﻿ 半径确定的最小二乘圆拟合法在电缆卷绕监测中的应用

# 半径确定的最小二乘圆拟合法在电缆卷绕监测中的应用Application of Radius-Determined Least Squares Circle Fitting Method in Cable Winding Monitoring

Abstract: Aiming at the problem of using only the least square method in the monitoring of the cable winding state and using insufficient cable specification information, an improved circle fitting method is designed: the least squares fitting method for radius determination. The effects of sensor angular resolution, cable diameter, cable outer surface material and other factors on the fitting accuracy of cable relative position were studied experimentally. The experimental results show that the main factor affecting the fitting accuracy is the accuracy of the measured data. For the cable with common black rubber appearance, when the random measurement error of the sensor is less than 7 mm, the algorithm can ensure that the fitting error of the relative position of the cable is less than 1.5 mm.

1. 引言

2. 数学原理

2.1. 一般的最小二乘圆拟合法

${\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}={r}_{0}^{2}$ (1)

$F\left({x}_{0},{y}_{0},{r}_{0}\right)=\underset{i=1}{\overset{N}{\sum }}{\left[{\left({x}_{i}-{x}_{0}\right)}^{2}+{\left({y}_{i}-{y}_{0}\right)}^{2}-{r}_{0}^{2}\right]}^{2}$ (2)

2.2. 半径确定的最小二乘圆拟合法

$G\left({x}_{0},{y}_{0}\right)=\underset{i=1}{\overset{N}{\sum }}{\left[{\left({x}_{i}-{x}_{0}\right)}^{2}+{\left({y}_{i}-{y}_{0}\right)}^{2}-{R}_{0}^{2}\right]}^{2}$ (3)

Figure 1. Comparison of fitting effects

3. 实验验证与分析

3.1. 角度分辨率对拟合精度的影响

(a) (b)(c)

Figure 2. Cable winding simulation diagram. (a) The experiment site; (b) Tubes with different surface materials; (c) Different distances for testing

Table 1. Relationship between sensor random error and reflectivity

Table 2. Relative position fitting errors at different angular resolutions

Figure 3. Fitting error of different angular resolutions

3.2. 电缆直径对拟合精度的影响

Figure 4. Fitting error of different cable diameters

Table 3. Relative position fitting errors for different cable diameters

3.3. 外表面材料对拟合精度的影响

Table 4. Relative position fitting errors of different outer surface materials

Figure 5. Fitting error of different outer surface materials

Figure 6. Reflectivity of different outer surface materials

Table 5. Comparison of fitting precision between white PVC and black rubber

4. 总结

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