## 3 Experimental Results  
### Experimental Setup  
To investigate the effect of the underactuated tail on both the speed of locomotion and energy efficiency of the fishbot, we conducted experiments with different degrees of underactuation, i.e., with different values of the maximum angle of underactuation \(\delta\). The experimental process with the setup value of the maximum angle of underactuation consists of running four runs (to account for the effect of environmental noise) of the fish robot with a different frequency (parameter \(f\) in Equations (1) and (2)) at the maximum amplitude of undulation of both servo motors (i.e., parameters \(A_{1}\) and \(A_{2}\) in Equations (1) and (2) are set to \(45^{\circ}\)) and no phase shift between the undulations of both servo motors (i.e., the parameter \(\beta\) in Equation (2) is set to 0). We have verified in our previous research that these values of amplitudes and the phase shift do indeed result in the maximum speed and energy efficiency of the robot [3; 12; 13]. We have tested five different frequencies ranging from 0.4 Hz to 2.0 Hz with a discretization step of 0.4 Hz. The maximum frequency of oscillation is set to a value that is close to but does not exceed the technical specification of the maximum angular speed (\(600^{\circ}\)/s) of the servo motors. The maximum angular velocity \(\omega_{MAX}\) that servo motors sustain during the undulation could be obtained from the maximum value of the derivative of the turning angles of undulation, as expressed in Equations (1) and (2):  
\[\omega_{max}=2\,\times\,\pi\,\times f\,\times A \tag{7}\]  
For \(f=2.0\) Hz and \(A=45^{\circ}\), we obtain \(\omega_{max}=565^{\circ}\)/s, which is well within the technical specification of the servo motors, with a safety margin (to \(600^{\circ}\)/s) of about 6%.  
Figure 6: Implementation of angular limiter of the limited underactuated joint of the tail fin.  
Figure 7: Kinematic diagram of the fishbot with a limited underactuated joint of the tail fin.  
The results of the four runs are aggregated based on the calculated mean of the achieved performances (e.g., speed and energy efficiency). The experimental system comprises the (i) fishbot (ii) and its control system, as illustrated in Figure 8.  
The control system consists of an abstract layer (AL) that generates the control signals with a sampling interval of 20 ms according to Equations (1) and (2), and a physical layer (PL) that converts the instant values of the turning angles of the servo motors to pulse-width-modulated (PWM) signals that govern the instant angular positions of the motors. Also, we connected a current sensor (CS) serially to the power supply (\(Ucc=5\) V) line of the robot in order to estimate the electrical power consumption of the robot during the trials. For hardware implementation of the PL and CS, we used Micro Maestro 6-Channel USB Servo Controller and ACS724 (both produced by Pololu Robotics and Electronics), respectively [20,21].  
In addition to the fishbot and its control system, the experimental setup also involves an optical tracker, which uses a video feed from an overhead camera to track the fishbot (via a bright red LED mounted on the latter) and estimate the instant speed (as the Euclidean distance between the consecutive positions of the robot, divided into precisely measured time intervals between these positions) of the robot during the 5 s trial. The maximum value of the instant speed is assumed to be the actual steady-state swimming speed of the fishbot. The details of the experimental setup are elaborated in [3].