Abstract and Figures

Continuum robots can traverse anatomical pathways to intervene in regions deep inside the human body. They are able to steer along 3D curves in confined spaces and dexterously handle tissues. Concentric tube robots (CTRs) are continuum robots that comprise a series of precurved elastic tubes that can be translated and rotated with respect to each other to control the shape of the robot and tip pose. CTRs are a rapidly maturing technology that has seen extensive research over the past decade. Today, they are being evaluated as tools for a variety of surgical applications, as they can offer precision and manipulability in tight workspaces. This review provides an exhaustive classification of research on CTRs based on their clinical applications and highlights approaches for modeling, control, design, and sensing. Competing approaches are critically presented, leading to a discussion of future directions to address the limitations of current research and its translation to clinical applications. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 5 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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From Theoretical Work to
Clinical Translation:
Progress in Concentric Tube
Zisos Mitros1,2, S.M.Hadi Sadati1,, Ross Henry1,
Lyndon Da Cruz2,3, and Christos Bergeles1
1Robotics and Vision in Medicine Lab, School of Biomedical Engineering &
Imaging Sciences, King’s College London, UK.;
2Wellcome/EPSRC Centre for Interventional and Surgical Sciences, University
College London, London, UK, W1W 7TY
3Moorfields Eye Hospital, London, UK, E1 4DG
corresponding email: smh_sadati@kcl.ac.uk
Xxxx. Xxx. Xxx. Xxx. YYYY. AA:1–25
https://doi.org/10.1146/((please add
article doi))
Copyright c
YYYY by Annual Reviews.
All rights reserved
Concentric Tube Robots, Continuum Robots, Surgical Robots
Continuum robots can traverse anatomical pathways to intervene at
regions deep inside the human body. They are able to steer along 3D
curves in confined spaces and dexterously handle tissues. Concentric
Tube Robots (CTRs) are continuum robots that comprise a series of
precurved elastic tubes that are translated and rotated with respect to
each other to control the shape of the robot and tip pose. CTRs are a
rapidly maturing technology that has seen extensive research over the
past decade. Nowadays CTRs are being evaluated as tools for a variety
of surgical applications as they can offer precision and manipulability in
tight workspaces. This manuscript delivers an exhaustive classification
of CTR research based on their clinical application, while also highlight-
ing the explored approaches on modelling, control, design, and sensing.
Competing approaches are critically presented, leading to a discussion
on future directions to address current research and research translation
Tube i
Tube 2
Tube 1
Tube 3
Figure 1
A representative CTR manipulator composed of three precurved NiTi tubes. Each tube can be
independently rotated and translated. Tubes are grasped at their respective proximal ends. The
actuation variables αi(t)and βi(t)denote the rotation, and translation of the i-th tube,
respectively. Variables uzi, and uxyi , denote the torsion, and bending of each tube, respectively.
Robot-assisted single port surgery is envisioned as the next step in minimally invasive
surgery (MIS) as it offers lower morbidity, improved cosmesis due to elimination of pe-
ripheral ports, reduced trauma, and shorter hospitalization time (1). To deliver optimal
therapies, especially when deep seated pathologies are targeted, single port surgery requires
instrumentation with increased dexterity and flexibility to overcome the challenges of con-
fined surgical workspace and lack of articulation.
Continuum robots show promise when anatomical pathways need to be traversed, as
they are able to reach regions that are inaccessible by conventional rigid robotic or manual
instruments, while requiring only a single incision/port. Concentric Tube Robots (CTRs),
also known as active cannulas, are continuum robots that possess a continuously flexible
backbone that comprises concentric precurved tubes made of super-elastic material, most
commonly nitinol (NiTi). The shape and tip pose of those miniature robotic manipulators
is controlled by the relative rotation and translation of each tube. Thus, CTRs are able
to steer without the need to exert force on tissue (2;3). Bending actuation arises due to
elastic interaction between the tubes. Since their actuation is only based on the flexing
of their own backbone, i.e., the robot’s outermost structure, and not on mechanisms like
tendon wires or pneumatic/hydraulic chambers, they are able to steer in hollow regions or
liquid filled cavities. CTRs can be very thin without sacrificing dexterity.
The first concentric tube style device was disclosed in a patent application filed in 1990s
(4) and was composed of a straight outer tube and a precurved NiTi wire. The device was
proposed for the localization of lesions within the body and, in particular, of non-palpable
lesions within the breast. Cuschieri et al. (5) highlighted for the first time the new di-
rections that concentric tube devices, made of precurved tubular NiTi components, can
open up in endoscopic surgeries. Few years later, Daum GmbH filed a patent application
for a deflectable needle assembly which included a telescoping cannula, a catheter and a
stylet (6). The catheter, made from NiTi, was curved at its distal end and axially ro-
2 Mitros et al.
               
 
 
 
- 1st CTR design
Furusho et al. (7)
- CTR Worldwide Recognition &
studies in The US
Sears et al. (9)
Webster et al. (8)
- Thorough Theoretical Studies
Dupont et al. (2)
Rucker et al. (15)
- Theory of External Loads
Rucker et al (69)
- Design Optimization Algorithms
Anor et al. (92), Bedell et al. (93)
- Sterilizable Design
Burgner-khars et al. (113)
- Visual Servoing
Li et al. (74)
- Learning Methods for CTRs
Bergeles et al. (88)
- Follow-The-Leader planning
Gilbert et al. (87)
- Hand-held Design
Wu et al. (111)
CTR Dynamics
-Till et al. (31)
-3D-Printed CTRs
Morimot et al. (127)
-In-vivo alive
Fagogenis et al. (67)
-Distributed Force Estimation
Aloi et al. (56)
Figure 2
Number of occurrence of keywords categories per year versus the number of publications. Selected
works in the research field of CTRs are shown.
tatable within the lumen of the cannula. The first motorized concentric tube system was
presented in (7) in 2005 and the first mechanics-based models were introduced by (8;9)
in 2006. Since then, energy minimization methods, Bernoulli-Euler beam theory, Cosserat
rod theory and, recently, data-driven learning of forward and inverse kinematics of a CTR
have all been explored as modelling approaches for effective robot control. Similar progress
has been achieved on the computational design of CTRs, incorporating task considerations
and patient-anatomy constraints. Eccentric arrangement of tubes is also being investi-
gated, while multi-arm CTRs with straight or flexible backbones have also appeared. The
former are already being considered for first-in-human evaluation and clinical translation,
(10). Startups leveraging this technology have started working on bringing systems into
the operating room (OR) (see Virtuoso Surgical (virtuososurgical.net) and EndoTheia, Inc.
(www.endotheia.com) ).
This review provides an extended literature survey on the aforementioned advancements
in the field of CTRs and a discussion on future opportunities to address current research
constraints. We critically discuss the robot prototypes that have been developed along
with their design principles. The theoretical contributions are identified, and a comparison
between the theoretical models and control theories is presented.
The scientific literature search was conducted by considering the flagship robotics con-
ferences e.g. ICRA, IROS, BioRob, RSS, journals e.g. IJRR, TRO, TMECH, TBME,
Frontiers in Robotics & AI, Annuals Reviews and previous review works (11;12;13). The
authors exhaustively went through each journal and conference dated back to the year that
the first paper related to CTRs, or the first issue of a journal, was published. At that stage,
the first search was performed using keywords: continuum robot, active cannula, CTR and
soft robot. Then, the collection of the papers that had been identified was further evaluated
with regards to its “forward” citations and “backwards” references. All papers were read by
the authors and categorized based on their focus. The main categories of keywords were:
application and design, analysis and modelling theory, experimental evaluation, and con-
trol, which are all echoed in the manuscript’s organisation. Figure 2shows the number of
occurrences of each category per year versus the number of publications. In the beginning,
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 3
biannual publication rise is observed probably due to the limited research groups working
on CTRs. Subsequently, a steady rise is depicted showing the spread of research. Moreover,
Fig. 2shows the increased number of publications on control of CTRs that progressively
takes place.
A CTR’s workspace is determined by the robot tubes’ precurvature, stiffness, and length.
The joint to task-space mapping for a CTR is complex, consisting of unstable equilibrium
points causing snapping (structural instability resulting in the robot’s rapid transitioning
from one configuration, i.e. equilibrium state, to another of lesser energy), and concentric
constraints resulting in non-commutativity of the robot input sequence (14;12). These
characteristics complicate the modeling, motion planning, and control of CTRs.
2.1. Modelling
Quasi-Static Models: Two of the most widely studied and referenced CTR mechanics
models are based on (i) curvature weighted superposition (15;16;2) and (ii) differential
(variable) curvature kinematics (17). Beam mechanics or Cosserat rod theories and Hooke’s
material law are utilized as the system conservational (system) and constitutional (material)
laws (mechanics) for both the aforementioned kinematic representations.
In weighted curvature superposition, the system kinematics (tubes’ local shape and
orientation) is based on weighted curvature superposition combined with torsional rigidity.
The curvature weights are the tubes’ bending and torsional structural stiffness values, which
form a simple to integrate Initial Value Problem (IVP) that is well-suited for real-time
numerical performance.
In differential curvature kinematics, the system kinematics is derived via a set of dif-
ferential equations for curvatures and torsion known as variable curvature kinematics in
continuum robotics. The system kinematics is usually combined with Cosserat rod theory
to derive the system governing equations (17). As a result, a Boundary Value Problem
(BVP) is formed with unknown boundary conditions at the base and tip of the robot back-
bone, which can be solved via numerical optimisation method, e.g. single shooting (17).
This method is favorable because of accuracy, generality, and robustness.
Other investigated models include: (iii) approximating the robot forward map via curve
(e.g. Truncated Fourier Series) fitting (18), (iv) forming a lookup table for the robot
input-output relation based on precollecting experimental or precomputing simulation data
collection, (v) simplifying the system BVP problem based on tube-base force sensing (19),
(vi) finite shell element modeling based on piecewise constant strain assumption (20;21),
(vii) Reduced-Order techniques based on curvature (22) and shape polynomial (23) ap-
proximation, and (viii) learning-based methods (24;25;26). The effect of external loads
are extensively studied based on Cosserat rod model (27;28). A summary of the techni-
cal aspects, implementation procedure, advantages and shortcomings of these methods are
presented in Table 1, where lis length, φis twist angle, κis curvature, Cis fitting curve
coefficient, Pis Cartesian position vector, ntis number of tubes, nsis number of overlap-
ping segments, neis number of finite elements, nψis number of shape functions in a fitting
method, npis polynomial order, and ij k are general numerators.
4 Mitros et al.
Table 1 Forward and inverse modeling techniques for CTRs.
Models Curvature Superposition Differential Curvature Approximate Forward Map Precomputed Lookup
(Piecewise Const. Curves) (Truncated Fourier Series) Table
Ref.s (9;29) (30;17;31) (18) (32)
States li, φi, κi, i = 1..nsli, κ(s), φi(s), i = 1..ntCi, i = 1..nψli, φi,Pt, i = 1..ns
[No.] [3ns] [] [nψ] [2nt+ 6]
Inv. Kin. Inv. Jacobian, Num. optimization Inv. Jacobian, Num. optimization Analytical Lookup table
or lookup table or lookup table
Dynamic NA (33) No No
IK Algoritm 1. Desired tip p osition Pt1. Form optimization problem 1. Forward kin. precomputation 1. Forward kin. precomputation
2. Piecewise CC fitting based on forward map 2. Fourier Series fitting 2. Forming look-up table
3. Inputs (li, φi) via inv. CC map 2. BCs initial guess for curvatures 3. Inv. kin. via the fitted mapping 3. Inv. kin. via table search
4. Mapping for external loads 3. Inv. Kin. via num. optimization
Error 4 & 1.9 mm (0.3 & 1.3%) 2-12.5 mm (1.5-3.9%) 0.03-4.2 mm (0.01-3.1%) 3-21 mm (17.5%)
(without & with load) 0.06 8.6
Pros. + Real-time + Inv. compliance map + Real-time + Fast inverse map
+ Inv. kin. map + Inv. kin. map + Inv. kin. & compliance maps + Inv. kin. & compliance maps
+ Simplest model + Exact formulation + Real-time adaptive implementation
+ Multiple solutions + Multiple solutions
Cons. - external load only - Num. inefficient - Pre-known external loading - No external loads
via separate mapping - Initial guess issues - Fixed lengths - No multiple solutions
- if High relative stiffness - Sever Convergence issues - Fitting convergence issues - Storage memory issue
- Only circular const. pre-curves - Hard to find multiple solutions - No multiple solutions
- Dataset richness limitations
Software (34) (33) NA (35)
Models Forward Integration by Discretization (finite shell or Reduced-Orde Kin. (Polunomial Deep learning &
Sensing Base Loads constant strain elements) Curve or Shape Approx.)) (Neural Network models)
Ref.s (36;37) (20;38;21) (22;23) (24;26)
States li, κ(s), φi(s), i = 1..ntli, κij, φij , i = 1..nt, j = 1..neCijk , i = 1..nt, j = 3or4, k = 0..npNN parameters
[No.] [] [3ntne] [(3or4) ×nt×np][No. of NN parameters]
Inv. Kin. Multi-layer Inv. Jacobian Inv. Jacobian, Optimization Neural Network
Dynamic No No Yes No
IK Algoritm 1. Cosserat ro d formulation 1. Discertized rod formulation 1. General form shape function 1. Large experimental dataset
2. Forward integration with 2. Forming an optimization problem 2. Substituting in governing eq. 2. NN training
known BCs at s= 0 3. Solve for system states (derive linear form) 3. Inv. kin. via the trained NN
3. Solve for Cij k
Error 1.5-4.7 mm (1-3%) 3.3 mm (1.3%, no load) 2.5 mm (1.3%) forward: 1.36-2.3 mm (0.6-1%), 1.1
2.5-4% (point & distributed load) inverse: 4 mm (1.7%), 8.3
Pros. + Real-time + Possible real-time implementation + Possible real-time implementation + Fast inverse map
+ Force observation + External loads + Inv. kin. & compliance maps + Inv. kin. map
+ Force compensation controller + Finite num. of states + Finite number of states + High accuracy
+ Multiple solutions + Multiple solutions
+ External loads
Cons. - Sensitivity to force sensor errors - Requires convergence guarantee - Complex derivation - Known loading condition
- Requires known loading condition - Computationally expensive - Estimation casues error - No multiple solutions
- Complex setup design - Complex inv. kin. formulation - Convergence guarantee needed - Dataset richness limitations
- no inv. kin. or compliance map
Software NA NA TMTDyn by (23) NA
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 5
Absolute (Euclidean distance of simulation predictions and robot tip position in ex-
periments) and relative (Absolute error value divided by the robot backbone curve length
expressed in percentage) error values are used for experimental verification of theoretical
studies. The state-of-the-art normalized error with respect to arc length for the modeling
tasks is approximately 1.52% (1.54.7mm) (27;39). The modeling and tracking error
can reach values as small as 0.02 mm for a carefully optimized tube parameters and sim-
ple trajectories (40). Adaptive frameworks, e.g. truncated Fourier series shape estimation
(3mm, 1.2% error) (41), and Kalman-filter-based model parameter estimation (2mm, 0.8%
error) (42) can improve the accuracy of the physics-based models. Medical applications of
CTRs usually call for positioning accuracy of 12mm (41), implying that models can per-
form on par with the clinical requirements. It must be noted, however, that most models
and controllers perform their best in unloaded conditions.
Elastic Stability & Snapping Motion: Relative rotation of the tubes results in
the accumulation of torsional energy up to an unstable point where this torsional energy
overcomes the bending energy. At these unstable points, the system’s torsional energy is
released causing the tubes’ rapid motion, known as snapping. Such sudden high energy
motion is usually undesirable in medical interventions and should be avoided by structural
design, motion control, or planning (43). Mathematically, this instability is associated
with a bifurcation point in the tube tip twist angle when plotted versus the base input
rotation angle. This is equivalent to the appearance of two stable attractors (minimas) in
the deformation energy plot versus the tubes’ base rotation angle (16). The system static
formulation has multiple solutions (equilibrium points) at this instance, known as a system
with cardinality greater than 1(44).
The system’s linearized governing equations can be used to analyze the system local
(arbitrary tube number) and global (two-tube systems) stability (45;44). Alternatively,
the relation between the tubes’ distal and proximal rotation angle (46), investigating the
momentum free condition at the distal tube ends (47;48), the bifurcation and elastic stabil-
ity theory (45;49), and employed optimal control theories based on second time-derivative
of elastic energy function (50) can be used. More recently, local stability of CTRs with
general precurved shapes, e.g. helical, were investigated as well (44).
Alternatively, snapping’s sudden energy release can be utilized to perform high energy
tasks such as driving a suturing needle through a tissue. Such efforts significantly benefit
from dynamic models able to capture the snapping transient dynamics (51;33;23).
Hysteresis Due to Tube Friction & Clearance: CTRs’ shape not only depends
on the actuation input values but also on their time history due to the tubes’ friction and
clearance. These are the main sources of hysteresis in the system. Early investigation based
on simple mechanical models by (52) suggested that the dominant effect of the frictional
force is associated with concentrated bending moment at the tube ends. More recently,
(53) reported the hysteresis due to distributed torsional friction as the main source of CTR
modeling inaccuracy compared to the tube clearances.
Dynamic Modeling: Existing quasi-static models evaluated independently or as part
of inverse kinematics approaches cannot capture the system transient dynamics, such as
vibration and overshoot due to snapping, high bandwidth maneuvers when the system
inertial forces matter (e.g. for a hyper-elastic structure robot (54) or due to sudden exertion
or release of external loads (33)). Furthermore, extra modeling layers are needed to capture
system hysteresis and Coulomb friction (52). Till et al. (33) presented the first dynamical
model for CTRs by extending their real-time solver for dynamical modeling of continuum
6 Mitros et al.
robots based on Differential Curvature kinematics and Cosserat rod mechanics (27;39).
Their model could capture a CTR’s transient vibrations after snapping and environmental
contact release. However, the employed Cosserat rod-based methods require an infinite
number of states and are not suitable for nonlinear control and observation design. Recently,
approaches based on modelling with reduced number of states attempt to address these
challenges (22;23).
2.2. State Estimation & Observation
Shape Estimation: Effective control usually requires real-time shape information of the
robot complex mechanics especially when external loads are present. Curve fitting tech-
niques, e.g. via Bezier curves (1.38 mm, 1.1% tip error) (55), and truncated Fourier series
(3mm, 1.2%) (41), trigonometric or data-driven learning of the basis function (3.7mm)
(56) are employed to observe the robot’s overall shape based on visual (57) or limited sensor
Force Observation: Safe intervention is not possible without force observation and
controlled force measurements. Tube deflections are used to formulate an inverse mechan-
ical problem based on Cosserat rod (58) and cantilever beam (59) models to estimate the
tubes’ distal forces in real-time via numerical optimisation and inverse Jacobian formulation
Work in (60) used sensor fusion based on recordings from a tip camera and a base-fixed
force sensor, while (61) employed Deep Direct Cascade Learning frameworks and showcased
2.1% tip force magnitude estimation error.
Distributed force estimation is closer to a clinical scenario. Aloi et al. (58) modelled
the point and distributed loads based on fitting Fourier series curves and Dirac Delta func-
tions respectively to comply with the observed backbone shape while satisfying the system
mechanics. They reported 6.7% force magnitude and 2% force location error values with
near zero overall shape estimation error.
2.3. Control
Visual Servoing: Model-free visual servoing techniques (known as eye-in-hand method)
based on recordings from a robot tip camera are the direct solutions for control of CTRs
(62;63;64;65). As a most notable example that included in vivo deployment, (60) utilized
this method for autonomous blood vessel navigation by following the arterial wall.
Sampling-Based Optimization & Precomputing Lookup Tables: Sampling-
based path planning methods via numerical optimization are widely investigated in the
literature (30;66;67). Obstacle avoidance, global goal convergence, avoiding unstable con-
figurations, spatial and temporal stability can be guaranteed by introducing required con-
straints in well-defined optimization problems alongside standard planning methods such as
Rapidly Exploring Random Trees (RRT) (68;46;69) and graph search (70). Alternatively,
active constraints are employed for trajectory tracking where a sliding-mode-like controller
brings the robot tip trajectory toward a safe desired trajectory, away from obstacles and
unstable configurations (35).
Inverse Kinematics & Control: Table 1presents the different inverse kinematic
solutions developed to control a CTR. Unconstrained optimization and inverse Jacobian
formulation are the most extensively investigated methods. A notable alternative is analyt-
ical solution when forward kinematics is approximated by Fourier series, which is favorable
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 7
for its simplicity, numerical performance, accuracy for the trained dataset. This, however,
comes at the cost of model generality and robustness (18). Alternatively, learning-based
controllers are proposed based on experimental estimation of the system inverse Jacobian
(71;72;65). Other research focuses on control of multi-arm configurations (73;74) and
practical issues with the system automation such as reproducible robot calibration (75).
The tip tracking accuracy of Jacobian-based and damped least-square curve fitting
control methods are around 3.2% and 2.5% respectively. The state of the art multi-layer
closed-loop control architectures can achieve errors of 0.9% (37) and 0.5% (76). Investigating
dexterity (77) and force-velocity manipulability (78) enables real-time planning (in an active
constraint and redundant task framework) and control (based on a Model Predictive closed-
loop Controller) of a CTR with tip errors of as small as 0.30.5% (0.50.8mm) (34).
Whole-Body & Follow-The-Leader Control: Follow-the-leader method has been
the main framework for CTR automation and whole-body control in medical applications
(79;80;81). It has been shown that CTRs cannot perfectly perform follow-the-leader tasks
except for paths consisting of constant curvature or helical curves and planar arrangement
of the tubes with relative high stiffness and carefully chosen tube precurvatures and lengths
(82;40). Gilbert et al. (82) proposed a measure for evaluating approximate follow-the-
leader performance showing larger errors for longer tubes and larger difference between the
tube relative rotation. More recently, (79) has shown that helical precurved tubes with
exponentially varying curvature value can be a candidate for follow-the-leader control.
Compliance Matrix & Stiffness Control: With force control in place, calcula-
tion and tuning of robot compliance matrix in task space can lead to stiffness tuning and
impedance control. Jacobian based methods based on derivative term propagation (31) and
multi-step mapping (83) are introduced to calculate and design force controllers based on
a CTR tip compliance matrix.
Rucker et al. (31) presented a calculation framework for Jacobian and compliance
matrices calculation along a CTR backbone by considering the necessary partial derivatives
in a Cosserat rod-based model. Work in (83) proposed a two step transformation map.
The CTR’s unloaded deformation based on curve superposition was followed by a map for
the tube deformation due to external loads based on beam theory. They achieved force
estimation, stability during environmental contact, and steady state dynamic performance
via the inverse Jacobian method. A formal stability analysis of their proposed controller
is yet to be investigated. Granna et al. (84) extended a similar framework for distal force
sensing to a controller for trajectory tracking under external load.
2.4. Planning & Automation
Planning deals with sequential control actions to provide reliable and repeatable safe robot
motions regardless of the operator experience and fatigue. Stability, goal convergence, and
task specification satisfaction such as impedance control, and obstacle avoidance are the
main objectives of a planning paradigm in medical scenarios.
Multi-Objective Path Planning & Optimization: Multi-task (74;85) or multi-
level (34;86) planning paradigms are presented to satisfy multiple control objectives such
as task specific planning, trajectory tracking, stability guarantee, self-collision prevention,
obstacle avoidance, and even robot structural design optimization.
Whole-Shape Planning: Whole-shape planning is needed for obstacle avoidance dur-
ing intraluminal navigation or self-collision avoidance of multi-arm CTRs (74). Zhang et al.
8 Mitros et al.
(87) showed that a RRT-smooth method based on the CTR kinematics can generate path
trees that are compliant with whole CTR shapes. Alternatively, precomputing a lookup ta-
ble of the configuration map is an effective method for selecting collision-free configurations
for a desired robot tip position (3mm, 1% tip position error) (88;35). Integration of further
tip correction steps reduces the error to 0.021 mm (0.07%), making this combination the
most accurate controller reported in the literature (88).
2.5. Learning-Based Methods: A Surging Paradigm in CTR Research
Advancements in neural networks and improved methods for precise data collection have
led to the exploration of Machine Learning (ML) as a alternative for kinematics modelling.
The most common architecture employed is the multilayer perceptron, (MLP), with a
varying degree of hidden layers, inputs, and outputs. In the forward kinematics case, the
input values can be tube rotations and extensions, or actuator inputs directly. Then, the
outputs explored are tip 3D position or full tip pose. In inverse kinematics, the inputs and
outputs are swapped.
One of the first papers that explored neural network based kinematics and inverse
kinematics in CTR was (89). The dataset comprised tube rotation/translation and tip
pose, generated by the mechanics model implementation of (80). The paper considered
MLP with a single hidden layer for forward kinematics, and two hidden layers for inverse
kinematics. Mean square error at the tip or robot configuration inputs was used as the loss
function that guided training. Data-driven inverse kinematics modelling throughout the
full robot workspace was showcased as a challenging problem to overcome in a repeatable
way. No experimental evaluation was carried out.
Grassmann et al. (25) also implemented a forward and inverse kinematics learning
model based on MLP networks with one, and two hidden layers, respectively. The dataset
was generated through software implementation of (39). The forward kinematics network
was able to determine the tip pose with 1% error w.r.t the length of the system. The
network responsible for inverse kinematics was trained and evaluated in a limited region
of the workspace only, and led to 2.8% error. The paper did not contain evaluation on a
robotic system, however.
Given the challenge in capturing inverse kinematics using MLPs, (90) explored Deep
Reinforcement Learning (DRL). DRL incorporates “agents” that dynamically learn by ad-
justing their actions to maximise a reward function, without requiring a full training dataset
to be available a priori. (90) report an error of 0.33 mm while following a circle pattern.
The effect of different agents was evaluated in (91). Newer work considered the effect of
curriculum in DRL for CTRs. As with the previously mentioned works, DRL in CTR has
still not been experimentally evaluated.
Notably, experimental evaluation was carried out in (26). An MLP with reduced recep-
tive fields was trained given segmentations of CTR shapes that were approximated using a
polynomial basis function. The MLP predicted the coefficients of the basis function, lead-
ing to full shape reconstruction. Grid search in hyperparameter space was performed to
fine-tune the MLP. The authors also reported improvement in error when using a combi-
nation of recorded and simulated datasets over purely simulated datasets. The conclude
that pre-training with simulated data followed by re-training with experimental data leads
to the best performance.
Other research considers the most appropriate presentation of network inputs to train
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 9
forward kinematics networks. The topic is thoroughly investigated in (25), in direct relation
to CTRs. Grassmann et al. (92) also investigated the effect of input data normalization.
Significant performance improvements was showcased when simulated data were used, but
this was not the case for experimental data. This may be attributed to the different order
of magnitude of points (500K vs 100K).
Recent work also explores learning of force application. (61) compared MLP, Extreme
Learning Machine (ELM) and Deep Direct Cascade Learning (DDCL) to estimate contact
forces. DDCL was found to be most effective with an accuracy of 3.8% error when a maximal
force of 2.29 N is considered.
2.6. Limitations & Future Directions
Available modeling and control methods can achieve tip positioning errors as small as 0.3mm
(0.5%) (34). It is hard, however, to compare the reported error values in the literature
given the difference in the stability of the experimental systems, the limited information
provided regarding the workspace coverage, and the difference in complexity of test cases.
Introducing metrics for comparing different setups’ stability, and results’ workspace coverage
and complexity help with justifying the reported achievements in the literature.
Despite many recent advances in modeling complementary phenomena such as hysteresis
(53), stability (44), and manipulability (78), a unified framework capturing all the necessary
elements of modeling and control of a CTR such as dynamics, hysteresis effects, snapping,
compliance analysis, force, position, and shape control in a numerically efficient way is yet
to be developed. In a similar vein, no combined learning of force estimation and shape
estimation exists, while it is an open research problem how networks trained on a specific
CTR robot can extrapolate to other CTRs, even if the same number of tubes is present.
Perhaps generalization beyond the observed data and incorporation of the non-linear
mechanics effects can be addressed by integrating data-driven approaches with physics-
based methods to achieve robustness to disturbances that are not present in the data.
Specific surgical tasks and the anatomical environment impose constraints on the design of
a CTR end-effector. Design methodologies and optimization frameworks have been devel-
oped to tailor the workspace, dexterity and task specificity of CTRs prior to an intervention.
Computational design methods aim to maximise performance metrics, such as manipulabil-
ity, stiffness, and stability, by fine tuning characteristics, such as tube diameters, lengths,
and curvatures through constrained optimisation. Constraints may include the requirement
to avoid contact with the anatomy, limit the approach angle of the robot to vessels, or push
for a reduced overall length and curvature. The state-of-the-art on computational CTR
design methods is summarised in Table 2.
Early research was presented in (93;94). In (93), a systematic approach to optimize
the design of CTRs for neurosurgical procedures with a focus on endoscopic choroid plexus
ablation was provided. This method for the first time identified the need for either fixed-
curvature versus variable-curvature sections, while in (94) the notion of robot navigation and
manipulation sections were presented. In both works, Global Pattern Search Optimization
determined the length and curvature of each tube to reach multiple targets while avoiding
contact with brain ventricles and heart wall, and minimizing tubes’ length and curvature.
10 Mitros et al.
An unconstrained nonlinear simplex search method in a skull base surgery scenario for
the minimization of the unreachable points was presented in (95), where the authors intro-
duced the concept of volume-coverage in robot design instead of specific point reachability.
A torsionally rigid robot kinematic model was considered. An optimal cannula tube design
was acquired when the end-effector’s tip was within the desired working volume. In (96), the
authors provided an in-depth discussion on the idea of volume-based design. They advanced
their previous work by using a mechanics-based model which include torsional compliance
and incorporated workspace constraints related to the robot’s entry path in approaching
the surgical site. Moreover, the authors introduced a new volume-based optimization met-
ric that penalized voids in the robot’s workspace. In similar vein, (97) added the number
of tubes, apart from the curvature, in the design process. The authors employed a brute
force and greedy algorithm to maximize workspace coverage for intracerebral hemorrhage
evacuation. A generalization of the previous works was presented in (85). The authors pro-
posed a novel computer-assisted design process which decomposes the problem into task-
and robot-specific design optimizations. The method was based on a multi-objective par-
ticle swarm optimization algorithm with variable length. The authors used the scenario
of laser-induced thermotherapy in the brain to validate their method. The robot-specific
design process determined the tube curvature and length of the CTR end-effector as well
as different configurations. The proposed algorithm was evaluated on real patient datasets.
Inspired from path-planning, (98) employed a rapid exploring random tree (RRT) to
acquire CTR designs to reach multiple sites within the bronchial tubes while avoiding the
anatomy. The method included for the first time the notion of design coherence, which is
based on the observation that collision-free configuration spaces of robots of similar design
are also similar. It was the first work to incorporate mechanics models with torsional
Bergeles et al. (80) introduced the elastic stability of CTRs in the design optimisation
by including heuristics that maximize robot stability. Moreover, the effect of section type
(variable or fixed curvature) on the boundaries of the workspace was discussed in detail and
affected the design methodology and optimization method. The authors used Nelder-Mead
Downhill Simplex as an optimisation algorithm, and explored the scenarios of hydrocephalus
treatment and beating heart surgery in simulation.
The work of (103) introduced the number of tubes as an additional optimization variable,
along with tubes’ curvature. They introduced geometry-based kinematics in CTR design,
which significantly reduced the computational time. The geometry-based method estimated
continuous circular curves while intermediate nodes, derived from the desired trajectory,
determined the number and type of subsections that the final design of the CTR comprised.
It should be noted that all the works mentioned above on the design of a CTR are
based on optimization variables and functions that the designer has decided to implement
in their method with little-to-no surgeon input to the final design. In some works, e.g.,
(93;46;85), a surgeon can select the initial entry point/vector and constraints for the
initial configuration, but their vast knowledge of the patient anatomy and procedure is not
fully leveraged. Morimoto et al. (104) created a surgeon design interface to design a CTR
for a specific patient and procedure so to enable the surgeon to give more input into the
design. The intuitive interface allowed the user to see the anatomical model of interest in
3D and initialize a CTR design by setting a number of via points. The user could alter
individual tube parameters until the desired configuration was acquired. Moreover, the
interface allowed the user to explore the environment and simulate CTR’s motion through
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 11
Table 2 Computational CTR Design
Name Method Obj. Function Opt. Variables
(93) Global Pattern Search 1) Min(L & κ) L & κ
(GPS) Optimization Method 2) Reach multiple targets
3) Obstacle Avoidance
(94) Global Pattern 1) Min(L & κ) L & κ
Search 2) Obstacle Avoidance
3) Reach all targets
(95) Unconstrained Nonlinear Min(unreachable points) L & κ
Simplex Search Method
(98) Rapidly Exploring 1) Reach multiple targets L & κ
Random Tree (RRT) 2) Obstacles avoidance
(96) Nelder-Mead Max(Workspace Coverage) L & κ
Simplex Algorithm
(48) Study of the Monotony Max(Stability) Precurvature funtion
(99) Adaptive Simulated Max(Workspace) L & κ
Annealing (ASA) & RRT
(80) Nelder-Mead 1) Anatomical Constraints Number of Tubes
Downhill Simplex Method 2) Stability 3) Workspace Bending Stiffness & κ
(100) Pareto Grid Reachability L & κ
Searching Method Elastic Stability
(101) fminsearch Max(Workspace Coverage) -
(97) Brute Force Max(Workspace Coverage) Number & Type
& Greedy Algorithms of Aspiration Tube
(102) Steepest Descent Max(Stability) Combined precurvature
(73) Particle Swarm Max(Collaborative L & κ
Optimization Configurations)
(85) Multi-Objective Particle 1) Max(Tumor’s Coverage) Ablation objects
Swarm Optimization 2) Min(ablation overlap) L & κ
(103) Mesh Adaptive 1) Min(Number of tub es) Number of Tubes
Direct Search 2) Max(Distance from organs) & κ
3) Follow Tra jectory
the body.
Design algorithms are not the only way to acquire optimal CTR designs. To minimize
the effect of instability, methods for anisotropic patterning of tubes has been studied. Sev-
eral works and experimental results have shown the promising employment of patterned
tubes to achieve higher curvatures while eliminating the problem of snapping. In (105),
the use of a cellular tube to minimize the ratio of bending-to-stiffness ration (BTSR) was
explored. Simulations using finite element analysis derived the optimal design of the cell
geometry via trial-and-error. Experiments using the derived optimal design showed that a
patterned tube can exhibit a smooth rotation without snap-through motion. Research in
(106) improved previous work by building a lumped analytical model and examining it with
finite-element analysis (FEA) and providing an in-depth discussion on patterning of NiTi
tubes. The developed experimental system shows that the tubes’ patterning can eliminate
snapping and decrease the BTSR ratio. Similarly, (107) studied the creation of nonuniform
pattern on coaxial tubes to enhance stiffness of a CTR via a continuously variable stiffness
mechanism. The stiffness change was validated via analytical modeling, FEM simulation
studies, and experimental results all of which showed an increase in stiffness. Finally, (108)
was the first work to employ topology optimization methods to acquire the optimal design
of patterning so to decrease the BTSR and resolve the snapping problem. The developed
method was validated through FEA as well as experimental testing.
Remarks & Limitations: Methods for computing optimal CTR designs to reach
specified positions have been derived by employing different optimization algorithms and
taking into account different design variables. Also, topology optimization methods and
FEA have been used to derive optimal designs with enhanced stability and stiffness. How-
ever, a unified framework taking into account all possible design variables and requirements
has not yet been released neither deployed in a real case scenario.
Moreover, new design variables can be included in future optimization methods. Design
variables can include metrics used in control e.g force/ velocity manipulability (109) or
characteristics such as triangulation (73) that describe cooperation in the case of multi-arm
systems. These metrics can also be employed as a unified comparison metric despite the
12 Mitros et al.
(g) (i)
(d) (e)
Figure 3
Concentric tube robots for various medical applications: a) Endonasal Surgery ((110)) b) Prostate
Surgery ((111)), c) Neurosurgical MIS ((103)), d) Intracardiac Surgery ((112)), e) Light Pipe for
Intraocular Procedures ((113)), f) Deep Orbital Interventions ((114)), g) Intracerebral
Hemorrhage Evacuation ((115)), h) Endonasal Tumor Removal ((116)), i) Partial Nephrectomy
((117)) and j) Handheld CTR for MIS ((118)).
fact that the community unofficially uses the error per unit length to compare ones’ work
to the state of the art. End-effector’s stiffness and anisotropy is a possible candidate for
future design optimization algorithms as there are limitations on the diameters of tubes
that can be manufactured. In addition, tissue properties should really be taken thoroughly
into consideration. Finally, a general analytical model for optimal patterning designs has
yet to be derived.
In this section, concentric tube robot prototypes are organised according to their surgical
application. The singled-out prototypes have been evaluated on realistic phantoms and/or
cadavers, or show innovative design characteristics. Figure 3provides a pictorial overview
of some of the evaluated systems.
Brain & Skull Base Surgery: Skull base surgery takes place near locations where neu-
rovascular structures enter and exit the brain. It is prescribed for a wide variety of neuro-
logical disorders, such as intracerebral hemorrhage (ICH), brain tumors, and epilepsy. Rigid
instruments limit the effectiveness of procedures as they must follow straight trajectories
resulting in increased danger to harm critical tissue and structures. When regions deep
inside the brain must be reached, conventional approaches can result in heavy trauma to
healthy brain tissues. CTRs promise to dexterously access regions within the brain and
skull base and deliver therapies to deep seated pathologies.
Burgner et al. (115) introduced a sterilisable and biocompatible robot with 3Degrees of
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 13
Freedom (DoFs) for intracerebral hemorrhage evacuation shown in Fig. 3(g). The prototype
was the first reusable, sterilizable, and operating room ready actuation unit for CTRs. All
its components were autoclavable and biocompatible, while the motors could be bagged
to ensure the sterility of the system. The robot comprised 2tubes, with the outer one
being straight and the inner one having a curved distal end. The inner aspiration tube was
interchangeable during the application. The motors could be attached or detached from the
transmission through Oldham couplings. Lead screws were used for the translation of the
tubes while the rotation of the inner aspiration tube was achieved via a square shaft that
interfaced with a gear train. The authors validated the concept by performing experiments
using a gelatin phantom. The phantom was placed in an acrylic box and was made from
10% by weight Knox gelatin (Kraft Foods Global, Inc., USA), while clots were made from
Jell-O gelatin.
Burgner et al. (116) developed a telerobotic system for endonasal skull base surgery.
The system comprised two concentric tube arms, made from NiTi, holding a ring curette and
a gripper [see Fig. 3(h)]. A straight manually operated endoscope was used for visualization.
The robotic system also comprised two 6DoF input devices and an Electro-Magnetic (EM)
tracking system. The translation of the tubes was based on the use of a worm gear which
rotates a nut that rides on a stationary lead screw. The worm gear driving the rotation,
rotated a spring collet which grasped the base of its respective tube. The robot and the
clinical concept were evaluated by performing a mock-up surgery on a human cadaver
head. The two manipulators entered the nasal passage of the cadaver through a single
nostril to show that they can successfully reach the pituitary gland. An updated system
[see Fig. 3(a)] included three arms to satisfy real-world surgical workflow requirements, see
(110). Each arm was a single interchangeable tube cassette, which was mounted on any of
the four module carriers of a base and was locked by a large handle. Motors were placed
outside of the module carriers to reinforce sterility. Tube translation was achieved via lead
screws, while spur gears and square shafts were used for the rotation. The tool module was
made of autoclavable and biocompatible materials. Visualization was achieved by a Karl
Storz EndoCAMeleon rod lens endoscope with adjustable lens direction. Four additional
motors located behind each module carrier translated the modules relative to the robot
base, enabling full system insertion or retraction.
Eye & Deep Orbital Interventions: CTRs offer compelling new solutions to chal-
lenging intraocular and orbit surgery, which require dexterous manoeuvres of sub-millimetre
surgical tools. The forces that relevant tissues can withstand without damage are minuscule,
while the constrained workspace introduces further complications.
The first use of CTRs for vitreoretinal surgery was proposed by (119;120). The au-
thors proposed a 16-DOF hybrid robotic system for applications that require fine dexterous
manipulation such as Internal Limiting Membrane pealing and treatment of severe retinal
detachments. The intraocular part of the robot was a 2 DoF CTR which comprised a
precurved NiTi tube that was extended from a straight cannula. The parallel part of the
robot provided global precise positing of the eye and the surgical instrument. Later on,
(121), and (113) developed one-arm CTRs with 4, and 3DoFs, respectively. The systems
were tested on custom-made phantoms while the robot developed by (113) was also eval-
uated on porcine eyes. These were the first systems wherein the concept of miniaturizing
the actuation design was considered. The system of (121) measured just 66mm x 52mm
x29.5mm, with a linear travel range of 30mm corresponding to the eye’s diameter. The
robot comprised two NiTi tubes with the diameter of the outer one being less that 23-gauge.
14 Mitros et al.
The inner tube housed a gripper, which comprised steel forceps with a diameter of 300µm
welded with a piece of 27-gauge stainless steel tube.
The prototype of (113), shown in Fig. 3(e), was 40mm x 40mm x 210mm. The outer
tube was in the 20-gauge range, while the inner tube, with a bending radius of 30mm, was
less than 23-gauge with inner diameter sufficient to house a 25-gauge light pipe. Hollow
shaft motors eliminated the need for gears or lead screws and avoided backlash. They were
controlled by a custom made joystick and buttons on the top side of the robot. The robot
was very light, with a total weight of 0.496kg, to enable handheld operation, being the first
that was proposed to be used in this fashion.
Mitros et al. (114) reported a multi-arm CTR system, shown in Fig. 3(f), for deep
orbital interventions with a focus on Optic Nerve Sheath Fenestration (ONSF). The robot
consisted of 3arms offering 12 DoFs in total. The CTR arms accessed the eye orbit and the
optic nerve by navigating periocularly, following the eye surface, to reach the position where
they collaboratively performed the intervention. One arm held a 1.2mm chip-on-tip camera
(Enable Inc., United States), while the others held a gripper and a cannula. Illumination
was achieved through optical fibres housed within the camera body. The robotic prototype
was tested on a bespoke phantom of the orbit manufactured based on patient data, as well
as on porcine eyes. It was the first multi-arm system designed for deep orbital interventions
and operations on the optic nerve.
In (10), the robotic system first presented in (110) for pituitary tumor removal surgery,
was used for the removal of tumours growing behind the eyes in the orbital apex region. The
authors integrated a sterile draping concept for non-sterile components, and a cartridge-
based tool change approach that eased instruments swaps. The robot was evaluated on a
silicone eye phantom housed in a portion of a skull 3D-printed in plastic. Two otolaryn-
cologists performed 10 orbital tumor resections in total while at the same time minimizing
unnecessary fat removal. The phantom of the tumor was made from silicone, and ballistics
gel was cast to simulate orbital fat and connective tissues. The authors developed a mod-
ular solution for the electrical hardware comprising of multiple units with microcontrollers
rather than a traditional computer, reducing in this way the cost and size. The distribution
of the computational load permits real-time, synchronous and closed-loop position control
of the motors. Regarding the surgeon interface console, it was a custom made mobile cart
which housed a high-level control computer, two human-machine interfaces (PHANTOM
Omni haptic devices, 3DSystems, Inc.) and a 42-inch high-definition monitor. This was
the first system that consisted of a complete, clinically practical system which permitted
intraoperative interchange of concentric tube instruments.
Fetoscopic Interventions: Fetoscopic interventions are unique in their requirement
to protect the welbeing of both the mother and the fetus. Delicate manoeuvres under
poor visualisation conditions are required, occasionally with oblique lines of sight. Dwyer
et al. (122) developed a 2DoF CTR with primary novelty its coupling to a KUKA LBR
iiwa 7R800 robotic arm constrained by a remote centre of motion (RCM). The robot was
designed for Fetoscopic Laser Photocoagulation (FLP), a minimally invasive surgery used
to treat Twin-Twin Transfusion Syndrome. The robot was evaluated by scanning a human
placenta, via the employment of a miniature camera (Naneye Stereo Camera, AWAIBA Lda,
Portugal) at the robot’s tip. The tubes are actuated by servo motors (Dynamixels MX-28)
with the use of square shafts and gears for the rotation and lead screws for the translation.
The robot was controlled via VxWorks (Wind River Systems Inc., United States) providing
a soft real-time system.
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 15
Work in (123) showcased a multi-arm CTR for fetal surgery with a focus on spina
bifida, which is one of the most common birth defects. The authors were the first to
explore the macro-micro concept during the robot’s design. The robot comprised four arms
housed within a rigid insertion sheath with an outer diameter of 11mm. The robot had 2
mirrored instruments arms, a camera arm (NanEye camera, AWAIBA Lda), and one arm
for suction and irrigation. Macromotion was achieved based on concentric tube technology,
while the micromotion part was a highly bending segment actuated by a miniaturized fluidic
McKibben muscle of 1.2mm outer diameter.
Cardiac Surgery: Heart surgery is an acute procedure usually requiring the cut and
spread of the sternum to expose the heart, and a cardiopulmonary bypass to perform the
final intervention in a non-beating heart. Interventions on a beating heart obviate the need
for the bypass but increase the risk of adverse effects and co-morbidity. CTRs have reported
potentially game-changing benefits in heart surgery.
Gosline et al. (112;124) used a CTR prototype, shown in Fig. 3(d), to deliver metal
microelectromechanical systems (MEMS) to intracardiac locations though the patient’s neck
to the right atrium of the heart. This work was the fist that proposed the use of CTRs as
a means to deliver micro-manufactured instruments rather than only as active cannulas or
dexterous manipulators. Validation in (125) demonstrated robotic percutaneous beating-
heart tissue removal through an in vivo atrial septostomy on a Yorkshire swine. Finally,
(60) demonstrated in a preclinical porcine in vivo model autonomous navigation of a CTR
inside the heart by ensuring low-force contact with the heart tissue and then following tissue
walls to reach a goal location.
Lung Interventions: CTRs have also been proposed for early detection of lung cancer.
While CTRs are compliant robotic systems, they are rigid and cannot safely conform to
the human anatomy unless they are patient-specific. To mitigate this, (126) developed a
three-stage steering system for lung biopsy and therapy delivery, shown in Fig. 3(i). The
system comprised a bronchoscope housing a CTR. The bronchoscope was used to reach the
bronchial wall, while the CTR was used to penetrate it, carrying also the tools necessary
for a biopsy. The robot was validated on phantom consisting of a bronchial tree (made from
plastic tubes) that was embedded in a phantom parenchyma (gelatin) and on ex vivo porcine
wall tissue. Amack et al. (127) improved the concept of accessing the peripheral lung by
designing a more compact, modular, multi-stage robot to retrieve biopsies from lesions in the
peripheral regions of the lung. The improved version included a quick-connect mechanism,
which allowed the rapid tool interchange employing a similar concept as (115). Each tool
was pre-configured with a spur gear. The robotic system featured two spring-loaded levers,
with adjustable sprint tension to minimize backlash, which were able to deflect to accept
the spur gear hub. Finally, the new design featured the first precise, systematic homing
protocol to acquire a repeatable home position for all DoFs of the robot achieving a homing
precision with a standard deviation of ±7.3µm and ±0.09.
Prostate Surgery: Transurethral laser prostate surgery was studied in (111). CTRs
were developed to facilitate holnium laser enucleation of the prostate (HoLEP), which is
a currently very challenging procedure. The authors demonstrated the use of a robotic
system with two concentric tube manipulators housed within a rigid endoscope, as shown
in Fig. 3(b). The hand-held system was suspended from a spring-loaded counter balanced
boom arm. The two manipulators possessed 9DoFs in total with one of them comprising
three precurved tubes (6DoFs) while the other included two tubes with the outer one being
straight (3DoFs). One arm facilitated tissue manipulation and retraction while the other
16 Mitros et al.
aimed the laser fiber. The linear motion of the tubes was achieved via lead screws which
drove tube carriers on ball screws, while rotation was achieved via square shafts, which
transmitted torque through a gear train to each tube. Key innovation of the work was that
the user teleoperated the concentric tube arms using joysticks mounted on the system itself.
The system has been demonstrated on phantom and cadaver experiments on a procedure
for benign prostatic hyperplasia.
4.2. Fabrication Techniques
CTRs are mostly made from NiTi due to its super-elastic capabilities. NiTi tubes can be
precurved using heat treatment either via the employment of a furnace (114) or via an
electric technique that uses Joule heating (14).
To shape setting NiTi tubes via an electric furnace in (114), an aluminum template (Al
2219) with grooves of the desired curvature was machined. It was experimentally found
that the template should be preheated to 520for approximately 10 mins to ensure its
uniform heating when the tubes are inserted. Next, the tubes were inserted in the preheated
template and the assembly was inserted in the oven at 510-514steady state temperature
for 30 mins. Afterwards, the template with the tube was rapidly quenched in cold water for
immediate cooling. Reliable shape setting was observed for curvatures ranging from 14.5
to 285.7m1, and diameters from 1to 2.8mm.
To decrease the manufacturing time, overall cost, and also achieve higher accuracy on
precurvature setting without the presence of relaxation, an electric technique that uses
Joule heating was proposed in (14). A complete system for closed loop, high temperature
resistance heating of NiTi tubes was presented. The template was made of inexpensive
medium density fiberboard (MDF) and an Arduino microcontroller board was employed
the regulate the on-ff state of the measured resistance of the heated part while a MOSFET
is used to control the flow of current, commanded by the microcontroller. The designed
system was evaluated on shape setting 10 wires with a target radius of curvature of r=
63.7mm. The mean radius of curvature of the resulting wires was 65.1mm with a standard
deviation of 1.7mm.
4.3. Motorised Systems for MIS
In the interest of brevity, we do not discuss in detail all the systems developed solely for
evaluation of theoretical works. However, systems which present a novel design and have
not presented above are discussed here.
Girerd et al. (118) designed a hand-held CTR, shown in Fig. 3(c), for MIS which can
provide increased accessibility and dexterity as large robotized devices while maintaining
the footprint of a traditional hand-held tool. The robot comprised 3NiTi tubes while
roller gears for the simultaneous rotation and translation of a tube, creating a lightweight
and easy to assemble system. It was controlled via a symmetric hand held interface which
enabled single-hand operation. Also, (128) designed the first 3D printed CTR weighted
490 gr and with dimensions 17.500 ×3.500 ×400. The robot was composed of 3tubes and 6
DoFs in total. It demonstrated through experiments the precision and accuracy that a 3
system can have. Childs et al. (54) presented the first continuum robot based on a pair of
concentric precurved bellows. Each bellow rotated axially at its base, allowing independent
control of the end-effector’s curvature and bending plane. The authors developed a 3D
printed system as a proof of concept and performed experiments to demonstrate payload
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 17
capacity and validate their theoretical work. It was the first proposition of a continuum
robot based on concentric tube technology able to be used as a soft gripper and generally
in non medical application.
4.4. Summary of CTR Applied Research in Medicine
Design variables that should be considered when designing a CTR are determined by the
application that the robot will be used on. They include the diameter of the manipulator,
the number of tubes, the material, the curvature and the stiffness of each tube. The DoFs
that the robot should possess, the output force range, the type of transmission which affects
the friction and the backdrivability of the system should be taken into account as well.
CTRs can either be a standalone robotic system or part of a hybrid system based on
the required DoF and the aimed intervention. Most of the CTRs are standalone systems to
take advantage of the miniature end-effector and the increased curvature they can possess.
On the other hand, the use of hybrid systems, e.g. combining either a parallel robot or a
flexible catheter with the a CTR, results in systems with no unstable configurations but at
the expense of end-effector’s dimensions. The overall system, however, end ups being more
complex when hardware design is considered.
Several challenges should be addressed prior to the employment of CTRs in the clinical
setting. Safety during in an intervention is one of the most important requirements. The
employment of several sensors and the use of the robot by an experiment surgeon can
minimize the risk of complications. Moreover, the size and the placement of a robot in
the operating room should not intervene with the surgical procedure as the space in an
operating room is limited. CTRs can be mounted either on robotics manipulators or on
passive articulated arms, which can be placed aside when they are not needed. Works on
the miniaturization of CTRs and the surgical tools are taken place so to minimize their
hardware footprint.
Remarks on the design of CTRs based on the prototypes that are presented above can
be summarized to the fact that components should be biocompatible and tools should be
interchangeable. The sterility of the system should be taken into consideration if the ulti-
mate goal is to deploy robots in the operating room. This can be achieved by biocompatible
materials and bagging the motors when they are close to the surgical area. Furthermore,
the ability of the system to possess a miniature camera at its end-effector to visualize the
surgical area is critical when visualisation through other modalities is not possible. Finally,
the research trend is to consider multi-arm systems because of the need to manipulate tissue
and deliver more than one tools at the same time in the surgical area.
It should be highlighted that startups leveraging this technology have appeared, show-
casing the promise the the technology encompasses. Virtuoso Surgical (virtuososurgical.net)
and EndoTheia, Inc. (www.endotheia.com) are two startup companies from the Medical
Engineering and Discovery Lab (MEDLab) of Vanderbilt University, with the aim to bring
concentric tube robots to operating rooms and commercialize steerable sheaths for flexible
endoscopy respectively.
Over the last decades, concentric tube robots have matured from a relatively niche area
of research to the point where modelling accuracy, control, system design, and application
18 Mitros et al.
have showcased impressive outcomes. This review paper attempted to capture the historical
transition from low fidelity models to complete incorporation of non-linear behaviours in
system dynamics, the evolution of computational robot design approaches, and the full
spectrum of innovative engineered robots. Each section identified relevant research gaps,
indicating potential future areas of investigation. Notably, data-driven modelling methods
are on the rise, while the systems are taking their first steps towards clinical translation.
The authors disclose no conflicts of interest. This research was supported by an ERC
Starting Grant [714562].
1. Vitiello V, Lee SL, Cundy TP, Yang GZ. 2013. Emerging Robotic Platforms for Minimally
Invasive Surgery. IEEE Reviews in Biomedical Engineering 6:111–126
2. Dupont PE, Lock J, Itkowitz B, Butler E. 2010. Design and Control of Concentric Tube
Robots. IEEE Transactions on Robotics 26:209–225
3. Webster RJ, Jones BA. 2010. Design and Kinematic Modeling of Constant Curvature Contin-
uum Robots: A Review. The International Journal of Robotics Research 29:1661–1683
4. Bates BL, Hall TA, Osborne TA. 1993. Guide for Localizing a Nonpalpable Breast Lension
5. Cuschieri A, Buess G. 1992. Operative Manual of Endoscopic Surgery
6. Daum WR, Schalgdorf N. 1995. Deflectable Needle Assembly, Patent No: US 6,572,593 b1
7. Furusho J, Ono T, Murai R, Fujimoto T, Chiba Y, Horio H. 2005. Development of a Curved
Multi-Tube (CMT) Catheter for percutaneous umbilical blood sampling and control methods
of CMT catheters for solid organs. International Conference on Mechatronics and Automation
8. Webster RJ, Okamura AM, Cowan NJNJ. 2006. Toward Active Cannulas: Miniature Snake-
Like Surgical Robots. In IEEE/RSJ International Conference on Intelligent Robots and Sys-
tems, pp. 2857–2863
9. Sears P, Dupont PE. 2006. A Steerable Needle Technology Using Curved Concentric Tubes. In
IEEE International Conference on Intelligent Robots and Systems, pp. 2850–2856
10. Bruns TL, Remirez AA, Emerson MA, Lathrop RA, Mahoney AW, et al. 2021. A Modular,
Multi-Arm Concentric Tube Robot System with Application to Transnasal Surgery for Orbital
Tumors. The International Journal of Robotics Research 40:521–533
11. Mahoney AW, Gilbert HB, Webster RJ. 2018. A Review of Concentric Tube Robots: Modeling,
Control, Design, Planning, and Sensing. In The Encyclopedia of Medical Robotics, no. 4.
12. Alfalahi H, Renda F, Stefanini C. 2020. Concentric Tube Robots for Minimally Invasive
Surgery: Current Applications and Future Opportunities. IEEE Transactions on Medical
Robotics and Bionics 3202:1–1
13. Gilbert H, Rucker DC, Webster R. 2013. Concentric Tube Robots: The State of the Art and
Future Directions. In ISRR
14. Gilbert HB, Webster RJ, Al. Ue. 2016. Rapid, Reliable Shape Setting of Superelastic Nitinol
for Prototyping Robots. IEEE Robotics and Automation Letters 1:98–105
15. Sears P, Dupont PE. 2007. Inverse Kinematics of Concentric Tube Steerable Needles. In IEEE
International Conference on Robotics and Automation, pp. 1887–1892
16. Webster RJ, Romano JM, Cowan NJ, Webster Robert J. III, Romano JM, Cowan NJ. 2009.
Mechanics of Precurved-Tube Continuum Robots. IEEE Transactions on Robotics 25:67–78
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 19
17. Rucker DC, Jones BA, Webster RJ. 2010. A Geometrically Exact Model for Externally Loaded
Concentric Tube Continuum Robots. IEEE Transactions on Robotics 26:769–780
18. Dupont PE, Lock J, Itkowitz B. 2010. Real-Time Position Control of Concentric Tube Robots.
In IEEE International Conference on Robotics and Automation, pp. 562–568
19. Xu R, Asadian A, Naidu AS, Patel RV. 2013. Position Control of Concentric-Tube Continuum
Robots using a Modified Jacobian-Based Approach. In IEEE International Conference on
Robotics and Automation, pp. 5813–5818
20. Baek C, Yoon K, Kim DN. 2016. Finite Element Modeling of Concentric-Tube Continuum
Robots. Structural Engineering and Mechanics 57:809–821
21. Pourafzal M, Talebi HA, Rabenorosoa K. 2021. Piecewise Constant Strain Kinematic Model
of Externally Loaded Concentric. Mechatronics 74:102502
22. Renda F, Messer C, Rucker C, Boyer F. 2021. A Sliding-rod Variable-strain Model for Con-
centric Tube Robots. IEEE Robotics and Automation Letters :1–8
23. Sadati S, Mitros Z, Henry R, Cruz LD, Bergeles C. 2020. Reduced-Order Real-Time Dynamics
of Concentric Tube Robots: A Polynomial Shape (PS) Parametrization. Tech. rep., King’s
College London
24. Fagogenis G, Bergeles C, Dupont PE. 2016. Adaptive Nonparametric Kinematic Modeling of
Concentric Tube Robots. In IEEE International Conference on Intelligent Robots and Systems,
pp. 4324–4329
25. Grassmann R, Modes V, Burgner-Kahrs J. 2018. Learning the Forward and Inverse Kine-
matics of a 6-DOF Concentric Tube Continuum Robot in SE(3). In IEEE/RSJ International
Conference on Intelligent Robots and Systems, pp. 5125–5132
26. Kuntz A, Sethi A, Webster RJ, Alterovitz R. 2020. Learning the Complete Shape of Concentric
Tube Robots. IEEE Transactions on Medical Robotics and Bionics 2:140–147
27. Rucker DC, Jones BA, Webster RJ. 2010. A Model for Concentric Tube Continuum Robots
Under Applied Wrenches. In IEEE International Conference on Robotics and Automation,
pp. 1047–1052
28. Lock J, Laing G, Mahvash M, Dupont PE. 2010. Quasistatic Modeling of Concentric Tube
Robots with External Loads. In IEEE/RSJ International Conference on Intelligent Robots and
Systems, pp. 2325–2332
29. Webster RJ, Romano JM, Cowan NJ. 2008. Kinematics and Calibration of Active Cannulas.
In IEEE International Conference on Robotics and Automation, pp. 3888–3895
30. Lyons LA, Webster RJ, Alterovitz R. 2009. Motion Planning for Active Cannulas. In
IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 801–806
31. Rucker DC, Webster RJ. 2011. Computing Jacobians and Compliance Matrices for Externally
Loaded Continuum Robots. In IEEE International Conference on Robotics and Automation,
pp. 945–950
32. Torres LG, Baykal C, Alterovitz R. 2014. Interactive-Rate Motion Planning for Concentric
Tube Robots. In IEEE International Conference on Robotics and Automation, pp. 1915–1921
33. Till J, Aloi V, Riojas KE, Anderson PL, Webster RJ, Rucker C. 2020. A Dynamic Model for
Concentric Tube Robots. IEEE Transactions on Robotics 36:1704–1718
34. Khadem M, O’Neill J, Mitros Z, Da Cruz L, Bergeles C. 2020. Autonomous steering of con-
centric tube robots via nonlinear model predictive control. IEEE Transactions on Robotics
35. Leibrandt K, Bergeles C, Yang Gz, Leibrandt BK, Bergeles C, Yang Gz. 2017. Concentric
Tube Robots: Rapid, Stable Path-Planning and Guidance for Surgical Use. IEEE Robotics &
Automation Magazine 24:42–53
36. Xu R, Patel RV. 2012. A Fast Torsionally Compliant Kinematic Model of Concentric Tube
Robots. In IEEE International Conference of Engineering in Medicine and Biology, pp. 904–
37. Xu R, Asadian A, Atashzar SF, Patel RV. 2014. Real-Time Trajectory Tracking for Exter-
20 Mitros et al.
nally Loaded Concentric-Tube Robots. In IEEE International Conference on Robotics and
Automation, pp. 4374–4379
38. Ha J, Dupont PE. 2017. Designing Stable Concentric Tube Robots Using Piecewise Straight
Tubes. IEEE Robotics and Automation Letters 2:298–304
39. Rucker DC, Webster RJ, Chirikjian GS, Cowan NJ. 2010. Equilibrium Conformations of
Concentric-tube Continuum Robots. The International Journal of Robotics Research 29:1263–
40. Girerd C, Rabenorosoa K, Rougeot P, Renaud P. 2017. Towards Optical Biopsy of Olfac-
tory Cells using Concentric Tube Robots with Follow-the-Leader Deployment. In IEEE/RSJ
International Conference on Intelligent Robots and Systems, pp. 5661–5887
41. Kim C, Ryu SC, Dupont PE. 2015. Real-Time Adaptive Kinematic Model Estimation of
Concentric Tube Robots. In IEEE/RSJ International Conference on Intelligent Robots and
Systems, pp. 3214–3219
42. Jang C, Ha J, Dupont PE, Park FC. 2016. Toward on-line parameter estimation of concentric
tube robots using a mechanics-based kinematic model. In IEEE/RSJ International Conference
on Intelligent Robots and Systems, pp. 2400–2405
43. Ha J, Park FC, Dupont PE. 2017. Optimizing Tube Precurvature to Enhance the Elastic
Stability of Concentric Tube Robots. IEEE Transactions on Robotics 33:22–37
44. Peyron Q, Rabenorosoa K, Andreff N, Renaud P. 2019. A Numerical Framework for the Sta-
bility and Cardinality Analysis of Concentric Tube Robots: Introduction and Application to
the Follow-the-Leader Deployment. Mechanism and Machine Theory 132:176–192
45. Hendrick RJ, Gilbert HB, Webster RJ. 2015. Designing Snap-Free Concentric Tube Robots: A
Local Bifurcation Approach. In IEEE International Conference on Robotics and Automation,
pp. 2256–2263
46. Bergeles C, Dupont PE. 2013. Planning Stable Paths for Concentric Tube Robots. In
IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3077–3082
47. Xu R, Atashzar SF, Patel RV. 2014. Kinematic Instability in Concentric Tube Robots: Mod-
eling and Analysis. In IEEE RAS/EMBS International Conference on Biomedical Robotics
and Biomechatronics, pp. 163–168
48. Ha J, Park FC, Dupont PE. 2014. Achieving Elastic Stability of Concentric Tube Robots
Through Optimization of Tube Precurvature. In IEEE/RSJ International Conference on In-
telligent Robots and Systems, pp. 864–870
49. Gilbert HB, Hendrick RJ, Webster RJ, Webster III RJ. 2016. Elastic Stability of Concentric
Tube Robots: A Stability Measure and Design Test. IEEE Transactions on Robotics 32:20–35
50. Ha J, Park FC, Dupont PE. 2016. Elastic Stability of Concentric Tube Robots Subject to
External Loads. IEEE Transactions on Biomedical Engineering 63:1116–1128
51. Riojas KE, Hendrick RJ, Webster RJ. 2018. Can Elastic Instability Be Beneficial in Concentric
Tube Robots? IEEE Robotics and Automation Letters 3:1624–1630
52. Lock J, Dupont PE. 2011. Friction Modeling in Concentric Tube Robots. In IEEE International
Conference on Robotics and Automation, pp. 1139–1146
53. Ha J, Fagogenis G, Dupont PE. 2019. Modeling Tube Clearance and Bounding the Effect of
Friction in Concentric Tube Robot Kinematics. IEEE Transactions on Robotics 35:353–370
54. Childs JA, Rucker C. 2020. Concentric Precurved Bellows: New Bending Actuators for Soft
Robots. IEEE Robotics and Automation Letters 5:1215–1222
55. Wang J, Lu Y, Zhang C, Song S, Meng MQ. 2017. Pilot Study on Shape Sensing for Continuum
Tubular Robot with Multi-Magnet Tracking Algorithm. In IEEE International Conference on
Robotics and Biomimetics, pp. 1165–1170
56. Lobaton EJ, Fu J, Torres LG, Alterovitz R. 2013. Continuous Shape Estimation of Continuum
Robots using X-Ray images. In IEEE International Conference on Robotics and Automation,
pp. 725–732
57. Vandini A, Bergeles C, Glocker B, Giataganas P, Yang GZ. 2017. Unified Tracking and Shape
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 21
Estimation for Concentric Tube Robots. IEEE Transactions on Robotics 33:901–915
58. Aloi VA, Rucker DC. 2019. Estimating Loads Along Elastic Rods. In International Conference
on Robotics and Automation, pp. 2867–2873
59. Wei W, Simaan N. 2012. Modeling, Force Sensing, and Control of Flexible Cannulas for Mi-
crostent Delivery. Journal of Dynamic Systems, Measurement, and Control 134:1–12
60. Fagogenis G, Mencattelli M, Machaidze Z, Rosa B, Price K, et al. 2019. Autonomous Robotic
Intracardiac Catheter Navigation Using Haptic Vision. Science Robotics 4:eaaw1977
61. Donat H, Lilge S, Burgner-Kahrs J, Steil JJ. 2020. Estimating Tip Contact Forces for Concen-
tric Tube Continuum Robots Based on Backbone Deflection. IEEE Transactions on Medical
Robotics and Bionics 2:619–630
62. Wu K, Wu L, Lim CM, Ren H. 2015. Model-Free Image Guidance for Intelligent Tubu-
lar Robots with Pre-Clinical Feasibility Study: Towards Minimally Invasive Trans-Orifice
Surgery. In IEEE International Conference on Information and Automation, pp. 749–754
63. Kudryavtsev AV, Chikhaoui MT, Liadov A, Rougeot P, Spindler F, et al. 2018. Eye-in-hand
visual servoing of concentric tube robots. IEEE Robotics and Automation Letters 3:2315–2321
64. Li X, Choi T, Chun H, Gim S, Lee S, et al. 2013. Active Cannula Robot with Misorientation
Auto-Recovery Camera: A Method to Improve Hand-Eye Coordination in Minimally Invasive
Surgery. In International Conference on Control, Automation and Systems, pp. 276–280
65. Wu L, Wu K, Ren H. 2016. Towards Hybrid Control of a Flexible Curvilinear Surgical Robot
with Visual/Haptic Guidance. In IEEE/RSJ International Conference on Intelligent Robots
and Systems, pp. 501–507
66. Lyons LA, Webster RJ, Alterovitz R. 2010. Planning Active Cannula Configurations through
Tubular Anatomy. In IEEE International Conference on Robotics and Automation, pp. 2082–
67. Kuntz A, Fu M, Alterovitz R. 2019. Planning High-Quality Motions for Concentric Tube
Robots in Point Clouds via Parallel Sampling and optimization. In IEEE/RSJ International
Conference on Intelligent Robots and Systems, pp. 2205–2212
68. Torres LG, Alterovitz R. 2011. Motion Planning for Concentric Tube Robots using Mechanics-
Based Models. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.
69. Baykal C, Alterovitz R. 2017. Asymptotically Optimal Design of Piecewise Cylindrical Robots
using Motion Planning. In Robotics: Science and Systems
70. Fu M, Kuntz A, Salzman O, Alterovitz R. 2019. Toward Asymptotically-Optimal Inspection
Planning Via Efficient Near-Optimal Graph Search. In Robotics: Science and Systems
71. Baran Y, Rabenorosoa K, Laurent GJ, Rougeot P, Andreff N, Tamadazte B. 2017. Prelim-
inary Results on OCT-based Position Control of a Concentric Tube Robot. In IEEE/RSJ
International Conference on Intelligent Robots and Systems, pp. 3000–3005
72. Lu Y, Zhang C, Song S, Meng MQ. 2017. Precise Motion Control of Concentric-Tube Robot
based on Visual Servoing. In IEEE International Conference on Information and Automation,
pp. 299–304
73. Chikhaoui MT, Granna J, Starke J, Burgner-Kahrs J. 2018. Toward Motion Coordination
Control and Design Optimization for Dual-Arm Concentric Tube Continuum Robots. IEEE
Robotics and Automation Letters 3:1793–1800
74. Sabetian S, Looi T, Diller ED, Drake J. 2019. Self-Collision Detection and Avoidance for
Dual-Arm Concentric Tube Robots. IEEE Robotics and Automation Letters
75. Modes V, Burgner-Kahrs J. 2020. Calibration of Concentric Tube Continuum Robots: Auto-
matic Alignment of Precurved Elastic Tubes. IEEE Robotics and Automation Letters 5:103–
76. Wu K, Zhu G, Wu L, Gao W, Song S, et al. 2019. Safety-Enhanced Model-Free Visual Servoing
for Continuum Tubular Robots Through Singularity Avoidance in Confined Environments.
IEEE Access 7:21539–21558
22 Mitros et al.
77. Leibrandt K, Bergeles C, Yang GZ. 2017. Implicit Active Constraints for Concentric Tube
Robots Based on Analysis of the Safe and Dexterous Workspace. In IEEE/RSJ International
Conference on Intelligent Robots and Systems, pp. 193–200
78. Khadem M, Cruz LD, Bergeles C. 2018. Force/Velocity Manipulability Analysis for 3D Con-
tinuum Robots. In IEEE/RSJ International Conference on Intelligent Robots and Systems,
pp. 4920–4926
79. Garriga-Casanovas A, Baena FRy, Rodriguez y Baena F. 2018. Complete Follow-the-Leader
Kinematics using Concentric Tube Robots. The International Journal of Robotics Research
80. Bergeles C, Gosline AH, Vasilyev NV, Codd PJ, del Nido PJ, Dupont PE. 2015. Concentric
Tube Robot Design and Optimization Based on Task and Anatomical Constraints. IEEE
Transactions on Robotics 31:67–84
81. Girerd C, Kudryavtsev AV, Rougeot P, Renaud P, Rabenorosoa K, Tamadazte B. 2020. SLAM-
Based Follow-the-Leader Deployment of Concentric Tube Robots. IEEE Robotics and Automa-
tion Letters 5:548–555
82. Gilbert HB, Neimat J, Webster RJ. 2015. Concentric Tube Robots as Steerable Needles:
Achieving Follow-the-Leader Deployment. IEEE Transactions on Robotics 31:246–258
83. Mahvash M, Dupont PE. 2011. Stiffness Control of Surgical Continuum Manipulators. IEEE
Transactions on Robotics 27:334–345
84. Granna J, Burgner J. 2014. Characterizing the Workspace of Concentric Tube Continuum
Robots. In International Symposium on Robotics, pp. 1–7
85. Granna J, Nabavi A, Burgner-Kahrs J. 2019. Computer-Assisted Planning for a Concentric
Tube Robotic System in Neurosurgery. International Journal of Computer Assisted Radiology
and Surgery 14:335–344
86. Morimoto T, Greer J, Hawkes E, Okamura A, Hsieh M. 2017. Design, Fabrication, and testing
of patient specific concentric tube robots for nonlinear renal access and mass ablation. Journal
of Urology 197:e817
87. Zhang D, Wang J, Yang X, Song S, Meng MQ. 2018. RRT*-Smooth Algorithm Applied to Mo-
tion Planning of Concentric Tube Robots. In IEEE International Conference on Information
and Automation, pp. 487–493
88. Torres LG, Kuntz A, Gilbert HB, Swaney PJ, Hendrick RJ, et al. 2015. A Motion Planning
Approach to Automatic Obstacle Avoidance During Concentric Tube Robot Teleoperation. In
IEEE International Conference on Robotics and Automation, pp. 2361–2367
89. Bergeles C, Lin FY, Yang GZ. 2015. Concentric Tube Robot Kinematics Using Neural Net-
works. In The Hamlyn Symposium on Medical Robotics, pp. 13–14
90. Iyengar K, Dwyer G, Stoyanov D. 2020. Investigating Exploration for Deep Reinforcement
Learning of Concentric Tube Robot Control. International Journal of Computer Assisted
Radiology and Surgery
91. Solberg FS. 2020. Effect of Demonstrations for Deep Reinforcement Learning-Based Control
of Concentric Tube Robots. Master’s thesis, Stanford
92. Grassmann R, Burgner-Kahrs J. 2019. On the Merits of Joint Space and Orientation Repre-
sentations in Learning the Forward Kinematics in SE(3). In Robotics: Science and Systems
93. Anor T, Madsen JR, Dupont P. 2011. Algorithms for Design of Continuum Robots using the
Concentric Tubes Approach: A Neurosurgical Example. In IEEE International Conference on
Robotics and Automation, pp. 667–673
94. Bedell C, Lock J, Gosline A, Dupont PE. 2011. Design Optimization of Concentric Tube
Robots Based on Task and Anatomical Constraints. In IEEE International Conference on
Robotics and Automation, pp. 398–403
95. Burgner J, Swaney PJ, Rucker DC, Gilbert HB, Nill ST, et al. 2011. A Bimanual Teleoper-
ated System for Endonasal Skull Base Surgery. In IEEE/RSJ International Conference on
Intelligent Robots and Systems, pp. 2517–2523
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 23
96. Burgner-Kahrs J, Gilbert HB, Webster RJ, Burgner J, Gilbert HB, Webster RJ. 2013. On the
Computational Design of Concentric Tube Robots: Incorporating Volume-Based Objectives.
In IEEE International Conference on Robotics and Automation, pp. 1193–1198
97. Granna J, Guo Y, Weaver KD, Burgner-Kahrs J. 2016. Comparison of Optimization Algo-
rithms for a Tubular Aspiration Robot for Maximum Coverage in Intracerebral Hemorrhage
Evacuation. Journal of Medical Robotics Research 02:1750004
98. Torres LG, Webster RJ, Alterovitz R. 2012. Task-Oriented Design of Concentric Tube Robots
using Mechanics-Based Models. In IEEE/RSJ International Conference on Intelligent Robots
and Systems, pp. 4449–4455
99. Baykal C. 2015. Design Optimization Algorithms for Concentric Tube Robots. Master’s thesis,
University of North Carolina at Chapel Hill
100. Boushaki MN. 2016. Design optimization and control for concentric tube robot in assisted
single-access laparoscopic surgery. Ph.D. thesis, Université Montpellier
101. Noh G, Yoon SY, Yoon S, Kim K, Lee W, et al. 2016. Expeditious Design Optimization of a
Concentric Tube Robot with a Heat-Shrink Plastic Tube. In IEEE International Conference
on Intelligent Robots and Systems, vol. 2016-Novem, pp. 3671–3676
102. Ha J, Dupont PE. 2017. Incorporating Tube-to-Tube Clearances in the Kinematics of Con-
centric Tube Robots. In IEEE International Conference on Robotics and Automation, pp.
103. Yang X, Song S, Liu L, Yan T, Meng MQ. 2019. Design and Optimization of Concentric Tube
Robots Based on Surgical Tasks, Anatomical Constraints and Follow-the-Leader Deployment.
IEEE Access 7:173612–173625
104. Morimoto TK, Greer JD, Hsieh MH, Okamura AM. 2016. Surgeon Design Interface for
Patient-Specific Concentric Tube Robots. In IEEE International Conference on Biomedical
Robotics and Biomechatronics, vol. 2016, pp. 41–48
105. Azimian H, Francis P, Looi T, Drake J. 2014. Structurally-Redesigned Concentric-Tube Manip-
ulators with Improved Stability. In IEEE/RSJ International Conference on Intelligent Robots
and Systems, pp. 2030–2035
106. Lee D, Kim J, Kim J, Baek C, Noh G, et al. 2015. Anisotropic Patterning to Reduce Instability
of Concentric-Tube Robots. IEEE Transactions on Robotics 31:1311–1323
107. Kim J, Choi WY, Kang S, Kim C, Cho KJ. 2019. Continuously variable stiffness mecha-
nism using nonuniform patterns on coaxial tubes for continuum microsurgical robot. IEEE
Transactions on Robotics 35:1475–1487
108. Xin Jue Luo KA, Looi T, Sabetian S, Drake J. 2018. Designing Concentric Tube Manipu-
lators for Stability Using Topology Optimization. In IEEE/RSJ International Conference on
Intelligent Robots and Systems, pp. 1764–1769
109. Khadem M, O’Neill J, Mitros Z, Cruz LD, Bergeles C. 2019. Autonomous Steering of Concen-
tric Tube Robots for Enhanced Force/Velocity Manipulability. In IEEE International Confer-
ence on Intelligent Robots and Systems, pp. 2197–2204
110. Gilbert HB. 2016. Concentric Tube Robots: Design, Deployment, and Stability. Ph.D. thesis,
Vanderbilt University
111. Hendrick RJ, Mitchell CR, Herrell SD, Webster III RJ. 2015. Hand-held Transendoscopic
Robotic Manipulators: A Transurethral Laser Prostate Surgery Case Study. International
Journal of Robotics Research 34:1559–1572
112. Gosline AH, Vasilyev NV, Veeramani A, Wu M, Schmitz G, et al. 2012. Metal MEMS tools
for Beating-Heart Tissue Removal. In IEEE International Conference on Robotics and Au-
tomation, pp. 1921–1926
113. Wu L, Tan BLW, Ren H. 2015. Prototype Development of a Hand-Held Robotic Light Pipe
for Intraocular Procedures. In IEEE International Conference on Robotics and Biomimetics,
pp. 368–373
114. Mitros Z, Sadati S, Seneci C, Bloch E, Leibrandt K, et al. 2020. Optic Nerve Sheath Fenestra-
24 Mitros et al.
tion with a Multi-Arm Continuum Robot. IEEE Robotics and Automation Letters 5:4874–4881
115. Burgner J, Swaney PJ, Lathrop RA, Weaver KD, Webster RJ. 2013. Debulking from Within:
A Robotic Steerable Cannula for Intracerebral Hemorrhage Evacuation. IEEE Transactions
on Biomedical Engineering 60:2567–2575
116. Burgner J, Rucker DC, Gilbert HB, Swaney PJ, Russell PT, et al. 2014. A Telerobotic System
for Transnasal Surgery. IEEE/ASME Transactions on Mechatronics 19:996–1006
117. Amanov E, Nguyen TD, Markmann S, Imkamp F, Burgner-Kahrs J. 2018. Toward a Flex-
ible Variable Stiffness Endoport for Single-Site Partial Nephrectomy. Annals of Biomedical
Engineering 46:1498–1510
118. Girerd C, Morimoto TK. 2020. Design and Control of a Hand-Held Concentric Tube Robot
for Minimally Invasive Surgery. IEEE Transactions on Robotics
119. Wei W, Goldman R, Simaan N, Fine H, Chang S. 2007. Design and Theoretical Evaluation
of Micro-Surgical Manipulators for Orbital Manipulation and Intraocular Dexterity. IEEE
International Conference on Robotics and Automation :3389–3395
120. Wei W, Goldman RE, Fine HF, Chang S, Simaan N. 2009. Performance Evaluation for Multi-
Arm Manipulation of Hollow Suspended Organs. IEEE Transactions on Robotics 25:147–157
121. Lin FY, Bergeles C, Yang GZZ. 2015. Biometry-Based Concentric Tubes Robot for Vitreoreti-
nal Surgery. In IEEE International Conference on Engineering in Medicine and Biology, pp.
122. Dwyer G, Chadebecq F, Amo MT, Bergeles C, Maneas E, et al. 2017. A Continuum Robot
and Control Interface for Surgical Assist in Fetoscopic Interventions. IEEE Robotics and Au-
tomation Letters 2:1656–1663
123. Vandebroek T, Ourak M, Gruijthuijsen C, Javaux A, Legrand J, et al. 2019. Macro-Micro
Multi-Arm Robot for Single-Port Access Surgery. In IEEE/RSJ International Conference on
Intelligent Robots and Systems, pp. 425–432
124. Gosline AH, Vasilyev NV, Butler EJ, Folk C, Cohen A, et al. 2012. Percutaneous Intracardiac
Beating-Heart Surgery using Metal MEMS Tissue Approximation Tools. The International
Journal of Robotics Research 31:1081–1093
125. Vasilyev NV, Gosline AH, Veeramani A, Wu MT, Schmitz GP, et al. 2014. Tissue Removal
Inside the Beating Heart Using a Robotically Delivered Metal MEMS Tool. The International
Journal of Robotics Research 34:236–247
126. Swaney PJ, Mahoney AW, Remirez AA, Lamers E, Hartley BI, et al. 2015. Tendons, Concen-
tric Tubes, and a Bevel Tip: Three Steerable Robots in one Transoral Lung Access System.
In IEEE International Conference on Robotics and Automation, pp. 5378–5383
127. Amack S, Rox MF, Emerson M, Webster RJ, Alterovitz R, et al. 2019. Design and Control of a
Compact Modular Robot for Transbronchial Lung Biopsy. In Medical Imaging: Image-Guided
Procedures, Robotic Interventions, and Modeling., vol. 1095
128. Morimoto TK, Hawkes EW, Okamura AM. 2017. Design of a Compact Actuation and Control
System for Flexible Medical Robots. IEEE Robotics and Automation Letters 2:1579–1585
www.annualreviews.org From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots 25
... Each tube can be independently rotated and translated to control the tip pose and overall robot shape. CTRs are being investigated for a breadth of interventions, e.g. for heart, brain, eye, kidney, prostate, and fetus [4], [5]. ...
... The system EOMs, derived in Matlab, are optimised and exported as C++ functions for possible real-time computational performance. We used trapezoidal method for spatial integration in (5) and SUNDIALS SVODE solver for temporal integration, for its superior computational performance. ...
... The state-of-the-art simulation error is 1%-6% mostly for quasistatic stable motions [5]. The proposed model simulations were about 1.16 (stable configurations) and 2.8-4.4 (unstable configurations) times more accurate compared to a CR static model while preserving real-time performance, correctly predicting the snapping instances and post snapping configurations. ...
... Due to their small size, navigability, and inherent compliance, CTRs can enable safer and more effective minimally invasive surgical procedures than current techniques. Though this class of robots has been studied for a number of years and numerous procedures have been performed in simulation, there are still many open research questions [1]. Work on modeling has slowed in the past decade, with the most common approach forming around the Cosserat rod model, which consists of a kinematic structure accounting for the internal tubes' rotation [2], [3]. ...
... The typical normalized tip error normalized for arc length in state of the art models is 1.5% to 2% and errors under 1-2 mm are generally tolerated in medical scenarios [1]. Compared to the experimentally validated model in [2], our variable-strain model had a mean error of 0.14%-0.23% ...
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In this work, we present Concentric Tube Robot Design and Path Plan (CTR DaPP), a Python application that utilizes the recently introduced Piecewise Variable Strain approach for the design and path planning of concentric tube robots (CTRs). The application provides a modular platform for implementing and testing combinations of path planners and design optimization algorithms. We apply the Randomly Exploring Rapid Tree and Nelder-Mead algorithms to test the 'follow-the-leader' behavior and demonstrate the potential benefits of variable strain tubes in path planning problems. This platform, paired with the variable-strain model, opens up new research avenues to investigate follow-the-leader behavior, elastic stability, and tube design of variable-strain CTRs through simulation.
... Continuum robots can be categorized based on their structural design and actuation architecture. For the purposes of this paper, notable continuum robots are the multibackbone continuum robots [1] and concentric tube robots (CTRs) [9]. Multibackbone systems were introduced in [10], and are typically composed of multiple elastic elements, rods or tubes, that run in parallel [11]. ...
... As the tubes' diameter decreases, the elastic range of robot curvature increases and with it robot dexterity. Therefore, CTRs can catalyse novel medical applications that require tip dexterity, as thoroughly reviewed in [9]. However, the range of shapes that a CTR can achieve is limited and determined by the tubes' design (precurvature and curved length). ...
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Continuum surgical robots can navigate anatom-ical pathways to reach pathological locations deep inside thehuman body. Their flexibility, however, generally comes with re-duced dexterity at their tip and limited workspace. Building onrecent work on eccentric tube robots, this paper proposes a newcontinuum robot architecture and theoretical framework thatcombines the flexibility of push/pull actuated snake robots andthe dexterity offered by concentric tube robotic end-effectors.We designed and present a prototype system as a proof-of-concept, and developed a tailored quasistatic mechanics-basedmodel that describes the shape and end-effector’s pose for thisnew type robotic architecture. The model can accommodate anarbitrary number of arms placed eccentrically with respect tothe backbone’s neutral axis. Our experiments show that theerror between model and experiment is on average 3.56% ofthe manipulator’s overall length. This is in agreement with stateof the art models of single type continuum architecture.
... Several approaches have been developed to try to solve this design optimization problem [15]. Many optimization algorithms use a torsionally-rigid model to optimize the tube length and curvature, while avoiding obstacles in the environment [16], [17]. ...
... Subsequently, a steady rise shows the spread of research.In this dissertation, the literature survey on the developed CTR systems, modelling theory and design optimization techniques is presented. The presented survey is part of an extensive literature review in the field of CTRs which was published on Annual Review of Control, Robotics, and Autonomous Systems entitled "From Theoretical Work to Clinical Translation: Progress in Concentric Tube Robots" in which the author of this dissertation is the first author[1]. ...
Continuum robots are snake-like systems able to deliver optimal therapies to pathologies deep inside the human cavity by following 3D complex paths. They show promise when anatomical pathways need to be traversed thanks to their enhanced flexibility and dexterity and show advantages when deployed in the field of single-port surgery. This PhD thesis concerns the development and modelling of multi-arm and hybrid continuum robots for medical interventions. The flexibility and steerability of the robot’s end-effector are achieved through concentric tube technology and push/pull technology. Medical robotic prototypes have been designed as proof of concepts and testbeds of the proposed theoretical works.System design considers the limitations and constraints that occur in the surgical procedures for which the systems were proposed for. Specifically, two surgical applications are considered. Our first prototype was designed to deliver multiple tools to the eye cavity for deep orbital interventions focusing on a currently invasive intervention named Optic Nerve Sheath Fenestration (ONSF). This thesis presents the end-to-end design, engineering and modelling of the prototype. The developed prototype is the first suggested system to tackle the challenges (limited workspace, need for enhanced flexibility and dexterity, danger for harming tissue with rigid instruments, extensive manipulation of the eye) arising in ONSF. It was designed taking into account the clinical requirements and constraints while theoretical works employing the Cosserat rod theory predict the shape of the continuum end-effector. Experimental runs including ex vivo experimental evaluations, mock-up surgical scenarios and tests with and without loading conditions prove the concept of accessing the eye cavity. Moreover, a continuum robot for thoracic interventions employing push/pull technology was designed and manufactured. The developed system can reach deep seated pathologies in the lungs and access regions in the bronchial tree that are inaccessible with rigid and straight instruments either robotically or manually actuated. A geometrically exact model of the robot that considers both the geometry of the robot and mechanical properties of the backbones is presented. It can predict the shape of the bronchoscope without the constant curvature assumption. The proposed model can also predict the robot shape and micro-scale movements accurately in contrast to the classic geometric model which provides an accurate description of the robot’s differential kinematics for large scale movements.
... As an example, Concentric Tube Robots (CTR) are continuum robots made of a series of pre-curved, elastic tubes where each tube can individually be rotated, as well as extended and recalled; interactions between each tube allows for turns and twists, giving control over the length and configuration of the robot. CTRs can assist in minimally invasive surgery (MIS) to access difficult to reach areas, due to the intricate human anatomy, with advantages including single-site entry and their malleable nature [1]. However, utilizing CTRs, and continuum robots in general, are not without their own challenges. ...
Conference Paper
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The concept of continuum and soft robotics has opened new abilities that were previously unachievable by rigid robotics alone, such as squeezing, growing, and morphing to their environments. As an example, Concentric Tube Robots (CTR) are continuum robots made of a series of pre-curved, elastic tubes where each tube can individually be rotated, as well as extended and recalled; interactions between each tube allows for turns and twists, giving control over the length and configuration of the robot. CTRs can assist in minimally invasive surgery (MIS) to access difficult to reach areas, due to the intricate human anatomy, with advantages including single-site entry and their malleable nature [1]. However, utilizing CTRs, and continuum robots in general, are not without their own challenges. Due to the complex nature of a continuum structure, fast and accurate simulations are still in development and require specific skills to operate. These simulations are not usually accurate due to the complex behaviours of the materials used and their deformable nature. SOFA (Simulation Open Framework Architecture) was introduced as an open-source platform to address some of the challenges with real-time physics-based simulation of interaction with deformable tissue in medical applications and later for modelling soft robots. More specifically, a BeamAdapter plugin2 was developed based on interpolation of a continuous geometry over multiple consecutive Timoshenko beam segments to address the simulation challenges of neurovascular interventions using interleaved catheters and guidewires [2]. The BeamAdapter plugin has been also utilized for interactive planning of coil embolization in brain aneurysms [3] and interactive training system for interventional electrocardiology procedures [4]. We have recently developed a Reduced-Order dynamic model for CTRs based on the shape interpolation of the robot backbone and showcased its real-time performance, correct estimation of the path-dependent motions and snapping instances, accurate simulations of stable and post-snapping motions in an experimental comparative study [5]. In this paper, we outline the process and the code on how a CTR model can be implemented into an example scene provided as a part of the SOFA-framework ‘BeamAdapter’ plugin [2].
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In this work, the Piecewise Variable-strain (PVS) approach is applied to the case of Concentric Tube Robots (CTRs) and extended to include the tubes sliding motion. In particular, the currently accepted continuous Cosserat rod model is discretized onto a finite set of strain basis functions. At the same time, the insertion and rotation motions of the tubes are included as generalized coordinates instead of boundary kinematic conditions. Doing so, we obtain a minimum set of closed-form algebraic equations that can be solved not only for the shape variables but also for the actuation forces and torques for the first time. This new approach opens the way to torque-controlled CTRs, which is poised to enhance elastic stability and improve interaction forces control at the end-effector.
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Minimally invasive surgery is of high interest for interventional medicine since the smaller incisions can lead to less pain and faster recovery for patients. The current standard-of-care involves a range of affordable, manual, hand-held rigid tools, whose limited dexterity and range of adoptable shapes can prevent access to confined spaces. In contrast, recently developed roboticized tools that can provide increased accessibility and dexterity to navigate and perform complex tasks often come at the cost of larger, heavier, and grounded devices that are teleoperated, posing a new set of challenges. In this article, we propose a new hand-held concentric tube robot with an associated position control method that has the dexterity and precision of large roboticized devices, while maintaining the footprint of a traditional hand-held tool. The device shows human-in-the-loop control performance that meets the requirements of the targeted application, percutaneous abscess drainage. In addition, a small user study illustrates the advantage of combining rigid body motion of the device with more precise motions of the tip, thus showing the potential to bridge the gap between traditional hand-held tools and grounded robotic devices.
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This paper presents a medical robotic system for deep orbital interventions, with a focus on Optic Nerve Sheath Fenestration (ONSF). ONSF is a currently invasive ophthalmic surgical approach that can reduce potentially blinding elevated hydrostatic intracranial pressure on the optic disc via an incision on the optic nerve. The prototype is a multi-arm system capable of dexterous manipulation and visualization of the optic nerve area, allowing for a minimally invasive approach. Each arm is an independently controlled concentric tube robot collimated by a bespoke guide that is secured on the eye sclera via sutures. In this paper, we consider the robot's end-effector design in order to reach/navigate the optic nerve according to the clinical requirements of ONSF. A prototype of the robot was engineered, and its ability to penetrate the optic nerve was analysed by conducting ex vivo experiments on porcine optic nerves and comparing their stiffness to human ones. The robot was successfully deployed in a custom-made realistic eye phantom. Our simulation studies and experimental results demonstrate that the robot can successfully navigate to the operation site and carry out the intervention.
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Minimally Invasive Surgery (MIS) evolved as an alternative to open surgical approaches, and resulted in reduced surgical trauma and faster recovery. However, the confined workspace, loss of depth perception, compromised hand-eye coordination, as well as extended learning curves precluded the surgeons from adapting this approach in all cases. Unlike rigid robots that are characterized with mechanical rigidity and limited degrees-of-freedom (DOFs), continuum robots were developed mainly for interventional medicine. A special class of continuum robots are Concentric Tube Robots (CTRs), that have diameters comparable to those of catheters and steerable needles. Since their introduction, huge effort has been dedicated to the kinematic modelling and real-time control. Due to their intrinsic actuation capability, miniaturization potential and controllable mechanical properties, they have been adopted for different surgical applications. Experimental tests based on phantom and cadaveric studies were performed to prove their feasibility. This review encompasses the generic design, modelling and control methods of CTRs. It also addresses, in detail, their different surgical applications. Finally, related main research limitations and future opportunities are briefly discussed.
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PurposeConcentric tube robots are composed of multiple concentric, pre-curved, super-elastic, telescopic tubes that are compliant and have a small diameter suitable for interventions that must be minimally invasive like fetal surgery. Combinations of rotation and extension of the tubes can alter the robot’s shape but the inverse kinematics are complex to model due to the challenge of incorporating friction and other tube interactions or manufacturing imperfections. We propose a model-free reinforcement learning approach to form the inverse kinematics solution and directly obtain a control policy.Method Three exploration strategies are shown for deep deterministic policy gradient with hindsight experience replay for concentric tube robots in simulation environments. The aim is to overcome the joint to Cartesian sampling bias and be scalable with the number of robotic tubes. To compare strategies, evaluation of the trained policy network to selected Cartesian goals and associated errors are analyzed. The learned control policy is demonstrated with trajectory following tasks.ResultsSeparation of extension and rotation joints for Gaussian exploration is required to overcome Cartesian sampling bias. Parameter noise and Ornstein–Uhlenbeck were found to be optimal strategies with less than 1 mm error in all simulation environments. Various trajectories can be followed with the optimal exploration strategy learned policy at high joint extension values. Our inverse kinematics solver in evaluation has 0.44 mm extension and \(0.3^{\circ }\) rotation error.Conclusion We demonstrate the feasibility of effective model-free control for concentric tube robots. Directly using the control policy, arbitrary trajectories can be followed and this is an important step towards overcoming the challenge of concentric tube robot control for clinical use in minimally invasive interventions.
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This article presents a model predictive controller (MPC) developed for the autonomous steering of concentric tube robots (CTRs). State-of-the-art CTR control relies on differential kinematics developed by local linearization of the CTRs mechanics model and cannot explicitly handle constraints on robot’s joint limits or unstable configurations commonly known as snapping points. The proposed nonlinear MPC explicitly considers constraints on the robot configuration space (i.e., joint limits) and the robot’s workspace (i.e., mixed boundary conditions on robot curvature). Additionally, the MPC calculates control decisions by optimizing the model-based predictions of future robot configurations. This way, it avoids configurations it cannot recover from, i.e., joint limits, singular configurations, and snapping. The proposed controller is evaluated via simulations and experimental studies with a variety of trajectories of increasing complexity. Simulation results demonstrate the capability of MPC to avoid singularities while satisfying robot mechanical constraints. Experimental results demonstrate that our solution enables following of trajectories unattainable by state-of-the-art controllers with mean error corresponding to $1\%$ of robot arclength.
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Inspection planning, the task of planning motions that allow a robot to inspect a set of points of interest, has applications in domains such as industrial, field, and medical robotics. Inspection planning can be computationally challenging, as the search space over motion plans grows exponentially with the number of points of interest to inspect. We propose a novel method, Incremental Random Inspection-roadmap Search (IRIS), that computes inspection plans whose length and set of successfully inspected points asymptotically converge to those of an optimal inspection plan. IRIS incrementally densifies a motion planning roadmap using sampling-based algorithms, and performs efficient near-optimal graph search over the resulting roadmap as it is generated. We demonstrate IRIS's efficacy on a simulated planar 5DOF manipulator inspection task and on a medical endoscopic inspection task for a continuum parallel surgical robot in cluttered anatomy segmented from patient CT data. We show that IRIS computes higher-quality inspection plans orders of magnitudes faster than a prior state-of-the-art method.
This paper presents a piecewise constant strain kinematic model for concentric tube robots (CTR) in externally loaded conditions. It discretizes the pre-curved tubes comprising the robot into a finite number of pieces and involves external effects as a set of wrench vectors exerted along the robot backbone. Constant strain lets us describe the pieces with helices in which shear deformation and elongation are neglected. The resulting piecewise helix is the simplest curve that can catch the torsion of tubes that play a crucial role in kinematic behavior. This approximation transforms the conventional boundary value problem (BVP) of CTRs models into a set of nonlinear equations that drastically decreases the model resolution time. The present method uses a Lyapunov function and torsional Jacobian to ensure the distal torsion constraint consistently and, as a result, the solution’s convergence. The paper’s primary purpose is to present a fast, numerically stable, and relatively accurate kinematic model not reliant on measurement data. Experimental results on a two-tube prototype and provided for different tip loading conditions reveal maintaining a balance between adequate accuracy and reasonable running time, about 7 ms for five pieces per section, for real-time applications in the presence of external load.
Concentric Tube Continuum Robots are among the smallest and most flexible instruments in development for minimally invasive surgery, thereby enabling operations in areas within the human body that are difficult to reach. Unfortunately, integrating state-of-the-art force sensors is challenging for these robots due to their small form factor, although contact forces are essential information in surgical procedures. In this work, we propose a novel data-driven approach based on Deep Direct Cascade Learning (DDCL) to create a virtual sensor for computing the tip contact force of Concentric Tube Continuum Robots. By exploiting the robot’s backbone’s inherent elasticity, deflection is used to estimate the respective external tip contact force. We evaluate our approach on different data representations for a single tube and apply it subsequently on a three-segment Concentric Tube Continuum Robot. Furthermore, we devise a novel transfer learning approach through DDCL to improve the estimation accuracy by pre-training a cascaded network with simulated data. Subsequently, we fine-tune the network based on a small real-world data set recorded from the physical robot.
Existing static and kinematic models of concentric tube robots are based on the ordinary differential equations of a static Cosserat rod. In this article, we provide the first dynamic model for concentric tube continuum robots by adapting the partial differential equations of a dynamic Cosserat rod to describe the coupled inertial dynamics of precurved concentric tubes. This generates an initial-boundary-value problem that can capture robot vibrations over time. We solve this model numerically at high time resolutions using implicit finite differences in time and arc length. This approach is capable of resolving the high-frequency torsional dynamics that occur during unstable “snapping” motions and provides a simulation tool that can track the true robot configuration through such transitions. Furthermore, it can track slower oscillations associated with bending and torsion as a robot interacts with tissue at real-time speeds. Experimental verification of the model shows that this wide range of effects is captured efficiently and accurately.