Emulating Snake Locomotion: A Bioinspired Continuum Robot with Decoupled Symmetric Control

Abstract

Inspired by the musculoskeletal structure of snakes, this study proposes a cable-driven continuum robotic system, comprising a dual-segment continuum arm and a linear feeding module. The continuum arm provides four joint degrees of freedom through coordinated cable actuation for snake-like bending, while the feeding module enables linear translation along the Z-axis, resulting in a total of five degrees of freedom. A constant-curvature kinematic model is developed, and a real-time inverse kinematics solution based on fifth-order Taylor expansion is proposed. To enhance postural stability, a master–slave teleoperation control framework is implemented that decouples translational motion from orientation control. Leveraging the geometric symmetry of its dual-segment design, the system achieves consistent end-effector orientation by coordinating bending angles and rotation directions between segments. Simulation and experimental results validate the accuracy of the kinematic model and demonstrate the robot’s capability for dexterous, stable movements in confined environments. The proposed continuum robot offers high positioning accuracy, structural adaptability, and strong potential for bioinspired applications in endoscopy and minimally invasive surgical procedures.

1. Introduction

Traditional rigid-link robotic manipulators have been widely adopted in industrial, medical, and service applications due to their mature kinematics, precise control, and structural simplicity [1]. However, their inherent reliance on discrete joints and bulky mechanical structures limits their dexterity, flexibility, and adaptability—especially in cluttered or constrained environments where smooth, spatially continuous motion is required [2]. These limitations have prompted the exploration of alternative designs that can offer greater compliance and environmental adaptability [3].

In contrast, nature offers a rich source of inspiration for soft and flexible actuation. In particular, snakes exhibit exceptional locomotion and manipulation capabilities by coordinating muscle contractions along their vertebral columns [4,5]. This distributed actuation system allows them to generate smooth body curvatures and navigate through highly complex, narrow environments. This musculoskeletal system integrates a segmented yet flexible backbone with directionally asymmetric muscle contractions, allowing coordinated, multi-modal locomotion while preserving head orientation and body stability [6,7].

Drawing on these biological principles, cable-driven continuum robots have emerged as promising candidates for tasks that demand compliant interaction, spatial adaptability, and precise control in confined settings [8,9]. These robots emulate the structural and functional properties of snake locomotion; their flexible backbones serve as artificial spines, while cable tension mimics the pulling effect of muscle contractions. As a result, continuum robots have been widely studied in diverse domains, including navigation [10], exploration [11], search and rescue [12,13,14], manufacturing and assembly [15,16], equipment maintenance and overhaul [17,18], and medical surgery [19]. Among them, snake-inspired continuum robots particularly excel for their potential to operate in constrained and tortuous environments, enabling biologically inspired robotic systems to adapt and perform in unstructured environments.

Continuum robots have been extensively explored in minimally invasive medical procedures as well as in other scenarios requiring high flexibility and dexterity, such as confined-space exploration and soft-environment interaction. For instance, Burgner et al. [20] developed a concentric tube robot for cerebral hemorrhage removal, enabling directional suction via a bendable inner tube while ensuring sterilization and modularity. Similarly, Hansen Medical’s Magellan system [21] adopts a tendon-driven catheter with coaxial wire actuation to facilitate cardiovascular interventions, with model-based control derived from beam mechanics.

A triple-arm continuum robot platform for single-port laparoscopic surgery was introduced by researchers at the Technical University of Munich [22], utilizing cable-driven NiTi alloy tubes to achieve dexterous manipulation through a compact 12 mm access port. Xu et al. [23] proposed a dual-arm endoscopic system with modular continuum segments supported by superelastic backbones, enabling flexible vision and operation in endonasal procedures.

In high-precision domains such as neurosurgery, Rosen et al. [24] designed a highly articulated robot with 12 degrees of freedom and integrated 3D vision, allowing tool deployment through narrow skull entry points. Across these platforms, key structural features—such as distributed compliance, modular construction, and tendon actuation—mirror biological musculoskeletal principles, especially those observed in snake locomotion. These designs not only support miniaturization and compliance, but also set the stage for advanced control strategies that exploit geometric symmetry and actuation asymmetry, as explored in this work.

Cable-driven continuum robots, owing to their semi-rigid backbones and structural simplicity, allow tractable kinematic modeling and are among the most widely adopted actuation mechanisms for continuum systems [25,26,27,28,29].

Recent studies have further explored shape reconstruction and feedback control in continuum robots through various sensing strategies. For example, hybrid systems combining displacement sensors with visual feedback [30] and flexible electronic textiles [31] have shown promise in enhancing configuration awareness and enabling real-time motion control. However, these approaches often face limitations in response latency, calibration complexity, and integration with multi-segment architectures, which restrict their practical deployment in highly flexible robotic systems.

Leveraging snake-inspired musculoskeletal principles, this work presents a novel cable-driven continuum robotic arm with a high aspect ratio and modular multi-segment architecture. The design emphasizes spatial flexibility, directional consistency, and structural symmetry, making it well-suited for navigation and manipulation tasks in confined curvilinear environments. While existing control approaches for continuum robots primarily focus on the position or velocity of the end-effector, they often overlook the optimization of the robot’s full-body configuration during motion. Recent advances have introduced learning-based strategies, such as reinforcement learning methods, to improve control flexibility [32]. However, these approaches often rely on simulation-based training, lack analytical interpretability, and do not explicitly guarantee smooth or stable configurations—factors that are critical for avoiding excessive curvature and ensuring mechanical safety in physical implementations. Moreover, few studies have addressed posture stability or ensured unique solutions in inverse kinematics, which are essential for real-time teleoperation in constrained environments [33]. A master–slave teleoperation framework is employed to control the robot’s distal tip position, with kinematic modeling and simulation validating the system’s accuracy and responsiveness. This continuum robotic system demonstrates ease of operation, structural scalability, and strong potential for applications in both bioinspired navigation and soft-environment interactions. In particular, the system leverages structural symmetry and directional asymmetry—key features in snake locomotion—to enhance posture stabilization and improve motion efficiency in constrained spaces.

The remainder of this paper is organized as follows. Section 2 introduces the design of the continuum robot, detailing its bioinspired principles and multi-segment structural configuration. Section 3 presents the kinematic modeling, including both single- and multi-segment formulations, workspace analysis, inverse kinematics simulations, and the master–slave control strategy. Section 4 describes the experimental validation of the continuum arm, focusing on its bending, rotational, and end-effector positioning performance. Section 5 provides a comprehensive discussion of the results, analyzes the system’s limitations, and outlines future research directions. Section 6 concludes the paper.

2. Continuum Robot Design

2.1. Bioinspired Design Principles

Drawing inspiration from the musculoskeletal system of snakes—where a flexible spine works in coordination with surrounding muscles to produce smooth, multi-directional curvatures (as illustrated in Figure 1)—the proposed continuum robot adopts a segmented, cable-driven structure that emulates these motion characteristics. This biologically inspired design allows the robot to perform continuous and compliant bending, enabling it to traverse long, narrow, and curved environments. The system’s structural symmetry and modular architecture further contribute to consistent directional control and enhanced scalability, making it well-suited for tasks requiring dexterity and adaptability in constrained spaces.

2.2. Structural Design of Multi-Segment Cable-Driven Continuum Robots

As shown in Figure 2, the robotic system comprises three main components: a flexible continuum robotic arm, a driving module, and a feeding module. The continuum arm replicates the segmented and flexible morphology of a snake’s body, enabling bending motions through coordinated cable actuation. A central flexible support rod, mimicking the snake’s spine, serves as the backbone of the arm and provides structural stiffness. The actuation cables emulate muscle contractions to produce smooth bending via a series of flexible segments that approximate joint-like behavior. Here, the term “joint” refers to each actively bendable segment of the continuum structure, serving as a unit that is functionally equivalent to joints in traditional rigid-link robots. The driving module generates the required tension for actuation, and the feeding module allows for the linear advancement of the entire robotic structure. The detailed structural design of the continuum module is presented in the following section.

The continuum arm consists of a flexible support rod, cable-guided discs, and actuating Nitinol (NiTi) wires. The support rod is made from a superelastic NiTi wire with a diameter of 1.2 mm and a length of 400 mm. One end of the rod is anchored to the central hole of a transition disc, offering axial support and enhancing the structural stiffness required for effective bending.

To ensure adequate flexibility, 20 wire-guided discs are uniformly distributed along the arm body at 20 mm intervals. These components provide multiple degrees of freedom, enhancing the robot’s adaptability to diverse application scenarios. As shown in Figure 2a, the first 10 discs constitute the first segment, while the remaining 10 form the second. Each disc, detailed in Figure 2b, features an outer diameter of 12 mm and a thickness of 2 mm. Six through-holes, each with a diameter of 0.6 mm, are arranged on a 10 mm pitch circle at 60° intervals, allowing for a symmetrically distributed three-wire actuation configuration at 120°. Importantly, each disc is rigidly fixed onto the flexible backbone rod, meaning it cannot rotate or slide relative to the backbone. This design ensures that the continuum structure cannot undergo axial torsion or twisting.

Each 0.4 mm-diameter NiTi actuation wire is fixed at one end by the crimp plate and slider of the driving module. The other end passes through the holes in each wire-guided disc and is anchored either at the 10th disc (for the first segment) or at the end disc (for the second segment), via the transition disc. By selectively pulling the actuation wires, the continuum arm bends in the corresponding direction. Adjusting the relative lengths of the wires results in the controlled bending of each segment, allowing for precise manipulation of the arm’s end-effector posture. The continuum arm incorporates two segments and offers four degrees of freedom in total.

The driving module comprises a lead screw motor assembly, a crimp plate, and a stainless steel enclosure, as illustrated in Figure 2c. The stainless steel plate forms a rectangular frame, within which the lead screw motor modules are securely mounted to the inner walls using bolts and nuts through pre-drilled holes, ensuring the precise positioning of each motor. A total of six lead screw motors are employed, arranged in two layers—three motors per layer. The motor positioned at the center of the top layer is designated as Motor No. 1. The remaining motors are numbered in a clockwise sequence based on their placement on the motor holder. Motors No. 2, 4, and 6 form the first actuation group, responsible for driving the first continuum segment. Motors No. 1, 3, and 5 comprise the second group, which controls the second continuum segment.

Each lead screw motor module primarily consists of a stepper motor, a screw, a slider, and a crimp plate. The crimp plate is used to clamp the initial section of the actuation wire, ensuring the effective transmission of motion.

The feeding module mainly consists of a stepper motor linear module and a base, as shown in Figure 2d. Since the accuracy of the feed direction of the continuum robot is determined by the feed unit, it adopts a ball screw with relatively high accuracy.

3. Kinematic Modeling of the Continuum Robot

3.1. Kinematic Modeling of a Single-Segment Continuum Arm

Terminology Clarification: Throughout this paper, the term joint refers to each actively bendable segment of the continuum robotic arm. Although the structure lacks discrete joints like those in rigid-link robots, each segment is independently actuated and exhibits controlled bending behavior. Therefore, it is functionally treated as a “joint” for the purposes of kinematic modeling and joint-space mapping.

The segmented constant curvature modeling approach is a widely adopted assumption in the design of continuum robots [34]. By approximating the continuum structure as a series of discrete arcs, this method reduces the complexity of the robot’s state vector and is extensively applied in real-time control algorithms and other scenarios demanding high-speed computation [35,36,37].

To precisely describe the kinematic behavior of a continuum robotic arm and enable steady-state control with minimal computational effort, the following assumptions are made based on the constant curvature model [36,38]:

(a)

The continuum arm has a uniform and symmetrical geometry;

(b)

External loading effects are negligible;

(c)

The arm’s trunk maintains a constant length during bending, and both its curvature and that of the driving cables are smooth curves with equal curvature;

(d)

Torsional effects along the longitudinal axis of the arm are negligible.

A simplified model of the continuum unit is shown in Figure 3a. θ ∈ [0, π] is the bending angle in the curvature plane, indicating the bending angle of the robotic arm. φ ∈ [0, 2π] is the directional angle, referring to the angle between the plane where the main trunk is located along the bending of the robotic arm and the positive X₀ axis, i.e., the rotation angle of the robotic arm. Any bending motion of the single-segment continuum arm in space can be described by combining θ and φ to obtain a two-degree-of-freedom model.

Firstly, the mapping relationship between the joint and operational space of the continuum arm is analyzed. When the robotic arm bends, the transformation from the base coordinate system {O₀} to the final coordinate system {O₁} can be achieved by the following four basic transformations:

(a)

Translate the origin O₀ to O₁, which represents the positional transformation of the end relative to the base coordinate system;

(b)

Rotate counter-clockwise about the Z-axis by an angle φ;

(c)

Rotate counter-clockwise about the Y-axis by θ;

(d)

Rotate −φ around the Z-axis to ensure that the bending cross-section is not affected by torsional effects. The three rotational transformations are attitude transformations of the end concerning the base coordinate system.

For ease of writing, cos and sin in the following equations are abbreviated as c and s, respectively. L is the length of an individual segment of the continuum. From the geometric relationship, the homogeneous transformation matrix of the origin of the end coordinate system with respect to the base coordinate system can be introduced.

$${}_{O_1}{}^O\mathit{T}=\left[\begin{array}{cccc}{\mathrm{c}}^2\phi \mathrm{c}\theta +{\mathrm{s}}^2\phi & \mathrm{c}\phi \mathrm{s}\phi \mathrm{c}\theta -\mathrm{c}\phi \mathrm{s}\phi & \mathrm{c}\phi \mathrm{s}\theta & {\displaystyle \frac{L}{\theta }}\left(1-\mathrm{c}\theta \right)\mathrm{c}\phi \\ \mathrm{c}\phi \mathrm{s}\phi \mathrm{c}\theta -\mathrm{c}\phi \mathrm{s}\phi & {\mathrm{s}}^2\phi \mathrm{c}\theta +{\mathrm{c}}^2\phi & \mathrm{s}\phi \mathrm{s}\theta & {\displaystyle \frac{L}{\theta }}\left(1-\mathrm{c}\theta \right)\mathrm{s}\phi \\ -\mathrm{c}\phi \mathrm{s}\theta & -\mathrm{s}\phi \mathrm{s}\theta & \mathrm{c}\theta & {\displaystyle \frac{L}{\theta }}\mathrm{s}\theta \\ 0& 0& 0& 1\end{array}\right]$$

(1)

The inverse mapping from the operation space to the joint space is intended to solve for the bending angle θ and the rotation angle φ for the given end position of the continuum robotic arm, i.e., to solve the inverse kinematics of the continuum robotic arm. Let n, o, and a be the unit vectors corresponding to the X, Y, and Z axes, respectively, and p be the position vector of the end of the robotic arm, after which the pose matrix of the endpoint T_p can be expressed as follows:

$${\mathit{T}}_p=\left[\begin{array}{cccc}{n}_x& o_x& a_x& p_x\\ n_y& o_y& a_y& p_y\\ n_z& o_z& a_z& p_z\\ 0& 0& 0& 1\end{array}\right]$$

(2)

When the homogeneous transformation matrix is equal to the pose matrix, a system of nonlinear equations for solving the inverse kinematics can be obtained ${}_{O_1}{}^O\mathit{T}={\mathit{T}}_p$, which can be solved by making the corresponding elements equal,

$$\left\{\begin{array}{l}\theta =\arccos a_z\\ \phi =\mathrm{atan}2\left(p_y,p_x\right)\end{array}\right.$$

(3)

where the bending angle is θ ∈ [0, π], and the rotation angle is φ ∈ [0, 2π].

Figure 3b shows a cross-section of Figure 3a along X₁O₁Y₁, where φ_i is the angle between the bending direction and the center of the hole of the ith actuation wire, r is the radius of curvature at the center axis of the main trunk, r_i is the radius of curvature at the center axis of the ith actuation wire, and d is the distance from the hole of the wire to the center axis of the main trunk.

It can be seen from the geometric relations in Figure 3b that $r_i-r=d\cos {\phi }_i,{\phi }_1=\phi ,{\phi }_2={\displaystyle \frac{4\pi }{3}}+\phi ,{\phi }_3={\displaystyle \frac{2\pi }{3}}+\phi$, the original length of the actuation wire of the continuum robotic arm without bending is $l=\theta r$, the length of the ith actuation wire in bending l_i is $l_i=\theta r_i$, and the change of the length of the ith wire Δl_i is

$$\Delta l_i=l_i-l=\theta d\cos {\phi }_i$$

(4)

Equation (4) is the mapping from the joint space to the drive space of the single-segment continuum.

3.2. Kinematic Modeling of a Multi-Segment Continuum Robot

Inspired by the S-shaped postures commonly exhibited by snakes during locomotion and spatial navigation, this system employs a motion strategy in which the two continuum segments bend at equal angles but with opposite bending directions. This configuration ensures that the distal tip maintains a consistent forward-facing orientation while the body achieves the necessary curvature to navigate confined or tortuous paths. The following kinematic model describes how this symmetric bending strategy is realized in the continuum robotic arm.

The coordinate system with the fixed continuum robotic arm system as the research object is established, as shown in Figure 4. {D₀} is the base coordinate system; {D₁} is the coordinate system at the end of the first continuum segment; {D₂} is the coordinate system at the end of the second continuum segment. The origin and axes are set with reference to the way the coordinate system was set up in the kinematic analysis of the single-segment continuum in the previous section.

The mapping relationship from the joint space to the operational space can be characterized through the kinematics analysis of the tandem robot. The position of the ith segment relative to the i − 1 segment of the continuum can be expressed by the homogeneous coordinates transformation ${}_i{}^{i-1}\mathit{T}$, and the bending direction and bending angle of each segment relative to the previous segment of the continuum can be expressed by φ_i and θ_i, respectively. Then,

$${}_i{}^{i-1}\mathit{T}=\left[\begin{array}{cccc}{\mathrm{c}}^2{\phi }_i\mathrm{c}{\theta }_i+{\mathrm{s}}^2{\phi }_i& \mathrm{c}{\phi }_i\mathrm{s}{\phi }_i\mathrm{c}{\theta }_i-\mathrm{c}{\phi }_i\mathrm{s}{\phi }_i& \mathrm{c}{\phi }_i\mathrm{s}{\theta }_i& {\displaystyle \frac{L}{{\theta }_i}}\left(1-\mathrm{c}{\theta }_i\right)\mathrm{c}{\phi }_i\\ \mathrm{c}{\phi }_i\mathrm{s}{\phi }_i\mathrm{c}{\theta }_i-\mathrm{c}{\phi }_i\mathrm{s}{\phi }_i& {\mathrm{s}}^2{\phi }_i\mathrm{c}{\theta }_i+{\mathrm{c}}^2{\phi }_i& \mathrm{s}{\phi }_i\mathrm{s}{\theta }_i& {\displaystyle \frac{L}{{\theta }_i}}\left(1-\mathrm{c}{\theta }_i\right)\mathrm{s}{\phi }_i\\ -\mathrm{c}{\phi }_i\mathrm{s}{\theta }_i& -\mathrm{s}{\phi }_i\mathrm{s}{\theta }_i& \mathrm{c}{\theta }_i& {\displaystyle \frac{L}{{\theta }_i}}\mathrm{s}{\theta }_i\\ 0& 0& 0& 1\end{array}\right]$$

(5)

Combined with the one feed degree of freedom in the Z-direction provided by the ball screw, the homogeneous transformation matrix ${}_2{}^0\mathit{T}$ of the coordinate system at the end of the second continuum segment with respect to the base coordinate system can be expressed as

$${}_2{}^0\mathit{T}\left({\phi }_1,{\theta }_1;{\phi }_2,{\theta }_2\right)={\mathit{T}}_Z·{}_1{}^0\mathit{T}\left({\phi }_1,{\theta }_1\right)·{}_2{}^1\mathit{T}\left({\phi }_2,{\theta }_2\right)$$

(6)

Among these,

$${\mathit{T}}_{\mathit{Z}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right]$$

(7)

Small deviations in the distal posture of the continuum robotic arm can result in significant changes to its overall shape, increasing the risk of unintended contact with surrounding tissues.

Inspired by the observation that snakes maintain a forward-facing head orientation despite complex body curvatures, we designed a system that adopts a decoupled control strategy—a consistent end-effector attitude is maintained by coordinating the bending angles and rotation directions of the continuum segments, while specific orientation adjustments are independently executed by a distal actuator, such as the da Vinci EndoWrist surgical instrument [39].

As illustrated in Figure 5, this approach allows the robot to preserve a stable orientation at the tip, even when its body undergoes significant postural changes.

Since the continuum robot only realizes the translation function, the direction of the X, Y, and Z axes in the end coordinate system of the continuum robot is always the same as the direction of the X, Y, and Z axes of the base plate coordinate system. Consequently, the rotation matrix representing the relationship between these two coordinate systems remains

$$\mathbf{R}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

(8)

The following relationship can be obtained:

$$\left\{\begin{array}{l}{\theta }_1={\theta }_2\hfill \\ {\phi }_1={\phi }_2-\pi \hfill \end{array}\right. \mathrm{or} \left\{\begin{array}{l}{\theta }_1={\theta }_2\hfill \\ {\phi }_1={\phi }_2+\pi \hfill \end{array}\right.$$

(9)

This equation indicates that the two segments of the continuum robotic arm bend at the same angle but with opposite rotation directions. As illustrated in Figure 6, such S-shaped postures represent a biologically effective solution for balancing flexibility and directional stability.

As a result, the end position [x₂, y₂, z₂] of the entire arm becomes twice the end position [x₁, y₁, z₁] of the first segment,

$$\left[\begin{array}{c}{x}_1\\ y_1\\ z_1-z_0\end{array}\right]={\displaystyle \frac{1}{2}}\left[\begin{array}{c}{x}_2\\ y_2\\ z_2-z_0\end{array}\right]$$

(10)

where z₀ is the absolute position of the ball screw feed in the Z-axis direction. Based on this, and in combination with Equation (1), a system of equations can be established whereby

$$\left\{\begin{array}{l}{\displaystyle \frac{x_2}{2}}={\displaystyle \frac{L}{{\theta }_1}}\left(1-\cos {\theta }_1\right)\cos {\phi }_1\hfill \\ {\displaystyle \frac{y_2}{2}}={\displaystyle \frac{L}{{\theta }_1}}\left(1-\cos {\theta }_1\right)\sin {\phi }_1\hfill \\ {\displaystyle \frac{z_2-z_0}{2}}={\displaystyle \frac{L}{{\theta }_1}}\sin {\theta }_1\hfill \end{array}\right.$$

(11)

Subsequently, the following equation can be obtained:

$${\phi }_1=\mathrm{atan}2\left({\displaystyle \frac{y_1}{x_1}}\right)=\mathrm{atan}2\left({\displaystyle \frac{y_2}{x_2}}\right)$$

(12)

To address the master–slave control requirement of the whole system, a stable and efficient real-time online solution method for θ₁ is necessary. In this paper, firstly, the Newton iterative method is employed to sequentially approximate each zero point,

$$\left\{\begin{array}{l}f\left({\theta }_1\right)={\displaystyle \frac{2L\cos {\phi }_1\left(1-\cos {\theta }_1\right)}{{\theta }_1}}-x_2\hfill \\ f^{\prime }\left({\theta }_1\right)={\displaystyle \frac{2L\cos {\phi }_1\left({\theta }_1\sin {\theta }_1+\cos {\theta }_1-1\right)}{{\theta }_1^2}}\hfill \end{array}\right.$$

(13)

where θ₁ is an iteration variable and x₂, L represents the known quantities. The iteration terminates when the error between the values of two consecutive iterations falls below 0.001, and the approximate value of the last iteration is assigned as a return value for the bending angle θ₁. The initial interval of θ₁ is [0, π], and the Newton iterative method can achieve local convergence only when the initial value of θ₁ is close enough to the solution. However, for the online inverse kinematics solution in master–slave control, both f(x) and the interval of the solution are constantly changing, increasing the difficulty of selecting the initial value. The wrong initial value selection leads to the non-convergence of the iteration, obtaining the wrong solution. Therefore, this paper seeks a more stable inverse kinematics solution method for the continuum robotic arm.

As θ₁ approaches zero, all expressions to the right of the middle sign in Equation (11) become zero divided by zero, resulting in a numerical singularity. In order to avoid such singularities throughout the analysis, a fifth-order Taylor expansion of θ₁ is performed [40], whereby

$$\cos {\theta }_1=1-{\displaystyle \frac{1}{2!}}{\theta }_1^2+{\displaystyle \frac{1}{4!}}{\theta }_1^4$$

(14)

$$\sin {\theta }_1={\theta }_1-{\displaystyle \frac{1}{3!}}{\theta }_1^3+{\displaystyle \frac{1}{5!}}{\theta }_1^5$$

(15)

The new equations corresponding to the end positions of the slave continuum robot arm and θ₁ and φ₁ are obtained.

$$\left\{\begin{array}{c}{\displaystyle \frac{x_2}{2}}={\displaystyle \frac{L}{24}}\left(-{\theta }_1^3+12{\theta }_1\right)\cos \left({\phi }_1\right)\\ {\displaystyle \frac{y_2}{2}}={\displaystyle \frac{L}{24}}\left(-{\theta }_1^3+12{\theta }_1\right)\sin \left({\phi }_1\right)\\ {\displaystyle \frac{z_2-z_0}{2}}={\displaystyle \frac{L}{120}}\left({\theta }_1^4-20{\theta }_1^2+120\right)\end{array}\right.$$

(16)

This fifth-order Taylor expansion approximation is used to analyze the error. The constant term in the equation is omitted to study the functional relationship between the end X-coordinate and θ₁. As shown in Figure 7, the images of the function before and after the expansion are compared. It can be seen that when ${\theta }_1={\displaystyle \frac{\mathrm{\pi }}{2}}$, the position error of the X-axis direction at the end of the continuum arm is 1.99%. When ${\theta }_1={\displaystyle \frac{3\pi }{5}}$, the position error of the X-axis direction at the end of the continuum arm is 4.42%.

The designed continuum robotic arm cannot completely bend the trunk to 180° in the actual bending movement, and the movement of the continuum robotic arm is limited to a very small range for general endoscopic surgery. Therefore, this paper restricts the range of bending angle movement for the continuum to [$0,{\displaystyle \frac{3\pi }{5}}$], ensuring both an ample workspace for the continuum robotic arm and a controllable error range. This value was chosen empirically to balance bending range and kinematic accuracy, based on simulation and experimental performance.

It is observed that the first term in Equation (16) is a univariate cubic equation lacking a quadratic term. By rearranging, we define

$$f\left({\theta }_1\right)={\theta }_1^3-12{\theta }_1+{\displaystyle \frac{12x_2}{L\cos {\phi }_1}},p=-12,q={\displaystyle \frac{12x_2}{L\cos {\phi }_1}}$$

(17)

Taking the derivative with respect to $f\left({\theta }_1\right)$, we find that it monotonically decreases over the domain ${\theta }_1\in \left[0,{\displaystyle \frac{3\pi }{5}}\right]$, and

$$f\left(0\right)·f\left({\displaystyle \frac{3\pi }{5}}\right)<0$$

(18)

Therefore, according to the Intermediate Value Theorem, the first term in Equation (16) has a unique real number solution within its domain of definition, and its solution within the complex number domain is

$$\left\{\begin{array}{l}{r}_1=\sqrt[3]{-{\displaystyle \frac{q}{2}}+\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}+\sqrt[3]{-{\displaystyle \frac{q}{2}}-\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}\hfill \\ r_2={\displaystyle \frac{-1+\sqrt{3}i}{2}}\sqrt[3]{-{\displaystyle \frac{q}{2}}+\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}+{\displaystyle \frac{-1-\sqrt{3}i}{2}}\sqrt[3]{-{\displaystyle \frac{q}{2}}-\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}\hfill \\ r_3={\displaystyle \frac{-1-\sqrt{3}i}{2}}\sqrt[3]{-{\displaystyle \frac{q}{2}}+\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}+{\displaystyle \frac{-1+\sqrt{3}i}{2}}\sqrt[3]{-{\displaystyle \frac{q}{2}}-\sqrt[2]{{\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}}}\hfill \end{array}\right.$$

(19)

Among them, r₁, r₂ and r₃ are the three roots of θ₁. The discriminant is

$$\Delta ={\displaystyle \frac{q^2}{4}}+{\displaystyle \frac{p^3}{27}}$$

(20)

If Δ > 0, r₁ is the only real root. If Δ = 0, all three solutions are real roots and r₂ = r₃. If Δ < 0, all three solutions are real roots and unequal. When all three solutions are real roots, the solution within the domain of the definition must be used as the unique solution to the equation. After determining the other three joint parameters, θ₁ is substituted into the third term of Equation (16) to find the absolute position feed z₀ of the ball screw.

According to the mapping relationship from the joint space to the drive space of the single-segment continuum in Equation (4), the length variation ΔL_ij of the three actuation wires of the first-segment continuum robotic arm is

$$\left\{\begin{array}{l}\Delta L_{11}=\Delta L_1={\theta }_{1}d\cos {\phi }_1\hfill \\ \Delta L_{12}=\Delta L_3={\theta }_{1}d\cos {\phi }_2={\theta }_{1}d\cos \left({\displaystyle \frac{4\pi }{3}}+{\phi }_1\right)\hfill \\ \Delta L_{13}=\Delta L_5={\theta }_{1}d\cos {\phi }_3={\theta }_{1}d\cos \left({\displaystyle \frac{2\pi }{3}}+{\phi }_1\right)\hfill \end{array}\right.$$

(21)

The second section of the continuum is affected by the bending variation of the first section, and the variation ΔL_ij of its three actuation wires is

$$\left\{\begin{array}{l}\Delta L_{21}=\Delta L_2={\theta }_{1}d\cos \left({\displaystyle \frac{5\pi }{3}}+{\phi }_1\right)+{\theta }_{2}d\cos \left({\displaystyle \frac{5\pi }{3}}+{\phi }_2\right)\hfill \\ \Delta L_{22}=\Delta L_4={\theta }_{1}d\cos \left(\pi +{\phi }_1\right)+{\theta }_{2}d\cos \left(\pi +{\phi }_2\right)\hfill \\ \Delta L_{23}=\Delta L_6={\theta }_{1}d\cos \left({\displaystyle \frac{\pi }{3}}+{\phi }_1\right)+{\theta }_{2}d\cos \left({\displaystyle \frac{\pi }{3}}+{\phi }_2\right)\hfill \end{array}\right.$$

(22)

3.3. Workspace Analysis

The workspace of a two-segment continuum robotic arm is simulated using the Monte Carlo method, and ${\theta }_1\in \left[0,{\displaystyle \frac{3\pi }{5}}\right]$, φ₁ ∈ [0, 2π], ${\theta }_2\in \left[0,{\displaystyle \frac{3\pi }{5}}\right]$ and φ₂ ∈ [0, 2π] are the ranges of joint values of the two continuum robotic arms. The z-axis feed is from 0 to 0.3 m. Then, 50,000 groups of values are randomly taken in its range, and each group of θ₁, φ₁, θ₂ and φ₂ is substituted into Equation (6). The calculated coordinates of the end positions are labeled by Matlab R2022b (The MathWorks, Natick, MA, USA) to generate a point cloud map of the workspace, as shown in Figure 8.

As shown in Figure 8, the workspace of the two-segment continuum is peach-shaped. After the feeding module is mounted, the workspace in the X-axis direction is approximated to be [−277.78, 277.78] mm, the Y-axis direction is approximated to be [−277.78, 277.78] mm, and the workspace in the Z-axis direction is approximated to be [201.82, 700] mm.

3.4. Continuum Inverse Kinematics Simulation Example and Analysis

In this paper, simulations are performed in Matlab to further verify the correctness and efficiency of the inverse kinematics solution method for the continuum mechanical arm based on the fifth-order Taylor expansion. A spatial helix is initially set as the continuum end trajectory, and the helix equation is shown in Equation (23). The spatial spiral trajectory is shown in Figure 9a.

$$\left\{\begin{array}{l}x={\displaystyle \frac{86400}{\pi t^3}}\left[1-cos\left({\displaystyle \frac{\pi t^3}{216}}\right)\right]cos\left({\displaystyle \frac{\pi t^3}{108}}\right)\hfill \\ y={\displaystyle \frac{86400}{\pi t^3}}\left[1-cos\left({\displaystyle \frac{\pi t^3}{216}}\right)\right]sin\left({\displaystyle \frac{\pi t^3}{108}}\right)\hfill \\ z={\displaystyle \frac{86400}{\pi t^3}}sin\left({\displaystyle \frac{\pi t^3}{216}}\right)\hfill \end{array}\left(0\le t\le 5\right)\right.$$

(23)

The starting position at the end of the continuum robot arm is (0, 0, 400), and the initial joint parameters are set as [θ₁, φ₁, θ₂, φ₂] = [0, 0, 0, 0]. The points on the helix are collected with the interval time t = 0.1 s. Subsequently, the position coordinates (x₀, y₀, z₀) of these points are input to the inverse kinematics solver module to determine the point positions at the end of the continuum and to solve for the four joint parameters [θ₁, φ₁, θ₂, φ₂]. Finally, the position of the continuum robot arm in 0–5 s is plotted, as shown in Figure 9b.

As shown in Figure 9b, the position and attitude at the end of the continuum robotic arm are separated using the method proposed in this paper, with the end position trajectory always falling on the helix. In addition, the end attitude remains vertically upward, consistent with the attitude of the base coordinate system.

3.5. Master-Slave Teleoperation Control Study

The Geomagic Touch device in the 3D systems is selected as the main controller to construct the master–slave teleoperation control system, as shown in Figure 10.

The end position of the manipulator pen is the zero point of the touch end position coordinates, as shown in Figure 11. The starting point of Link 1 is defined as the origin O of the base coordinate system, the Z-axis is perpendicular to the ground in an upward direction, the Y-axis points to the operator, and the X-axis is determined based on the right-hand rule. β₁ is the rotation angle around the Z-axis, β₂ is the rotation of link 1 around the X-axis, and β₃ is the rotation of link 2 around the X-axis. L₁ and L₂ are the lengths of link 1 and link 2, respectively, and L₃ and L₄ are the offsets at the end of the manipulator pen with respect to the origin of the coordinates, O (the plane illustrated is the YOZ-plane). L₁ = L₂ = 133.35 mm, L₃ = 23.35 mm, and L₄ = 168.35 mm.

From the geometric relationship, we can determine the following:

$$\left\{\begin{array}{l}x=-sin{\beta }_1\left(L_{2}sin{\beta }_3+L_{1}cos{\beta }_2\right)\hfill \\ y=-L_{2}cos{\beta }_3+L_{1}sin{\beta }_2+L_3\hfill \\ z=L_{2}cos{\beta }_{1}sin{\beta }_3+L_{1}cos{\beta }_{1}cos{\beta }_2-L_4\hfill \end{array}\right.$$

(24)

The range of motion for both the Touch stylus end and the continuum robotic arm end along the X, Y, and Z axes is summarized in

Table 1. Range of motion of Touch stylus end and continuum robotic arm end in the X, Y, and Z directions.

Direction	Touch Stylus End Movement Range/m	Range of Motion at the End of the Continuum Arm/m
X	−0.2666 to 0.1670	−0.27778 to 0.27778
Y	−0.2055 to 0.2536	−0.27778 to 0.27778
Z	−0.4351 to 0	0.20182 to 0.7

. This information helps to compare the workspace of the master device and the robotic arm, which is essential for scaling during teleoperation.

In order to enhance the teleoperation experience, an appropriate set of scaling coefficients was determined through multiple experimental trials. These coefficients were designed to proportionally enlarge the motion range of the master device (Touch stylus) so that it could effectively span the reachable workspace of the continuum robotic arm. The final values for the scaling coefficients k_x, k_y, and k_z—which relate the positional increments of the Touch stylus in the X, Y, and Z directions to the increments of the single-segment continuum’s joint variables—are listed in

Table 2. Selection of scale factor.

${\mathit{k}}_{\mathit{x}}$	${\mathit{k}}_{\mathit{y}}$	${\mathit{k}}_{\mathit{z}}$
2 (rad/m)	1.3 (rad/m)	1.3

.

Therefore, the mapping rule can be obtained as shown in Equation (25) below.

$$\left[\begin{array}{c}{x}_s\\ y_s\\ z_s\end{array}\right]=\left[\begin{array}{ccc}2& 0& 0\\ 0& 1.3& 0\\ 0& 0& 1.3\end{array}\right]·\left[\begin{array}{c}{x}_m\\ y_m\\ z_m\end{array}\right]$$

(25)

where S = [x_s, y_s, z_s] is the desired position vector of the Touch stylus end position obtained from the end of the continuum robotic arm by proportional scaling, and M = [x_m, y_m, z_m] is the actual position vector at the end of the Touch stylus. It should be noted that these scaling coefficients are not fixed constants. Rather, they were chosen according to the specific workspace characteristics of the master and slave devices, as well as the requirements of the current experimental setup. In practical scenarios, these factors can be flexibly adjusted to accommodate different workspace configurations and to further enhance control accuracy and user experience.

To validate the effectiveness of this mapping method, we compared the scaled workspace of the Touch stylus with the actual reachable workspace of the continuum robot’s end-effector, as illustrated in Figure 12. The blue point cloud shows the scaled master-side workspace, while the red point cloud represents the slave-side reachable workspace.

It should be noted that the operational workspace of the Touch stylus end exhibits a figure-eight shape in the Y–Z plane, consisting of two primary lobes. During the master–slave mapping process, we deliberately selected the larger lobe to align with the reachable workspace of the continuum robotic arm as closely as possible. Although the smaller lobe does not fully overlap with the slave workspace, this design choice ensures stable and intuitive control while avoiding unreachable poses or singular configurations.

4. Continuum Arm Motion Experiment

Based on the assumption of constant curvature, the kinematic model of the continuum module is simulated and analyzed, the mapping relationship among its drive space, joint space, and operation space is investigated, and its inverse kinematic solution process is provided. In order to verify the rationality of its model and its feasibility on real objects, an experiment is designed to test the kinematic performance of the continuum robotic arm. This experiment mainly includes four parts—a bending experiment of the single-segment continuum ranging from 0° to 90°, a rotation experiment of the single-segment continuum ranging from 0° to 360°, a positioning experiment at the end of the continuum to reach the target position, and a master–slave teleoperation simulation probe experiment. The deviations between the change in the actuation wire Δl_ij and the bending angle and rotation angle θ_i and φ_i of the continuum module are analyzed, as well as the overall deviation of the system to reach the target position.

4.1. Continuous Body Mechanical Arm Bending Performance Experiment

The first segment of the continuum mechanical arm was selected as the experimental object, and the bending angle and rotation angle of each joint were set to 0°. Additionally, the angle of the continuum mechanical arm to the ground was 0°, representing its initial position. Under the control of the theoretical actuation wires, the bending angle θ₁ of the first segment of the continuum increased by 10°, which was repeated 9 times. The rotation angle φ₁ constantly remained at 0° in the process, as shown in Figure 13. The actual bending angle of the first segment continuum each time is shown in Figure 14a, and the actual angle deviation values are shown in Figure 14b.

As shown in Figure 13, feeding in the change in the drive line derived from the theoretical bending angle results in a corresponding bending deformation of the continuum, which becomes more pronounced in that direction with increasing theoretical bending angle. The theoretical and actual bending angles are compared, as shown in Figure 14. The deviation between them increases with increases in the theoretical bending angle. When the theoretical bending angle is 10° and 20°, the actual bending angle is slightly smaller than the theoretical bending angle, with a deviation of less than 1°. When the theoretical bending angle is 90°, the maximum deviation angle of the maximum reaches 7.6°. The main reason for this phenomenon is that when the drive continuum is deformed to a larger theoretical bending angle, the larger the elastic strain of the superelastic rod is, the larger the resulting elastic stress is, and the larger the friction between the driving wires and the guide module is, all of which lead to an increase in the required driving force, resulting in axial strain in the driving wires, and the actual changes in the driving wires are smaller than the theoretical changes. In addition, the preload force applied to the actuation wires by the driving module itself is insufficient, there is installation error and deformation of the discs during movement, and pulses from the stepping motor are lost, etc., all of which directly or indirectly cause the deviation of the actual angle.

4.2. Experiments on the Rotational Performance of a Continuum Robot Arm

The first segment of the continuum robotic arm was selected as the experimental object. As the initial position, the bending angle of the first segment continuum was set to 30°, and the rotation angle was set to 0°. For the second segment continuum, the bending and rotation angles were set to 0°. The rotation angle of the first segment continuum was increased by 45° in the clockwise direction with changes in the ideal actuation wire, and this was repeated 8 times. The bending angle θ₁ remained at 30° during the process, as shown in Figure 15. The actual rotation angle of the first segment continuum each time is shown in Figure 16a, and the deviations of the actual angle are shown in Figure 16b.

As shown in Figure 15, the rotation experiment effectively achieves the intended objective. The continuum mechanical arm can rotate in the correct direction, and the continuum rotation angle is in the range of [0, 2π]. Compared to the actual deviation in the bending angle, the deviation in the rotation angle is relatively small. As shown in Equation (21), this is because the rotation angle φ₁ is determined by the coordinated actuation of the three driving wires in the first segment of the continuum. By differentially adjusting the displacement of each wire, the segment bends in specific directions, resulting in a controlled spatial orientation of the tip, which defines the azimuth angle. This further validates the accuracy of the proposed kinematic model for the continuum structure.

Moreover, compared with the bending angle, the rotation angle is less affected by the elastic deformation of the flexible backbone of the continuum robotic arm. As a result, the rotation error does not accumulate with increasing angle. The main source of rotation angle error stems from gravitational influence on the continuum module during the actual operation process.

As can be seen from Figure 16, the actual deviations of the rotation angle are symmetrically distributed around 180°, primarily attributed to the slight gravitational influence causing a minor descent of the continuum module. The maximum deviation is 11° when the continuum is rotated to 135°. At a rotation angle of 225°, the deviation reaches an extreme value of 6°. This phenomenon is due to the movement of the continuum module to 90° and 270°, which aligns it horizontally and exposes it to the maximum gravitational influence on its rotation angle.

4.3. Continuum Robot Arm End Positioning Experiment

The end localization experiments of the continuum robot were performed under the condition that the rotation matrix of the end coordinate system {D₂} of the continuum robot with respect to the base coordinate system {D₀} is always Equation (8).

The coordinates of the continuum end were measured using the RealSense D435i depth camera from Intel. Ten randomly selected points were measured in the {D₀} base coordinate system to measure the actual position coordinate values reached by the end of the continuum robot using the proposed kinematic inverse solution algorithm. The comparison between the actual location and the theoretical coordinate values is shown in Figure 17.

It can be seen from Figure 15 that the error is relatively large when the bending angle of the continuum is large, which is consistent with the conclusion obtained from the continuum bending performance test in Section 3.1. In addition, the error at the end of the first continuum segment is superimposed on the second continuum segment, resulting in a more significant change in position.

4.4. Continuum Robotic Arm Simulation Probe Experiments

In order to simulate the process of probing the lesion area in endoscopic surgery, this paper aims to realize the precise control on the end of the continuum robotic arm using the master–slave integrated control strategy. Specifically, the robotic arm starts from the center point of the upper rectangular frame and moves along a 5 × 8 grid rectangular frame on the grid paper in a clockwise direction, and finally returns to the vertex of the upper frame. The master–slave workspace mapping is selected, followed by the incremental mapping method based on the operation space, and the master–slave scale mapping coefficient is set to 1:1. Figure 18 shows the complete process of master–slave control, and Figure 19 demonstrates the process of end trajectory calibration of the continuum robotic arm.

During the experiment, the operator can easily control the end of the continuum robot arm to achieve precise positioning by touch, demonstrating the effectiveness of the master–slave integrated control strategy based on the incremental mapping of the operation space. However, during the control process, the end of the continuum occasionally rubs against the black-and-white grid paper due to the operator’s unconscious movement in the Z-axis direction at the end of the Touch stylus. Therefore, the operator needs to focus on controlling the Touch device to correct the unconscious deviation in the Z-axis direction.

In this experiment, the slave end robotic arm does not exhibit an obvious lag phenomenon when following the movement of the Touch device, and there is no jitter phenomenon at the end. The overall delay of the control system is low, which proves the effectiveness of the continuum robotic arm system and its integrated master–slave control strategy in performing endoscopic surgical probing. However, there are still many shortcomings in the sensory aspects given to the operator during the actual operation. In particular, the operator experiences uncertainty when controlling the robot in the complex tubular environment of the human body. Therefore, more compact microsensors need to be designed for the slave robot control system to improve the accuracy of the overall system and its surgical safety performance.

5. Discussion

In this study, we addressed two key challenges in continuum robot control: postural stability and the computational complexity of inverse kinematics. To this end, we proposed a unified S-shaped symmetric configuration strategy that ensures a stable end-effector posture during motion. This symmetry constraint not only prevents excessive curvature and potential structural stress, but also simplifies the kinematic mapping, resulting in a unique and well-conditioned solution space for inverse kinematics. Building on this, we introduced a real-time inverse kinematics algorithm based on a fifth-order Taylor series expansion. This method significantly reduces the computational load while maintaining sufficient accuracy for practical tasks. The combination of structural symmetry and lightweight computation enables fast and stable control, making the proposed approach particularly well-suited for teleoperation scenarios, where real-time responsiveness and reliability are essential.

While the constant curvature assumption greatly simplifies the kinematic modeling of continuum robots and enables efficient real-time control, it inherently introduces limitations in accurately capturing the robot’s actual deformation, especially under large bending angles or external loading conditions.

In practice, factors such as non-uniform bending, elastic deformation of the backbone, friction between cables and guiding structures, and mechanical imperfections cause deviations from the ideal constant curvature behavior. As observed in our experiments, as the bending angle (and thus curvature) increases, the continuum arm is more prone to exhibiting non-uniform bending and elastic deformation along its length. These effects arise from factors such as material elasticity, friction between cables and guide structures, and mechanical tolerances. Consequently, the actual shape of the arm deviates progressively from the idealized constant curvature assumption, causing cumulative errors between the theoretical bending angle predicted by the model and the actual measured joint angles.

To address these challenges and improve the robot’s performance in delicate tasks like surgical operations, integrating real-time sensing technologies (e.g., fiber optic shape sensors, strain gauges, or electromagnetic trackers) is essential. These sensors can provide accurate feedback on the robot’s actual shape and pose, enabling closed-loop control schemes that compensate for model inaccuracies and external disturbances.

Future work will focus on integrating advanced sensor fusion techniques—such as the combination of fiber-optic sensing, inertial measurement units (IMUs), and vision-based feedback—to provide more robust and real-time shape estimation across multiple segments. Additionally, we aim to explore enhanced modeling approaches beyond the constant-curvature assumption, including variable curvature models and data-driven learning-based representations (e.g., neural kinematic models), to improve the fidelity of kinematic and dynamic predictions. These improvements are expected to significantly enhance the control accuracy, configuration stability, and safety of continuum robots, particularly in complex, dynamic, or human-interactive environments such as minimally invasive surgery, disaster response, and soft-tissue manipulation. Furthermore, future efforts will also investigate online calibration methods, model uncertainty quantification, and adaptive control schemes to ensure long-term robustness and usability in real-world deployments.

Compared to more complex approaches that rely on optimization-based or learning-based inverse kinematics, our method prioritizes computational efficiency and real-time feasibility, which are critical in teleoperation and time-sensitive applications. While this comes at the cost of reduced shape fidelity under certain conditions, it offers a practical and scalable control solution for multi-segment continuum robots. This trade-off reflects a different design philosophy that emphasizes speed, robustness, and stability over exhaustive modeling detail, and is particularly suitable for scenarios where responsiveness is more critical than exact geometric reconstruction.

6. Conclusions

This study presents a bioinspired, cable-driven continuum robotic system tailored for operation in constrained and tortuous environments, such as those encountered in minimally invasive surgical procedures. The system comprises a modular two-segment continuum arm with a superelastic NiTi backbone, providing four degrees of freedom through smooth and compliant bending, as well as a linear advancement module that adds a fifth degree of freedom for depth control. A constant-curvature-based kinematic model is developed, alongside a fifth-order Taylor-series inverse kinematics algorithm that balances computational efficiency and control accuracy. To improve postural stability, a decoupled master–slave control strategy is implemented, allowing the independent management of translational and rotational motion while maintaining consistent end-effector orientation. Simulation and experimental results demonstrate the system’s effectiveness in executing precise and stable movements. Overall, the proposed approach offers a promising solution for teleoperated applications requiring flexible navigation and dexterous manipulation. Future work will explore enhanced sensing integration and real-time control optimization for broader clinical applicability.