# Bio-Inspired Design of an Underwater Robot Exploiting Fin Undulation Propulsion
## 2 Materials and Methods
### Kinematic Synthesis of the Fin Mechanisms  
The mechanisms actuating the fins were designed with the objective of accurately mimicking the fin deformation that is responsible for thrust generation. The motion of the fin is a travelling wave with an amplitude increasing towards the tip. Since the number of waves on the fin is about 0.4 [4], three mechanisms are enough to reproduce the travelling wave, and they are positioned near the head, in the centre, and near the tail [9]. The equations representing the deformation of the fin can be obtained from the work of [PERSON] et al. [20], who analysed the skeletal structure of a cownose ray and related it to the kinematics of the fin. Assuming that the deformation is continuous along the fin, the equations representing the deformation of a section of the fin are as follows:  
\[y(s,t)=\frac{1-\cos\left[\theta_{max}s\sin\left(\phi x-\omega t\right)\right] }{\theta_{max}\sin\left(\phi x-\omega t\right)}, \tag{1}\]  
\[z(s,t)=\frac{\sin\left[\theta_{max}s\sin\left(\phi x-\omega t\right)\right]}{ \theta_{max}\sin\left(\phi x-\omega t\right)}, \tag{2}\]  
where \(y\) and \(z\) are depicted in Figure 3, \(\omega\) is the frequency of fin flapping, \(\phi\) is the wave number along the fin, \(\theta_{max}\) is the angle formed by the fin at its tip, and \(s\) is the curvilinear abscissa along the fin span. Figure 3: Reference system on the fish. Figure 2: (**a**) Dimensions of the external surface of the robot and (**b**) cross sections of a fin. A kinematic diagram of the mechanism actuating the central part of the fin is presented in Figure 4. The four-bar linkage actuated by the motor is connected to the first link, which moves the second and third links through sliders and hinges. In the first stage, the four-bar linkage O-H-K is not considered in kinematic synthesis of the mechanism. The lengths of the links are selected using an optimization method to approximate as accurately as possible the deformed shape of the fin using the minimum area error method [21]. This method consists of finding the optimal combination of link lengths that minimize the area between the real deformed curve and the shape obtained with the three links. The optimization procedure is performed considering an adimensional problem and considering the overall length of the fin span in the central section. The area below the real curve is obtained by integrating Equation (1) along \(z\):  
\[A=\frac{z_{max}}{\theta_{max}\sin\left(\phi x-\omega t\right)}\mp\frac{1}{2} \Bigg{(}z_{max}\sqrt{\frac{1}{(\theta_{max}\sin\left(\phi x-\omega t\right))^ {2}}-z_{max}^{2}}\pm\frac{\arcsin\left(z_{max}\theta_{max}\sin\left(\phi x- \omega t\right)\right)}{(\theta_{max}\sin\left(\phi x-\omega t\right))^{2}} \Bigg{)}. \tag{3}\]  
The area below the links, instead, is as follows:  
\[\tilde{A}=\frac{L_{1}^{2}\sin\alpha\cos\alpha}{2}+\frac{(2L_{1}\sin\alpha+L_{2 }\sin\beta)L_{2}\cos\beta}{2}+\frac{(2L_{1}\sin\alpha+2L_{2}\sin\beta+L_{3} \sin\gamma)L_{3}\cos\gamma}{2}. \tag{4}\]  
The angle \(\alpha\) is a sinusoidal function of time, and the variable \(\theta\) can be introduced as a cyclical variable. The amplitude of the oscillation of \(\alpha\) is unknown, and it is determined together with the other parameters of the optimization:  
\[\alpha=X\sin\left(\phi x-\omega t\right)=X\sin\theta. \tag{5}\]  
The angles \(\beta\) and \(\gamma\) can be written as functions of \(\alpha\):  
\[\beta=\arctan\left(\frac{L_{1}\sin\alpha}{L_{1}\cos\alpha-a}\right), \tag{6}\]  
Figure 4: Kinematic scheme of the mechanism actuating the central part of the fin.