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多芯光栅阵列三维形状重构方法的前提在于有栅测点区域的曲率和挠率的精确监测,继而采用插值与坐标解算实现整体形状重构,解算过程共包括五个阶段,如图1所示。首先通过解调仪获取所有有栅测点未形变前与形变后的中心波长数据,根据每个FBG的波长漂移量进行曲率和挠率的计算,然后对这些FBG点之间的空白无栅区域的曲率和挠率利用三次样条插值法进行插值,再利用齐次矩阵法进行连续坐标解算,并整合至同一坐标系,最后绘制出光纤的整体三维形状。以下将重点对曲率和挠率计算以及齐次矩阵法进行阐述。
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如图2(a)所示,当七芯光纤朝某一个方向弯曲时,边芯(顺时针编号2~7)和中芯(编号1)两纤芯的中心之间的距离为d,一般称为芯间距。当光纤发生弯曲后,假设光纤弯曲的曲率半径为R,N为垂直于弯曲方向的中性轴,则任意边芯的轴向应变ε可以表示为:
$$ \varepsilon =\frac{d}{R}\cdot \mathrm{cos}\theta $$ (1) 式中:θ为横截面上偏芯和弯曲方向之间的夹角。处于中性面的外侧的边芯为拉伸状态,处于中性面的内侧的边芯为压缩状态。由于中间芯位于中性面上,其中心波长可视为不受光纤弯曲的影响,可以进行轴向应变和温度的补偿。光纤弯曲曲率与波长漂移之间的线性关系可以表示为:
$$ \Delta {\lambda }_{n}-\Delta {\lambda }_{1}=k\cdot d\cdot \rho \cdot {\lambda }_{n}\cdot \mathrm{cos}{\theta }_{n}\text{,}n=2,3,\cdot \cdot \cdot ,7 $$ (2) 式中:Δλn为偏芯波长漂移量;Δλ1为中芯光栅中心波长漂移量;ρ=1/R为弯曲的曲率;k=(1−Pε)为标准石英光纤纵向应变敏感系数,通常为0.784;λn为多芯光纤光栅中偏芯在自由平直无弯曲状态下的初始波长。实验过程中,当检测到波长漂移时,利用中芯和任意两个边芯的数据分别代入公式(2)联立起来,就可以推导出每个芯与弯曲方向的夹角以及弯曲的曲率。如图2(b)所示,以边芯2芯为例,可以求出相邻光栅的2芯与弯曲方向的夹角
$ {\theta }_{i} $ ,再根据公式(3)就可以求出各相邻FBG点之间的挠率大小。图 2 七芯光纤传感原理示意图。(a)曲率;(b)挠率
Figure 2. Schematic diagram of the seven-core optical fiber sensing principle. (a) Curvature; (b) Torsion
$$\begin{split} \\ {\tau }_{12}={\theta }_{2}-{\theta }_{1}\text{;}{\tau }_{23}={\theta }_{3}-{\theta }_{2}\cdots \end{split} $$ (3) 根据公式(1)、(2)分析七芯光纤光栅的波长漂移量,可以计算出FBG处的弯曲曲率与弯曲方向,再根据公式(3)可以计算出挠率的大小。显然由于七芯光纤的六个边芯是对称分布的,所有方向上都具有很好的弯曲分辨能力,同时相较于只有两个边芯且互呈90°分布的多芯光纤,七芯光纤多出来的边芯可以用作取平均值,这样可以减小偶然误差对实验结果的影响,提高测量精度。
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根据1.1节中计算出的曲率与挠率,利用齐次变换矩阵法(公式(4)~(6))将离散的点坐标解算出来。
$$ \tilde S_{i + 1}^0 = M_1^0M_2^1 \cdot \cdot \cdot M_i^{i - 1}M_{(i + 1)}^i\tilde S_{i + 1}^{i + 1} = M_{i + 1}^0\tilde S_{i + 1}^{i + 1} $$ (4) $$ \begin{split} \tilde S_{i + 1}^i = \left[ {\begin{array}{*{20}{c}} {s_{i + 1,x}^i} \\ {s_{i + 1,y}^i} \\ {s_{i + 1,z}^i} \\ 1 \end{array}} \right] = M_{i + 1}^i\tilde S_{i + 1}^{i + 1} = \left[ {\begin{array}{*{20}{c}} {R_{i + 1}^i}&{{\text{o}}_{i + 1}^i} \\ O&1 \end{array}} \right]\tilde S_{i + 1}^{i + 1} \end{split}$$ (5) $$ M_{i + 1}^i = \left[ {\begin{array}{*{20}{c}} {\cos {\alpha _i}\cos {\varphi _i}}&{ - \sin {\alpha _i}}&{\cos {\alpha _i}\sin {\varphi _i}}&{{R_i}(1 - \cos {\varphi _i}){\cos} {\alpha _i}} \\ {\sin {\alpha _i}\cos {\varphi _i}}&{\cos {\alpha _i}}&{\sin {\alpha _i}\sin {\varphi _i}}&{{R_i}(1 - \cos {\varphi _i}){\sin} {\alpha _i}} \\ { - \sin {\varphi _i}}&0&{\cos {\varphi _i}}&{ {R _i}\sin {\varphi _i}} \\ 0&0&0&1 \end{array}} \right] $$ (6) 式中:
$ {\tilde{S}}_{i+1}^{i} $ 是以FBGi为坐标系来表示的FBGi+1的齐次坐标点;$ {s}_{i+1,k}^{i} $ 中的k={x, y, z}是$ {\tilde{S}}_{i+1}^{i} $ 的位置分量;${{M}}_{{i+1}}^{{i}}$ 是从坐标系i+1到坐标系i的齐次变换矩阵;$ {\tilde{S}}_{i+1}^{i+1} $ 是FBGi+1在FBGi+1坐标系下的齐次坐标,即$ {{[0 \; 0\; 0\; 1]}}^{\text{T}} $ ;$ {\alpha }_{i} $ 为弯曲方向角的变化量;$ {\phi }_{i} $ 为光纤圆弧所对应的圆心角度数。齐次变换矩阵可以看成由四个矩阵组合而成,每个矩阵发挥不同的作用,其中${{R}}_{{i+1}}^{{i}}$ 是一个3×3的旋转矩阵,其作用为坐标系对齐,${{o}}_{{i+1}}^{{i}}$ 是i+1坐标系的原点在i坐标系下的坐标,${{O}}_{\text{3}}^{{\text T}}$ 是一个1×3的零向量。因此,在形状重构期间将采用公式(6)这种坐标变换方法来计算以起始坐标系表示的所有FBG光栅点位置。齐次变换矩阵将每个独立坐标系中的点转换到同一坐标系下,将这些点按照顺序连接起来即可实现三维空间曲线的重构,如图3所示。 -
根据波长漂移量计算出各光栅点的曲率和挠率,实验先后进行了五次测量,取其平均值,利用三次样条插值法对空白处的曲率和挠率进行插值,再结合齐次变换矩阵重建算法得到了三维重建效果图,如图7(a)所示。其中红色曲线为光栅间距为10 cm的实验重建三维螺旋线,蓝色曲线为光栅间距为5 cm的实验重建三维螺旋线,黑色曲线为真实三维螺旋线。实验结果表明,重构出的三维形状与真实形状符合得较好。图7(b)是对重建图形的误差分析,从图中可以看出,整体误差随着光纤长度的增加呈现出一种增长的趋势,最大误差出现在尾点处。表1为实验值与模拟值的数据对比,从表中可以看出,光栅间距5 cm时的最大误差为1.15 cm,占全长的1.4%,平均误差为0.62 cm,占全长的0.8%。光栅间距10 cm时的最大误差为2.56 cm,占全长的3.2%,平均误差为1.32 cm,占全长的1.7%。第3节得出模拟值的最大误差占全长的2.7%和0.6%,实验值与模拟值符合得较好,两组实验值与模拟值的相对误差分别为0.5%和0.8%。误差产生的主要原因有:(1)通过显微镜观察发现,实验所用的七芯光纤的六个边芯并非完全按正六边形排布的,而是有一定的误差,这对计算弯曲曲率会产生一定的影响,从而导致实验误差;(2)实验中采用的是分辨率为10 pm的解调仪解调FBG的峰值波长,当波长变化小于10 pm时无法检测到,存在系统误差。
图 7 三维形状传感实验。(a)重构曲线与真实曲线;(b)误差
Figure 7. 3D shape sensing experiment. (a) Reconstructed curve and true curve; (b) Error
表 1 实验值与模拟值对比
Table 1. Comparison of experimental and simulated values
Grating pitch,
L/cmMaximum error
/cmError percentage Simulation error percentage 5 1.15 1.4% 0.6% 10 2.56 3.2% 2.7%
Research on shape sensing performance and reconstruction error of multi-core fiber grating
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摘要: 多芯光纤光栅形状传感技术利用空分复用以及应变监测的优势,结合不同的栅点布设方案,实现待测对象的连续曲率和形状传感。首先介绍了多芯光纤光栅曲率和挠率传感原理,提出采用齐次矩阵变换的三维重构算法实现光纤的三维形状重构。为了探究不同光栅密度对实验精度的影响,利用算法编程模拟了不同光栅间距下的三维形状重构精度,依据模拟仿真的结果,建立了不同光栅间距与三维重构误差之间的关系。三维形状传感实验使用光栅间距为10 cm和5 cm的七芯光纤光栅串。实验结果表明,最大误差出现在尾点处,分别为2.56 cm和1.15 cm,占全长的3.2%和1.4%,平均误差为1.32 cm和0.62 cm,占全长的1.7%和0.8%。实验结果与仿真值比较接近,说明可以依据仿真结果对不同光栅间距下的三维形状误差进行预测。结合具体的应用场景合理配置测点资源,在较低的成本范围内实现高性能的检测。Abstract:
Objective At present, shape sensing technology based on multi-core fiber grating is mainly divided into two categories, one is the optical frequency domain reflectometry (OFDR) demodulation technology based on the all-same weak grating. The other is multi-core grating or integrated grating shape sensing technology based on wavelength demodulation mode, and the multi-core grating array sensing system based on wavelength demodulation mode has the advantages of small size, high signal-to-noise ratio, fast acquisition speed and high real-time demodulation. In view of the diverse shape sensing needs in the actual application, the multi-core grating array multi-core sensing system using wavelength demodulation can be flexibly configured according to the actual needs of the number and spacing of the grating, which can realize the shape monitoring of the measured object with more flexible sensing distance and more diverse shape change span, which has a wider application prospect. However, because the multi-core grating measurement point cannot be continuously in space, the shape of the blank grating area between the measurement points cannot be obtained, and there is a blind zone of the measurement point. In practical applications, in order to improve the accuracy of shape sensing, it is necessary to study the shape deformation characteristics of the object to be measured, and design a reasonable grating distribution configuration scheme under the premise of taking into account the measurable key points and the full range of measurable areas to be measured. Methods Based on the strain sensing characteristics of seven-core fiber gratings, this paper adopts a three-dimensional shape reconstruction method based on homogeneous transformation matrix to reconstruct the shape of two seven-core fiber gratings with different grating spacing, solve the curvature and torsion of the region where the grating measurement point is located by the relative change value of the wavelength of the grating point, and obtain the curvature and torsion of the blank grating region between the FBG measurement points by cubic spline interpolation, and finally integrate the curvature and torsion of all points into the same coordinate system. The three-dimensional shape reconstruction of the object to be measured is realized. In order to explore the influence of grating spacing on shape reconstruction accuracy, the three-dimensional shape reconstruction error under different grating spacing based on this algorithm principle is simulated by algorithm programming, and the experimental verification is completed by building a three-dimensional shape sensing system, and the rationality of the error and simulation error in the experiment is discussed. Results and Discussions The simulation calculation selects the raster spacing L of 12.5 cm, 10 cm, 8 cm, 5 cm and 1 cm, and plots the 3D reconstruction curve and the real curve under these raster spacing. Both the simulation assumes that the curvature and torsion measurement error of the gate measurement point is 0. The results show that when the grating spacing L is 12.5 cm, 10 cm, 8 cm, 5 cm and 1 cm, the spatial position error shows a gradual increasing trend with the length of the fiber, and the larger the grating spacing is, the larger the error is. The maximum reconstruction error falls at the end point of the analog length, which is 7.75 cm, 4.35 cm, 2.63 cm, 0.94 cm and 0.25 cm, accounting for 4.8%, 2.7%, 1.6%, 0.6% and 0.16% of the total length (Fig.4). In the experiment, a grating string with a grating spacing of 5 cm and 10 cm was selected to carry out a three-dimensional shape sensing experiment. The experimental results show that the reconstructed three-dimensional shape matches the real shape well (Fig.7). The maximum error at a raster spacing of 5 cm is 1.15 cm, accounting for 1.4% of the total length, and the average error is 0.62 cm, accounting for 0.8% of the total length. The maximum error at a raster spacing of 10 cm is 2.56 cm, accounting for 3.2% of the total length, and the average error is 1.32 cm, accounting for 1.7% of the total length (Tab.1). Conclusions Whether it is simulation or experimental verification, the overall trend of error value and relative length of the object to be measured is in line with the trend of linear growth. The maximum error points all occur at the end point. By establishing the variation relationship between the raster spacing and the slope of the linear fit, the correspondence between different raster spacing and the 3D reconstruction error can be explored. According to this relationship, the three-dimensional shape reconstruction error of any grating spacing and any length of optical fiber under similar shapes can be predicted, so that the appropriate grating spacing and demodulation method can be selected in combination with specific application scenarios, reasonable allocation of measurement point resources, and improvement of detection performance in a lower cost range. -
Key words:
- fiber shape sensing /
- multi-core fiber /
- Bragg grating /
- numerical simulation /
- grating pitch
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表 1 实验值与模拟值对比
Table 1. Comparison of experimental and simulated values
Grating pitch,
L/cmMaximum error
/cmError percentage Simulation error percentage 5 1.15 1.4% 0.6% 10 2.56 3.2% 2.7% -
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