## 2 Materials and Methods  
### Structural Design of Fiber-Constrained Flexible Actuator  
The fiber-constrained actuator generally consisted of an elastomer (with an embedded chamber), a strain limiting layer and a restraining fiber (Figure 1a). The elastomer deformed under a certain pressure to complete the bending movement by limiting the deformation of the actuator surface through the strain limiting layer. The restraint fiber limited the radial expansion of the elastomer, making the actuator efficient in changing the pressure into the bending deformation of the actuator, and thereby reducing its expansion volume. More complex movements of the actuator could be achieved by varying the density and angle of the restraining fibers.  
A fiber-constrained annular flexible actuator (Figure 1b) was designed based on the fiber-constrained flexible actuator drive principle. In this actuator, the left and right ventricular side chambers were used to squeeze the left and right ventricles, and a strain limiting layer was attached to the outside. The ventricular septal side chamber was used to adjust the deformation of the annular actuator, and the strain-limiting layer was attached to the outside of the chamber, while the constraint fibers were wound around it. Then, this actuator was arranged around the heart and secured with a silicone shell in order to obtain a fiber-constrained direct ventricular assist device (Figure 1c).  
According to the shape and size of human heart and the existing research theories of fiber-constrained actuators, the structures and parameters of three kinds of the annular flexible actuator were determined (Figure 1d).  
Some studies showed that when the same pressure was applied to the fiber-constrained pneumatic flexible actuator, the thinner the balloon wall was, the larger the bending angle became. In order to ensure the strength of the flexible actuator, the wall thickness of the chamber t = 1 mm was selected. In this study, the radius \(r_{1}\) of actuator 1 and the angle \(\alpha_{1\text{-}1}\) of the ventricular chamber satisfied the following relationship (Equation (1)):  
\[2r_{1}\sin(\frac{\alpha_{1}-1}{2})=h_{1}-2\delta \tag{1}\]  
where \(h_{1}\) is the length of the ventricular septum at the point where it is wrapped by the annular flexible actuator 1 as \(h_{1}=57\) mm, and \(\delta\) is the thickness of the left ventricular wall as \(\delta=9\) mm.  
The calculation showed that \(\alpha_{1\text{-}1}=58^{\circ}\). As the left ventricle resembled a cone, actuator 2 and actuator 3 were reduced in equal proportions, and their lateral ventricular chamber  
Figure 1: Structural design of fiber-constrained flexible actuator. (**a**) Fiber-constrained flexible actuator; (**b**) fiber-constrained annular flexible actuator; (**c**) arrangement of fiber-constrained annular flexible actuators and direct ventricular assist devices; (**d**) structural parameters of the fiber-constrained annular flexible actuator, where \(\alpha_{1}\) is the ventricular balloon circumferential angle, \(\alpha_{2}\) is the septal balloon circumferential angle, \(r\) is the actuator radius, a is the cross-sectional width, b is the section length, t is the balloon wall thickness, \(r_{0}\) is the balloon circumferential radius, \(\beta\) is the actuator inclination angle, \(\gamma\) is the fiber inclination angle and \(\rho\) is the fiber spacing.  
rounding angle were rounded at the same angles \(\alpha_{1\text{-}2}\) and \(\alpha_{1\text{-}3}\) as \(\alpha_{1\text{-}1}\), such that \(\alpha_{1\text{-}2}=\alpha_{1\text{-}3}\) = \(\alpha_{1\text{-}1}=\alpha_{1\text{-}1}=58^{\circ}\).  
The three annular flexible actuators dimension parameters are shown in Table 1; the other parameters (fiber spacing, fiber inclination, interventricular septum side balloon rounding angle, actuator cross-sectional width) were to be optimized in Section 2.2.