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目前对光纤形状传感器的设计和测量方法的研究可以分为两大类: 一类是基于光纤布喇格光栅(fiber Bragg grating,FBG)的形状传感器;一类是基于分布式光纤传感(distributed fiber optic sensing,DFOS)的形状传感器。
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以光纤为传输介质的FBG传感技术,是通过观察光纤传输过程中FBG中心波长的漂移量,来感知外界温度或应变等待测参数的变化[6]。由模式耦合理论得到FBG中心波长λB, 可用下式表示[7]:
$ {\lambda _{\rm{B}}} = 2{n_{{\rm{eff}}}}\mathit{\Lambda } $
(1) 式中,neff为有效折射率,Λ为FBG的光栅周期。因为neff和Λ都是温度和应变的函数,当FBG工作的环境温度和所受应变发生改变时,neff(T, ε)和Λ(T, ε)就会发生变化,相应地,FBG的中心波长就会发生漂移,其中心波长漂移量可用下式表示[8-9]:
$ \begin{gathered} \Delta \lambda_{\mathrm{B}}=2 \Delta n_{\mathrm{eff}} \mathit{\Lambda }+2 n_{\mathrm{eff}} \Delta \mathit{\Lambda }= \\ \lambda_{\mathrm{B}}\left[\left(1-P_{\mathrm{e}}\right) \varepsilon+(\alpha+\xi) \Delta T\right] \end{gathered} $
(2) 式中,Pe为FBG的有效弹光系数,ε为FBG所受到的应变,α为光纤的热膨胀系数,ξ为光纤的热光系数,ΔT为温度变化量。假设外界温度恒定,即ΔT=0,则可以忽略温度对λB的影响,(2)式就可以简化为:
$ \Delta \lambda_{\mathrm{B}}=\lambda_{\mathrm{B}}\left(1-P_{\mathrm{e}}\right) \varepsilon $
(3) 已有的研究表明,光纤所受应变和其曲率之间是成线性关系的[10-12],从(3)式可以看出,通过测量FBG形状传感器中心波长的漂移量,就可以计算得到传感器所受的应变量,从而得到FBG传感器所在光纤段的曲率信息。
光纤在感受到弯曲形变后,光纤上刻写的FBG会因为应力产生相应地中心波长漂移,基于FBG的形状传感将光纤中FBG产生的波长漂移量相应的转换为该位置的曲率,再将光纤按空间分辨率量级分成若干段,计算出每个光纤段的曲率,通过一定的还原算法就可以在电脑端还原整条光纤的形状。
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由于光纤介质的不均匀性,光在光纤中传输会产生散射信号[13-14],分布式形状传感器通过将光纤中产生的散射信号和本地参考信号做对比处理,从而得到光在光纤中传输产生的参数变化量,来确定光纤所受到的应变,并通过数据处理,对应变进行定位。以采用多芯光纤(multi-core fiber,MCF)[15-16]为传输介质的形状传感器为例,则分布式形状传感器的简化理论模型如图 1所示[17]。图中,R为光纤的弯曲半径,L是传感器的初始长度,d为纤芯和传感器中性轴的距离,ΔL为纤芯被拉伸/压缩的长度。
设L1和L2侧所受应力为ε1和ε2,则两侧应力变化量应有如下关系式:
$ \Delta \varepsilon_1=\frac{d}{R}=-\Delta \varepsilon_2 $
(4) 以基于布里渊散射信号的系统为例,其布里渊频移(Brillouin frequency shift,BFS)受温度和应变两种变量的影响,其中温度系数为CT,应变系数为Cε,则其变化量可以表示为[18]:
$ \Delta \nu_{\mathrm{B}}=C_T \Delta T+C_{\varepsilon} \Delta \varepsilon $
(5) 假设多芯光纤在同一位置上所受温度相同,则选用多芯光纤中任意两个芯的布里渊频移量相减,就可以消除温度对BFS的影响,得到下式:
$ \begin{gathered} \Delta \nu_{\mathrm{B}, 2}-\Delta \nu_{\mathrm{B}, 1}= \\ C_T \Delta T_2+C_{\varepsilon} \Delta \varepsilon_2-\left(C_T \Delta T_1+C_{\varepsilon} \Delta \varepsilon_1\right)= \\ C_T\left(\Delta T_2-\Delta T_1\right)+C_{\varepsilon}\left(\Delta \varepsilon_2-\Delta \varepsilon_1\right)= \\ -C_{\varepsilon} \frac{2 d}{R}=-2 C_\kappa \kappa \end{gathered} $
(6) 式中, Cκ=Cεd, 为光纤的曲率系数,κ=1/R, 为光纤的曲率。
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空间曲线其实可以看作是同个空间下的任意一个粒子或无数个粒子的运动轨迹进行组合的结果,弗莱纳(Frenet-Serret)公式是对3维空间中连续可微曲线上粒子运动的描述,即对空间曲线的切向、法向、副法向方向之间关系的表示[19]:
$ \left\{\begin{array}{l} \frac{\mathrm{d} \boldsymbol{T}(s)}{\mathrm{d} s}=\kappa(s) \boldsymbol{N}(s) \\ \frac{\mathrm{d} \boldsymbol{N}(s)}{\mathrm{d} s}=-\kappa(s) \boldsymbol{T}(s)+\tau(s) \boldsymbol{B}(s) \\ \frac{\mathrm{d} \boldsymbol{B}(s)}{\mathrm{d} s}=-\tau(s) \boldsymbol{N}(s) \end{array}\right. $
(7) 式中,s代表空间中曲线的长度,T(s)为单位切向量且指向粒子运动的方向,N(s)为单位法向量,B(s)为单位副法向量,κ(s)为曲线曲率,τ(s)为曲线挠率。
光纤形状传感的还原重构就是将整条光纤看作若干个光纤段,这些光纤段在空间中就是若干个空间曲线,将这些空间曲线结合各种空间坐标算法拼接在一起,就可形成一条完整的有形态的光纤。在基于弗莱纲框架的基础上,出现了很多空间坐标算法,如多项式拟合[20]、三次样条插值算法[21]、曲率差值算法[22]等。
基于光纤传感的形状传感发展研究
Research on the development of shape sensing based on optical fiber sensing
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摘要: 光纤传感技术是近年来新兴的一种传感技术, 在众多领域中得到了广泛的关注和研究。归纳总结了光纤形状传感技术研究的主要进展, 讨论了基于光纤光栅传感和分布式传感的光纤形状传感技术的基本原理, 介绍了形状重构的理论框架——Frenet-Serret公式, 分析了分布式光纤形状传感技术研究中的关键问题, 并在此基础上对分布式光纤形状传感技术的研究前景进行了展望。Abstract: Fiber optic sensing technology is a kind of sensing technology emerging in recent years, which has received extensive attention and research in many fields. The main progress of optical fiber shape sensing technology was summarized, and the basic principle of fiber shape sensing technology based on fiber grating sensing and distributed sensing was discussed. The theoretical framework of shape reconstruction (Frenet-Serret formula) was introduced, and the key problems in the research of distributed fiber shape sensing technology were summarized and discussed. On this basis, the research prospect of distributed fiber shape sensing technology is prospected.
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图 2 ShapeTape的重建弯曲轮廓[25]
图 3 集成了光纤形状传感器后的Ion系统进入肺部神经末梢的过程[33]
图 4 柔性结构空间形变测量装置和结果[36]
图 5 柔性基材曲线重构结果[37]
图 6 基于七芯MCF的分布式BOTDA系统原理框图[39]
图 7 七芯D形光纤剖面结构[42]
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