## 3 Principle of Sensing and Measurement of Embedded FBGs  
According to the sensing principle of embedded FBG sensor, the relationship between the reflected wavelength and the grating period of FBG sensor can be expressed as [5; 10]:  
\[\lambda_{B}=2n_{eff}\cdot\Lambda \tag{10}\]  
where \(n_{eff}\) represents the effective index of the fiber's refractive, \(\Lambda\) represents the grating period. The grating period of FBG sensor will be changed due to the changes in the strain of the grating and the temperature, and then the reflected wavelength will be changed. The relation between the strain, temperature change, and the wavelength shift is governed by [6; 7]:  
\[\Delta\lambda_{B}/\lambda_{B}=(1-P_{\varepsilon})\cdot\varepsilon+(\alpha_{T} +\xi)\cdot\Delta T \tag{11}\]  
where \(\Delta\lambda_{B}\) is the change of wavelength, \(\varepsilon\) is the strain of soft actuator, \(P_{\varepsilon}\) is a constant value calculated from the photo-elastic coefficient of the fiber material, \(\alpha_{T}\) is the grating thermal coefficient of expansion, \(\xi\) is the grating thermo-optical coefficient, \(\Delta T\) is the change of temperature. Considering the soft surgical actuators are commonly used at room temperature with minor variation, so the influence of the temperature is ignored in this paper. Then Equation (11) can be written as Equation (12). The strain of soft actuator can be obtained from the wavelength shift of the FBGs and the photo elastic coefficient of the optical fiber [19; 20; 21; 22];  
\[\Delta\lambda_{B}/\lambda_{B}=(1-P_{\varepsilon})\cdot\varepsilon \tag{12}\]  
In this study, two FBG sensors were integrated into the polyimide strain limited layer which was embedded into the soft actuator, the relative position and bending deformation of the FBG sensors in a microstructure of the soft actuator under an actuating force is schematically shown in Figure 5. The length and thickness of the microstructure is assumed as \(L\) and \(h\), as Figure 5a shows. The pure bending model is an extensively used model representing the bending moment [10]. In this paper, an optical fiber with two FBG sensors embedded in an offset position from the neutral line, which is depicted in Figure 5b, is the dashed line E-F. The distance from the FBG sensor to the outer arc of the soft silicone sheet is \(h_{1}\), and the distance from the FBG sensor to the inner arc of the soft silicone sensing sheet is \(h_{2}\). When the optical fiber sensor is bending, as shown in Figure 5b,  
Figure 4: 3D printed mold of the soft actuator. the relationship between the bending radius, bending angle and the length of the neutral line can be expressed as:  
\[L_{EF}=R\cdot\theta \tag{13}\]  
where \(L_{EF}\) represents the constant value of the length of the polyimide tape in neutral line, \(R\) represents the radius of the neutral line, \(\theta\) represents the central angle of the neutral line. As the pneumatic actuating forces are applied on the soft actuator, the inner side of the micro structure is pressed, shortening the length, and the outer side of the microstructure is pulled and extending the length. Under the bending condition, the length variation of both inner and outer side is assumed as the same, and the length of the neutral layer does not change. In this paper, the FBG sensors and the polyimide sheet are considered as undergoing the same bending, due to the optical fiber being very thin and glued together with the polyimide tape. The FBG sensor is compressed during bending, so the offset location (\(x\)) of the FBG sensor leads to a slight change in the bending radius and is expressed as \(R-x\), the length of FBG (\(L_{AB}\)) under an applied curvature can be written as:  
\[L-\Delta L=L_{AB}=(R-x)\cdot\theta=(R-(h_{1}+h_{2}/2))\cdot\theta \tag{14}\]  
\[L+\Delta L=L_{GH}=(R+x)\cdot\theta=(R+(h_{1}+h_{2})/2)\cdot\theta \tag{15}\]  
\[x=h_{1}-(h_{1}+h_{2})/2 \tag{16}\]  
where \(x\) represents the distance from the FBG to the neutral line.