FBG-based Shape Sensing Tubes for Continuum Robots

Seok Chang Ryu, Member, IEEE and Pierre E. Dupont, Fellow, IEEE

Abstract— Fiber Bragg gratings (FBG)-based optical sensorsare a promising real-time technique for sensing the 3D curva-ture of continuum robots. Existing implementations, however,have relied on embedding optical fibers in small-diameter metalwires or needles. This paper proposes polymer tubes as analternative substrate for the fibers. This approach separatesthe sensors from the robot structural components while usinga minimal amount of the robot’s tool lumen and providing thepotential of inexpensive fabrication. Since the fibers are stifferthan the polymer substrate, however, design challenges arisein modeling strain transfer between the fibers and the tubesubstrate. To investigate the potential of this approach, a straintransfer model is derived and validated through simulation andexperiment.

I. INTRODUCTION

Continuum robots take the shape of three dimensionalcurves and are able to change their shape through a com-bination of bending, rotation and extension or contraction oftheir structural components. Because of these capabilities,continuum robots are ideally suited for applications such asminimally invasive surgery. Their flexibility, however, leadsto uncertainty in the shape of their backbone curve as wellas in the location of their tip.

Approaches to real-time sensing that have been studiedinclude imaging, electromagnetic (EM) tracking and forcesensing for tendon-based actuation. These techniques all haveshortcomings, however. For example, drawbacks with imag-ing include limited resolution (ultrasound), risk of ionizingradiation (x-ray or CT), and slow speed (MRI). Tendonlength-based shape estimation is limited to single bends andits accuracy depends highly on the robot kinematic model.

In contrast to these approaches, a real-time sensing tech-nology that could be easily inserted and removed fromthe robot is preferred because of its direct measurement,instant adaptability to the continuous shape change andease of replaceability. Curvature sensing using fiber Bragggratings (FBG) has received recent attention, due to itssmall size, biocompatibility and high sensitivity. Comparedto EM sensors [1], FBG-based sensors are smaller, immuneto EM noise and can contain multiple sensors along thelength of a fiber. For example designs using attached [2]or embedded [3], [4] fibers have been implemented on 1mmdiameter metal wires. Both approaches reasonably assumed aperfect strain transfer from the wire to FBG. While the smallwire diameter had minimal effect on structural stiffness andmade it possible to accommodate large curvatures (gratings

Seok Chang Ryu and Pierre Dupont are with the Departmentof Cardiovascular Surgery, Boston Children’s Hospital, HarvardMedical School, Boston, MA 02115, USA seokchang.ryu,pierre.dupont@childrens.harvard.edu

(a)

Mi f

( )

Tool control wire

Micro forceps

Shape sensinpolymer tubep y

(b)

Shape sensing tube inside orking channelinside working channel

ContinuumTool control wire

10 mm

ng wire braid e

Micro forceps

m robot

Fig. 1. Use example of a shape sensing tube: (a) incorporated shape-sensingcapability in the wire braid polymer actuation tube of 1mm diameter forceps,(b) shape sensing and tool actuation tube inserted into a continuum robot.

can tolerate 0.8% strain [5]), a solid wire sensor runningthrough the center of a robot’s lumen can interfere with theinsertion, removal and control of tip-mounted tools.

An alternative approach is to employ a tubular substratemade of a low modulus material. Such a design woulduse a minimal amount of the robot lumen and, despite alarger diameter, its compliance in bending would tend not tochange the curvature of the robot into which it is inserted.An additional benefit of a compliant substrate is the strainreduction experienced by embedded optical fibers [6]. Usingthis effect, a sensing tube can be designed to experiencelarger bending strains than the optical fibers could otherwisetolerate.

A potential problem of a low-modulus homogeneous tubu-lar substrate, however, is that compliance in bending isaccompanied by compliance in torsion. Thus, any twistingof the sensing tube, arising either from sensor insertion orfrom robot motion, would confound robot shape estimationsince the orientation of the sensors with respect to the crosssection would be unknown. This issue has been recognizedin the literature [7] and has been addressed using fiberoptic torsion sensing, although this has been limited tolarge diameter devices due to the minimum fiber bendingradius [8], [9]. Twisted-core fibers have also been developed(http://lunainc.com/), which sense bending and twisting si-multaneously.

An inexpensive alternative solution is to employ a compos-ite material substrate that is compliant in bending and stiff in

2014 IEEE International Conference on Robotics & Automation (ICRA)Hong Kong Convention and Exhibition CenterMay 31 - June 7, 2014. Hong Kong, China

978-1-4799-3684-7/14/$31.00 ©2014 IEEE 3531

A A section view

Wire braided polymer tube

Surface bonded optical fibers

A-A section view

Adhesive

Sensing points,composed of three FBGs

per location

A

A

Channel for tool control wire

per location

Tool can be attached

Fig. 2. Schematic of wire braided polymer tube with surface mountedoptical fibers.

torsion. Such tubular elements are commonly used in medicaldevices, e.g., catheters and are composed of biocompatiblepolymer with an embedded wire braid. With a high braid an-gle, torsional rigidity is high while flexural stiffness remainslow compared to that of the wire material [10]. Ovalizationand the resultant buckling are another concern in polymertube bending [11], but the resistance against it, called kinkresistance, is also increased with a high braid angle [12]. Asa result, the circular tube cross section is maintained even athigh curvature, facilitating sensor modeling. Finally, a metalwire braid can be expected to reduce viscoelasticity in thepolymer substrate.

An additional benefit of such a sensing tube is that it canbe integrated with various tools or device delivery systemsand not take up any additional area of the robot lumen. Forexample, Fig. 1 (a) depicts 1mm diameter forceps that ourgroup uses with our concentric tube continuum robots. Theforceps are controlled by relative translation between a wire-braided polymer tube and a wire running through the tube.This shape-sensing and tool-actuation tube occupies a singleworking channel and conforms to the continuum robot shape(Fig. 1 (b)).

The contribution of this paper is to develop and vali-date a mechanics-based model for these sensing systemsso as to identify the important design variables. Section IIpresents the sensor design and reviews the related mechanicsliterature. A model describing strain transfer between thecomposite tube and the optical fibers is derived in section IIIand analyzed through simulation. Experimental results usinga sensor prototype are provided in section IV and conclusionsappear in section V.

II. SENSOR DESIGN

As shown in Fig. 2, the design considered here utilizesthree surface mounted optical fibers. Although not consideredhere, an alternative design could embed the fibers in thetube cross section. Multiple sensing locations, distributedalong the tube, are used to estimate the bending curvature atdiscrete values of arc length. From these values, the shapeof the backbone curve can be reconstructed as in [13].

Fig. 3 shows the detailed cross section at one sensing

Bending or κ FBG1

Do = 1.05 mm

α

D ≈ 1.40 mmBending plane

direction ε1

D 120°

Di= 0.75 mm

α

FBG2FBG3

Df≈ 1.22 mm

125 µm coatingdiameter fiber

2ε2

ε3PTFE liner

SS-304V wire braid Polyimide tube

Fig. 3. Cross sectional sensor configuration with three fiber Bragg gratings.Dimensions correspond to prototype of section IV.

location. It is composed of three equally spaced optical fibers.To provide a sense of scale, the dimensions of the prototypeconsidered later in the paper are provided here. The tube,composed of wire-braided (SS-304V) polymer, has an outerdiameter of 1.05mm and an inner diameter of 0.75mm. Theoptical fibers have 125µm and 80µm coating and claddingdiameters.

A. Curvature-Strain Model

Shape estimation starts from the local curvature calcula-tion using the measured fiber strains. In [4], a simple linearmodel was used to relate fiber strain and the curvature basedon beam mechanics, but the coefficients were calibration-based rather than model-based. Thermal strains were com-pensated in the same way. Improved linear curvature-strainmodels providing better understanding of the mechanics weredeveloped in [2], [13], in which the coefficient was thefiber distance from the neutral axis. Also thermal strainswere included in the analytical model and compensated byassuming uniformity of the cross section.

In contrast, the composite polymer substrate presents twocomplicating factors owing to its susceptibility to axial load-ing and its non-unity strain transfer ratio. These phenomenawere negligible for prior metal wire sensors because of theirhigh elastic modulus. These factors can be incorporated intothe model of [13] by introducing the strain transfer ratio fromthe tube surface to the fiber, η , and the axial strain, εa, as

ε1 = η · {εmax · sin(α)+ εa}+ εt

ε2 = η · {εmax · sin(α +120◦)+ εa}+ εt (1)ε3 = η · {εmax · sin(α +240◦)+ εa}+ εt

where εmax, α and εt are the maximum bending strain in thetube, the angle of FBG1 from the bending plane and thermalstrain, respectively.

In Eq. 1, η is only applied to the mechanically inducedstrains, which is reasonable by assuming negligible thermalexpansion, because thermal strain is not an actual strain, buttemperature-induced changes in optical properties like therefractive index of gratings [14]. Also εt and εa are assumeduniformly distributed on the cross section.

3532

Then, because of the equally spaced fiber arrangement, thecombined uniform strain is calculated, independent of α , as

εuni = εt +η · εa =ε1 + ε2 + ε3

3(2)

Solving Eq. 1 about α and εmax using Eq. 2, yields,

α = arctan{√

3 · (ε1− εuni)

ε2− ε3} (3)

εmax =1η· sgn{(ε1− εuni) · sinα} ·

√23

3

∑i=1

(εi− εuni)2

(4)

= κ · Do

2(5)

where κ is tube curvature and Do is used instead of fiberdistance D f , as opposed to [13], because of the definitionof η . Compared to perfect strain transfer with η = 1, thesensing tube can undergo up to 1

η≥ 1 times larger curvature

for a given tensile strength of FBG and sensor size.In order to solve for the magnitude of curvature, κ , using

Eq. 4 and Eq. 5, the value of η is needed. Although a simplecalibration technique could be used to estimate its value fora particular sensor, a mechanics-based model will providedesign insights into the most important variables relating toits value. In this way, it can be possible to select designparameters to trade off maximum curvature versus sensitivityto meet desired performance specifications.

B. Strain Transfer Reduction Effects

As shown in Fig. 4 (b), η < 1 is caused by two strainreduction effects: the shearing effect in the soft interlayer andthe fiber reinforcement effect on the low modulus substrate.

The first is related to the shear deformation in the adhesiveand in the fiber coating. This effect has been studied usingshear-lag models for embedded fibers [6], assuming that thelayers only carry pure shear stress when they (at most, a fewGPa) are much softer than the silica glass (72 GPa) used forthe fiber core and cladding. Although this shear-lag modeldoes not consider transverse shear stress causing bendingdeformation, the insensitivity of the FBG to this stress allowsits reasonable use for sensing tube bending.

For surface bonded fibers, a modified shear-lag modelwas developed in which the component of η due to shearlag, ηSL, is dominated by the fiber bonding length and thebottom adhesive thickness while insensitive to the side ortop adhesive [15]. This analytical model, however, assumedan unrealistic axisymmetry for the adhesive. In [16], thisaxisymmetry was discarded, however, a simplified modelgeometry was assumed in which the substrate had a sig-nificantly larger cross section than the fiber, an assumptionunsuitable for thin polymer tubes.

In addition, these shear-lag models in fibers assume a uni-form strain applied to the long bonded fibers [15] or requirethe strain profile in order to obtain the fiber strain [16], whichis not feasible for a sensing tube experiencing a tortuouscurve along a fixed fiber length.

l fibrc model boundary

θhfz

yadhesive

tube

glass fiberrgcoating

hf – rcsinθ, assuming

z substratets

(a)

2rc

f cflat tube surface(a)

Lg/2x

y A halfgratingsσg + Δσg

Eg

σg

τc

τgεg

εchf – rcsinθ

γcGc

γaGa

Strain reductiondue to shear, ɳSL

2rgrc

τaεs = εe - Δεs

hf rcsinθ γaGa

Ests

Δεs due to reinforcement by f tsurface stress τa , ɳFR(b)

Fig. 4. Schematic for modeling strain reduction in a surface bonded fiber(refer to Table. I for variables). (a) Cross sectional view, (b) Side view forhalf the length of an FBG sensor.

The second strain reduction effect is due to the highmodulus glass fibers acting as reinforcing elements for thesoft substrate. The resulting bending strain experienced bythe substrate is reduced by ∆εs from the expected strainwithout fibers, εe. This reinforcement effect on a low mod-ulus substrate has been studied in both surface bonded andembedded strain gauges [17], where the interfacial surfaceshear stress was transferred into the normal stress in a semi-infinite substrate.

For fiber optic sensors with a relatively long bondinglength, the η due to the fiber reinforcement effect, ηFR,was obtained using simple composite theory [16], which is,however, based on an assumption of equivalent strains forall components. Values of η estimated with this model wereconsistently less than the experimental results, though thecauses were not discussed.

These works do provide, however, a starting point fordeveloping a strain transfer model for the proposed sensingtube as derived in the next section. Following [15], both topand side adhesives can be neglected. An analytical model fora surface bonded fiber can then be formulated assuming acircular glass fiber with a bottom half soft coating, a bottomadhesive layer and a relatively flat polymer tube substrate(the white-dashed region in Fig. 4 (a)). Using symmetry, halfthe length of an FBG sensor can be considered (Fig. 4 (b)).

III. STRAIN TRANSFER MODEL

The strain transfer model is comprised of an interfacialshear model and a fiber reinforcement model. For modelingshear lag, axisymmetry is only applied within the thin fibercoating and the circular geometry between the coating and

3533

TABLE INOMENCLATURE

rg radius of fiber glass core and cladding layersrc radius of fiber coating layerLg length of fiber Bragg gratingts thickness of thin tube substrateh f fiber distance from tube surfaceθ angle from horizontal axisEg Young’s modulus of fiber core and claddingEs Young’s modulus of thin tube substrateGc shear modulus of fiber coating layerGa shear modulus of adhesive layerγc shear strain in fiber coating layerγa shear strain in adhesive layerug displacement at fiber cladding and coating interfaceuc displacement at fiber coating and adhesive interfaceus displacement at adhesive and tube interfaceεg normal strain of fiber core and claddingεc normal strain at fiber coating and adhesive interfaceεs normal strain at adhesive and tube interfaceεe expected normal strain at tube surface without fiberσg normal stress of fiber core and claddingτg shear stress at fiber cladding and coating interfaceτc shear stress in fiber coating layerτa shear stress in adhesive layerη strain transfer ratio from substrate to fiber

ηSL η due to shear lag effectηFR η due to fiber reinforcement effectα shear lag parameterβ fiber reinforcement parameter

adhesive is considered. On the other hand, for the rein-forcement effect, the surface shear stress at the adhesive-tube interface, τa obtained from the shear-lag based model,is assumed to create ∆εs on the surface, which allows thepossibility of a strain difference between components, incontrast to [16]. Each model is described below. All variablesare summarized in Table. I.

A. Interfacial Shear Effect

From the force equilibrium for each layer in Fig. 4 (b),the following equations are obtained for the glass fiber (coreand cladding), the coating and the adhesive,∫

π

0τg(rg,θ ,x) · rgdθ = ∆σg ·πrg

2 = Esεg ·πrg2 (6)

∫π

0τc(r,θ ,x) · rdθ =

∫π

0τg(rg,θ ,x) · rgdθ (7)

∫ rc

−rc

τa(x,−h f ,z)dz =∫

π

0τc(rc,θ ,x) · rcdθ (8)

where σ , τ and r are normal stress, shear stress and radius,respectively (subscripts g, c and a indicate glass core, coatingand adhesive.). εg is the glass fiber strain, measured by theFBG.

From Eq. 7 and the assumed axisymmetry within thebottom half of the coating layer, the shear stress in thecoating, at radial distance r, is written as,

τc(r,θ ,x) =rc

rτc(rc,θ ,x) for rg ≤ r ≤ rc (9)

Using Eq. 9 and the assumption of linear elasticity andpure shear strain transfer in the soft layers, we also obtain

the following relationships for the two interlayers,

uc(x)−ug(x) =∫ rc

rg

γc(r,θ ,x)dr (10)

=rc

Gcτc(rc,θ ,x) · ln

rc

rg(11)

us(x)−uc(x) = (h f − rc sinθ)γa(rc,θ ,x) (12)

= (h f − rc sinθ)τa(rc,θ ,x)

Ga(13)

where u, γ , G and h f are displacement at interfaces, shearstrain, shear modulus and fiber distance from the outersurface of substrate (subscript s indicates substrate.).

Combining Eq. 11 with Eq. 13, the shear stress of the ad-hesive at the interface with the coating, τa(θ ,x), is obtained.Because this stress should be equivalent with that of thecoating at the interface, which can be calculated using Eq. 9at r = rc, the interfacial shear stress at coating, τc(rc,θ ,x) isexpressed as,

τc(rc,θ ,x) = τa(rc,θ ,x) =us(x)−ug(x)

(h f−rc sinθ)

Ga+ rc

Gcln( rc

rg)

(14)

which is linearly proportional to the displacement of the fiberwith respect to the substrate. In the denominator of Eq. 14,the first term is related to the shear of the adhesive layerwhile the second term is related to the axisymmetric shearwithin the bottom half of the coating layer.

Assuming uniform strains in the Lg/2 half length ofthe FBG, reasonable with fiber bonding length exceeding11 mm [18], the displacements are replaced with ug(x) =Lg/2 · εg, uc(x) = Lg/2 · εc and us(x) = Lg/2 · εs. Then, theη due to the shear-lag effect, ηSL, is defined as the constantratio of εg to εs.

Based on these assumption, substituting Eq. 14 into Eq. 7,then again into Eq. 6, ηSL is obtained as,

ηSL =εg

εs=

α

1+α(15)

where the shear parameter α is

α =Lg

2πrg2Eg

∫π

0

1(h f−rc sinθ)

Ga+ rc

Gcln( rc

rg)

dθ (16)

which contains both mechanical and geometrical propertiesof the soft interlayers.

Due to the positive α , ηSL is always less than unity, butfor a significantly large value of α , caused by either harderinterlayers or a thin bottom adhesive, ηSL approaches 1.

B. Fiber Reinforcement Effect

To calculate the fiber reinforcement effect, τa at the tubesurface is replaced with an uniform normal stress on the thinsubstrate piece under the gratings. Then, substituting Eq. 14into Eq. 8 and adopting Eq. 15, the strain reduction, ∆εs, is

3534

Es = 25 GPa

1

hf = 0.875 m(a) (b)0 998

0.8

0.9

1

1.1

in tr

ansf

er r

atio

SL 0.4

0.6

0.8

in tr

ansf

er r

atio

0.952

0.9

10.950

0.9960.998

0.07 0.08 0.09 0.1 0.110.5

0.6

0.7

hf [mm]

Stra

i

FR

20 40 600

0.2

Es [GPa]

Stra

i

5 10 15

0.8

mm (c)

3.5hf = 0.875 mm

with E = 2.5 GPa

SLFR

1.5

2

2.5

3

a & c [M

Pa]

a sa with Es = 70 GPac with Es = 2.5 GPac with Es = 70 GPa

Slope= 226.7

Slope=155.7

Slope=145 1

0 80 100

20

0 0.005 0.01 0.0150

0.5

1

e

a

Slope=99.6

=145.1

Fig. 5. Strain transfer reduction model simulation : (a) η with different adhesive thickness, (b) η with different substrate modulus, (c) maximum τa andτc with varying εe

TABLE IIMATERIAL PROPERTY AND MODEL GEOMETRY FOR SIMULATION

rg[mm] rc[mm] h f1[mm] ts[mm] Lg[mm]

0.04 0.0625 0.0625−0.1125 0.15 5

Eg[GPa] Gc2[GPa] Ga

3[GPa] Es[GPa]

72.0 0.933 1.379 0.1−100.0

obtained with the following relationship,

∆εs =1

2rcEsts

∫ rc

−rc

τa(z,x)dz (17)

=1

2Ests

∫π

0τc(rc,θ ,x)dθ (18)

= βεs (19)

where the reinforcement parameter β is

β =Lg(1−ηSL)

4Ests

∫π

0

1(h f−rc sinθ)

Ga+ rc

Gcln( rc

rg)

dθ (20)

Using superposition, εs is the sum of εe with −∆εs, thatis,

εs = εe−∆εs = εe−βεs (21)

Thus, ηFR is written as,

ηFR =εs

εe=

11+β

(22)

When the Es or ts is sufficiently large, β becomes negli-gible, resulting in approximately unity ηFR.

Combining Eq. 22 with Eq. 15 yields the relationship ofεg with εe as

εg = ηSL ·ηFR · εe = η · εe (23)

For the case of bending strain only in FBG1 (Fig. 3),εe = κ · sinα · Do

2 = εmax · sinα .

1the available range acquired from the sensor dimension2estimated from the properties of Polyimide with 2.5 GPa of Young’s

modulus and 0.34 of Poisson’s ratio3htt p : //www.henkelna.com/

C. Simulation

To demonstrate the value of the model, the effects ofadhesive thickness, i.e., h f − rc, and Es on η are studied.In addition, interfacial shear stress is estimated for differentvalues of εe. The mechanical and geometrical properties usedfor the simulation are summarized in Table. II.

As shown in Fig. 5 (a) and (b), ηSL is approximatelyunity in the given dimension, because the thin adhesive layerresults in a large α (Eq. 15), such that Es has a dominanteffect on η . Thus, in this dimension, it is reasonable toassume the same η for all fibers as in Eq. 1, regardless ofthe adhesive amount. In each FBG, however, non-uniformadhesive along the gratings can distort the spectral peak socare should be taken to distribute adhesive evenly along thelength of each set of gratings.

A substrate with a modulus higher than 10 GPa transfersmore than 90% of its strain. Thus, in this range, lowsensitivity is not a concern. Instead, when larger curvaturemeasurements are required in a given sensor size, tube mod-ulus should be modified rather than changing the adhesives.According to [10], low modulus is obtainable by furtherincreasing the braid angle, by using a softer polymer, orby using different wire materials such as Kevlar 49 fibers(112 GPa). For such softer sensors, differential sensing usingfour orthogonally arranged fibers as in [2] can be adoptedif necessary to double the sensitivity, though it increases itsflexural rigidity as well as the cost.

It can also be seen that maximum τa and τc at θ = π/2,tend to be slightly reduced for a softer substrate, probablydue to the buffering effect (Fig. 5 (c)). They are expectedto be less than 5 MPa at both interfaces. The slopes ofτ increase in proportion to (1−ηSL)ηFR as expected fromEq. 14, 15 and 22.

This enables the use of a low shear strength (3-20 MPadepending on adherence) super glue instead of a higher-viscosity epoxy, the latter being more difficult for micro-scalefabrication. However, at larger diameters, epoxy is effectivewhen high strain transfer is necessary.

3535

braid angle θB ≈ 62°

fiber holder

θB

tube with

fibers at vertices hole in the h ld

tube axial direction

grooved metal frame

fibers

holder

tubesubstrate

grooved metal framefor alignment usingthe holder bumps at bottom

Fig. 6. Surface bonding process for optical fibers on a wire braid polymertube.

circular hole with a 1.5 mmdiameter

curvature template covered with Acrylic plate

sensing tube

sensing tube

sensing tubewith fibers rotation at both ends

(b)(a)

Fig. 7. Sensing tube calibration. (a) Prototype using three optical fibers,(b) Calibration process using curvature templates.

IV. EXPERIMENTS

A low-cost sensor prototype was constructed using oneFBG fiber and two identical fibers without FBGs (Fig. 6) toinvestigate the effect of the soft substrate on the curvatureestimation in single-plane bending. A 3D-printed fiber holderwith a curved triangular hole holds the fibers (TechnicaSA, Switzerland) and a wire braided Polyimide tube witha braid angle of 62◦ (295-VI, MicroLumen, USA). Thelongitudinal Young’s modulus of the tube was measured asaround 24.79 GPa, with an expected η ≈ 0.951.

Based on Sec. III-C, a medical device super glue (Loc-tite 4013, Henkel, USA) was applied to slightly tensionedstraight fibers. After curing, excess glue was carefully re-moved using Acetone, to reduce flexural rigidity and size. Asshown in Fig. 7 (a), the prototype smoothly passes through a1.5mm circular hole so that D≈ 1.4mm since its minimum,Do + 4rc, is 1.3mm. In this dimension, the effect of non-uniform adhesive thickness on η , an expected manufacturingerror, is negligible as anticipated in Sec. III-C. This wasalso supported by the sensor’s sharp spectral peak for allcurvatures, indicating sufficiently uniform adhesive along thegratings

A. Calibration and Shape Sensing Capability

The sensor prototype was manually rotated 2π withingrooves of a curvature template covered with a transparentacrylic plate (Fig. 7 (b)) and sensor signal was recorded

0.8 (a) (

0 2

0

0.2

0.4

0.6

g [%

]

g

Max g

Min g

Mean

ɳt = 0.91ɳm = 0.93

(a) (

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.8

-0.6

-0.4

-0.2

e [%]

Mean

ɳc = 0.95

(b) (10(b) (

0

5

10

re e

rror

[%]

m * tt * tm * cc * c

2 4 6 8 10 12 1410-3

-15

-10

-5

Template curvature [1/mm]

Cur

vatu

r

Mean : -1.32 %STD : 3.96 %

x 10 3 , Template curvature [1/mm]

y Position error P = | P P |(a)

PTIP

PTIP EST

OTIP

OTIP_ESTκ

y Position error, Perr = | PTIP - PTIP _EST |Orientation error, Oerr = | OTIP – OTIP_EST |

(a)

TIP_EST

x

L

(b)

L long constant curvaturerobot

4.5(b)

2

2.5

3

3.5

4

4.5

r [m

m] o

r [d

eg]

Position error [mm]Orientation error [deg]

L = 80mm

2 4 6 8 10 12 14x 10-3

0

0.5

1

1.5

2

[1/mm]

Tip

erro

r

x 10

Fig. 8. Sensor calibration results. (a) εe versus compensated εg, (b)Curvature error produced by tensile, compressive and mean values of η .

with an optical interrogator at 1 kHz (sm130, Micron Optics,USA). In the process, the clearance between the grooves andthe sensing tube, necessary for the smooth rotation, can causecalibration error.

As shown in Fig. 8 (a), for each groove with a fixedcurvature, all εg, after axial and thermal strain compensation,are plotted on a vertical line. The maximum and minimumvalues in each line correspond to the maximum tension andcompression at the given curvature.

Although the results exhibit fair linearity and symmetryin tension and compression, the η obtained from the linearfit of each set of the maxima (εt ) and the minima (εc) isslightly different from each other, i.e., 0.91 (ηt ) and 0.95(ηc), respectively. The mean (ηm) is given by 0.93. Whilethese values are close to the expected value of 0.951, thevariation is large enough to produce significant error incurvature as shown in Fig. 8 (b).

To investigate the effect of curvature error on robot tipposition error, Perr, and tip orientation error, Oerr, a robotof constant curvature and length 80mm is assumed. Thislength corresponds to the actively controlled portion of acontinuum robot used for intracardiac surgery [19]). At eachcurvature, the FBG can measure strain either in tension orin compression. Both cases acquired from the calibrationwere considered, but the constant ηm was used in curvatureestimation. The resulting tip errors are plotted in Fig. 9.These are given by Perr=0.84±0.62 mm (96.7% less than2 mm error) and Oerr=1.21±0.91◦ (80.7% less than 2◦ error).

Although the required sensor accuracy depends on ap-

3536

0.8 (a) (

0 2

0

0.2

0.4

0.6

g [%

]

g

Max g

Min g

Mean

ɳt = 0.91ɳm = 0.93

(a) (

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.8

-0.6

-0.4

-0.2

e [%]

Mean

ɳc = 0.95

(b) (10(b) (

0

5

10

re e

rror

[%]

m * tt * tm * cc * c

2 4 6 8 10 12 1410-3

-15

-10

-5

Template curvature [1/mm]

Cur

vatu

r

Mean : -1.32 %STD : 3.96 %

x 10 3 , Template curvature [1/mm]

y Position error P = | P P |(a)

PTIP

PTIP EST

OTIP

OTIP_ESTκ

y Position error, Perr = | PTIP - PTIP _EST |Orientation error, Oerr = | OTIP – OTIP_EST |

(a)

TIP_EST

x

L

(b)

L long constant curvaturerobot

4.5(b)

2

2.5

3

3.5

4

4.5

r [m

m] o

r [d

eg]

Position error [mm]Orientation error [deg]

L = 80mm

2 4 6 8 10 12 14x 10-3

0

0.5

1

1.5

2

[1/mm]

Tip

erro

r

x 10

Fig. 9. Estimated position and angle error for an 80mm long constant-curvature robot. (a) Error definition schematic, (b) Tip errors produced bythe mean value of η .

plications, tip position errors of 1-2 mm and orientationerrors of 0.5-2◦ are considered sufficient in most presentminimally invasive surgical applications [20]. In addition,the other errors from the registration and organ or patientmovements are often much larger, so sensor improvementover the requirements has little effect on the system accuracy.

Therefore, while these tip errors can likely be reducedthrough improved fabrication and calibration processes, theyindicate that the proposed approach holds promise.

V. CONCLUSIONS

The proposed composite tubular shape sensor providessufficient torsional rigidity and flexural compliance throughthe use of wire braid reinforcement. Additionally, this designprovides a spacious tool channel. As a first step toward de-veloping this concept, a mechanics-based model has been de-rived to characterize the strain transfer between the complianttube and the stiff glass fibers. This model incorporated boththe shear lag as well as the fiber reinforcement effects andprovides a means to identify the important design variables.This approach was illustrated through model simulation anda simple prototype was presented to validate the model. Incurrent research, manufacturing tolerances and calibrationprocedures are being improved to improve sensor accuracy.

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