## 4 Aquatic Locomotion Model
Moreover, in this configuration, the centrifugal force due to the rotating mass is pointing upwards, so the submerged part of the feet is less than in the static equilibrium configuration (rotating mass not moving). In the screenshot (e) (which is a dual configuration with respect to (a)), the rotating link is vertical and pointing downward and the feet of the robot are moving backward. In this configuration, the maximum (positive, i.e., directed forward) horizontal drag forces are applied to the feet of the robot (see \(F_{f,x}\) and \(F_{h,x}\) at \(t=0.17\) s in Figure 7). In this configuration, the centrifugal force due to the rotating mass is pointing downward, so the submerged part of the feet is more than in the static equilibrium configuration. Therefore, the submerged area of the feet subject to horizontal drag forces is higher than in (a), and this contributes to have a higher value (in modulus) of the forward force with respect to the backward force related to (a). Nevertheless, the main reason why the absolute value of the maximum forward force (for each foot) is higher than the absolute value of the minimum (backward) force is because \(c_{x,b}>c_{x,f}\). \[F_{f,x}=-\frac{1}{2}\rho_{w}c_{x,h}L_{p}y_{f}\dot{x}_{f}^{2}+F_{f,l},\text{ if }\dot{x}_{f}<0\]2 Screenshots of robot configuration for a WM simulation with \(\omega\) = 24 rad/s. Similarly, in all the other configurations, the robot motion is dominated by the corresponding direction of the centrifugal force generated by the rotating mass. In particular, in configuration (b), the minimum value of \(F_{f,y}\) is reached, as it can be verified in Figure 7 for \(t=0.08\) s and, indeed, the direction of the centrifugal force (i.e., the direction of the rotating link) is such that it reduces the vertical load on the front foot (and, thus, reduce the buoyancy force and \(F_{f,y}\)). In configuration (c), the maximum backward inclination of the robot (about \(-2.5\) deg) is reached and the centrifugal force is horizontal so that the maximum destabilizing torque on the robot is achieved. In configuration (d), the maximum value of \(F_{h,y}\) is reached, as it can be verified in Figure 7 for \(t=0.14\) s, and the direction of the centrifugal force (i.e., the direction of the rotating link) is such that it increases the vertical load on the hind foot (and, thus, increases the buoyancy force and \(F_{h,y}\)). Exactly the same considerations can be done for configurations (f), (g), and (h), in which the centrifugal force makes the vertical load on the front foot increase (together with the related buoyancy force), makes the robot achieve its maximum forward inclination (about \(-2.5\) deg), and makes \(F_{h,y}\) (and the related buoyancy force) reach its minimum value, respectively. It can be concluded that the main reason for robot net forward motion is that \(c_{x,b}>c_{x,f}\) due to the asymmetric shape of the feet. In the periodic forward-backward motion of the feet with respect to the water (due to the centrifugal force of the eccentric mass that pulls forward and backward the robot), a positive integral force due to horizontal drag forces allows the robot to advance. This can be easily verified looking at the profile of \(F_{f,x}\) and \(F_{h,x}\) in \[F_{f,x}=-\frac{1}{2}\rho_{w}c_{x,h}L_{p}y_{f}\dot{x}_{f}^{2}+F_{f,l},\text{ if }\dot{x}_{f}<0\]3 The area of the positive part of the plot is higher than the area of the negative part of the plot for both feet. This phenomenon is similar to the one exploited in vibration conveying [48], in which the inclination of the inertia force caused by vibrations increases the static friction force in a direction and decreases the static friction force in the opposite direction. The dynamic behavior of the robot using a constant motor torque as an input has also been investigated. In particular, simulations have been carried out with torques (directed counterclockwise) from 0.00001 Nm to 0.01 Nm. For low torque values, i.e., from 0.00001 to 0.00004 Nm, the torque is not sufficient to rotate the eccentric mass.