## 2 Design and Modeling of Soft Actuator with Embedded FBG Sensors  
### Structure Design and Analytical Modeling  
Through adjusting some dimension parameters of the soft actuator, including its length, the diameter of the hemi-circle chamber, the pneumatic chamber's wall thickness, the fiber-winding angle and orientation, etc., The design details of the soft actuator are presents in Figure 1[16]. The dimensions of the hemi-circular shape are achieved (see Figure 2a) using the cross-sectional area of \(\pi a^{2}/2\), and the wall thickness of the actuator is \(t=a/4\). With an input air pressure of \(P_{in}\), the bending torques (\(M_{a}\)) of internal air pressure relative to the closed port of the soft actuator are expressed as [17; 18]:  
\[M_{a}^{HC}=0.34a^{3}P_{in} \tag{1}\]  
Considering the characteristics of the hyper-elastic material of silicone and the dimension of the soft actuator; the relationship between the input pressure and the bending angle were analyzed. The variables in the model were actual actuator dimensions and material properties, that could be either measured or obtained from calibrations. From the model of incompressible Neo-Hookean (NH) material [4], the strain energy of the soft actuator can be written as:  
\[W=\frac{\mu}{2}(I_{1}-3)=\frac{\mu}{2}\Big{(}\lambda_{1}^{2}+\lambda_{2}^{2}+ \lambda_{3}^{2}-3\Big{)} \tag{2}\]  
where \(\mu\) represents the initial shear modulus of the silicone, \(I_{1}\) represents the first constant of axial, circumferential and radial principal. \(\lambda_{1}\) represents the primary stretch in the axial direction of the actuator, \(\lambda_{2}\) (=1) represents the circumferential strain, and it was ignored due to the fiber reinforcement constraint, and \(\lambda_{1}\lambda_{2}\lambda_{3}\) = 1 due to the incompressibility of the material, then it was obtained that:  
\[\lambda_{1}=\lambda\ \,\ \ \lambda_{2}=1\ \,\ \ \lambda_{3}=\frac{1}{\lambda} \tag{3}\]  
When supplying the pressure (\(P_{1}>P_{atm}\)) into the air chamber, the top wall of the actuator extended and the non-retractable layer restricted its bottom layer, the actuator bent with a radius \(R\) and angle \(\theta\) toward the bottom layer (see Figure 2). In addition, the fiber-reinforced structure was regarded as a  
Figure 1: Design of the proposed Fiber Bragg Grating (FBG) soft pneumatic actuator. (**a**) FBG-based polyimide sensing layer and its location. (**b**) Geometrical parameters and design variables of soft fiber-reinforced actuator. rigid constraint to the soft actuator, and the actuator was simplified as a homogeneous incompressible NH material [4] whose initial shear modulus is \(\overline{\mu}\). In the model of the actuator, some dynamics pressurization effects were ignored, and the actuator had an identical bending curvature. The nominal main strain \(s_{i}\) can be defined by \(W\), \(\lambda i\). Where \(p\), the Lagrange multiplier is:  
\[s_{1}=\frac{\partial W}{\partial\lambda_{1}}-\frac{p}{\lambda_{1}}=\overline{ \mu}\bigg{(}\lambda-\frac{1}{\lambda^{3}}\bigg{)} \tag{4}\]  
\[s_{2}=\frac{\partial W}{\partial\lambda_{2}}-\frac{p}{\lambda_{2}}=\overline{ \mu}\bigg{(}1-\frac{1}{\lambda^{2}}\bigg{)} \tag{5}\]  
\[s_{3}=\frac{\partial W}{\partial\lambda_{3}}-\frac{p}{\lambda_{3}}=0 \tag{6}\]  
Within the range of stretches considered in soft actuator application (\(1\leq\lambda<1.5\)), the axial stretch \(s_{1}\) this paper only considered was denoted as \(s\). The bending moment was induced by the internal strain of the soft actuator, the bending torque and the combined moment at each bending state can be expressed as:  
\[M_{a}=2(P_{1}-P_{atm})\int\limits_{0}^{\frac{\pi}{2}}(a\sin v+b)a^{2}\cos^{2} vdv=\frac{4a^{3}+3\pi a^{2}b}{6}(P_{1}-P_{atm}) \tag{7}\]  
\[M_{\theta}=\int\limits_{0}^{b}2s_{\beta}(a+t)L\beta d\beta+2\int\limits_{0}^{ t}\int\limits_{0}^{\frac{\pi}{2}}s_{\tau,\phi}\Big{(}(a+\tau)^{2}\sin\phi+b(a+ \tau)Ld\phi\Big{)}d\tau \tag{8}\]  
where \(M_{a}\) is the strain of the soft actuator, \(M_{\theta}\) is the combined moment of the strain.