IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, …research.vuse.vanderbilt.edu/medlab/sites/default/files/papers/... · curved concentric tubes to achieve a wide variety ... Gravagne

IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010 769

A Geometrically Exact Model for Externally LoadedConcentric-Tube Continuum Robots

D. Caleb Rucker, Student Member, IEEE, Bryan A. Jones, Member, IEEE, and Robert J. Webster, III, Member, IEEE

Abstract—Continuum robots, which are composed of multipleconcentric, precurved elastic tubes, can provide dexterity at diam-eters equivalent to standard surgical needles. Recent mechanics-based models of these “active cannulas” are able to accuratelydescribe the curve of the robot in free space, given the preformedtube curves and the linear and angular positions of the tube bases.However, in practical applications, where the active cannula mustinteract with its environment or apply controlled forces, a modelthat accounts for deformation under external loading is required.In this paper, we apply geometrically exact rod theory to pro-duce a forward kinematic model that accurately describes largedeflections due to a general collection of externally applied pointand/or distributed wrench loads. This model accommodates arbi-trarily many tubes, with each having a general preshaped curve.It also describes the independent torsional deformation of the in-dividual tubes. Experimental results are provided for both pointand distributed loads. Average tip error under load was 2.91 mm(1.5%–3% of total robot length), which is similar to the accuracyof existing free-space models.

Index Terms—Active cannula, concentric-tube robot, continuumrobot, Cosserat-rod theory.

I. INTRODUCTION

CONCENTRIC-TUBE continuum robots, which are alsocalled active cannulas due to their promise in interven-

tional medicine, use the geometry and elastic interaction of pre-curved concentric tubes to achieve a wide variety shaft curvesand end-effector poses. As shown in Fig. 1, the shape of thecannula’s telescoping backbone can be changed by axially ro-tating and translating each individual tube at its base. Harnessingprecurvature in this manner enables a larger variety of shapesat smaller diameters than is possible with continuum robotsactuated by mechanisms external to the backbone (e.g., cables,pushrods, etc.). These characteristics have led to many proposedminimally invasive surgical applications for active cannulas, in-

Manuscript received September 12, 2009; revised March 29, 2010; acceptedJuly 23, 2010. Date of publication August 30, 2010; date of current versionSeptember 27, 2010. This paper was recommended for publication by AssociateEditor S. Hirai and Editor K. Lynch upon evaluation of the reviewers’ com-ments. This work was supported in part by the National Science Foundationunder Grant 0651803 and in part by the National Institutes of Health underGrant R44 CA13416. This paper was presented at the 2010 IEEE InternationalConference on Robotics and Automation, Anchorage, AK.

D. C. Rucker and R. J. Webster, III are with Vanderbilt Univer-sity, Nashville, TN 37235, USA (e-mail: [email protected];[email protected]).

B. A. Jones is with the Mississippi State University, Mississippi State, MS39762, USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2010.2062570

Fig. 1. Active cannula composed of three telescoping Nitinol tubes, which isactuated by rotating and translating the tubes at their bases.

cluding use in fetal procedures [2], the lung [3], the heart [4],and in transnasal and transgastric surgeries [5], among otherinterventions. While some of these applications, such as manip-ulating a fiber-optic laser in the lung [6], may be approachablewith free-space kinematic models that do not include externalloading, in many other foreseeable applications, it will be usefulfor the cannula to intentionally manipulate tissue by retractingit, cutting it, dissecting it, traveling through it like a needle, etc.Furthermore, as the cannula approaches the area in which it isto work, it is likely that tissue will contact it at one or morepoints along its shaft. Gravity can also cause some (albeit typi-cally small) deflection in an active cannula. To enable accuratecontrol of cannula position and applied forces under these con-ditions, it is essential to have a model that describes cannulashape under externally applied point and distributed forces andmoments.

The value of modeling external loading has recently beendemonstrated in larger scale pneumatically actuated continuumrobots where the robot sags significantly under the self-weightof the arm. Trivedi et al. used geometrically exact Cosserat-rodtheory to describe the shape of the OctArm under load, therebyreducing model errors from 50% to 5% [7]. With respect tosmall-scale continuum robots for medical applications, Xu andSimaan have recently demonstrated intrinsic-force sensing us-ing a flexible push–pull-rod-actuated multibackbone robot andapplied it to palpation [8]. The model that we present here mayone day enable intrinsic-force sensing with active cannulas.

With respect to tendon-driven continuum robots, the effect ofexternal loading has not yet been a focus, but rod theory has beenapplied to model backbone shape under tendon loads. Much ofthis work has built on Chirikjian’s application of continuummodels to hyperredundant robots [23]. Li and Rahn formulateda geometrically exact model for the nonlinear deformation ofindividual sections of the backbone due to tendon loads [9].

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770 IEEE TRANSACTIONS ON ROBOTICS, VOL. 26, NO. 5, OCTOBER 2010

Gravagne et al. used geometrically exact rod theory to derivethe large-deflection dynamics of a planar continuum manipula-tor and control its vibration [10]. Camarillo et al. modeled boththe bending and compression of a constant-curvature continuumrobot [11]. Recently, Jones et al. explored the real-time solutionof the static Cosserat-rod equations for a tendon-driven con-tinuum robot under applied loads [12]. While our work in thispaper draws upon similar geometrically exact rod theory as hasbeen used in these studies, our concentric precurved tube designpresents a fundamentally different problem because there aremany elastica to consider rather than just one. While concentrictubes do share a common backbone shape, each can undergotorsion independent of the others, which precludes the use ofany existing model.

While significant prior work has been done in active-cannulamodeling and sophisticated free-space models exist, none haveyet accounted for external forces or moments. The modelingframeworks that exist today have been developed in parallelby several groups. The simplest possible model of an activecannula by Furusho et al. [2] makes the assumption that theoutermost tube in any given section of the robot has infinitestiffness compared with all tubes within it. Webster et al. [5] andSears and Dupont [4] provided initial beam mechanics modelsthat accounted for tube interaction and, thereby, achieved bet-ter accuracy. The importance of torsion was also recognizedand initially modeled in straight sections of the device [3], [5],before being extended (in closed form) to curved sections ina two-tube cannula [13], [14]. The latest (numerically evalu-ated) models generalize to arbitrarily many tubes and variableprecurvature [13]–[15]. While these are useful and general mod-els, they are only able to consider cannulas in free space. Ourpurpose in this paper is to extend them to describe the shapeof an active cannula under external loading. Such a model isa necessary prerequisite to future development in areas suchas design based on compliance, manipulation of objects, andintrinsic-force sensing and control.

A. Contributions

Our contributions in this paper are as follows: 1) We presentan extension of the classical, geometrically exact Kirchoff rodtheory from one rod to many precurved concentric tubes un-der arbitrary external point and distributed wrench loading, and2) we apply the theory to a specific active-cannula continuumrobot and validate it experimentally.

Some results in this paper were presented in preliminary formin [1]. Noteworthy additions/enhancements in the current paperinclude the following:

1) an enhanced treatment of relevant results from classicalCosserat-rod theory;

2) derivation of the fundamental torsional component of ourmodel using both Cosserat-rod methods and energy meth-ods, thus demonstrating their consistency;

3) an expanded experimental discussion and new figures il-lustrating the cannula workspace, our experimental appa-ratus, and experimental error;

4) new experimental results for distributed loading;

5) a discussion of unmodeled effects including friction, elon-gation, and transverse shear, and how they could be (andwhether they may need to be) included in future active-cannula models.

II. MECHANICS-BASED MODEL

Our derivations in this section draw on Antman’s thoroughwork on nonlinear elasticity [16, chs. 4 and 8]. In the interest ofclarity to those familiar with robotics, we have replaced some ofthe geometric nomenclature with the more-compact kinematicnotation familiar to the robotics community (e.g., the use ofrotation matrices rather than directors).

This section is organized as follows. After discussing mod-eling assumptions in Section II-A, we proceed in Section II-Band C to introduce the kinematics and constitutive behavior ofa single tube. Section II-D gives a condensed derivation of theCosserat equilibrium equations for a precurved rod under load,the results of which provide the basis for the multitube derivationin Section II-E and F.

A. Assumptions

A theory of rod deformation is termed “geometricallyexact” if it makes no approximations with respect to kinematicvariables [17]. The nonexact methods typically used to predictthe deformation of structural beams often employ two “small-deflection” approximations (either of which removes geometricexactness) to enable closed-form solutions: 1) The deformedshape is assumed “close” to the initial shape when computing in-ternal stresses, and 2) some approximate formula is used for thebeam’s curvature in calculating the elastic curve. In this paper,our approach is based on the geometrically exact Cosserat-rodtheory, which makes neither approximation.

With respect to constitutive behavior, we use the standard as-sumptions of the classical elastic-rod theory of Kirchoff, whichis a special case of Cosserat-rod theory [16]. The assumptionsof Kirchoff are 1) inextensibility and no transverse shear strainand 2) linear constitutive equations for bending and torsion.Inextensibility and shearlessness are generally regarded to begood assumptions for long thin rods, such as the tubes in activecannulas (e.g., the prototype described in Section III). To illus-trate the validity of the inextensibility assumption, we providethe following calculation. The maximum insertion force for theinner tube of our prototype was measured to be 10.1 N using anATI Nano17 force sensor, which is a number that also exceedsany of the applied loads in our experiments. If this load wasapplied at both ends of a straight tube with the same dimensionsas the inner tube used in our experiments, the total elongationwould be less than 50 µm. Thus, we can be confident that bend-ing and torsion will dominate the deformation behavior of anactive cannula. For simplicity, we adopt the linear constitutiveequations, but our overall approach does not require it. Activecannulas often remain below 3% strain in practical use, whichis in the linear range of Nitinol [18].

We also neglect friction in this paper, as do all active-cannulamodels to date. While friction is a worthy topic of future mod-eling and compensation efforts, the fact that we can achieve less

RUCKER et al.: GEOMETRICALLY EXACT MODEL FOR EXTERNALLY LOADED CONCENTRIC-TUBE CONTINUUM ROBOTS 771

Fig. 2. The shape of a precurved rod is defined by an arc-length parame-terized frame along the rod’s length. When external loads are applied, the roddeforms under a combination of bending and torsion. Since transverse shear andelongation are neglected, the frame’s z-axis remains tangent to the deformedcurve.

than 3 mm average tip error without modeling friction indicatesthat its effects do not dominate the behavior of our experimentalprototype. Furthermore, in our experimental procedure, we ex-plored hysteresis and noted no discernible effects, i.e., we couldnot induce the cannula to reach a different final position underload when the load was allowed to oscillate. We will commentfurther upon how frictional effects might be added to our mod-eling framework in Section IV.

B. Kinematics of a Single Precurved Tube

Let the unloaded precurved shape of a tube be defined by anarc-length parameterized curve r∗(s).1 We assign frames con-tinuously along r∗(s) and, by convention, choose the z-axes ofthese frames to always be tangent to the curve (see Fig. 2 foran illustration). The well-known Frenet–Serret apparatus pro-vides closed-form equations, which can be used to generate suchframes as long as r∗(s) is twice differentiable. In this conven-tion, one axis is always aligned with the plane of instantaneousgeometric curvature. Perhaps more intuitive, but less analyti-cally simple, are rotation minimizing, or Bishop frames, whichcan be thought of as “sliding” along the curve without under-going any rotation about the tangent z-axis. Both these framingconventions have been used in prior active-cannula modeling.Regardless of the convention choice (the subsequent analysisaccommodates either), framing the initial tube curve createsa rotation matrix R∗(s) at every arc-length location s on theundeformed curve.

Thus, a continuous homogeneous transformation g∗(s) is es-tablished, consisting of the position and orientation of an arc-

1Note that throughout the paper, we use the ∗ to denote variables associatedwith undeformed individual tube shapes. Thus, r∗(s) indicates the preset shapea single tube has in the absence of any other tubes or external loads.

length-parameterized reference frame along the curve

g∗(s) =[

R∗(s) r∗(s)

0T 1

]. (1)

We can obtain a local curvature vector by using the well-known relationship

u∗(s) = (R∗T(s)R∗(s))∨.

Here and throughout the paper, the dot denotes a derivative withrespect to arc length s. The ∨ operator denotes conversion of anelement of so(3) (the Lie algebra of SO(3)) to its correspond-ing element in R

3 . Following convention, we “overload” the ∨

notation so that it also indicates the mapping from se(3) (theLie algebra of SE(3)) to R

6 . The operator denotes the inverseoperation in both cases. For an in-depth discussion on this nota-tion, see [19]. Thus, the original arc-length-parameterized curver∗(s) could be reconstructed by integrating

g∗(s) = g∗(s)ξ∗(s)

where ξ∗(s) = [eT3 u∗T(s)]T , and e3 = [0 0 1]T .

In Cosserat-rod theory, any deformation of a tube from itsinitial state g∗(s) to a new state g(s) can be described by acorresponding change from ξ∗(s) to ξ(s), which we denote∆ξ(s) = ξ(s) − ξ∗(s). The three components of ∆u(s) (i.e.,the change in u(s)) correspond to bending strains about the x-and y-axes of the attached local reference frame, and torsionalstrain about the z-axis. In general, transverse shear strain andelongation can be similarly captured by changes in the firstthree components of ξ(s). Since we neglect these effects in thispaper (for the modeling assumptions, see Section II-A), ∆u(s)completely captures the deformation of the tube. The deformedbackbone shape of the tube g(s) is then defined differentially by

g(s) = g(s)ξ(s)

where ξ(s) = [eT3 uT (s) ]T , or equivalently

r(s) = R(s)e3 , R(s) = R(s)u(s) (2)

where u(s) = u∗(s) + ∆u(s) is the curvature vector of thedeformed backbone curve.

Note that in the kinematic formulation above, one can makethe following analogy to rigid-body motion: As a “body frame”angular velocity ω describes how a rotation matrix R(t) changeswith respect time [19], a local curvature vector u describes howa rotation R(s) changes with respect to the arc length of the rod.Thus, the expressions for the elastic energy stored in a deformedrod are of the same form as those for the kinetic energy of atumbling rigid body. This is termed Kirchoff’s kinetic analog,as discussed in [20]. This analogy can be helpful for those withexperience in robot dynamics to gain intuition about the modelin this paper.

C. Constitutive Relationships

We use a linear constitutive law to describe the relationshipof the strains to the internal moment vector (expressed in globalframe coordinates) at s

m(s) = R(s)K(s)∆u(s) (3)


Fig. 3. Section of rod from s to the free end � subject to distributed forces andmoments. The internal force and moment are also shown.

where

K(s) =

E(s)I(s) 0 0

0 E(s)I(s) 0

0 0 G(s)J(s)

,

E(s) is Young’s modulus, I(s) is the second moment of areaof the tube cross section, G(s) is the shear modulus, and J(s)is the polar moment of inertia of the tube cross section. Weuse this linear relationship because it is notationally convenientand accurate for our robot, but our approach does not require it.One could easily incorporate a nonlinear constitutive law here(i.e., m = f(∆u, s)). The following model derivation woulddiffer only in the computation of the arc-length derivative andthe inverse of m.

D. Equilibrium Equations: A Single Tube Under Load

We now give a brief development of the fundamental equi-librium equations needed to obtain the deformed shape of oneprecurved rod under a prescribed load. Note that this exposi-tion directly parallels the development of the multitube case inSection II-F. In [16, ch. 4.1], by writing a moment balance on asection of a rod under load then taking the derivative of the re-sult with respect to arc length, Antman derives the equations ofequilibrium for the special theory of Cosserat rods. We providea modified, condensed version of that derivation here, whichparallels the more complex development of the multitube casethat follows. We will also use certain results from the single-tubeanalysis in the multitube development.

Consider a cantilevered precurved rod, the kinematics ofwhich are described in Section II-B, extending from arc lengths = 0 to s = �, and subject to an arbitrary combination of dis-tributed forces f(s) and moments l(s) along its length. We nowcut a section at an arbitrary arc-length location s, as shown inFig. 3. By convention, we denote the internal force, which thematerial of [s, �] exerts on the material of [0, s), as n(s). Simi-larly, the internal moment that the material of [s, �] exerts on thematerial of [0, s) is m(s). Summing the forces on the portion[s, �], we obtain ∫ �

s

f(σ)dσ − n(s) = 0. (4)

Similarly, summing the moments on the portion [s, �] about theworld frame origin, we obtain∫ �

s

(r(σ) × f(σ) + l(σ)) dσ

− m(s) − r(s) × n(s) = 0. (5)

The unknown, deformed shape r(s) is present in both (5)and in the differential equation (2). Thus, it is not generallypossible to obtain either the deformed curve or the curvaturealgebraically. The classical approach to solve this problem is totake derivatives of the force and moment balances with respect toarc length and then use the constitutive law to obtain derivativesof the kinematic variables that can be numerically integratedsimultaneously with the kinematic equation (2) as a boundaryvalue problem. This is the approach that we adopt here.

Taking the derivative of (4) with respect to s, we obtain

n(s) + f(s) = 0. (6)

Similarly, taking the derivative of (5) with respect to s andsubstituting (6) into it yields

m(s) + r(s) × n(s) + l(s) = 0. (7)

Equations (6) and (7) are the classical forms of the equations ofequilibrium for a special Cosserat rod, as given in [16]. In orderto obtain the deformed shape of a single rod under load, they canbe expanded using (3) to obtain equations for the local curvaturederivatives, which can then be integrated simultaneously with(2) (as is done, e.g., in [7] and [12]).

We now proceed to do this by expressing (6) and (7) interms of the local curvature and kinematic framework fromSection II-B. Taking the derivative of (3) and substituting (2)into it yields

m = R(K(u − u∗) + (uK + K)(u − u∗)). (8)

For simplicity, we have dropped the (s) notation and will con-tinue to do so, except where needed for clarity, for the remainderof the paper. Using (2), (4), and (8), we can rewrite (7) in termsof the curvature and the applied loads. Premultiplying by RT

and K−1 and then solving for u, we obtain

u = u∗ − K−1(

(uK + K)(u − u∗)

+ e3RT

∫ �

s

f(σ)dσ + RT l

). (9)

This equation, combined with (2), results in a system of dif-ferential equations (u and g in terms of u, g, and the appliedloads). Typical boundary conditions for this simple case wouldbe g(0) = g0 , and mi(�) = 0 in the absence of any point mo-ments at the tip.

E. Kinematics of a Collection of Concentric Tubes

We now consider a robot composed of a collection of pre-curved concentric tubes subject to a set of external forces andmoments along its length. In this section, we give the geomet-ric constraints appropriate to such a collection, and in the next


section, we derive the equations of equilibrium that can be in-tegrated to obtain the final shape of the tubes. For simplicityin deriving the fundamental equations, we consider n tubes ofthe same length, which are fully overlapped. In Section II-G,we will discuss how to apply the model to the case where tubesbegin and end at different locations.

We assume each tube has its own arc-length-parameterizedtransformation g∗i (s), as in (1), and associated precurvature vec-tors u∗

i (s), as set forth in Section II-B. If the tubes are concen-tric, then each of their final deformed curves must be equal, i.e.,r1(s) = r2(s) = · · · = rn (s). We designate this common de-formed curve as r(s). This does not imply that each gi(s) mustbe equal. The tubes are free to twist independently during de-formation, and this prohibits us from considering the collectionof concentric tubes to be a single, precurved rod.

The equality of the deformed curves can be equivalently ex-pressed by the two statements r1(0) = r2(0) = · · · = rn (0)and r1(s) = r2(s) = · · · = rn (s). Thus, their tangent vectorsare equal along the length. Recalling from (2) that ri(s) =Ri(s)e3 , this implies that the third columns of each Ri(s) areequal. Therefore, each Ri(s) differs from the others by a rota-tion about the local tangent z-axis. We introduce an angle θi(s)to parameterize this difference as

Ri(s) = R1(s)Rθi(10)

where Rθi= ee3 θi (s) denotes a rotation about the z-axis by

θi(s), and θ1 ≡ 0 by definition. We use this to obtain a rela-tionship between the tube curvature vectors that enforces thetangency constraint. Applying the definition of ui , we obtain

ui =(RT

i Ri

)∨ = RTθi

u1 + θie3 . (11)

Interpreted geometrically, this equation says that the local xand y curvatures of each deformed tube are equal when ex-pressed in a common reference frame. The torsional z compo-nents are free to vary independently for each tube. The variableθi provides a parameterization of this variance as

θi = ui,z − u1,z . (12)

F. Equilibrium Equations: A Collection of Tubes Under Load

We now proceed to derive the equilibrium equations for a col-lection of concentric tubes under applied external forces and mo-ments. This section will parallel the derivation of Section II-D.We begin by writing a force balance on a section of the cannulafrom arc length s to the end �∫ �

s

n∑i=1

f i(σ)dσ −n∑

i=1

ni = 0. (13)

Similarly, summing the moments on the portion [s, �] about theworld frame origin, we obtain∫ �

s

(r(σ) ×

n∑i=1

f i(σ) +n∑

i=1

li(σ)

)dσ

−n∑

i=1

(mi(s) − r(s) × ni(s)) = 0 (14)

where f i and li are external force and moment distributions,respectively, applied specifically to tube i.

Taking the derivative of (13) with respect to s, we obtain

n∑i=1

(ni + f i) = 0. (15)

Similarly, taking the derivative of (14) with respect to s andsubstituting (15) into it yields

n∑i=1

(mi + r × ni + li) = 0. (16)

Equations (15) and (16) are the multitube analogs of the Cosseratequilibrium equations (6) and (7). Indeed, they are merely thesums of (6) and (7) over all the tubes. The substance of ourapproach lies in the following application of the concentric-tube kinematic constraints (allowing independent tube torsion)to these equations in order to obtain a set of differential equationssimilar to the single-tube case while introducing the minimumnumber of additional variables.

We now wish to expand (16) and solve for u1 in terms ofthe precurvatures, the applied loads, and the state variables. Asin the single-tube case, we use the constitutive law (3) and thekinematic relationship (2) for each tube to obtain

n∑i=1

mi(s) =n∑

i=1

Ri(Ki(ui − u∗i )

+ (uiKi + Ki)(ui − u∗i )).

We use r = R1e3 and (13) to obtain the last two terms of (16)in terms of the applied loads

n∑i=1

(r × ni + li) = (R1e3) ×∫ �

s

f(σ)dσ + l

where f(s) =∑n

i=1 f i(s), and l(s) =∑n

i=1 li(s). Then, afterpremultiplying by RT

1 (s) and recalling that Rθi= RT

1 (s)Ri(s)from (10), (16) becomes

n∑i=1

Rθi(Ki(ui − u∗

i ) + (uiKi + Ki)(ui − u∗i ))

+ e3 × RT1

∫ �

s

f(σ)dσ + RT1 l = 0. (17)

We wish to obtain an expression for u1 in terms of f , l, R1 ,and u1 , . . . ,un ; therefore, we apply the derivative of (11) asfollows:

ui(s) = θi

dRTθi

dθiu1 + RT

θiu1 + θie3 (18)

to eliminate u2 , . . . , un from (17). This substitution enables usto solve (17) for the first two components u1,x and u1,y in terms


of the state variables[u1,x

u1,y

]= −K−1

n∑i=1

Rθi

(Ki

(θi

dRTθi

dθiu1 − u∗

i

)

+ (uiKi + Ki)(ui − u∗i )

)∣∣∣∣x,y

− K−1(

e3RT1

∫ �

s

f(σ)dσ + RT1 l

)∣∣∣∣x,y

(19)

where K =∑n

i=1 Ki , and |x,y denotes selection of only thefirst two components of a vector.

There is not enough information in (17) and (18) to similarlyobtain u1,z . This is because our kinematic constraints allow thetorsional strains of the individual tubes, i.e., ui,z , to be different,where θi parameterizes this difference. Thus, we now return tothe single-tube formulation, as described in Section II-D, toobtain the behavior of the individual torsional strains.

Equation (9) describes the curvature of a single tube undera general load. Therefore, it remains true for every individualtube in an arrangement of many concentric tubes. We generallydo not know a priori what forces and moments the tubes in aconcentric collection apply to one another. However, noting thatthe third column of the matrix e3 is all zeros, we find that thethird component of u is actually independent of any forces andnonaxial moments. Writing this third component for tube i, weobtain

ui,z = u∗i,z +

EiIi

GiJi(ui,xu∗

i,y − ui,y u∗i,x)

+˙(GiJi)

GiJi(u∗

i,z − ui,z ) −1

GiJieT

3 RTi li . (20)

The only external load appearing in this equation is the com-ponent of the distributed moment at s about the local z-axis.In a collection of concentric tubes, the assumption of no staticfriction between tubes implies that the tubes cannot apply suchaxial-moment distributions on one another. Therefore, in thisequation, li represents a purely external torsional moment dis-tribution applied specifically to tube i.2

This torsional behavior completes our multitube model sothat we now have first-order state equations for the variable set{g1 ,u1 , u2,z , . . . , un,z , θ2 , . . . , θn}. The equations that definetheir derivatives are (2), (12), (19), and (20). The intermediatevariables ui can be calculated algebraically from u1 at everystep using (11).

G. Model Implementation

Challenges in practical implementation of our model includenumerically dealing with point loads and the issue of tubes be-ginning and ending at different arc lengths. In an active cannula,all tubes are clamped to actuators at their bases and are subjectto a set of applied loads along their length. Since the distributedforce is integrated in (19), finite-point forces can be conveniently

2Since energy methods have been used in prior unloaded active-cannulamodels, we also present an energy-based derivation of (20) in the Appendix todemonstrate the connection of our present study to prior modeling approaches.

Fig. 4. Two-tube cannula showing transition points where continuity of shapeand internal moment must be enforced. The constrained point of entry into theworkspace is designated as the arc-length zero position.

included in the model through the use of Dirac delta functions inthe distribution. However, including finite-point moments usingthis method would cause (19) to become infinite at a point, andmost numerical techniques are ill equipped to handle this. Thesame difficulty arises if tubes end at different arc lengths or ifthe precurvature of a tube undergoes a step change (i.e., if r∗(s)is not twice differentiable at a point).

All such occurrences should ultimately result in a step changein the deformed curvature. This can be accounted for by solvinga series of continuous systems bounded by the discontinuoussolution points while enforcing appropriate boundary conditionsat the junctions. Discontinuous solution points for a typical two-tube cannula are illustrated by gray lines perpendicular to thecannula in Fig. 4, which break it into sections. The boundaryconditions to be enforced across each transition point betweensections (at arc length s) are as follows: 1) The position andorientation of each tube must be continuous across the boundary,and therefore

gi(s−) = gi(s+)

and 2) static equilibrium requires that the sum of the internalmoments carried by the tubes just before the end of the sectionequal the sum of the internal moments carried by the tubes justafter the end of the section plus the sum of the applied pointmoments at the boundary (a point moment applied to tube i isdenoted by lp,i(s)), i.e.,

n∑i=1

mi(s−) =n∑

i=1

mi(s+) +n∑

i=1

lp,i(s).

This enforcement of static equilibrium across discontinuousboundaries is also required in application of Cosserat theoryto other types of continuum robots, and forms of these sameconditions are also given in [7].

One must also consider the boundary conditions of the entirecannula for practical implementation. In many physical proto-types of active-cannula robots, all the tubes are constrained topass through a fixed entry point, which we designate as s = 0,as shown in Fig. 4. The actuators grasp the bases of the tubesbehind this point at negative arc-length values s = −Di . At theproximal end of each tube, gi(−Di) is determined by the actua-tion inputs, namely the translation and axial rotation of the baseof the tube. At the distal end of the robot, the static equilibrium


Fig. 5. Measured curvatures of the preset tube shapes expressed in a rotation-minimizing frame (note that rotation-minimizing frames, i.e., u∗

1 ,z and u∗2 ,z ,

are zero by definition).

condition becomesn∑

i=1

mi(s) =n∑

i=1

lp,i(s).

Also, at the distal end of each tube, we have a natural boundarycondition for the local axial component of the internal momentof each tube

eT3 RT

i mi(�i) = eT3 RT

i (�i)lp,i(�i)

where �i denotes the arc length at which tube i ends. This arisesbecause in the absence of friction, the tubes cannot apply axialmoments to one another.

The tubes are straight for s < 0, and therefore, the unknownboundary conditions on the proximal side of the robot are theset of initial curvature values {u1,x(0),u1,y (0),u1,z (−Di),u2,z (−D2), . . . ,un,z (−Dn )}. In our experiments described inthe next section, we employ a standard shooting method to solvefor the values of these unknown initial curvatures which satisfythe boundary conditions on the distal side while enforcing theconditions given above at each transition point.

III. EXPERIMENTS

In order to validate the model developed in Section II, a set ofexperiments was performed for a collection of two Nitinol tubeswith general precurvatures (see Fig. 5) in various configurationsand under various loading conditions.

A. Tube Properties and Measurement Procedures

The physical properties of the tubes used are given inTable I. Each tube has an initial straight length followed bya curved section, the curvature of which is shown in Fig. 5. Inour experiments, the outer tube was held stationary in its fullyextended position, while the base of the inner tube was trans-lated to five different positions (by the actuation unit shown inFig. 6), which are given in Table II. At each of these translational

TABLE IMEASURED AND ASSUMED PHYSICAL QUANTITIES FOR EXPERIMENTAL TUBES

Fig. 6. Manual actuation unit used to precisely position the bases of the tubes.

TABLE IITRANSLATIONAL ACTUATOR CONFIGURATIONS

TABLE IIIROTATIONAL ACTUATOR POSITIONS APPLIED AT EACH TRANSLATIONAL

CONFIGURATION SHOWN IN TABLE II

positions, the inner tube was rotated to eight evenly spaced an-gular positions, which are given in Table III. Thus, the tubeswere actuated to 40 different workspace locations which evenlyspan the set of angular and linear differences of tube base po-sitions (see Fig. 7). The rest of the configuration space couldbe generated by a rigid rotation of the experimentally sampledspace about the base frame z-axis; therefore, this set of tubepositions evenly samples the unique—from the perspective ofthe model—configuration space locations.

As shown in Fig. 8, at each of these configurations, a setof 3-D points along the backbone was determined via imagestaken from a calibrated pair of stereo cameras (Sony XCD-X710 Firewire cameras with a resolution of 1024 × 768 pixels)mounted above the robot. The fiducial markers shown in theinset image in Fig. 8 enabled determination of point correspon-dences for stereo triangulation after they had been identified inimage coordinates by manually clicking on the center of theblack bands in each image with MATLAB’s ginput command.The cameras were calibrated using a camera calibration tool-box for MATLAB [21], and the transformations between thestereo-camera coordinate frames and the robot base frame were


Fig. 7. Cannula in all 40 experimental configurations. One can span the entireworkspace by rigidly rotating this collection about the z-axis, which can beaccomplished by rotating the base of each tube by the same amount whilekeeping their angular differences the same. Thus, the above illustrates a samplingof all unique configuration space locations from the model’s point of view. Foreach configuration, backbone data were collected in the unloaded state and witha force applied to the tip of the cannula.

initially estimated by triangulating a grid of points with knownlocations in the base frame and performing rigid point cloudregistration [22]. The mean, max, min, and standard deviationof the euclidean registration errors were 0.57, 1.30, 0.11, and0.32 mm, respectively. Directionally, the mean registration er-rors along the x-, y-, and z-axes were 0.50, 0.12, and 0.15 mm,respectively, where the x-axis points toward the cameras and thez-axis points along the robot axis at the base. These numbersencompass the error in the process of manually identifying thepixel coordinates of the points, as well as any error intrinsicto the stereo-camera system. We take this to be a rough esti-mate of the effective accuracy of our vision-based triangulationsystem.

In each of the 40 actuator configurations, a point force wasalso applied to the tip of the cannula by a wire tied through ahole in the tip of the inner tube, and backbone data were takenin the robot’s loaded state. The direction of the force vectorapplied by this wire was also determined by triangulating pointsmarked along its length, as shown in Fig. 8. As can also be seenin the figure, the wire was run over a pulley and attached to amass ranging from 100 to 500 g, as detailed in Table II. Fromthe perspective of the cannula, the applied tip-load vector was ina different direction in each experiment because the robot wasin a different configuration in each.

The pulley was mounted to a 6-degree-of-freedom (DOF)manually adjustable frame made from standard 80-20 Inc. parts.In each of the 40 robot configurations, before taking data, the

Fig. 8. Experimental setup. Tube bases were translated and rotated preciselyby manual actuators. Three-dimensional backbone points were triangulated byidentifying corresponding markers along the cannula in stereo images. Thevector of the applied force was measured by triangulating positions along thewire that connects the cannula tip (via the pulley) to the applied weight.

location and angle of the pulley was adjusted as needed to makesure that the cable was orthogonal to the pulley axis. At thistime, we also checked for hysteresis due to pulley friction bydisplacing the mass up and down (the flexible cannula actingas a spring) and noting that the pulley always returned to thesame equilibrium angle when the weight was released. A sub-sequent experiment was carried out using this same procedure,in which the cable tension was measured using a six-axis loadcell (Nano17, ATI, Inc.). The resolution of the cable tensionwas found to be ±0.009 N for the 0.981 N load, ±0.020 N forthe 1.962 N load, and ±0.088 N for the 4.905 N load. Thus, weconclude that any unmodeled pulley effects did not significantlyaffect the loads transmitted to the cannula.

The two tubes have general precurvatures u∗1(s) and u∗

2(s).To obtain these, we began by capturing points along them (in-dividually before inserting one into the other) using the stereotriangulation system in the manner described previously. Wethen fit a parametric polynomial curve to these points and gath-ered curvature data from these smooth fits. The components ofprecurvature for each tube are plotted in Fig. 5 (the tube curveswere framed using Bishop’s frames; therefore, there are only xand y curvatures). Note that the curvatures are not constant overs, and all prior prototypes reported in the literature have hadconstant-curvature preset tube shapes.


B. Model Performance and Calibration

Since model error nearly always increases with the arc lengthalong the robot toward the tip, we use tip location differ-ence (called “tip error” henceforth) as a metric for comparingpredictions to experiments, as has been done in many previousstudies with active cannulas and other continuum robots. In ourparticular experiments, we also visually verified that the tip wasthe point of greatest deviation between model and experimentaldata by plotting the two together for all experimental positions.

Using the nominal parameter values (those that were directlymeasured or appeared on data sheets; for further discussion ofactive cannula nominal parameters, variances, and error propa-gation in model parameters, see [3]) listed in Table I, the meantip error over all 80 experiments was 5.94 mm. Since actual val-ues for the moduli of Nitinol tubes are highly uncertain (Young’smodulus is listed as 41–75 GPa on data sheets from the man-ufacturer, NDC, Inc.), the values of each tube’s bending andtorsional stiffness were subsequently calibrated by finding theset of tube parameters that minimized the sum of the positionalerrors at three locations along the robot: the base (s = 0), thetip of the outer tube (s = �2), and the tip of inner tube (thetip of the device, s = �1). To reduce uncertainty in the regis-tration of the robot base frame to the stereo-camera frames,we included small changes to the base-frame position (trans-lations δx, δy, δz) and to the orientation (XYZ Euler anglerotations δα, δβ, δγ) as additional parameters to be calibrated.Our calibration process is accomplished by solving an uncon-strained nonlinear optimization problem for the parameter setP = {E1I1 , E2I2 , J1G1 , J2G2 , δx, δy, δz, δα, δβ, δγ}:

Pcalibrated = argminP

(80∑

k=1

ek (0) + ek (�2) + ek (�1)

)

where ek (s) = ‖rmodel(s) − rdata(s)‖k is the Euclidean dis-tance between the model backbone prediction and the datain experiment k. To implement this minimization, we usedthe Nelder–Meade simplex algorithm, as implemented byMATLAB’s fminsearch function.

The parameters resulting from this model-fitting procedureare shown in comparison to their nominal counterparts inTable IV. The base-frame parameters showed only smallchanges during optimization, with XYZ Euler angles changingby −0.06◦, 0.67◦, and −0.58◦, while the frame origin translated2.0 mm. Using these calibrated parameters, the mean error overall experiments was 2.91 mm, as shown in Table VI.

Fig. 9 shows the unloaded and loaded states of the cannulafor actuator values of D1 = 154.7 mm, D2 = 30.7 mm, α1 =135◦, and α2 = 0◦. Experimental data points are overlaid on themodel prediction, and the model shown in the figure uses thecalibrated parameters. The configuration shown is typical of all80 experimental positions in that the tip error in both cases isabout 3 mm, while the mean for all experiments is 2.91 mm. Therest of the shape is also typical of the 80 experimental runs inthat the experimental data lie very close to the model predictionalong the entire backbone, and the applied forces were sufficientto cause large deflection in all cases.

TABLE IVNOMINAL AND CALIBRATED PARAMETERS

Fig. 9. Comparison of model prediction and experimentally determined back-bone points for the unloaded and loaded cases where actuators are set toD1 = 154.7 mm, D2 = 30.7 mm, α1 = 135◦, and α2 = 0◦. The directionof the 0.981 N applied force is shown by an arrow at the tip of the deformedmodel prediction. These examples are representative of our dataset—their tiperrors (approximately 3 mm) are near the 2.91-mm mean tip error over all 80experiments.

C. Distributed Load Experiment

In order to demonstrate capability of the model of Section IIto handle distributed loads, an experiment was conductedwhere a force distribution was applied along the length ofthe cannula. The actuator configuration was D1 = 122.7 mm,D2 = 30.7 mm, and α1 = α2 = 0◦. Note that the cannula’s ownweight is not sufficient to cause appreciable gravitational deflec-tion. Therefore, we added additional weights along its length.As shown in Fig. 10, we approximated a distributed load byplacing a large number of nuts along the shaft of the cannula(on both the outer tube and the portion of the inner tube whichextended out from the outer tube). The nuts were spaced evenlyalong the shaft and had a total mass of 56.96 g. Stereo-point


Fig. 10. Active cannula under a distributed load represented by nuts equallyspaced along its length.

Fig. 11. Comparison of loaded and unloaded model predictions with exper-imentally determined backbone points for a distributed load. The tip error is4.54 mm.

TABLE VTIP ERROR STATISTICS—NOMINAL PARAMETERS

correspondences were determined based on manually clickingbackbone points between the nuts. The tip of the device was cov-ered by the last nut, thus making it impossible to locate the tipin stereo images; therefore, the tip was considered to be the lastvisualizable section of the backbone, and the model arc lengthwas reduced by one nut width. The model and experimental datafor this loading condition are shown in Fig. 11, along with theunloaded model for the same configuration. The tip error was4.54 mm.

D. Statistical Analysis

Error statistics for both nominal and fitted parameter setsare given in Tables V and VI. The dataset corresponds well tomodel predictions, with a mean tip error of 2.91 mm. An errorhistogram for all 80 cases shows that 75% of the errors werebelow 3 mm, and 85% were below 4 mm (see Fig. 12).

A statistical outlier with 15.20 mm of tip error occurred in theloaded state with D1 = 130.7 mm, D2 = 30.7 mm, α1 = 270◦,

TABLE VITIP ERROR STATISTICS—CALIBRATED PARAMETERS

Fig. 12. Histogram of tip error for all 80 experiments using fitted modelparameters. A total of 75% of the errors are below 3 mm, and 85% are below4 mm.

and α2 = 0◦. The error in this case may have been increased bya procedural error (e.g., an incorrect α1 value being recorded)or simply a worst-case compounding of unmodeled phenomenaand measurement uncertainty. It is also worth noting that in thisconfiguration, the cannula was fully extended, and this errorcorresponds to only 7.68% of the arc length.

We note that for long, slender continuum robots, tip error ishighly dependent on the total arc length, since errors tend toincrease from the base of the cannula to its tip, as mentionedpreviously. The total arc length of the active cannula in our ex-periments ranged from 105.9 to 197.9 mm over the experimentaldataset. Thus, an average tip error of 2.91 mm is approximately1.5%–3% of the arc length.

E. Error Sources

The unmodeled phenomena of transverse shear strain andelongation could potentially be accounted for by allowing ∆ξ(s)to include the first three components, but the kinematic con-straints in this case become more complex, and shear effects areknown to be negligible for long, thin beams. Additional torsiondue to friction is likely a more significant effect and could poten-tially be included in the model by additions to or modifications of(20), which describes the axial torque along a tube. Dependingon cannula design (intratube tolerances, precurvature functions,arc lengths, etc.), we believe that friction can become more pro-nounced than it was in our experimental prototype, and futuremodel extensions accounting for it may be useful, despite theminimal hysteresis observed in our experiments.


An unmodeled hardware detail that may be important is thefact that there is a small amount of clearance between the outsidediameter of the inner tube and the inside diameter of the outertube. This means that the tube tangents are not quite coincidentwhere one exits the other, as the model assumes. In terms ofdesign, this effect can be reduced by choosing tighter intratubetolerances at the cost of increasing frictional effects. Alterna-tively, this effect could be modeled by modifying the continuityof position and orientation across transition points to include asmall rotational and translational displacement, where one tubeexits another.

However, the fact that small tip errors were achieved withoutmodeling any of the above effects indicates that they are largelynegligible in our prototype. The model presented in this papereffectively captures the main structural features of concentric-tube continuum robots because it allows independent tube tor-sion during deformation. Whether it will be necessary to modelany of these effects in the future will likely depend upon thedesign of the active cannula and the accuracy required by theapplication. Many clinical applications (e.g., needle biopsy orthermal ablation) can tolerate 3-mm tip errors. Other potentialapplications, such as retinal microsurgery, will require higheraccuracy. In many cases, implementation of closed-loop con-trol using this model is likely to significantly increase the op-erational accuracy and may render additional modeling detailunnecessary.

IV. CONCLUSION

In this paper, we have presented an extension of the geo-metrically exact Cosserat-rod theory to analyze a collection ofconcentric precurved tubes under a general set of distributedand point wrenches. We then tested this model in a set of exper-iments using an active-cannula robot composed of two Nitinoltubes with general preset curved shapes, subject to both tiploads and distributed loads along the length of the device. Withparameter fitting, the model achieved an average tip error of2.91 mm across all 80 experimental positions, which span themodel-unique configuration space.

Our modeling work in this paper provides a theoretical foun-dation from which to understand active-cannula shape underload and from which to begin to explore many future applica-tions. We expect that this model will be a valuable design tool insimulating proposed active-cannula designs, thereby enablingaccurate compliance and kinematic analysis for cannulas in-tended to interact with tissue under environmental constraints.We also expect that it may facilitate use of the cannula’s flexibil-ity to sense and control contact forces. Intrinsic-force sensing isdesirable for thin continuum medical robots where inclusion of aforce sensor could significantly affect device function [8]. Forceinformation has the potential to enable tissue-property estima-tion to locate lesions via palpation or provide haptic feedbackto the surgeon in a teleoperated system.

APPENDIX

Prior derivations of active-cannula shape have proceededfrom both the perspective of energy minimization and Cosserat-

rod theory. In this Appendix, we derive (20) using energy meth-ods to demonstrate the consistency of the two approaches.

Using Euler’s linear constitutive bending equation (3), theelastic energy stored in the n-tube deformations is

E =12

n∑i=1

∫ �

0(ui − u∗

i )T Ki(ui − u∗

i )ds.

Substituting (11) into this, we have

E =12

∫ �

0uT

1 K1u1 − 2uT1 K1u

∗1 + u∗T

1 K1u∗1

+n∑

i=2

(uT

1 Kiu1 + 2uT1 Kiθie3 − 2uT

1 RθiKiu

∗i

+ u∗Ti Kiu

∗i − 2u∗T

i Kiθie3 + θ2i eT

3 Kie3)ds. (21)

To find the functions θi , which minimize the stored elastic en-ergy, we apply the following Euler–Lagrange equation to thefunctional n − 1 times, once with respect to each θi :

∂f

∂θi− d

ds

(∂f

∂θi

)= Q′

i . (22)

Note that the right-hand side is not zero, as is the case in priorfree-space models [15]. Instead, we have Q′

i , which representsany generalized forces associated with θi . This corresponds toan arc-length-dependent torque per unit length applied aboutthe e3-axis of tube i. Applying (22) to the integrand of (21), weobtain

− uT1

∂Rθi

∂θiKiu

∗i − (u1 − u∗

i )T Kie3 − (u1 − u∗

i )T Kie3

− θieT3 Kie3 − θie

T3 Kie3 = e3Ri(s)li(s). (23)

We can use (11) and (18) to replace u1 and u1 and then solvefor ui,z to obtain (20).

REFERENCES

[1] D. C. Rucker, B. A. Jones, and R. J. Webster, III, “A model for concentrictube continuum robots under applied wrenches,” in Proc. IEEE Int. Conf.Robtics Autom., 2010, pp. 1047–1052.

[2] J. Furusho, T. Katsuragi, T. Kikuchi, T. Suzuki, H. Tanaka, Y. Chiba,and H. Horio, “Curved multi-tube systems for fetal blood sampling andtreatments of organs like brain and breast,” J. Comput. Assist. Radiol.Surg., vol. 1, pp. 223–226, 2006.

[3] R. J. Webster III, J. M. Romano, and N. J. Cowan, “Mechanics ofprecurved-tube continuum robots,” IEEE Trans. Robot., vol. 25, no. 1,pp. 67–78, Feb. 2009.

[4] P. Sears and P. E. Dupont, “A steerable needle technology using curvedconcentric tubes,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2006,pp. 2850–2856.

[5] R. J. Webster III, A. M. Okamura, and N. J. Cowan, “Toward activecannulas: Miniature snake-like surgical robots,” in Proc. IEEE/RSJ Int.Conf. Intell. Robots Syst., 2006, pp. 2857–2863.

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[9] C. Li and C. Rahn, “Design of continuous backbone, cable-driven robots,”ASME J. Mech. Des., vol. 124, no. 2, pp. 265–271, 2002.


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[13] D. C. Rucker, R. J. Webster, III, G. S. Chirikjian, and N.J. Cowan, “Equilibrium conformations of concentric-tube continuumrobots,” Int. J. Robot. Res., to be published. [Online]. Available:http://ijr.sagepub.com/content/early/2010/04/01/0278364910367543

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[21] J.-Y. Bouguet. (2010, Mar.). Camera calibration toolbox for Matlab[Online]. Available: http://www.vision.caltech.edu/bouguetj/calib_doc/index.html

[22] K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-squares fitting of two3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-9,no. 5, pp. 698–700, Sep. 1987.

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D. Caleb Rucker (S’07) received the B.S. degreein engineering mechanics and mathematics fromLipscomb University, Nashville, TN, in 2006. Heis currently working toward the Ph.D. degree inmechanical engineering with Vanderbilt University,Nashville.

In 2007, he was an Engineer with Jacobs Technol-ogy Group, Tullahoma, TN. His current research in-terests include medical robotics, continuum robotics,mechanics-based modeling, and control.

Bryan A. Jones (S’00–M’00) received the B.S.E.E.and M.S. degrees in electrical engineering from RiceUniversity, Houston, TX, in 1995 and 2002, respec-tively, and the Ph.D. degree in electrical engineeringfrom Clemson University, Clemson, SC, in 2005.

He is currently an Assistant Professor with theMississippi State University, Mississippi State, MS.From 1996 to 2000, he was a Hardware Design Engi-neer with Compaq, where he specialized in board lay-out for high-availability redundant array of indepen-dent disks (RAID) controllers. His current research

interests include robotics, real-time control-system implementation, rapid proto-typing for real-time systems, and modeling and analysis of mechatronic systems.

Robert J. Webster, III (S’97–M’08) received theB.S. degree in electrical engineering from ClemsonUniversity, Clemson, SC, in 2002 and the M.S. andPh.D. degrees in mechanical engineering from theJohns Hopkins University, Baltimore, MD, in 2004and 2007, respectively.

In 2008, he joined the Faculty of Vanderbilt Uni-versity, Nashville, TN, as an Assistant Professor ofmechanical engineering, where he is currently theDirector of the Medical & Electromechanical De-sign Laboratory. His current research interests in-

clude medical robotics, image-guided surgery, human–machine interfaces, andcontinuum robotics.

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