# Analysis of Distributed Measurement Method for Array Antenna Position
## 2 Scheme and Principle
### Fiber Bragg Grating (FBG) Sensing Technology  
According to the different characteristics of the material and study from the research of [PERSON] et. al. [15], the classification of optical fiber sensors is shown in Figure 2.  
The FBG sensor has been applied in different engineering fields for its advantages of light weight, small volume and anti electromagnetic interference [15; 16; 17; 18], and FBG sensing technology is used in the paper. The resolution of deformation displacement measurement of FBG sensing technology can reach 0.3 mm. The sensing schematic diagram is shown in Figure 3.  
In the calibration experiment [19], we used SM130 interrogation (Micron Optics, Inc., Atlanta, GA, USA) and the Electro-mechanical Universal Testing Machines (EUTM) (S  
Figure 3: Schematic diagram of working principle of fiber Bragg grating (FBG) sensor. Figure 2: Classification of optical fiber strain sensors [15]. stock Co., Ltd, Shenzhen, China) to test the strain transfer characteristics of the FBG sensor, and the result showed that the strain transfer characteristics of the FBG sensor and strain gauge have good consistency. The researchers found that the appropriate type of interrogation can realize aircraft wing deformation monitoring based on the FBG sensor in flight, that is to say, the interrogation can normally obtain the data of the FBG sensor in flight state [20,21]. The working principle of the FBG sensor on the aircraft wing is to use the bending deformation of the wing, and the grating period or the refractive index of the core will change, so that the central wavelength of the Bragg grating will change. Through the mathematical model, we can obtain the change of physical parameters such as stress and strain, and then get the displacement change of each node of the FBG sensor. The wavelength of the FBG sensor obeys the following formula [22]:  
\[\lambda_{B}=2n_{eff}\Lambda \tag{1}\]  
where, \(\lambda_{B}\) is the center wavelength of the fiber grating, \(n_{eff}\) is the effective refractive index of the fiber grating area, \(\Lambda\) is the period of refractive index in grating region. The main influencing factors of the center wavelength offset of the FBG sensor on the aircraft wing are temperature and strain, and the relationship is as follows:  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=(1-P_{e})\varepsilon+(\alpha+\zeta)\Delta T \tag{2}\]  
where, \(P_{e}\) is the photoelastic coefficient, \(\varepsilon\) is strain, \(\alpha\) is the coefficient of thermal expansion, \(\zeta\) is the thermal optical coefficient, \(\Delta T\) is the temperature change. There are many methods of temperature compensation [23]. In order to study the relationship between strain and wavelength, the paper assumes that only constant temperature measurement is considered, so the formula can be simplified as follows:  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=(1-P_{e})\varepsilon \tag{3}\]  
Assuming that the deformation of the wing is pure bending, and the wing is composed of several micro elements, according to the pure bending theory of flexible beam, the transformation relationship between the curvature and strain of each node can be expressed as:  
\[k=1/\rho=\frac{2}{h}\varepsilon \tag{4}\]  
where, \(k\) is the curvature, \(\rho\) is the radius of curvature, \(h\) is the thickness of the micro element. Therefore, the relationship between the curvature \(k\) and the wavelength variation \(\Delta\lambda_{B}\) can be expressed as:  
\[k=\frac{2}{h}\frac{\Delta\lambda_{B}}{\lambda_{B}(1-P_{e})} \tag{5}\]  
The \(\lambda_{B}\), \(P_{e}\) and \(h\) of FBG sensors encapsulated on aircraft wings are all constant, so the curvature \(k\) is linear with \(\Delta\lambda_{B}\). The flexible plane of an aircraft wing can be regarded as a collection of multiple plane curves. The reconstruction of deformed wing surface can be realized by continuous reconstruction of multiple plane curves. First, the relationship between curvature and arc length is determined.