Development of a Tendon-Driven Continuum Robot for Medical Applications

Abstract

This paper presents the design, kinematics, and development of a tendon-driven continuum robot for surgical applications. The continuum robot has a flexible and adaptable construction that imitates the movements of natural organisms. The robot’s unique structure comprises disk members, springs, and a continuum backbone member, enabling it to bend, contract, and deform in complex ways. The robot is operated by pulling tendons, giving it the agility and flexibility necessary to bend in confined spaces. This study discusses the main design considerations and challenges in creating a tendon-driven continuum robot, including the kinematics of the four-tendon mechanism. The developed tendon-driven continuum robot is categorized into two modules: the distal end and the proximal end. The distal end consists of the continuum robot structure, whereas the proximal module consists of the actuating unit that actuates the distal end. The experimental results demonstrate the continuum robot’s ability to be used in medical fields and pipe inspections because of the miniaturized design of the distal end, which allows it to enter confined spaces. This paper provides valuable insights into the design, kinematics, and appropriate materials to build a tendon-driven continuum robot; its bending and deformation capabilities can be used in many fields, especially surgical applications and confined space explorations.

1. Introduction

Robots are rapidly increasing in many parts of modern life, from the manufacturing industry and healthcare to exploration. Robots can be divided into two main categories: flexible and rigid. Rigid robots are constructed of hard materials like metals or plastics, and their mechanical designs limit their motions [1,2]. Moreover, due to their rigid structures, they are limited to working in confined spaces and need special modifications to meet the flexibility requirements for many precise applications, such as minimally invasive surgery. Rigid robots have constraints in flexibility, especially when performing operations on internal organs, since it is quite difficult to access the circuitous parts of the human body. These limitations can be overcome by using flexible robots, which solve this problem. Continuum robots adjust their shape at any point along their length, enabling them to function or operate in confined areas and complex environments inaccessible to traditional rigid link robots. Continuum robots are under-actuated systems where degrees of freedom (DOF) can be varied based on design and actuation mechanisms [3,4]. Continuum robots are broadly categorized into three types: tendon-driven continuum manipulators (TCMs), tendon-driven serpentine manipulators (TSMs), and concentric tube manipulators (CTMs). A detailed description of them can be found in [5].

Tendon-driven continuum robots are actuated from the base. Upon increasing the number of tendons/cables, the DOF of a continuum robot can be varied [6,7]. As DOF are variable and do not consist of links and joints, D-H parameters to solve kinematics are not applicable; hence, the mathematical modeling of continuum robot kinematics is complex [8]. Researchers have developed many models of continuum robots based on their applications [9,10]. The kinematics of continuum robots depends on their design considerations [11]. A method to derive kinematics has been developed by assuming constant curvature bending [12,13]. Many designs have been proposed [14,15,16] to model continuum robots, considering their applications in many fields. Experimental validations have been done assuming a constant curvature approach [17]. Different methods to derive constant curvature kinematics have been discussed and compared [18]. It was stated that a smaller diameter-length ratio is required in minimally invasive surgeries to follow torturous paths [19]. Workspace analysis of TCMs and CTMs has been compared by researchers [5]. Material selection and various actuation methods have been described to actuate the continuum robots [20]. Necessary tests are needed to find the bending, deformation capabilities, tip position tracking, workspace analysis, and actuation mechanism for enhanced use in confined space explorations and surgical applications [21].

The advantage of choosing cable/tendon-driven continuum robots rather than concentric tube manipulators is that they adhere to the design [22]. Concentric tube robots, often formed of super-elastic nitinol (NiTi) tubes, comprise several nested pre-curved tubes. The tubes are overlapped to create CTMs. The translation and rotation of each tube control the position and orientation of the distal end [23,24,25]. The discussion above summarizes that the continuum robot should have the following features:

• It should bend in any direction.
• It should be as small as possible for medical pipe inspection applications to explore confined and contour environments during its operation.
• The backbone material should be strong and flexible enough for structural rigidity.
• Overall, the structure should be flexible enough to achieve a compliant mechanism.

Many studies have been conducted that aim to create the best designs that are feasible and suitable for applications including medical, search, and rescue operations due to their compliance, flexibility, dexterity, and adaptability to confined spaces and environments. The realization of DOFs and kinematics is also necessary.

2. Literature Study

Several researchers have proposed many designs and methods to solve the kinematics of tendon-driven continuum robots for different applications. The background study performed in this research mainly focused on two aspects: design, the kinematics of continuum robots, and the research gap.

2.1. Design

The number of DOF of a continuum robot can be varied based on designs and actuation mechanisms. Several studies have proposed many designs to model tendon-driven continuum robots for different applications. Ouyang et al. in 2016 developed a design consisting of a three-segment continuum robot for minimally invasive surgery. The continuum robot design employs a single super-elastic nitinol rod as the backbone and concentric disks assembled on the backbone for the tendon attachment [14]. Li et al. in 2017 designed a novel tendon-driven continuum robot with two modules/segments and a compliant/flexible backbone formed by helical springs. In this design, each module is driven by four parallel arranged tendons to implement a redundant actuation mechanism that gives dexterous motions to the robot [15]. Castledine et al. in 2019 proposed a design consisting of a flexible core combined with rigid interlocking vertebrae evenly distributed along its length. This design allows bending in two DOF while minimizing torsional movement. The proposed design segment is actuated by two antagonistic tendon pairs (one pair contracts and the other pair relaxes or lengthens). This design also features a large hollow central bore, which could be used for suction-assisted gripping of objects [16].

2.2. Kinematics

Several approaches to derive the kinematics of a tendon-driven continuum robot using several tendons in the design are available in the literature. Jones et al. in 2006 coined a concept in which they fit a conceptual conventional rigid link manipulator to the continuum backbone of a continuum robot [12]. Mosqueda et al. in 2018 derived kinematics using a constant curvature bending approach for three-tendon-driven actuations. The continuum robot developed has one segment with seven sections, each driven by three cables/tendons [17]. Bhattacherjee et al. in 2018 derived the kinematics of a continuum robot driven by four tendons that are 90° apart. The kinematics was solved by taking the parameters of the bent section (primary backbone) and corresponding tendons of the section [26]. Wu et al. in 2022 proposed a new geometric approach to solve the IK of the continuum robot consisting of two segments [27].

2.3. Research Gap

It is evident that a continuum robot with variable-length behavior has yet to be developed, which is our main objective. A constant curvature kinematic approach [12] will be followed in this research to solve the kinematics of the continuum robot. Two stepper motors will drive four tendons, making the continuum robot an under-actuated system. The proposed designs so far from the literature study include either tendons for the actuation of the continuum robot [12,14,16,17] or springs [15] for creating the backbone structure of the continuum robot with structural rigidity [28,29]. Minimal work was done using both tendons and springs for modeling the continuum robots, and the same can be justified from the literature.

3. Methodology

The current research focuses on developing a tendon-driven continuum robot driven by four tendons 90° apart due to their symmetry in architecture, and it has added advantages over three-tendon-driven continuum robots in terms of design, kinematics, and actuation. An under-actuation system drives the tendons of the robot. First, the kinematics of the robot are solved using the constant curvature approach [12]. Then, the design of the robot is carried out, considering all the components necessary to build the model, including mathematical modeling. The structural members are manufactured using digital light processing (DLP) 3D printing. Testing is carried out once prototyping is done by integrating the actuation system into the structural members of the robot. The overall workflow of this research is represented in Figure 1.

3.1. Kinematics

3.1.1. Forward Kinematics

Deriving the kinematics of a continuum using D-H parameters will not provide a solution for this research. An assumption is made that the continuum robot follows constant curvature bending to derive the kinematics, which contradicts practical implementations, as the load varies from the end to the base structure, thereby leading to a non-linear distribution of the load. Studies have also proposed various methods to derive the kinematics of continuum robots [12,17,18,26], as mentioned in the previous section. The kinematics of a continuum robot can be developed by considering a bent section [15].

The bent section, as shown in Figure 2, creates an arc of length (s) with curvature (k), making an angle (Φ). The kinematic mapping from the actuation space to the task space can be decomposed into two parts, which map from the actuator lengths ([l₁l₂l₃l₄]^T) to the arc parameters (s, k, Φ), and from the arc parameters to the robot tip position $\left({\left[xyz\right]}^T\right)$. The mapping from the actuator lengths to the arc parameters for a bent section is given by following

Table 1. Forward kinematics parameters.

Sr No	Parameter	Description
1	Arc Length ($s)$	$\left({\displaystyle \frac{2n}{k}}\right)\mathrm{asin}\left({\displaystyle \frac{k\left(l_1+l_2+l_3+l_4\right)}{8n}}\right)$	$\left(1\right)$
2	Curvature ($k)$	${\displaystyle \frac{\left(l_1-3l_2+l_3+l_4\right)\left(\sqrt{{\left(l_4-l_2\right)}^2+{\left(l_3-l_1\right)}^2}\right)}{\left.\mathrm{r}(l_1+l_2+l_3+l_4\right)(l_4-l_2)}}$	(2)
3	Angle of Curvature ($\Phi )$	$atan{\displaystyle \frac{\left(l_4-l_2\right)}{(l_3-l_1)}}$	$\left(3\right)$

.

The homogeneous transformation matrix (A) of the 4-tendon-driven continuum robot gives the forward kinematics and can be referred from Equation (4) below. Here, the 3 × 3 matrix from A represents the orientation of the tip, and the 3 × 1 vector from A represents the position of the tip.

$$A=\left[\begin{array}{cccc}{\mathit{cos}\left(\Phi \right)}^2(\mathit{cos}\left(ks\right)-1)+1& \mathrm{s}\mathrm{i}\mathrm{n}\left(\Phi \right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\Phi \right)\left(\mathrm{c}\mathrm{o}\mathrm{s}\right(ks)-1)& -\mathrm{c}\mathrm{o}\mathrm{s}\left(\Phi \right)\mathrm{s}\mathrm{i}\mathrm{n}\left(ks\right)& (\mathit{cos}\left(\Phi \right)\mathit{cos}\left(ks\right)-1))/k\\ \mathrm{s}\mathrm{i}\mathrm{n}\left(\Phi \right)\mathit{cos}\left(\Phi \right)\left(\mathit{cos}\left(ks\right)-1\right)& {\mathit{cos}\left(\Phi \right)}^2\left(1-\mathit{cos}\left(ks\right)\right)+\mathit{cos}\left(ks\right)& -\mathit{sin}\left(\Phi \right)\mathit{sin}\left(ks\right)& (\mathit{sin}\left(\Phi \right)\mathit{cos}\left(ks\right)-1))/k\\ \mathit{cos}\left(\Phi \right)\mathit{sin}\left(ks\right)& \mathit{sin}\left(\Phi \right)\mathit{sin}\left(ks\right)& \mathit{cos}\left(ks\right)& {\displaystyle \frac{\mathit{sin}\left(ks\right)}{k}}\\ 0& 0& 0& 1\end{array}\right]$$

(4)

The kinematic simulation was performed using OCTAVE software (version 9.3.0). Figure 3a indicates the tip position in the XZ plane. The same representation in the 3D XYZ plane is indicated in Figure 3b below. Figure 3c shows the continuum robot’s tip coordinates (x, y, z) for all bent sections with varying tendon lengths, which gives the visualization of the robot’s workspace.

3.1.2. Inverse Kinematics

The inverse kinematics of a continuum robot driven by four tendons are obtained by mapping the lengths of the tendons of a bent segment to the arc parameters. The equations to find the tendon lengths in terms of arc parameters will differ by 90° between l₁l₂l₃ and l₄, since all four tendons are arranged 90° apart.

$$l_1=\left(2n\right)\sin \left({\displaystyle \frac{ks}{2n}}\right)\left(\left({\displaystyle \frac{1}{k}}\right)-d\mathit{cos}\left(\phi \right)\right)$$

(5)

$$l_2=\left(2n\right)\sin \left({\displaystyle \frac{ks}{2n}}\right)\left(\left({\displaystyle \frac{1}{k}}\right)-d\mathit{cos}\left({\displaystyle \frac{\pi }{2}}-\phi \right)\right)$$

(6)

$$l_3=\left(2n\right)\sin \left({\displaystyle \frac{ks}{2n}}\right)\left(\left({\displaystyle \frac{1}{k}}\right)-d\cos \left(\pi -\phi \right)\right)$$

(7)

$$l_4=\left(2n\right)\sin \left({\displaystyle \frac{ks}{2n}}\right)\left(\left({\displaystyle \frac{1}{k}}\right)-d\cos \left(3\left({\displaystyle \frac{\pi }{2}}\right)-\phi \right)\right)$$

(8)

where

n is the number of disks considered = 10 for this research, and

d = 3.6 mm (distance from tendon to continuum backbone).

Figure 4 maps the arc parameters (s, k, Φ) to the tendon lengths (l₁, l₂, l₃, l₄). This gives the inverse kinematics by using Equations (5)–(8).

3.2. Kinematic Validation

Kinematic validation is also performed for both the forward and inverse kinematics of a 4-tendon-driven continuum robot. To validate the forward kinematics for a particular bent section, the lengths of tendons [l₁, l₂, l₃, l₄] are found to be [5,17,19,30], the arc parameters are calculated (s, k, Φ) = (0.1098, 8.0026, 0.3218) and the end position coordinates (x, y, z) = (−3.1263, −1.0421, 7.0118) are found. A similar approach is followed to validate the inverse kinematics by taking the arc parameters (s, k, Φ) to be (0.1098, 8.0026, 0.3218) of a particular bent section and calculating the lengths of the tendons [l₁l₂l₃l₄] = [5.0001, 6.9999, 11, 9.0002] for the above-derived end position coordinates. A note is made that at a particular bending angle, the lengths of the tendons become [l₁l₂l₃l₄] = [9.9998, 9.9998, 10.9998, 10.9998].

l₁ = l₂; l₃ = l₄ when the robot bends with 45°.

3.3. Design

In this study, many designs are explored [14,15,16,30,31] to model a continuum robot with a very small form factor suitable for confined space explorations. The developed design, as shown in Figure 5, includes structural members like disks, backbone, tendons, and springs responsible for the change in the robot’s shape under actuation. The use of springs in the design will help the continuum robot to achieve contraction along its length. A provision in the disk has been given for the spring to sit on the disk. The main goal is to design a continuum robot driven by four tendons that are 90° apart. So, a guide for each tendon is provided on the disks, along with a central guide for the backbone member. Provision for springs is given on the top and bottom faces of the disks.

The reason for this small dimension is for the robot to be used in surgical applications. Once disks are designed, springs are added to the design. The number of disks considered is limited to 10 in the design, and nine springs will accommodate the disks. A backbone design member and tendons are attached to the design. The guides provided in the disks for the tendons and backbone maintain clearance, enabling them to route easily. A base plate is also designed to let the continuum robot’s distal end sit on it. The final assembly achieved can be referred to in Figure 6.

All four tendons and the continuum backbone are crimped at the face of the top disk using an rj45 crimping tool, restricting them from sliding through their respective guides. The tendons are routed from the distal end to the proximal end for actuation.

3.4. Prototype Development

The 4-tendon-driven CR model has tendons arranged 90° apart, allowing two tendons to be in a horizontal direction (x-axis) and the other two in a vertical direction (y-axis). This configuration allows the opposite tendons to act as antagonistic pairs to achieve pitch and yaw-bending motions. All the components necessary to build the prototype of a 4-tendon-driven continuum robot were manufactured using DLP 3D printing (disks and base plate) and procured (compression springs, tendons, and NiTi backbone). The components include disks, springs, the continuum backbone, tendons, and the actuation unit at the proximal end. Two NEMA-17 stepper motors actuate four tendons. Motor drives for both stepper motors are connected, and the proximal module is controlled by Arduino UNO and run by a 12 V DC power supply.

Development of the prototype involves two stages:

• Distal end;
• Proximal module.

3.4.1. Distal End Development

The distal end of the developed continuum robot consists of an assembly of disks, springs, tendons, and a backbone, as shown in Figure 7. All four tendons and backbone are crimped on the face of the top disk to restrict their sliding through their respective guide holes, as mentioned in Section 2. All the components of the distal end are assembled carefully and crimped. An identical distance is maintained between each disk because of the spring, which means that all disks are equidistant along the length of the continuum backbone.

3.4.2. Proximal Module

The proximal module contains the actuation unit that actuates the continuum robot. The actuation unit includes 2 Nema-17 stepper motors, 2 L298N motor drivers, an Arduino UNO controller, and splice connectors. Necessary connections are given between all electronic components in the proximal module to actuate the distal end.

The built proximal module is integrated with the distal end, and the continuum robot prototype is built, as shown in Figure 8. Before connecting, the tendons are categorized into two pairs. Each pair has opposite-faced wires, making each pair responsible for bending in the respective direction. This is often referred to as an antagonistic pair of tendons. This arrangement allows the distal end of the prototyped continuum robot to bend in pitch (y-axis) and yaw (x-axis). Also, using springs in the distal end enables it to contract along its length, giving 3 DOF to the developed continuum robot.

3.5. Materials

The use of super-elastic nitinol for the backbone and tendons provides flexibility to the robot. Nitinol is biocompatible as it does not harm human tissue [32] when used in building this robot for surgical applications. A practical application of nitinol is its use in stents for cardiovascular surgeries. Stainless steel 304 compressive springs are used. SS304 is known for its excellent low carbon content and high corrosion resistance, making it suitable for use in medical applications.

4. Testing

After building the prototype, the continuum robot is tested to check its bending capabilities. The testing of the continuum robot is carried out by connecting a 12 V power supply to the proximal module of the continuum robot to run both stepper motors, which in turn actuate the distal end, allowing it to bend in both the pitch and yaw directions and contract along its length.

Testing has been carried out to find the following:

• Bending angle in pitch;
• Bending angle in yaw;
• Percentage contraction in length.

Both stepper motors are coded in a way that they rotate two revolutions in both the clockwise and counterclockwise directions. Based on motor control, the continuum robot’s bending can be achieved along with its distal end’s length contraction.

4.1. Pitch Motion

Bending in the pitch axis is achieved by connecting a pair of tendons to one stepper motor. To measure the bending angle, graph paper is placed behind the distal end, taking the vertical axis as a reference for bending. The tendon pair is wound on the motor shaft in the opposite direction, making them an antagonistic pair. When the motor rotates, one tendon gets wound, and the other tendon gets unwound, achieving bending.

To achieve negative pitch, motor 1 is made to rotate clockwise for two revolutions, allowing the distal end to bend. The bending angle is measured from the base of the vertical axis to the tip of the continuum backbone and is represented in Figure 9a,b. To achieve a positive pitch, the same motor is made to rotate anti-clockwise for two revolutions, and the corresponding bending angle is measured in the positive pitch axis, represented by Figure 9c.

The bending achieved in the positive pitch is equal to the bending achieved in the negative pitch due to the prototype’s symmetry in the pitch axis.

4.2. Yaw Motion

When the motor is made to rotate counterclockwise, the tendon that is wound on the shaft gets pulled, and this bends the robot in a negative yaw direction, as shown in Figure 10a. The same way is used to measure the bending angle in the yaw axis, as shown in Figure 10b.

To achieve a positive yaw, the same motor is made to rotate clockwise for two revolutions, and the corresponding bending angle is measured in the positive yaw axis, as shown in Figure 10c. The bending achieved in both the positive and negative yaw axes is the same due to the model’s symmetry in the yaw axis.

4.3. Contraction

A change in the length of the distal end is achieved by operating both motors simultaneously in directions opposite to each other. The use of compression springs in the prototype enables the distal end to compress along its length by pulling all four tendons. The initial length of the continuum robot’s distal end is calculated manually under no actuation condition. Then, the motors are made to run, which pulls the tendons by exerting force on the springs, allowing them to compress, resulting in overall contraction of the distal end. The percentage change in length is found to be 46%. The lengths of the robot before and after actuation are represented in Figure 11a and Figure 11b, respectively.

5. Results and Discussion

After testing the bending capabilities and contraction length of the developed prototype, the following is found:

• Bending in yaw direction = ±45°;
• Bending in pitch direction = ±42°;
• Max. contraction in length = 46%.

Necessary relations are developed to relate to the motor rotation, pull in tendon length, and bending angles, and the values are tabulated in

Table 2. Relations between motor rotation, length of tendons, and corresponding bending angles.

Motor Turning Angle (°)	Length Pulled by Motor (mm)	Bending Angle (°)
720	75.36	45
540	56.52	33.75
360	37.68	22.5
270	28.26	16.875
180	18.84	11.25
90	9.42	5.625
45	4.71	2.8125

.

• Two revolutions of motor = 400 steps = 720° (1.8° rotation for each step);
• Pull in tendon length by motor = 75.36 mm.

It can be said that if the motor rotates 720° (400 steps), this rotation will pull the tendons by 75.36 mm, resulting in a 45° bending of the continuum robot’s distal end. The same follows for the remaining tabular values. It represents the relation between the bending angles and tip positions of the continuum robot. The tip coordinates of the continuum robot in a 2D plane are given by the following:

$$x=\left({\displaystyle \frac{l}{\theta }}\right)\left(1-\cos \left(\theta \right)\right)$$

(9)

$$y=\left({\displaystyle \frac{l}{\theta }}\right)\sin \left(\theta \right)$$

(10)

where

l = the length of the arc when the robot is bent and

$\theta$ = the bending angle.

The main relation to finding the tip coordinates of the distal end from the proximal module is given in a flow, as shown in Figure 12.

The arc lengths of the continuum robot bent section for the corresponding bending angles are tabulated in

Table 3. Arc lengths for corresponding bending angles.

Bending Angle (θ) in Deg.	Arc Length (l) in mm
80	26.54
70	35.14
60	43.73
50	53.33
45	56.6
42	60
40	60.92
37.5	63.1
35	65.22
30	69.5
27	72
21	75
15.75	81.5
15	82.4
12	84.98
10.5	88
5.25	90.78
1	94.4

.

The tip coordinates are measured using Equations (10) and (11) and are tabulated for the corresponding angles given in

Table 4. Coordinates in a 2D plane.

Bending Angle (θ) in Deg.	X (mm)	Y (mm)
80	15.70719202	18.71910252
70	18.92513333	27.02789145
60	20.87953698	36.1644189
50	21.82984873	46.81426167
45	21.10745474	50.95790349
42	21.02388207	54.76908527
40	20.41530632	56.09059313
37.5	19.92274212	58.69050572
35	19.30852	61.23880169
30	17.78314784	66.36761127
27	16.65297642	69.36463886
21	13.59129014	73.33203936
15.75	11.13138122	80.47745788
15	10.72466952	81.46195259
12	8.866602522	84.36008786
10.5	8.040879517	87.50826009
5.25	4.156166816	90.65302157
1	0.823774495	94.39520743

.

Figure 13 shows the relation between the tip coordinates of the continuum robot’s distal end for the corresponding bending angles.

6. Conclusions

In this research, a single-module tendon-driven continuum robot has been developed. The use of super-elastic nitinol as the backbone provided the bending capabilities, giving a 45° bending angle to the distal end, allowing it to interact with unstructured environments. From the experiments, it is evident that the developed continuum robot achieved three DOF. Also, it can be said that the developed tendon-driven continuum robot bends with a constant curvature and has symmetric bending in both the pitch and yaw directions. Mathematical modeling of the kinematics was derived for fa our-tendon-driven continuum robot, and the results comply with each other in terms of tip positions. Deformation has been achieved with 46% contraction of the distal end along its length.

7. Future Scope

Programming can be performed to solve the kinematics, path planning of the continuum robot, and its control in minimally invasive surgical procedures. Sensors like flex sensors to measure the bending and optical sensors to track the tip positions of the continuum robot can be used to receive feedback for better closed-loop control. As for the under-actuated system, DC servo motors could offer advantages such as smoother motion and higher torque. Hence, DC servo motors can be employed in place of stepper motors for enhanced accuracy and precise control because of their feedback nature.