This may be the author’s version of a work that was submitted/acceptedfor publication in the following source:

Wu, Leo, Crawford, Ross, & Roberts, Jonathan(2016)Dexterity analysis of three 6-DOF continuum robots combining concentrictube mechanisms and cable driven mechanisms.IEEE Robotics and Automation Letters, 2(2), pp. 514-521.

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https://doi.org/10.1109/LRA.2016.2645519

IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016 1

Dexterity Analysis of Three 6-DOF ContinuumRobots Combining Concentric Tube Mechanisms

and Cable Driven MechanismsLiao Wu1, Ross Crawford1, and Jonathan Roberts1

Abstract—Continuum robots are increasingly used in minimal-ly invasive surgeries. To date, concentric tube mechanism andcable driven mechanism have been two prevalent mechanismsfor constructing continuum robots. As these two mechanismscomplement each other, it is worth exploring the possibilityof combining them together. In this paper, we investigate thedexterity of three continuum robots combining both mechanisms.Indices based on the concept of orientability are introduced toanalyze the distribution of the dexterity. A Monte Carlo method isused to calculate the dexterity distribution across the workspace.Particularly, directional dexterity indices are put forward todescribe the dexterity along different axes. Results imply thatevenly allocating degrees of freedom (DOFs) among the segmentsachieves the best workspace and dexterity. Otherwise, assigningmore DOFs to the proximal segment tends to enlarge theworkspace and adding more DOFs to the distal segment tends toimprove the dexterity. In addition, the dexterity along differentaxes can vary significantly and thus requires special attentionwhen applying the robot to specific surgical procedures.

Index Terms—Mechanism Design of Manipulators, MedicalRobots and Systems, Concentric Tube Robots, Cable DrivenRobots.

I. INTRODUCTION

CONTINUUM robots are rapidly emerging to become aclass of useful tools in many areas including industrial

and medical applications [1], [2]. Different from conventionalserial link robots that are driven by discrete joints, continuumrobots have the ability to continuously deform into differentshapes. Usually actuated at the remote proximal base, contin-uum robots can be made very compact along the part from theactuators to the tip. As a consequence, they are increasinglyused in minimally invasive surgeries where their compactnessenables agile manipulation in confined environment inside thehuman body. During the last decade, much research has beencarried out in exploring the application of continuum robotsto many minimally invasive procedures. Examples includetransoral/transnasal procedures [3]–[6], urology surgeries [7],

Manuscript received: September 10, 2016; Revised November 8, 2016;Accepted December 15, 2016.

This paper was recommended for publication by Editor Paolo Rocco uponevaluation of the Associate Editor and Reviewers’ comments. This work wassupported by the Vice-Chancellor’s Research Fellowship from QueenslandUniversity of Technology under Grant 322450-0096/08 awarded to Dr. LiaoWu.

1The authors are with the Australian Centre for Robotic Vision, Scienceand Engineering Faculty, Queensland University of Technology, Brisbane,Australia. liao.wu@qut.edu.au/dr.liao.wu@ieee.org;r.crawford@qut.edu.au; jonathan.roberts@qut.edu.au

Digital Object Identifier (DOI): see top of this page.

intracardiac procedures [8], ophthalmology [9], and orthopedicsurgeries [10].

To date, two prevalent mechanisms have been adopted forcontinuum robots in minimally invasive surgeries: concentrictube mechanism [3]–[5], [7]–[13] and cable driven mechanism[6], [14]–[17]. Concentric tube robots use a set of telescopictubes to achieve spatial movability. As each tube can beprecurved within its elastic deformation zone, the interactionbetween the tubes, while they are being rotated or translatedwith respect to each other, results in variation of both thecurvature of the combined tubes and the position of the tip.The main advantage of a concentric tube mechanism lies inits capability of being scaled down to submillimeter, whichis favorable in many minimally invasive surgical applica-tions. However, since each tube is precurved, the variationof the resultant curvature is limited. Therefore, optimizationof parameters of the tubes based on the specific task isusually required before applying the robot to any surgicalprocedure [4]. The combined segments can be endowed withvariable curvature when the contributing tubes are comparablein stiffness [18], but the snapping problem occurring when therelative rotation angle exceeds a certain critical point preventsthe natural curvature of each tube from being large, limitingthe variation range of the combined segments.

A cable driven mechanism, as the name suggests, use cablesto actuate the deformation of the robot. It usually consists ofbackbones as the main deformation elements, cables as theactuators, and vertebrae, which distribute along the backbones,as the “joints” to transmit and restrict the actuation. The mainadvantage of a cable driven mechanism is its flexibility andvariable curvature which is determined by the elasticity ofthe backbone and the space between each vertebra. Typically,a bending segment comprises at least three cables so that itcan bend from a straight segment to a curve in any direction.Multiple bending segments usually share the same backbone,but have different groups of cables for individual actuation[16]. Nevertheless, control of multiple bending segments issomewhat challenging as there is a coupling between eachsegment [15]. It is also difficult to scale down a cable drivenmechanism to the level of a concentric tube mechanism; atthe same scale, the cable driven mechanism lacks the rigidityexisting in a concentric tube mechanism that facilitates forcetransmission.

As can be seen from the above, concentric tube mechanismsand cable driven mechanisms complement each other to someextent, which has also been discussed in [19]. Therefore, it

2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016

t1

t3

r1

r2

r3

t1

t2

t3

t1

t2r1

r2

r3

r3

r4

r1

r2

Robot 1 Robot 2 Robot 3

Cable Driven Mechanism

Concentric Tube Mechanism

Segment 1

Segment 2

Segment 3

t2

Fig. 1. Structure of three 6-DOF continuum robots investigated in this paper. t stands for translational DOF and r stands for rotational DOF.

is natural to explore the possibility of combining these twomechanisms. Several researchers have investigated this idea,including integrating concentric tubes with an endoscope [20],using a cable driven mechanism as the end effector of aconcentric tube robot [21], and constraining a cable drivenrobot with a tube [14]. In this paper, we investigate threecontinuum robots combining concentric tube mechanisms andcable driven mechanisms from the viewpoint of workspaceand dexterity. Workspace is conventionally defined as the setof all the spatial positions that the tip of a robot can reach. Itis an important performance index for evaluating robots, andgeometrical [22], analytical [23] and discrete [24] methodshave been proposed to calculate the workspace. Volume andboundary are usually used to describe the size and shape of theworkspace, but they do not contain the information of quality,i.e., how dexterous the robot is within its workspace. There-fore, researchers proposed another performance index, dex-terity. For surgical robots, dexterity is particularly importantbecause the robots are not only required to reach the surgicalsites, but also required to perform complicated operations. Twoclasses of indices have been put forward to define the dexterity:the manipulability [24], [25] describing the isotropy of themovability at the tip position of a robot, and the orientability[26], [27] representing how many orientations a robot can holdwhen its tip points to a certain position inside the workspace.As the orientability is more intuitive in evaluating the dexterityof a robot, we will use this performance index for dexterityanalysis in this paper.

The contributions of this paper are as follows. First, we pro-pose three structures combining a concentric tube mechanismand a cable driven mechanism and present the comparisonresults of the dexterity of these three structures. The conclu-sions drawn in this paper not only apply to the investigatedrobots but also provide generic guidelines for designing othercontinuum robots. Second, we introduce new indices forevaluating the dexterity of robots, especially for examiningthe distribution of dexterity along different directions. Thedirectional dexterity can be used to analyze other types ofrobots as well.

The rest of the paper is organized as below. Section IIintroduces the structures and kinematic models of the threecontinuum robots investigated in this paper. Then we describethe definition and calculation of the dexterity indices used to

evaluate each robot in Section III. Two groups of simulationsare conducted to compare the three robots and graphical andtabular results are presented in Section IV. Section V and VIdiscuss and conclude the paper, respectively.

II. KINEMATIC MODELS OF THREE 6-DOFCONTINUUM ROBOTS

In this paper, we investigate three 6-DOF (Degree of Free-dom) continuum robots, whose structures are shown in Fig. 1.All these robots have three segments, with Segment 1 iden-tically being a straight tube and having a translational DOF.This segment serves as a delivery section which fits the entrypath of many surgical applications such as intracardiac surgery[8], nasopharyngeal biopsy [5], and intracerebral hemorrhageevacuation [28].

The configurations of the other two segments are differentamong the three robots. In Robot 1, the second segment is con-figured with variable curvature. This can be achieved by usinga cable driven mechanism, such as the flexible endoscope usedin [20]. However, note that there is another way to endowthe segment with variable curvature by using two concentrictubes of the same length and initial curvature, and comparablestiffness. By independently rotating both concentric tubes, thecurvature of the resultant segment can vary, ideally, betweenzero and the initial curvature due to the interaction of the twotubes [18]. An example of this configuration can be foundin [8]. In addition to rotation, Segment 2 can be translatedthrough the lumen of Segment 1. Therefore, Segment 2 hastwo rotational DOFs and one translational DOF. The thirdsegment is constructed by a tube that can translate and rotateabout its axis, but its curvature is fixed, meaning that Segment3 has only one rotational DOF and one translational DOF.

In Robot 2, the DOF configurations of Segment 2 andSegment 3 are switched compared with Robot 1. Segment 2 isformed by a curved tube that has one rotational DOF and onetranslational DOF. Segment 3, however, has one additionalrotational DOF as it adopts a cable driven mechanism. Anexample of this structure can be seen in Fig. 2 [21].

In Robot 3, both Segment 2 and Segment 3 are made bycable driven mechanisms. Typically, these two segments insuch a configuration share the same backbone while differentgroups of cables are used to control each segment separately[16]. As a result, the two segments have only one translational

WU et al.: DEXTERITY ANALYSIS OF THREE 6-DOF CONTINUUM ROBOTS 3

Fig. 2. Example of Robot 2. Segment 2 is a curved tube with fixed curvatureand Segment 3 is constructed by a cable driven mechanism.

X

Y

Z

f

s

r=1/k

Fig. 3. Kinematic parameters of a single segment.

DOF in total. However, each segment has two rotational DOFsas they can bend in both directions, making the total numbersof DOF six as well as the Robot 1 and 2.

The kinematics of constant curvature continuum robotscan be decomposed into two mappings, a robot-independentmapping between configuration space and task space, and arobot-specific mapping between actuator space and configu-ration space [29]. In this paper, we only focus on the robot-independent mapping. As shown in Fig. 3, a single segment ofa continuum robot can be modelled by three variables underconstant curvature assumption. κ is the curvature of the arc,φ means the angle between the plane of the arc and the X-Zplane, and s stands for the length of the arc. The position andorientation at the distal end of the segment can be calculatedby the following homogeneous transformation [29],

Ti = cos2 φ(cosκs−1)+1 sinφ cosφ(cosκs−1) cosφ sinκs cosφ(1−cosκs)κ

sinφ cosφ(cosκs−1) cos2 φ(1−cosκs)+cosκs sinφ sinκs sinφ(1−cosκs)κ

−cosφ sinκs −sinφ sinκs cosκs sinκsκ

0 0 0 1

.

(1)

Depending on the configuration of the robot, the threevariables have different embodiments. For example, as the firstsegments of all the three robots are straight, κ1 and φ1 areconstant zero. Segment 3 of Robot 1 and Segment 2 of Robot2 have fixed curvature, and therefore κ3 of Robot 1 and κ2of Robot 2 are constant (non-zero). In Robot 3, the lengths ofSegment 2 and Segment 3 are dependent. As a result, s2 and

Configuration 1

Configuration 2 Service Sphere

Service Regions

dh

dq

X

YZ

Radial

CircumferentialAxial

Base Frame

Fig. 4. Dexterity at a spatial position.

s3 are related: if s2 is nonzero, then s3 can only be the fulllength, meaning that Segment 3 is fully exposed; if s3 is lessthan the full length, then s2 can only be zero, meaning thatSegment 2 is completely invisible.

The general kinematics of the robot with three segments isexpressed as follow,

T = T1(κ1,φ1,s1)T2(κ2,φ2,s2)T3(κ3,φ3,s3). (2)

III. DEXTERITY INDICES

The dexterity indices developed in this paper are basedon those introduced in [27], which adopted the concept ofdexterous solid angle [26], as shown in Fig. 4. For a robotwith more than three DOFs, there can be multiple possibleconfigurations with different orientations when reaching aspecific spatial position. Constructing a unit sphere with itscenter placed at the spatial position, which is named theService Sphere, all the possible areas on the sphere that canbe orientated to by the tip of the robot are referred to as theService Regions. Hence, the dexterity at this spatial positionis defined as

D(P) =AR(P)

AS=

NO(P)Nθ Nh

∈ (0,1] (3)

where AR(P) is the area of the Service Regions at positionP and AS is the total surface area of the unit sphere. As inthis paper, we will use a discretization method to obtain thedexterity indices, the surface of the unit sphere is discretizedinto Nθ by Nh patches by a series of longitude meridians andlatitude lines, with the angle between adjacent meridians beingδθ and the height interval of latitude lines being δh (Fig. 4).Letting NO(P) represent the number of orientable patches, thedexterity D(P) equals to NO(P) divided by Nθ Nh. It is notedthat all the patches should be equal in area so as to representequal probability. Under our discretization method, this canbe readily achieved by letting θ and h evenly divided becauseeach patch has an area of δθδh. Also note that, while thenorth pole of the Service Sphere can be arbitrarily oriented,we restrict it to be along the positive direction of the Z axis ofthe base frame for all the evaluation in this paper. Moreover,all the axes mentioned in the following part of this paper,unless specifically explained, refer to those of the base frameof the robot.

Workspace of a robot is a set of all the possible positionsthat are reachable. In this paper, we use a Monte-Carlo method

4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016

TABLE ICONDITIONS OF THE SIMULATIONS.

Segment 1 Segment 2 Segment 3κ1/mm−1 φ1/rad s1/mm κ2/mm−1 φ2/rad s2/mm κ3/mm−1 φ3/rad s3/mm

Group I

Robot 1 0 0 [0,100] [0, π

200 ] [0,2π] [0,100] π

200 [0,2π] [0,100]

Robot 2 0 0 [0,100] π

200 [0,2π] [0,100] [0, π

200 ] [0,2π] [0,100]

Robot 3 0 0 [0,100] [0, π

200 ] [0,2π]0

[0, π

200 ] [0,2π][0,100]

[0,100] 100

Group II

Robot 1 0 0 [0,100] [0, π

100 ] [0,2π] [0,100] π

200 [0,2π] [0,100]

Robot 2 0 0 [0,100] π

200 [0,2π] [0,100] [0, π

100 ] [0,2π] [0,100]

Robot 3 0 0 [0,100] [0, π

100 ] [0,2π]0

[0, π

100 ] [0,2π][0,100]

[0,100] 100

to calculate the workspace, that is, randomly sample thejoint space, calculate the forward kinematics, and statisticallyanalyze all the reachable positions. It is noted that all thethree robots investigated in this work have rotational symmetryabout the Z axis of the base frame. Therefore, we will onlycalculate the workspace within the right half of the X-Z plane(X+) of the base frame. By evenly discretizing the plane intopatches with side lengths of δx and δ z, the workspace can becalculated by

W =∫

P∈WdP = NPδxδ z (4)

where NP is the number of reachable patches.Having the workspace and dexterity at any position inside

the workspace, we can now define the total dexterity over theworkspace as

Dt =∫

P∈W

D(P)dPW

=∑

NPi=1 Ni

ONPNθ Nh

∈ (0,1]. (5)

Note that this is an index between 0 and 1. The higher theindex is, the more dexterity the robot has. If the robot has atotal dexterity of 1, that means the robot can be oriented toall directions at any position inside its reachable workspace.

We can further define the dexterous workspace as a productof dexterity and workspace,

W D = DtW =∑

NPi=1 Ni

Oδxδ zNθ Nh

. (6)

This is an integral index showing how large and dexterous theworkspace is.

In addition, we want to investigate the distribution propertyof the dexterity along different axes of the base frame. As theworkspace of the robot has a rotational symmetry about itsZ axis, any position within the workspace can be regarded ason the surface of a cylinder that aligns its axis with the Zaxis. Therefore, we introduce three new dexterity indices, Dr,Dc, and Da to represent the radial, circumferential, and axialdexterity, respectively. Each directional dexterity is evaluatedover the sectional workspace, i.e., over the right half of theX-Z plane. Therefore, the indices are defined in (7) - (9), with~o j being the jth orientation vector at the tip and ~x, ~y, and ~z

TABLE IIDEXTERITY ANALYSIS RESULTS.

Group I Group IIRobot 1 Robot 2 Robot 3 Robot 1 Robot 2 Robot 3

Dt 0.16 0.15 0.17 0.17 0.36 0.41Dr 0.074 0.085 0.095 0.085 0.18 0.20Dc 0.065 0.056 0.067 0.072 0.18 0.21Da 0.099 0.092 0.10 0.10 0.18 0.20W 31375 32000 37900 50675 36750 58225

W D 4901 4955 6608 8664 13142 23879

Workspaces (W and W D) are in mm2.

being the unit vectors along the X, Y, and Z axes of the baseframe, respectively.

Dr =∫

P∈W

1W

∫A∈AR

|d~A ·~x|AS

dP =∑

NPi=1 ∑

NiO

j=1 |~o j ·~x|NPNθ Nh

(7)

Dc =∫

P∈W

1W

∫A∈AR

|d~A ·~y|AS

dP =∑

NPi=1 ∑

NiO

j=1 |~o j ·~y|NPNθ Nh

(8)

Da =∫

P∈W

1W

∫A∈AR

|d~A ·~z|AS

dP =∑

NPi=1 ∑

NiO

j=1 |~o j ·~z|NPNθ Nh

(9)

As can be seen from (7) to (9), the radial, circumferential,and axial dexterity are actually the integral component of thetotal dexterity along the X, Y, and Z axes of the base frame,respectively.

IV. RESULTS

In this section, we present the dexterity analysis results ofthe three robots. Two groups of simulations, of which theconditions are listed in Table I, are carried out in MATLAB2016a on a WINDOWS 7 64 bit platform with Intel Core i7-5600U 2.60 GHz CPU and 16.0 GB RAM. In each group, thevariables of each robot are given a certain variation range, and100,000,000 samples are randomly collected from the variationranges. Then the kinematic model introduced in Sec. II is usedto calculate the position and orientation of the tip in each

WU et al.: DEXTERITY ANALYSIS OF THREE 6-DOF CONTINUUM ROBOTS 5

0 50 100 150

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Ro

bo

t 1

Ro

bo

t 2

Ro

bo

t 3

Total Radial Circumferential

Dexterity Distribution

Axial

0.150.12

0.19

0.35 0.23 0.15 0.18

0.36 0.23 0.150.19

Fig. 5. Dexterity distribution of the three robots in Group I. The color indicates the dexterity at each position patch. The maximum dexterity across theworkspace is marked out. The radial, circumferential, and axial dexterity distributions are also compared. The X and Z axes are all defined in the base frame.

sample configuration. If the position of the tip is not withinthe right half of the X-Z plane, a rotation about the Z axisis exerted onto both the position and the orientation vectors,so that the tip of the robot shifts to the right half of the X-Z plane. Then the positions and orientations are tagged withfour patch numbers in terms of X , Z, θ and h, and thosehaving exactly the same four tags are merged as one pair ofposition and orientation. The discretization of position is setas δx = δ z = 5mm, and for orientation Nθ = 60 and Nh = 30are selected. After the merging, the number of pairs having thesame position (X and Z) are added up to get the number oforientable patches at this position. All the reachable positionsare plotted in a scatter diagram to show the workspace and acolor bar is used to indicate the dexterity distribution acrossthe workspace.

The difference between Group I and Group II is the variationrange of the curvature. In Group I, all the bending segmentswith variable curvature have a range of [0,π/200]. Having

a maximum length of 100mm, this means that the segmentcan bend at most 90◦. In Group II, however, the range ofvariable curvature increases to [0,π/100], meaning that themaximum bending angle is 180◦. The results of Group I andII are depicted in Fig. 5 - 8 and quantitatively represented inTable II.

From Fig. 5, we can see the workspaces of the three robotsunder the conditions of Group I have different shapes. Theworkspaces of Robot 1 and 3 are closer to cones, whilethe workspace of Robot 2 is more like a cylinder (the 3Dworkspace is formed by rotating the sectional shape about theZ axis). Table II shows that the volume of the workspace ofRobot 2 is slightly larger than Robot 1, though both of themare much smaller than Robot 3.

The total dexterity of the three robots is almost the same,but its distribution across the workspace is quite different. Theposition with the maximum dexterity appears in the middle ofthe workspace of Robot 1, while for Robot 2 and 3, the most

6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016

Ro

bot 1

Robot 2

Ro

bo

t 3

X-Y View X-Z View

Fig. 6. The service regions of each robot in Group I at the position with themaximum total dexterity.

dexterous point locates near the edge of the workspace. Themaximum dexterity of Robot 2 and 3 is also greater than Robot1.

The dexterity distribution along different axes also evincesdifferent characteristics. For Robot 1, the dexterity along theaxial direction is generally the best while the circumferentialdexterity is the smallest. The most dexterous positions inradial and circumferential directions are located identicallyas the total dexterity. However, the axial dexterity has itsmaximum point placed near the Z axis. For Robot 2, thecircumferential direction is also the least dexterous. However,the radial dexterity outperforms the axial dexterity in termsof the maximum value, which is quite different from Robot1. All the three directional dexterity has the summit occurringnear the edge of the workspace, which is different from Robot1 as well. The dexterity distribution of Robot 3 is most similarto Robot 2 in terms of the extreme value of each directionaldexterity. Yet the locations of these extreme value points in theworkspace more resemble Robot 1. It is clear that the mostaxially dexterous position of Robot 1 appears closer to the Zaxis rather than the edge of the workspace.

The differences of dexterity distribution among these threerobots can also be found in Fig. 6, which shows the serviceregions of each robot at the position with the maximum totaldexterity. It can be easily concluded that Robot 1 has moredexterity along the Z axis while Robot 2 and 3 have betterradial dexterity. This finding conforms to the conclusions wehave drawn from Fig. 5.

Fig. 7 and Fig. 8 display the results of simulation GroupII. Recall that the only difference of Group II from GroupI is the increase of variation range of those segments that

have variable curvature, i.e., Segment 2 of Robot 1, Segment3 of Robot 2, and both Segment 2 and Segment 3 of Robot3. The immediate effect of this change is the enlargement ofworkspace, especially at the part below the X axis. Comparingthe data in Table II, it can be seen that the workspaces of Robot1 and 3 have significantly increased. The workspace of Robot2 also expands, but relatively more moderately.

In contrast to Group I, the total dexterity of Robot 2 and 3has dramatically improved, whereas that of Robot 1 does notchange too much. However, the maximum dexterity of Robot1 increases significantly from 0.29 to 0.41. The locations ofthe extreme value points all occur approximately in the middleof the workspace this time, which is different from Group I.

With regard to the dexterity distribution along different axes,Robot 1 basically retains the same characteristics, whereasRobot 2 and 3 evince evenly distributed dexterity among thethree directions in Group II. This is also supported by Fig.8, which shows that Robot 1 has more service regions in theaxial and radial directions at the position with the maximumtotal dexterity, while Robot 2 and 3 has their service regionsalmost covering all the surface of the service sphere.

V. DISCUSSION

The configurations of the three robots can be summarizedas:

1) Robot 1 has three DOFs in Segment 2 and two DOFsin Segment 3;

2) Robot 2 has two DOFs in Segment 2 and three DOFsin Segment 3;

3) and Robot 3 has two and a half DOFs in both Segment2 and 3, since they share the same translational DOFs.

Therefore, the main difference between these three robots isthe different allocation of DOFs. The results of simulationGroup I and Group II suggest that this different allocationaffects both the shape of workspace and the distribution ofdexterity. Comparing Robot 1 and 2 in Group I and Group II,we can see that augmentation of Segment 2 tends to enlargethe workspace while augmentation of Segment 3 tends toimprove the dexterity. In both groups, Robot 3 achieves thelargest workspace and the best dexterity, which implies thatevenly allocate DOFs among the segments is the best solution.In addition, with enhancement in variable curvature, Robot 1follows Robot 3 closely in terms of workspace and Robot 2is comparable with Robot 3 in terms of dexterity.

It is also worth noting that the distribution of dexterityalong different axes can represent various characteristics. Forexample, the axial dexterity of Robot 1 seems better thanthe radial and circumferential dexterity. If there is room foradjustment of the entry point or angle of the robot whenapplying it to a specific task, the indices suggest that weshould place the robot where operations in the axial directionare the dominant during the task. The position where themaximum dexterity appears also varies among different robottypes and along different directions, and also with changingvariation range of curvature. Based on the dexterity analysis,we should, if possible, let the task site be close to themaximum dexterity position rather than near the edge of the

WU et al.: DEXTERITY ANALYSIS OF THREE 6-DOF CONTINUUM ROBOTS 7

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Fig. 7. Dexterity distribution of the three robots in Group II. The color indicates the dexterity at each position patch. The maximum dexterity across theworkspace is marked out. The radial, circumferential, and axial dexterity distributions are also compared. The X and Z axes are all defined in the base frame.

workspace. In summary, it is suggested that attention shouldbe paid to the dexterity distribution when robots are applied toscenarios where directional dexterity is critical, such as mostsurgical applications.

VI. CONCLUSIONS

In this paper, we proposed three structures of combiningconcentric tube mechanisms and cable driven mechanisms,and introduced a set of indices based on the orientability toanalyze the dexterity of these three continuum robots. A MonteCarlo method was used to calculate the dexterity distributionacross the workspace. Particularly, directional dexterity indiceswere proposed to describe the dexterity along different axes.Results have implied that evenly allocating DOFs among thesegments achieves best workspace and dexterity. Otherwise,assigning more DOFs to the proximal segment tends to enlargethe workspace and adding more DOFs to the distal segmenttends to improve the dexterity. In addition, the dexterity

along different axes can vary significantly and thus requiresspecial attention when applying the robot to specific surgicalprocedures.

The analyses conducted in this paper were purely from theviewpoint of kinematics. In practice, many other factors willaffect the design of the structures. For example, in Robot 1,the inner tube of Segment 3 will restrict the variation range ofSegment 2 due to snapping problem. In Robot 3, the couplingof actuators when driving all the segments also impose chal-lenges in control. Robot 2 avoids the snapping problem andcan decouple the control of different segments, but to scale itdown is quite challenging. In addition, compared with analyticapproaches, the numeric method used in this paper makes itdifficult to give the optimized design parameters. Even though,the dexterity analysis performed in this paper provides goodguidelines for designing this kind of continuum robots and canbe adopted for evaluation of other robots as well.

8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016

Robot 2

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Fig. 8. The service regions of each robot in Group II at the position with themaximum total dexterity.

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