基于光纤传感器的平面曲线重建算法建模

Modeling of Plane Curve Reconstruction Algorithm Based on Optical Fiber Sensor

摘要: 为推动光纤传感技术的多领域应用，解决其平面曲线重建建模的核心难题，本文构建了基于四阶龙格–库塔法的平面曲线重构模型，提出了一套完整的曲率计算与曲线拟合方案。针对离散测点连续曲率分布的获取问题，首先依据曲率与波长的关联式，借助传感器波长测量数据，计算出两个初始状态下FBG1-FBG6的曲率；采用三次样条插值法拟合曲率曲线，经SPSS软件验证，模型相关系数R²均近似为1，精准明确了曲率与横坐标x的量化关系，并实现对应位置曲率的可靠估算。为完成平面曲线的精准重构，基于上述曲率数据构建微分方程，代入曲率函数后采用四阶龙格–库塔法求解微分方程近似解，通过Python编程迭代得到重构平面曲线，并分析了重构曲线的核心特征。此外，为优化曲线重建的采样策略并探究误差来源，以平面曲线y = x³ +x (0 ≤ x ≤ 1)为研究对象，通过Python程序计算其弧长，设定采样点数量n = 10并确定各采样点的横坐标与弧长间隔；求解各采样点的一、二阶导数，结合曲率公式计算曲率值，再通过迭代法求解弧长、切线方向变化量等参数以补充坐标点，最终绘制重构曲线与原曲线的对比图像。分析表明，重构过程中的误差主要来源于采样设计、离散数据处理、曲率近似计算及数值求解等环节。

Abstract: To promote the multi-field applications of fiber optic sensing technology and address the core challenge of planar curve reconstruction modeling, this paper constructs a planar curve reconstruction model based on the fourth-order Runge-Kutta method and proposes a complete set of solutions for curvature calculation and curve fitting. To obtain the continuous curvature distribution from discrete measurement points, first, based on the correlation between curvature and wavelength, the curvatures of FBG1-FBG6 in two initial states are calculated using the sensor’s wavelength measurement data. Then, the cubic spline interpolation method is employed to fit the curvature curve. Verified by SPSS software, the model’s correlation coefficient R² is approximately 1, accurately defining the quantitative relationship between curvature and the horizontal coordinate x, and enabling reliable estimation of curvature at corresponding positions. To achieve precise reconstruction of the planar curve, a differential equation is constructed based on the above curvature data. After substituting the curvature function, the fourth-order Runge-Kutta method is used to solve the approximate solution of the differential equation. Through Python programming iteration, the reconstructed planar curve is obtained, and its core characteristics are analyzed. Additionally, to optimize the sampling strategy for curve reconstruction and explore error sources, the plane curve y = x³ + x (0 ≤ x ≤ 1) is taken as the research object. Using a Python program, its arc length is calculated, the number of sampling points n = 10 is set, and the horizontal coordinates and arc length intervals of each sampling point are determined. The first and second derivatives at each sampling point are solved, and curvature values are calculated in combination with the curvature formula. Then, iterative methods are used to solve parameters such as arc length and tangent direction changes to supplement coordinate points, ultimately drawing a comparative image of the reconstructed curve and the original curve. Analysis shows that the errors in the reconstruction process mainly originate from sampling design, discrete data processing, approximate curvature calculation, and numerical solution.