to the component of the velocity at any point on the rigid body in the direction of the spin axis.

If a rotating joint induces the motion of the rigid body, namely, there are only rotational inputs considered at this point, the spiral motion becomes pure rotation, at which point the motion can be described by the velocity rotation with a spin distance $h=0$:

$$\mathbf{T}=\left[\begin{array}{c}\omega \\ v\end{array}\right]=\dot{\theta }\left[\begin{array}{c}\mathit{s}\\ r\times \mathit{s}\end{array}\right]$$

(6)

If the motion of the rigid body is caused by the kinematic pair, the motion can be described by the velocity spin with infinite spin distance. In this case, its velocity spin is

$$\mathbf{T}=v\left[\begin{array}{c}0\\ \mathit{s}\end{array}\right]$$

(7)

By characterizing the positional relationship of the moving platform to the static platform by the homogeneous transformation matrix and converting it to the exponential form, the homogeneous transformation matrix of the end position to the initial position after one helical motion can be obtained as

$$\mathbf{H}=\left[\begin{array}{cc}\mathbf{R}& \mathbf{P}\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}{e}^{\left[\omega \right]\theta }& G\left(\theta \right)v\\ 0& 1\end{array}\right]$$

(8)

where $\left[\omega \right]$ denotes the zero-product matrix of $\omega$. Assuming $\omega ={\left[{\omega }_1,{\omega }_2,{\omega }_3\right]}^T$, the expression for $\left[\omega \right]$ is

$$\left[\omega \right]=\left[\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ {-\omega }_2& {\omega }_1& 0\end{array}\right]$$

(9)

$$G\left(\theta \right)=\mathit{I}\theta +(1-\cos \theta )\left[\omega \right]+(\theta -\sin \theta ){\left[\omega \right]}^2$$

(10)

where $\mathit{I}$ denotes the unit matrix.

The position change matrix of the moving platform to the static platform is $\mathbf{H}={\mathbf{H}}_{\mathbf{I}\mathbf{I}}{\mathbf{H}}_{\mathbf{I}}$, where ${\mathbf{H}}_{\mathbf{I}}$ and ${\mathbf{H}}_{\mathbf{I}\mathbf{I}}$ denote the homogeneous transformation matrix of Motion I and Motion II, respectively. It can be derived that

$${\mathbf{T}}_{\mathbf{I}}=\left[\begin{array}{c}{\omega }_1\\ v_1\end{array}\right]=\dot{\theta }\left[\begin{array}{c}{\mathit{s}}_{\mathbf{I}}\\ r\times {\mathit{s}}_{\mathbf{I}}\end{array}\right]$$

(11)

$${\mathbf{T}}_{\mathbf{I}\mathbf{I}}=v\left[\begin{array}{c}0\\ {\mathit{s}}_{\mathbf{I}\mathbf{I}}\end{array}\right]$$

(12)

For Motion I, the direction of the helix axis is the vector $\mathit{s}$ in parallel to the line $\mathit{L}$. For Motion II, the helix axis is a unit vector in the direction of $O_0{O}_e$, and its direction can be determined by the following equation:

$${\mathit{s}}_{\mathbf{I}}=\mathit{s}={\left[\begin{array}{ccc}\sin \alpha & -\cos \alpha & 0\end{array}\right]}^{\mathrm{T}}$$

(13)

$${\mathit{s}}_{\mathbf{I}\mathbf{I}}={\left[\begin{array}{ccc}\sin {\displaystyle \frac{\theta }{2}}\cos \alpha & \sin {\displaystyle \frac{\theta }{2}}\sin \alpha & \cos {\displaystyle \frac{\theta }{2}}\end{array}\right]}^{\mathrm{T}}$$

(14)

Substituting into Equations (11) and (12), it yields that

$${\mathbf{T}}_{\mathbf{I}}={\left[\begin{array}{ccc}\sin \alpha & -\cos \alpha & \begin{array}{ccc}0& 0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\right]}^{\mathrm{T}}$$

(15)

$${\mathbf{T}}_{\mathbf{I}\mathbf{I}}={\left[\begin{array}{ccc}0& 0& \begin{array}{ccc}0& \sin {\displaystyle \frac{\theta }{2}}\cos \alpha & \begin{array}{cc}\sin {\displaystyle \frac{\theta }{2}}\sin \alpha & \cos {\displaystyle \frac{\theta }{2}}\end{array}\end{array}\end{array}\right]}^{\mathrm{T}}$$

(16)

Substituting the obtained results into Equation (8), the following can be obtained:

$${\mathbf{H}}_{\mathbf{I}}=\left[\begin{array}{cccc}{\cos }^2\alpha \cos \theta -{\sin }^2\alpha & -\sin \alpha \cos \alpha \left(1+\cos \theta \right)& \cos \alpha \sin \theta & 0\\ -\sin \alpha \cos \alpha \left(1-\cos \theta \right)& {\cos }^2\alpha -{\sin }^2\alpha \cos \theta & \sin \alpha \sin \theta & 0\\ \cos \alpha \cos \theta & \sin \alpha \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right]$$

(17)

$${\mathbf{H}}_{\mathbf{I}\mathbf{I}}=\left[\begin{array}{cccc}1& 0& 0& r\cos \alpha \sin {\displaystyle \frac{\theta }{2}}\\ 0& 1& 0& r\sin \alpha \sin {\displaystyle \frac{\theta }{2}}\\ 0& 0& 1& r\cos {\displaystyle \frac{\theta }{2}}\\ 0& 0& 0& 1\end{array}\right]$$

(18)

$$\mathbf{H}=\left[\begin{array}{cccc}{\cos }^2\alpha \cos \theta -{\sin }^2\alpha & -\sin \alpha \cos \alpha \left(1+\cos \theta \right)& \cos \alpha \sin \theta & r\cos \alpha \sin {\displaystyle \frac{\theta }{2}}\\ -\sin \alpha \cos \alpha \left(1-\cos \theta \right)& {\cos }^2\alpha -{\sin }^2\alpha \cos \theta & \sin \alpha \sin \theta & r\sin \alpha \sin {\displaystyle \frac{\theta }{2}}\\ \cos \alpha \sin \theta & \sin \alpha \sin \theta & \cos \theta & r\cos {\displaystyle \frac{\theta }{2}}\\ 0& 0& 0& 1\end{array}\right]$$

(19)

Based on Equation (19), the forward kinematic relationship of the SLPM unit can be established. However, since the drive wire is only directly connected to the two uppermost and lowermost connecting plates, the motions of different units within the same module cannot be fully constrained. Therefore, when the kinematic relationship is extended from an SLPM unit to a module, the assumption of equal curvature is required; namely, the different units in the same module have the same deformation forms:

$$\left\{\begin{array}{c}{\alpha }_{j,1}={\alpha }_{j,2}={\alpha }_{j,3}\\ {\theta }_{j,1}={\theta }_{j,2}={\theta }_{j,3}\\ r_{j,1}=r_{j,2}=r_{j,3}\end{array}\right.$$

(20)

where ${\alpha }_{j,i},{\theta }_{j,i},r_{j,i}$ denote the conformal angles and distances between the end moving platform of the $i$th $(i=1, 2, 3)$ cell in the $j$th $(j=1, 2, 3)$ module and the origin static platform. According to Equation (19) the homogeneous transformation matrices among the three cells in the same module are identical. It can be obtained that

$${\mathbf{H}}_{\mathbf{j}}={\mathbf{H}}_{\mathrm{j},1}{\mathbf{H}}_{\mathrm{j},2}{\mathbf{H}}_{\mathrm{j},3}={{\mathbf{H}}_{\mathrm{j},1}}^3$$

(21)

where ${\mathbf{H}}_{\mathbf{j}}$ denotes the transformation matrix of the $j$th module, and the forward kinematic relation of a single module can be obtained according to Equations (2) to (21).

For different modules, their configuration space variables $\alpha$, $\theta$, and $r$ will take different values, but the forms all satisfy Equation (19). Substituting different values of $\alpha$, $\theta$, and $r,$ respectively, the overall homogeneous transformation matrix of the robot can be derived as

$$\mathbf{H}={\mathbf{H}}_1={\mathbf{H}}_2={\mathbf{H}}_3$$

(22)

3.2. Workspace

To analyze and predict the robot’s workspace to guide the experiments, the reachable space of the robot’s end needs to be derived. The transformation matrix of each cell’s end position relative to the base coordinates has been derived above as in Equation (19). The end position of each cell is determined by the corresponding three configuration parameters of the deflection angle $\alpha$, bending angle $\theta ,$ and chord length $r$. The maximum bending angle $\theta$, depending on the structure of the cell, is an important factor limiting the robot workspace, as shown in Figure 8.

The Monte Carlo method [43] is used to solve the robot workspace. $\left\{{\alpha }_k,{\theta }_k,r_k\right\}\left(1\le k\le 9\right)$ are randomized for each cell separately, and there is no correlation between each set of parameters. Therefore, solving the reachable space of multiple units in series requires multiplying the transformation matrices of multiple units to obtain the final cumulative transformation matrix.

In Matlab R2016b, the cumulative transformation matrix of the robot is obtained by multiplying the transformation matrices of each cell to obtain the kinematic normal solution equation. Set the random variables of each joint to be substituted into the normal solution equation to obtain the endpoint pose. Set the number of cycles $n$ to obtain the end-point pose under this cycle number, and finally output the end-point pose to obtain the spatial shape.

The cycle is randomized 200,000 times to obtain the single-module workspace result, as shown in Figure 9. The approximation of the outer boundary is in the range of $x\in [-80,80$]$mm$, $y\in \left[-80,80\right] mm$, and $z\in \left[0,160\right] mm$. The motion limit position of the single-module is shown in Figure 9c.

The transformation matrix of the robot is obtained by multiplying the transformation matrices of the three modules, and the workspace of the robot can be solved, as shown in Figure 10. The approximation of the outer boundary of the workspace is in the range of $x\in \left[-300,300\right] mm, y\in \left[-300,300\right] mm,z\in \left[0,390\right] mm$.

3.3. Inverse Kinematics

3.3.1. Inverse Kinematics of Single Module

The motion of a single module can be regarded as a superposition of the motions of three units, and its inverse kinematic relation can be solved by the projection of a spatial folding line to the rotation plane. As mentioned before, the motion of the SLPM unit moving platform to the static platform can be decomposed into a rotational motion around the origin of the static platform and a translational motion in the plane of rotation. Since the driving force will be provided by the drive wire during the moving process, the drive wire can be considered to be in a straight state; namely, the length of the drive wire in any SLPM unit is equal to the distance between the drive wire arrangement points of the upper and lower platforms, and the length of the drive wire in any module is equal to the sum of the distances between the drive wire arrangement points of the three SLPM units in the module. According to the spatial geometric relationship, the relationship between each drive rope $l_i$ and the rotating plane is and only belongs to the plane or is parallel to the plane. The single-module inverse kinematic model of the robot is shown in Figure 11.

Using the constant curvature model for the analysis, the relationship between the equivalent radius of curvature of the SLPM unit bend and the chord length corresponding to the center of the moving platform is satisfied as

$$2R\times \sin {\displaystyle \frac{\beta }{2}}=l$$

(23)

The corresponding radius of the curvature of $l_i$ is satisfied after projecting it onto the rotation plane:

$$R_i=R-d\cos \left(-\alpha +\left(i+1\right)\times {\displaystyle \frac{2\pi }{3}}\right)\left(i=\mathrm{1,2},3\right)$$

(24)

where $d$ is the distance from the center of the triangular rigid plate to the drive wire hole, namely the distance between the points $O_0$ and $A_0$.

As mentioned above, the drive wire is always under force during motion; namely, it always stays straight and its length is equal to the length of the corresponding chord of $R_i$. Each module contains three units, so that

$$r_i=2\times R_i\times \mathit{sin}{\displaystyle \frac{{\theta }_i}{2}}$$

(25)

$$l_i=3\times R_i=6\times R_i\times \mathit{sin}{\displaystyle \frac{{\theta }_i}{2}}$$

(26)

According to Equations (23) to (26), the mapping relationship from the robot configuration space to the drive space can be established. The length $l_i$ of each drive wire can be calculated by the two bending angles $\alpha$ and $\theta$ of the module and the chord length l corresponding to the origin of the moving platform.

To further derive the correspondence between the robot workspace and the length of the drive wire for the localization control of the robot end, the $x$, $y$, and $z$ coordinates of point $O_d$ are denoted by $O_{dx}$, $O_{dy}$, and $O_{dz}$, respectively. According to the geometrical relationship, it is obtained that

$$\beta =2{\cos }^{-1}\left({\displaystyle \frac{O_{dz}}{\sqrt{O_{dx}^2+O_{dy}^2+O_{dz}^2}}}\right)$$

(27)

$${\theta }_i={\displaystyle \frac{\beta }{3}} \left(i=\mathrm{1,2},3\right)$$

(28)

The angle $\alpha$ and the distance $l$ satisfy that

$$\alpha ={\mathit{tan}}^{-1}{\displaystyle \frac{O_{dy}}{O_{dx}}}$$

(29)

$$l=\sqrt{O_{dx}^2+O_{dy}^2+O_{dz}^2}$$

(30)

According to Equations (27) to (30), a correspondence from the robot configuration space to the workspace can be established. By substituting it into Equations (23) to (26), the correspondence between the robot’s module end positions and the length of the drive wire can be determined.

3.3.2. Inverse Kinematics of Robot

When the robot expands to three modules, due to the increase in degrees of freedom, the drive space variables of the robot will not be uniquely determined by the end position, and there are multiple sets of inverse solutions for the robot for the same position. In this paper, we take a set of feasible solutions in the robot’s set of solutions for a certain end position.

As shown in Figure 12, ${\mathit{h}}_{\mathit{i}}$ denotes the position vector at the end of the $i$th $(i=1,2,3)$ module, ${\mathit{r}}_{\mathit{i}}$ denotes the vector pointing to the end position from the start position of the ith module, and ${\mathit{O}}_{\mathit{d}\mathit{i}}$ denotes the end of the $\mathit{i}$th module in triangle ${\mathit{O}}_{\mathbf{0}}{\mathit{O}}_{\mathit{d}\mathbf{1}}{\mathit{O}}_{\mathit{d}\mathbf{2}}$ and triangle ${\mathit{O}}_{\mathbf{0}}{\mathit{O}}_{\mathit{d}\mathbf{2}}{\mathit{O}}_{\mathit{d}\mathbf{3}}$, respectively. According to the geometrical relationship, it is derived that

$$\left\{\begin{array}{c}‖{\mathit{h}}_{\mathbf{2}}-{\mathit{r}}_{\mathbf{1}}‖\le ‖{\mathit{r}}_{\mathbf{2}}‖\\ ‖{\mathit{h}}_{\mathbf{3}}-{\mathit{h}}_{\mathbf{2}}‖\le ‖{\mathit{r}}_{\mathbf{3}}‖\end{array}\right.$$

(31)

The geometric significance of $‖{\mathit{r}}_{\mathit{i}}‖$ is the length from the static platform of the ith module to the center of the moving platform, and it should satisfy $‖{\mathit{r}}_{\mathit{i}}‖\le {\mathit{h}}_{\mathbf{0}}$, where ${\mathit{h}}_{\mathbf{0}}$ is the distance from the static platform to the moving platform in the initial state. Therefore, when ${\mathit{r}}_{\mathbf{3}}$ is given, ${\mathit{r}}_{\mathbf{1}}$ and ${\mathit{r}}_{\mathbf{2}}$ can take any value within a reasonable range as specified in Equation (25), which corresponds to multiple solutions of inverse kinematics. In this paper, a specific solution in the actual control process is taken to uniquely determine the lengths of each drive wire.

The end of the former module is equivalent to the origin of the latter. Equations (23) to (30) can be used with the transformation of the coordinate value from the ${\mathit{O}}_{\mathbf{0}}$ coordinate system to the ${\mathit{O}}_{\mathit{e}}$ coordinate system, and then substitute it into Equations (23) to (30). The transformation is

$${\mathbf{h}}_{\mathbf{2}}^{{\mathbf{O}}_{\mathbf{0}}}={\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{1}}$$

(32)

$${\mathbf{h}}_{\mathbf{2}}^{{\mathbf{O}}_{\mathbf{e}}}={}_{{\mathbf{O}}_{\mathbf{0}}}{}^{{\mathbf{O}}_{\mathbf{e}}}\mathbf{R}\times {\mathbf{h}}_{\mathbf{2}}^{{\mathbf{O}}_{\mathbf{0}}}$$

(33)

where ${\mathit{h}}_{\mathbf{2}}^{{\mathit{O}}_{\mathbf{0}}}$, ${\mathit{h}}_{\mathbf{2}}^{{\mathit{O}}_{\mathit{e}}}$ denote the values of ${\mathit{h}}_{\mathbf{2}}$ in the ${\mathit{O}}_{\mathbf{0}}$ and ${\mathit{O}}_{\mathit{e}}$ coordinate systems, respectively, and ${}_{{\mathit{O}}_{\mathbf{0}}}{}^{{\mathit{O}}_{\mathit{e}}}\mathit{R}$ denotes the rotation change matrix for the transformation of the coordinate system ${\mathit{O}}_{\mathbf{0}}$ to the coordinate system ${\mathit{O}}_{\mathit{e}}$, which has a rotation axis of $\mathit{s}$ and a rotation angle of $-3\theta$.

Rodrigue’s rotation formula is used for obtaining a new vector after a given axis of rotation and angle of rotation in the system of three-dimensional rotation theory, which can be rewritten in the form of a matrix, and is the basic formula for rigid body motion. The formula has two expressions:

$$\mathrm{R}=\left[\mathbf{I}+\left(1-\cos \mathsf{\theta }\right)\ast {\mathbf{N}}^2+\sin \mathsf{\theta }\ast \mathbf{N}\right]$$

(34)

$$\mathrm{R}=\cos \mathsf{\theta }\ast \mathbf{I}+\left(1-\cos \mathsf{\theta }\right)\ast n\ast n^{\mathrm{T}}+\sin \mathsf{\theta }\ast n$$

(35)

where $\mathbf{I}$ is the unit matrix, $\mathsf{\theta }$ is the rotation angle, $\mathbf{N}$ is the antisymmetric matrix form of the rotation vector, and $n$ is the rotation vector. Substituting the angle of rotation $-3\theta$ and the antisymmetric matrix $\left[\mathit{s}\right]$ of $\mathit{s}$ into Equation (34) yields

$${}_{{\mathbf{O}}_{\mathbf{0}}}{}^{{\mathbf{O}}_{\mathbf{e}}}\mathbf{R}=\mathbf{I}-\sin 3\theta \times \left[\mathit{s}\right]+\left(1-\cos 3\theta \right)\times \left[\mathit{s}\right]\times \left[\mathit{s}\right]$$

(36)

Similarly, the transformation from the second to the third module can be derived, which determines the distance between the two platforms’ drive wire arrangement points above and below the SLPM unit in each module, and thus the lengths of all drive wires.

4. Finite Element Simulation

To test the performance of the improved SLPM structure, Ansys 2021 R1 is used to perform finite element analysis of the SLPM unit to test its ability to resist lateral and torsional forces as well as its stiffness characteristics by loading transverse displacements, rotational angles, and axial displacements.

For structures with small thicknesses relative to other geometrical dimensions, shell units are often used for finite element analysis instead of 3D solid units. As shown in

Table 1. Material properties.

	Material	Density (g/cm3)	Poisson’s Ratio	Thickness/mm
Connection Plate	carbon fiber	1.80	0.1	2.0
PET Film	PET	1.38	0.384	0.1
Rigid Plate	PLA	$1.26$	0.42	1.0

, the thicknesses of the PET membrane and rigid part are set, and the material properties of each part are assigned. The mesh size is 1.0 mm × 1.0 mm, and the meshing results and quality are shown in Figure 13. The average quality of the mesh is much larger than 0.7, which meets the requirements.

Select the static platform as the fixed support and apply a displacement to the moving platform in the y-direction [44] for a distance of 3 mm. Try the calculation with the default settings and reset the parameters according to the results. Turn on automatic time-stepping; set the initial sub-step to 200, the minimum sub-step to 200, and the maximum sub-step to 500. To make the calculation more accurate, turn on the weak spring to balance the small force, turn on the large deformation switch to take into account the stiffness changes generated by the unit displacements and deformations, and then re-generate a new stiffness matrix at each iteration.

Taking the displacement as the horizontal coordinate and the force as the vertical coordinate, the displacement–force curve is shown in Figure 14. When the maximum displacement reaches 3.0 mm, the force applied to the unit is 24.73 N.

The static platform is selected as the fixed support, and a rotational displacement is applied to the moving platform to rotate it 7° around the z-axis, turning on the large deformation switch and the weak spring.

Taking the rotation angle as the horizontal coordinate and the moment as the vertical coordinate, the angle–moment curve is obtained as shown in Figure 15. When the maximum rotation angle reaches 7°, the moment is about 2068 N·mm.

We conduct simulation analysis of the model unit to obtain the unit’s stiffness characteristics and the axial force required for the complete compression of a single unit. The initial state of the unit model is not subject to external forces but needs to consider the self-weight and spring pressure, so the unit is not fully elongated under this circumstance, and without taking the thickness of the connection plate into account, the height is 46 mm; the final state of the unit model is fully compressed, and without taking the thickness of the connection plate into account, the height is 14 mm. The unit before and after the compression is shown in Figure 16.

Taking the axial displacement as the horizontal coordinate and the force as the vertical coordinate, the displacement–force curve is obtained as shown in Figure 17. The linear slope of the curve, namely, the unit stiffness, is approximately constant; 6.13 N of axial force is required for the unit to be fully compressed, and 18.39 N of axial force is required for a single module to be fully compressed. A total of 6.13 N of tension is required to be provided by each drive wire, from which the required motor torque can be calculated.

5. Control System and Experiment

5.1. Control System

The correspondence from robot workspace to drive space is established by inverse kinematic analysis, and the basic robot control system framework is designed, as shown in Figure 18.

Under the constant curvature model, the robot has nine degrees of freedom and needs nine motors for control. In this paper, the motor is selected referring to the required driving force during the robot movement and the corresponding motor output torque. According to the measured results, we select 42 stepping motors with a rated torque of 0.26 N·m, which can provide the required driving force for the robot in any state, and set the fine score to eight, considering the response speed, operating noise, and safety at the same time.

To improve the space utilization rate, reduce the volume of the control box, and avoid the instability of the wires caused by small-angle routing, the control box is designed as a three-layer structure. All the threading holes are arranged to minimize the possibility of loose wires or interference. The overall assembly of the control box is shown in Figure 19. The heights of the three layers are set to 60 mm, 40 mm, and 60 mm, respectively.

Control according to the instructions of the host computer can directly determine the robot’s end position, and the operation process is more intuitive; however, a wired connection is needed, and the applicability is relatively poor. Therefore, this paper uses the handle to control the robot. In this control program, the handle receiver is directly connected to the lower computer, and the information is transmitted through the SPI communication protocol.

5.2. Experiment of Continuum Robot

5.2.1. Folding Performance Evaluation

The main application scenarios of the robots proposed in this paper are narrow, complex, and unstructured spaces, and thus require high folding ratios to reduce the risk of accidental collision damages and improve space utilization. In this paper, the folding ratio refers to the ratio of the folded length of the robot to its pre-folded length. Theoretically, the robot is fully compressed, namely, each unit is fully compressed, when the upper and lower spring frames of each unit are in contact with each other, as shown in Figure 20.

It can be concluded that the minimum length of each unit is the height of two spring frames plus the thickness of two connecting plates. Since the robot designed in this paper consists of a total of nine units, and adjacent units share one connecting plate, it is known that the height of the robot is 545.6 mm before folding and 146 mm after folding.

Thus, the theoretical folding ratio is

$${\mu }_t={\displaystyle \frac{h_1-h_2}{h_1}}={\displaystyle \frac{545.6-146}{545.6}}\times 100\mathrm{\%}=73.24\mathrm{\%}$$

(37)

Experiments were conducted on the robot to test its folding ratio. Considering gravity and other factors, the robot cannot be completely stretched when it is upright and completely folded when it is inverted, which lowers the folding ratio. In this paper, experiments are carried out separately under the two states of upright and inverted, as shown in Figure 21.

When the robot is upright, the folding ratio is

$${\mu }_{r1}={\displaystyle \frac{420-165}{420}}\times 100\mathrm{\%}=60.71\mathrm{\%}$$

(38)

When the robot is inverted, the folding ratio is

$${\mu }_{r2}={\displaystyle \frac{536-205}{536}}\times 100\mathrm{\%}=61.75\mathrm{\%}$$

(39)

The folding ratio of the robot is nearly the same in both states, around 60%; for comparison, that of the deployable robot [34] is 50% and that of the flexible robot [45] is 56%. The continuum robot designed in this paper performs better regarding the actual folding rate.

5.2.2. Bending Performance Evaluation

Bending performance is the key to the flexible operation of continuum robots. Excellent bending performance enables the robot to have high adaptability and flexibility in unstructured environments, easily bypassing obstacles to achieve precise grasping, manipulating, and detecting, which can significantly improve the success rate and operation stability. Meanwhile, the excellent bending performance can enable the robot to perform more types of tasks and scenarios and improve its environmental adaptability. In this paper, the robot was tested for bending, as shown in Figure 22. The robot achieved a bending angle of 180° with good bending performance.

In addition, the robot was inverted, and a workspace with obstacles was designed to test its bending performance, as shown in Figure 23.

5.2.3. Accuracy Testing and Optimisation

In this paper, tracking experiments for specific trajectories were also carried out on a single module to test the motion accuracy of the continuum robot. During the experiments, the end of the robot adopted the NOKOV 3D motion capture system. Twelve MARS series motion capture cameras were installed, and three reflective markers were arranged at the center and corners of the moving platform at the end of the robot to construct a rigid body model of it. When the robot moves, the camera reads the position information of the robot end by detecting the markers and transmits the data to the host computer in real time. The reflective markers are shown in Figure 24.

The experimental results are shown in Figure 25. The robot end trajectory conforms to the planning value. The average error of each path point is 5.0630 mm, the maximum error is 9.2577 mm, and the error percentage is within 10% of the length of the robot arm.

Installation errors of drive wires, cumulative errors after operation times, etc., lead to errors in continuum robots. To improve the motion accuracy of the robot, a motion error feedback link needs to be introduced to compensate for the robot’s motion.

The upper computer receives the actual position information of the robot collected by the motion capture system. It solves the error in the length of the drive wire through the inverse kinematics model, which is converted into the motor angular deviation and transmitted to the lower computer. It sends pulses to the stepper motor driver to compensate for the error, as shown in Figure 26.

After feedback compensation, the robot moves again along the straight-line trajectory planned in the previous section. The results are compared with the theoretical and uncompensated values, as shown in Figure 27.

After feedback compensation, the average position error of the robot along the planned trajectory is reduced from 5.0930 mm to 1.7936 mm, and the maximum position error is reduced from 9.2577 mm to 3.3481 mm, which is a 64.8% reduction in the average position error.

To further verify the accuracy of the feedback compensation to the robot through the robot’s end position and attitude, square and circular trajectories were further planned based on the above experiments. The square trajectory, parallel to the $xoy$ plane, has a side length of 60 mm; the center is located at the point (0,0,100), and the four edges are parallel to the $x$ and $y$ axes, with a step size of 5 mm. The circular trajectory, parallel to the $xoy$ plane, has a diameter of 60 mm; the center of the circle is located at the point (0,0,100), and the step size is 9°. The experimental results are shown in Figure 28.

As shown in Figure 27 and Figure 28, the motion accuracy of the robot is significantly improved after the introduction of the feedback compensation link. Quantitatively analyzing the above three trajectories, the mean and standard deviation of the motion accuracy of the robot before and after compensation can be obtained, as shown in

Table 2. Mean error values before and after compensation for different trajectories.

Trajectory	Mean Error before Compensation/mm	Mean Error after Compensation/mm	Average Error Reduction Rate
Linear	5.0630	1.7926	64.6%
Circular	4.5760	1.5634	65.8%
Square	4.1516	1.4532	65.0%

and

Table 3. Standard deviation of error values before and after compensation for different trajectories.

Trajectory	Standard Deviation before Compensation/mm	Standard Deviation after Compensation/mm	Average Error Reduction Rate
Linear	2.4591	0.7580	69.2%
Circular	2.3087	0.3608	84.4%
Square	1.6345	0.5207	68.1%

.

From the experimental results, it can be concluded that the error of the robot’s end motion position is controlled within 1.8 mm after compensation, the error is reduced by about 65%, and the standard deviation is greatly reduced. It is proved that the motion compensation method can not only improve the motion accuracy of the robot but also improve the motion stability of the robot.

6. Conclusions

In this paper, the SLPM unit is optimized by using a PET membrane instead of flexible hinges. A hole–shaft structure instead of a fastener connection effectively avoids the shortcomings of easily falling off and the short lifespan of the SLPM unit. The forward and inverse kinematic models of the robot were established by using the screw theory, and finite element analysis was carried out on the SLPM cell to evaluate its performance of resisting lateral and torsional forces as well as its stiffness characteristics. Finally, the control system and robot prototype were constructed, and its folding performance and bending performance were tested. In the folding test, the folding rate reached 60.71% in the upright state and 61.75% in the inverted state. In the bending test, the robot can reach a 180° bending angle in both upright and inverted states, and can move flexibly in the workspace with obstacles. Finally, this study also tested the motion accuracy of the robot and performed error analysis and optimization; the optimized robot position error is reduced by about 65%, and the standard deviation is greatly reduced, effectively improving the motion accuracy and stability of the robot.