Kinematic and static analysis of a rigid-flexible hybrid robot with combined elastomer configuration of driving shafts and spinal rods

Abstract

Flexible shafts, exhibiting excellent tensile and torsional transmission characteristics, serve as an ideal power transmission medium for the design of flexible continuum robots. This study introduces a novel rigid-flexible hybrid robotic mechanism employing six flexible shaft actuators to achieve seven degrees of freedom (DOFs). The design integrates a dual-segment flexible body with a single-segment rigid body. A comprehensive kinematic model was derived to describe the hybrid structure’s motion. Concurrently, a static model was developed, incorporating driving forces, external loads, gravitational effects, and the combined elastic forces from the flexible driving shafts and spinal rods. Based on this static model, 182 combinations of flexible shaft pulling forces and distal loads were applied to control the robot’s posture. Experimental results indicated that the maximum positional errors for the single- and dual-segment configurations were 2.67% and 5.97% of the manipulator length, respectively. The maximum root mean square error for spatial configuration repeatability is 0.4 and 5.1 mm for single- and dual-segment flexible robots, respectively. The kinematic and static models developed in this research accurately predict the spatial configuration of the flexible continuum robot, which features an integrated elastic arrangement of driving shafts and spinal rods, thereby facilitating rapid deployment in similar flexible robotic systems.

Introduction

Minimally invasive surgery (MIS) is widely utilized in surgical treatments due to its advantages of smaller incisions, less blood loss, and shorter hospital stays. Surgical robots, leveraging robotic technology, not only retain the benefits of traditional MIS but also enable more precise surgical movements, reduce the workload on surgeons, and minimize postoperative complications for patients. ¹ Minimally invasive surgical techniques guided by endoscopic or laparoscopic imaging primarily encompass transoral surgery, thoracoscopic and laparoscopic surgery, as well as transvaginal, transurethral, and transrectal procedures. These techniques effectively meet the requirements for various surgical procedures in thoracic surgery, ² general surgery,^3,4 urological surgery, ⁵ and gynecological surgery.^6,7

In the aforementioned surgical methods, there is typically a certain distance from the patient’s body surface to the surgical site. Additionally, the posture of the surgical instrument as it enters the body varies depending on the surgical site, which imposes stringent requirements on the capability to flexibility and dexterity. Current surgical robots, particularly those based on continuum structures, still face significant challenges. A primary limitation lies in their actuation mechanisms. Many rely on traditional tendon/cable drives, which can only transmit tensile forces, ⁸ making it difficult to actuate complex distal mechanisms requiring torque, such as rotational wrists or graspers.

To overcome these limitations, rigid-flexible hybrid mechanisms have garnered increasing research attention. These designs seek to synergize the adaptability of flexible segments with the load-bearing capacity and operational precision of rigid segments. However, a review of prior studies reveals a common challenge in achieving a truly integrated and compact system. For instance, Li et al. ⁹ explored hybrid structures by combining novel composite materials with rigid elements, yet such approaches are limited in output force and control accuracy. Paternò et al. ¹⁰ integrated flexible actuators within a rigid frame to balance compliance and output force, but their dependence on external pneumatic power sources and complex fabrication processes restricts broader adoption. Furthermore, Wu et al. ¹¹ integrated a parallel mechanism with a tendon-driven continuum segment to balance positioning precision and posture adaptability; however, the auxiliary parallel mechanism increases the complexity and bulk of the actuation system, and the distal flexible manipulator, limited by tendon-driven actuation, often possesses insufficient degrees of freedom for versatile tasks. In summary, while research on rigid-flexible hybrid mechanisms has achieved notable progress, a common characteristic in these designs is that the actuation systems for the rigid and flexible parts remain largely separate and decoupled, leading to increased structural complexity. This study proposes a novel flexible robot driven by the tensile-torsional synergistic actuation of flexible shafts. Unlike traditional tendons, this design ingeniously leverages the dual functionality of the flexible shaft: serving as a medium for tensile-bending actuation of the flexible segments and concurrently transmitting torque to the distal rigid joint. This approach simplifies the robot’s drive system, obviates the need for auxiliary drive chains for the rigid joint, and reduces structural complexity while enhancing the robot’s overall dexterity.

An accurate mathematical model is essential to better design, analyze, and control flexible robots driven by flexible shaft pull-twist systems. ¹² Given the lightweight design of this robot and its application in slow-moving conditions, static and kinematic studies are prioritized over dynamics analysis. ¹³ The kinematic and static modeling methods for continuum mechanisms primarily include the constant curvature model, ¹⁴ the piecewise constant curvature model,^15–17 the Cosserat rod model, ¹⁸ and the pseudo-rigid body model.^19,20 Among these, the piecewise constant curvature method is favored for its high computational efficiency, convenient modeling, and broad applicability, has been successfully implemented in numerous kinematic and static models.^21,22 For instance, a real-time modeling and verification scheme for continuum robots using piecewise constant curvature kinematics was introduced in Barrientos-Diez et al., ²³ achieving real-time control and operation of continuum robots in various scenarios. Similarly, model-based feedforward position control method was proposed in Falkenhahn et al., ²⁴ where the continuum robot was precisely controlled using the piecewise constant curvature model, demonstrating its efficiency and accuracy in applications. Despite the widespread success and proven utility of the piecewise constant curvature method, its accuracy often relies on a key simplifying assumption: the actuation elements (e.g. cables or tendons) are treated as ideal, one-dimensional strings that contribute to the robot’s motion only through tension, while possessing no intrinsic bending stiffness themselves. In these models, the robot’s overall bending resistance is attributed solely to the central elastic backbone. This simplification, while computationally convenient, overlooks the contribution of actuation elements with non-negligible cross-sections (such as the flexible shafts used in this study) to the overall structural rigidity. This can lead to inaccuracies in predicting the robot’s shape, especially under significant external loads.

To accurately describe the shape or motion of flexible robots driven by the coordinated pull-twist of flexible shafts, this paper proposes a statics model with comprehensive consideration of various factors, including the driving forces on the flexible shafts, the interaction forces at the manipulator’s end-effector, the gravitational forces and moments of the spacer disks and elastic backbones, as well as the elastic forces and moments. Powder metallurgy oil-impregnated bearings are used to reduce the frictional force when the flexible shafts pass through the spacer disks. This unique design eliminates influence of friction in the statics model. Compared to existing research, the main contributions of this paper are as follows:

1. This study presents a multi-segment rigid-flexible hybrid dexterous robot structure driven by a novel “pull-twist” synergistic actuation mechanism. Unlike conventional designs where actuation for flexible and rigid parts are decoupled, our approach ingeniously uses the dual functionality of flexible shafts: the “pulling” motion actuates the bending of the continuum segments, while the “twisting” motion of the same shafts concurrently drives the distal multi-DOF rigid joint. This highly integrated design, proven feasible by a physical prototype, obviates the need for separate drive systems, significantly reducing mechanical complexity and size.

2. For the first time, this study incorporates the elastic properties of the driving flexible shafts into the static model of a rigid-flexible hybrid flexible robot. A comprehensive static model of the flexible body is proposed, taking into account multiple factors, and a kinematic model of the robot is established. The static model of the integrated elastomer configuration, which includes the elastic flexible shafts and elastic spinal rods, more accurately describes the spatial pose and shape of the flexible robot. This model addresses the deformation behavior of the robot under different external loads and drive combinations and systematically evaluates the impact of drive parameters on the shape and stability of the flexible robot.

3. The constructed pose description model and static model of the rigid-flexible hybrid robot demonstrate good versatility. They can be quickly adapted for the deployment of static models for flexible robots with various power transmission media, such as elastic flexible shaft drive-elastic spinal support rod and inelastic cable drive-elastic spinal support rod. Additionally, the number of spacer disks in the flexible joints, the number of routing holes, the number of flexible joints, and the degree of freedom configuration of the rigid joints can all be freely defined.

The structure of the paper is as follows: the flexible robot system driven by the coordinated pull-twist of the flexible shafts is described first. Next, the kinematic model of the rigid-flexible hybrid manipulator and the static model are constructed, considering driving force, external environmental forces, gravitational forces, and elastic forces, including the flexible shaft’s elastic properties. Then, the experimental setup and validation work are detailed, followed by a discussion on error analysis. Finally, the main conclusions and future extensions of this work are presented.

System description

The flexible robot system, incorporating the combined elastomer of the drive shafts and spinal rods, is illustrated in Figure 1, primarily consists of a drive module, shaped guide tubes, a rigid-flexible hybrid manipulator, a transition plate, and force sensors. The rigid-flexible hybrid manipulator is composed of a 4-DOF dual segment flexible joint and a 3-DOF rigid joint, powered by 9 brushless DC motors (CBL2040-2416F, Techrobots, China). Additionally, a linear joint is developed to actuate the translational reciprocating motion of the entire rigid-flexible hybrid manipulator. Miniature self-tightening chucks are used to secure the flexible shafts to the motor output shafts. The flexible shaft originates from the motor output shaft, passes through the shaped guiding tube channel, the hollow channel of the transition plate, and the manipulator’s mounting plate, ultimately reaching rigid-flexible hybrid manipulator base.

Figure 1.

Flexible robot system with elastic configuration of flexible shafts and spinal rods.

The rigid-flexible hybrid manipulator comprises two flexible joints and one rigid joint. Each flexible joint consists of multiple spacer disks, an elastic backbone, and several flexible shafts. The spacer disks are used to guide the flexible shafts along the motion path. The spacer disks and flexible shafts are fabricated from 304 stainless steel, while the elastic backbones are constructed using a NiTi alloy. A series of modular spacer disks, with identical in thickness but different in aperture design, are designed and manufactured. These disks allow for quick assembly and disassembly through different stacking schemes, facilitating flexible changes in the manipulator’s shape configuration. Friction between the flexible shaft and the spacer disk is significantly reduced by embedding a powder metallurgy oil-impregnated bearing within the wiring aperture of the spacer disk. When using the constant curvature method to construct the static model of a flexible continuum robot, in order to reduce the posture error of the static model, the preferred ratio of distance between two spacer disks to the distance from the wiring aperture to the elastic backbone should be 0.4. ²⁵ Given that the distance from the wiring aperture to the elastic backbone is 5 mm in this manipulator, the distance between two spacer disks is determined to be 12.5 mm.

Modeling

Kinematic analysis

In this section, the kinematic analysis of the rigid-flexible hybrid robot, is performed based on the combined use of the piecewise constant curvature method and Denavit–Hartenberg (D-H) parameters method. The notation used in this paper is provided in Table 1. The elastic backbone is assumed to be composed of a series of mutually tangent segments, each of which is approximated as an arc with constant curvature. Supposed the torsion of the elastic backbone is negligible, therefore, only the deformation of the elastic backbone in the plane will be considered. The unique rigid-flexible hybrid robot consists of a dual-segment flexible joint and a rigid joint, driven by flexible shafts through pull-twist coaxial coordination. The wiring coordinates of the flexible shafts passing through the spacer disks are illustrated in Figure 2(a), where paths 1, 2, and 3 represent the wiring paths of the distal segment, and paths 4, 5, and 6 represent the wiring paths of the proximal segment. Furthermore, the numbers of these wiring paths are consistent with the motor identifiers, which will be mentioned in the following sections. The angular positions of these wiring coordinates in the manipulator coordinate system are defined in equation (1) as follows:

$$\begin{array}{c}{\sigma }_{1i}=-\frac{\pi }{6}+(i-1)\beta \hfill \\ {\sigma }_{2i}=\frac{\pi }{6}-(i-1)\beta \hfill \\ i=1,2,3;\beta =\frac{2\pi }{3}.\hfill \end{array}$$

(1)

Table 1.

Nomenclature.

Symbol	Description
$(x,y,z)$	Base coordinate system of the manipulator
$(x_i,y_i,z_i)$	Local coordinate system fixed on the $i^{\mathrm{th}}$ spacer disk (equivalent to fixed on the $i^{\mathrm{th}}$ segment of the flexible joint)
${\sigma }_{1i}$	Azimuth angle of the wiring path of the first flexible joint segment, $i=1,2,3$
${\sigma }_{2i}$	Azimuth angle of the wiring path of the second flexible joint segment, $i=1,2,3$
$\beta$	Angle between adjacent wiring paths in each flexible joint segment
${\mathbf{P}}_s$	Position vector of any point on the arc at $s$
${\kappa }_i$	Curvature of the $i^{\mathrm{th}}$ flexible joint segment, $i=1,2,\dots ,10$
${\varphi }_i$	Angle between the bending plane of the $i^{\mathrm{th}}$ segment and the reference axis, i.e. azimuth angle
${\kappa }_{\mathrm{ix}}$ , ${\kappa }_{\mathrm{iy}}$	Curvature components along the $x$ and $y$ axes in the local coordinate system of the $i^{\mathrm{th}}$ segment
${\theta }_i$	Rotation angle of the $i^{\mathrm{th}}$ flexible joint segment around the $y$ axis
$R_z\left({\varphi }_i\right)$	Rotation matrix of the $i^{\mathrm{th}}$ flexible joint segment around the $z$ axis
$R_y\left({\theta }_i\right)$	Rotation matrix of the $i^{\mathrm{th}}$ flexible joint segment around the $y$ axis
${\epsilon }_i$	Additional torsion variable of the $i^{\mathrm{th}}$ flexible joint segment
${\theta }_{R1}$	Pitch angle of the rigid joint
${\theta }_{R2}$	Rotation angle of the rigid joint
iP fs	Position matrix of the flexible shaft in the $i^{\mathrm{th}}$ coordinate system
$n$	Total number of spacer disks, 10
${}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}$	Driving force vector exerted by the flexible shaft on the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{P}}_{\mathrm{fs},i+1}$	Position matrix of the wiring coordinates on the ${(i+1)}^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${\mathbf{F}}_{\mathrm{fs}\_i}$	Driving force vector along the directions of the $i^{\mathrm{th}}$ segment of the flexible shafts.
${}^{i-1}{\mathbf{M}}_{\mathrm{fs},i}$	Driving torque vector exerted by the flexible shaft on the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^j{\mathbf{F}}_{\mathrm{ex}}$	External load force vector in the $j^{\mathrm{th}}$ coordinate system
${}^j{\mathbf{M}}_{\mathrm{ex}}$	External load torque vector in the $j^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{F}}_{\mathrm{sd}\_g}$	Gravity vector of the spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
$\mathbf{A}$	Unit gravity column vector in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{P}}_{\mathrm{cfs}\_i}$	Position vector of the centroid of the elastic backbone in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{F}}_{\mathrm{bb}\_g}$	Gravity vector of the elastic backbone in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{F}}_{\mathrm{net}\_i}$	Net force vector exerted on the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^{i-1}{\mathbf{M}}_{\mathrm{net}\_i}$	Net torque vector exerted on the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system
${}^i{\mathbf{F}}_{\mathrm{ite}\_i+1}$	Force vector exerted on the ${(i+1)}^{\mathrm{th}}$ spacer disk in the $i^{\mathrm{th}}$ coordinate system
${\mathbf{M}}_{b\_i}$	Bending moment vector of the $i^{\mathrm{th}}$ manipulator
${\mathbf{M}}_{t\_i}$	Torsion moment vector of the $i^{\mathrm{th}}$ manipulator
$I_{\mathrm{fs}}$	Moment of inertia of the flexible shaft about the centroidal axis of the manipulator
$I_{\mathrm{self}}$	Moment of inertia of the flexible shaft about its own centroidal axis
$d$	Distance from the wiring aperture to the elastic backbone
$A_{\mathrm{fs}}$	Cross-sectional area of a single flexible shaft
$E_{\mathrm{bk}}$	Young’s modulus of the elastic backbone
$E_{\mathrm{fs}}$	Young’s modulus of the flexible shafts
$I_{\mathrm{bk}}$	Moment of inertia of the elastic backbone
$\mathrm{EI}$	Overall bending stiffness of the manipulator
$G$	Overall shear modulus of the manipulator
$l_i$	Distance between the $i^{\mathrm{th}}$ spacer disk and the ${(i-1)}^{\mathrm{th}}$ spacer disk in the upright position
$\mathbf{M}$	Static equilibrium equation
${}^{i-1}{\mathbf{P}}_{\mathbf{s}\_i}$	Position vector of the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system

Figure 2.

Schematic diagram of the wiring coordinates on the spacer disks: (a) definition of wiring coordinates on the spacer disks and (b) spacer disks with powder metallurgy oil-impregnated bearings for the proximal and distal segments.

The nomenclature of the coordinate system for the flexible joint is illustrated in Figure 3. And this naming convention is also applicable to each locally tangent segment. The elastic backbone is defined by $[{\kappa }_i,{\varphi }_i]$ , where ${\kappa }_i$ represents the curvature of the $i^{\mathrm{th}}$ segment, and ${\varphi }_i$ denotes the angle between the bending plane of the $i^{\mathrm{th}}$ segment and the reference axis, that is, the azimuth angle. The coordinates of any segment of the flexible robot in the task space can be determined using existing methods, including the arc geometry method, exponential coordinate method, DH parameter method, Frenet-Serret frame, and the integration formula assuming zero curvature. All these representations have been derived and yield the same mapping between configuration and task space. ¹⁴ Therefore, the arc geometry method is adopted to describe the manipulator’s posture in this study, and the position of arbitrary point on the arc of the $i^{\mathrm{th}}$ segment at $s$ relative to the base coordinate system can be expressed by the following equation:

$${\mathbf{P}}_s=\left[\begin{array}{c}\frac{\cos {\varphi }_i}{{\kappa }_i}(1-\cos \left({\kappa }_{i}s\right))\hfill \\ \frac{\sin {\varphi }_i}{{\kappa }_i}(1-\cos \left({\kappa }_{i}s\right))\hfill \\ \frac{1}{{\kappa }_i}\sin \left({\kappa }_{i}s\right)\hfill \end{array}\right],$$

(2)

where ${\kappa }_{i}s$ represents the rotation of the $i^{\mathrm{th}}$ flexible joint segment around the y-axis, which is simplified as ${\theta }_i$ . The bending curvature ${\kappa }_i$ and the azimuth angle ${\varphi }_i$ are expressed as:

$$\begin{array}{cc}\multicolumn{1}{c}{{\kappa }_i}& =\sqrt{{\kappa }_{\mathrm{ix}}^2+{\kappa }_{\mathrm{iy}}^2}\hfill \\ \multicolumn{1}{c}{{\varphi }_i}& =\mathrm{atan}2({\kappa }_{\mathrm{iy}},{\kappa }_{\mathrm{ix}}),\hfill \end{array}$$

(3)

where ${\kappa }_{\mathrm{ix}}$ and ${\kappa }_{\mathrm{iy}}$ represent the bending curvatures along the $x$ and $y$ axes, respectively, in the local coordinate system of the $i^{\mathrm{th}}$ segment.

Figure 3.

Nomenclature of the coordinate system in flexible joints.

The coordinates of arbitrary point at the $i^{\mathrm{th}}$ segment arc in the corresponding coordinate system can be described based on the combined use of a rotation around the $z$ axis by ${\varphi }_i$ and a rotation around the $y$ axis by ${\theta }_i$ within the $\mathrm{xz}$ plane. In order to map the $i^{\mathrm{th}}$ segment coordinate system to the base coordinate system, a transformation matrix relative to the coordinate system of the ${(i-1)}^{\mathrm{th}}$ segment is expressed as:

$${}_i{}^{i-1}T\left(s\right)=\left[\begin{array}{cc}{R}_z\left({\varphi }_i\right)R_y\left({\theta }_i\right)R_z(-{\varphi }_i)\hfill & {\mathbf{P}}_s\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right],$$

(4)

where matrices $R_z\left({\varphi }_i\right)$ and $R_y\left({\theta }_i\right)$ are expressed as follows:

$$\begin{array}{cc}{R}_z\left({\varphi }_i\right)\hfill & =\left[\begin{array}{ccc}\cos \left({\varphi }_i\right)\hfill & -\sin \left({\varphi }_i\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin \left({\varphi }_i\right)}& \cos \left({\varphi }_i\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{R_y\left({\theta }_i\right)}& =\left[\begin{array}{ccc}\cos \left({\theta }_i\right)\hfill & 0\hfill & \sin \left({\theta }_i\right)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \left({\theta }_i\right)}& 0\hfill & \cos \left({\theta }_i\right)\hfill \end{array}\right].\hfill \end{array}$$

Considering the torsion of the elastic backbone, the variable ${\epsilon }_i$ is introduced. ²⁶ The transformation matrix relative to the $(i-1)$ coordinate system is then adjusted as follows:

$${}_i{}^{i-1}T\left(s\right)=\left[\begin{array}{cc}{R}_z\left({\varphi }_i\right)R_y\left({\theta }_i\right)R_z({\epsilon }_i-{\varphi }_i)\hfill & {\mathbf{P}}_s\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right].$$

(5)

Thus, we can calculate the pose information of any point on the elastic backbone in the two-segment flexible joint, including the pose information of the 10 spacer disks. To ensure the continuity of the position and orientation at the rigid-flexible coupling, the continuity condition is defined as:

$$T_f^{\mathrm{end}}=T_r^{\mathrm{start}},$$

(6)

where $T_f^{\mathrm{end}}$ is the transformation matrix of the flexible joint’s end, and $T_r^{\mathrm{start}}$ represents the transformation matrix at the initial position of the rigid joint. This continuity condition is achieved at the rigid and flexible joint connections through the use of screw fastening.

To better describe the pose of the rigid joint attached to the second segment of the flexible joint, the coordinate system at the distal of the flexible joint is defined as $x_{10}{y}_{10}{z}_{10}$ , which is illustrated in Figure 4, where $a_0=13.63\mathrm{mm}$ , $a_1=14.7\mathrm{mm}$ , and $a_2=7.36\mathrm{mm}$ . The homogeneous transformation matrix of the rigid joint’s gripper relative to the base coordinate system of the manipulator is described in equation (7). The coordinate system of the rigid-flexible hybrid manipulator driven by coordinated pull-twist of the flexible shafts is depicted in Figure 5, where the green, red, and black segments correspond to $a_0$ , $a_1$ , and $a_2$ , respectively.

$$\begin{array}{c}\multicolumn{1}{c}{{}_{14}{}^{0}T{=}_{10}^{0}T{·}_{11}^{10}T{·}_{13}^{11}T{·}_{14}^{13}T}\\ {=}_{10}^{0}T·\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & a_2\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ \left[\begin{array}{cccc}{c}_1{c}_2\hfill & -c_1{s}_2\hfill & s_1\hfill & a_1{s}_1\hfill \\ \multicolumn{1}{c}{s_2}& c_2\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-c_2{s}_1}& s_1{s}_2\hfill & c_1\hfill & a_1{c}_1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ \left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & a_0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right],\hfill \end{array}$$

(7)

where $c_1=\cos \left({\theta }_{R1}\right)$ , $s_1=\sin \left({\theta }_{R1}\right)$ , $c_2=\cos \left({\theta }_{R2}\right)$ , and $s_2=\sin \left({\theta }_{R2}\right)$ .

Figure 4.

Naming convention and coordinate system definition for the rigid joint.

Figure 5.

Nomenclature of the manipulator coordinate system: (a) ${\theta }_{R1}=0$ , ${\theta }_{R2}=0$ and (b) ${\theta }_{R1}=\mathrm{pi}/6$ , ${\theta }_{R2}=\mathrm{pi}/6$ .

Static analysis

In this section, a static analysis is conducted on the rigid-flexible hybrid robot driven by the flexible shafts through pull-twist coaxial coordination, as shown in Figure 6. The flexible continuum robot comprises a series of identical adjacent units. For instance, the $i^{\mathrm{th}}$ unit refers to the part of the flexible continuum robot enclosed between the ${(i-1)}^{\mathrm{th}}$ spacer disk and the $i^{\mathrm{th}}$ spacer disk. The $z$ -axes direction of the ${(i-1)}^{\mathrm{th}}$ , $i^{\mathrm{th}}$ , and ${(i+1)}^{\mathrm{th}}$ spacer disks are tangent to the elastic backbone, and that of the $x$ - and $y$ -axes are defined as shown in Figure 2(a). In order to improve the accuracy of static analysis of the flexible continuum robot, the following four aspects are involved under comprehensive consideration, including: the driving forces on the flexible shafts, the interaction forces and moments with the external environment, the gravitational forces and moments of the spacer disks and elastic backbones, and the elastic forces and moments generated by the elastic backbones and flexible shafts.

Figure 6.

Schematic diagram of the static analysis of the manipulator, where the flexible shafts between each two adjacent spacer disks are in a straight-line configuration.

Tensile force on the flexible shafts

The position matrix of the wiring channels in the $i^{\mathrm{th}}$ coordinate system are expressed in the ${(i-1)}^{\mathrm{th}}$ coordinate system as follows:

$${}_i{}^{i-1}T{·}^i{\mathbf{P}}_{\mathrm{fs}}{=}^{i-1}{\mathbf{P}}_{\mathrm{fs}},(i<n+1),$$

(8)

where $n$ represents the total number of spacer disks, and ⁱP _fs is the coordinate matrix for the six wiring channels on the spacer disk in the $i^{\mathrm{th}}$ coordinate system. Combining with equation (1), it can be written as:

$$\begin{array}{cc}{}^i{\mathbf{P}}_{\mathrm{fs}}=\hfill & {\left[\begin{array}{cccc}r\cos {\sigma }_{11}\hfill & r\sin {\sigma }_{11}\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{r\cos {\sigma }_{12}}& r\sin {\sigma }_{12}\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{r\cos {\sigma }_{13}}& r\sin {\sigma }_{13}\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{r\cos {\sigma }_{21}}& r\sin {\sigma }_{21}\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{r\cos {\sigma }_{22}}& r\sin {\sigma }_{22}\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{r\cos {\sigma }_{23}}& r\sin {\sigma }_{23}\hfill & 0\hfill & 1\hfill \end{array}\right]}^T,\hfill \end{array}$$

where $r$ represents the distance between the wiring channel of the spacer disk and the elastic backbone. Then, the driving forces exerted by the flexible shafts on the spacer disks can be expressed as:

$$\{\begin{array}{c}{}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}=\frac{{}^i{\mathbf{P}}_{\mathrm{fs},i}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i}}{\Vert {}^i{\mathbf{P}}_{\mathrm{fs},i}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i}\Vert }·{\mathbf{F}}_{\mathrm{fs}\_i},(i=n)\hfill \\ {}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}=\frac{{}^i{\mathbf{P}}_{\mathrm{fs},i}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i}}{\Vert {}^i{\mathbf{P}}_{\mathrm{fs},i}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i}\Vert }·{\mathbf{F}}_{\mathrm{fs}\_i}\hfill \\ +\frac{{}^i{\mathbf{P}}_{\mathrm{fs},i+1}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i+1}}{\Vert {}^i{\mathbf{P}}_{\mathrm{fs},i+1}-{}^{i-1}{\mathbf{P}}_{\mathrm{fs},i+1}\Vert }·{\mathbf{F}}_{\mathrm{fs}\_i+1},(i<n)\hfill \end{array}$$

(9)

where ${}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}$ represents the driving forces exerted by the flexible shafts on the $i^{\mathrm{th}}$ spacer disks in the ${(i-1)}^{\mathrm{th}}$ coordinate system. ${}^{i-1}{\mathbf{P}}_{\mathrm{fs},i+1}$ denotes the position matrix of the wiring coordinates on the ${(i+1)}^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system. ${\mathbf{F}}_{\mathrm{fs}\_i}$ and ${\mathbf{F}}_{\mathrm{fs}\_i+1}$ represent the force and reaction force applied by the flexible shaft at each spacer disk, respectively. Together, they exert a moment on the elastic backbone.

Since powder metallurgy oil-impregnated bearings are embedded in the wiring channels of the spacer disks, the friction between the flexible shafts and the spacer disks can be ignored by means of the fine lubrication characteristics of the bearings. Thus, the force perpendicular to the spacer disk is zero. Furthermore, the orthogonal forces exerted by the flexible shafts on the spacer disks are eliminated when $i\ne n$ or $i\ne n/2$ (the end spacer disk of each segment). That is:

$${}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}{=}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}-\left({}_i{}^{i-1}T{(:,3)}^T{·}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}\right){·}_i^{i-1}T(:,3).$$

(10)

The torques generated by the forces exerted by the flexible shafts on the spacer disks can be described as follows

$${}^{i-1}{\mathbf{M}}_{\mathrm{fs},i}{=}^{i-1}{\mathbf{P}}_{\mathrm{fs},i}{\times }^{i-1}{\mathbf{F}}_{\mathrm{fs},i}.$$

(11)

Interaction force with the external environment

The interaction forces with the external environment, representing as the force vectors applied to the end of the manipulator in the base coordinate system, can be described as follows:

$${}^0{\mathbf{F}}_{\mathrm{ex}}{=}_j^{0}T{·}^j{\mathbf{F}}_{\mathrm{ex}},(j=n),$$

(12)

where ${}^j{\mathbf{F}}_{\mathrm{ex}}$ represents the external three-dimensional load vector in the $j$ coordinate system. Considering the influence of the external moment, the torque in the $j$ coordinate system can be described as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{}^j{\mathbf{M}}_{\mathrm{ex}}={\left({}_j{}^{0}T\right)}^{-1}{·}^0{\mathbf{M}}_{\mathrm{ex}}}\\ +({}_j{}^{j-1}T·{\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right]}^T){\times }^j{\mathbf{F}}_{\mathrm{ex}},(j=n),\hfill \end{array}$$

(13)

where ${}^0{\mathbf{M}}_{\mathrm{ex}}$ represents the torque vector applied to the distal of the flexible robot in the base coordinate system.

Gravitational force and moment

The gravitational forces primarily include the both gravity of the spacer disks and the elastic backbones. In this study, the direction of gravity is aligned with the negative y-axis. Therefore, the gravitational forces of the spacer disks in the ${(i-1)}^{\mathrm{th}}$ coordinate system can be described as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{}^{i-1}{\mathbf{F}}_{\mathrm{sd}\_g}={\mathbf{m}}_{\mathrm{sd}}g·\mathbf{A}}\\ \multicolumn{1}{c}{\mathbf{A}={\left({}_{i-1}{}^{0}T\right)}^{-1}·{\left[\begin{array}{cccc}0\hfill & -1\hfill & 0\hfill & 0\hfill \end{array}\right]}^T,}\end{array}$$

(14)

where $\mathbf{A}$ represents the unit gravity column vector in the $(i-1)$ coordinate system.

For the elastic backbones, the centroid position of the $i^{\mathrm{th}}$ segment of the elastic backbones in the ${(i-1)}^{\mathrm{th}}$ coordinate system can be described as follows:

$${}^{i-1}{\mathbf{P}}_{\mathrm{cfs}\_i}=\frac{1}{{\kappa }_i}\left[\begin{array}{c}(1-\frac{\sin ({\theta }_i/2)}{{\theta }_i/2}·\cos ({\theta }_i/2))·\cos {\varphi }_i\hfill \\ \multicolumn{1}{c}{(1-\frac{\sin ({\theta }_i/2)}{{\theta }_i/2}·\cos ({\theta }_i/2))·\sin {\varphi }_i}\\ \multicolumn{1}{c}{\frac{\sin ({\theta }_i/2)}{{\theta }_i/2}·\sin ({\theta }_i/2)}\end{array}\right],$$

(15)

where $\frac{\sin (\theta /2)}{\theta /2}$ is the correction term for the centroid position. The gravitational forces of the elastic backbones in the ${(i-1)}^{\mathrm{th}}$ coordinate system can be described as follows:

$${}^{i-1}{\mathbf{F}}_{\mathrm{bb}\_g}={\mathbf{m}}_{\mathrm{bb}}g·\mathbf{A}.$$

(16)

Thus, the forces and moments acting on the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system can be described as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{}^{i-1}{\mathbf{F}}_{\mathrm{net}\_i}{=}^{i-1}{\mathbf{F}}_{\mathrm{fs},i}{+}^{i-1}{\mathbf{F}}_{\mathrm{sd}\_g}{+}^{i-1}{\mathbf{F}}_{\mathrm{bb}\_g}}\\ {+}^j{\mathbf{F}}_{\mathrm{ex}}{+}_i^{i-1}T{·}^i{\mathbf{F}}_{\mathrm{ite}\_i+1}\hfill \\ \multicolumn{1}{c}{{}^{i-1}{\mathbf{M}}_{\mathrm{net}\_i}{=}^{i-1}{\mathbf{M}}_{\mathrm{fs},i}+({}_i{}^{i-1}T·\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\right]){\times }^{i-1}{\mathbf{F}}_{\mathrm{sd}\_g}}\\ {+}^{i-1}{\mathbf{P}}_{\mathrm{cfs}\_i}{\times }^{i-1}{\mathbf{F}}_{\mathrm{bb}\_g}{+}^j{\mathbf{M}}_{\mathrm{ex}}\hfill \\ +({}_i{}^{i-1}T·\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\right])\times \left({}_i{}^{i-1}T{·}^i{\mathbf{F}}_{\mathrm{ite}\_i+1}\right),(j=n),\hfill \end{array}$$

(17)

where ${\mathbf{F}}_{\mathrm{ite}}$ represent the forces acting on the ${(i+1)}^{\mathrm{th}}$ spacer disk in the $i$ coordinate system.

Elastic force and moment

The bending moment along the axial direction of the elastic backbone is simplified as a linear relationship proportional to its curvature by using piecewise constant curvature method. ²⁷ Supposed that both the elastic backbone and the flexible shafts are inextensible, thereby only deformations caused by bending movement are considered. In this study, the flexible shafts are uniformly distributed on the spacer disks with a circumference of certain radius. Therefore, the equivalent moment of inertia of the flexible shaft in the robot can be expressed as:

$$I_{\mathrm{fs}}=I_{\mathrm{self}}+A_{\mathrm{fs}}·d^2,$$

where $I_{\mathrm{fs}}$ denotes the moment of inertia of the flexible shaft relative to the centroidal axis of the flexible continuum robot, $I_{\mathrm{self}}$ represents the moment of inertia of the flexible shaft relative to its own centroidal axis, $d$ indicates the distance between the wiring aperture and the elastic backbone, and $A_{\mathrm{fs}}$ represents the cross-sectional area of a single flexible shaft. Therefore, the overall bending stiffness of the flexible continuum robot is expressed as:

$$\begin{array}{c}\{\begin{array}{c}E{I}_{\mathrm{pro}}=E_{\mathrm{bk}}{I}_{\mathrm{bk}}+6·E_{\mathrm{fs}}{I}_{\mathrm{fs}},i\le 5\hfill \\ \multicolumn{1}{c}{E{I}_{\mathrm{dis}}=E_{\mathrm{bk}}{I}_{\mathrm{bk}}+3·E_{\mathrm{fs}}{I}_{\mathrm{fs}},i>5}\\ \multicolumn{1}{c}{\mathrm{EI}=E{I}_{\mathrm{pro}}+E{I}_{\mathrm{dis}}}\end{array},\hfill \end{array}$$

(18)

where $i$ denotes the number of spacer disks, $E_{\mathrm{bk}}$ represents the Young’s modulus of the elastic backbone, $E_{\mathrm{fs}}$ represents the Young’s modulus of the flexible shafts, $I_{\mathrm{bk}}$ denotes the moment of inertia of the elastic backbone, $E{I}_{\mathrm{pro}}$ and $E{I}_{\mathrm{dis}}$ represent the bending stiffness of the robot’s proximal and distal segments, respectively, and $\mathrm{EI}$ denotes the overall bending stiffness of the robot. Then, the bending moment of the $i^{\mathrm{th}}$ segment of the robot is described as follows:

$${\mathbf{M}}_{b\_i}=\left[\begin{array}{cc}\cos {\varphi }_i-\sin {\varphi }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin {\varphi }_i}& \cos {\varphi }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]·\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\kappa }_i\mathrm{EI}}\\ \multicolumn{1}{c}{0}\end{array}\right].$$

(19)

Suppose that the flexible robot rotates around the $z$ -axis by angle ${\epsilon }_i$ . The torque of the $i^{\mathrm{th}}$ segment of the elastic backbone and the flexible shafts can be described as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{M}}_{t\_i}{=}_i^{i-1}T·{\left[\begin{array}{ccc}00\hfill & \frac{2\mathrm{IG}{\epsilon }_i}{l_i}\hfill & 0\hfill \end{array}\right]}^T}\\ \multicolumn{1}{c}{G=G_{\mathrm{fs}}+G_{\mathrm{bk}}}\end{array},$$

(20)

where $G$ represents the overall shear modulus of the robot, $G_{\mathrm{fs}}$ and $G_{\mathrm{bk}}$ denote the shear modulus of the flexible shafts and the elastic backbone, respectively, $l_i$ is the distance between the $i^{\mathrm{th}}$ spacer disk and the ${(i-1)}^{\mathrm{th}}$ spacer disk in the straight state, and ${\epsilon }_i$ is the additional rotation angle of the $i^{\mathrm{th}}$ segment around the $z$ -axis.

The comprehensive static equilibrium equation of the flexible continuum robot can be expressed as follows:

$$\mathbf{M}=\left[\begin{array}{c}{\mathbf{M}}_{b\_1}+{\mathbf{M}}_{t\_1}-\sum _{i=n}^1{}^{i-1}{\mathbf{M}}_{\mathrm{net}\_i}\hfill \\ \multicolumn{1}{c}{\dots }\\ \multicolumn{1}{c}{{\mathbf{M}}_{b\_n-1}+{\mathbf{M}}_{t\_n-1}-({}^{n-1}{\mathbf{M}}_{\mathrm{net}\_n}+{}^{n-2}{\mathbf{M}}_{\mathrm{net}\_n-1})}\\ \multicolumn{1}{c}{{\mathbf{M}}_{b\_n}+{\mathbf{M}}_{t\_n}-{}^{n-1}{\mathbf{M}}_{\mathrm{net}\_n}}\end{array}\right]$$

(21)

By setting $\mathbf{M}=0$ , the nonlinear static equilibrium equations equation (21) are solved using MATLAB’s fsolve() function with the trust-region-dogleg algorithm. The fsolve() function uses a tolerance of $1\times {10}^{-6}$ and an initial guess of 0.001 for ${\kappa }_{\mathrm{ix}}$ , ${\kappa }_{\mathrm{iy}}$ , and ${\epsilon }_i$ to ensure accurate solutions.

The solutions for ${\kappa }_{\mathrm{ix}}$ , ${\kappa }_{\mathrm{iy}}$ , and ${\epsilon }_i$ are used to compute the position vectors of the manipulator’s distal end in the base coordinate system:

$${}^0{\mathbf{P}}_{\mathbf{s}\_n}=\sum _{i-1}^{0}R{·}^{i-1}{\mathbf{P}}_{\mathbf{s}\_i},$$

(22)

where ${}^{i-1}{\mathbf{P}}_{\mathbf{s}\_i}$ represents the position vector of the $i^{\mathrm{th}}$ spacer disk in the ${(i-1)}^{\mathrm{th}}$ coordinate system. Meanwhile, the pose matrix of the end-effector can be obtained using equation (4).

Experimental evaluation

Experimental setup

A robot prototype is constructed to evaluate the static performance and shape prediction accuracy of the proposed flexible continuum robot driven by flexible shafts, which is shown in Figure 7. The pulling action of the flexible shafts is used to adjust the spatial posture of the two flexible joints, while the twisting motion of flexible shafts 1, 2, and 3 controls the pitch, rotation, and opening-closing functions of the rigid joint. Brushless DC motors (CBL2040-2416F, Techrobots, China) are used to provide the driving force for the flexible shafts. Tension-compression force sensors (RDF-TM10, Ryder, China), which are fixed to the motor’s output shaft, are used to measure the tension on the flexible shafts, while providing a comprehensive accuracy of 0.1% of full scale. An optical dynamic capture system is used to sense the postural information of the flexible continuum robot through identifying the position of the optical markers, which are placed at spacer disks along the elastic backbone. The optical dynamic capture system comprises four motion capture cameras (1280 × 1024 resolution, MARS1.3H, Nokov, China), achieving a maximum frequency of 240 FPS, and a positional accuracy of $\pm 0.2\mathrm{mm}$ . At the beginning of motion capture experiments, the entire 3D motion capture system will be calibrated in three aspects, including: determination of the coordinate origin, field sweeping, and error calculation. Then, the robot system will control the motors to drive flexible shafts for motion implementation. The material properties and geometric dimensions of the flexible robot prototype are listed in Table 2.

Figure 7.

Experimental platform and functional demonstration of the rigid-flexible hybrid robot: (a) the overall experimental setup, comprising the drive mechanism, optical motion capture system, and the manipulator and (b) functional demonstration of the robot, showing the rigid joint performing pitch, rotation, and opening-closing motions, and the flexible joints performing multi-segment bending.

Table 2.

The materials and geometric parameters of the flexible robot prototype.

Name	Value
Diameter of spacer disk	15 mm
Distance from wiring hole to elastic backbone, $d$	5 mm
Distance between adjacent spacer disks in upright configuration	12.5 mm
Length of single-segment manipulator	67.5 mm
Length of dual-segment manipulator	140.75 mm
Young’s modulus of elastic backbone, $E_{\mathrm{bk}}$	24 GPa
Young’s modulus of flexible shaft, $E_{\mathrm{fs}}$	9 GPa
Poisson’s ratio of elastic backbone	0.3
Poisson’s ratio of flexible shaft	0.32
Diameter of elastic backbone	0.92 mm
Diameter of flexible shaft	0.92 mm
Moment of inertia of elastic backbone and flexible shafts, $I$	$0.5627{\mathrm{mm}}^4$
Gravitational acceleration, $g$	$9.81\mathrm{m}/s^2$
Mass per unit length of elastic backbone	0.0046 kg/m
Weight of spacer disk	1.2 g
Weight of proximal segment end disk	11.2 g
Weight of distal segment end disk + rigid joint	20.5 g

Shape control validation

Shape perception experiment of single-segment flexible robot

A single-segment flexible robot consists of five spacer disks, one elastic backbone, and three flexible shafts. Reflective markers with a diameter of 8 mm are placed at the right above of the placer disks, and the three-dimension positions of the reflective markers are collected by using the optical dynamic capture system. At the initial condition, the traction on the flexible shafts is zero, while applying no external load at the distal of the flexible robot. The three-dimension positions of the reflective markers obtained from the optical dynamic capture system are shown in Figure 8. The black dots indicate the experimental data of the center position of the spacer disks, while the red solid line represents the simulation results of the elastic backbone under the static model. The Euclidean distance between the simulation data and the experimental data at the reflective marker position is calculated, therefore, the maximum Euclidean distance among these five spacer disks is 0.9 mm, accounting for 1.33 mm of the length of the flexible robot, and the average Euclidean distance among these five spacer disks is 0.6 mm, equivalent to 0.89 mm of the length of the flexible robot, indicating that the static model of the flexible robot has good tracking accuracy when the flexible robot is at its initial state.

Figure 8.

Zero driving force and external load on the single-segment flexible robot.

The shape of the flexible robot based on static model under the traction by a single flexible shaft is analyzed. As shown in Figure 2, the $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ flexible shafts are symmetrically located on both sides of the $\mathrm{yoz}$ plane, therefore, traction of 2.71, 5.30, 7.54, 10.42, 12.40 N, and traction of 2.75, 5.42, 7.99, 10.37, 13.61 N are applied on the $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ flexible shafts, respectively. When the assigned flexible shaft bends the flexible continuum robot through tractive action, the other flexible shafts in the robot will remain a force of 2 N for pretension. Shape data of the single-segment flexible robot under different configurations of traction on the $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ flexible shafts are presented in Figures 9 and 10. The results demonstrated that the shape of the flexible robot under the proposed static model can accurately fit the shape of the flexible robot in actual environment. The ratio of the maximum displacement error to robot length for each traction configuration of the $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ flexible shafts are shown in Figure 11, with maximum error ratios of 2.67% and 2.52%, respectively.

Figure 9.

Shape data of the single-segment flexible robot under different tractions of the $4^{\mathrm{th}}$ flexible shaft.

Figure 10.

Shape data of the single-segment flexible robot under different tractions of the $6^{\mathrm{th}}$ flexible shaft.

Figure 11.

Shape error data of the single-segment flexible robot under different tractions of the $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ flexible shafts.

The 3D shape of the flexible robot based on static model under the traction by two flexible shafts is analyzed. The $4^{\mathrm{th}}$ and $5^{\mathrm{th}}$ flexible shafts are actively applied traction under different tension configuration, while the $6^{\mathrm{th}}$ flexible shaft is passively applied traction of 2 N to remain pretension. Experimental and simulation results are presented in Figure 12. The experimental results of the 3D shape of the flexible continuum robot show good consistency with simulation under the proposed static model. The ratio of the maximum and average displacement error to robot length among the five traction configurations are 2.22% and 1.04%, respectively.

Figure 12.

3D shape data of the single-segment flexible robot under tractions of two flexible shafts: (a) three-dimensional spatial shape data and (b) bending state of the flexible robot under configurations 4, 5 (11, 8.65 N).

External load testing for the single-segment flexible robot

In this section, the shape of the flexible continuum robot under external loads is analyzed. Different standard weights (1, 2, 5, 10, 20, 50, and 200 g) are hung at the end of the single-segment flexible robot, while the direction of gravity of the weight is consistent with the −y axial direction. Experimental and simulation results of the flexible robot’s shape under different loads are presented in Figure 13, where the red solid line represents simulation results of the elastic backbone, and black dots indicate experimental results at the corresponding spacer disks. The ratio of the maximum displacement error to robot length under different loads are shown in Figure 14. The experimental results demonstrate that the maximum error ratio is 3.56% when the external load is less than 50 g. The maximum error ratio increases to 6.81% when the external load is 200 g.

Figure 13.

Shape data of the single-segment flexible robot under different loads.

Figure 14.

Evolution of model accuracy with different external loads at different spacer disks.

Shape perception experiment of dual-segment flexible robot

In this section, the performance of the proposed static model is further investigated by using a dual-segment flexible continuum robot. The proximal segment of the flexible robot comprises four identical spacer disks with six routing paths, while the distal segment consists of four identical spacer disks with three routing paths. Shape data of the dual-segment flexible continuum robot are obtained by using 11 reflective markers located at the spacer disks along the axial direction of the robot’s elastic backbone. The overall length of the dual-segment flexible continuum robot is 140.75 mm. In the initial condition, the traction of each flexible shaft is set as 0, and no external load is applied on the flexible robot. The experimental and simulation results are shown in Figure 15. The ratio of the average and maximum displacement errors to robot length are 1.49% and 2.42%, respectively.

Figure 15.

Driving force and external load are both 0 on the dual-segment flexible robot.

The planar shape analysis of the dual-segment flexible continuum robot under the pulling action of two flexible shafts are conducted on different traction configurations. The six traction configurations are as follows: 2 and 5 (6.96, 15.22 N), 2 and 5 (7.56, 15.07 N), 2 and 5 (9.96, 17 N), 3 and 4 (8.37, 15.97 N), 3 and 4 (8.55, 14.88 N), and 3 and 4 (7.82, 10.11 N). When a set tensile force is applied to the two specified flexible shafts, a pretension force of 2 N will be applied to other flexible shafts. The flexible shaft needs to pass through a shaped conduit from the motor output to the base of the flexible robot, therefore, the traction at the base of the flexible robot and the traction at the output end of the corresponding motor need to be converted. For example, under the configuration of 2 and 5 (6.96, 15.22 N), the traction on the corresponding motor output end is 2 and 5 (10, 50 N), and the real-time force data at the motor output end are shown in Figure 16. The shapes of the dual-segment flexible robot under traction by two flexible shafts 2, 5 and 3, 4 are presented in Figures 17 and 18, respectively. The ratios of the average and maximum displacement errors to robot length under different traction configurations by combination of the $2^{\mathrm{nd}}$ and $5^{\mathrm{th}}$ flexible shafts are 0.57% and 1.35%, respectively. Meanwhile, to the $3^{\mathrm{rd}}$ and $4^{\mathrm{th}}$ flexible shafts, the corresponding ratios are 1.71% and 4.69%, respectively. As shown in Figure 2, the $2^{\mathrm{nd}}$ and $5^{\mathrm{th}}$ flexible shafts are located in the vertical plane, while the $3^{\mathrm{rd}}$ and $4^{\mathrm{th}}$ flexible shafts are located in a plane with an intersection angle of 30° to the vertical plane, Furthermore, the reflective markers are placed directly above the spacer disks, and also within the vertical plane. Therefore, the displacement errors of the reflective markers generated by the traction of the $2^{\mathrm{nd}}$ and $5^{\mathrm{th}}$ flexible shafts are smaller than that by the traction of the and $4^{\mathrm{th}}$ flexible shafts. The main reason is that the position error caused by the assembly of the reflective markers has a smaller impact in the vertical plane.

Figure 16.

Real-time force data at flexible shafts 2 and 5 (7.61, 15.73 N).

Figure 17.

In-plane shape data of the dual-segment flexible robot driven by flexible shafts 2 and 5.

Figure 18.

In-plane shape data of the dual-segment flexible robot driven by flexible shafts 3 and 4.

The 3D shape of the dual-segment flexible continuum robot under the traction of three flexible shafts is analyzed. The detailed combination of three flexible shafts is shown in Figure 19. The blue solid line with circle markers represents experimental data, and the red solid line represents simulation data for the elastic backbone. The ratios of the average and maximum displacement errors to robot length under different traction configurations by the three flexible shafts are 1.78% and 5.97%, respectively.

Figure 19.

The shape error data of a dual-segment flexible robot under out-of-plane tension tests on three flexible shafts: (a) three-dimensional spatial shape data and (b) bending states of the robotic arm under configurations 1, 2, and 5 (5.21, 8.04, 12.71 N).

External load testing for the dual-segment flexible robot

The load capacity of the dual-segment flexible robot is verified when the $2^{\mathrm{nd}}$ and $5^{\mathrm{th}}$ flexible shafts under traction configuration of (6.96, 15.22 N). Seven different standard weights are used to hang at the distal of the dual-segment flexible robot, respectively. The experimental results are shown in Figure 20. The ratios of the average and maximum displacement errors to robot length are 0.78% and 2.13% respectively, when the external load is less than 50 g. The ratios of the average and maximum displacement errors to robot length increase to 0.92% and 3.41% respectively, when the external load rises to 200 g.

Figure 20.

Evolution of model accuracy with different spacer disks under varying external loads.

Positioning repeatability experiments

To assess the positioning repeatability of the proposed flexible continuum robot, traction configurations by flexible shafts are conducted for robot’s shape planning. In this experiment, the flexible shaft as the transmission medium for tensile force includes two situations: the first situation is that the flexible shafts used as driving source come from the same flexible segment, and the second situation is that the flexible shafts used as driving source come from the two different flexible segments. In the first situation, 15 traction configurations including 4(2.71), 4(5.3), 4(7.54), 4(10.42), 4(12.4), 6(2.75), 6(5.42), 6(7.99), 6(10.37), 6(13.61), 4, 5(3.39, 3.51), 4, 5(12.28, 5.9), 4, 5(11, 8.65), 5, 6(5.96, 12.2), and 5, 6(8.79, 10.33 N) are implemented. In the second situation, 12 traction configurations including 2, 5(6.96, 15.22), 2, 5(7.56, 15.07), 2, 5(9.96, 17), 3, 4(8.37, 15.97), 3, 4(8.55, 14.88), 3, 4(7.82, 10.11), 1, 2, 5(2.22, 8.17, 14.24), 1, 2, 5(5.21, 8.04, 12.71), 2, 3, 5(8.53, 3.18, 13.22), 2, 3, 5(7.62, 3.9, 12.2), 2, 5, 6(3.11, 10.55, 3.56), and 2, 5, 6(2.46, 8.63, 4.57 N) are implemented. All the traction configurations are repeated implemented five times, and the root mean square error (RMSE) for each traction configuration is calculated, as shown in equation (23).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{RMS}{E}_x=\frac{1}{M}\sum _1^M\left(\sqrt{\frac{1}{N}\sum _1^N{(x_i-\hat{x})}^2}\right)}\\ \multicolumn{1}{c}{\mathrm{RMS}{E}_y=\frac{1}{M}\sum _1^M\left(\sqrt{\frac{1}{N}\sum _1^N{(y_i-\hat{y})}^2}\right)}\\ \multicolumn{1}{c}{\mathrm{RMS}{E}_z}& =\frac{1}{M}\sum _1^M\left(\sqrt{\frac{1}{N}\sum _1^N{(z_i-\hat{z})}^2}\right)\hfill \end{array},$$

(23)

where $M$ denotes the number of reflective markers attached to the spacer disks, and $N$ denotes the number of repetitions. The RMSE in $x$ , $y$ , and $z$ directions for each traction configuration are shown in Figures 21 and 22, where blue and orange histograms indicate the robot’s shape perception in-plane and out-of-plane, respectively. In the first traction configuration experiments, the maximum RMSE in $x$ , $y$ , and $z$ directions are: 0.4, 0.3, and 0.2 mm, respectively. For the second traction configuration experiments, the maximum RMSE in $x$ , $y$ , and $z$ directions are: 2.7, 5.1, and 1.8 mm, respectively. It is observed that the traction configuration for out-of-plane shape perception shows larger RMSE in $x$ , $y$ , and $z$ directions than that generated by in-plane shape perception, regardless of which traction configuration they are involved.

Figure 21.

Repeatability experiments of single-segment flexible robot.

Figure 22.

Repeatability experiments of dual-segment flexible robot.

Discussion

In this study, a flexible continuum robot driven by flexible shafts is constructed and shape perception of the flexible continuum robot based on static model is analyzed. Tensile force on the flexible shafts, interaction force with external environment, gravitational force and moment, elastic force generated by spinal rods and flexible shafts, are comprehensive considered in the static analysis. The shape perception of flexible continuum robot under traction configurations by flexible shafts are performed, it shows that the simulation results of the pose and shape of the flexible continuum robot based on the static model have good consistency with the experimental results. The features of the flexible continuum robot based on the static model are summarized as follows.

Pull-twist coaxial collaboration actuation by flexible shafts

The proposed robot comprises of a rigid-flexible hybrid robotic mechanism. The rigid-flexible hybrid robotic mechanism consists of a 4-DoF dual-segment flexible continuum part and a 3-DoF rigid part, while, the distal of the flexible continuum part and the rigid part share the same mechanism. Six flexible shafts enter the robot’s body from the root, with three shafts terminating at the end of the first segment flexible joint, and the other three flexible shafts passing through the first segment flexible joint, terminating at the end of the second segment flexible joint, and connected to the driving rod of the rigid joint. The pulling effect of the flexible shaft can control the universal bending motion of the flexible joints, and the twist of the flexible shafts connected to the rigid joint can control the rigid joint to achieve the functions of forceps’s pitch, rotation, and opening-closing. This new approach can decrease the number of the mechanical components in the robot body for dexterous structure, and reduce the mechanical complexity of robot body.

Static model under comprehensive consideration of elastic components

The shape accuracy of the flexible continuum robot is related to the pull force of the actuator, external interaction force and its moment caused by external load, gravity and moment of each robot component, and the elastic force and its moment generated from bending movement of the elastic components. Compared with rope pulled flexible robot, the shape accuracy of the developed flexible robot driven by flexible shafts is not only affected by the traction and moment generated by the pulling effect of the flexible shaft, but also by the additional elastic force and moment generated by the elastic bending deformation of the flexible shaft. In this study, the flexible shafts are uniformly distributed on the spacer disks with a circumference of certain radius, while the elastic spinal rod is located at the center of the spacer disks. Therefore, the equivalent bending moment of the elastic comprehensive combination of the flexible shafts and the spinal rods can be calculated using parallel axis theorem. The theoretical computation was introduced in Elastic Force and Moment, and the shape perception of the flexible continuum robot was presented in Section Motion Performance Validation. The experimental results are consistent with the simulation results based on the theoretical analysis.

To model static shape estimation, we employed the piecewise constant curvature approach due to its high computational efficiency and suitability for the complex force interactions in our robot design. However, we acknowledge the limitations of PCC in capturing variable curvature characteristics under certain conditions. In high external load scenarios (e.g. 200 g), the maximum error ratio reaches 6.81% for single-segment robots and 3.41% for dual-segment robots. For non-planar deformations, such as dual-segment three-axis traction configurations, the maximum error ratio increases to 5.97%. The underlying reason for this discrepancy lies in the inherent limitations of the piecewise constant curvature model when dealing with non-uniform, large deformations. When a significant, concentrated load is applied at the distal end, the robot’s physical shape is no longer a smooth, constant-curvature arc. Instead, it exhibits a highly non-uniform bending profile, characterized by a sharp, high-curvature bend near the load, while the proximal section remains less deformed. The piecewise constant curvature model approximates this entire non-uniform segment with a single curvature value, and this averaging process leads to a geometric deviation from the true shape, resulting in the observed positional errors. To mitigate this, we optimized the segmentation (using 5 spacer disks for single-segment robots and 10 for dual-segment robots) to enhance accuracy within acceptable bounds. In future work, we plan to explore variable curvature models, such as Cosserat rod theory, to further improve shape estimation accuracy under high external loads and non-planar deformations.

In the current static analysis, only the influence of elastic force and moment generated by bending deformation under the traction of the flexible shaft on the shape accuracy of the flexible continuum robot has been considered, however, the influence of the flexible shaft caused by the torsional deformation has not been taken into account. Although the twist of the flexible shaft is intended to transmit power to the distal rigid joint of the rigid-flexible hybrid mechanism, the pull-twist coaxial collaboration will cause motion coupling on the flexible robot. Therefore, it is necessary to comprehensively consider the impact of the pull-twist coaxial collaboration of the flexible shafts in future work.

Rigid-flexible hybrid structure in the static model

The developed static model in this study takes into account the characteristics of the hybrid structure integrating the flexible and rigid joints. Based on the structural feature that the base of the rigid joint is the distal of the flexible joint, an entire kinematic model of the rigid-flexible hybrid mechanism is established, therefore, the position accuracy of the distal of the rigid-flexible hybrid robot under the used of the developed static model is guaranteed.

Conclusions

This paper presents a shape model for a novel flexible robot with elastic configuration consisting of the flexible shafts for power transmission and spinal rod for stem supporting. The entire static model is constructed under comprehensive consideration, including tensile force on the flexible shafts, interaction force with the external environment, gravitational force and moment, and the elastic force and moment in the elastic combination of the flexible shafts and spinal rods. The shape accuracy of the proposed robot under the kinematic and static models are validated through a total of 182 experiments. For single-segment flexible robot, the ratios of the average and maximum displacement errors to robot length in planar shape perception are 2.67% and 2.52%, respectively. And they become 2.22% and 1.04% when performing 3D shape perception. For dual-segment flexible robot, the corresponding ratios in planar shape perception are 1.71% and 4.69%, respectively. And they become 1.78% and 5.97% when performing 3D shape perception. Furthermore, the shape accuracy of the flexible robot with external loads were assessed, it demonstrated that the experimental results are consistent to the calculated value based on the static model when the external load is less than 200 g. The experimental results show that the flexible robot with the static model owns good shape accuracy in potential application scenarios.

In the future, the influence of the pull-twist coaxial collaboration of the flexible shafts will be considered into the static model, aiming to construct precise modeling of the rigid-flexible hybrid mechanism. Meanwhile, artificial intelligence methods, such as machine learning, will be used to establish the mapping relationship between various inputs and the robot’s shape information, including the tensile force and torsion, translational displacement, and twist angle of the flexible shafts. To deploy the improved mathematical models in real-time control systems, quantitative analysis on calculation time, robustness, and other factors will be conducted to ensure the effectiveness and responsiveness of the models in practical applications.