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Reconstruction error model of distributed shape sensing based on the reentered frame in OFDR

Sheng Li, Peidong Hua, Zhenyang Ding, Kun Liu, Yong Yang, Junpeng Zhao, Ming Pan, Haohan Guo, Teng Zhang, Li Liu, Junfeng Jiang, and Tiegen Liu

Open Access Open Access

Abstract

At present, the reconstruction error of optical fiber shape sensing is commonly represented by Euclidean distance error. However, the Euclidian error of shape reconstruction will be dependent on the shape complexity, which depends on length, curvature and torsion. In this paper, we establish a reconstruction error model of distributed shape sensing in optical frequency domain reflectometry (OFDR) based on the Frenet-Serret frame and the error delivering theory, which illustrates the relationship between the reconstruction error and parameters such as curvature, torsion, fiber length and strain measurement error. We experimentally verify the feasibility and applicability of the proposed reconstruction error model by distributed optical fiber shape sensing system based on OFDR. The proposed reconstruction error model can provide a prediction of the maximal reconstruction error when the estimated range of curvature, torsion, fiber length of a shape needs to be reconstructed and strain measurement errors of OFDR system are known. It is very useful to judge whether the shape reconstruction error meets the requirement according to the shape to be reconstructed.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, optical fiber shape sensing (OFS) has become a hot research direction in the field of optical fiber sensing. This outstanding technology has a series of advantages such as compact structure, high flexibility, resistance to harsh environments and corrosion [1], and being able to multiplex compared with other shape reconstruction technologies [2–4]. OFS has great application potential in civil, mechanical, aerospace, biological, medical and other fields [5–7].

Optical fiber shape sensors mainly consist of fiber optic cables with multiple cores and embedded strain sensors. The basic principle is the following: The curvature and bending direction of the deformed optical cable at different positions are calculated by measuring the strain in different cores at some specific cross-sectional positions, and the shape is reconstructed through numerical integration combined with specific algorithms. Fiber shape sensors can be divided into fiber Bragg grating (FBG) [8], Rayleigh scattering [9] and Brillouin scattering according to the different strain measurement technology [10]. Optical frequency domain reflectometry (OFDR) is an ideal approach to realize a distributed strain sensing with high spatial resolution and high sensitivity simultaneously based on Rayleigh backscattering spectra shift. Duncan et al. firstly applied OFDR to shape sensing [9]. In recent years, various shape reconstruction algorithms have been proposed to realize the transformation from the strain in the multi-path fiber to the spatial position coordinates for different types of fiber shape sensors. The spatial differential geometric reconstruction method based on the Frenet-Serret frame is the most widely applicable method at present [11–13]. Frenet-Serret formulas between the tangent vector, the normal vector and the bi-normal vector is solved to obtain the three-dimensional spatial coordinates of the multi-core fiber.

The reconstruction error of OFS is commonly represented by Euclidean distance error [10,11,14–16]. However, researches about reconstruction error model of distributed shape sensing are limited. In recent years, many researchers have carried out theoretical analysis and simulation of various factors affecting the shape reconstruction for different types of optical fiber shape sensors. In 2008, Askins et al. first studied the effect of external twist on shape reconstruction results and proposed a method to estimate fiber twist from internal strain state based on FBG [17]. In 2014, Kirsten et al. established a shape sensing model based on FBG for medical surgical needle tracing shape sensors, and the influence of factors such as wavelength measurement accuracy, sensor position accuracy, sensor configuration and interpolation method on the position accuracy of the surgical needle was analyzed by simulation [18]. In 2019, Floris et al. simulated the effects of strain measurement uncertainty and fiber core position accuracy on the curvature and bending direction measurements respectively through the Monte Carlo method [19,20]. In the same year, Floris et al. experimentally explored the effect of fiber grating length on the shape reconstruction results by passing two fiber gratings with different writing lengths and pointed out that the longer fiber grating has higher shape reconstruction accuracy [21]. Although several studies have investigated the accuracy of fiber optic shape sensors, the model between Euclidean distance error and shape parameters such as curvature, torsion, length and strain measurement error are not studied. If the reconstruction error model is established, we can estimate the reconstruction error of an arbitrary shape based on the strain measurement error and shape parameters, which is very useful for determining the shape reconstruction error whether or not agree with demanding based on the shape trace.

In this paper, we establish a reconstruction error model of distributed shape sensing in OFDR based on the Frenet-Serret frame and the error delivering theory, which illustrates the relationship between the reconstruction error and parameters such as curvature, torsion, fiber length and strain measurement error. We experimentally verify the feasibility of the proposed reconstruction error model in different determined constant strain measurement error by distributed optical fiber shape sensing system based on OFDR. In addition, we also verify the applicability of the reconstruction error model under different maximal margin of random strain measurement errors by changing strain measurement resolution of OFDR system. Maximal margin of random strain measurement errors is calculated from contrast measurement between OFDR and piezo-nanometer stage in different strain variations.

2. Three-dimensional shape sensing error model

Distributed strain measurement is the basis and premise of shape measurement. Obviously, a better spatial resolution and a higher strain measurement accuracy will result in a more accurate shape reconstruction. In fact, the sensing spatial resolution of OFDR in the OFS is usually set to the minimum value based on the signal to noise ratio (SNR) of OFDR system and the effect of the spatial resolution on the reconstruction error is negligible in this case. So, it is of great significance to establish a three-dimensional shape sensing error model between shape reconstruction error and strain measurement error.

The differential geometry reconstruction algorithm based on the Frenet-Serret frame is the most widely applicable method [11–13,22], which obtains the spatial coordinates of the curve by solving the Frenet formula between the tangent vector, normal vector and binormal vector [23]. The flow scheme and error propagation process of shape reconstruction algorithm based on Franet-Serret frame are shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Flow scheme and error propagation process of shape reconstruction algorithm based on Franet-Serret frame. The curvature κ and the bending direction Δb at each position of the fiber can be obtained according to the strain in each core εi. The discrete local κ and θb sets are converted into curvature and bend direction functions κ(s) and τ(s) using a curve interpolation algorithm. The tangent vector T(s) can be solved by solving the Frenet-Serret formula, and then the spatial position coordinate x(s), y(s), z(s) of the curve is obtained by the method of numerical integration.

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The error propagation process of the shape reconstruction algorithm based on the Franet-Serret frame is as follows: Strain measurement errors Δε cause curvature error Δκ and torsion error Δτ, which further lead to deviations in the calculation of the tangent vector ΔT. ΔT accumulates continuously during the integration process and eventually leads to reconstruction errors. The reconstruction error is usually measured by Euclidean distance error ΔE at the corresponding position of the reconstructed curve and the standard curve as

$$\Delta E = \sqrt {{{({x_{ac}} - {x_{sc}})}^2} + {{({y_{ac}} - {y_{sc}})}^2} + {{({z_{ac}} - {z_{sc}})}^2}} ,$$
where xac, yac, zac represent the spatial position coordinate of the actual reconstructed curve, and xsc, ysc, zsc represent the spatial coordinate of the standard reference curve. According to the error propagation process of shape reconstruction based on the Frenet-Serret frame, we can establish the relationship between ΔE and Δε.

2.1 Relationship between curvature error and strain measurement error

For a symmetrical distributed three-core fiber with an angle of 120° to each other, the bending curvature of the fiber κ at a certain position can be expressed as [23]

$$\kappa = \frac{{2\sqrt {{{\left( {\sum\limits_{i = 1}^3 {{\varepsilon_i}\cos{\theta_i}} } \right)}^2} + {{\left( {\sum\limits_{i = 1}^3 {{\varepsilon_i}\sin{\theta_i}} } \right)}^2}} }}{{3r}},$$
where εi are the magnitude of the strain in the different cores when i takes different values, θi are the angles between the line connecting each core and the neutral axis of the fiber and the coordinate axis at each location. r is the core spacing of the multi-core fiber. Equation (2) can be simplified and eliminated θi to obtain the relationship between $\kappa$ and strain in the three cores of the multi-core fiber ε1, ε2, ε3 as
$$\kappa = \frac{{2\sqrt {{\varepsilon _1}^2 + {\varepsilon _2}^2 + {\varepsilon _3}^2 - {\varepsilon _1}{\varepsilon _2} - {\varepsilon _1}{\varepsilon _3} - {\varepsilon _2}{\varepsilon _3}} }}{{3r}}.$$

According to the synthetic formula of the systematic error [24], the relative curvature error $\frac{{\Delta \kappa }}{\kappa }$ caused by Δεi can be expressed as

$$\frac{{\Delta \kappa }}{\kappa } = \left|{\frac{{\partial \ln \kappa }}{{\partial {\varepsilon_1}}}} \right|\Delta {\varepsilon _1} + \left|{\frac{{\partial \ln \kappa }}{{\partial {\varepsilon_2}}}} \right|\Delta {\varepsilon _2} + \left|{\frac{{\partial \ln \kappa }}{{\partial {\varepsilon_3}}}} \right|\Delta {\varepsilon _3}.$$

We assume that the strain measurement error in each core is the same as Δε and substitute Eq. (3) into Eq. (4) to get

$$\begin{aligned} \frac{{\Delta \kappa }}{\kappa } &= \frac{1}{2}\left( {\left|{\frac{{2{\varepsilon_1} - {\varepsilon_2} - {\varepsilon_3}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|+ \left|{\frac{{2{\varepsilon_2} - {\varepsilon_1} - {\varepsilon_3}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|} \right.\\ &+ \left. {\left|{\frac{{2{\varepsilon_3} - {\varepsilon_1} - {\varepsilon_2}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|} \right)\Delta \varepsilon . \end{aligned}$$

Strain in each core satisfies the relation as [23]

$${\varepsilon _i} ={-} \kappa r\cos ({\theta _b} - {\theta _i}),$$
where θb is the fiber bending direction. We substitute Eq. (6) into Eq. (5) and simplify Eq. (5) to obtain curvature error Δκ as
$$\Delta \kappa = \frac{2}{{3r}}\left( {\left|{\cos ({\theta_b} - {\theta_1} - \frac{{2\pi }}{3})} \right|+ |{\cos ({\theta_b} - {\theta_1})} |+ \left|{\cos ({\theta_b} - {\theta_1} + \frac{{2\pi }}{3})} \right|} \right)\Delta \varepsilon ,$$

We set the core initial angle θ1= 0, Eq. (7) is transformed as

$$\Delta \kappa = \frac{2}{{3r}}\left( {\left|{\cos ({\theta_b} - \frac{{2\pi }}{3})} \right|+ |{\cos ({\theta_b})} |+ \left|{\cos ({\theta_b} + \frac{{2\pi }}{3})} \right|} \right)\Delta \varepsilon .$$

We can know from Eq. (8) that Δκ caused by Δε is related to θb, which makes it difficult for us to deduce the relationship between ΔE and Δε later. To simplify the model, we assume a f (θb) as:

$$f({\theta _b}) = \left|{\cos ({\theta_b} - \frac{{2\pi }}{3})} \right|+ |{\cos ({\theta_b})} |+ \left|{\cos ({\theta_b} + \frac{{2\pi }}{3})} \right|.$$

From Eq. (9), f (θb) has a minor numerical range of $\left[ {\sqrt 3 ,2} \right]$. We assume that Δκ of each point on the fiber is the same and calculate the maximum Δκ under the same $\Delta \varepsilon$ by substituting $f({\theta _b}) = 2$ into Eq. (8). Equation (8) can be approximated as

$$\Delta \kappa = \frac{4}{{3r}}\Delta \varepsilon ,$$

Equation (10) shows the relationship between $\Delta \kappa$ and $\Delta \varepsilon$, which is proportional to the magnitude of $\Delta \varepsilon$ and inversely proportional to r. In addition, $\Delta \kappa$ is independent of $\tau$.

2.2 Relationship between torsion error and strain measurement error

We can start with θb to get the relationship between Δτ and Δε. According to the shape reconstruction algorithm based on Frenet-Serret frame, θb can be calculated as [23]

$${\theta _b} = \arctan \left( {\frac{{\sum\limits_{i = 1}^3 {\frac{{{\varepsilon_i}}}{r}\sin {\theta_i}} }}{{\sum\limits_{i = 1}^3 {\frac{{{\varepsilon_i}}}{r}\cos {\theta_i}} }}} \right).$$

Equation (11) can be simplified by eliminating ${\theta _2}$ and ${\theta _3}$ according to the relationship between θi :

$${\theta _b} = \arctan \left( {\frac{{\sin {\theta_1}({2{\varepsilon_1} - {\varepsilon_2} - {\varepsilon_3}} )+ \sqrt 3 \cos {\theta_1}({{\varepsilon_2} - {\varepsilon_3}} )}}{{\cos {\theta_1}({2{\varepsilon_1} - {\varepsilon_2} - {\varepsilon_3}} )- \sqrt 3 \sin {\theta_1}({{\varepsilon_2} - {\varepsilon_3}} )}}} \right).$$

The bending direction error $\Delta {\theta _b}$ caused by Δεi can be expressed as

$$\Delta {\theta _b} = \left|{\frac{{\partial {\theta_b}}}{{\partial {\varepsilon_1}}}} \right|\Delta {\varepsilon _1} + \left|{\frac{{\partial {\theta_b}}}{{\partial {\varepsilon_2}}}} \right|\Delta {\varepsilon _2} + \left|{\frac{{\partial {\theta_b}}}{{\partial {\varepsilon_3}}}} \right|\Delta {\varepsilon _3},$$

Similarly, we assume that the strain error in each core is the same as $\Delta \varepsilon $ and substitute Eq. (12) into Eq. (13) to get the relation:

$$\begin{aligned} \Delta {\theta _b} &= \frac{{\sqrt 3 }}{2}\left( {\left|{\frac{{{\varepsilon_1} - {\varepsilon_2}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|} \right. + \left|{\frac{{{\varepsilon_1} - {\varepsilon_3}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|\\ &\left. { + \left|{\frac{{{\varepsilon_2} - {\varepsilon_3}}}{{{\varepsilon_1}^2 + {\varepsilon_2}^2 + {\varepsilon_3}^2 - {\varepsilon_1}{\varepsilon_2} - {\varepsilon_1}{\varepsilon_3} - {\varepsilon_2}{\varepsilon_3}}}} \right|} \right)\Delta \varepsilon . \end{aligned}$$

We substitute Eq. (6) into Eq. (14) and set θ1 = 0, then Eq. (14) is simplified to

$$\Delta {\theta _b} = \frac{2}{{3\kappa r}}\left( {\left|{\sin ({\theta_b} - \frac{{2\pi }}{3})} \right|+ |{\sin ({\theta_b})} |+ \left|{\sin ({\theta_b} + \frac{{2\pi }}{3})} \right|} \right)\Delta \varepsilon .$$

According to $\Delta \tau = \frac{{d(\Delta {\theta _b})}}{{ds}}$, we take the derivative of both sides of Eq. (15) to get the relation as

$$\Delta \tau = \frac{{2\tau }}{{3\kappa r}}\left( {\left|{\cos ({\theta_b} - \frac{{2\pi }}{3})} \right|+ |{\cos ({\theta_b})} |+ \left|{\cos ({\theta_b} + \frac{{2\pi }}{3})} \right|} \right)\Delta \varepsilon .$$

Similarly, to simplify the model, we assume that $\Delta \tau$ of each point on the fiber is the same. We calculate the maximum $\Delta \tau$ under the same $\Delta \varepsilon$ by substituting f (θb) = 2 into Eq. (16). Equation (16) can be approximated as

$$\Delta \tau = \frac{{4\tau }}{{3\kappa r}}\Delta \varepsilon .$$

Comparing Eq. (10) and (17), the expressions for $\Delta \kappa $ and $\Delta \tau$ have a high degree of similarity. By contrast, $\Delta \tau$ is also related to the complexity of the reconstructed shape, which is proportional to $\tau$ and inversely proportional to $\kappa$.

2.3 Relationship between Euclidean distance error and strain measurement error

We can start with helices to construct the relationship between ΔE and Δε because any complex curve can be regarded as composed of arc micro-segments with constant curvature and torsion. Parametric equation for the tangent vector $\vec{T}$ with a constant κ and τ can be acquired by taking the derivation of a standard helix equation [25] as

$$\vec{T} = \left( {\frac{{ - \kappa }}{{\sqrt {{\kappa^2} + {\tau^2}} }}\sin (\sqrt {{\kappa^2} + {\tau^2}} s),\frac{\kappa }{{\sqrt {{\kappa^2} + {\tau^2}} }}\cos (\sqrt {{\kappa^2} + {\tau^2}} s),\frac{\tau }{{\sqrt {{\kappa^2} + {\tau^2}} }}} \right).$$

Then the tangent vector deviation due to Δκ and Δτ is

$$\Delta \vec{T} = \frac{{\partial \vec{T}}}{{\partial \kappa }}\Delta \kappa + \frac{{\partial \vec{T}}}{{\partial \tau }}\Delta \tau .$$

We calculate the partial derivatives of $\vec{T}$ with respect to $\kappa $ and τ respectively and substitute Eq. (10) and (17) into Eq. (19) to get

$$\Delta \vec{T} = \left( { - s\cos (\sqrt {{\kappa^2} + {\tau^2}} s), - s\sin (\sqrt {{\kappa^2} + {\tau^2}} s),0} \right) \cdot \frac{4}{{3r}}\Delta \varepsilon .$$

Then the deviation of the reconstructed curve is

$$\begin{aligned} \Delta \vec{r} &= \int_0^s {\Delta \vec{T}} ds\\ &= \left( {\frac{{ - 1}}{{({\kappa^2} + {\tau^2})}}\cos (\sqrt {{\kappa^2} + {\tau^2}} s) - \frac{s}{{\sqrt {{\kappa^2} + {\tau^2}} }}\sin (\sqrt {{\kappa^2} + {\tau^2}} s) + \frac{1}{{{\kappa^2} + {\tau^2}}}} \right., \\ &\left. {\frac{{ - 1}}{{({\kappa^2} + {\tau^2})}}\sin (\sqrt {{k^2} + {\tau^2}} s) + \frac{s}{{\sqrt {{\kappa^2} + {\tau^2}} }}\cos (\sqrt {{\kappa^2} + {\tau^2}} s), \textrm{0}} \right) \cdot \frac{4}{{3r}}\Delta \varepsilon . \end{aligned}$$

Then the relationship between the $\Delta E$ and $\Delta \varepsilon$ is

$$\begin{aligned} \Delta E &= |{\Delta \vec{r}} |\\ &= \frac{4}{{3r}}\Delta \varepsilon \sqrt {\frac{2}{{{{({{\kappa^2} + {\tau^2}} )}^2}}} + \frac{{{s^2}}}{{{\kappa ^2} + {\tau ^2}}} - \frac{2}{{{{({{\kappa^2} + {\tau^2}} )}^2}}}\cos \left( {\sqrt {{\kappa^2} + {\tau^2}} s} \right) - \frac{{2s}}{{{{\left( {\sqrt {{\kappa^2} + {\tau^2}} } \right)}^3}}}\sin \left( {\sqrt {{\kappa^2} + {\tau^2}} s} \right)} . \end{aligned}$$

Equation (22) expresses the relationship between $\Delta E$ and $\Delta \varepsilon$ in a three-dimensional curve with a constant κ and τ. From Eq. (22), we know that $\Delta E$ caused by Δε is related to the complexity of the reconstructed shape and fiber length and inversely proportional to r.

Particularly, when τ = 0, that's for a two-dimensional curve. Equation (22) can be simplified as

$$\Delta E = \frac{4}{{3r}}\Delta \varepsilon \sqrt {{s^2}{R^2} + 2{R^4} - 2s{R^3}\sin \frac{s}{R} - 2{R^4}\cos \frac{s}{R}} ,$$
where $R\textrm{ = }1/\kappa$, which is the radius of curvature. Equation (22) expresses the relationship between $\Delta E$ and $\Delta \varepsilon$ in a two-dimensional curve.

For a general curves, both κ and τ are variable, then Eq. (22) can be written as follows:

$$\begin{aligned} \Delta E &= \left|{\int_0^s {\Delta \vec{T}ds} } \right|\\ &= \frac{4}{{3r}}\Delta \varepsilon \left|{\left( {\int_0^s { - s\cos (\sqrt {{\kappa^2}(s) + {\tau^2}(s)} \cdot s)} ds,\int_0^s { - s\sin (\sqrt {{\kappa^2}(s) + {\tau^2}(s)} \cdot s)} ds,0} \right)} \right|. \end{aligned}$$

Equation (24) reveals the relationship between $\Delta E$ and $\Delta \varepsilon$ in an arbitrary three-dimensional curve. To simply the model of Eq. (24), Eq. (20) is calculated by Eq. (10) and (17), so an approximate error is put forward to the model of Eq. (24). When f (θb) = 2, the model is completely precise. When f (θb) ≠ 2, the model has a minor approximate error.

It should be pointed out that Eq. (24) only shows the reconstruction error caused by Δε rather than total error. In fact, fiber twist is also an important factor causing reconstruction error, which affects the reconstruction result by directly changing the torsion of the curve [22]. The shape reconstruction error caused by fiber twist can be reduced by using a spun multi-core fiber [12]. Here the reconstruction error caused by fiber twist is not discussed in detail.

From Eq. (24), we can estimate the reconstruction error of any complex curve when Δε and parameters of fiber shape including κ, τ and s are determined. However, in practical, Δε will not be the same at every sensing point, which is a random variable due to the diversity and uncertainty of error sources. We can obtain the maximum strain measurement error Δεmax of OFDR by referring to the data sheet of OFDR products [26] or calibrating by piezo-nanometer stage in different strain variations. Then we can use Eq. (24) to estimate the maximum $\Delta E$ of the shape sensing system based on Δεmax.

3. Reconstruction error model verification and analysis

3.1 Experimental setup

An OFDR-based distributed shape sensing system we built is shown in Fig. 2. A tunable light source (TLS, New focus TLB-8800) is divided into two paths by a 1:99 fiber coupler. Among them, 1% of the light enters an auxiliary interferometer composed of a circulator, a 50:50 coupler and a delay fiber, and then a beat signal is generated at the balanced photodetector (BPD), which is used to provide an external clock to data acquisition card (DAQ). The clock trigger signal (f-clock) is used to sample the beat signal output by the main interferometer at equal optical frequency intervals, so as to compensate the nonlinear frequency sweep effect of TLS [27]. 99% of the light enters the main interferometer section, 20% of the light enters the reference arm and 80% of the light enters the measurement arm after a 20:80 coupler. The optical switch is connected to the fan out device (Yangtze Inc. FAN-7-42) used to switch the fiber core to be tested. Optical signal of the measurement arm enters multi-core fiber after passing through the circulator, optical switch and the fan out device, and the Rayleigh scattering light of the multi-core fiber and the reference light form beat interference in the coupler. BPD converts the beat-frequency optical signal into an electrical signal and then collected by a four-channel DAQ. The sweep rate and the starting wavelength of TLS are set to 80 nm/s and 1550 nm respectively, and the number of sampling points is set to 7.5M Samples. The length of the delay fiber of the auxiliary interferometer is 136 m. The actual scanning wavelength range of the laser is 45 nm (5600 GHz) corresponding to a system spatial resolution of 18 μm for a data point in the spatial domain. The multi-core fiber is an ordinary 7-core fiber (Yangtze Inc. MCF 7-42/250SM), the diameter of the fiber cladding is 150 μm, the distance from the surrounding fiber core to the central axis (core spacing) is 42.5 μm, and the center wavelength is 1550 nm. Three cores numbered 2, 4, and 6 are used in the experiment and the angle between each two cores is 120° as shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Experimental setup. TLS: tunable light source; FRM: faraday rotating mirror; PC: polarization controller; BPD: balanced photodetector; DAQ: data acquisition card.

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3.2 Experiment design and calibration

In order to verify the effectiveness of the proposed reconstruction error model in Eq. (24), we performed multi-core fiber shape sensing experiments based on OFDR. We divide the experiment into two parts. In the first part, we verify the relationship between the reconstruction error and the parameters in the model by variable-controlling approach. In the second part, we verify the same applicability of the model to random strain error by calibrating the maximum strain error of an OFDR system. In order to eliminate the influence of the spatial resolution on experimental, the spatial resolution of all experiments is set to a minimum of 3.6 mm based on the SNR of OFDR system.

How to choose a reasonable shape to verify the model is a key point. The complexity of a shape cannot be quantitatively defined. Any complex shape can be regarded as composed of arc micro-segments with constant κ and τ. If we have verified that the model is correct for the shape with a constant curvature and torsion, this model would be true for any complex shape. The reason is that a complex shape is an integral process by many simple shapes. In order to control the κ and τ of the curve to a constant value, we combine 3D printing technology to design and print a series of phantoms containing grooves with known κ and τ curves as shape carriers for multi-core fibers in the experiment. In addition, we choose a shorter fiber for experiments to make the fluctuations of κ and τ of the curve as small as possible. Part of SOLIDWORKS’ 3D drawing and photographs of the 3D printing phantoms are shown in Fig. 3. In order to minimize the influence of twist on the reconstruction results, in our reconstruction experiments, adhesive tapes are attached to the beginning and end of an optical fiber to ensure that the relative angle of the beginning and end of the optical fiber are fixed.

 figure: Fig. 3.

Fig. 3. SOLIDWORKS’ 3D drawing and photograph of the 3D printing phantoms. (a) SOLIDWORKS’ 3D drawings of spiral phantom with κ=10 m-1, τ=1 rad/m. (b) Photographs of helical molds with different torsions. (c) SOLIDWORKS’ 3D drawings of spiral phantom with κ=10 m-1, τ=8 rad/m. (d) Photographs of phantom with R = 0.1 m and 0.2 m respectively. € SOLIDWORKS’ 3D drawings with R = 0.3 m. (f) Photographs of phantom with R = 0.5 m and 1.0 m respectively.

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Before applying the verification experiment, we need to calibrate the coefficient between strain variation and Rayleigh scattering spectral shift Kε. We paste the multi-core fiber to be tested on the stretching piezo-nanometer stage, and adjust the distance between these two working stages to 40 cm [28]. The piezo-nanometer stage is driven to generate strains of 5 με, 10 με, 15 με, 20 με, and 25 με, respectively. For each strain, nine consecutive experimental data acquisitions are performed. OFDR is used to measure the Rayleigh scattering spectral shifts at the stretched position and calculate average value as the spectral shift at this strain point. A linear fitting is performed on the experimental data and Kε = 8.549 με/GHz according to the slope of the fitting curve shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. Calibrating the coefficient between strain variation and Rayleigh scattering spectral shift.

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3.3 Validation of reconstructed error models under determined constant strain measurement error

3.3.1 Relationship between reconstruction error and torsion

We experimentally verify the relationship between reconstruction error and torsion of reconstructed error model. We design and print six standard 3D printing phantoms with κ = 0.1 m and τ = 0 rad/m, 2 rad/m, 4 rad/m, 6 rad/m, 8 rad/m and 10 rad/m respectively. During the experiment, the multi-core fiber is wound in grooves on the surface of a 3D printing model to perform nine repeated shape-reconstruction experiments. In order to reduce the influence of κ and τ fluctuation on the experimental verification, we select a set of experimental results whose κ and τ are the closest to the theoretical values we designed. Shape reconstruction is performed again after artificially adding a determined constant $\Delta \varepsilon$ of 5 με on the distributed strain curve along the fiber length shown in Fig. 5(a). We calculated $\Delta E$ at the ends of the two reconstructed curves before and after adding the determined constant $\Delta \varepsilon$ under the conditions of s = 0.2 m and 0.3 m. We compared measured $\Delta E$ with the theoretical $\Delta E$ based on the reconstruction error model shown in Fig. 5(b). From Fig. 5 (b), we know that the theoretical and measured $\Delta E$ are basically consistent. The $\Delta E$ curve decreases gradually with increasing $\tau$. When τ is small, $\Delta E$ does not change significantly with $\tau$. When $\tau$ is large, $\Delta E$ is more and more affected by $\tau$. By comparing the relationship between the $\Delta E$ and $\tau$ under different s, we can find that the trend of the $\Delta E$ decreasing with $\tau$ is more obvious under the longer s.

 figure: Fig. 5.

Fig. 5. Torsion verification results. (a) Reconstructed curves with κ=10 m-1 and τ=2 rad/m, 6 rad/m and 10 rad/m respectively, where Rm represents the average curvature of the actual reconstructed curve in the length of 0.3 m and Tm represents the average torsion. (b) Comparison of measured and theoretical results of ΔE as a function of τ.

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From Fig. 5(b), we find that a relatively large deviation between the theoretical and the measured $\Delta E$ occurs when $\tau$ is greater than 4 rad/m, which is caused by an approximation error in Eq. (24). In Eq. (9), ${\theta _b} = \int_0^s \tau ds$[23]. When $\tau $= 0, θb = 0 and $f({\theta _b}) = 2$. There is no approximation error in Eq. (24). Along with an increasing of $\tau $, the approximation error is increased. As a results, from Fig. 5(b), the deviation is small when $\tau $ is closed to 0. The deviation is increased when $\tau $ is increased.

3.3.2 Relationship between reconstruction error and curvature

We experimentally verify the relationship between reconstruction error and curvature of the reconstructed error model. From the relationship between ΔE and τ shown in Fig. 5(b), the curve with a zero τ has the largest ΔE when $\kappa$ is constant. Therefore, we use two-dimensional plane curves to verify the relationship between ΔE and $\kappa$. We design and print six 3D printing phantoms with R = 0.05 m, 0.1 m, 0.2 m, 0.35 m, 0.5 m, and 1.0 m respectively. The design and real photographs of part of 3D printing phantoms are shown in Fig. 3. Similarly, we add the determined constant $\Delta \varepsilon$ of 5 με on the distributed strain curve along the fiber length. Shape reconstruction results before and after artificially adding a determined constant $\Delta \varepsilon$ are shown in Fig. 6(a). The measured $\Delta E$ at the ends of the two reconstructed curves with different radius of curvature at s = 0.2 m and 0.3 m are calculated and compared with the theoretical $\Delta E$ based on the reconstruction error model separately as shown in Fig. 6 (b). The theoretical and measured $\Delta E$ are highly coincident, which verifies the reliability of the reconstruction error model. The $\Delta E$ increases with R and when R is small, $\Delta E$ increases rapidly, when R is large, $\Delta E$ basically remains unchanged. And by comparing the two sets of results with s = 0.2 m and 0.3 m, the shorter the length of the fiber, the smaller the radius of curvature at which ΔE begins to stabilize.

 figure: Fig. 6.

Fig. 6. Curvature verification results. (a) Reconstructed curves with theoretical curvature radius of 0.05 m, 0.1 m, 0.2 m, 0.5 m and 1.0 m respectively, where Rm represents the average radius of curvature of the actual reconstructed curve in the length of 0.3 m. (b) Comparison of measured and theoretical results of ΔE as a function of R.

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3.3.3 Relationship between reconstruction error and fiber length

We experimentally verify the relationship between reconstruction error and fiber length of the reconstructed error model. From Eq. (24), we know that the relationship between ΔE and s is related to the complexity of the shape. In order to make the verification more convincing, we reconstruct a shape of the helix with s = 0.55 m, R = 0.052 m, and τ=3.8 rad/m before and after adding a $\Delta \varepsilon$ of 5 με on the distributed strain curve along the fiber length shown in Fig. 7(a). The experimental and theoretical ΔE have a high degree of agreement even for complex three-dimensional curves shown in Fig. 7(b). $\Delta E$ grows nonlinearly along with s, which is the result of error accumulation during the shape reconstruction process.

 figure: Fig. 7.

Fig. 7. Verification results of the relationship between reconstruction error and fiber length when R = 0.052 m, τ=3.8 rad/m. (a) Reconstructed curve of the helix before and after adding a determined constant Δε. (b) Comparison of theoretical and measured results of ΔE as a function of s.

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3.3.4 Relationship between reconstruction error and strain measurement error

We experimentally verify the relationship between reconstruction error and strain measurement error of the reconstructed error model. We use a 0.3 m multi-core fiber and bend it into an arc with R = 1.0 m for the shape reconstruction. Determined constant $\Delta \varepsilon$ of 1 με, 2 με, 3 με, 4 με and 5 με are added on the distributed strain curve along the fiber length. ΔE at the ends of the two reconstructed curves before and after adding different $\Delta \varepsilon$ are calculated and shown in Fig. 8. Experiment results verify the reconstruction error model very well. When $\Delta \varepsilon$ is a constant along the fiber, ΔE and Δε has a linear relationship.

 figure: Fig. 8.

Fig. 8. Validation results of the relationship between reconstruction error and determined strain measurement error.

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3.4 Verification of reconstruction error model under random strain measurement error

We have previously demonstrated the feasibility of the reconstruction error model under constant strain error, which is significant for us to analyze the influencing factors of optical fiber shape sensing and correct the shape reconstruction results in some cases. In this section, we will further verify the applicability of the reconstruction error model to random strain measurement error by calibrating the maximum strain error Δεmax of the OFDR system.

How to construct shape sensing systems with different Δεmax is a key point to the experiment. As we know, for a measurement system, the limited measurement resolution is one of the most important factors that limit the strain measurement accuracy. Therefore, we consider changing Δεmax of the system by changing the theoretical strain resolution εr, which is related to the number of window data points N and the number of zero padding M when windowing the distance domain signal as [29],

$${\varepsilon _r} = {K_\varepsilon }\frac{{\Delta \nu }}{{M + N}},$$
where $\Delta \nu $ is the tuning range of TLS and $\Delta \nu$ = 5600 GHz in this experiment. N is a quantity related to the spatial resolution, which is usually set to be as small as possible to reduce the effect of spatial resolution on reconstruction results. Here we take N = 200, and the corresponding effective spatial resolution is 3.6 mm. Therefore, we can construct a shape sensing system under different εr by changing M.

We calibrated Δεmax of the system by contrast measurements between OFDR and piezo-nanometer stage in different strain variations. The strain calibration range is from 0 to 25 με and the calibration is performed every 5 με. We take 80 strain measurement points for each strain and calculate the maximum strain measurement error between the measured and the true strain values., as shown in Table 1. The maximum measurement error of all measurement points under different strains variation is the calibration result of Δεmax.

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Table 1. Calibration process of strain measurement error under different theoretical strain resolution

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From Table 1, we know that Δεmax decreases as εr becomes smaller. However, Δεmax will not decrease along with εr decreasing all the time. When M > 10000 corresponding εr < 4.69 με, Δεmax remains basically unchanged. The reason is that Δεmax reaches the maximum strain measurement accuracy of the system under the experimental conditions, which is limited by various factors such as the wavelength stability of TLS, SNR and the surrounding environment [29].

We perform shape reconstruction experiments to verify the applicability of the reconstruction error model under different Δεmax. In order to minimize the reconstruction error caused by twist, we choose an arc with a s = 0.3 m and R = 1 m for the experiment. In addition, to highlight the randomness of the experimental results, we bend the multi-core fiber from a straight line to an arc with R = 1 m ten times continuously and collect data. The shape reconstruction is carried out under the conditions that Δεmax is 6.19 με, 6.22 με, 8.41 με and 10.04 με respectively and the measured ΔE is calculated by compared the reconstructed curve with the standard curve. The relationship between measured ΔE and s is shown and compared with the theoretical $\Delta E$ based on the reconstruction error model as shown in Fig. 9. The measured $\Delta E$ of each shape reconstruction is different even under the same experimental conditions due to the uncertainty of $\Delta \varepsilon$. It should be noted that the difference between theoretical and measured $\Delta E$ observed in Fig. 9 are relatively large. Whereas, the measured $\Delta E$ of multiple experiments does not exceed the theoretical maximum $\Delta E$ calculated by the reconstruction error model. Since the strain measurement accuracy of OFDR system can be evaluated in advance, so the theoretical Δεmax can be used to guide us to choose a suitable strain measurement accuracy of OFDR system in the design of fiber optic shape sensor and to predict the shape reconstruction accuracy.

 figure: Fig. 9.

Fig. 9. Results of ten shape reconstruction experiments under different maximum strain measurement errors. The reconstruction Euclidean distance error for each experiment is uncertain due to the randomness of the strain measurement error, and the maximum reconstruction error will not exceed the error calculated by the reconstruction error model. (a) Δεmax = 6.19 με. (b) Δεmax = 6.22 με. (c) Δεmax = 8.41 με. (d) Δεmax = 10.04 με.

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4. Conclusion

Based on the shape sensing model of the Franet-Serret frame and the theory of systematic error propagation, a three-dimensional shape sensing error model between the reconstruction error and the strain measurement error is established. We acquire the functional relationship between reconstruction error and curvature, torsion, fiber length, strain measurement error and fiber core spacing. Two parts of experiments are designed to distinguish and verify the feasibility and applicability of the reconstruction error model. Through theoretical analysis and experimental verification in OFDR, we came to the following conclusions:

  • 1) For an optical fiber shape sensor, compared with the shape with large curvature and torsion, the reconstruction of the shape with small curvature and torsion is usually more difficult under the premise of only considering the inherent strain error of the system. This can be derived from the reconstruction error versus curvature and torsion.
  • 2) The shape reconstruction error increases with the increase of the fiber length and the strain measurement error, which is consistent with the conclusion obtained from the error accumulation effect in shape sensing.
  • 3) In practical applications, the strain measurement accuracy of the shape sensing system is usually provided by OFDR manufacturer, so the reconstruction error model proposed can be used to estimate the maximum reconstruction error of the shape sensor in a specific application scenario.

The establishment of the 3D shape sensing error model not only gives us a clearer understanding of the factors that affect the shape reconstruction results, but also provides some theoretical guidance for the design and performance evaluation of fiber optic shape sensing systems in different application scenarios. For example, when the shape measurement accuracy requirements and the range of curvature and torsion of the shape to be reconstructed have been clarified, reconstruction error model proposed can be used to calculate the minimum strain measurement error that meets the requirements. Therefore, it can effectively guide the designer to choose a suitable strain measurement system for the design of the shape sensor.

Limited by the experimental conditions, the verification of the reconstruction error model under random strain error in this paper only considers a two-dimensional arc shape with a large radius of curvature. The reason is that a complex shape will increase many other factors that affect the shape reconstruction results such as twist of the fiber, which will make verification experiments extremely difficult. In future studies, we will consider using a spun multi-core fiber to estimate the influence of twist to further validate the proposed reconstruction error model.

Funding

National Natural Science Foundation of China (61975147, 61735011, 61635008, 61505138); Key Technologies Research and Development Program (2019YFC0120701).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. Flow scheme and error propagation process of shape reconstruction algorithm based on Franet-Serret frame. The curvature κ and the bending direction Δb at each position of the fiber can be obtained according to the strain in each core εi. The discrete local κ and θb sets are converted into curvature and bend direction functions κ(s) and τ(s) using a curve interpolation algorithm. The tangent vector T(s) can be solved by solving the Frenet-Serret formula, and then the spatial position coordinate x(s), y(s), z(s) of the curve is obtained by the method of numerical integration.

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Fig. 2.
Fig. 2. Experimental setup. TLS: tunable light source; FRM: faraday rotating mirror; PC: polarization controller; BPD: balanced photodetector; DAQ: data acquisition card.

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Fig. 3.
Fig. 3. SOLIDWORKS’ 3D drawing and photograph of the 3D printing phantoms. (a) SOLIDWORKS’ 3D drawings of spiral phantom with κ=10 m-1, τ=1 rad/m. (b) Photographs of helical molds with different torsions. (c) SOLIDWORKS’ 3D drawings of spiral phantom with κ=10 m-1, τ=8 rad/m. (d) Photographs of phantom with R = 0.1 m and 0.2 m respectively. € SOLIDWORKS’ 3D drawings with R = 0.3 m. (f) Photographs of phantom with R = 0.5 m and 1.0 m respectively.

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Fig. 4.
Fig. 4. Calibrating the coefficient between strain variation and Rayleigh scattering spectral shift.

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Fig. 5.
Fig. 5. Torsion verification results. (a) Reconstructed curves with κ=10 m-1 and τ=2 rad/m, 6 rad/m and 10 rad/m respectively, where Rm represents the average curvature of the actual reconstructed curve in the length of 0.3 m and Tm represents the average torsion. (b) Comparison of measured and theoretical results of ΔE as a function of τ.

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Fig. 6.
Fig. 6. Curvature verification results. (a) Reconstructed curves with theoretical curvature radius of 0.05 m, 0.1 m, 0.2 m, 0.5 m and 1.0 m respectively, where Rm represents the average radius of curvature of the actual reconstructed curve in the length of 0.3 m. (b) Comparison of measured and theoretical results of ΔE as a function of R.

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Fig. 7.
Fig. 7. Verification results of the relationship between reconstruction error and fiber length when R = 0.052 m, τ=3.8 rad/m. (a) Reconstructed curve of the helix before and after adding a determined constant Δε. (b) Comparison of theoretical and measured results of ΔE as a function of s.

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Fig. 8.
Fig. 8. Validation results of the relationship between reconstruction error and determined strain measurement error.

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Fig. 9.
Fig. 9. Results of ten shape reconstruction experiments under different maximum strain measurement errors. The reconstruction Euclidean distance error for each experiment is uncertain due to the randomness of the strain measurement error, and the maximum reconstruction error will not exceed the error calculated by the reconstruction error model. (a) Δεmax = 6.19 με. (b) Δεmax = 6.22 με. (c) Δεmax = 8.41 με. (d) Δεmax = 10.04 με.

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Tables (1)

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Table 1. Calibration process of strain measurement error under different theoretical strain resolution

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Equations (25)

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Δ E = ( x a c − x s c ) 2 + ( y a c − y s c ) 2 + ( z a c − z s c ) 2 ,
κ = 2 ( ∑ i = 1 3 ε i cos ⁡ θ i ) 2 + ( ∑ i = 1 3 ε i sin ⁡ θ i ) 2 3 r ,
κ = 2 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 3 r .
Δ κ κ = | ∂ ln ⁡ κ ∂ ε 1 | Δ ε 1 + | ∂ ln ⁡ κ ∂ ε 2 | Δ ε 2 + | ∂ ln ⁡ κ ∂ ε 3 | Δ ε 3 .
Δ κ κ = 1 2 ( | 2 ε 1 − ε 2 − ε 3 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | + | 2 ε 2 − ε 1 − ε 3 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | + | 2 ε 3 − ε 1 − ε 2 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | ) Δ ε .
ε i = − κ r cos ⁡ ( θ b − θ i ) ,
Δ κ = 2 3 r ( | cos ⁡ ( θ b − θ 1 − 2 π 3 ) | + | cos ⁡ ( θ b − θ 1 ) | + | cos ⁡ ( θ b − θ 1 + 2 π 3 ) | ) Δ ε ,
Δ κ = 2 3 r ( | cos ⁡ ( θ b − 2 π 3 ) | + | cos ⁡ ( θ b ) | + | cos ⁡ ( θ b + 2 π 3 ) | ) Δ ε .
f ( θ b ) = | cos ⁡ ( θ b − 2 π 3 ) | + | cos ⁡ ( θ b ) | + | cos ⁡ ( θ b + 2 π 3 ) | .
Δ κ = 4 3 r Δ ε ,
θ b = arctan ⁡ ( ∑ i = 1 3 ε i r sin ⁡ θ i ∑ i = 1 3 ε i r cos ⁡ θ i ) .
θ b = arctan ⁡ ( sin ⁡ θ 1 ( 2 ε 1 − ε 2 − ε 3 ) + 3 cos ⁡ θ 1 ( ε 2 − ε 3 ) cos ⁡ θ 1 ( 2 ε 1 − ε 2 − ε 3 ) − 3 sin ⁡ θ 1 ( ε 2 − ε 3 ) ) .
Δ θ b = | ∂ θ b ∂ ε 1 | Δ ε 1 + | ∂ θ b ∂ ε 2 | Δ ε 2 + | ∂ θ b ∂ ε 3 | Δ ε 3 ,
Δ θ b = 3 2 ( | ε 1 − ε 2 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | + | ε 1 − ε 3 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | + | ε 2 − ε 3 ε 1 2 + ε 2 2 + ε 3 2 − ε 1 ε 2 − ε 1 ε 3 − ε 2 ε 3 | ) Δ ε .
Δ θ b = 2 3 κ r ( | sin ⁡ ( θ b − 2 π 3 ) | + | sin ⁡ ( θ b ) | + | sin ⁡ ( θ b + 2 π 3 ) | ) Δ ε .
Δ τ = 2 τ 3 κ r ( | cos ⁡ ( θ b − 2 π 3 ) | + | cos ⁡ ( θ b ) | + | cos ⁡ ( θ b + 2 π 3 ) | ) Δ ε .
Δ τ = 4 τ 3 κ r Δ ε .
T → = ( − κ κ 2 + τ 2 sin ⁡ ( κ 2 + τ 2 s ) , κ κ 2 + τ 2 cos ⁡ ( κ 2 + τ 2 s ) , τ κ 2 + τ 2 ) .
Δ T → = ∂ T → ∂ κ Δ κ + ∂ T → ∂ τ Δ τ .
Δ T → = ( − s cos ⁡ ( κ 2 + τ 2 s ) , − s sin ⁡ ( κ 2 + τ 2 s ) , 0 ) ⋅ 4 3 r Δ ε .
Δ r → = ∫ 0 s Δ T → d s = ( − 1 ( κ 2 + τ 2 ) cos ⁡ ( κ 2 + τ 2 s ) − s κ 2 + τ 2 sin ⁡ ( κ 2 + τ 2 s ) + 1 κ 2 + τ 2 , − 1 ( κ 2 + τ 2 ) sin ⁡ ( k 2 + τ 2 s ) + s κ 2 + τ 2 cos ⁡ ( κ 2 + τ 2 s ) , 0 ) ⋅ 4 3 r Δ ε .
Δ E = | Δ r → | = 4 3 r Δ ε 2 ( κ 2 + τ 2 ) 2 + s 2 κ 2 + τ 2 − 2 ( κ 2 + τ 2 ) 2 cos ⁡ ( κ 2 + τ 2 s ) − 2 s ( κ 2 + τ 2 ) 3 sin ⁡ ( κ 2 + τ 2 s ) .
Δ E = 4 3 r Δ ε s 2 R 2 + 2 R 4 − 2 s R 3 sin ⁡ s R − 2 R 4 cos ⁡ s R ,
Δ E = | ∫ 0 s Δ T → d s | = 4 3 r Δ ε | ( ∫ 0 s − s cos ⁡ ( κ 2 ( s ) + τ 2 ( s ) ⋅ s ) d s , ∫ 0 s − s sin ⁡ ( κ 2 ( s ) + τ 2 ( s ) ⋅ s ) d s , 0 ) | .
ε r = K ε Δ ν M + N ,
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