# Bio-Inspired Design of an Underwater Robot Exploiting Fin Undulation Propulsion
## 2 Materials and Methods
### Dynamic Analysis and Motor Sizing
#### 2.4.2 Multibody Analysis  
Once the forces acting on a fin are known, they can be used to investigate the dynamics of the robot. The objective of this multibody analysis is to understand the maximum torque required by motors to produce this travelling wave on the fin, so that a propulsive force is generated. Including both fins in the analysis, it is possible to investigate forward swimming and turning maneuves. If the wave travels in the same direction on both fins, the generated forces are equal and the robot swims in a straight direction. Conversely, if the waves on the two fins travel in opposite directions, the generated force are opposite and a turn can be achieved, as represented in Figure 14.  
In this multibody analysis, the motion laws of the robot fins are prescribed: there is a phase shift between the three mechanisms consistent with the travelling wave present in the CFD analysis, which features a frequency of \(1\,\mathrm{Hz}\) and a wavelength of 0.4 body lengths per second.  
The motors move the links in a vertical and in a span-wise direction, and there is no relative motion of the links with respect to the base in a longitudinal direction. Therefore, only vertical and span-wise forces have an effect on motor torque; the propulsive force acting in a longitudinal direction is only a consequence of the vertical and span-wise deformation of the fin. Hence, the yaw rotation, which arises as a result of the application of two opposite longitudinal forces, does not directly affect the motor torque.  
Figure 14: Travelling waves directions and propulsive forces generated by the fin motion.  
Figure 13: Subdivision of the fin in smaller regions and computation of the forces.  
The forces and moments acting on each region of the fin are presented in Figures 15 and 16, where the number of regions correspond to the numbers indicated in Figure 13. The presented forces and moments refer to the right fin, the one present in the simulation; on the left fin, the vertical force and the pitch moment are equal, whereas the force on the span-wise axis and the roll moment have an opposite sign.  
The force acting in the vertical direction has the same frequency as the fin motion; conversely, the frequency of the force acting in the span-wise direction is double because, when the fin tip approaches both the maximum positive amplitude and minimum negative amplitude, the fluid is pushed towards the central body of the robot and it is moved outwards when the fin returns to the median position.  
The resulting torque required by each motor and the power consumed to move each fin are plotted in Figure 17.  
Figure 16: Moments acting on each mechanism.  
Figure 17: Torque required by the motors (**a**) and power consumed by the robot (**b**).  
Figure 15: Forces acting on each mechanism.  
Thanks to the phase shift between the three motors, the peaks of requested torque are not simultaneous. This is advantageous in terms of instantaneous power, which remains smaller than it would have been with no phase shift. The servomotors are chosen so that their maximum torque limit is never exceeded. The selected motors are Feetech FB5311m, able to generate a torque of 2 N m rotating at 60 rpm; the controlled variable is velocity, and they provide an angular position feedback.