IEEE TRANSACTIONS ON ROBOTICS 1

Concentric Tube Robot Design and OptimizationBased on Task and Anatomical Constraints

Christos Bergeles, Andrew H. Gosline,Nikolay V. Vasilyev, Patrick J. Codd, Pedro J. del Nido and Pierre E. Dupont

Abstract—Concentric tube robots are catheter-sized continuumrobots that are well suited for minimally invasive surgery insideconfined body cavities. These robots are constructed from setsof pre-curved superelastic tubes and are capable of assumingcomplex 3D curves. The family of 3D curves that the robotcan assume depends on the number, curvatures, lengths andstiffnesses of the tubes in its tube set. The robot design probleminvolves solving for a tube set that will produce the family ofcurves necessary to perform a surgical procedure. At a minimum,these curves must enable the robot to smoothly extend intothe body and to manipulate tools over the desired surgicalworkspace while respecting anatomical constraints. This paperintroduces an optimization framework that utilizes procedure-or patient-specific image-based anatomical models along withsurgical workspace requirements to generate robot tube setdesigns. The algorithm searches for designs that minimize robotlength and curvature and for which all paths required for theprocedure consist of stable robot configurations. Two mechanics-based kinematic models are used. Initial designs are sought usinga model assuming torsional rigidity. These designs are thenrefined using a torsionally-compliant model. The approach isillustrated with clinically relevant examples from neurosurgeryand intracardiac surgery.

I. INTRODUCTION

While in a few important cases, anatomical constraints canbe obviated, e.g., by insufflation of the abdominal cavity,there are many sites within the body for which reducingprocedural invasiveness requires inserting instruments alongtortuous paths in a follow-the-leader fashion and manipulatingtip-mounted tools inside small body cavities. Such situations,involving coordinated control of an instrument’s many degreesof freedom to navigate in complex 3D geometries, are wellsuited to robotic solutions using continuum-type (continuouscurve) architectures [2]–[6].

In some interventions, such as those performed by cathetersor endoscope, the robotic devices are passive along much oftheir length and rely on contact with the surrounding tissueto guide their advance through passageways of the body. Anycompliance introduced to limit contact forces, however, alsoreduces tip stiffness and consequently limits what tasks canbe performed at a robot’s tip. Furthermore, reliance on tissue

This work was supported by the National Institutes of Health under grantsR01HL073647, R01HL087797 and R01HL124020.

A portion of this work has been presented at the 2011 IEEE Int. Conf.Robot. Autom. [1].

C. Bergeles, A. H. Gosline, N. V. Vasilyev, P. J. del Nido, and P. E. Dupontare with the Department of Cardiovascular Surgery, Boston Children’s Hospi-tal, Harvard Medical School, Boston, MA, 02115, USA. P. Codd is with theDepartment of Neurosurgery, Boston Children’s Hospital, Harvard MedicalSchool, Boston, MA, 02115, USA. C. Bergeles is now with the HamlynCentre, Imperial College London, London SW7 2AZ, UK.

Correspondence: P. E. Dupont (pierre.dupont@childrens.harvard.edu)

Delivery Canula Tube 1& 2

Tube 3

Tube 4

θ1, θ2

θ3

θ4φ1 = φ2

φ3 φ4

û1 = û2, k1, L1= L2

û3, k3, L3

û4, k4, L4

[ [[ [û2 = û1, k2, L2= L1

[ [

[ [z

x

y

zx

y

z

x

y

Fig. 1: Concentric tube robot comprised of three curved telescopingsections that can be rotated and translated with respect to each other.The first section, comprising two tubes, is a variable curvature section.

contact for steering can result in damage to sensitive tissues.Thus, the distribution of degrees of freedom along a robot’slength together with selection of materials and desired stiffnessare closely coupled to the clinical application.

Concentric tube robots are one type of continuum robot, asshown in Fig. 1, with cross sections comparable to needlesand catheters. They are capable of actively-controlled lateralmotion and force application along their entire length. Further-more, the lumen of the tubes can act as a tool delivery channeland can house additional tubes and wires for controllingarticulated tip-mounted tools. They can be fabricated from avariety of materials in order to achieve a range of compliancesfor a given diameter.

While not considered here, they can also be used as steer-able needles. In this way, if anatomical constraints precludereaching a surgical site entirely through body lumens, theycan be steered through through a combination of tissue andfluid-filled spaces to reach a target.

While concentric tube robots are a recent innovation, sub-stantial progress has been made in formulating the underlyingtheory and in adapting the technology for specific medicalapplications [1], [7]–[16]. Design principles have been formu-lated [8] and mechanics-based kinematic and quasistatic forcemodels have been derived [8]–[11]. Since robot shape dependson elastic deformation of the component tubes, the stability ofsolutions obtained from these models has also been studied[8], [9]. A variety of model-based approaches to real-timecontrol have been formulated [8], [12], [17]. Path planningalgorithms are also being developed to enable robot navigationwithin anatomical constraints [13], [18]. Clinical applicationsconsidered to date include neurosurgery [7], [19], lung surgery[13], [14], [18] and cardiac surgery [1], [15], [16], including invivo demonstrations of percutaneous beating-heart intracardiacsurgery in an animal model [16], [20].

A topic that has received less attention is how to designa concentric tube robot to meet the constraints imposed bya specific surgical task and anatomical environment [1], [7],

IEEE TRANSACTIONS ON ROBOTICS 2

TABLE I: Nomenclature, pt. 1

Symbol Descriptiong

c

3D curve for follow-the-leader extensiont Time instances during follow-the-leader extensions

c

Physical location of a robot that follows curve s

N Set of nonnegative integersR Set of real numbersn Number of tubes in a robot designm Number of sections in a robot designm

n

Number of sections in robot navigation portionm

m

Number of sections in robot manipulation portionv Number of robot variable curvature sections⇢ Total number of arrangements of v variable

curvature sectionst Number of tip task frames for specific procedureV Binary m-vector specifying variable curvature sectionsV

n

Binary m

n

-vector specifying variable curvature sectionsin navigation portion of robot

V

m

Binary m

m

-vector specifying variable curvature sectionsin manipulator portion of robot

V

p

Binary vector specifying variable curvature sectionsextending from straight proximal sections

s Arc length along centerline of tube or tube setL

i

Total length of tube i

ix

(s) Bending pre-curvature of ith tube about xas function of arc length, s

iy

(s) Bending pre-curvature of ith tube about yas function of arc length, s

u

j

Bending pre-curvature of jth tube or sectionu ∈ Rm Vector of section pre-curvaturesu

n

∈ Rmn Vector of navigation section pre-curvaturesu

m

∈ Rmm Vector of manipulation section pre-curvatures

[14], [21]. The robot design problem is of high computa-tional complexity since evaluation of each candidate solutioninvolves solving a path planning problem for a robot whosekinematic model is derived as the solution to a 3D beam-bending problem with split boundary conditions.

Tractability of the design problem can be achieved byprescribing design guidelines that constrain the free (tube) pa-rameters, but this is challenging since, while the mathematicalkinematic model and stability results for a pair of tubes areknown, by themselves they do not provide any intuition aboutwhat the workspace of a specific robot will look like nor wherein its workspace it will be stable.

The main contribution of this paper is a design methodologyand optimization framework based on anatomical and surgicaltask constraints that considerably reduces the dimensionalityof the design space while still providing a rich solution set.Surgical tasks are prescribed as regions of the robot workspacerepresented as sets of tip coordinate frames. Robot-anatomyinteraction constraints are specified with respect to image-based 3D models of the anatomy. Path planning is performedimplicitly by defining a sufficiently dense set of tip coordinateframes in the task description. Computational tractability isachieved using a simplified (torsionally rigid) kinematic modelduring the initial tube parameters search. Model refinementis then performed using the torsionally compliant kinematicmodel.

This paper provides a number of contributions beyond theinitial design optimisation approach presented in [1]. In sectionIIA, geometric conditions for follow-the-leader insertion arederived to motivate the design rules. The effect of the design

TABLE II: Nomenclature, pt. 2

Symbol Description�

i

Relative extension of the ith tube or section�

i

Maximum relative extension of the ith tube or section�

p

, �p

Extension variables for proximal section�

d

, �d

Extension variables for distal sectionk

ix

(s) Bending stiffness of ith tube about xas function of arc length, s

k

iy

(s) Bending stiffness of ith tube about yas function of arc length, s

k

B

Bending stiffness of distal manipulation sectionk

A

Bending stiffness of distal navigation sectionD Stiffness ratio of a robot section with respect to

proximal section✓

i

Rotation of the ith tube↵

i

Relative rotation of the ith tube with respect to tube 1Tu

Unconstrained robot tube setT Robot tube set satisfying design rules of Section II-Aq Set of kinematic input variablesRoC Radius of curvature, equivalent to 1

r Radius of a tubeB

i

, B The set of surgical task frames, B = {Bi

, i = 1, . . . , t}E Frame of entry into the anatomyE

g

Initial guess for frame of entry into the anatomyA Frame of the manipulator baseA

g

Initial guess for manipulator base frameR(E,A) Clinician selected regions for frames E and A

e

B

x

x-axis vector for frame B

�, �m

, �n

Representation of the anatomy�, �

n

Penalty function for the anatomy⌫ Poisson’s ratioR

z

(✓) Rotation matrix of ✓ around the z-axis⌦ Occupancy volume for anatomical model⌦

r

Occupancy volume eroded by radius r

U

r

Distance map corresponding to ⌦r

S

r

Spherical structural element of radius r

P

i

Point on robot centerline, i = 1, . . . , o

rules in reducing the number of design variables and thussimplifying the minimization problem is presented in sectionIIC. Moreover, in section IID, this paper examines for the firsttime the effect of section type (variable or fixed curvature) andarrangement of section types on the workspace of a concentrictube robot and defines the boundaries of the workspace interms of the section variables. This leads to counterintuitiveresults crucial for understanding the robot design problem.

This is also the first paper to include elastic stabilityin the concentric tube robot design process. To do so, theoptimization function has been adapted to include heuristicsthat maximise robot stability. It is demonstrated that designsexhibiting instabilities can be used as long as unstable config-urations are avoided.

Another improvement is that while [1] considered a set oftip targets, it had not addressed whether the robot could reachthose targets from its entry point in the anatomy nor whetherit could safely move between them, i.e., path planning to thetargets was not considered. Here, implicit path planning isperformed by introducing a sequence of waypoints startingfrom the entry location (section IIE and examples). This is acrucial issue when elastic instability is considered.

While our prior work utilized a simplified kinematic model,the approach presented here uses both simplified and com-plete models to speed the design process without sacrificingaccuracy (section IV). Also, [1] had implemented anatomical

IEEE TRANSACTIONS ON ROBOTICS 3

interference as a binary decision function, which necessitatedthe use of computationally intensive global optimization tech-niques. Our current submission substitutes potential fields andso greatly reduces the computational time involved in solvingfor a design (section IVB).

Finally, the clinical design examples presented in sectionV are more sophisticated and complete than prior publishedresults. In particular, the neurosurgical example solves for arobot design that can safely navigate through both ventri-cles from a single insertion point while prior designs wereconstrained to navigating within a single ventricle [7]. Wehave also added an experimental validation of a robot designfor intracardiac PFO closure by comparing it against a robotsuccessfully employed in beating-heart procedures [16], [20].

This paper is structured as follows. Sec. II presents our robotdesign methodology that is based on the architecture of Fig.1 in which tube sets are constrained so that the robot takesthe form of a telescoping concatenation of fixed and variablecurvature sections. The effect of section type and arrangementof sections on robot shape, workspace and solution stabilityis also explored. This section also introduces a decomposi-tion of the design problem in which the distal sections arefirst designed to achieve the desired surgical workspace and,subsequently, the proximal sections are designed to navigateand position the distal sections at the surgical site. Thedesign optimization framework is presented in Sec. III andimplementation details are provided in Sec. IV. The designapproach is validated for two challenging clinical proceduresin Sec. V and conclusions appear in Sec. VI. All variablenames used in the paper are listed in Table I and Table II.

II. ROBOT DESIGN

In contrast to standard robots possessing rigid links anddiscrete joints, concentric tube robots are continuum robots.When their constituent pre-curved tubes are inserted insideeach other, their common axis conforms to a mutual resultantcurvature. By controlling relative translations and rotations ofthe tubes at their proximal ends, the shape and length of therobot can be varied. Thus, the tubes act as both links andflexure joints. By extending these robots telescopically, theyoffer the potential to act as steerable needles following curvedpaths through tissue while also being capable of manipulatingtools inside body cavities.

Unlike hyper-redundant continuum robots, however, that areoften modeled using large numbers of independently actuatedrevolute or universal joints that are closely spaced with respectto arc length [22], [23], concentric tube robots possess a muchsmaller number of degrees of freedom equal, at most, to twicethe number of tubes comprising the robot. Furthermore, it isdifficult to predict the workspace and arm motions producedby a robot constructed from tubes of arbitrary pre-curvatureand relative stiffness since the effect of rotating or translatingany individual tube is not localized in arc length and maychange the shape along the entire length of the robot.

By focusing on the desired capabilities, it is possible toconstrain the design space to those tube sets most likelyto produce clinically relevant solutions. In particular, thefollowing properties are desired:

Retracted Portion of Robotg0

sr=0 sc=0

{ gr(sr )

v

sr ,sc

Extended Portion of Robot

{gr(Lr)

sr=Lrsc=vt

gc(sc )

Desired Curve

sc=Lc

Delivery Cannula

Fig. 2: Follow-the-leader robot extension. Robot cross sections,described by g

r

(sr

), move along desired curve, described by g

c

(sc

)with arc length velocity, v.

1) The ability to follow curved paths through tissue whileexerting minimal lateral forces and to navigate throughnarrow curved body passages, and

2) The ability to perform complex tissue manipulations atthe interventional site while moving only distal sections.

The first property corresponds to follow-the-leader insertionas a robot extends along a desired 3D curve, typically to reacha desired site inside the body. The second property providesfor the proximal portion of the robot to be used primarily fornavigation to an interventional site while the distal portion isused, independently, for tissue interaction. As shown below,the design guidelines to achieve follow-the-leader insertionalso provide this property.

A. Follow-the-Leader Extension

In follow-the-leader extension, a 3D curve is defined usinga coordinate frame, g

c

(sc

), parameterized by curve arc length,s

c

, as shown in Fig. 2. The initial frame of the curve, gc

(0) =g

0

is defined at the base of the robot and the curve itself isgiven by the solution to the differential equation

dg

c

ds

c

= gc

(sc

) � u

c

(sc

) v

c

(sc

)0 0

� , g

c

(0) = g0

(1)

in which u

c

(sc

) ∈R3 is the body-frame curvature vector andv

c

(sc

) = [0 0 1]T . Since robot cross sections slide along thecurve during extension, robot arc length, s

r

∈ [0, Lr

] is definedindependently with s

r

= 0 at the proximal end and s

r

= Lr

atthe distal end.

Assuming constant velocity extension, v, the robot crosssection, s

r

, is physically located along the curve at time t at

s

c

(sr

, t) = sr

− (Lr

− vt), ∀t ≥ 0,∀ s

r

∈ [Lr

− vt,Lr

] (2)

in which L

r

− vt is the length of the retracted portion of therobot. For follow-the-leader extension, at each instant of time,t, every robot cross section in the interval of s

r

must satisfy(1) such that

g

r

(sr

, t) = gc

(sc

(sr

, t))u

r

(sr

, t) = uc

(sc

(sr

, t))v

r

(sr

, t) = vc

(3)

Furthermore, each robot cross section must bend with timeas it slides along the curve with arc length velocity, v. The

IEEE TRANSACTIONS ON ROBOTICS 4

temporal variation in robot curvature is

du

r

(sr

, t)dt

= @u

c

(sc

)@s

c

ds

c

dt

= v@uc

(sc

)@s

c

,

∀t ≥ 0,∀ s

r

∈ [Lr

− vt,Lr

](4)

Recognizing that the time dependence of u

r

is through thekinematic input variables, q, this equation can be rewritten as

@u

r

(sr

, q)@q

dq

dt

= v@uc

(sc

)@s

c

,

∀t ≥ 0,∀ s

r

∈ [Lr

− vt,Lr

] (5)

To satisfy this equation for all values of the continuous variables

r

∈ [Lr

− vt,Lr

] at any time t ≥ 0 would require q to be ofinfinite dimension. Consequently, follow-the-leader extensionalong an arbitrary curve can only be performed by a robotwith infinite degrees of freedom.

An alternative approach for robots with finite degrees offreedom is to constrain the set of curves to be followed.One important set of curves are those in which curvature isindependent of arc length, corresponding to the trivial solutionof (5) given by

@u

c

(sc

)@s

c

= 0, ∀sc

∈ [0, vt] (6)

By (4), this implies that robot curvature is independent of timeand by (3) yields the solution

u

r

(sr

, q) = uc

(sc

) = const, ∀t ≥ 0,∀ s

r

∈ [Lr

− vt,Lr

] (7)

This constant-curvature solution is comprised of arcs (whenthe z-component of u

c

is zero) and helices (when the z-component is non-zero). Notice that this solution does notimply that dq�dt = 0 in (5) since, for example, some kinematicvariables control extension.

Thus, any robot architecture that can extend with con-stant curvature can perform follow-the-leader extension alongcurves comprised of arcs or helices. This result can be gener-alized if the robot design enables the trivial solution of (5) tobe applied over m subintervals of s

c

∈ [0, Lc

], each of whichcan be taken as constant curvature yielding an overall curveof piecewise constant curvature,

u

r

(sr

) = uc

(sc

) =�����������������

c

1

, ∀sc

∈ [0, sc1

]c

2

, ∀sc

∈ (sc1

, s

c2

]c

3

, ∀sc

∈ (sc2

, s

c3

]. . .

c

n

, ∀sc

∈ (sc,m−1, Lc

](8)

As shown below, applying these geometric results for follow-the-leader extension on concentric tube robots is straightfor-ward. Follow-the-leader conditions are also considered in [24].

B. Design Guidelines

By constraining the parameter space, concentric tube robotdesigns can be made to provide the two desired clinical prop-erties of follow-the-leader insertion and independent motion ofthe distal sections. These are achieved through the followingthree design rules.

1) The pre-curvature of each tube is piecewise constant.2) The bending stiffness of each telescoping section dom-

inates that of all distal sections.

3) Each telescoping section is designed to be of either fixedcurvature or of varying curvature.

The first two rules, taken together, enable a design toapproximately satisfy (8) for a specific piecewise-constant-curvature curve. The third rule enables a single design tosatisfy (8) for a parameterized family of piecewise-constant-curvature curves. The second and third rules also provide thesecond desired clinical property - the ability to perform tissuemanipulation at the robot tip while moving only distal sections.

The first rule is based on the result that concentrically com-bined tubes of piecewise constant curvature yield a telescopingshape that is also approximately piecewise constant. This hasbeen considered in detail for arcs in, e.g., [8] and initial resultsfor helices appear in [24]. Without loss of generality and tofurther reduce the number of design parameters, only arcs areconsidered in the remainder of this paper.

To satisfy (8), it must also be true that, during telescopicinsertion, extension proceeds from the most proximal sectionto the most distal and, as each constant-curvature sectionextends, the proximal sections should not be displaced laterallyfrom the desired curve. The same must be true to performtissue manipulations using only the distal sections.

This can be achieved by selecting the bending stiffness (and,consequently, the torsional stiffness) of each section to besubstantially larger than the combined stiffness of the distalsections. The design examples in this paper use a stiffnessratio of 10 between adjacent sections, but ratios of 6-8 haveproven sufficient in practice. In addition to follow-the-leaderextension, this rule is also advantageous since it produces anapproximate kinematic decoupling between each telescopingsection of the robot.

The third design rule prescribes each telescoping sectionto be of either fixed or variable curvature. A single tube isrequired to construct a constant curvature section while twotubes are needed to construct a variable curvature section [8].

A fixed curvature section extends along its pre-curved curva-ture when extended from its stiffer preceding section. In con-trast, the extended portion of a variable curvature section cantake on a continuous range of curvature magnitudes usuallyranging between zero (straight) and a maximum value. Thesecan be interpreted as continuum-robot analogs to prismatic androtary joints, respectively.

In follow-the-leader extension, a section of fixed curvaturecan only assume its pre-curved value over its interval of arclength in (8) while a variable curvature section can assume anycurvature in its permissible range, e.g., ��c

i

�� ∈ [0, ��ci,max

��],enabling extension along a family of curves parameterized bythe curvatures of these sections. For tissue manipulation usingthe distal sections, fixed and variable curvature sections canbe combined to produce the task-prescribed workspace whilerespecting anatomical constraints.

Using these rules, the design problem is to solve for atelescoping arrangement of fixed- and variable-curvature robotsections in which the proximal sections are predominantlyused for follow-the-leader navigation to the interventional siteand the distal sections are used to perform the intervention.The effect of these rules on reducing the dimension of thedesign space is detailed below.

IEEE TRANSACTIONS ON ROBOTICS 5

C. Design Variables

The unconstrained robot design problem consists of solvingfor the discrete variable, n, defining the number of tubes, andfor the curvature and bending stiffness of each tube as continu-ous functions of arc length, s. Using Bishop coordinate framesfor each tube as shown in Fig. 1, the unknown pre-curvaturefunctions are given by [

ix

(s),

iy

(s)]T , s ∈ [0, Li

], i =1,2, . . . , n. Assuming circular cross sections for the tubes, thebending stiffnesses will equate in the x and y directions suchthat there is a single unknown continuous stiffness functionfor each tube, k

ix

(s) = k

iy

(s), s ∈ [0, Li

], i = 1,2, . . . , n.Together, the variables define an unconstrained robot tube set,denoted by:T

u

= {n ∈ N, (s) ∈ R2×n, k(s) ∈ R2×n

, L ∈ Rn} (9)

By imposing the design rules of the preceding subsection,solving for these continuous functions is reduced to solvingfor a set of discrete parameters for each tube. To identify thisset, consider first that constant curvature sections have twokinematic input variables, {�

i

, ✓

i

} corresponding to sectionextension length and tube rotation. Variable curvature sec-tions consist of two tubes of equal bending stiffness whichundergo identical translations but individual rotations. Thesesections possess three independent kinematic input variables{�

i

= �i+1, ✓i, ✓i+1}. The angles {✓

i

, ✓

i+1} control rotation andcurvature of the section and �

i

controls extension arc length.Given that the robot comprises m telescoping sections, the

pre-curvatures of the tubes comprising a section are given by

[jx

(s),

jy

(s)]T = � [0,0]T, s ∈ [0, Lj

−�j

][0, uj

]T , s ∈ [Lj

−�j

, L

j

] (10)

in which j = 1,2, . . . ,m, m ≤ n, and u

j

is the pre-curvatureover the distal length �

j

of the jth section composed oftubes with total length of L

j

. Note that L

j

are dependentparameters since they can be computed from the maximumsection lengths, �

j

.If the bending stiffnesses of the sections are selected accord-

ing to a single stiffness ratio, D >> 1 then the free parametersassociated with stiffness reduce to two discrete values – the ac-tual bending stiffness of any one section and the ratio, D. Forexample, it is often useful to specify the stiffness of the mostdistal section, k

d

, since it is the most compliant. Naturally,the stiffness, radius, and maximum possible pre-curvature ofa tube are related through its mechanical properties.

The design rules also replace the selection of tube number,n, with the selection of section number m along with selectionof the number of variable curvature sections, v. These arerelated by:

n =m + v, v ≥ 0 (11)

If v > 0 then the location of the variable curvature sectionsalong the length of the robot must also be specified. Thenumber ⇢ of arrangements is given by the permutations ofm sections taken v at a time:

⇢ =m!�v!(m − v)! (12)

Due to the exponential nature of the equation, there is adrastic difference, for example, between using 3 sections (8

combinations) and 5 sections (32 combinations). Eq. (12)underlines this combinatorial explosion for the general robotdesign problem that follows the guidelines provided in thispaper.

In summary, the design rules replace solving for con-tinuous functions of curvature and bending stiffness for n

individual tubes, as well as their lengths, with solving forthe 2m + 2 parameters corresponding to the curvature andmaximum extension length of each section along with twostiffness parameters. Together with the number and locationof the variable curvature sections, these provide a completedescription of the robot tube set,T = {m ∈ N, V ∈ Nv

, u ∈ Rm

,� ∈ Rm

,D ∈ R, kd

∈ R} (13)

in which the V ∈ Nv specifies the variable curvature sections.To potentially prune the search space so as to avoid con-

sidering all 2m possible combinations of fixed and variablecurvature sections, it is worthwhile to gain insight into howthe number and arrangement of variable curvature sectionsaffect robot workspace and section stability. These topics areconsidered in the following subsection.

D. Effect of Section Type

While workspaces of standard robot architectures, such asSCARA or PUMA arms, are well known, there are no priorresults for concentric tube robots. Since the number, typeand arrangement of robot sections are inputs to the designprocess, only by understanding the achievable workspacescan one intelligently select these inputs. For example, whilevariable curvature sections possess an extra degree of freedomcompared to fixed curvature sections, they also require an addi-tional tube. This can increase both the cost and diameter of thetubes comprising a robot design and potentially introduce aninstability associated with straightening the variable curvaturetube pair. In order to guide the design process, the effects ofsection type and arrangement on workspace and stability aredeveloped below.

1) Workspace: To gain such an understanding, four two-section concentric tube robots are considered here. Listingthe section type from base to tip, these are: (1) fixed-fixedcurvature, (2) fixed-variable curvature, (3) variable-fixed cur-vature and (4) variable-variable curvature. These designs arecomprised of two, three (two designs) and four tubes andpossess four, five (two designs) and six degrees of freedom,respectively.

Using the elastically stable parameter set of Design 1 inTable III, the workspaces, comprising the sets of reachabletip positions, are compared in Fig. 3. It is assumed that thetwo sections are extended from a straight rigid vertically-oriented cannula whose tip is located at the origin. Due to thecylindrical symmetry of the workspaces, only the xz plane isplotted.

The plots were created using Monte-Carlo simulation togenerate 2 million kinematic configurations using the torsion-ally compliant model of [8] through uniform sampling of eachkinematic variable. All configurations were rotated about thez axis to place the robot tip on the xz plane. This resulted in a

IEEE TRANSACTIONS ON ROBOTICS 6

TABLE III: Parameters of robot design examples.

Base location [mm] [0,0,0]TEntry vector [0,0,1]TSection stiffness ratio D = 10Design 1 - StableSection 1�i [mm] �i [mm] Li [mm]proximal 60 40 40distal 25 20 60

Design 2 - UnstableSection 1�i [mm] �i [mm] Li [mm]proximal 40 40 40distal 10 20 60

60

50

40

30

20

10

0

z-ax

is [m

m]

0 5 10 15 20 25 30x-axis [mm]

A

B

C

D

E

(a) Fixed-fixed case

60

50

40

30

20

10

0

z-ax

is [m

m]

0 5 10 15 20 25 30x-axis [mm]

A

B

C

D

E

(b) Fixed-variable case

60

50

40

30

20

10

0

z-ax

is [m

m]

0 5 10 15 20 25 30x-axis [mm]

A

B

C

DE

(c) Variable-fixed case

Fig. 3: Tip position workspace for robot Design 1 of Table II showingxz-plane slices. Complete workspace is generated by rotation of sliceabout z axis. (a) Fixed-fixed curvature sections, (b) fixed-variablecurvature sections, (c) variable-fixed curvature sections. Dark shadedarea in each plot is workspace of variable-variable curvature design.Red ○ are tip positions used to generate Fig. 4.

dense workspace point cloud, which was subsequently binnedinto 250 × 250µm clusters. Alternative efficient methods forcalculation of this workspace density can be found in [25],[26].

The curves forming the workspace boundaries are describedin Table IV. Except for EA, these curves are generated aslimiting values of section extension. Thus, while specificparameter values were used to generate these plots, they arerepresentative of their designs, and researchers can use thistable to compute the workspace of their robot without perform-ing Monte-Carlo simulations, clustering, and visualisation.

Several important observations can be made in comparingworkspaces. First, the workspace of the variable-variable de-sign, depicted as the dark shaded area in each subfigure, is asuperset of all other workspaces and so provides a benchmarkfor comparing the other workspaces. Second, while the fixed-variable design is comprised of three tubes, its workspaceis very close to that of the fixed-fixed design that requiresonly two tubes. Furthermore, the workspace of the three-tubevariable-fixed design is close to that of the four-tube variable-variable design. In particular, it eliminates the central voidlocated along the longitudinal z-axis.

Since these robot designs possess 4−6 degrees of freedom,it is also worthwhile to consider the range of orientations

that can be achieved at each tip position in the workspace.The families of solutions for the labeled points of Fig. 3 aredepicted in Fig. 4. The xz-plane views on the top show asubset of solutions for clarity. To illustrate the 3D geometryof the solution sets, the intersections of the robot configurationsets with cut planes are also plotted in the figure. The cut planeviews illustrate the range of robot shapes associated with a tipposition that can be used to satisfy anatomical and stabilityconstraints. Smaller cut plane sets provide fewer solutions forsatisfying these constraints. The variable-fixed design can beseen to provide the largest set of shapes.

In summary, for two-section robot designs, the 3-tubevariable-fixed curvature section design offers advantages bothin workspace size and range of possible orientations at eachpoint within the workspace. This design possesses five degreesof freedom (DOF) with the missing DOF corresponding toa roll rotation at the tip. Roll can easily be added to a tip-deployed tool through an inner rotating straight tube. Thus,the variable-fixed design can be a good choice for the manip-ulation portion of a robot design when using the navigationand manipulation decomposition described in Section II-E.Moreover, these results demonstrate that, counterintuitively, adistal variable curvature section provides minimal benefit overa fixed curvature section in terms of workspace.

2) Stability: When two or more curved tubes undergorelative rotation at their base, elastic energy is stored andreleased through twisting and bending of the tubes. As thecurvatures and lengths of the tubes increase, the mappingfrom kinematic input variables (base rotations and extensionsof tubes) to robot tip frame can fail to be injective with theextra solutions corresponding to elastically unstable solutions[8]. To uniquely describe all solutions, a robot configurationis defined here by both the kinematic input variables and bythe associated tip frame.

Since the instability occurs only for specific configurationsof the tubes, such designs can still be used as long asthe unstable configurations are avoided. For example, Fig. 5illustrates the case for the variable-fixed section arrangementof Design 2 in Table III. For this tube set, there is aninstability associated with rotating the distal fixed curvaturesection while partially retracted into the curved balanced pair.Two configurations associated with the same tip position, butdifferent extensions are shown in Fig. 5. In configuration 1, thedistal curved section is substantially retracted into the proximalsection and oriented so that the curvatures oppose each other.This configuration is unstable. In contrast, configuration 2achieves the same workspace position as configuration 1, butit is stable since the distal section is substantially extendedand the curvatures of the two sections are aligned.

Such instabilities can be graphically visualized for specificvalues of relative section extension as shown in Fig. 5. Therelative twist angles at the tips of the tubes, ↵

i

(Li

) are plottedwith respect to the relative twist angles at the proximal end ofthe robot, ↵

i

(0), with ↵

i

defined by:

↵

i

= ✓i

− ✓1

, i = 2, . . . , n (14)

A configuration can be unstable if multiple values of ↵

i

(Li

)correspond to the same value of ↵

i

(0). Graphically, this occurs

IEEE TRANSACTIONS ON ROBOTICS 7

TABLE IV: Workspace Boundary Curves for Fig. 3.

Boundary Curve Section Variables Boundary CurvatureAB �

p

= 0 0 ≤ �d

≤ �d

u

d

= ud

u

p

= up

u

d

BC 0 ≤ �p

≤ �p

�

d

= �d

u

d

= ud

u

p

= up

u

p

CD �

p

= �p

�

d

= �d

0 ≤ ud

≤ ud

0 ≤ up

≤ up

- -DE 0 ≤ �

p

≤ �p

�

d

= �d

u

d

= ud

0 ≤ up

≤ up

u

p

EA 0 ≤ �p

≤ �p

0 ≤ �d

≤ �d

0 ≤ ud

≤ ud

0 ≤ up

≤ up

0

0 10 20 300

5

10

15

20

25

30

35

40

45

50

z-ax

is [m

m]

x-axis [mm]

0 2 4 6 8 10 12 14 16-4

-2

0

2

4

x-axis [mm]

y-ax

is [m

m]

cut #1

cut #1

cut #2

cut #2

cut #3

cut #3

(a) Fixed-fixed case

0 10 20 300

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12 14 16-4

-2

0

2

4

x-axis [mm]

y-ax

is [m

m]

z-ax

is [m

m]

x-axis [mm]

cut #1 cut #2 cut #3

cut #1

cut #2

cut #3

(b) Fixed-variable case

0 10 20 300

5

10

15

20

25

30

35

40

45

50

x-axis [mm]

z-ax

is [m

m]

0 2 4 6 8 10 12 14 16-4

-2

0

2

4

x-axis [mm]y-

axis

[mm

]

cut #1 cut #2 cut #3

cut #1

cut #2

cut #3

(c) Variable-fixed case

Fig. 4: Solution sets of orientations for three tip positions labeled in Fig. 3. Cut planes show cross sections of solution sets. (a) Fixed-fixeddesign, (b) Fixed-variable design, (c) Variable-fixed design.

when the planar cuts of the surfaces resemble s-shaped curves.For the stable configuration of Fig. 5(a), the retracted distalfixed curvature section possesses a single solution for ↵

2

(L)[Fig. 5(b)], but has multiple solutions associated with ↵

3

(L)[Fig. 5(c)]. In contrast, for the stable configuration of Fig.5(a), the substantially larger distal section extension producesunique twist angle solutions as shown in Figs. 5(d), (e). Inthis paper, this approach was used to evaluate the stabilityof specific robot configurations. A configuration was deemedunstable if any of the directional derivatives of ↵

i

(Li

) withrespect to ↵i(0) were negative.

Since each tip position in the workspace may be reachablethrough multiple tube configurations (associated with differentorientations; see Fig. 4), the workspace can be divided asshown in Fig. 5 into sets comprising tip positions that arestable for all configurations and those that are stable for someconfigurations. Path planning through these positions involvessolving for stable configurations.

The following heuristics can be defined to guide an op-timization toward stable configurations. The first two aremotivated by the examples above while the third follows frominequality (38) in [8] which relates the existence of unstableconfigurations to the length of a variable curvature section.

As shorthand below, one configuration is defined as more orless stable than another based on their relative distance in thespace of kinematic variables to an unstable configuration.● Variable curvature sections are most stable at maximum

curvature.● The stability of adjacent constant curvature sections in-

creases as the distal section is extended (assuming thatthe retracted transmission portion of the extended sectionhas zero curvature).

● If a variable curvature section extends from a straightdominating proximal section, stability of the variablecurvature section increases as it is retracted into thestraight proximal section.

E. Navigation and Manipulation Design Decomposition

As depicted in Fig. 6, minimally invasive surgery mayinvolve navigating through narrow body lumens to reachsurgical targets and, subsequently, deploying and manipulatingtools in confined spaces to perform the procedure. In thecase of concentric tube robots, navigation to the surgical siteinvolves telescopic extension and steering from the entry pointon the body, defined by coordinate frame E, to the entrypoint into the body lumen where the surgery will occur,

IEEE TRANSACTIONS ON ROBOTICS 8

0 5 10 15 20 25 30 35x-axis [mm]

0

10

20

30

40

50

60

z-ax

is [m

m]

(a)

Configuration 1

Configuration 2

0 1 2 3 4 5 60

2

4

60

1

2

3

4

5

6

α 2(L

) [ra

d]

0

2

4

60 1 2 3 4 5 6

0

1

2

3

4

5

6

a31basea21base

a31t

ip

α3 (0) [rad]

α2(0) [rad]

0

2

4

60 1 2 3 4 5 6

0

1

2

3

4

5

6

α3(0) [rad]

α2 (0) [rad]

α 3(L

) [ra

d]

(b) (c)

0 1 2 3 4 5 60

2

4

60

1

2

3

4

5

6

α2(0) [rad]

α3 (0) [rad]

α 2(L

) [ra

d]

(d) (e)

01

23

45

60 1 2 3 4 5 6

0

1

2

3

4

5

6

α 3(L

) [ra

d]

α2 (0) [rad]

α3(0) [rad]

Fig. 5: Workspace of variable-fixed design. (a) Configuration 1 (dotted) is unstable and snaps. Configuration 2 is stable. Region of workspacecontaining unstable configurations is indicated in blue. (b)-(c) S-surfaces of Configuration 1, (d)-(e) S-surfaces of Configuration 2.

denoted by coordinate frame A. Once inside this body lumen(e.g., a chamber of the heart), it is often desirable to controlthe position and orientation of the instrument’s distal tipto manipulate tools, e.g., to reach the set of tip coordinateframes, B

i

, i = 1, . . . , t, while holding relatively immobilethe proximal length responsible for navigation.

For concentric tube robots, this leads to a natural decom-position over the length of the robot in which the proximalsections are responsible for navigation and the distal sectionsare responsible for tissue manipulation [see Fig. 6(b)]. Manyinterventions fit this decomposition, such as those inside theheart, the fluid-filled spaces of the central nervous system,the throat, the lungs and the kidneys. Accordingly, the robotdesign problem can be decomposed into a sequence of twosimpler problems in which the distal manipulation sectionsare designed first and the navigation sections subsequently.

As shown in Fig. 6, the navigation portion of the robotextends between coordinate frames E and A, and the manip-ulation portion of the robot extends from frame A to the setof tip task frames, B. This set of t tip task frames are selectedby the clinician to define the region (i.e., curve, surface orvolume) of anatomical locations that the robot tip must reachto perform a procedure. This set may also include waypointsto enable safe or stable navigation of the robot tip from frameA to B

1

and also between various task frames as needed.Depending on the surgical task, different components of the B

i

may be unspecified, e.g., only tip position may be important.While the tip task frames, B

i

, are selected as specificlocations with respect to the anatomy, there is usually somefreedom in locating the navigation frames E and A. Conse-quently, the clinician selects regions, labeledR(E) andR(A),in which these frames can be located and the robot design

algorithm selects the specific frames within these regions.All of these frames and regions must be defined with respect

to an anatomical model that is derived from images generated,e.g., using Magnetic Resonance Imaging (MRI), ComputedTomography (CT) or 3D ultrasound, together with softwaretools that enable user-guided organ segmentation and render-ing, e.g. ITK-Snap. Given this anatomical information, termed�, the anatomical constraints, termed �(�), may be specifiedby the clinician in accordance with the various types of tissuelocated along the length of the robot. For example, in thecontext of intracardiac surgery, constraints on the navigationportion of the robot passing through the vasculature shouldbe defined to avoid puncture or large deflections. In contrast,constraints on the manipulation portion of the robot inside theheart should be defined to avoid contact with the heart wall.

Using this terminology, the overall robot design problemconsists of solving sequentially the manipulation and naviga-tion design problems as defined below.1) Manipulation Design Problem: Given● a region R(A) and an initial guess A

g

,● a set of tip task frames, B = {B

i

, i = 1, . . . , t}, and● a manipulator robot architecture specifying the number of

sections in the manipulator portion of the robot, mm

, thenumber and location of variable curvature sections in themanipulator, V

m

, tip tube stiffness, kB

and dominatingstiffness ratio, D,

Solve for the coordinate frame, A ∈ R(A), tube curvatures,u

m

∈ Rmm and extension lengths, �m

∈ Rmm that minimize:● the curvatures of the manipulator sections, u

m

and● the extension lengths, �

m

such that:

IEEE TRANSACTIONS ON ROBOTICS 9

E E

EEA

(1)

(3)

(2)

(4)

(a)

Manipulation section

B1

E

Navigation section

Body lumen

Targets

R(E)R(A)

B2B3

A

(b)

Fig. 6: Navigation and manipulation tasks. (a) Navigation - telescopicextension and steering of proximal sections from entry frame, E, toframe A. (b) Manipulation - distal sections move from A to set oftip task frames, B

i

located at surgical sites.

● the tip task frames, B = {Bi

, i = 1, . . . , t} lie in theworkspace of the robot, and

● the robot satisfies the anatomical constraints, �m

(�).2) Navigation Design Problem: Given● a region R(E) and an initial guess E

g

,● the coordinate frame A obtained from solving the manip-

ulation problem, and● a navigation robot architecture specifying the number of

sections in the navigation portion of the robot, mn

, thenumber and location of variable curvature sections, V

n

,desired stiffness of the distal navigation section, k

A

anddominating stiffness ratio, D,

Solve for the coordinate frame, E ∈ R(E), tube curvatures,u

n

∈ Rmn , and extension lengths, �n

∈ Rmn that minimize:● the curvatures of the navigation sections, u

n

and● the extension lengths, �

n

such that:● the robot satisfies the anatomical constraints, �

n

(�).The resulting robot design is given by the combined solu-

tions to the manipulation and navigation problems,

T =������������������ m

n

m

m

�������������������������������m

, � V

Tn

V

Tm

��������������������������V

, � u

Tn

u

Tm

��������������������������u

, � �Tn

�Tm

�����������������������������

, D, k

d

�����������������(15)

III. ROBOT DESIGN OPTIMIZATION

The algorithm is used to solve both the navigation andmanipulation design problems, each of which can be posed assets of nested, simpler optimization problems in which subsetsof the design variables are held constant. The two constitu-tive optimization problems are: (1) solving the anatomically-constrained inverse kinematics problem for a given robot

design and base location, (2) solving the optimal robot designand base location problem. These are defined below.

A. Anatomically Constrained Inverse Kinematics

For a given concentric tube robot architecture, T , theproblem involves solving for the vector of robot tube kinematicvariables, q = {�, ✓}, that position and tangentially align therobot tip with coordinate frame B ∈ B, given that its base islocated at frame A and imposed anatomical constraints �(�)are respected. Using homogeneous coordinates to representcoordinate frames, frame B can be written as:

B = � e

B

x

e

B

y

e

B

z

p

B

0 0 0 1� (16)

Note that eBx

and e

B

y

are irrelevant since only the tangentialvector will be considered. We denote the forward-kinematicsmapping as: F ∶ (q,A,T )→ Btip (17)

where Btip is the coordinate frame of the tip. Using a penaltymethod to represent the tip configuration and anatomicalconstraints, a cost function, c, can be defined as follows, withoverbars indicating fixed parameters:

c

�q,T ,A,B,�� = �

1

�pF(q,A) − pB�������������������������������������������������������������������������������tip position error+ �

2

�eF(q,A)z

× eBz

������������������������������������������������������������������������������tip orientation error+ �

3

�(q,T ,A,�)�������������������������������������������������������������������anatomical constraints

(18)

The first two terms penalize the tip position and tangentdirection. Note that an additional tube can be added to performtip roll as needed. The third term employs the function �that computes the anatomical constraints, e.g., the interferencebetween the robot and the anatomy. The scalar constants�

1

,�

2

,�

3

are weighting factors. Minimization of this costfunction results in the kinematic variable vector q

� that bestsolves the anatomically-constrained inverse-kinematics prob-lem:

q

� = argminq

c(q,T ,A,B,�) (19)

Alternate formulations of the cost function c can be useful.For example, in some applications including the neurosurgicalexample discussed later in the paper, the tip tangent directionmay not be clinically important. Furthermore, cost criteria maybe included to utilize kinematic redundancy to avoid unstabletube configurations. For example, the three rules for avoidinginstabilities that are included at the end of Section II-D canbe included as given in the cost function below in which thescalars �

s1 ,�s2 ,�s3 are weighting factors and ✏ > 0 is included

IEEE TRANSACTIONS ON ROBOTICS 10

in the last term to avoid singularity.

c

s

�q,T ,A,B,�� = c

�q,T ,A,B,��

+ �

s1

m�i=1,i∈V

[↵i1(0) − ↵i2(0)]2����������������������������������������������������������������������������������������������������������������������������������������������������������

relative rotation of tubes+ �

s2

m�i=1,i∈Vp

�

i������������������������������������extension of variable curvature sections

from straight proximal sections

+ �

s3

m�i=1,i∉V

(�i

+ ✏)−1���������������������������������������������������������������������������������������extension of fixed curvature sections

(20)

B. Robot Design and Base Location Optimization

This problem involves solving for the optimal robot designthat can reach a workspace defined by the set B of tipcoordinate frames, while satisfying anatomical constraints.Simultaneously, the optimization solves for the base coordinateframe A. Since material properties place limits on tube cur-vature, robot sections with smaller curvatures are preferred.In addition, robot length should be minimized in order tomaximize robot stiffness. These considerations lead to thefollowing design cost function, f , that can be written as afunction of the inverse-kinematics cost, c:

f

�T ,A,B,�� = n�i=1

�

1i

u

i��������������������������curvature penalty+ n�

i=1�

2i

�i����������������������������

maximum extension length penalty

+ t�j=1

c(q�j

,T ,A,B

j

,�)���������������������������������������������������������������������������������������������������������������������������inverse kinematics cost function

(21)

Here, {�1i

}, and {�2i

} are scalar weights on section curvaturesand lengths. In practice, the number of weights can bereduced, e.g., one can assign a single weight per designvariable type. The examples of Sec. V discuss this in detail.

The optimal design satisfies:[{u�i

,��i

},A�] = arg min{ui,li},Af(T ,A,B,�) (22)

where i = 1, ..., n. The manipulation design problem of Sec. IIcan be solved directly with this formulation. For the navigationproblem, the kinematic cost function involves a single tipframe, A.

IV. IMPLEMENTATION

A block diagram of the design optimization appears in Fig.7. The optimization algorithm is initialized with the robotarchitecture, task description, anatomical constraints and stiff-ness parameters. Starting with the torsionally rigid kinematic

model, the robot design and base location optimization routineuses (21) to compare prospective designs.

When the optimization routine either converges or meetsiteration limits, the routine switches to the torsionally compli-ant kinematic model and uses the solution from the torsionallyrigid model as its initial guess. For the cases we have consid-ered, the design obtained from the torsionally rigid model isclose to meeting the anatomical and task constraints and sofewer iterations are needed for this second optimization pass.

The main code components are associated with computingthe inverse kinematics and with evaluating the anatomicalconstraints. The kinematic and anatomic models are describedin the following together with the optimization algorithm.

A. Robot Kinematic Model

Current models based on tube mechanics are boundaryvalue problems (BVP) comprised of differential equations withrespect to robot arc length that have their boundary conditionssplit between the robot base and tip [8], [9]. These modelsassume that the tubes are rigid longitudinally and with respectto shear of the cross section. Each tube, however is free tobend and twist about its axis.

For design optimization, a fast inverse kinematic solveris critical. The approach taken here is to implement inversekinematic solvers of the both the BVP and of an approximatealgebraic model that treats the tubes as torsionally rigid [8].Both models are solved by root finding.

During the design process, the optimization routine arrivesat a preliminary design using the simplified kinematic model.This design is used as the initial guess for the BVP model.This approach is intended to achieve computational efficiencywhile still providing the accuracy obtained from the completemodel. If more accurate models are introduced in the future,they can be easily incorporated into this framework.

B. Anatomical Model

The anatomical model is generated from MRI or CT imagesby image segmentation and is represented as a triangulatedsurface. Computationally efficient encodings of spatial rela-tionships can be achieved using KD-trees [27]. Thus, thevertices of the anatomy are used to populate a KD-tree,and the tree can be queried for the proximity and geometricrelationship of the robot to the anatomy.

For fast collision detection, a linear-time algorithm wasdeveloped. First, a binarization step creates an anatomical oc-cupancy map indicating forbidden and allowed robot regions.Second, the allowed occupancy volume is shrunk by erosionoperations with spherical elements of radii corresponding tothe cross sections of the concentric tube robot elements:

⌦r

= ⌦⊖ S

r

(23)

where S

r

is the structural element corresponding to radius r, ⌦is the allowed occupancy volume, and ⌦

r

the eroded volume.For the cross sections of all the tubes comprising a concentrictube robot, (23) results in a pyramidal occupancy map thatcan be used for collision detection using only the discretizedcenterline/skeleton of the concentric tube robot.

IEEE TRANSACTIONS ON ROBOTICS 11

Stiffness ratio,

Normalizing stiffness,

InputsRobot architecture,

Base region,

Robot base estimate,

Surgical tasks,

Anatomy,

Anatomical constraints,

OutputsBase frame,

Extensions lengths,

Tube curvatures,

Robot Design

Anatomical constraints: Eq. (17)

Robot and base location optimization

Inverse kinematics solver

Task and

anatomical constraints

met?

NO

END

YES

Kinematic model =torsionally compliant

Kinematics models

Torsionally rigid (Algebraic)

Torsionally compliant (BVP)

YES

NO

Kinematic model = torsionally compliantKinematic model =

Torsionally rigid

Optimization criteria: Eq. (20)

Fig. 7: Robot design optimization framework.

The binarized anatomical model is used to extract a Eu-clidean distance map that simplifies the anatomically con-strained inverse kinematics problem. The distance d

r

( �P ) tothe anatomy boundary is calculated for each point �P ∈ ⌦

r

. Apotential function [28] is calculated as:

U

r

( �P ) = 1

d

r

( �P ) + ✏ (24)

where U

r

( �P ) is the function’s value at �P , and ✏ ensures a non-zero denominator. The inverse kinematics should be calculatedsuch that the concentric tube robot maximizes its distance fromanatomical boundaries, similar to [13]. This can be satisfiedby minimizing the values of U

r

( �P ) along the centerline of therobot, where, depending on the radius, r, of the section underexamination, the appropriate U

r

is selected:

�(q,T ,A,�) = p�i=1

U

r

( �Pi

) (25)

where { �Pi

}, i = 1, . . . , p is the robot centerline. The introduc-tion of the anatomical distance-based functional smooths thecost function f of (21) and allows efficient optimization.

Querying U

r

for values of 1�✏ allows collision detection inO(n), where n are the points on the discretized concentrictube robot centerline. The number of points is held constantfor each robot configuration during the evaluation of thekinematics to avoid discretisation bias. The collision detector’scomplexity is lower than O(nlogk), which is the expectationfor n nearest-neighbor queries on a KD-tree with k nodes and,consequently, is used for collision detection.

The anatomical constraints are implemented as soft con-straints. While interference and constraint violation (e.g.,greater than maximum allowable deflection) can be treatedin a binary fashion wherein a solution is abandoned wheninterference is detected, a soft implementation enables implicitconstruction of a smooth minimisation “error map” rather thanone that contains “unmapped” areas of abandoned solutions.Moreover, the selected weighting functions provide an elementof robustness to model error in contrast to binary decisionfunctions since they drive the inverse kinematic solutions awayfrom the anatomical boundaries.

C. Optimization Algorithm

Preliminary implementations of our framework in [1], [7]required optimization using Generalized Pattern Search GPSmethods [29], as the cost function was nonsmooth and non-linear. GPS methods are effective in optimizing nonsmoothproblems, since they do not require any differentiation [30].Due to their sampling approach, however, they are computa-tionally inefficient.

The introduction of (25) smooths the cost function, andenables the use of faster optimization methods like the Nelder-Mead downhill simplex method [31]. The implementationof the Nelder-Mead method provided by the OptimizationToolbox of Matlab® was used in the following examples.

V. CLINICAL EXAMPLES

Two examples are presented here to showcase the perfor-mance of the proposed design algorithm. The first is a neu-rosurgical example that involves choroid plexus cauterizationfor hydrocephalus treatment. The second example considersclosure of a patent foramen ovale inside the beating heart.

A. Choroid Plexus Cauterization

Cerebrospinal fluid (CSF) is a watery fluid that surroundsthe brain and spinal cord. Formed by the choroid plexus(CP), it fills the ventricular spaces within the brain (Fig. 8).Hydrocephalus is the pathologic imbalance of CSF productionand absorption leading CSF accumulation. This can lead toelevations in intracranial pressure and compression of braintissue resulting in neurologic dysfunction and even death.

Standard treatment of hydrocephalus involves diversion ofCSF from the ventricles through a catheter that drains this fluidfrom the ventricles to another absorptive cavity in the body(typically the peritoneal cavity or pleural cavity). Alternativemethods of CSF diversion and production decrease includethird ventriculostomy combined with cauterization of the CP[32]–[34]. Endoscopic third ventriculostomy involves creatinga burr hole in the skull, inserting a straight endoscope, andpuncturing the floor of the third ventricle to create a naturalbypass for CSF drainage (see Fig. 8). CP cauterization (CPC)is performed by monopolar cautery using a Bugbee wire. TheCP covers portions of the two lateral ventricles and the thirdventricle (see Fig. 8). Conventional tools, flexible endoscopes

IEEE TRANSACTIONS ON ROBOTICS 12

Cortical tissue

3rd ventriculostomy region

Robot Straight cannula

3rd Ventricle

ChoroidPlexus

Lateral Ventricles

Fig. 8: Robotic cauterization of the choroid plexus. Robot enters rightlateral ventricle and also crosses over into left ventricle to performcauterization. Red ○ define the tip task frame set B indicating thecauterization points in the right lateral ventricle.

included, cannot perform a thorough cauterization since theydo not possess the necessary flexibility and dexterity [35].

Concentric tube robots can be employed during the cauteri-zation process to deliver the wire to the challenging locations,and a concentric-tube-robot-based surgical platform is cur-rently under investigation [35]. The optimal robot architectureand parameters, however, are unknown. With the frameworkproposed in this paper, a variety of designs using differentarchitectures can be created and evaluated.

The brain ventricles can be reliably visualized with MRIusing T1- and T2-weighted sequences. High-resolution imagestacks were used to produce a model of the ventricular systemof a hydrocephalic 10 month old male child. The robotmust enter the ventricles along specific paths through thebrain tissue in order to avoid passing through critical brainregions. Consequently, coordinate frames E and A, definingthe navigation portion of the robot are clinician specified andthis robot section consists of a single straight tube as shown inFig. 8. Thus, for this example, the entire portion of the robotinside the ventricles comprises the manipulation section.

Clinician-specified anatomical targets, covering the CP onthe lateral ventricles were specified (see Fig. 9) along withwaypoints selected to guide the robot safely from the straightinsertion tube to the CP point set, essentially coupling therobot design problem with implicit path-planning. Togetherthese sets of points form the set of target points, B. Since thesurgical task to be performed is cauterization, which is largelycontact angle independent, only the reachability of the targetsis evaluated.

To avoid tissue damage, the inserted length of the manip-ulation section should only contact the brain at its tip andonly at those locations where cauterization is to occur. Inconsequence, anatomical collisions are assigned a high penaltyin the anatomical constraint function, �.

1) Manipulation Section Design: The design algorithmrequires the number, type and arrangement of robot sectionsas inputs. The Bugbee wire for cauterization acts as the distalrobot section and behaves as a straight constant curvaturesection that flexes when retracted into a stiffer curved tube,

Entry waypointsCauterization points

R(A)

Entry vector

Fig. 9: Cauterization targets and entry waypoints specified on theanatomical model of the hydrocephalic ventricles.

(a) Variable-variable-fixed case (b) Variable-fixed-fixed case

Intersection with anatomy

(c) Fixed-variable-fixed caseIntersection with anatomy

(d) Fixed-fixed-fixed case

Fig. 10: Architecture-dependent optimized robot designs. (a)Variable-variable-fixed curvature, (b) Variable-fixed-fixed curvature,(c) Fixed-variable-fixed curvature, (d) Fixed-fixed-fixed curvature.Red lines in (c) and (d) indicate violation of anatomical constraints.

but returns to zero curvature when extended. The geometryof the ventricles shown in Fig. 8 indicates that at least twocurved sections are needed to reach the most distal targets.Consequently, the design algorithm was run for the four robotarchitectures consisting of three sections: (a) variable-variable-fixed curvature, (b) variable-fixed-fixed curvature, (c) fixed-variable-fixed curvature and (d) fixed-fixed-fixed curvature.

The distal fixed section corresponds to the zero curvaturecauterization wire. Owing to its extreme flexibility, the stiff-ness ratio for the two distal sections was taken to be D = 20while the ratio for the two proximal sections was specified asD = 10. For the purposes of this design example, the bendingstiffness of the Bugbee wire was normalized to k

d

= 1. Toproperly expose the wire for cauterization, a minimum sectionextension of 10mm was also specified for the distal section.

The design variable weights of (18), (20), (21), were se-

IEEE TRANSACTIONS ON ROBOTICS 13

TABLE V: Robot Design Parameters for CPC.

Robot Architecture 1�i [mm] �i [mm] Li [mm](a) (successful)variable curvature 18 54 54variable curvature 18 52 106fixed curvature ∞ 10 116

(b) (successful)variable curvature 19 58 58fixed curvature 19 58 116fixed curvature ∞ 10 126

(c) (unsuccessful)fixed curvature 20 37 37variable curvature 22 65 102fixed curvature ∞ 10 112

(d) (unsuccessful)fixed curvature 22 35 35fixed curvature 26 68 103fixed curvature ∞ 10 113

lected to be:�������������������������������������������������

�

1

= ��������pF(q,A) − pB� , for �pF(q,A) − pB� < 1.5mm

exp�104 �pF(q,A) − pB�� otherwise

�

2

= 0�

3

= 1�

s1

= 103�

s2

= 104�

s3

= 104�

1i

= �2i

= 1, i = 1, . . . , n(26)

where �pF(q,A) − pB� is the tip position error from (18).The weight on tip position error, �

1

, was set so thaterrors greater than 1.5mm are penalized exponentially, where1.5mm corresponds to anticipated coagulation area given bythe diameter d = 1mm of the Bugbee wire. This weightingheavily penalizes target errors larger than 1.5mm, which wouldprevent execution of the surgical task, while also ensuringthat small changes in displacement about target points do notdominate the cost function. This discontinuity encodes in theoptimization the importance of respecting the task constraints.

Since tip orientation is unimportant for cauterization, �2

isset to zero. For the anatomical constraint function, a weightof �

3

= 1 proved sufficient. It was also sufficient to specifyunit weights for all robot section curvatures and lengths.The stability-related weighting factors, �

si

, were initially allset to unity and then increased by powers of ten until astable solution set of configurations was found. Increasing theweights further increased the tip position errors.

The design code was run for four of the eight possible 3-section architectures with results provided in Table V and Fig.10. All optimizations commenced from a robot design thatdid not satisfy the anatomical constraints. As shown, only thetwo architectures with a proximal variable curvature sectioncan reach all of the target points in B while respecting theanatomical constraints. From the table, it is observed that both

rigid model

compliant model

(a)

rigid model

compliant model

(b)

Fig. 11: Comparison of torsionally rigid and torsionally compliantmodels: (a) Front view, and (b) Side view.

Fig. 12: Stable and unstable configurations for a target point.Unstable configuration is shown dotted.

designs require very similar section curvatures. This is perhapsnot surprising since the workspace analysis of Section IIdemonstrated that variable-fixed and variable-variable designsshare comparable workspaces. Note that the total lengths ofthese two designs (116mm versus 126mm) are similar andthat the Bugbee wire length was minimum for all designs.

Since the robot architecture composed of variable-fixed-fixed sections uses fewer tubes and yet satisfies the anatomicaland surgical task constraints, it is the preferred design. Thisdesign optimisation converged in 2h and 24 mins, after434 design iterations. A C++ implementation would decreasecomputation time by a factor of ten to about 15 mins.

2) Effect of Kinematic Model: Recall from Fig. 7 thatthe optimization algorithm first utilizes a simplified algebraickinematic model and then refines the design, as needed, usingthe complete torsionally compliant BVP model. In the case ofthe variable-fixed-fixed model, it was observed that the designparameters obtained using the simplified model also satisfiedthe BVP model constrains. Thus, the algorithm ran for a singleiteration of the torsionally compliant model. Small differencesin the inverse kinematic solution joint variables as well as therobot shape were present as shown in Fig. 11.

3) Configuration Stability: It can be shown that thevariable-fixed-fixed curvature design of Table V can exhibittwo types of instability. The first is associated with straight-ening the variable curvature section while the second arisesfrom rotating the distal curved section while it is substan-tially retracted inside the proximal variable curvature section.

IEEE TRANSACTIONS ON ROBOTICS 14

Consequently, it is important to ensure that each of the targetpoints and waypoints can be reached through stable config-urations. The cost function of (20) is designed to guide theinverse kinematic solver away from unstable configurations.Furthermore, these configurations were explicitly tested forstability as a post-processing step using the graphical methoddescribed earlier. A path planning algorithm that explicitlyconsiders stability can also be employed [36].

Figure 12 shows that set B did include tip positions associ-ated with unstable configurations. For each of these positions,however, there were also stable configurations. For the exam-ple illustrated, stability was achieved by further extension ofthe middle fixed-curvature section. The inverse kinematic costfunction was effective in finding these stable configurations.

B. Robotic Closure of a Patent Foramen Ovale

The goal of this example is to investigate whether the designalgorithm can reproduce a previously validated robot architec-ture and design that was successfully used on a sequence ofpigs as described in [16] and [20]. This example serves notonly to validate the algorithm, but also to evaluate the conceptof developing a single robot design that can accommodate agroup of “patients” instead of the more costly approach ofhaving to produce a specific design for each patient.

A patent foramen ovale (PFO) is a heart defect characterizedby a channel between the layers of the septum that separatesthe right and left atria. This channel occurs naturally in thefetus and normally seals shortly after birth. If not sealed, itcan allow blood returning from the body to be recirculated tothe body without filtration and oxygenation by the lungs [37].

Recently, intracardiac beating-heart repair with concentrictube robots has been successfully demonstrated in a porcinemodel under fluoroscopic and ultrasound imaging [16]. Whileall trials were performed in pigs, Fig. 13 depicts the equivalenthuman procedure in which the robot is introduced into theright atrium percutaneously via the internal jugular vein usingtelescopic extension to navigate through the internal jugularvein, the right brachiocephalic vein, and the superior vena cavainto the right atrium. The actual porcine anatomy considered isshown in Fig. 16. Once the robot has reached the right atrium,the proximal sections are held fixed and the distal sections areused to manipulate the septal tissue and to deploy a tissueapproximation device to seal the PFO channel.

The robot design used in these procedures was developedusing postmortem measurements. It consisted initially of twofixed curvature sections for telescopic extension into the rightatrium and a distal variable-fixed curvature architecture operat-ing within the right atrium. During procedure development, thedesign was simplified to include a single section for vascularnavigation resulting in a fixed-variable-fixed robot design.

The optimization algorithm was employed to solve forthe section parameters for the two experimentally evalu-ated design architectures (fixed-fixed-variable-fixed and fixed-variable-fixed) as described below using a stiffness ratio ofD = 10 and a distal section stiffness normalized to k

d

= 1.An anatomical model of the vasculature and cardiac cham-

bers was obtained by MRI for a 40kg Yorkshire swine.

Contrast agent was used together with respiratory and cardiacgating to obtain a sequence of 1mm thick MRI slices. The 3Dgeometry was generated by threshold segmentation to eachslice, followed by triangulation of the enclosed volume andGaussian smoothing. The resulting model is shown in Fig. 16.

1) Manipulation Section Design: Sealing any specific PFOrequires reaching a region on the septal ridge as shown in Fig.13 [16]. In order to create a design that can accommodatea range of patient sizes, a set of target points has beendefined that encloses an enlarged region overlapping the septalridge, B, as shown in Figs. 13 and 16. In contrast to CPablation, orientation of the robot tip tangent with respect tothe septum normal vector is important for device deployment.Consequently, a tip tangent constraint was specified to allowa maximum angle difference of 30○.

The allowable region for coordinate frame A, R(A), wasselected to be at the ostium of the superior vena cava, with abounding box covering the full vein diameter. While the sidesof the bounding box are aligned with the image coordinatedirections, the z axis of frame A was constrained to be parallelto the vein’s centerline.

When operating in the right atrium, it is important that therobot avoids contact with the cardiac wall. As a result, colli-sions are heavily penalized by the optimization algorithm. Thiscriterion is encoded in the anatomical constraints function �.

The section parameters for a variable-fixed manipulationsection architecture were calculated from a random initialconfiguration as shown in the inset of Fig. 16 with convergencein 45mins after 473 iterations. Notice how the location offrame A together with the shape of the manipulator sectionsenable the tip to achieve the desired orientation with respectto the septal surface. For the weights given below, the designparameters are provided in Table VI.�����������������������������

�

1

= ��������pF(q,A) − pB� , for �pF(q,A) − pB� < 1mm

exp�104 �pF(q,A) − pB�� otherwise

�

2

= 105�

3

= 1�

1i

= �2i

= 1, i = 1, . . . , n(27)

The weights are similar to those used for the neurosurgicalexample. The exponential tip error weighting threshold wasreduced to 1mm based on clinical tolerances and a hightip orientation error weight was introduced to achieve thetip tangent constraint of 30○. Having previously manuallydesigned and tested tube sets for this example that all proved tobe globally stable, we anticipated that algorithmically-obtaineddesigns would also be stable for all configurations and sodid not employ the stability cost function for this example.Anatomical and section parameter constraints match those ofthe previous example.

2) Navigation Section Design: Using the optimized manip-ulator base frame A, the entry frame of the navigation section,E, is selected to be in the jugular vein, at the level of the neck,with a bounding box covering the full vein diameter as shownin Fig. 16. The z axis of frame E is prescribed to be parallel

IEEE TRANSACTIONS ON ROBOTICS 15

to the centerline of the jugular vein.The geometry of the vessels, shown in Fig. 16, suggests that

multiple curved robot sections may be needed for navigation.It is possible, however, to substantially straighten and laterallymove the vessels during robot insertion. This observation isencoded in two anatomical navigation constraints (see Fig. 14):● The vessels can be laterally displaced up to � ≤ 1cm,● During telescopic extension, the angle between the tip

tangent vector and the vascular tangent vector should beµ

i

≤ 25○ in order to avoid puncture of the vessel wall.To avoid computation of tissue deformation during opti-

mization, these criteria can be approximated by comparing theshape of the extended robot with the undeformed shape of thevessels. Furthermore, the manipulation section was assumed tobe fully retracted from the navigation tubes during telescopicextension to simplify kinematic calculations.

Initial design optimizations considered a single fixed cur-vature section for navigation, however, it was not possible tosatisfy the design constraints using a single curved section.Subsequently, the design algorithm was executed for a fixed-fixed curvature architecture resulting in the design shown inFig. 16 and detailed in Table VI. It can be seen that thereis significant difference between the two section curvatures,which is expected, considering that a single fixed curvaturesection was inadequate.

The design weights of (27) with their heavy penalties ontip position and orientation were used to ensure that the distalpoint of the navigation section matches the proximal point ofthe manipulation section in position and orientation. Stabilityas well as sections curvature and length were also penalisedas in the preceding examples.

The anatomical constraint function of (25), however, wasmodified to reflect the deformation constraints of Fig. 14.Since lateral deformation up to 1 cm is allowed, the KD-tree is queried for the distances of all robot points from theanatomy and exponentially penalises the maximum to be under1 cm. Similarly, the robot points closer to the anatomy areinvestigated for their angle-of-attack to the anatomy, limitingthem close to a prescribed 25○. As shown in Fig. 14, theentry and exit angles, µ

1

and µ

2

are limited to 25○, and themaximum displacement marked as � must be less than 1 cm.

The navigation design problem starts from a random initialconfiguration, as shown in Fig. 16, and converges in 2h and32minutes, after 1396 iterations.

As in the neurosurgical example, waypoints were used toguide the navigation section through the anatomy in an implicitpath-planning fashion; these waypoints corresponded to thecentreline of the jugular vein. The solution provided by thetorsionally rigid kinematic model was found to also satisfy thetorsionally compliant model constrains. Telescopic extensionalong the centerline of the vasculature using the two kinematicmodels is depicted in Fig. 15. Note that while the sectionparameters and centerline points are the same, the depictedconfigurations are obtained by solving the anatomically con-strained inverse kinematics problem independently for eachkinematic model.

For both the manipulation and navigation portions of therobot, the design algorithm was able to utilize the simplified

Fig. 13: Percutaneous robotic PFO closure. Inset: Target pointsintended to allow treatment for a range of anatomical sizes.

μ2

μ1 #1#2vein

λ

Fig. 14: Anatomical constraints for vascular navigation.

algebraic kinematic model during parameter optimization andthen verify the solution using the BVP model during a finaliteration. In addition, it was verified that the overall robotdesign is stable for all configurations.

For comparison with the experimentally validated designof [16], the actual tube parameters are listed in Table VI.This robot design was used successfully for PFO closure inYorkshire pigs varying between 45−65kg. In these procedures,the robot was able to position its tip on the atrial septal ridgeand perform the required tissue manipulation before deliveryof the PFO closure device [16].

From the table, it can be observed that the curvatures ofthe optimized variable-fixed design closely match those usedexperimentally. While the lengths of the experimental sectionsare significantly longer than the minimum required lengthscomputed by the algorithm, this is not surprising since theexperimental tube set was deliberately constructed to be longerthan required. Furthermore, it was observed that neither curvedsection needed to be fully extended during any surgery.

Unlike what is shown in Fig. 16, however, the manipulatorsections were not configured in an ”S” shape during surgery.Instead the curvatures were aligned as shown in Fig. 13. Thisconfiguration was necessary since the surgeon rotated the pig’sheart to the left within the chest cavity, displacing the septalridge to the left, in order to reproduce the orientation of thehuman heart within the chest.

For the navigation section, the algorithm was unable tosolve for a single curved section that satisfied the anatomicalconstraints. Our initial experimental tube set also includedtwo fixed curvature sections. During surgery, however, itwas discovered that by first inserting a plastic introducersheath through the vasculature, it was possible to displaceand straighten the vasculature more substantially than wasanticipated during robot insertion without causing damage.

IEEE TRANSACTIONS ON ROBOTICS 16

(a) Rigid model (b) Compliant model

Fig. 15: Telescopic extension through the vasculature using the optimized navigation sections. Configurations shown are solutions to theanatomically-constrained inverse kinematics problem using the: (a) torsionally rigid model, (b) torsionally compliant model.

Initial configuration

Optimized configuration

Targets

Optimizedconfiguration

Initialconfiguration

Septal normal

Navigation sections

Manipulationsections

R(A)R(E)

B

Fig. 16: Anatomical model together with initial and optimizeddesigns for the navigation and manipulation portions of the robot.Inset: close up of manipulation sections and target points.

Consequently, the pair of sections was simplified to consist of asingle section with a 600mm radius of curvature. Thus, whilethe algorithm provided a solution that closely fit the anatomy,our selected anatomical constraints proved conservative.

VI. CONCLUSIONS AND DISCUSSION

In contrast to standard robots for which the same robotdesign is used for all procedures, the set of tubes com-prising a concentric tube robot can be easily customized tomeet the task requirements and anatomical constraints of aspecific procedure or class of procedures. While this designproblem need only be solved once for any given procedure,the unconstrained design problem is high dimensional andcomputationally intensive.

To address this , this paper presents a design methodologyand optimization framework that considerably reduce the

TABLE VI: Robot Design Parameters for PFO Closure

Manipulation SectionsSection type [Algorithm] 1�i [mm] �i [mm] Li [mm] *variable curvature 75 28 28fixed curvature 22 28 56

Section type [Experiment] 1�i [mm] �i [mm] Li [mm]*variable curvature 80 45 45fixed curvature 24 35 80* Measured from frame A

Navigation SectionsSection type [Algorithm] 1�i [mm] �i [mm] Li [mm]fixed curvature 436 89 89fixed curvature 117 109 198

Section type [Experiment] 1�i [mm] �i [mm] Li [mm]fixed curvature 600 170 170

dimensionality of the design space while still providing arich solution set. In this framework, robots are constructedof telescoping sections of either fixed or variable curvature.Furthermore, it is shown how the design problem can oftenbe decomposed into two lower dimensional problems of nav-igation to the surgical site and manipulation at the site.

As with any set of design rules, those proposed hererepresent tradeoffs. For example, implementing the sectionalstiffness dominance of rule 2 requires larger tube diameters,and the variable curvature sections of rule 3 do not utilizethe relative translation between the tube pair. These rulesdo, however, provide a systematic means to arrive at designswith the clinically-desirable capabilities of follow-the-leaderinsertion and kinematic decoupling of the each section fromits proximal sections.

To provide insight for guiding the design process, the paperalso compares the tip workspace that can be achieved by vari-ous arrangements of fixed and variable curvature sections. Fortwo-section robots, the superiority of variable-fixed designs isclearly demonstrated.

Furthermore, it is shown that the workspace can be de-composed into regions according to the elastic stability of theconfigurations within the regions. In this context, it is demon-strated that tube sets exhibiting elastic instabilities withintheir workspace can still be safely employed by ensuring that

IEEE TRANSACTIONS ON ROBOTICS 17

alternate stable configurations are available within the desiredtask space that satisfy the anatomical constraints.

These concepts are illustrated through design examples fromneurosurgery and intracardiac surgery. Since the intracardiacdesign is compared to previously performed in vivo robotictrials [16], it provides strong validation of the approach.

Beyond the framework presented here, there are additionaldesign issues that must be considered. These include selectingsuch variables as tube diameters, thicknesses and materials.The design process can be started, for example, by selectingthe inner diameter of the innermost tube to be just large enoughto deliver all tools and devices needed for a procedure. Therequired robot tip stiffness can then be used to compute thethickness of the innermost tube. Dimensions of outer tubescan be subsequently computed.

This design process is often iterative since tube diameter andmaterial type determine maximum bending curvature, whichmay or may not enable the desired workspace. Mechatronicissues also must be considered, such as the need for tubes toextend out of the body and into the drive system. While thesetransmission lengths can decrease torsional stiffness, this issuecan be addressed by constructing the transmission lengths froman alternate stiffer material, e.g., stainless steel.

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