# Design and Fabrication of a Fiber Bragg Grating Shape Sensor for Shape
Reconstruction of a Continuum Manipulator

Golchehr Amirkhani  Anna Goodridge  Mojtaba Esfandiari  Henry Phalen  Graduate
Student Member, IEEE  Justin H. Ma  Graduate Student Member, IEEE  Iulian
Iordachita  Senior Member, IEEE  and Mehran Armand  Member, IEEE This work was
supported in part by the National Institutes of Health (NIH) under Grant R01
EB016703 and R01 AR080315.Golchehr Amirkhani, Mojtaba Esfandiari, Henry
Phalen, Justin H. Ma, and Iulian Iordachita are with the Department of
Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218 USA, and
also with the Laboratory for Computational Sensing and Robotics, Johns Hopkins
University, Baltimore, MD, 21218 USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>iordachita@jhu.edu).
Anna Goodridge is with the Laboratory for Computational Sensing and Robotics,
Johns Hopkins University, Baltimore, MD, 21218 USA (e-mail:
anna.goodridge@jhu.edu). Mehran Armand is with the Department of orthopaedic
Surgery, the Department of Mechanical Engineering, the Department of Computer
Science, Johns, and also with the Laboratory for Computational Sensing and
Robotics, Johns Hopkins University, Baltimore, MD, 21218 USA (e-mail:
marmand2@jhu.edu).

###### Abstract

Continuum dexterous manipulators (CDMs) are suitable for performing tasks in a
constrained environment due to their high dexterity and maneuverability.
Despite the inherent advantages of CDMs in minimally invasive surgery, real-
time control of CDMs’ shape during non-constant curvature bending is still
challenging. This study presents a novel approach for the design and
fabrication of a large deflection fiber Bragg grating (FBG) shape sensor
embedded within the lumens inside the walls of a CDM with a large instrument
channel. The shape sensor consisted of two fibers, each with three FBG nodes.
A shape-sensing model was introduced to reconstruct the centerline of the CDM
based on FBG wavelengths. Different experiments, including shape sensor tests
and CDM shape reconstruction tests, were conducted to assess the overall
accuracy of the shape sensing. The FBG sensor evaluation results revealed the
linear curvature-wavelength relationship with the large curvature detection of
0.045 mm at a 90∘ bending angle and a sensitivity of up to 5.50
$\nicefrac{{nm}}{{mm}}$ in each bending direction. The CDM’s shape
reconstruction experiments in a free environment demonstrated the shape
tracking accuracy of 0.216±0.126 mm for positive/negative deflections. Also,
the CDM shape reconstruction error for three cases of bending with obstacles
were observed to be 0.436±0.370 mm for the proximal case, 0.485±0.418 mm for
the middle case, and 0.312±0.261 mm for the distal case. This study indicates
the adequate performance of the FBG sensor and the effectiveness of the model
for tracking the shape of the large-deflection CDM with nonconstant-curvature
bending for minimally-invasive orthopaedic applications.

Fiber Bragg grating (FBG), continuum manipulator, shape reconstruction, large
curvature, shape sensing, minimally invasive robotic surgery.

##  1 Introduction

In recent years, continuum manipulators have shown great potential to improve
steerability and dexterity in minimally invasive robotic surgery (MIRS)
procedures. The compliant structure of CDMs offer flexibility for reaching the
targeted site of treatment in a constrained environment as their bodies can
conform to their surroundings [1, 2, 3, 4, 5]. Despite many advantages of CDMs
for medical applications, accurate estimation of its position and shape is a
challenging task, especially when operating within unknown/unmodeled
environments.

Approaches for the shape reconstruction of CDMs in surgical scenarios may
include model-based, image-based, sensor-based and/or their combination. The
model-based approach employs the mechanics and kinematics of CDMs for shape
reconstruction [6, 7, 8, 9, 10]. Due to the compliance of CDMs, these models
may be complex and require a long computational time, making them challenging
to use in real-time applications [11, 6]. The accuracy of the model-dependent
CDM shape sensing relies on the system parameter identification and
assumptions when dealing with unknown disturbances and environmental
constraints [12, 13]. The image-based approaches, on the other hand, may be
more accurate since they do not rely on the mechanics of the CDM [14, 15].
However, radiation (i.e. fluoroscopy) or lower-quality images (i.e ultrasound
imaging) introduce specific challenges for the real-time application of MIRS
[16, 17].

The literature reports a number of shape-sensing approaches for real-time
control and shape reconstruction of CDMs, including the use of piezoelectric
polymers, electromagnetic (EM), and optical sensors [18, 19, 20]. EM and
optical sensors are more commonly used, because of their smaller size, in
real-time localization and the shape tracking of CDMs for surgery [21, 22,
23]. EM tracking systems, however, are susceptible to errors arising from
magnetic field distortions caused by the conductive objects in the work field
[24].

The thin, flexible, lightweight and biocompatible structure of optical shape
sensors with the added advantage of electromagnetic immunity from external
devices has received increasing attention in the sensor-based shape sensing of
the flexible instruments. By measuring the reflected wavelength from each
fiber, strain and curvature sensing of FBG sensors enable accurate shape-
tracking without relying on the kinematics and mechanics of the robotic
systems [25, 26, 19, 27]. Optical fibers such as FBG with different
configurations have been employed for the shape reconstruction of flexible
instruments such as catheters and biopsy needles as well as CDMs [28, 29, 30,
31]. FBG shape sensors, in particular, are classified into two main
configurations: A bundle of single-core fibers; and multicore fibers (MCFs).
Roesthuis et. al [32] fabricated triplet optical fibers, each with four FBG
nodes. Fibers were attached to a three-groove NiTi rod, placed in the backbone
of the continuum manipulator. The three-dimensional shape of the manipulator
has been reconstructed in terms of the curvature and strain relationship,
derived from axial strain measurements of FBG nodes. Liu et. al [23] developed
an FBG shape sensor by attaching a fiber with three FBG nodes to two NiTi rods
in a triangular configuration and passing the sensor assembly through the
sensor channel of a continuum manipulator. FBG wavelength-curvature
calibration was used to find the curvature at discrete locations and then
reconstruct the shape of the continuum manipulator. Sefati et. al [33] built
an FBG sensor assembly by embedding a fiber and two NiTi rods into a three-
lumen polycarbonate tube with circular cross-section. The sensor assembly was
passed through the side channel of a CDM. The design was later changed to
insert three fibers, each with three FBG nodes, into the three-groove NiTi
rod. A data-driven approach was developed to track the tip position of the CDM
using FBG wavelength measurements [34, 35]. Moore et. al [36] reconstructed
the three-dimensional shape using MCFs by combining the elastic rod theory and
differential geometry. Khan et. al [37] employed several MCFs for sensing the
shape of a flexible medical instrument. The shape of the instrument is
reconstructed using the curvature and torsion calculated from the FBG
wavelength data. Cao et. al [38] integrated MCFs into a continuum robot to
track the shape of the robot in the free space.

A major limitation of the previous work on the bundle of single-core fibers is
the challenge and fabrication time associated with inserting and keeping
fibers in the grooves of the sensor’s substrate due to the thin and delicate
nature of fibers [28, 32, 34]. When the sensing unit is embedded into the
channel within the wall of the continuum manipulator, the glue amount applied
to the grooves of the sensor’s substrate is an important factor. Finding a
sufficient amount of glue to not exceed the channel diameter constraint makes
the building procedure difficult [32, 34]. Furthermore, attaching the fiber to
the outer wall of the substrate is a complex manufacturing process. The sensor
configuration becomes trial-dependent, making reproducibility a challenge
[23]. The enclosed substrate is an appropriate option, especially if the
sensor assembly is routed through the channels within the wall of the CDM.
However, in the previous study, the shape sensor suffered from low sensitivity
to small curvature changes, resulting in a low signal-to-noise ratio [33]. The
shape sensor with MCFs, on the other hand, has a smaller cross-section and
more accurate FBG alignment [36, 37]. Nevertheless, light coupling into each
core is difficult, and MCFs are more expensive than single-core fibers due to
the draw tower fabrication procedure [39, 40]. Moreover, for applications in
minimally invasive orthopedic interventions, a bundle of single core fibers is
sufficient for detecting the CDM centerline [33, 34, 23, 35]. Most of the
prior work have introduced the FBG-based shape reconstruction algorithm for
the case that the shape sensor is directly attached to the flexible medical
instrument [28, 32, 36, 37, 26]. The free movement of the sensing unit inside
the wall channel of the CDM makes shape tracking difficult. The CDM centerline
detection has rarely been developed in previous studies, and the sensing model
did not account for the friction effect between the sensor and the CDM
channel’s wall [34, 23, 33, 38].

The purpose of this study is to design and build a novel FBG-based shape
sensor that is inexpensive, can be easily fabricated and integrated with a CDM
undergoing large bending and deflections during orthopaedic procedures.
Another contribution includes the formulation of the shape reconstruction
model for the CDM, that can significantly lessen the influence of internal
twist and friction. The performance of the model is assessed in both free and
constrained environments.

##  2 Design

###  2.1 FBG Sensor Design Requirements

As shown in Fig.1a, the cable-driven CDM employed in the orthopaedic
procedures has been constructed from a Nitinol (NiTi) rod with an outside
diameter (OD) of 6 mm and several notches along its 35 mm flexible length to
reach compliance in the bending direction while achieving high stiffness in
the perpendicular direction to the bending plane [10]. The CDM also consists
of a 4 mm diameter open lumen as an instrument channel for passing flexible
debriding tools[41] (Fig.1b). The overall length of the CDM is 70 mm, and the
CDM wall contains four lengthwise channels such that each pair of channels is
along the two opposite sides of the wall (Fig.1b). As shown in Fig.1c, the
actuating cable is passed through its channel with a 0.5 mm diameter to create
the bidirectional planar bending; the sensor channel with a diameter of 0.6 mm
is for embedding the FBG-based shape sensing unit into the CDM and tracking
the 2D-shape of the CDM in real-time[42].

Figure 1: (a) Schematics of the CDM while bending. (b) Cross-section of the
CDM tip, including sensor, cable, and instrument lengthwise channels. (b)
Detailed view of the CDM flexible segment with the shape sensing unit and the
actuating cable.

The design requirements of the sensor assembly are derived from the CDM
specifications and its applications. For orthopaedic applications such as core
decompression of the hip [41] and the treatment of osteolysis behind the
acetabular implant [42], the outer diameter of the CDM must be less than 6 mm.
The CDM has, therefore, a thin wall thickness (1 mm), with sensor channels of
less than 0.5 mm [43]. Considering the major challenges of the previous work
as well as the CDM design constraints, the development of an FBG-based shape
sensing unit needs to meet some crucial requirements: (1) easy fabrication and
integration into the CDM; (2) inexpensive compared to multi-core optical
fibers; (3) high wavelength shift to curvature ratio (sensitivity) for the
small local deformations, (4) measure high deformations without surpassing the
FBG strain limit, (5) less than 0.6 mm diameter to ensure the easy sliding of
the shape sensing unit through the sensor channel, and (6) an enclosed
structure to protect optical fibers and prevent adhesive wear due to the
sensing unit slide inside the sensor channel.

###  2.2 FBG Sensor Configuration

To address the aforementioned requirements, a polycarbonate tube with an outer
diameter of 500±15 $\mathrm{\SIUnitSymbolMicro m}$ and three symmetric 150±5
$\mathrm{\SIUnitSymbolMicro m}$ lumens was developed (Paradigm Optics,
Inc.)[33]. Three lumens are 120∘ radially apart on a circle that has a 100
$\mathrm{\SIUnitSymbolMicro m}$ diameter. The three-lumen polycarbonate tube
as a substrate meets requirements 1 and 5 to ensure the simple sensor
assembly, consistently repeatable fabrication, and unchallenging placement of
the sensing unit into the CDM. The flexible enclosed structure of the
polycarbonate tube fully covers optical fibers, which fulfills requirement 6.
The strain of FBG fibers should not exceed their allowable range. The sensing
unit can achieve high sensitivity as well as large deformations if the
orthogonal distance between the FBG sensor and the neutral axis of the sensor
assembly, called sensor orthogonal distance, is neither too large nor too
small[44, 45] (requirements 3 and 4). Hence, our sensing unit is designed with
two optical fibers and one NiTi rod with a 125 $\mathrm{\SIUnitSymbolMicro m}$
diameter (Fig.2b). NiTi rod is chosen due to its super elastic property, and
the capability of undergoing large deflections. Also, the NiTi rod can
increase the rigidity of the sensor assembly and, hence, its sensitivity. An
optical fiber with an 80 $\mathrm{\SIUnitSymbolMicro m}$ cladding diameter and
a 120 $\mathrm{\SIUnitSymbolMicro m}$ coating diameter (Technica Optical
Components, LLC., Atlanta, GA) offers a small size for insertion into the
three-lumen polycarbonate tube. The strain range of the optical fiber is up to
1.5%. Fig.2a shows the arrangement of FBG nodes inside the CDM. The sensing
unit is along the wall of the sensor channel which is $d_{offset}=$ 2.45 mm
away from the CDM centerline. Since each fiber consists of three FBG nodes
with a 5 mm length, the sensing unit includes three active areas
(${AA}_{1}-{AA}_{3}$) in total. ${N}_{kj}$ is the $j^{th}$ FBG node on the
fiber $k\in\\{1,2\\}$ and ${AA}_{j}$ is the $j^{th}$ active area.
$j\in\\{1,2,3\\}$ is the subscript that indicates the number associated with
the active area as well as the corresponding FBG nodes. The center of the
active areas are 10 mm apart, and the distance between the center of the
active area ${AA}_{1}$ and the distal end of the CDM is 5 mm. This arrangement
of active areas provides adequate coverage for reconstructing the CDM
centerline.

Figure 2: (a) Arrangement of FBG nodes, including two fibers and one NiTi rod,
inside the CDM. Three sets of active areas along the length of the sensing
unit are denoted by: ${AA}_{1}$, ${AA}_{2}$, ${AA}_{3}$. (b) Shape sensing
unit in the cross-section view. The two cores of fibers are labeled by Fiber1
and Fiber2.

###  2.3 Neutral Plane of the Sensing Unit

The neutral axis is normally positioned at the location of the geometric
centroid. We assume that the sensing unit is a composite beam consisting of
different materials at each cross-section. As shown in Fig.2b, the sensing
unit cross-section is symmetric about the $z_{n}$-axis of the local frame {n},
which is located at the center of the NiTi rod. Since the neutral plane always
passes through the geometric centroid of the sensor assembly, $O_{c}$, the
y-coordinate of the geometric centroid is zero, $[y_{c}]_{\\{n\\}}=0$, at each
cross section. The z-coordinate of the geometric centroid, $z_{c}\in R^{>0}$,
can be derived from the equilibrium equation of forces in the local frame {n}:

$\sum_{i=1}^{4}\int_{A_{i}}\sigma_{i}dA_{i}=0,\>\sigma=\frac{E_{i}(z-z_{c})}{\rho}.$
(1)

where $\rho$ corresponds to the radius of the curvature. The i subscript
indicates the components of the sensing unit including the polycarbonate tube,
NiTi rod, FBG fibers, and lumens of the polycarbonate tube. Also,
$\sigma_{i}$, $E_{i}$, and $A_{i}$ are the stress, Young’s modulus, and the
cross-section area of the $i^{th}$ component of the sensor assembly,
respectively. Using (1), the $z_{c}$ in the local frame {n} can be obtained
by:

$[z_{c}]_{\\{n\\}}=\frac{(3E_{f}D_{f}^{2}+E_{t}(D_{t}^{2}-3D_{l}^{2}))r}{E_{nw}D_{nw}^{2}+2E_{f}D_{f}^{2}+E_{t}(D_{t}^{2}-3D_{l}^{2})},$
(2)

where components of the sensor assembly namely polycarbonate tube, NiTi rod,
FBG fibers, and lumens of the polycarbonate tube are represented as subscripts
t, nw, f, and l, respectively. D denotes the diameter of the sensor assembly
component and $r\in R^{>0}$ denotes the radius of the circle in which the
center of three lumens of the polycarbonate tube is located. The Young’s
modulus of the polycarbonate tube, NiTi rod and FBG fiber are 2.6 GPa, 75 GPa,
and 70 GPa, respectively. By substituting the properties and dimensions of the
sensor assembly components into (2), the value of the $[z_{c}]_{\\{n\\}}$
becomes 0.095 mm.

Changing the bending direction of the sensing unit affects the sensor’s
orthogonal distance and hence the sensor sensitivity. To satisfy requirements
3 and 4 for both optical fibers, the sensing unit’s neutral plane should be
perpendicular to the CDM bending plane, making the sensor’s orthogonal
distance to the two fibers equal.

###  2.4 FBG Sensor Assembly

Since the core of the fiber alone is not affected by the bending strain, using
the polycarbonate tube as a substrate can provide an offset between the center
of the fiber and the neutral axis. Also, the fixed geometry of the three-lumen
polycarbonate tube can help with the precise placement of sensor components.
Two fibers were first passed through the two lumens while the tube was kept
straight. After embedding and aligning both fibers in the longitudinal
direction of the tube, the UV glue, which has a low viscosity, was passed
through the lumens by submerging one end in the UV glue and applying suction
to the other end. The glue injection was continued until all trapped air
bubbles which are a potential source of the sensor error were removed. The
NiTi rod was then passed through the third lumen, and finally, the UV glue
inside the lumens was cured using a UV spot gun. Fig.3a illustrates the
segment of the shape sensing unit after assembly. The new approach makes the
sensor fabrication process far easier, less time-consuming ( 4 hours of
assembly), and low-cost (requirement 2) compared to previously reported
designs [46, 34, 37].

After assembly, as described in section II-C, the sensing unit is passed
through the CDM sensor channel such that its neutral plane is kept
perpendicular to the CDM’s bending plane (Fig.3b). The sensor tip is then
glued at the distal end of the CDM. The sensor, hence, can move freely as the
CDM bends or straightens.

Figure 3: (a) Section of the FBG sensor assembly. (b) FBG sensor assembly
embedded in the CDM.

##  3 methods

###  3.1 The Principal of FBG-based Shape Sensing

The FBG wavelength shift is linearly dependent on the mechanical strain,
$\varepsilon$, and the temperature, T, which is given by[31, 47]:

$\frac{\Delta\lambda_{B}}{\lambda_{B_{0}}}=K_{\varepsilon}(\varepsilon-\varepsilon_{0})+K_{T}(T-T_{0}),$
(3)

where $\Delta\lambda_{B}$ is the FBG wavelength shift and $\lambda_{B_{0}}$ is
the reference FBG wavelength at the reference strain, $\varepsilon_{0}$, and
the reference temperature, $T_{0}$. $K_{\varepsilon}$ and $K_{T}$ refer to the
constant coefficients associated with the strain and the temperature,
respectively. The strain coefficient is directly related to the photo-elastic
coefficient by $K_{\varepsilon}=1-p_{\varepsilon}$, where the photo-elastic
coefficient is set to 0.22 [48]. In this study, the reference wavelength is
collected when the fiber is straight, the reference strain, thus, can be
assumed to be zero. Also, due to the proximity of fibers, the temperature
variation of FBG nodes at each active area is the same [49]. In the case of
the constant temperature, by measuring the FBG wavelength, strain can be
calculated using (3).

###  3.2 Curvature Calculation of the Shape Sensor

The curvature and the bending direction of the shape sensor can be obtained by
strain calculation at each active area, referred to ${AA}_{j}$. Fig.4
illustrates the cross-section of the shape sensing unit at ${AA}_{j}$.
{$ss_{j}$} is the frame of the shape sensor centerline at the $j^{th}$ active
area. The location of each active area and its associated strain, curvature,
and bending direction are parameterized by the arc length, $s$, which passes
through the geometric centroid of the sensing unit. The arc length parameter
is for reconstructing the shape sensor centerline. Axis $y_{ss_{J}}$ is
tangential to the shape sensor centerline, and the direction of the axis
$x_{ss_{J}}$ is from the origin of the frame {$ss_{j}$} to the fiber 1 at the
$j^{th}$ active area. Considering the pure bending for the sensing unit,
equations which relate the axial strain, $\varepsilon_{kj}$, of the $j^{th}$
FBG node on the fiber $k\in\\{1,2\\}$ to the corresponding curvature and
bending direction are as follows[32]:

$\displaystyle\varepsilon_{1j}(s)$
$\displaystyle=-\kappa^{{}^{\prime}}(s)\delta_{1j}$ (4)
$\displaystyle=-\kappa^{{}^{\prime}}(s)[z_{c}\text{sin}(\phi^{{}^{\prime}}(s))-r_{1j}\text{sin}(\theta_{1j}(s)-\phi^{{}^{\prime}}(s))],$
$\displaystyle\varepsilon_{2j}(s)$
$\displaystyle=-\kappa^{{}^{\prime}}(s)\delta_{2j}$ (5)
$\displaystyle=-\kappa^{{}^{\prime}}(s)[z_{c}\text{sin}(\phi^{{}^{\prime}}(s))+r_{2j}\text{sin}(\theta_{2j}(s)+\phi^{{}^{\prime}}(s))].$

where $\phi^{{}^{\prime}}$ indicates the bending direction which is the angle
between the axis $z_{ss_{j}}$ of the local frame {$ss_{j}$} and the neutral
axis. $\kappa^{{}^{\prime}}$ is the curvature, $\delta_{kj}$ denotes the
sensor orthogonal distance of the $j^{th}$ FBG node on the $k^{th}$ fiber,
$r_{kj}$ denotes the radial distance from the $j^{th}$ FBG node on the
$k^{th}$ fiber to the center of the polycarbonate tube, $\theta_{kj}$ is the
angular offset from the negative axis $z_{ss_{j}}$ to the core of $j^{th}$ FBG
node on the $k^{th}$ fiber.

Figure 4: Section of the shape sensing unit at ${AA}_{j}$ with the cross-
section view at FBG nodes labeled by $N_{1j}$ and $N_{2j}$.

By assuming the reference strain to be zero, the following sets of equations
can be derived from substituting (4) and (5) into (3):

$\displaystyle\frac{\Delta\lambda_{1j}}{\lambda_{0_{1j}}}=$ $\displaystyle-
K_{\varepsilon}\kappa^{{}^{\prime}}(s)[z_{c}\text{sin}(\phi^{{}^{\prime}}(s))$
(6) $\displaystyle-
r_{1j}\text{sin}(\theta_{1j}(s)-\phi^{{}^{\prime}}(s))]+K_{T}\Delta T_{j},$
$\displaystyle\frac{\Delta\lambda_{2j}}{\lambda_{0_{2j}}}=$ $\displaystyle-
K_{\varepsilon}\kappa^{{}^{\prime}}(s)[z_{c}\text{sin}(\phi^{{}^{\prime}}(s))$
(7)
$\displaystyle+r_{2j}\text{sin}(\theta_{2j}(s)+\phi^{{}^{\prime}}(s))]+K_{T}\Delta
T_{j}.$

where the term $\Delta T_{j}$ is the temperature variation at the $j^{th}$
active area. It is assumed that the term $K_{T}\Delta T_{j}$ is the same for
both fibers due to the proximity of the FBG nodes at each active area.
$\Delta\lambda_{kj}$ and $\lambda_{0_{kj}}$ are the wavelength shift and the
reference wavelength associated to the $j^{th}$ FBG node on the $k^{th}$
fiber, respectively. Equations (6) and (7) can be simplified and then written
in a matrix form:

$\displaystyle\Lambda_{j}=A_{j}B_{j},$ (8)

where:

$\displaystyle\Lambda_{j}=\begin{bmatrix}\frac{\Delta\lambda_{1j}}{\lambda_{0_{1j}}}\\\
\frac{\Delta\lambda_{2j}}{\lambda_{0_{2j}}}\end{bmatrix},B_{j}=\begin{bmatrix}B_{j_{1}}\\\
B_{j_{2}}\\\
B_{j_{3}}\end{bmatrix}=\begin{bmatrix}\kappa^{{}^{\prime}}(s)\text{sin}(\phi^{{}^{\prime}}(s)\\\
\kappa^{{}^{\prime}}(s)\text{cos}(\phi^{{}^{\prime}}(s)\\\ K_{T}\Delta
T_{j}\end{bmatrix}$ $\displaystyle
A_{j}=\begin{bmatrix}-K_{\varepsilon}(z_{c}+r_{1j}\text{cos}(\theta_{1j}(s)))&K_{\varepsilon}r_{1j}\text{sin}(\theta_{1j}(s))&1\\\
-K_{\varepsilon}(z_{c}+r_{2j}\text{cos}(\theta_{2j}(s)))&-K_{\varepsilon}r_{2j}\text{sin}(\theta_{2j}(s))&1\end{bmatrix}.$

$B_{j}$ from (8) can be obtained as:

$\displaystyle B_{j}=A_{j}^{{\dagger}}\Lambda_{j},$ (9)

where ${\dagger}$ is the Moore-Penrose pseudo-inverse symbol. By solving (9),
the curvature and the bending direction at the $j^{th}$ active area can be
determined as:

$\displaystyle\kappa^{{}^{\prime}}(s)=\sqrt{(B_{j_{1}}^{2}+B_{j_{2}}^{2})},$
(10) $\displaystyle\phi^{{}^{\prime}}(s)=\text{atan2}(B_{j_{1}},B_{j_{2}}).$
(11)

As shown in Fig.5, the CDM can only bend in the XY plane of the planar bending
which is aligned with the frame {$cm_{p}$}. {$cm_{p}$} is the frame of the CDM
centerline at the proximal end of the CDM. Axis $y_{cm_{p}}$ is tangential to
the CDM centerline, and the direction of the axis $x_{cm_{p}}$ is from the
origin of the frame {$cm_{p}$} to the sensing unit. The wavelength shift sign
of the sensing unit can simply specify the moving direction of the sensing
unit as well as the CDM deflection direction; if the wavelength shifts of the
FBG nodes on fiber 1 and fiber 2 are positive and negative, respectively, the
CDM is undergoing positive deflection and the sensing unit is moving forward
to the distal end of the CDM, if the situation is opposite, the CDM is
undergoing negative deflection and the sensing unit is moving back to the
proximal end of the CDM. Of note, the friction between the shape sensor and
the wall of the CDM sensor channel affects the FBG wavelength measurements. To
compensate for the influence of the friction, we consider a calibration
coefficient at the $j^{th}$ active area, represented by $C_{j}$:

$\displaystyle\kappa[j]=C_{j}\kappa^{{}^{\prime}}[j],\quad j=1,2,3$ (12)

Figure 5: Moving direction of the shape sensing unit at (a) CDM negative
deflection and (b) CDM positive deflection. ${AA}_{1_{0}}$, ${AA}_{2_{0}}$,
and ${AA}_{3_{0}}$ are the initial locations of the active areas when the CDM
is straight and ${AA}_{1}$, ${AA}_{2}$, and ${AA}_{3}$ are the active areas
locations after the CDM bending.

where $\kappa[j]$ is the compensated curvature at the $j^{th}$ active area
after applying the calibration coefficient, and $\kappa^{{}^{\prime}}[j]$ is
the curvature at the $j^{th}$ active area before applying the calibration
coefficient. Furthermore, the sensing unit may exhibit internal twists which
result in the out-of-plane CDM centerline reconstruction. The effect of the
twist on the bending direction is compensated at each active area by:

$\displaystyle\phi[j]=\phi^{{}^{\prime}}[j]-\phi_{t}[j],\quad j=1,2,3$ (13)

where $\phi[j]$ is the compensated bending direction at the $j^{th}$ active
area, and $\phi_{t}[j]$ is the twisted bending direction at the $j^{th}$
active area. Two steps are required to find $\phi_{t}[j]$: 1) the shape
sensing unit is lying straight in the XY plane of the bending plane such that
the NiTi rod is facing the positive Z direction. The wavelengths at each
active area are recorded as the reference wavelength, $\lambda_{0_{kj}}$; 2)
the shape sensing unit is inserted in a constant curvature groove which
implies bending in the XY plane of the bending plane. Then, the twisted
bending direction at each active area due to the internal twist can be
computed using (9) and (11). These values are constant and do not change
during the bending of the CDM, embedded with the sensing unit.

###  3.3 Sensor Assembly Shape Reconstruction

Figure 6: The side cross-section view of the CDM for shape reconstruction of
the sensor assembly and the CDM. $\\{{ss}_{p}\\}$ and $\\{{ss}_{d}\\}$ are the
frames of the shape sensor centerline at proximal and distal ends,
respectively. Axes $\\{y_{{ss}_{p}}\\}$ and $\\{y_{{ss}_{d}}\\}$ are
tangential to the curve of the shape sensor centerline. The direction of axes
$\\{x_{{ss}_{p}}\\}$ and $\\{x_{{ss}_{d}}\\}$ are from the origin of their
associated frames to the fiber 2 and fiber 1, respectively.

As described in section III-B, the sensor assembly moves freely, while CDM
bends (Fig.5). Since the sliding amount of the sensing unit is unknown, the
shape of the sensing unit is reconstructed from the distal end rather than the
proximal end, where the arc lengths of all active areas remain constant. As
shown in Fig.6, some coordinate systems are defined for reconstructing the
shape sensor centerline. The measured FBG wavelengths are utilized to
determine the curvature and the bending direction at the location of each
active area, using (10)-(13). By knowing the arc length of each active area in
frame $\\{{ss}_{d}\\}$ as well as the corresponding curvature and bending
direction, spline interpolation can be performed to compute the continuous
curvature, $\kappa(s)$, and the bending direction, $\phi(s)$. The interval of
the arc length of the shape sensor centerline which is denoted by $s$, is from
the origin of the frame $\\{{ss}_{d}\\}$ to the last active area, $AA_{3}$.

As shown in Fig.7, the centerline can be parametrized by the arc length as
$r(s)$ in the frame $\\{i\\}$. Using the Frenet-Serret apparatus, the tangent
of the curve, $t(s)$, is given by:

$\displaystyle{}_{\\{i\\}}=\frac{dr(s)}{ds}=\begin{bmatrix}\frac{dx(s)}{ds}&\frac{dy(s)}{ds}&0\end{bmatrix}^{T},$
(14)

Figure 7: Centerline representation which is parameterized by a planar curve,
$r(s)$, with Frenet-Serret vectors including the tangent vector, $t(s)$, and
the normal vector, $n(s)$.

where x(s) and y(s) represent the 2D position of a point along the centerline
in the frame $\\{i\\}$. For the sensor assembly shape reconstruction, frame
$\\{{ss}_{d}\\}$ is considered as the frame $\\{i\\}$. The slope of the
centerline is related to the curvature by:

$\displaystyle\theta(s)=\int_{0}^{s}\kappa(s)ds+\theta_{0},$ (15)

where $\theta(s)$ is the slope of the centerline which is the angle between
the tangent vector at a point along the centerline and the positive direction
of the $y_{\\{i\\}}$-axis in the frame $\\{i\\}$. $\theta_{0}$ is the initial
value of the slope of the centerline at s=0. By taking the integral of $t(s)$
in (14), x and y positions of the centerline in term of the arc length can be
derived as:

$\displaystyle{}_{\\{i\\}}=\begin{bmatrix}(\int_{0}^{s}\text{sin}(\theta(s))ds+x_{0})\text{cos}(\phi(s))\\\
\int_{0}^{s}\text{cos}(\theta(s))ds+y_{0}\\\ 0\end{bmatrix},$ (16)

where $x_{0}$ and $y_{0}$ are the initial deflections in the x and y
directions, respectively. Finally, the coordinates of the shape sensor
centerline in the frame $\\{{ss}_{d}\\}$ can be obtained using (15) and (16).

###  3.4 CDM Shape Reconstruction

The shape sensor centerline obtained in section III-C should be translated to
the CDM centerline, which has a constant offset, $d_{offset}$, from the shape
sensor centerline (Fig.6). $\\{{cm}_{d}\\}$ and $\\{{cm}_{j}\\}$ are the
frames of the CDM centerline at the distal end and the corresponding cross-
section of the $j^{th}$ active area, respectively. Axes $y_{{cm}_{j}}$ and
$y_{{cm}_{d}}$ are tangential to the curve of the CDM centerline. The
direction of axes $x_{{cm}_{j}}$ and $x_{{cm}_{d}}$ are the same as axes
$x_{{ss}_{j}}$ and $x_{{ss}_{d}}$, respectively. The coordinate of the
$j^{th}$ active area on the CDM centerline in the frame $\\{{cm}_{d}\\}$ is
determined by:

$\displaystyle\begin{bmatrix}x_{O_{cm_{j}}}\\\ y_{O_{cm_{j}}}\\\ 0\\\
1\end{bmatrix}_{\\{{cm}_{d}\\}}=\begin{bmatrix}R^{{ss}_{d}}_{{ss}_{j}}(\theta[j])&V^{{cm}_{d}}_{{ss}_{j}}\\\
[0\quad 0\quad 0]&1\end{bmatrix}\begin{bmatrix}x_{O_{cm_{j}}}\\\
y_{O_{cm_{j}}}\\\ 0\\\ 1\end{bmatrix}_{\\{{ss}_{j}\\}},$ (17)

where $x_{O_{cm_{j}}}$ and $y_{O_{cm_{j}}}$ denote the coordinates of the
origin of the frame $\\{cm_{j}\\}$, and $R^{{ss}_{d}}_{{ss}_{j}}$ is a
3$\times$3 rotation matrix about the z-axis from the frame $\\{{ss}_{d}\\}$ to
the frame $\\{{ss}_{j}\\}$ by an angle $\theta[j]$. $V^{{cm}_{d}}_{{ss}_{j}}$
which is a 3$\times$1 translation vector from the frame $\\{{cm}_{d}\\}$ to
the frame $\\{{ss}_{j}\\}$ is found using (18):

$\displaystyle V^{{cm}_{d}}_{{ss}_{j}}=[x[j]-d_{offset}\quad y[j]\quad
0]^{T}_{\\{cm_{d}\\}},$ (18)

where $x[j]$ and $y[j]$ are the coordinates of the $j^{th}$ active area on the
shape sensor centerline. After finding the coordiantes of active areas on the
CDM centerline in the frame $\\{{cm}_{d}\\}$, it is possible to perform spline
interpolation to find the arc length of each active area in the frame
$\\{{cm}_{p}\\}$:

$\displaystyle
s^{{}^{\prime}}[j]=L-\int_{0}^{x_{O_{cm_{j}}}}\sqrt{1+(\frac{df(x_{cm})}{dx_{cm}})^{2}}\,dx_{cm},$
(19)

where $f(x_{cm})$ is the function associated with the fitted curve of the CDM
centerline in frame $\\{cm_{d}\\}$, $s^{{}^{\prime}}$ denotes the arc length
of the CDM centerline which its interval is from the origin of the frame
$\\{cm_{p}\\}$ to the origin of the frame $\\{cm_{d}\\}$, and $L=35$ mm is the
total arc length of the CDM centerline. Next, the curvature of each active
area on the shape sensor centerline needs to be translated to the CDM
centerline by:

$\displaystyle\kappa_{cm}[j]=\frac{\kappa[j]}{1+d_{offset}\kappa[j]\text{cos}(\phi[j])},$
(20)

where $\kappa_{cm}[j]$ is the curvature of the $j^{th}$ active area on the CDM
centerline. Using (19) and (20), the continuous curvature of the CDM center
curve, $\kappa_{cm}(s^{{}^{\prime}})$, can be determined by spline
interpolation. Finally, considering the frame $\\{{cm}_{p}\\}$ as the frame
$\\{i\\}$, and $s^{{}^{\prime}}$ instead of $s$, the coordinates of the CDM
centerline can be found using (15) and (16).

##  4 EXPERIMENTS AND RESULTS

###  4.1 Experimental Setup

Figure 8: The experimental setup, including the actuation apparatus, the
stereo camera system, the FBG shape sensor.

The experimental setup included actuation apparatus, the shape sensing unit
inside the CDM, and the stereo camera system (Fig.8). The actuation apparatus
consisted of the linear actuator for driving the actuating cable of the CDM,
and a holding block to keep DC motors and the CDM’s chunk. The linear actuator
consisted of two graphite brushed DC-motors (4.5W, RE 16, Maxon, Switzerland)
with a high-precision encoder (MR, Type M, 512 CPT, Maxon, Switzerland) and an
integrated gear box-ball screw mechanism (Spindle Drive GP 16S, Maxon,
Switzerland), as well as a position controller (EPOS2, Maxon, Switzerland)
[50]. While the CDM was bending, the FBG sensor wavelengths were measured by
an optical sensing interrogator (sm130, Micron Optics Inc., Atlanta, GA) at a
frequency of 100 Hz. Also, two FL2-08S2C cameras (Point Grey Research Inc.,
Richmond, BC, Canada), attached to the height-adjustable camera stand mount,
captured the CDM images at a frequency of 30 Hz. To always make the CDM
flexible segment visible to both cameras, the cameras were placed 30 cm above
the CDM working area. The measurements for FBG wavelengths were performed
using a C++ code, developed using the C++ CISST-SAW libraries[51]. The images
of both stereo cameras and FBG wavelengths were recorded, and equations
(10)$-$(20) were applied to reconstruct the CDM centerline using the measured
wavelengths.

###  4.2 Image Processing

Figure 9: Extraction of 3-D locations of the CDM centerline from stereo images
using color-based segmentaion and triangulation.

A pair of stereo cameras with a resolution of 1024×768 was used to track red
markers attached to the CDM centerline. To find the cameras’ intrinsic and
extrinsic matrices, the stereo camera pair was calibrated using the stereo
camera calibration toolbox in MATLAB. The resulting overall mean error for the
stereo pair calibration was 0.23 pixels. The measurement uncertainty of the
stereo cameras after camera calibration was 0.253 mm, which was obtained by
finding the Euclidean distance between two markers with predetermined
positions on the calibration jig. The 3D locations of the centroids of the
markers were calculated by developing a computer algorithm for color-based
segmentation and triangulation of the images (Fig.9). The reconstructed CDM
centerline from stereo images was then used as a ground truth to validate the
CDM shape reconstruction model.

###  4.3 Calibration

To validate the linear relationship between the curvature and FBG wavelength,
3D-printed calibration jigs with discrete constant curvature grooves ranging
from $-90^{\circ}$ to $90^{\circ}$ at $5^{\circ}$ interval and constant arc
length were designed (Fig.10). In this experiment, the sensing unit was
examined at the optimal value of the sensor orthogonal distance, discussed in
section II-C. The sensing unit was fixed between two clamps to maintain the
sensing unit’s orientation. The wavelengths of FBG nodes were sampled five
times at each curvature groove, and then the mean value of data points was
calculated. The sampling time was set to 3 seconds. Fig.11 shows the
calibration results of three FBG nodes at each fiber. The wavelength-curvature
data points were well fitted with lines, with R squared ($R^{2}$) in the range
of 0.996-0.998.

Figure 10: Experimental setup for FBG sensors calibration using calibration
jigs of positive and negative deflections with known curvatures.

Figure 11: Linear correlation between the Curvature and wavelength of the
shape sensing unit at (a) fiber1 and (b) fiber2.

Due to the manual fabrication of the sensing unit and the limited accuracy in
the building process, the exact values of the position, $r_{kj}$, and
orientation, $\theta_{kj}$, of each FBG node are unknown. To find these
unknown parameters, the data points of the wavelength-curvature experiment of
the shape sensor for curvature grooves ranging from $-90^{\circ}$ to
$90^{\circ}$ at $10^{\circ}$ interval were used. Considering the pure bending
at a constant temperature, the calibration process at the $j^{th}$ active area
was accomplished by a least square optimization problem:

$\displaystyle\min_{P_{j}}\sum_{m=1}^{N}re_{m_{j}}^{2}=\min_{P_{j}}\|y_{model_{j}}-y_{gt_{j}}\|_{2}^{2},$
(21)

where $re_{m_{j}}$ is the residual error of the $m^{th}$ observation, and
$P_{j}$ is the set of unknown parameters for the positions and orientation of
FBG nodes. $y_{gt_{j}}\in R^{N\times 2}$ is a stack of N ground truth data
points for the curvature and bending direction, $y_{model_{j}}\in R^{N\times
2}$ is a stack of N observation data points for the curvature and bending
direction, obtained by (10) and (11). This minimization was implemented using
least squares method. Results are shown in Table 1.

Table 1: Calibrated ParameterS for Positions ($r_{K1}$, $r_{K2}$, $r_{K3}$) and Orientations ($\theta_{K1}$, $\theta_{K2}$, $\theta_{K3}$) of FBG Nodes on Each Fiber (K=1,2) Fiber index (k) | Position (mm) | Orientation (degree)
---|---|---
$r_{k1}$ | $r_{k2}$ | $r_{k3}$ | $\theta_{k1}$ | $\theta_{k2}$ | $\theta_{k3}$
1 | 0.159 | 0.150 | 0.155 | 60.848 | 60.790 | 60.733
2 | 0.159 | 0.158 | 0.154 | 60.848 | 60.790 | 60.733

To validate the accuracy of the obtained positions and orientations of FBG
nodes, the data points of the wavelength-curvature experiment of the shape
sensor for curvature grooves ranging from $-85^{\circ}$ to $85^{\circ}$ at
$10^{\circ}$ interval were used. Results demonstrate that the model of the
sensing unit with equivalent position and orientation of each FBG node can
predict the curvature and bending direction with the mean error of
4.704$e^{-4}$±5.730$e^{-4}$ $mm^{-1}$ and 0.057±0.019 degree, respectively.

Based on the deflection direction as well as the inconsistent effect of the
hindered friction at interactions between the sensing unit and the CDM sensor
channel [52], the wavelength shifts at each active area for positive and
negative deflections are exactly the same. As discussed in section II-B, two
sets of calibration coefficients are defined at each active area; $C_{p_{j}}$
and $C_{n_{j}}$ are the calibration coefficients of $j^{th}$ active area for
positive and negative deflections, respectively. 3D-printed calibration jigs
with discrete curvature grooves ranging from $-90^{\circ}$ to $90^{\circ}$ at
$5^{\circ}$ interval were designed (Fig.12). The curvature and total arc
length of each groove are constant. The CDM was fixed at its proximal end by
two clamps, which inserted into the slots of the stationary plate for
maintaining the CDM’s bending plane parallel to the surface of the stationary
plate. Five trials, one each with 3 seconds sampling time, were conducted at
each groove. The wavelengths of FBG nodes from all five trials were averaged.
The calibration process was performed by a least square optimization problem
to find the set of coefficients that minimizes the curvature error at each
active area. Experimental results of the CDM with embedded sensing unit for
curvature grooves ranging from $-90^{\circ}$ to $90^{\circ}$ at $10^{\circ}$
interval were used for the least square problem which is given by:

$\displaystyle\min_{C_{j}}\sum_{m=1}^{N}re_{m_{j}}^{2}=\min_{C_{j}}\|\kappa_{model_{j}}-\kappa_{gt_{j}}\|_{2}^{2},$
(22)

where $C_{j}$ is the unknown coefficient of either $C_{p_{j}}$ or $C_{n_{j}}$,
based on the bending direction at the $j^{th}$ active area.
$\kappa_{gt_{j}}\in R^{N}$ is a stack of N ground truth data points for the
curvature, $\kappa_{{model}_{j}}\in R^{N}$ is a stack of N observation data
points for the curvature, obtained by (10) and (12). Results are shown in
Table 2.

Figure 12: Experimental setup for FBG sensors calibration inside the CDM using calibration jigs of positive and negative deflections with known curvatures. Table 2: Calibration Coefficients for Positive ($C_{p_{1}}$, $C_{p_{2}}$, $C_{p_{3}}$) and Negative ($C_{n_{1}}$, $C_{n_{2}}$, $C_{n_{3}}$) Deflections of the Sensing Unit Inside the CDM (j=1,2,3) Active area index (j) | Calibration coefficient
---|---
|  $C_{p_{j}}$ |  $C_{n_{j}}$
1 |  1.024 |  0.917
2 |  0.945 |  0.836
3 |  0.985 |  0.655

To find the accuracy of the obtained coefficients for curvature detection, the
wavelength-curvature experimental results of the sensing unit inside the CDM
for curvature grooves ranging from $-85^{\circ}$ to $85^{\circ}$ at
$10^{\circ}$ interval were used. Results indicated that the model with the
equivalent calibration coefficients can determine the curvature of positive
and negative deflections with the mean error of 4.295$e^{-4}$±4.585$e^{-4}$
$mm^{-1}$ and 4.414$e^{-4}$±3.842$e^{-4}$ $mm^{-1}$, respectively. As
discussed in section III-B, the sign of the wavelength shifts at each active
area can be used to specify the corresponding calibration coefficient of the
model.

###  4.4 Static Experiment Conducted in Free Environment

Two sets of experiments in a free environment including positive and negative
deflections were conducted. Each set of experiments consists of two cycles:
bending, and straightening. The bending cycle started from zero deflection at
which the CDM was straight. Then, for each direction of deflection, the
corresponding actuating cable was driven by the linear actuator, in the range
of 0$-$5 mm, at 1 mm increments. When the actuating cable reached the maximum
displacement, the straightening cycle was initiated such that the cable was
released at 1 mm increments to return the CDM to its reference position. At
each increment, the FBG wavelengths as well as the camera images of the CDM
were recorded. Each set of experiments contains data points for 9 cable
displacements, and repeated five times to assess the repeatability of the
results.

The CDM’s centerlines obtained from the reconstruction model and stereo images
are illustrated in Fig.13. Each curve represents the average of five sampling
data points. To compare the CDM centerlines with ground truth data, the mean
absolute error and standard deviation of the CDM centerline and tip pose in
addition to the maximum absolute error of the CDM centerline were calculated
at each cable displacement. The errors are presented in Table 3. Results
indicated a good agreement between the CDM centerline of the model and image
in bending and straightening cycles with the overall mean error of 0.175±0.081
mm for positive deflection, 0.262±0.144 mm for negative deflection, and
0.216±0.126 mm for positive/negative deflections. There was also a good
agreement between the CDM tip pose of the model and image in bending and
straightening cycles with the overall mean error of 0.197±0.094 mm for
positive deflection, 0.339±0.202 mm for negative deflection, and 0.268±0.172
mm for positive/negative deflections.

Figure 13: Shape reconstruction of the CDM in the free environment for (a) positive deflection and (b) negative deflection. Table 3: Shape Reconstruction and Tip Position Errors of the CDM in the Free Environment |  Cable displacement (mm) | Positive deflection | Negative deflection
---|---|---|---
Cycle | Shape error | Tip error | Shape error | Tip error
|  Mean (mm) |  Std. (mm) |  Max (mm) |  Mean (mm) |  Std. (mm) |  Mean (mm) |  Std. (mm) |  Max (mm) |  Mean (mm) |  Std. (mm)
| 1 | 0.126 | 0.065 | 0.261 | 0.204 | 0.003 | 0.227 | 0.122 | 0.407 | 0.302 | 0.001
| 2 | 0.167 | 0.072 | 0.281 | 0.153 | 0.007 | 0.218 | 0.129 | 0.467 | 0.064 | 0.006
Bending | 3 | 0.163 | 0.063 | 0.280 | 0.211 | 0.028 | 0.252 | 0.118 | 0.488 | 0.201 | 0.014
| 4 | 0.166 | 0.066 | 0.283 | 0.250 | 0.017 | 0.291 | 0.125 | 0.511 | 0.390 | 0.037
| 5 | 0.191 | 0.077 | 0.305 | 0.249 | 0.019 | 0.357 | 0.159 | 0.617 | 0.559 | 0.110
| 4 | 0.271 | 0.071 | 0.292 | 0.187 | 0.003 | 0.296 | 0.190 | 0.656 | 0.656 | 0.004
| 3 | 0.139 | 0.064 | 0.276 | 0.188 | 0.009 | 0.214 | 0.093 | 0.386 | 0.213 | 0.003
Straightening | 2 | 0.169 | 0.057 | 0.263 | 0.184 | 0.001 | 0.204 | 0.104 | 0.377 | 0.170 | 0.004
| 1 | 0.142 | 0.008 | 0.267 | 0.185 | 0.004 | 0.210 | 0.105 | 0.361 | 0.238 | 0.003

###  4.5 Static Experiment Conducted in Presence of Obstacles

To validate the shape reconstruction model in the case the contact forces
applied along the CDM, two sets of experiments were conducted in the presence
of obstacles for positive and negative deflections. As shown in Fig.14,
3D-printed obstacles were placed at three different locations along the length
of the CDM which was: 1) near the proximal end; 2) in the middle segment; 3)
near the distal end.

In each case, the segment of the CDM which interacted with the obstacle was
obstructed from free bending and enforced the CDM to conform to more complex
shapes. Each set of experiments was repeated five times. Fig.15-(c) and
Fig.15-(f) show the comparison between the ground truth data extracted from
stereo images and the reconstruction model in the constrained environment for
positive and negative deflections, respectively. The mean absolute error and
standard deviation of the CDM centerline and tip as well as the maximum
absolute error of the CDM centerline for three obstacle-interaction cases are
given in Table 4. The overall mean tracking accuracy of the CDM centerline for
positive/negative deflections was 0.436±0.370 mm, 0.485±0.418 mm, and
0.312±0.261 mm, respectively, for proximal, middle, and distal cases of CDM
bending with obstacles. There was also a good accuracy between the CDM tip
position derived from the model and image in positive/negative deflections
with the overall mean error of 0.811±0.383 mm for the proximal case,
1.040±0.327 mm for the middle case and 0.680±0.422 mm for the distal case.

##  5 Discussion

This study presented a novel technique for building a thin, large deflection
FBG-based shape-sensing unit that can be integrated into minimally invasive
surgical systems. Our primary motivation is to develop a relatively
inexpensive, easy-to-fabricate, and fast-fabricating (4 hours vs. days [34,
46]) shape sensing for a CDM that is designed for orthopaedic applications
[43, 53]. The CDM, therefore, is constrained to bend in one plane and provide
maximum stability and resistance to bending in the plane orthogonal to the
bending plane. The proposed sensing method, however, can be extended to the
CDM designs with 3D bending. The shape reconstruction model was introduced to
relate the wavelengths of FBG nodes to the curvature and bending direction of
the CDM. The model was then implemented on the CDM and its efficacy in free
and constrained environments for positive and negative deflections were
assessed.

Figure 14: Experimental setup of the CDM bending in constrained environment
for (a-c) positive deflection and (d-f) negative deflection. Obstacles were
placed at three different locations on each side of the CDM: (a), (d)
proximal; (b), (e) middle; (c), (f) distal.

Figure 15: Shape reconstruction of the shape sensing unit in the constrained environment for (a-c) positive deflection and (d-f) negative deflection in three different obstacle-interaction cases: (a), (d) proximal; (b), (e) middle; (c), (f) distal. Table 4: Shape reconstruction and tip position errors of the CDM in the constrained environment Obstacle status |  Cable displacement (mm) | Positive deflection | Negative deflection
---|---|---|---
|  | Shape error | Tip error | Shape error | Tip error
|  |  Mean (mm) |  Std. (mm) |  Max (mm) |  Mean (mm) |  Std. (mm) |  Mean (mm) |  Std. (mm) |  Max (mm) |  Mean (mm) |  Std. (mm)
Proximal |  1 | 0.486 | 0.412 | 1.162 | 1.158 | 0.027 | 0.171 | 0.114 | 0.362 | 0.360 | 0.011
|  2 | 0.682 | 0.415 | 1.298 | 1.267 | 0.023 | 0.219 | 0.163 | 0.548 | 0.438 | 0.018
|  3 | 0.753 | 0.436 | 1.477 | 1.321 | 0.108 | 0.210 | 0.121 | 0.539 | 0.478 | 0.038
|  4 | 0.700 | 0.375 | 1.360 | 1.117 | 0.168 | 0.245 | 0.121 | 0.606 | 0.533 | 0.148
|  5 | 0.620 | 0.337 | 1.264 | 0.996 | 0.277 | 0.276 | 0.161 | 0.812 | 0.547 | 0.217
Middle |  1 | 0.512 | 0.466 | 1.323 | 1.315 | 0.055 | 0.335 | 0.200 | 0.841 | 0.837 | 0.112
|  2 | 0.768 | 0.567 | 1.440 | 1.554 | 0.059 | 0.387 | 0.220 | 0.716 | 0.896 | 0.114
|  3 | 0.675 | 0.401 | 1.256 | 1.157 | 0.074 | 0.275 | 0.254 | 0.820 | 0.798 | 0.107
|  4 | 0.643 | 0.384 | 1.267 | 1.124 | 0.107 | 0.283 | 0.187 | 0.790 | 0.635 | 0.120
Distal |  1 | 0.235 | 0.144 | 0.617 | 0.614 | 0.012 | 0.391 | 0.167 | 0.542 | 0.540 | 0.011
|  2 | 0.479 | 0.328 | 1.210 | 1.199 | 0.019 | 0.242 | 0.175 | 0.274 | 0.366 | 0.021

The present technique for building the enclosed sensing unit was found to be
easy, time-saving, repeatable, and cost-effective [37, 54]. The linear
wavelength-curvature relationship of the developed shape-sensing unit
(Fig.11), had a pattern similar to previous studies [46, 34, 33, 55, 56]. The
sensing unit has a high sensitivity, up to 5.50 $\nicefrac{{nm}}{{mm}}$, at
the $90^{\circ}$ bending angle compared to Sefati [34], up to 3
$\nicefrac{{nm}}{{mm}}$, and Liu [46], up to 4 $\nicefrac{{nm}}{{mm}}$. The
sensitivity-enhancing property enables the sensing unit to detect small
changes in curvature.

The results of the static experiments in the free environment (Fig.13 and
Table 3) and constrained environment (Fig.15 and Table 4) indicate that the
reconstruction model for tracking the CDM centerline is comprehensive and
effective and justifies the building approach of the FBG-based shape sensing
unit employed in our study. The reconstruction model could track the CDM
centerline, which is extracted from stereo images, with a maximum shape
deviation of 0.305 mm for positive deflection and 0.656 mm for negative
deflection in the free environment (Table 3), both below 2 percent of the CDM
length, confirming the CDM is symmetric with respect to the deflection. The
good consistency of the reconstruction model results in the constrained
environment, implied by the maximum shape deviation of 1.477 mm for positive
deflection and 0.841 mm for negative deflection (Table 4), suggests that the
developed sensing unit and the reconstruction model are efficient for shape
tracking of the CDM, eliminating the need for integrated multi-core optical
fibers. Nevertheless, for the same actuating cable displacement, the mean
errors of the CDM shape tracking in obstacle-interaction cases were slightly
higher than free bending. This may be due to the clearance between the sensing
unit and the CDM sensor channel. Considering the free movement of the sensing
unit inside the CDM sensor channel with its tip fixed at the distal end and
the frictional effects within the channel, the rigor of the model was higher
than the models developed for the sensing unit directly attached through the
length of the medical instruments [26, 57, 58]. Moreover, compared to the
data-driven approach which estimated the CDM centerline by solving a
constrained optimization problem associated with the C-shaped bending [59],
the proposed reconstruction model can be applied to S shape CDMs.

A limitation of the approach included the asymmetrical friction and local
twist of the sensing unit affecting the shape reconstruction model by causing
error in shape sensing. Although these errors due to the free motion of the
sensing unit inside the CDM sensor channel were reduced by applying the
calibration coefficients, the small clearance between the sensing unit and the
CDM sensor channel is still a source of error for the curvature estimation.
Local twist measurement and incorporating its effects remain for the future
studies. In addition, small errors in curvature estimation especially at the
proximal end of the CDM can cause error accumulation through the CDM
arclength. Depending on the accuracy requirements for a specified application,
the number of FBG nodes at each shape sensing unit and/or equipping the CDM
with two shape sensors may improve the shape reconstruction model.

Hysteresis is another source of the error for CDM shape tracking (Fig.13).
Hysteresis behavior in the CDM is due to the factors including friction
between the actuating cable and the CDM cable channel, backlash of the
actuating cables, and the hysteresis property of the NiTi. The effect of
hysteresis may require additional modeling and must be considered in the
process of designing feedback control paradigms for the CDM.

In our study, the thermal conductivity of the NiTi rod used in the shape
sensing unit and the CDM body reduces the temperature gradient through the
sensor length, which can decrease the influence of the temperature on the
wavelength shifts of FBG nodes. Although this assumption in general is
compatible with FBG-based shape sensors [38], if needed, the shape
reconstruction model can be further extended to account for changes in
temperature.

##  6 Conclusion

The goal of the study was to design, implement, and validate a novel shape
reconstruction model, based on a CDM equipped with a novel FBG-based sensing
unit developed in-house. The fabrication process of the FBG shape senor was
easy and fast, using a polycabonate tube with three lumens as a flexible
enclosed substrate. The sensing model accounted for the internal twist
compensation and the hindered friction compensation between the sensing unit
and the CDM sensor channel. To reconstruct the CDM centerline, the wavelength
measurements of the FBG nodes were utilized to determine the curvature and
bending direction of the CDM. The results of experiments demonstrated the
linear wavelength-curvature relationship with high sensitivity and large
bending capability. Moreover, it was shown in static experiments that the
shape reconstruction model can accurately track the CDM centerline in free and
constrained environments. It was concluded that the shape sensing of the CDM
in the presence of non-constant curvature bending is feasible using the
proposed sensing model.

## Acknowledgment

The author would like to thank Dr. S. Sefati for his contribution with the
development of the software infrastructure.

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[]Golchehr Amirkhani received the B.Sc. degree in Mechanical Engineering from
Shahid Beheshti University, Tehran, Iran, in 2015, and the M.Sc. degrees in
Mechatronics from Sharif University of Technology, Tehran, Iran, in 2018. She
is currently working toward the Ph.D. degree in Mechanical Engineering with
Johns Hopkins University, Baltimore, MD, USA.

Her research interests include sensing and control of continuum manipulators
with applications in surgical robotics and autonomous systems.

[]Anna Goodridge received the B.S. degree in Mechanical Engineering from Johns
Hopkins University, Baltimore, MD, USA, in 2017 and is currently working for
the Laboratory for Computational Sensing and Robotics as a research engineer.
She has previously worked in research and development teams in the consumer
product industry and her current work is with applications in surgical
robotics and hardware design.

[]Mojtaba Esfandiari received the B.Sc. degree in Mechanical Engineering from
Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in
2014, and the M.Sc. degree in Mechanical Engineering from Sharif University of
Technology, Tehran, Iran, in 2017. He is currently pursuing the Ph.D. degree
in Mechanical Engineering at Johns Hopkins University, Baltimore, MD, USA. He
also worked as a research assistant at the Project neuroArm, the University of
Calgary, Calgary, AB, Canada, during 2018-2021.

His research interests include modeling, optimization, and control of surgical
robotic systems.

[]Henry Phalen received a B.S. degree in Bioengineering from the University of
Pittsburgh, Pittsburgh, PA, USA in 2018 and a M.S.E. degree in Robotics from
the Johns Hopkins University (JHU), Baltimore, MD, USA in 2021. He is
currently a Ph.D. candidate in Mechanical Engineering at JHU.

His research interests include state estimation and control of dexterous
robotic systems in constrained and safety-critical environments with a focus
on demonstrating integrated systems for use in surgical procedures.

[]Justin H. Ma received the B.S degree in Mechanical Engineering from Georgia
Institute of Technology, Atlanta, GA, USA, in 2019, and the M.S degree in
Mechanical Engineering from Johns Hopkins University, Baltimore, MD, USA, in
2022. He is currently pursuing the Ph.D. degree in Mechanical Engineering at
Johns Hopkins University, Baltimore, MD, USA.

His research interests include design and control of continuum manipulators
and mechanisms for minimally invasive surgical robotics in orthopaedics. He
has previously worked in the medical device industry developing implants for
spinal fusion.

[]Iulian Iordachita (Senior Member, IEEE) received the M.Eng. degree in
industrial robotics and the Ph.D. degree in mechanical engineering from the
University of Craiova, Craiova, Romania, in 1989 and 1996, respectively.

He is currently a Faculty Member with the Laboratory for Computational Sensing
and Robotics, Johns Hopkins University, Baltimore, MD, USA, and the Director
of the Advanced Medical Instrumentation and Robotics Research Laboratory. His
current research interests include medical robotics, image guided surgery,
robotics, smart surgical tools, and medical instrumentation.

[]Mehran Armand (Member, IEEE) received the Ph.D. degree in mechanical
engineering and kinesiology from the University of Waterloo, Waterloo, ON,
Canada, in 1998.

He is currently a Professor of Orthopaedic Surgery, Mechanical Engineering,
and Computer Science with Johns Hopkins University (JHU), Baltimore, MD, USA,
and a Principal Scientist with the JHU Applied Physics Laboratory (JHU/APL).
Prior to joining JHU/APL in 2000, he completed postdoctoral fellowships with
the JHU Orthopaedic Surgery and Otolaryngology-Head and Neck Surgery. He
currently directs the Laboratory for Biomechanical- and Image-Guided Surgical
Systems, JHU Whiting School of Engineering. He also directs the AVICENNA
Laboratory for advancing surgical technologies, Johns Hopkins Bayview Medical
Center. His laboratory encompasses research in continuum manipulators,
biomechanics, medical image analysis, and augmented reality for translation to
clinical applications of integrated surgical systems in the areas of
orthopaedic, ENT, and craniofacial reconstructive surgery.