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IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016 1
Dexterity Analysis of Three 6-DOF Continuum
Robots Combining Concentric Tube Mechanisms
and Cable Driven Mechanisms
Liao Wu1, Ross Crawford1, and Jonathan Roberts1
Abstract—Continuum robots are increasingly used in minimal-
ly invasive surgeries. To date, concentric tube mechanism and
cable driven mechanism have been two prevalent mechanisms
for constructing continuum robots. As these two mechanisms
complement each other, it is worth exploring the possibility
of combining them together. In this paper, we investigate the
dexterity of three continuum robots combining both mechanisms.
Indices based on the concept of orientability are introduced to
analyze the distribution of the dexterity. A Monte Carlo method is
used to calculate the dexterity distribution across the workspace.
Particularly, directional dexterity indices are put forward to
describe the dexterity along different axes. Results imply that
evenly allocating degrees of freedom (DOFs) among the segments
achieves the best workspace and dexterity. Otherwise, assigning
more DOFs to the proximal segment tends to enlarge the
workspace and adding more DOFs to the distal segment tends to
improve the dexterity. In addition, the dexterity along different
axes can vary significantly and thus requires special attention
when applying the robot to specific surgical procedures.
Index Terms—Mechanism Design of Manipulators, Medical
Robots and Systems, Concentric Tube Robots, Cable Driven
Robots.
I. INTRODUCTION
CONTINUUM robots are rapidly emerging to become a
class of useful tools in many areas including industrial
and medical applications [1], [2]. Different from conventional
serial link robots that are driven by discrete joints, continuum
robots have the ability to continuously deform into different
shapes. Usually actuated at the remote proximal base, contin-
uum robots can be made very compact along the part from the
actuators to the tip. As a consequence, they are increasingly
used in minimally invasive surgeries where their compactness
enables agile manipulation in confined environment inside the
human body. During the last decade, much research has been
carried out in exploring the application of continuum robots
to many minimally invasive procedures. Examples include
transoral/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 upon
evaluation of the Associate Editor and Reviewers’ comments. This work was
supported by the Vice-Chancellor’s Research Fellowship from Queensland
University of Technology under Grant 322450-0096/08 awarded to Dr. Liao
Wu.
1The authors are with the Australian Centre for Robotic Vision, Science
and 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 orthopedic
surgeries [10].
To date, two prevalent mechanisms have been adopted for
continuum robots in minimally invasive surgeries: concentric
tube mechanism [3]–[5], [7]–[13] and cable driven mechanism
[6], [14]–[17]. Concentric tube robots use a set of telescopic
tubes to achieve spatial movability. As each tube can be
precurved within its elastic deformation zone, the interaction
between the tubes, while they are being rotated or translated
with respect to each other, results in variation of both the
curvature of the combined tubes and the position of the tip.
The main advantage of a concentric tube mechanism lies in
its capability of being scaled down to submillimeter, which
is favorable in many minimally invasive surgical applica-
tions. However, since each tube is precurved, the variation
of the resultant curvature is limited. Therefore, optimization
of parameters of the tubes based on the specific task is
usually required before applying the robot to any surgical
procedure [4]. The combined segments can be endowed with
variable curvature when the contributing tubes are comparable
in stiffness [18], but the snapping problem occurring when the
relative rotation angle exceeds a certain critical point prevents
the natural curvature of each tube from being large, limiting
the variation range of the combined segments.
A cable driven mechanism, as the name suggests, use cables
to actuate the deformation of the robot. It usually consists of
backbones as the main deformation elements, cables as the
actuators, and vertebrae, which distribute along the backbones,
as the “joints” to transmit and restrict the actuation. The main
advantage of a cable driven mechanism is its flexibility and
variable curvature which is determined by the elasticity of
the backbone and the space between each vertebra. Typically,
a bending segment comprises at least three cables so that it
can 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 is
somewhat challenging as there is a coupling between each
segment [15]. It is also difficult to scale down a cable driven
mechanism to the level of a concentric tube mechanism; at
the same scale, the cable driven mechanism lacks the rigidity
existing in a concentric tube mechanism that facilitates force
transmission.
As can be seen from the above, concentric tube mechanisms
and cable driven mechanisms complement each other to some
extent, 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
t2
r1
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. tstands for translational DOF and rstands for rotational DOF.
is natural to explore the possibility of combining these two
mechanisms. Several researchers have investigated this idea,
including integrating concentric tubes with an endoscope [20],
using a cable driven mechanism as the end effector of a
concentric tube robot [21], and constraining a cable driven
robot with a tube [14]. In this paper, we investigate three
continuum robots combining concentric tube mechanisms and
cable driven mechanisms from the viewpoint of workspace
and dexterity. Workspace is conventionally defined as the set
of all the spatial positions that the tip of a robot can reach. It
is an important performance index for evaluating robots, and
geometrical [22], analytical [23] and discrete [24] methods
have been proposed to calculate the workspace. Volume and
boundary are usually used to describe the size and shape of the
workspace, 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 important
because the robots are not only required to reach the surgical
sites, but also required to perform complicated operations. Two
classes of indices have been put forward to define the dexterity:
the manipulability [24], [25] describing the isotropy of the
movability at the tip position of a robot, and the orientability
[26], [27] representing how many orientations a robot can hold
when its tip points to a certain position inside the workspace.
As the orientability is more intuitive in evaluating the dexterity
of a robot, we will use this performance index for dexterity
analysis in this paper.
The contributions of this paper are as follows. First, we pro-
pose three structures combining a concentric tube mechanism
and a cable driven mechanism and present the comparison
results of the dexterity of these three structures. The conclu-
sions drawn in this paper not only apply to the investigated
robots but also provide generic guidelines for designing other
continuum robots. Second, we introduce new indices for
evaluating the dexterity of robots, especially for examining
the distribution of dexterity along different directions. The
directional dexterity can be used to analyze other types of
robots as well.
The rest of the paper is organized as below. Section II
introduces the structures and kinematic models of the three
continuum robots investigated in this paper. Then we describe
the definition and calculation of the dexterity indices used to
evaluate each robot in Section III. Two groups of simulations
are conducted to compare the three robots and graphical and
tabular results are presented in Section IV. Section V and VI
discuss and conclude the paper, respectively.
II. KINEMATIC MODELS OF THREE 6-DOF
CONTINUUM 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 entry
path of many surgical applications such as intracardiac surgery
[8], nasopharyngeal biopsy [5], and intracerebral hemorrhage
evacuation [28].
The configurations of the other two segments are different
among the three robots. In Robot 1, the second segment is con-
figured with variable curvature. This can be achieved by using
a cable driven mechanism, such as the flexible endoscope used
in [20]. However, note that there is another way to endow
the segment with variable curvature by using two concentric
tubes of the same length and initial curvature, and comparable
stiffness. By independently rotating both concentric tubes, the
curvature of the resultant segment can vary, ideally, between
zero and the initial curvature due to the interaction of the two
tubes [18]. An example of this configuration can be found
in [8]. In addition to rotation, Segment 2 can be translated
through the lumen of Segment 1. Therefore, Segment 2 has
two rotational DOFs and one translational DOF. The third
segment is constructed by a tube that can translate and rotate
about its axis, but its curvature is fixed, meaning that Segment
3 has only one rotational DOF and one translational DOF.
In Robot 2, the DOF configurations of Segment 2 and
Segment 3 are switched compared with Robot 1. Segment 2 is
formed by a curved tube that has one rotational DOF and one
translational DOF. Segment 3, however, has one additional
rotational DOF as it adopts a cable driven mechanism. An
example of this structure can be seen in Fig. 2 [21].
In Robot 3, both Segment 2 and Segment 3 are made by
cable driven mechanisms. Typically, these two segments in
such a configuration share the same backbone while different
groups 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 curvature
and 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 DOFs
as they can bend in both directions, making the total numbers
of DOF six as well as the Robot 1 and 2.
The kinematics of constant curvature continuum robots
can be decomposed into two mappings, a robot-independent
mapping between configuration space and task space, and a
robot-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 of
a continuum robot can be modelled by three variables under
constant curvature assumption. κis the curvature of the arc,
φmeans the angle between the plane of the arc and the X-Z
plane, and sstands for the length of the arc. The position and
orientation at the distal end of the segment can be calculated
by the following homogeneous transformation [29],
Ti=
cos2φ(cosκs−1)+1 sinφcosφ(cosκs−1)cosφsinκscosφ(1−cos κs)
κ
sinφcosφ(cosκs−1)cos2φ(1−cosκs)+cosκssinφsinκssinφ(1−cos κs)
κ
−cosφsinκs−sinφsin κscos κssinκs
κ
0 0 0 1
.
(1)
Depending on the configuration of the robot, the three
variables have different embodiments. For example, as the first
segments of all the three robots are straight, κ1and φ1are
constant zero. Segment 3 of Robot 1 and Segment 2 of Robot
2 have fixed curvature, and therefore κ3of Robot 1 and κ2
of Robot 2 are constant (non-zero). In Robot 3, the lengths of
Segment 2 and Segment 3 are dependent. As a result, s2and
Configuration 1
Configuration 2 Service Sphere
Service Regions
dh
dq
X
Y
Z
Radial
Circumferential
Axial
Base Frame
Fig. 4. Dexterity at a spatial position.
s3are related: if s2is nonzero, then s3can only be the full
length, meaning that Segment 3 is fully exposed; if s3is less
than the full length, then s2can only be zero, meaning that
Segment 2 is completely invisible.
The general kinematics of the robot with three segments is
expressed 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 based
on those introduced in [27], which adopted the concept of
dexterous solid angle [26], as shown in Fig. 4. For a robot
with more than three DOFs, there can be multiple possible
configurations with different orientations when reaching a
specific spatial position. Constructing a unit sphere with its
center placed at the spatial position, which is named the
Service Sphere, all the possible areas on the sphere that can
be orientated to by the tip of the robot are referred to as the
Service Regions. Hence, the dexterity at this spatial position
is defined as
D(P) = AR(P)
AS
=NO(P)
NθNh
∈(0,1](3)
where AR(P)is the area of the Service Regions at position
Pand ASis the total surface area of the unit sphere. As in
this paper, we will use a discretization method to obtain the
dexterity indices, the surface of the unit sphere is discretized
into Nθby Nhpatches by a series of longitude meridians and
latitude lines, with the angle between adjacent meridians being
δθand the height interval of latitude lines being δh(Fig. 4).
Letting NO(P)represent the number of orientable patches, the
dexterity D(P)equals to NO(P)divided by NθNh. It is noted
that all the patches should be equal in area so as to represent
equal probability. Under our discretization method, this can
be readily achieved by letting θand hevenly divided because
each patch has an area of δ θ δ h. Also note that, while the
north pole of the Service Sphere can be arbitrarily oriented,
we restrict it to be along the positive direction of the Z axis of
the 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 frame
of the robot.
Workspace of a robot is a set of all the possible positions
that are reachable. In this paper, we use a Monte-Carlo method
4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016
TABLE I
CONDITIONS OF THE SIMULATIONS.
Segment 1 Segment 2 Segment 3
κ1/mm−1φ1/rad s1/mm κ2/mm−1φ2/rad s2/mm κ3/mm−1φ3/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 the
joint space, calculate the forward kinematics, and statistically
analyze all the reachable positions. It is noted that all the
three robots investigated in this work have rotational symmetry
about the Z axis of the base frame. Therefore, we will only
calculate the workspace within the right half of the X-Z plane
(X+) of the base frame. By evenly discretizing the plane into
patches with side lengths of δxand δz, the workspace can be
calculated by
W=ZP∈W
dP =NPδxδz(4)
where NPis the number of reachable patches.
Having the workspace and dexterity at any position inside
the workspace, we can now define the total dexterity over the
workspace as
Dt=ZP∈W
D(P)dP
W=∑NP
i=1Ni
O
NPNθNh
∈(0,1].(5)
Note that this is an index between 0 and 1. The higher the
index is, the more dexterity the robot has. If the robot has a
total dexterity of 1, that means the robot can be oriented to
all directions at any position inside its reachable workspace.
We can further define the dexterous workspace as a product
of dexterity and workspace,
WD=DtW=∑NP
i=1Ni
Oδxδz
NθNh
.(6)
This is an integral index showing how large and dexterous the
workspace is.
In addition, we want to investigate the distribution property
of the dexterity along different axes of the base frame. As the
workspace of the robot has a rotational symmetry about its
Z axis, any position within the workspace can be regarded as
on the surface of a cylinder that aligns its axis with the Z
axis. Therefore, we introduce three new dexterity indices, Dr,
Dc, and Dato represent the radial, circumferential, and axial
dexterity, respectively. Each directional dexterity is evaluated
over the sectional workspace, i.e., over the right half of the
X-Z plane. Therefore, the indices are defined in (7) - (9), with
~ojbeing the jth orientation vector at the tip and ~x,~y, and ~z
TABLE II
DEX TE RIT Y AN ALYS IS R ES ULTS .
Group I Group II
Robot 1 Robot 2 Robot 3 Robot 1 Robot 2 Robot 3
Dt0.16 0.15 0.17 0.17 0.36 0.41
Dr0.074 0.085 0.095 0.085 0.18 0.20
Dc0.065 0.056 0.067 0.072 0.18 0.21
Da0.099 0.092 0.10 0.10 0.18 0.20
W31375 32000 37900 50675 36750 58225
WD4901 4955 6608 8664 13142 23879
Workspaces (Wand WD) are in mm2.
being the unit vectors along the X, Y, and Z axes of the base
frame, respectively.
Dr=ZP∈W
1
WZA∈AR
|d~
A·~x|
AS
dP =∑NP
i=1∑Ni
O
j=1|~oj·~x|
NPNθNh
(7)
Dc=ZP∈W
1
WZA∈AR
|d~
A·~y|
AS
dP =∑NP
i=1∑Ni
O
j=1|~oj·~y|
NPNθNh
(8)
Da=ZP∈W
1
WZA∈AR
|d~
A·~z|
AS
dP =∑NP
i=1∑Ni
O
j=1|~oj·~z|
NPNθNh
(9)
As can be seen from (7) to (9), the radial, circumferential,
and axial dexterity are actually the integral component of the
total dexterity along the X, Y, and Z axes of the base frame,
respectively.
IV. RESULTS
In this section, we present the dexterity analysis results of
the three robots. Two groups of simulations, of which the
conditions are listed in Table I, are carried out in MATLAB
2016a on a WINDOWS 7 64 bit platform with Intel Core i7-
5600U 2.60 GHz CPU and 16.0 GB RAM. In each group, the
variables of each robot are given a certain variation range, and
100,000,000 samples are randomly collected from the variation
ranges. Then the kinematic model introduced in Sec. II is used
to 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
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.2
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.2
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 50 100 150
X /mm
0
50
100
150
200
250
300
Z /mm
0.05
0.1
0.15
0.29
Robot 1Robot 2
Robot 3
Total Radial Circumferential
Dexterity Distribution
Axial
0.15 0.12
0.19
0.35 0.23 0.15 0.18
0.36 0.23 0.15
0.19
Fig. 5. Dexterity distribution of the three robots in Group I. The color indicates the dexterity at each position patch. The maximum dexterity across the
workspace 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 within
the right half of the X-Z plane, a rotation about the Z axis
is 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 with
four patch numbers in terms of X,Z,θand h, and those
having exactly the same four tags are merged as one pair of
position and orientation. The discretization of position is set
as δx=δz=5mm, and for orientation Nθ=60 and Nh=30
are selected. After the merging, the number of pairs having the
same position (Xand Z) are added up to get the number of
orientable patches at this position. All the reachable positions
are plotted in a scatter diagram to show the workspace and a
color bar is used to indicate the dexterity distribution across
the workspace.
The difference between Group I and Group II is the variation
range of the curvature. In Group I, all the bending segments
with variable curvature have a range of [0,π/200]. Having
a maximum length of 100mm, this means that the segment
can bend at most 90◦. In Group II, however, the range of
variable curvature increases to [0,π/100], meaning that the
maximum bending angle is 180◦. The results of Group I and
II are depicted in Fig. 5 - 8 and quantitatively represented in
Table II.
From Fig. 5, we can see the workspaces of the three robots
under the conditions of Group I have different shapes. The
workspaces of Robot 1 and 3 are closer to cones, while
the workspace of Robot 2 is more like a cylinder (the 3D
workspace is formed by rotating the sectional shape about the
Z axis). Table II shows that the volume of the workspace of
Robot 2 is slightly larger than Robot 1, though both of them
are 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. The
position with the maximum dexterity appears in the middle of
the workspace of Robot 1, while for Robot 2 and 3, the most
6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016
Robot 1Robot 2
Robot 3
X-Y View X-Z View
Fig. 6. The service regions of each robot in Group I at the position with the
maximum total dexterity.
dexterous point locates near the edge of the workspace. The
maximum dexterity of Robot 2 and 3 is also greater than Robot
1.
The dexterity distribution along different axes also evinces
different characteristics. For Robot 1, the dexterity along the
axial direction is generally the best while the circumferential
dexterity is the smallest. The most dexterous positions in
radial and circumferential directions are located identically
as the total dexterity. However, the axial dexterity has its
maximum point placed near the Z axis. For Robot 2, the
circumferential direction is also the least dexterous. However,
the radial dexterity outperforms the axial dexterity in terms
of the maximum value, which is quite different from Robot
1. All the three directional dexterity has the summit occurring
near the edge of the workspace, which is different from Robot
1 as well. The dexterity distribution of Robot 3 is most similar
to Robot 2 in terms of the extreme value of each directional
dexterity. Yet the locations of these extreme value points in the
workspace more resemble Robot 1. It is clear that the most
axially dexterous position of Robot 1 appears closer to the Z
axis rather than the edge of the workspace.
The differences of dexterity distribution among these three
robots can also be found in Fig. 6, which shows the service
regions of each robot at the position with the maximum total
dexterity. It can be easily concluded that Robot 1 has more
dexterity along the Z axis while Robot 2 and 3 have better
radial dexterity. This finding conforms to the conclusions we
have drawn from Fig. 5.
Fig. 7 and Fig. 8 display the results of simulation Group
II. Recall that the only difference of Group II from Group
I is the increase of variation range of those segments that
have variable curvature, i.e., Segment 2 of Robot 1, Segment
3 of Robot 2, and both Segment 2 and Segment 3 of Robot
3. The immediate effect of this change is the enlargement of
workspace, especially at the part below the X axis. Comparing
the data in Table II, it can be seen that the workspaces of Robot
1 and 3 have significantly increased. The workspace of Robot
2 also expands, but relatively more moderately.
In contrast to Group I, the total dexterity of Robot 2 and 3
has dramatically improved, whereas that of Robot 1 does not
change too much. However, the maximum dexterity of Robot
1 increases significantly from 0.29 to 0.41. The locations of
the extreme value points all occur approximately in the middle
of 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, whereas
Robot 2 and 3 evince evenly distributed dexterity among the
three directions in Group II. This is also supported by Fig.
8, which shows that Robot 1 has more service regions in the
axial and radial directions at the position with the maximum
total dexterity, while Robot 2 and 3 has their service regions
almost covering all the surface of the service sphere.
V. DISCUSSION
The configurations of the three robots can be summarized
as:
1) Robot 1 has three DOFs in Segment 2 and two DOFs
in Segment 3;
2) Robot 2 has two DOFs in Segment 2 and three DOFs
in Segment 3;
3) and Robot 3 has two and a half DOFs in both Segment
2 and 3, since they share the same translational DOFs.
Therefore, the main difference between these three robots is
the different allocation of DOFs. The results of simulation
Group I and Group II suggest that this different allocation
affects both the shape of workspace and the distribution of
dexterity. Comparing Robot 1 and 2 in Group I and Group II,
we can see that augmentation of Segment 2 tends to enlarge
the workspace while augmentation of Segment 3 tends to
improve the dexterity. In both groups, Robot 3 achieves the
largest workspace and the best dexterity, which implies that
evenly allocate DOFs among the segments is the best solution.
In addition, with enhancement in variable curvature, Robot 1
follows Robot 3 closely in terms of workspace and Robot 2
is comparable with Robot 3 in terms of dexterity.
It is also worth noting that the distribution of dexterity
along different axes can represent various characteristics. For
example, the axial dexterity of Robot 1 seems better than
the radial and circumferential dexterity. If there is room for
adjustment of the entry point or angle of the robot when
applying it to a specific task, the indices suggest that we
should place the robot where operations in the axial direction
are the dominant during the task. The position where the
maximum dexterity appears also varies among different robot
types and along different directions, and also with changing
variation range of curvature. Based on the dexterity analysis,
we should, if possible, let the task site be close to the
maximum dexterity position rather than near the edge of the
WU et al.: DEXTERITY ANALYSIS OF THREE 6-DOF CONTINUUM ROBOTS 7
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.1
0.2
0.3
0.4
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.2
0.4
0.6
0.8
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.1
0.2
0.3
0.4
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.1
0.2
0.3
0.4
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.1
0.2
0.3
0.4
Total Radial Circumferential
Dexterity Distribution
Axial
Robot 2
Robot 3
0.84 0.40 0.46 0.44
0.96 0.46 0.49 0.49
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.05
0.1
0.15
0.2
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 50 100 150
X /mm
-100
0
100
200
300
Z /mm
0.05
0.1
0.15
0.2
Robot 1
0.41 0.23 0.18 0.22
Fig. 7. Dexterity distribution of the three robots in Group II. The color indicates the dexterity at each position patch. The maximum dexterity across the
workspace 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 should
be paid to the dexterity distribution when robots are applied to
scenarios where directional dexterity is critical, such as most
surgical applications.
VI. CONCLUSIONS
In this paper, we proposed three structures of combining
concentric tube mechanisms and cable driven mechanisms,
and introduced a set of indices based on the orientability to
analyze the dexterity of these three continuum robots. A Monte
Carlo method was used to calculate the dexterity distribution
across the workspace. Particularly, directional dexterity indices
were proposed to describe the dexterity along different axes.
Results have implied that evenly allocating DOFs among the
segments achieves best workspace and dexterity. Otherwise,
assigning more DOFs to the proximal segment tends to enlarge
the workspace and adding more DOFs to the distal segment
tends to improve the dexterity. In addition, the dexterity
along different axes can vary significantly and thus requires
special attention when applying the robot to specific surgical
procedures.
The analyses conducted in this paper were purely from the
viewpoint of kinematics. In practice, many other factors will
affect the design of the structures. For example, in Robot 1,
the inner tube of Segment 3 will restrict the variation range of
Segment 2 due to snapping problem. In Robot 3, the coupling
of actuators when driving all the segments also impose chal-
lenges in control. Robot 2 avoids the snapping problem and
can decouple the control of different segments, but to scale it
down is quite challenging. In addition, compared with analytic
approaches, the numeric method used in this paper makes it
difficult to give the optimized design parameters. Even though,
the dexterity analysis performed in this paper provides good
guidelines for designing this kind of continuum robots and can
be adopted for evaluation of other robots as well.
8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DECEMBER, 2016
Robot 2
Robot 3
X-Y View X-Z View
Robot 1
Fig. 8. The service regions of each robot in Group II at the position with the
maximum total dexterity.
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