## Abstract

An extensible continuum manipulator (ECM) has specific advantages over its nonextensible counterparts. For instance, in certain applications, such as minimally invasive surgery or pipe inspection, the base motion might be limited or disallowed. The additional extensibility provides the robot with more dexterous manipulation and a larger workspace. Existing continuum robot designs achieve extensibility mainly through artificial muscle/pneumatic, extensible backbone, concentric tube, and base extension, etc. This article proposes a new way to achieve this additional motion degree-of-freedom by taking advantage of the rigid coupling hybrid mechanism concept and a flexible parallel mechanism. More specifically, a rack and pinion set is used to transmit the motion of the i-th subsegment to drive the (i+1)-th subsegment. A six-chain flexible parallel mechanism is used to generate the desired spatial bending and one extension mobility for each subsegment. This way, the new manipulator can achieve tail-like spatial bending and worm-like extension at the same time. Simplified kinematic analyses are conducted to estimate the workspace and the motion nonuniformity. A proof-of-concept prototype was integrated to verify the mechanism’s mobility and to evaluate the kinematic model accuracy. The results show that the proposed mechanism achieved the desired mobilities with a maximum extension ratio of 32.2% and a maximum bending angle of 80 deg.

## 1 Introduction

Inspired by nature, continuum robots, especially continuum manipulators, are developed to achieve animal-like compliant-to-object property. This property is thought critical for certain applications that require passive compliance, for instance, medical robots that need to interact with human tissues, manipulation robots that need to handle fragile objects, or exploration robots that need to go through unexpected narrow passages. Traditional solutions for this type of robot focus on using deformable materials (e.g., an elastic backbone) and/or deformable actuation (e.g., tendon or rod driven). Existing examples using this technology include the Elephant trunk [1], the Tentacle robot [2], and the DDU [3], etc. Another solution is to use a hyperredundant structure, which is not theoretically a continuum robot but can behave like one. The proposed design utilizes traditional serially connected rigid link structures and usually distributes/transmits the actuation on each joint. The typical representations for this category are snake-like robots [46] and multilink tail robots [79].

Limited by the mechanical structure and the actuation technology, the aforementioned solutions are usually not extensible (which are referred to as tail-like spatial bending mobility in this article). However, for certain applications where the manipulator base motion is constricted or disabled, an additional worm-like extensibility (self-extension mobility) can significantly augment its manipulability and dexterity.

Motivated by this observation, this article proposed a new mechanism that enables the continuum manipulator to bend and extend simultaneously. An extensible continuum manipulator (ECM) based on this mechanism was designed and integrated. Both the mechanism and the manipulator are the first ones of their kind, which use motion propagation mechanisms to achieve extension mobility. An overview of the prototype is shown in Fig. 1. This new manipulator is envisioned to have potential applications in minimally invasive surgery, tentacle-like manipulations, as well as for search and rescue tasks (as a novel snake robot design).

Fig. 1
Fig. 1
Close modal

The following sections are organized as follows. Section 2 reviews the existing technological background and the design motivations of this study. Section 3 describes the mechanical design of the robot system. Section 4 formulates the kinematics based on constant curvature bending assumption and conducts workspace analysis as well as motion nonuniformity evaluation accordingly. Section 5 presents the prototyping details and experimental results. Section 6 recaps the main points of this article and discusses the future work.

## 2 Background and Design Motivation

This section reviews the existing technologies for extensible continuum manipulator designs and presents the previously proposed rigid coupling hybrid mechanism (RCHM) concept, which is the foundation of the new mechanism presented in this paper.

### 2.1 Extensible Continuum Manipulator Review.

To realize the extension mobility on traditional continuum robots, the easiest way might be converting the nonextensile backbone structure into an extensible one. The existing approaches that apply this idea include the NASA tendril type robots [1013], which use extension and compression springs as the backbone, the tendon-driven continuum robot [14], which takes advantage of the magnetic repulsion force for backbone extension, the extensible continuum robot [15] using origami modules, and the concentric tube robot [16]. However, except for the concentric tube robot, the extension motion of these manipulators is usually passive or not actuated (similarly, actuated extension is referred to as active), especially when a tendon-driven system is used. This shortcoming causes the manipulator stiffness to decrease significantly as the manipulator extends. The stiffness of the concentric tube robot, although having active extension mobilities, is still uncontrollable due to the precurved tubes. Therefore, besides making the backbone extensible, a more straightforward way is to eliminate the backbone structure and use extensible actuators directly, such as a pneumatic actuator or an artificial muscle [17]. This approach avoids the passive extension disadvantages but introduces other shortcomings, such as the need for a heavy and unstable pneumatic actuators. More importantly, without the backbone, the manipulator becomes a tentacle, which has limited usages for carrying a load or applying force control.

Therefore, looking for a new continuum robot mechanism that has both a stronger backbone and an active extension, mobility becomes a promising direction for continuum robotics research. To achieve this goal, one existing solution is to use the RCHM concept, which was originally proposed to design spatial curvature bending mechanisms based on rigid links [18]. These types of mechanisms take advantage of the traditional hybrid mechanism structure [19], but use specific transmission mechanisms to couple adjacent subsegment mechanisms. Following this novel mechanism design idea, extension mobility for continuum manipulators could be added by designing special subsegment parallel mechanisms (PMs).

### 2.2 Rigid Coupling Hybrid Mechanism.

Fig. 2
Fig. 2
Close modal

The RCHM has two main advantages in comparison with the traditional cable-driven hyper-redundant designs: (1) the RCHM usually has good rigidity due to the parallel mechanism used for each subsegment, which is known to have higher stiffness, higher precision, and better load-bearing capability than its serial counterpart; (2) the usage of rigid link transmission in the RCHM avoids the commonly observed cable-driven issues, such as the unidirectional driving problem and the cable tension control problem. These two features, together, provide the RCHM with good rigidity and enable the mechanism to respond to high-frequency input, which is critical for applications that need high-speed or high dynamic motion. Second, since the RCHM has centralized actuation, the weight of the robot itself could be significantly reduced. As a result, the motion accuracy of the robot could be increased, and the controller could be simplified.

## 3 Mechanical Design

This section details the mechanical design of the new extensible manipulator. Since the RCHM design is based on rigid link transmission, to meet the mobility requirement, the PM is first synthesized using rigid links and then modified to use flexible links.

### 3.1 Parallel Mechanism Design Using Flexible Parallel Mechanism.

Since the “Driving PM” is already fully constrained, the three additional chains cannot exert more constraints onto the system. Therefore, three spherical-spherical-prismatic (SSP) chains are selected to guarantee enough degrees-of-freedom for the “Measuring PM.” Figure 3(a) shows one potential subsegment design based on this mechanism configuration, and Fig. 3(b) shows the corresponding kinematic diagram. The overall mobility can be verified by the Grübler–Kutzbach criterion (G-K criterion) [22] as follows:
$M=6n−∑i=1j(6−fi)=6×13−3×(5+3+5)−3×(3+3+5)=6$
(1)
where n is the number of moving bodies, j is the number of joints, and fi is the corresponding degree-of-freedom (DOF) of joint i. Although the calculation shows the mechanism having 6DOF, three of them are actually internal DOF (self-rotation with respect to the axis connecting the two ball joint centers) induced by the spherical-spherical (SS) chains, which do not affect the overall mobility. Therefore, the actual mobility of the 3PSR-3SSP mechanism is 3.
Fig. 3
Fig. 3
Close modal

The PM together with the rack and pinion transmission forms the basic motion propagation mechanism. For instance, referring to Fig. 3(a), if an input motion (indicated by the solid arrow) is exerted on Rack Mi,1, Rack Di,1 is pushed right through the gear. This motion causes the clockwise rotation of Link i + 1, which further induces the relative motion of Mi+1,1 (indicated by the dashed arrow). Because of the gear, this relative motion continues to be transmitted onto Rack Di+1,1, which becomes the driving motion for the next subsegment.

Using rigid links and joints provides the advantages of being able to bear a larger load and having higher stiffness. The disadvantages include complicated mechanical structure that makes the manufacturing process more challenging in terms of manufacturing tolerance control problems (e.g., the backlash is rapidly amplified due to the motion propagation characteristics of this type of mechanism). Therefore, flexible parallel mechanisms [23,24] (FPMs) are proposed to replace the rigid link-based PM. The flexible structure facilitates the manufacturing process significantly and increases the accuracy by avoiding backlash (i.e., the deformation of the material itself does not induce backlash). Moreover, the flexible structure has the same compliant-to-obstacle benefit as traditional continuum robots.

As shown in Fig. 4, the modified subsegment design uses flexible rods to replace the original rigid links and joints. Similarly, the six-chain FPM is subdivided into one “Driving FPM” and one “Measuring FPM.” After changing to flexible rods, the mounting and connection among parts become easier too. For instance, the rods could be easily connected with the racks and the links using glue. The rack and pinion sets are also placed internally to achieve better assembly accuracy.

Fig. 4
Fig. 4
Close modal

### 3.2 System Assembly.

Figure 5 shows the overall design of the new ECM, where the ECM body is composed of four serially connected subsegments. Customized housing covers are designed to mount three linear actuators. The connection between the actuation module and the ECM body is achieved by a specifically designed first link and three special racks. To facilitate adding more subsegments to the ECM, the rest of the links and racks are designed to be identical. This modular design feature also allows extending the one segment design to multiple segments design, which could significantly increase the manipulator’s workspace and enhance its dexterity.

Fig. 5
Fig. 5
Close modal

It is worth to note that the size of the ECM body is mainly constrained by the gear teeth modulus. Using a smaller gear modulus allows designing smaller links without introducing significant twist effects (will be discussed in Sec. 4.4) and thus reduces the overall manipulator thickness. The current design uses the common 0.5 modulus gears, which limits the manipulator thickness to be 25 mm. In future designs, if smaller module gears (e.g., 0.2 modulus gears) are used, the manipulator thickness could be reduced to 15 mm, which would enable the ECM to be a good candidate for minimally invasive surgical applications.

## 4 Kinematic Analysis

For the preliminary kinematic analysis, certain assumptions could be made to simplify the computation. Due to the similar mobility as in the traditional extensible continuum manipulator, the circular arc bending assumption [25] is made here. That is, each subsegment is regarded as a constant curvature bending continuum robot section, and each rod together with its rack is regarded as the driving cable/rod for that continuum robot section. This way, each subsegment shape is fully defined by the three chains in the “Driving FPM,” and the three chains in the “Measuring FPM” only measure the corresponding arc length and transmit it to the next subsegment.

### 4.1 Subsegment Kinematics.

Figure 6 illustrates the subsegment kinematic model based on the circular arc bending assumption, where the driving arcs (in the section view, each of these arcs corresponds to the dot on the counterclockwise direction of each dot pair) are the abstraction of the driving chains with length di,j and the measuring arcs (corresponding to the dots on the clockwise direction of the dot pairs in the section view) are the abstraction of the measuring chains with length mi,j. i ∈ {1, 2, 3, 4} represents the i-th link and j ∈ {1, 2, 3} represents the jth chain in one subsegment. Body fixed frame $∑Ci=(Ci,xi,yi,zi)$ is placed at the center of the i-th link. li, κi, ri, and θi denote the arc length, curvature, radius, and central angle for the central bending arc (in dash-dot line), respectively. φi is the angle of the bending plane from $xi$ axis, and R is the distance of the driving/measuring arcs from the central arc. Based on mathematical definitions, the following relationships are self-satisfied:
$li=θiri$
(2)
$ri=1/κi$
(3)
Fig. 6
Fig. 6
Close modal
Therefore, with three arc lengths di,j, the bending shape is fully determined. The forward kinematics is obtained in the same way as in Ref. [25]:
$li=di,1+di,2+di,33$
(4)
$κi=2di,12+di,22+di,32−di,1di,2−di,1di,3−di,2di,3R(di,1+di,2+di,3)$
(5)
$φi=−atan2(di,3+di,2−2di,1,3(di,3−di,2))+e2R$
(6)
where the second term in Eq. (6) is the angle shift due to the mounting point offset of the driving arc on the sectional view plane (the dots are not exactly located on the $zi$ axis).
Knowing the bending shape, the three measuring arc lengths could be obtained by inverse kinematics as follows:
$mi,j=li−liκiRcos(φi+e2R+7π6−2π3j)$
(7)
For the (i+1)-th subsegment, the driving arc length should be replaced by the measuring arc length from the i-th subsegment. That is,
$di+1,j=mi,j$
(8)

Note that the aforementioned and the following equations do not include the κi = 0 cases, which could be easily handled in actual programming by manually assigning values to all the variables.

### 4.2 Overall Kinematics.

The overall kinematic model could be easily obtained as long as the subsegment wise kinematics is known. That is, with li, κi, and φi known, the vector from Ci to Ci+1 is obtained as follows:
$pi,i+1=risinθiyi+(ri−cosθiri)(cosφixi−sinφizi)$
(9)
The rotation from $∑Ci$ to $∑Ci+1$ is formulated as follows:
$iRi+1=eθiξ^$
(10)
where $ξ=−sinφixi−cosφizi$ is the rotation axis vector and the hat above ξ indicates the skew-symmetric expansion. Equation (10) could be easily evaluated by Rodrigues’ formula [26] as follows:
$iRi+1=I+sinθiξ^+ξ^2(1−cosθi)$
(11)
With local displacement $pi,i+1$ and $iRi+1$ known, the global displacement of $∑Ci$ can be obtained recursively
$pi=pi−1+pi−1,i$
(12)
$Ri=i−1RiRi−1$
(13)
with the initial displacement of $p1=0$ and $R1=I$.

### 4.3 Workspace Analysis.

The workspace of the new ECM is defined by all the points that the manipulator tip can reach in 3D space. Based on the prototype measurements (R = 25 mm, d is from 42 mm to 62 mm, and e = 2.3 mm) and aforementioned kinematic analysis, the workspace is generated and presented in Fig. 7. As shown in this figure, the ECM workspace is a volume with three ridges appearing on both the concave and the convex surfaces. These ridges correspond to the cases that one or two actuators are in their extreme positions. As a contrast, the nonextensible counterpart (2DOF manipulator) of the ECM can only generate a surface for the workspace. More details of this case could be found in Ref. [18].

Fig. 7
Fig. 7
Close modal

The workspace shows that the fully shortened manipulator has a length of 176 mm, and the fully extended case has a length of 256 mm. The maximal extension ratio (for what percentage the ECM can extend the most) is (256 − 176)/176 = 45.45%. Note that since the workspace is generated purely based on the simplified kinematic model, the workspace analysis accounting for static metrics (e.g., pose accuracy, load capacity) is not applicable here.

### 4.4 Motion Nonuniformity Evaluation.

As discussed in Sec. 3.1, due to the rod mounting angle shift e/R ≠ 0, the “Measuring PM” cannot exactly copy the “Driving PM” motion. This fact leads to a twist motion along the manipulator axial direction, which breaks the desired uniform motion for each subsegment. To evaluate the nonuniformity induced by this phenomenon, different angle shift e/R values are tested and the corresponding manipulator configurations with the same inputs (d1,1 = 42 mm, d1,2 = 52 mm, and d1,3 = 52 mm) are plotted in Fig. 8, in which five additional subsegments (in black) are added to make the twist motion more visible. The other colors indicate the four subsegments in the actual design.

Fig. 8
Fig. 8
Close modal

As shown in Fig. 8, the twist effect becomes quite significant as e/R is beyond 10 deg and more subsegments worsen the situation significantly. Therefore, for practical design purposes, reducing e to a value as small as possible and choosing fewer subsegments help reduce the undesired twist motion. For the existing design with a minimized e value (2.3 mm), the twist effect is also evaluated for different manipulator configurations. The nonuniformity is defined by the difference between the last subsegment bending plane angle φ4 and the first subsegment bending plane angle φ1. The numerical calculation was conducted and plotted in Fig. 9, which surprisingly shows that the nonuniformity (the value in the figure is $15.81deg$) is not affected by the manipulator configuration.

Fig. 9
Fig. 9
Close modal
This can be verified analytically by substituting Eq. (8) into Eq. (6), which yields
$−tan(φi+1−e2R)=mi,3+mi,2−2mi,13(mi,3−mi,2)$
(14)
Substituting Eq. (7) into Eq. (14) and evaluating, Eq. (14) is simplified to
$φi+1−φi=eR$
(15)
which means that the twist effect only depends on the rod mounting angle shift e/R and the subsegment number.

## 5 Prototyping and Experiments

To verify the proposed mobility of the new mechanism, a proof-of-concept prototype was built using acrylonitrile butadiene styrene-based 3D printing. Three Actuonix linear actuators (L12-30-210-6-P) with corresponding controller boards were used to drive the manipulator. For the rack and pinion transmission, off-the-shelf 0.5 modulus nylon gears were utilized, and customized racks were 3D printed. The flexible rods were made from Trik Fish lines with 1.35 mm diameter.

As shown in Fig. 10, the prototype exhibits the proposed 2R1 T mobility, for which the most shortened length is measured as 177 mm and the most extended length as 234 mm. The extension ratio is computed to be 32.2%, which is smaller than the ratio predicted by the workspace analysis. This is partially due to the smaller range applied on the linear actuator to avoid potential damage to the prototype. The maximal bending angle was measured to be around 80 deg. The prototype was also tested for its load capacity using a scale with 1 g precision. For the fully shortened configuration, the manipulator tip can generate a radial force of up to 59 g (around 0.58N) and an axial force of up to 671 g (around 6.6N) before significant buckling failure occurs.

Fig. 10
Fig. 10
Close modal

To further evaluate the accuracy of the kinematic model, two sets of experiments were conducted. The first set measured three arbitrary shapes of the ECM on the $y1C1z1$ plane using one stationary camera (Samsung HMX-F90 with 1696 × 954 pixels). The camera was placed 1.5 m away from a vertical pegboard (25 mm hole pattern), on which the manipulator base was fixed. The camera pose was adjusted carefully, so that the camera frame is parallel to the pegboard plane. The second set measured another three static shapes on the $x1C1y1$ plane using the same camera setup. Due to the relatively large distance between the camera and the measuring plane, the distortion effect of the camera was neglected. Therefore, the world coordinates of the manipulator points could be measured by manually finding out the pixel coordinates on the image [27]. After calibrating with a checkerboard, this method could reach an accuracy of ±3 pixels, which corresponds to ±2 mm in the world coordinates on the measuring plane. Figure 11 shows the measurement results, where the three shapes of each experiment set were combined in one image. The estimated shapes based on the kinematic model are also plotted in the same figure. Note that the superimposition of images from the experiments and the images from the theoretical computations are rough and only meant to provide intuitive impressions on how large the error is. For accurate image composition, a rigorous process of mapping the theoretical data in the world coordinate into the pixel coordinates in the camera frame would be needed. The detailed measurement results are reported in Table 1, where the driving arc lengths are measured indirectly by recording the corresponding rack positions using a caliper. Since the single camera-based measurement cannot provide depth information, the unmeasurable components in the second column are set to zero and other components are rounded to the nearest integer. The error norm was also computed only for the measurable components. Based on the experimental data, the current kinematic model shows good accuracy in estimating the relative positions of the manipulator (bending shapes) but requires improvements in the absolute position estimation. The mismatch between the theoretical predictions and the experimental results is partly due to the fact that the flexible rods bend in a more complicated way when the manipulator approaches its extreme poses. This complicated bending behavior violates the circular arc assumption and thus causes the error. Another important reason contributing to the mismatch is the large friction that exists in the prototype (due to the usage of plastic gears and the lack of bearings). The friction forces the rods to bend more than the ideal case. Therefore, as more efforts will be directed to improve the prototype in the future, the friction could be reduced, and the kinematic model would be expected to have better prediction results.

Fig. 11
Fig. 11
Close modal
Table 1

Experimental data

ShapeDriving arc length (d1,1, d1,2, d1,3) (mm)Measured C5 position (mm)Computed C5 position (mm)Error norm (mm)
1(57.3, 56.9, 57.2)(0, 229, −20)T(2.7, 228.6, −3.4)T16.6
2(51.7, 56.9, 57.2)(0, 220, 67)T(10.8, 209.1, 60.8)T12.5
3(47.8, 56.9, 57.2)(0, 178, 118)T(15.4, 182.8, 96.7)T21.8
4(54.8, 54.0, 55.9)(10, 220, 0)T(19.3, 218.5, −0.8)T9.4
5(52.9, 54.1, 50.8)(−53, 202, 0)T(−32.5, 207, −1.6)T21.1
6(53.8, 56.8, 48.1)(−102, 178, 0)T(−81.1, 189.2, −3.8)T23.7
ShapeDriving arc length (d1,1, d1,2, d1,3) (mm)Measured C5 position (mm)Computed C5 position (mm)Error norm (mm)
1(57.3, 56.9, 57.2)(0, 229, −20)T(2.7, 228.6, −3.4)T16.6
2(51.7, 56.9, 57.2)(0, 220, 67)T(10.8, 209.1, 60.8)T12.5
3(47.8, 56.9, 57.2)(0, 178, 118)T(15.4, 182.8, 96.7)T21.8
4(54.8, 54.0, 55.9)(10, 220, 0)T(19.3, 218.5, −0.8)T9.4
5(52.9, 54.1, 50.8)(−53, 202, 0)T(−32.5, 207, −1.6)T21.1
6(53.8, 56.8, 48.1)(−102, 178, 0)T(−81.1, 189.2, −3.8)T23.7

Although the prototype demonstrates consistent bending shapes in general, the first subsegment was observed to have larger bending angles than the rest. The reason was partly due to the nonuniform motion effect that was discussed in Sec. 4.4. But more importantly, the nonuniformity for the first subsegment comes from its large driving force. As shown earlier, the new manipulator mechanism utilizes motion propagation as a way to transmit motion from the base to the tip link. Based on conservation of energy, this method will accumulate and amplify the driving force from each subsegment onto the first subsegment, which makes its flexible rods to deform more than those of the rest. If this force becomes too large, the flexible rods may not be able to stably transmit the axial force and may result in a sudden loss of stability, also known as the buckling phenomenon. Therefore, to analyze these behaviors and to better estimate the manipulator shape, a more accurate statics-based kinematic model (such as the modeling method used in Ref. [24]) is required for future work.

## 6 Conclusion and Future Work

By leveraging the rigid coupling hybrid mechanism concept and the flexible parallel mechanisms, a new 3DOF extensible continuum manipulator with spatial bending (2R) and one axial extension (1 T) mobility was proposed. The core idea lies in using the motion of the i-th link to drive the (i+1)-th link, so that the local motion can be copied and propagated from the base link to the tip link. To achieve this design goal, flexible parallel mechanisms were used to realize the basic 2R1 T subsegment motion, and rack and pinion sets were used to couple the adjacent subsegments. This way, the 2R1 T motion is copied by each subsegment and the entire manipulator achieves spatial bending and one extension mobility. To calculate the configuration of this new mechanism, a simplified kinematic model was formulated. The workspace analysis based on the kinematic model showed that the new manipulator is able to generate a volumetric workspace in comparison to the superficial workspace that its nonextensible counterparts generate. A small-scale proof-of-concept prototype was manufactured to validate the proposed mobility. Preliminary tests showed that the current design is able to extend 32% of its original length and bend over 80 deg. The manipulator can also generate a radial force of up to 0.58N and an axial force of up to 6.6N at its tip. Additional experiments on position accuracy showed that the simplified kinematic model can correctly predict the bending shape of the manipulator but has a noticeable error on the absolute position estimation.

For future work, since the neglection of the static effects caused error in the kinematic analysis, one important focus is to develop a more accurate kinematic model based on the Coserrat rod theory, which will take into account the gravity, friction, and external loads. Moreover, improving the mechanical design to reduce the prototype friction (e.g., using metal gears and racks with smaller modulus) will also be an important part of the future work. Considering the potential applications of the new manipulator on medical robotics, the team will also work on the miniaturization of the mechanical design.

## Funding Data

• National Science Foundation (Grant No. 1906727; Funder ID: 10.13039/100003187).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

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