Content uploaded by Liao Wu
Author content
All content in this area was uploaded by Liao Wu on Aug 15, 2021
Content may be subject to copyright.
A Static Model for a Stiffness-Adjustable Snake-Like Robot
Di Shun Huang, Jian Hu, Liuchunzi Guo, Yi Sun, and Liao Wu, Member, IEEE
Abstract— In minimally invasive surgery, miniaturisation and
in situ adjustable stiffness of robotic manipulators are desired
features. Previous research proposed a simple and effective
tendon-driven curve-joint manipulator design using a variable
neutral-line mechanism, which highly satisfies both criteria. A
kinematic model was developed for such a manipulator based on
the geometry of the structure. However, such a model assumes
that joint angles are all equal between disks without a rigorous
derivation, and fails if not all the shapes of the disks are
identical. Moreover, the model does not involve an analysis of
the tension of each tendon. This paper suggested a static model
for predicting the articulation of such a manipulator given the
applied tensions on driving tendons. It validates the assumption
of equally distributed joint angles and works for manipulators
with more general configurations of disks and tendons. It also
sets a foundation for further development of tension based
control and external force estimation. Simulations on Adams
were conducted to prove the correctness of the proposed model.
A video demonstrating the simulation results can be found via
https://youtu.be/MXhL1LGwLtw
I. INTRODUCTION
Snake-like robots have gained popularity for surgical ap-
plications as they become a highly promising solution to
the design of flexible instruments for minimally invasive
surgery. The special structure enables the distal segments
of the instrument to present a curved shape with varying
curvatures. This substantially enhances the maneuverability
within the human body and thus reduces the required sizes
and number of incisions, leading to less post-operative pain
and quicker recovery [1]–[6].
A traditional snake-like robot consists of a flexible back-
bone or multiple universal joints [7], whose stiffness is not
adjustable during operation. As a result, such a device cannot
adapt to changing requirements of the stiffness in a surgical
procedure: low stiffness for safe and compliant deployment
and high stiffness for high-force and precise operations [8].
To address this issue, different stiffness-adjustable mech-
anisms have been proposed. An initial and simple approach
was to have a set of tendons travelling across the manipulator,
making its stiffness controllable by adjusting tensions applied
to these tendons [9]. However, this induces tremendous
pressure on the backbone, making it likely subject to fracture.
Also, its structural integrity is difficult to maintain after
miniaturisation due to the presence of disproportionally thin
or tiny components, especially on a manipulator with pivot
joints [10]. Others included the use of advanced materials
D. Huang, J. Hu, L. Guo, and L. Wu are with the School of Mechanical
and Manufacturing Engineering, University of New South Wales, Sydney,
Australia. dr.liao.wu@ieee.org
Y. Sun is with the Sydney Institute for Robotics and Intelligent Systems
and Australian Centre for Field Robotics, The University of Sydney, Sydney,
Australia
(a) (b)
Fig. 1: (a) A 2-segment stiffness-adjustable manipulator, with
the exposed view of the intermediate disks on the left. Four
tendons are knotted at the 3rd disk’s top while the rest are
fixed at the distal end disk’s top. (b) The ith joint of a 4-
tendon segment.
such as SMA [11], [12] and PMA [13] to switch between
different states of the materials, which, however, requires
complex design and integration for effective bi-directional
transition between the states.
A tendon-driven curve-joint manipulator was proposed
which achieves adjustable stiffness while being simple in
design and fabrication [14], [15]. As illustrated in Fig. 1a,
the design features curved surfaces between adjacent disks
connected by groups of tendons. Due to the variation of a
neutral line during bending, the stiffness of the manipula-
tor can be adjusted through changing the tensions of the
tendons. The applied tensions are endured directly by the
disks instead of a backbone or pivot joints. Therefore, the
maximum stiffness is purely dictated by the compressive
strength and stiffness of the disks, allowing high tension
magnitudes. Meanwhile, it is also easier to be miniaturised
[16] compared to pivot-joint based manipulators, since there
are no small components required. These features make it
an appropriate driving mechanism for minimally invasive
surgical instruments.
A kinematic model between the bending angle and the
displacements of the tendons has been developed for such a
manipulator in [14]. While the results showed the effective-
ness of the model, it was purely established on the geometry
of the manipulator and an assumption that equal angles will
be distributed among the joints (an arc-like shape), without a
rigorous derivation. This assumption fails if not all the shapes
of the disks are identical, external loads exist, or gravity
is not negligible. Moreover, the model did not involve a
relationship between the bending angles of the joints and the
tensions applied to each tendon. Since stiffness adjustment
is a key feature of this manipulator, a model that allows
a direct mapping between the shape of the manipulator
and the tensions of the tendons are much desired. It will
facilitate a stiffness/compliance control algorithm based on
direct control of the tensions [17], [18], and also enable an
estimation of external forces based on the measurement of
the tensions.
This paper presents a static model for estimation of every
joint angle, and subsequently the pose of this manipulator,
at static equilibrium depending on the tensions applied to
the tendons. It verifies the equal joint angle hypothesis
in [14] and its applicability extends to multiple-segment
manipulators.
II. STATIC MODEL
In this section, a model is developed aiming to identify the
joint angles and the tip pose of a stiffness adjustable snake-
like manipulator given the tension applied to each tendon.
A. Manipulator Geometry
A single-segment 2-degree-of-freedom (2-DoF) manipula-
tor with four tendons is shown in Fig. 1b. A base disk is
fixed to the ground, and the intermediate disks, followed by
a distal end disk, are joined together by tendons travelling
through their respective tendon guides. Each tendon has one
end forming a knot at the distal end disk’s top and the other
end connected to an actuator.
For a manipulator with two or more segments (see Fig.
1a), in each segment, there is a separate set of tendons with
their ends knotted and placed at the slots of the segment’s
distal end disk so that the knots will not interfere with the
joint motion.
B. Assumptions
For the physical behaviour, it is assumed that no gravita-
tional force is acted on the model. The manipulator is always
either at static or quasi-static state. All disks are rigid bodies
and thus no deformation occurs on them. The tendons are
assumed slender bodies such that there are negligible shear
forces and torques throughout their bodies. Friction between
every contacting curved surface pair is always sufficient to
prevent slipping motion, while friction between tendons and
tendon guides are assumed negligible because it is subject to
their mechanical properties and geometries which are beyond
the scope of this paper.
In terms of its geometry, firstly, the contacting curved sur-
face between every two disks are perfectly circular. Secondly,
there is no clearance between tendon and tendon guides such
that the tendon sections within tendon guides are always
parallel to the corresponding disks’ central axis. This ensures
that all tendon segments’ orientations at the same joint are
always identical.
C. Reference Frame
Each disk’s reference frame is located halfway between
its top end and the bottom end along its center axis. The z-
axis is co-linear with the respect disk’s center axis, whereas
its x-axis is parallel to the disk’s bottom curved surface’s
rotational axis, except for the base disk whose reference
frame’s x-axis is parallel to its top surface’s rotational axis.
D. Evaluation Procedure
To determine the manipulator’s static pose, every joint
angle must be computed. This is achievable with a single-
disk model, whose free body diagram is shown in Fig. 2a
and 2b. Each consists of the disk and the tendon segments
within its tendon guides.
The tension magnitudes of the tendons are constant
throughout the manipulator due to the zero-friction assump-
tion and thus the corresponding force vectors are the function
of their respective joints’ angles. The contact geometry
between curved surfaces is a line parallel to the rotational
axis of the curved surfaces due to rigid body assumption. The
distributed reactions can be summarised into three orthogonal
point force reactions and two pure moment reactions which
are about the orthogonal axes perpendicular to the contacting
line, namely the y and z axes of the disk’s reference frame.
Initially, there are twelve unknowns for all intermediate
disk models, and six for the distal end disks, which is the
only deterministic model. Also, the six unknowns at the
top joint of an intermediate disk are directly derivable from
the six unknowns at the bottom joint of the disk above it.
Therefore, the distal end disk model is first evaluated so that
the disk proximal to it becomes deterministic. Such a process
is repeated until the state of the disk above the base disk is
computed. The pose of the manipulator’s tip can then be
confirmed.
E. Variable Definition
The ith disk has a center axis length Li, a bottom curved
surface radius ri, and a z-axis orientation θmi, with respect
to the global coordinate system when the manipulator is not
bent. For compliance, its top curved surface radius and z-
axis orientation with respect to the base disk must be ri+1
and θmi+1respectively, and thus its top curved surface z-axis
orientation with respect to its reference frame is θi+1,i=
(θmi+1−θmi). Its bottom joint angle is defined as φi.
The jth tendon’s z-axis orientation with respect to the base
disk is τmjand its perpendicular distance from every disk’s
center axis is dj. Therefore its orientation with respect to the
ith disk’s reference frame is τj,i= (τmj−θmi). The tension
applied to this tendon is Tj. All geometrical variables are
depicted in Fig. 3a.
F. Vector Definition
For computation, the following vectors are all with refer-
ence to the reference frame. The force vector applied at the
bottom curved surface by the jth tendon on the ith disk is:
f
f
ftb,i,j=
0
−Tjsin φi
2
−Tjcos φi
2
(1)
and its displacement vector from the disk center is:
r
r
rtb,i,j=
djcos(τj,i)
djsin(τj,i)
−qr2
i−(djsin(τj,i))2+ri−Li
2
.(2)
Similarly, the force vector applied at the top curved surface
by the jth tendon on the ith disk is:
f
f
ftt,i,j=
Tjsin φi+1
2sin(θi+1,i)
−Tjsin φi+1
2cos(θi+1,i)
Tjcos φi+1
2
(3)
and its displacement vector from the disk center is:
r
r
rtt,i,j=
djcos(τj,i)
djsin(τj,i)
qr2
i+1−(djsin(τj,i))2+Li
2−ri+1
.(4)
The defined point force reaction acts at the center of each
contacting line. Therefore, the bottom contact displacement
from the disk’s center is:
r
r
rcb,i=
0
−risin φi
2
−ricos φi
2+ri−Li
2
(5)
and the top contact displacement is:
r
r
rct,i=
ri+1sin φi+1
2sin(θi+1,i)
−ri+1sin φi+1
2cos(θi+1,i)
ri+1cos φi+1
2+Li
2−ri
.(6)
All displacement vectors are illustrated in Fig. 3b.
G. Constraint equations
Let f
f
fcb,iand f
f
fct,ibe the contact reaction force vectors, and
M
M
Mcb,iand M
M
Mct,ibe the contact reaction pure moment vectors,
at the bottom and top contacting surfaces, respectively. For
static equilibrium, one must satisfy:
∑F
F
F=0
0
0=
j
∑(f
f
ftb,i,j+f
f
ftt,i,j) + f
f
fcb,i+f
f
fct,i(7)
∑M
M
M=0
0
0=
j
∑(r
r
rtb,i,j×f
f
ftb,i,j+r
r
rtt,i,j×f
f
ftt,i,j)+
r
r
rcb,i×f
f
fcb,i+r
r
rct,i×f
f
fct,i+M
M
Mcb,i+M
M
Mct,i
(8)
Since the contacting surface is a line between two rigid
convex surfaces, the pure moment contact reaction about the
x-axis of the reference frame (M
M
Mcb,i)xis zero.
Finally, the relationship of contact reaction vectors acted
on a disk in its reference frame and that on its bottom disk’s
in its bottom disk’s reference frame is as follows:
f
f
fct,i=R
R
Rθi+1,iR
R
Rφi+1(
(
(−
−
−f
f
fcb,i+1)
)
)(9)
M
M
Mct,i=R
R
Rθi+1,iR
R
Rφi+1(
(
(−
−
−M
M
Mcb,i+1)
)
)(10)
where R
R
Rθi+1,iand R
R
Rφi+1are the rotational matrices of θi+1,i
and φi+1about z and x-axis, respectively. For the distal end
disk, f
f
fct,iand M
M
Mct,iare both zeros.
(a) The bottom of the ith disk.
(b) The top of the (i−1)th disk.
Fig. 2: Force vectors acted on the bottom of the ith disk and
the top of the (i−1)th disk corresponding to Fig. 1b.
H. Solving methods
Eq. (7) and (8) can be solved with a numerical approach
such as bisection method. However, due to the surface con-
tacts being convexly circular, these equations can be solved
directly. For every disk, given the top reaction force vector
f
f
fto p =∑j(f
f
ftt,i,j) + f
f
fct,iand Eq. (7), the bottom reaction
vector is:
f
f
fcb,i=−f
f
fto p −
j
∑f
f
ftb,i,j=
−(f
f
fto p)x
∑jTjsin φi
2−(f
f
fto p)y
∑jTjcos φi
2−(f
f
fto p)z
(11)
Substituting M
M
Mcb,ix=0 and Eq. (11) into Eq. (8), the first
row becomes:
αsin φi
2+βcos φi
2+δ=0 (12)
where:
α=ri(f
f
fto p)z+ ( Li
2−ri)
j
∑Tj+
j
∑((r
r
rtb,i,j)zTj)
β=−ri(f
f
fto p)y−
j
∑((r
r
rtb,i,j)yTj)
δ=( f
f
fto p)y(ri−Li
2)+
[
j
∑(r
r
rtt,i,j×f
f
ftt,i,j) + r
r
rct,i×f
f
fct,i+M
M
Mct,i]x
which can be directly solved to obtain φiif and only
if δ
√α2+β2∈[−1,1]. Additionally, the contacts between
disks must occur only at their curved surfaces, thus φi∈
[−arctan(ci
ri),arctan(ci
ri)] must be valid, where cidenotes the
ith disk’s cylindrical radius. The ith disk’s bottom contact
(a) (b)
Fig. 3: Definition of variables and vectors on the ith disk
(4-tendon).
(a) Variables Definition. For tendon guide orientation, only
the orientation of the second tendon is demonstrated and
those of the rest are omitted. Also, centers of curved surfaces
must be on center axis but not necessarily at the center.
(b) End positions of the displacements r
r
rtb,i,j,r
r
rtt,i,j,r
r
rcb,iand
r
r
rct,irepresented as dots. The starting position is at the disk
center.
Fig. 4: Planar contact between two disks (Top disk is semi-
translucent so that overlapping area is shown in light gray
color at the center).
reaction can then be deduced from Eq. (7) and Eq. (8) and
converted to the top contact reaction of its consecutively
proximal disk with Eq. (9) and Eq. (10). The same evaluation
process is repeated on every proximal disk to acquire the
bending angle at every joint of the manipulator, from which
the transformation matrix of the manipulator‘s tip can be
determined.
III. VALIDATION WITH SIMULATION
Simulations were performed on Adams View 2020 to
validate the correctness of the static model. Multiple disks are
initially stacked together with slight vertical overlap. On each
disk, tension vectors, corresponding to the components f
f
ftt
and f
f
ftb in Fig. 2, are located at the tendon guide ends’ cen-
ters and directed towards the centers of their paired tendon
guide ends to which the same tendons travel at the same joint,
respectively. Their magnitudes equal the tensions applied to
their corresponding tendons. Contact reaction components
on curved surfaces, namely f
f
fct ,M
M
Mct ,f
f
fcb, and M
M
Mcb in Fig. 2,
are computed by the built-in contact module in Adams View,
with high stiffness and frictional coefficient for rigid body
behaviour and purely rolling motion. Besides, the base disk
is fixed to the ground. Static simulation was performed to
ignore inertia. For each loading case, the applied tensions’
Fig. 5: Disk arrangements for simulation: (a) 1st manipulator
(b) 2nd and 3rd manipulators (c) 4th and 5th manipulators (d)
6th manipulator. Each arrow points to the joint’s top surface
at which a set of tendons’ ends are fixed.
magnitudes are all initially set to the average of their loads
and change towards their respective loads linearly at each
step for a smooth transition between static states because
joint angles increase with applied tension difference.
There are some limitations to the simulation. Firstly, the
contact module requires disks to overlap with their contacting
peers, which violates the rigid body pre-condition and the
contact is no longer linear (see Fig. 4). Therefore, (M
M
Mcb,i)x
at every joint becomes non-zero which is against the static
model constraint. Secondly, the geometry overlap may make
the tension vectors’ orientations inaccurate if the tendon
guide end pairs are too close (In Fig. 1b, the tendon guide
end pair of the 1st tendon at the ith joint coincides, and thus
their tension vectors’ orientations are undefined given the
aforementioned simulation setup). Therefore, all the tendon
guide ends defined in the simulation are displaced by a
fixed margin from their respective curved surfaces. Both
limitations induce deviation to the results.
Simulations were conducted on six configurations (see Fig.
5) under various loading cases. Configurations were selected
randomly to prove the model was applicable regardless of
their complexities. The 1st contains one segment with two
tendons, oriented at 90 and 270 degrees respectively. The
2nd also contains one segment but with four tendons evenly
distributed around the center axis starting from 0 degrees
and joint orientations alternate by 90 degrees, starting from
0 degrees. The 3rd has the identical joint orientation as the
2nd but there are three evenly distributed tendons. All tendons
travel throughout the manipulators and are fixed at the distal
end disk’s top.
The 4th and 5th ones have two segments each. Each
segment contains four 90-degree alternating joints. The most
proximal joints’ orientations of the bottom and the top
segments are 0 and 45 degrees respectively. The 4th one
has four evenly distributed tendons, where one of them
is oriented at 0 degrees, fixed at the 5th disk’s top and
another four evenly distributed tendons, where one of them
is orientated at 45 degrees, fixed at the distal end disk’s
top. The 5th one has three evenly distributed tendons for
each segment instead. The set fixed at the 5th disk’s top has
the starting orientation at 0 degrees, whereas the other set’s
Fig. 6: Simulation of the last case in Table I. (a) Manipula-
tor’s shape (b) Tension vectors, represented by red arrows,
applied at the 2nd disk’s bottom surface. (c) Tension vectors
applied at the 1st disk’s top surface.
TABLE I: Simulation results
No.1Applied tensions (N)2Max. joint angle error (degree) Displacement error (mm)3
11,1 0.22 2.01
2,1 0.57 2.99
2
1,1,1,1 0.08 0.41
1.5,1,1,1 1.32 6.5
2,2,1,1 1.4 7.95
2,2,2,1 0.25 1.12
3
1.5,1,1 1.27 5.49
1,1.5,1 0.51 2.11
1.5,1.5,1 1.9 7.87
1,1.5,1.5 0.37 1.46
41.5,1.5,1,1,1,1,1,1 2.03 5.92
2.5,2,1.5,1,1,2.5,2,1.5 2.18 9.73
51.5,1,1,1.5,1,1 0.67 0.72
2.5,2,1.5,1,2,2.5 2.58 4.99
6
1,1,1,1,1,1,1,1 0.32 1.55
1.5,1.5,1,1,1,1,1,1 2.16 8.07
2.5,2,1.5,1,1,2.5,2,1.5 2.25 11.2
1Referred to Fig. 5.
2Arranged in counterclockwise order starting at 0 degree
orientation of the base disk’s reference frame.
3Manipulator’s length is 108 mm for the 4th and 5th config-
urations and 120 mm for the rest.
starting orientation is 60 degrees.
The 6th one has three segments, each containing three 90-
degree alternating joints. The most proximal joints’ orienta-
tions of the bottom, the middle and the top segments are 0, 45
and 0 degrees, respectively. Four evenly distributed tendons,
where one of them is oriented at 0 degrees, are fixed at the
distal end disk’s top. Another four evenly distributed tendons,
where one of them is oriented at 45 degrees, are fixed at the
7th disk’s top.
Simulation results are presented in Table Iand the portrait
of the final case is displayed in Fig. 6as an example. For
equal tension cases, the first loading cases of the 1st, 2nd
and 6th configurations, the joint angles differ slightly from
that of the static model. Theoretically, the aforementioned
limitations have no effect on such deviation. It may be
induced by the numerical error within the built-in algorithm
and the disk geometries. Although the error increases with
the tension differences, the maximum displacement error is
11.2mm which is 9.33% of the manipulator length. There-
fore, in conclusion, the manipulator pose computed from the
simulation match that of the static model to a great extent.
IV. DISCUSSION
A. Equal joint angle distribution
According to Table II, the angles of all joints of a 1-DoF
segment (the 1st manipulator) and that of the alternating
joints of a 2-DoF segment (the rest of the manipulators)
are equal regardless of the number of segments between the
TABLE II: Joint Angles of Robots Under Their Respective
Last Loading Cases in Table I, Computed by Static Model
Joint
No. 123456789
1-26.93 -26.93 -26.93 -26.93 -26.93 -26.93 -26.93 -26.93 -26.93
2-10.36 0-10.36 0-10.36 0-10.36 0-10.36
30-9.06 0-9.06 0-9.06 0-9.06 0
46.34 7.87 6.34 7.87 5.11 -5.43 5.11 -5.43
55.56 12.69 5.56 12.69 15.96 7.66 15.96 7.66
66.34 7.87 6.34 9.89 1.1 9.89 5.27 15.71 5.27
Same values are highlighted with the same color on each row.
base disk and itself. Therefore, the assumption utilised in the
kinematic model proposed by [14] is valid, given that the the
shapes of the disks are identical, no external loads exist, and
gravity is negligible.
B. Relationship between disk length and joint angle
By further simplifying Eq. (12), the distal end disk’s bot-
tom joint angle is confirmed to be independent of its length.
Although this is false for intermediate disks, the dependency
can be considered negligible according to the results solved
on the 2-segment manipulator in both numerical and direct
methods with the disk length as the independent variable. The
results revealed that even though the joint angles difference
increases with the disk lengths’ difference, these are all
insignificant to the minimal joint angle 0.059rad. Such
property decoupling helps simplify the design process of the
manipulator. However, validation is required for manipulator
design finalisation as it is an empirical conclusion.
C. External loads
Tensions applied to tendons are the only external loads
considered in the model. However, if there exist other
external loads, such as gravity, they can be included in the
static model by appending them into Eq. (7) and (8). If every
external load vector remains constant, in the reference frame
of the disk on which the load is acted, regardless of that
disk’s orientation, then the loading case can be solved with
the direct method. Otherwise, it must be solved numerically.
D. Static stability
It is proven that there is always a physically possible
solution provided that all boundary conditions are satisfied.
Static stability has not been validated in the current model.
However, in real-world scenario, the friction between tendons
and disks are sufficient damping forces to stabilise the
manipulator’s orientation given that the tendons’ tensions are
large enough due to the proportionality between the tensions
and the frictions.
E. Model applicability to different configurations
The model applies to any configurations provided that the
following conditions are met. All the disks’ curved surfaces
must be circular and symmetrical with their contacting pairs,
about the tangential plane at their contact. Only convexly
rolling motion occurs between each contacting pair. All the
tendon segments at the same joint must have an identical
orientation about manipulator’s center axis in a neutral state.
Therefore, the model can be applied regardless of the disk
and tendon numbers.
The curved surface can even have any curvature, not
necessarily symmetrical to its counterpart, or can be inte-
grated with gears. However, the orientations of the tendon
segments belonging to the same joint may differ, and thus
the model may not be solvable with the direct method. It
has to be computed numerically on Eq. 7and 8with custom
displacement vectors.
F. Static model application
Although friction between tendon and tendon guides is
ignored in the proposed model, its estimation is still mean-
ingful as it is a rough average of final poses for the same
input tensions regardless of the initial poses due to the
symmetrical influence of backlash [14], [15]. It can be used
for designing such kinds of manipulators, or integrated into
feedback-based pose control systems [15], [19] for enhanced
reliability, or used for training or quick validation of the
outputs from machine learning based control systems [17].
V. CONCLUSION
This paper presents a static model to acquire the pose of a
tendon-driven curve-joint manipulator using variable neutral-
line mechanism if the applied tensions on the driving tendons
are known. It assumes that the disks are rigid, tendons are
slender bodies, and no friction between links and tendons.
The solution was proven unique and obtainable without using
numerical methods.
Simulations were performed to verify the static model.
Multiple cases with varying disk and tendon arrangements
as well as applied tensions were examined. Its maximum
displacement difference with that computed from the static
model is 9.33% of the manipulator length. The errors might
be caused by the geometry overlap between links for contact
simulation and offsets of the positions at which tensions
are applied to each link. Other sources of errors include
numerical errors in the convergence algorithm and the disks’
geometries. However, the manipulator’s articulation in sim-
ulation is mostly aligned with that of the static model.
The static model results revealed that the angles of all
joints in a 1-DoF segment and alternating joints in a 2-DoF
segment are equal, and this verified the kinematic model
developed in [14]. Also, the disk lengths were empirically
discovered to have negligible impact on the joint angles. Such
decoupling can be taken advantages of in manipulator design.
Moreover, the static model is applicable on manipulators
with modified geometries and with external forces or torques
acting on the manipulator body.
The static model assumes no friction between the tendons
and the tendon guides since this is subject to the material
properties and additional geometric features. However, it was
reported as a major cause for inaccurate estimation of the
manipulator orientation [15], [16], [19]. Therefore, it is still
insufficient to be a standalone model used in practice, but
can serve as the basis for the design of more advanced
control models and its outcome can be treated as a quickly
computable reference in the actual use cases.
REFERENCES
[1] A. L. Orekhov, C. Abah, and N. Simaan, “Snake-like robots for
minimally invasive, single-port, and intraluminal surgeries,” The En-
cyclopedia of Medical Robotics, p. 203–243, Aug 2018.
[2] J. Burgner-Kahrs, D. C. Rucker, and H. Choset, “Continuum robots
for medical applications: A survey,” IEEE Transactions on Robotics,
vol. 31, no. 6, pp. 1261–1280, 2015.
[3] L. Paul, T. Chant, R. Crawford, J. Roberts, and L. Wu, “Prototype
development of a hand-held steerable tool for hip arthroscopy,” in
2017 IEEE International Conference on Robotics and Biomimetics
(ROBIO). IEEE, 2017, pp. 989–995.
[4] L. Wu, S. Song, K. Wu, C. M. Lim, and H. Ren, “Development of a
compact continuum tubular robotic system for nasopharyngeal biopsy,”
Medical and Biological Engineering and Computing, vol. 55, no. 3,
pp. 403–417, 2017.
[5] Z. Li, L. Wu, H. Ren, and H. Yu, “Kinematic comparison of sur-
gical tendon-driven manipulators and concentric tube manipulators,”
Mechanism and Machine Theory, vol. 107, pp. 148–165, 2017.
[6] S. Angioni, A. Pontis, F. Sedda, T. Zampetoglou, V. Cela, L. Mereu,
and P. Litta, “Single-port versus conventional multiport access pro-
phylactic laparoscopic bilateral salpingo-oophorectomy in high-risk
patients for ovarian cancer: A comparison of surgical outcomes,”
OncoTargets and therapy, vol. 8, pp. 1575–80, 06 2015.
[7] S. Shaikh and C. Thompson, “Natural orifice translumenal surgery:
Flexible platform review,” World journal of gastrointestinal surgery,
vol. 2, pp. 210–6, 06 2010.
[8] A. Loeve, P. Breedveld, and J. Dankelman, “Scopes too flexible... and
too stiff,” IEEE pulse, vol. 1, no. 3, pp. 26–41, 2010.
[9] X. Dong, M. Raffles, S. Guzman, D. Axinte, and J. Kell, “Design
and analysis of a family of snake arm robots connected by compliant
joints,” Mechanism and Machine Theory, vol. 77, p. 73–91, 07 2014.
[10] D. Noonan, V. Vitiello, J. Shang, C. Payne, and G. Yang, “A modular,
mechatronic joint design for a flexible access platform for mis,” 2011
IEEE/RSJ International Conference on Intelligent Robots and Systems,
pp. 949–954, 2011.
[11] S. Jiang, B. Chen, F. Qi, Y. Cao, F. Ju, D. Bai, and Y. Wang, “A
variable-stiffness continuum manipulators by an sma-based sheath in
minimally invasive surgery,” The International Journal of Medical
Robotics and Computer Assisted Surgery, vol. 16, no. 2, p. e2081,
2020.
[12] Y. Zhong, R. Du, L. Wu, and H. Yu, “A novel articulated soft robot
capable of variable stiffness through bistable structure,” in 2020 IEEE
International Conference on Robotics and Automation (ICRA). IEEE,
2020, pp. 2939–2945.
[13] J. Wang, S. Wang, J. Li, X. Ren, and R. M. Briggs, “Development
of a novel robotic platform with controllable stiffness manipulation
arms for laparoendoscopic single-site surgery (less),” The International
Journal of Medical Robotics and Computer Assisted Surgery, vol. 14,
no. 1, p. e1838, 2018.
[14] Y.-J. Kim, S. Cheng, S. Kim, and K. Iagnemma, “A stiffness-adjustable
hyperredundant manipulator using a variable neutral-line mechanism
for minimally invasive surgery,” Robotics, IEEE Transactions on,
vol. 30, pp. 382–395, 04 2014.
[15] J. Lee, J. Kim, K.-K. Lee, S. Hyung, Y.-J. Kim, W. Kwon, K. Roh,
and J.-Y. Choi, “Modeling and control of robotic surgical platform for
single-port access surgery,” 09 2014, pp. 3489–3495.
[16] X. He, V. Geirt, P. Gehlbach, R. Taylor, and I. Iordachita, “Iris: In-
tegrated robotic intraocular snake,” Proceedings - IEEE International
Conference on Robotics and Automation, vol. 2015, pp. 1764–1769,
06 2015.
[17] D. Jakes, Z. Ge, and L. Wu, “Model-less active compliance for
continuum robots using recurrent neural networks,” in 2019 IEEE/RSJ
International Conference on Intelligent Robots and Systems (IROS),
2019, pp. 2167–2173.
[18] R. E. Goldman, A. Bajo, and N. Simaan, “Compliant motion control
for multisegment continuum robots with actuation force sensing.”
IEEE Trans. Robotics, vol. 30, no. 4, pp. 890–902, 2014.
[19] M. Wang, X. Dong, W. Ba, A. Mohammad, D. Axinte, and A. Norton,
“Design, modelling and validation of a novel extra slender continuum
robot for in-situ inspection and repair in aeroengine,” Robotics and
Computer-Integrated Manufacturing, vol. 67, 08 2020.
