the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Design and kinematics of a lightweight cruciform continuum robot

### Pan Zhou

### Jiantao Yao

### Hongyu Zhang

### Xuanhao Zhang

### Shuaiqi kong

### Kunming Zhu

The design of new lightweight and dexterous configurations is a major research focus for continuum robotics. This work proposes a cruciform continuum robot. Its unique feature is that it is formed by multiple cruciform-arranged elastic sheets with a single dimension of motion connected in series, and thus it has low-coupling motion characteristics. In addition, the cruciform continuum robot has the advantages of lighter weight (65 g), better dexterity, and higher motion accuracy. In this paper, the forward and inverse kinematics models of the cruciform continuum robot are established by geometric methods based on the assumption of constant curvature, and its workspace is analysed. It is experimentally verified that the tip position errors are less than 1 mm, and the cable length errors are less than 0.4 mm. Further, the cruciform continuum robot is successfully used for the nucleic acid detection simulation experiment, which confirms its good dexterity and man–machine safety. The main contribution of this paper is to provide a new configuration for the lightweight and dexterous continuum robots, and to further provide a reference method for improving their modelling accuracy from the perspective of structure.

Different sorts of soft manipulators have been developed to deal with diverse operational tasks and work environments (Rus and Tolley, 2015; Krovi and Nie, 2008; Zhao et al., 2020; Dong et al., 2016; Yarbasi and Samur, 2018; Yu et al., 2018). Continuum robots are known for their flexibility and dexterity, which have significant performance advantages in dealing with unstructured environments and non-cooperative target operations, making them always active at the forefront of robotics research. They can be applied in various fields such as disaster relief (Tsukagoshi et al., 2001; Polygerinos et al., 2017), minimally invasive surgery (Kato et al., 2015; Burgner-Kahrs et al., 2015; Zhu et al., 2021; Wang et al., 2019; Yuan et al., 2017; Bajo and Simaan, 2016), space inspection (Tonapi et al., 2014; Guo et al., 2022), nuclear fusion vessel maintenance (Buckingham and Graham, 2012), and manipulation (Yang et al., 2018; Laschi et al., 2016; Godage et al., 2019; McMahan et al., 2005). The existing continuum robots have two classic structure forms, which are the single-backbone configuration (Kato et al., 2015; Burgner-Kahrs et al., 2015; Yuan et al., 2017; McMahan et al., 2005; Roesthuis and Misra, 2016) and the multi-backbone configuration (Wang et al., 2019; Bajo and Simaan, 2016; Godage et al., 2019). The former is usually driven by cables and the latter often uses rods and variable-length soft actuators as both the backbone and actuation of continuum robots. Among them, the cable-driven single-backbone continuum robots are very popular in medical examinations and minimally invasive surgery due to their compact structure, high motion and positioning accuracy, and relatively large load capacity (Kato et al., 2015; Burgner-Kahrs et al., 2015).

Continuum robots can be regarded as long cantilever beam structures. Thus, their deformation shape is easily affected by their gravity and the load on their tip. This phenomenon becomes more and more significant as length and load increases, which greatly reduces their motion accuracy. One way to tackle this negative deformation is to incorporate variable stiffness techniques into the development of continuum robots, such as the jamming method (Jiang et al., 2014; Bamotra et al., 2019; Miller-Jackson et al., 2019), actuation coupling (Al-Fahaam et al., 2018), solid–liquid conversion (Yoshida et al., 2018), and variable stiffness material (Zhang et al., 2019). However, variable stiffness structures have difficulty quickly responding to the motion of continuum robots and will increase their weight, size, and complexity of control. The lightweight and stable design of continuum structures can effectively alleviate this negative deformation from the root, which has always been the focus of research in the field of robotics.

In order to develop ultra-lightweight continuum robots, researchers have tried to fabricate continuum structures through origami technology (Childs and Rucker, 2021; Santoso and Onal, 2021) and airtight films (Hofer et al., 2021; Hawkes et al., 2017), prominent among which is an ultra-long growth robot based on an airtight film structure (Hawkes et al., 2017). However, the origami continuum structures are generally slightly insufficient in terms of compactness, and the thin film continuum structures are usually not easy to be accurately driven. The development of continuum robots adopting non-contact actuation (such as magnetic – Boyvat et al., 2017; Kim et al., 2019; and light responsive actuation – Qian et al., 2018) is another potential approach to lightweight design because they no longer require actuating components. For example, a long thin continuum robot based on magnetic actuation can move in human organs (Kim et al., 2019), and the hybrid drive of magnetic fields and tendons enables the development of a millimetre-scale soft continuum robot capable of high-precision manipulation (Zhang et al., 2021). However, the workspace of such continuum robots is strictly limited by the medium fields (such as magnetic and light fields), which makes it difficult to develop a robot system that has a large-scale operation range. Tendon-driven continuum robots can have a larger workspace and a lightweight and compact robotic arm. The classic single-backbone continuum robots driven by tendons usually take a flexible member with a circular section as the bending section (Kato et al., 2015; Burgner-Kahrs et al., 2015). Unlike these designs, this paper proposes a novel lightweight continuum robot (named the cruciform continuum robot) using a flexible member with a rectangular section as the bending section. Multiple flexible members are combined to make the cruciform arrangement, which enables the multi-degree-of-freedom motion and the bending coupling reduction in different directions of the continuum robot. These design features give the continuum robot the basic characteristics of large load-bearing beams in the direction without deformation and better bending characteristics in the deformation direction.

Compared with traditional rigid robots, the modelling of continuum robots is relatively complex and difficult, and there is no mature theoretical system yet. Although scholars have put forward the constant curvature hypothesis (Webster and Jones, 2010), Cosserat rod theory (Till and Rucker, 2017; Alqumsan et al., 2019), finite element (Bieze et al., 2018), intelligent algorithm (Braganza et al., 2007), and other methods to build the theoretical model of continuum robots, their high-precision modelling remains a difficulty, which is closely related to their structural configuration. Most existing multi-backbone continuum robots are composed of multiple flexible parallel mechanisms in series, which can be referred to as series-parallel mechanisms (Wang et al., 2019; Bajo and Simaan, 2016; Godage et al., 2019), and existing single-backbone continuum robots are mostly pure-series mechanisms (Kato et al., 2015; Burgner-Kahrs et al., 2015; Yuan et al., 2017; McMahan et al., 2005; Roesthuis et al., 2016). Compared with series-parallel mechanisms, pure-series mechanisms are simpler to model in terms of kinematics and mechanics. The cruciform continuum robot adopts a pure-series cruciform configuration. This configuration can greatly reduce the motion coupling between different bending sections connected in series, and it enables the continuum robot to establish a simpler and more accurate theoretical model. Therefore, the cruciform continuum robot provides a solution to improve the modelling accuracy of continuum robots from the perspective of structure and further lays a foundation for their precise control.

First, the lightweight cruciform continuum robot is discussed, and its special structural advantages are analysed. Then, the kinematic model is derived using geometric methods, and its accuracy is verified experimentally. Finally, the nucleic acid detection experiment fully demonstrates the dexterity, compliance, and good adaptability of the robot.

As shown in Fig. 1, the manipulating system of the continuum robot is mainly composed of two series units in a cruciform arrangement, and each series unit consists of two bending sections in a cruciform arrangement. The bending section is composed of an elastic piece connected with two connectors through an interference fit. The elastic piece as the middle backbone is made of 65 MN spring steel which has good bending properties, and its bending is more regular and controllable. The connector was fabricated by 3D printing technology to reduce the weight of the continuum robot. Each bending section has 1 degree of freedom, which is driven by two evenly distributed cables. As shown in Fig. 1a, the driving cables of the tip bending section thread through the holes in the lowermost disc, each connector in the middle, and its upper disc in turn, with the cables being tied at the upper disc. We route the driving cables of the other bending sections in the same way, with each cable being driven independently by a stepper motor. The robot can perform dexterous manipulation when cables are driven by motors, as shown in Fig. 1b.

Unlike most existing single-backbone continuum robots that use rod-like structures as the backbone, the cruciform continuum robot uses sheet-like structures in a cruciform arrangement as the backbone for the following reasons. The cruciform arrangement of the sheet-like structures enables each bending section to have a larger bearing capacity in the non-deformation direction and better bending characteristics in the deformation direction. In addition, the width direction of the first elastic piece is arranged to be consistent with its gravity direction. Due to the much larger moment of inertia along the axis perpendicular to the sheet, this arrangement can reduce the initial deformation caused by its gravity and external load. Thirdly, the pure-series connection and cruciform arrangement of the continuum robot can greatly reduce the coupling between bending motions in different directions. For example, the series unit can bend up and down or left and right independently, which avoids the movement coupling of two bending directions.

As shown in Fig. 2, we chose a thickness of 0.2 mm for the tip elastic piece and a thickness of 0.3 mm for the other three elastic pieces in the prototype. The thickness of the tip elastic piece is slightly smaller than that of the other three elastic pieces, to reduce the influence of its driving cables on the other elastic pieces when bending the tip elastic piece. The robot is 255 mm long and has a radius of 21 mm. In terms of length, its average mass per millimetre is 0.255 g, and from its cylindrical envelope volume, its average mass per cubic millimetre is 0.0002 g. Its weight is light, which is 65 g. The continuum robot can move automatically or by manual remote control. The working range and motion capability of the continuum robot are effectively improved when it is installed on an omnidirectional mobile platform. The proposed continuum robot has 4 degrees of freedom, but by adding the series units, it can get longer and have more degrees of freedom, enabling it to complete more complex operations.

## 3.1 Forward kinematics

In order to obtain the forward kinematics solution of the series unit, we
first have to get the forward kinematics solution of the bending section.
Due to the good bending properties of the elastic piece, its forward
kinematics solution can be found using geometric methods based on the
assumption of constant curvature. As shown in Fig. 3, the tip coordinate of
the first bending section in the task space can be expressed by the lengths
of its driving cables. Since the arc length *S* of the bending curve of the
bending section is constant, the position of the bending section can be
expressed by the curvature *k*_{1} of the bending curve. The bending
direction of the first bending section about the *y*_{0} axis shown in Fig. 3
is defined as a positive bending direction. The maximum bending angle
*α* of the elastic piece is limited to less than 85^{∘}, and it
can exhibit good bending characteristics.

We first can obtain

where *l*_{0} is the chord length of the bending curve of the first bending
section, *l*_{1} and *l*_{2} are the driving cable lengths of the first
bending section, and *d*_{1} is the thickness of the tip connector of the
first bending section.

As shown in Fig. 4, we get

where *r*_{1} is the radius of the tip connector of the first bending section, and *R*_{1} is the radius of the first bending section.

Substituting Eqs. (1) and (2) into Eq. (4) yields *R*_{1} in terms
of cable lengths as

As shown in Fig. 5, using the same method, we obtain

where *R*_{2} is the radius of the second bending section, *l*_{32} and
*l*_{42} are a section of the driving cables of the second bending section
which is between the top plane of the lower connector and the top plane of
the upper connector of the second bending section, and *d*_{2} and *r*_{2} are the thickness and radius of the tip connector of the second bending section, respectively.

The bending of the second bending section about the *y*_{3} axis shown in
Fig. 5 is defined as a positive bending, and we get

where *l*_{3} and *l*_{4} are the total lengths of the two driving cables of the second bending section, and *l*_{31} and *l*_{41} are a section of the driving cables of the second bending section between the top plane of the lower connector and the top plane of the upper connector of the first bending section.

When *α*_{1}≠0 and *α*_{2}≠0, the expression of the tip
coordinate of the series unit obtained from the geometry in Fig. 5 is

where *x*, *y*, and *z* represent the tip coordinate of the series unit in the task space, and *α*_{1} and *α*_{2} represent the bending angles of the first and second bending sections, respectively.

When ${\mathit{\alpha}}_{\mathrm{1}}={\mathit{\alpha}}_{\mathrm{2}}=\mathrm{0}$,

The bending angle *α* of the bending curve can be expressed as

where *S* represents the arc length of the first and second bending sections.

Equations (1), (5), (6), (7), (8), (9), (14), and (15) are substituted into Eqs. (10), (11), and (12), and the tip coordinate of the series unit
can be expressed by cable lengths when *α*_{1}≠0 and *α*_{2}≠0 as follows:

where $A=\frac{\left({l}_{\mathrm{1}}+{l}_{\mathrm{2}}-\mathrm{2}{d}_{\mathrm{1}}\right){r}_{\mathrm{1}}}{{l}_{\mathrm{2}}-{l}_{\mathrm{1}}}$, $B=\frac{S\left({l}_{\mathrm{2}}-{l}_{\mathrm{1}}\right)}{({l}_{\mathrm{2}}+{l}_{\mathrm{1}}-\mathrm{2}{d}_{\mathrm{1}}){r}_{\mathrm{1}}}$, and $C=\frac{({l}_{\mathrm{3}}+{l}_{\mathrm{4}}-{l}_{\mathrm{1}}-{l}_{\mathrm{2}}-\mathrm{2}{d}_{\mathrm{1}}-\mathrm{2}{d}_{\mathrm{2}}){r}_{\mathrm{2}}}{{l}_{\mathrm{4}}-{l}_{\mathrm{3}}}$.

To replace the cable lengths with the cable length variations, we first get

where Δ*l*_{1}, Δ*l*_{2}, Δ*l*_{3}, and Δ*l*_{4} are the variations of cables *l*_{1}, *l*_{2}, *l*_{3}, and *l*_{4}.

Then, when Δ*l*_{1} and Δ*l*_{3} are both less than zero, we
get the following according to the geometric relationship in Fig. 3:

The shape of each bending section is determined by one cable. Δ*l*_{2} can be computed by Δ*l*_{1}. According to Eqs. (23) and
(24), we use MATLAB software to draw up a relationship graph between Δ*l*_{1} and Δ*l*_{2} when *α*_{1} is in the range of
0 to 85^{∘}, which is fitted as

Further, we can get the following according to the geometric relationship in Fig. 3:

where Δ*l*_{0} is the chord length of the first bending section
minus the length *S* of the first elastic piece.

According to Eqs. (23) and (26), we draw up a relationship graph between Δ*l*_{0} and Δ*l*_{1} when *α*_{1} is in the
range of 0 to 85^{∘}, which is fitted as

where Δ*l*_{31} and Δ*l*_{41} are the variations of cables 3
and 4 when driving cable 1.

Further, we can get

Similarly, according to Eqs. (29) and (30), the relationship between
Δ*l*_{32} and Δ*l*_{42} when *α*_{2} is in the range
of 0 to 85^{∘} is fitted as

One can then substitute Eqs. (25), (27), (28), (31), (32), and (33) into Eqs. (19), (20), (21), and (22) to obtain the cable lengths *l*_{1}, *l*_{2}, *l*_{3}, and *l*_{4} expressed by the cable variations Δ*l*_{1} and Δ*l*_{32}, which can be put into Eqs. (16), (17), and (18) to get the tip coordinate of the series unit.

## 3.2 Inverse kinematics

Similarly, in order to obtain the inverse solution of the series unit, we first need to get the inverse solution of the first bending section by using geometric methods. We can get this from the geometry in Fig. 3:

According to Eqs. (7), (8), and (9), we can get

And then we can get

Substituting Eqs. (34), (39), and (40) into Eqs. (37) and (38) yields the cable lengths *l*_{3} and *l*_{4} as

Substituting ${R}_{\mathrm{1}}=S/{\mathit{\alpha}}_{\mathrm{1}}$ and ${R}_{\mathrm{2}}=S/{\mathit{\alpha}}_{\mathrm{2}}$ into Eqs. (10), (11), and (12) yields

According to Eq. (44), we use MATLAB to draw up a relationship graph
between *y* and *α*_{2} when *α*_{2} is in the range of
0 to 85^{∘}, which is fitted as

The relationship between *x*, *y*, and *α*_{1} is also drawn up according to Eqs. (43) and (45) when *α*_{1} and *α*_{2} are both
in the range of 0 to 85^{∘}, and the fitting equation is obtained as follows:

Substituting ${R}_{\mathrm{1}}=S/{\mathit{\alpha}}_{\mathrm{1}}$, ${R}_{\mathrm{2}}=S/{\mathit{\alpha}}_{\mathrm{2}}$, Eqs.(46), and (47) into Eqs. (35), (36), (41), and (42), one can obtain the lengths of the four cables represented by the tip coordinates of the series unit.

## 3.3 Workspace of the series unit

The maximum bending angle *α* of each bending section is limited to
85^{∘} where it can exhibit good bending characteristics. The physical parameters of the robot are *S*=50 mm, *d*_{1}=10 mm, *d*_{2}=14 mm, and ${r}_{\mathrm{1}}={r}_{\mathrm{2}}=\mathrm{15}$ mm. Thus, through the Eqs. (35), (36), (41), and (42), the length ranges of cables *l*_{1} and *l*_{2} can be
obtained, which are both 35.61–75.47 mm, and the length ranges of cables *l*_{3} and *l*_{4} are both 95.15–135.01 mm. The workspace of the series unit is shown in Fig. 6, which is found using the forward kinematics equations. The *x*, *y*, and *z* are respectively in the range
of −104.48 to 104.48 mm, −44.71 to 44.71 mm, and 40.03 to 124 mm.

## 4.1 Forward kinematics verification

The tip position coordinates of the series unit obtained from the experiment
are compared with the theoretical model results to verify the accuracy of
the theoretical model. The verification test device of the series unit is
shown in Fig. 7. The cable length changes Δ*l*_{1} and Δ*l*_{3} can be measured by two hand-operated mobile stands equipped with
digital rulers ① and ② having a resolution of 0.01 mm, respectively. The
tip coordinates of the series unit can be measured by the other two
hand-operated mobile stands equipped with laser locators and digital rulers,
and digital rulers ③, ④, and ⑤ are used to record the values of *x*, *y*, and
*z*, respectively. First, cables 1 and 3 were tightened. Then, the cable length
*l*_{1} was reduced by 1.22 mm in turn, and meanwhile the cable length
*l*_{3} was reduced by a corresponding length which was calculated by
combining Eqs. (27) and (28). Further, the cable length *l*_{3} was
reduced by 1.22 mm in turn. This process performs 20 times, and the bending
angle of each bending section reaches the maximum value (85^{∘}).

As shown in Fig. 8, 20 sets of tip position coordinates obtained in this
process were compared with the theoretical values. The theoretical values of
*x* and*y* are generally smaller than the experimental values, and the theoretical values of *z* are generally larger than the experimental values. This is mainly due to the small deformation of the two elastic sheets in advance caused by the tensioning of cables. The formula $\sqrt{({P}_{\mathrm{3}}-{P}_{s\mathrm{3}}{)}^{T}({P}_{\mathrm{3}}-{P}_{s\mathrm{3}})}$ is used to express the normalised errors which are shown in Fig. 8d, where *P*_{3} and *P*_{s3} represent the experimental and
theoretical tip position coordinates, respectively. With the increase of the
cable length variation, the normalised errors also increase, but the maximum
error does not exceed 1 mm. It shows that when the bending angles of the
bending sections are within 85^{∘}, the tip coordinates of the
theoretical model are in good agreement with the experimental data. The
excellent bending characteristics of the elastic sheet deteriorate as its
bending angle becomes larger, and thus the position coordinate errors of the
theoretical model with constant curvature increase. However, the model
errors are within the desired range, which verifies that the theoretical
model has good accuracy.

## 4.2 Inverse kinematics verification

The theoretical values of Δ*l*_{1} and Δ*l*_{3} are obtained
by substituting the above experimental tip position coordinates into inverse
kinematics equations. As shown in Fig. 9a and b, the Δ*l*_{1} and
Δ*l*_{3} results of theory and experiment have a good consistency.
The actual length reductions of cables 1 and 3 are generally slightly
smaller than the theoretical length reductions when obtaining the same tip
coordinates. This trend is more pronounced as the bending angle increases.
There are three main reasons for this phenomenon. Firstly, the tip position
of the series unit is changed in advance because of the beforehand
deformation of the elastic sheets due to the tensioning of the cables.
Secondly, the errors of the inverse kinematics model with constant curvature
will also increase when the excellent bending characteristics of the elastic
sheets deteriorate as their bending angle becomes larger. Thirdly, the
theoretical bending angles obtained from the fitting equations are slightly
larger, and the deviation increases as the bending angles climb. As a
result, the theoretical variations of cable length are also higher. As shown
in Fig. 9c and d, the errors of Δ*l*_{1} and Δ*l*_{3} are less than 0.38 and 0.32 mm, respectively. The errors of Δ*l*_{1} are generally larger than those of Δ*l*_{3} because the fitting errors of *α*_{1} are greater than those of *α*_{2}. In general, the errors of Δ*l*_{1} and Δ*l*_{3} are both within the desired ranges, which verifies that the inverse kinematics model has good accuracy.

The bending coupling of the cruciform continuum robot in different directions is low. The up–down and left–right movements of the tip of the robot can be realised by driving its fourth and third bending sections, respectively, as shown in Fig. 10a. In addition, the cruciform continuum robot has good dexterity, allowing it to easily avoid obstacles such as the ball in Fig. 10b. The cruciform continuum robot can be installed in a mobile vehicle, and its working range and operation ability are improved. Besides this, the robot can be applied for manipulation and exploration. In an example demonstration, we install a suction cup and camera at the tip of the robot, as shown in Fig. 10c. The robot can complete the nucleic acid detection procedure, including grasping the cotton swab, taking a sample, putting the pharyngeal swab into a test tube, and breaking it off. The biggest advantage of using the robot for nucleic acid detection is the natural compliance of the continuum manipulator, which allows the robot to adaptively deform when colliding with the human body or the environment to minimise damage to the patient or the robot itself. The experiment fully demonstrated its advantages of good safety, flexibility, and compliance, and the robot is expected to be applied in practice.

One of the research focuses in the field of continuum robotics is to realise
the lightweight and dexterity design of continuum robots. Considering the
basic structural forms of continuum robots and their equivalent load beam
types, this paper proposes a cruciform continuum robot, which is formed by
multiple elastic sheets with rectangular beam structures arranged in a
cruciform shape. The cruciform layout structure of the rectangular beams
enables the bending coupling reduction of the continuum robot in different
directions and good bending characteristics in the deformation direction.
Meanwhile, this unique structure ensures sufficient bearing capacity in the
non-deformation direction and reduces the materials required for robot
manufacture. The robot has a mass of 65 g. In terms of length, its average
mass per millimetre is 0.255 g, and from its cylindrical envelope volume,
its average mass per cubic millimetre is 0.0002 g. This paper presents a new
structural design for lightweight dexterous continuum robots. The cruciform
continuum robot has good constant curvature characteristics when each
bending section has a bending angle smaller than 85^{∘}, which
provides a good hardware foundation for its high-precision kinematic
modelling. Further, its forward and inverse kinematics models with constant
curvature are established using geometric methods, and their accuracy is
experimentally verified. The maximum value of the prediction error of the
tip position coordinates is less than 1 mm, and the maximum value of the
prediction error of the cable length is less than 0.4 mm. In addition, the
nucleic acid detection simulation demonstration proves the dexterity and
friendly man–machine interaction of the cruciform continuum robot, which is
expected to be applied in man–machine interactive fields.

In the future, we will add a protective cover for the continuum robot, and thus the exposed driving cables and elastic backbone will not collide with other objects in the environment. We will further optimise the robot prototype to make it have better operational performance in practical applications. For example, we will make the elastic sheet gradually become thicker from the tip to meet the stress distribution law of the cantilever beam structure. In addition, our future work will mainly focus on the following two aspects. On the one hand, we will continue to create new configurations and achieve the lightweight design of continuum robots. On the other hand, we will develop an ultra-long continuum robot which has more degrees of freedom and a more compact structure.

All the data used in this paper can be obtained upon request from the corresponding author.

ZP and JY conceived the presented idea. ZP established an overall paper research framework and the model. HZ and XZ assisted with the theory. SK and KZ assisted with the experiment. All the authors read and approved the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study has been funded by the National Natural Science Foundation of China (grant nos. U2037202 and 51975505) and the International Cooperative Research and Development Project of Intelligent Fire Fighting Robot (grant no. 19391825D). The authors sincerely thank the editors and reviewers for their insight and comments, which further improved the quality of this paper.

This research has been supported by the National Natural Science Foundation of China (grant nos. U2037202 and 51975505) and the International Cooperative Research and Development Project of Intelligent Fire Fighting Robot (grant no. 19391825D).

This paper was edited by Daniel Condurache and reviewed by two anonymous referees.

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