## 2 Materials and Methods
### The Concept of Limited Underactuation  
Unlike actuated joints, underactuated joints do not use any actuators (servomotors). This kind of joint can be compared to a completely loose (free) joint that will move according to the environmental forces exerted on it. Our approach consists of implementing a new joint on the fishbot tail segment in order to reduce the hydrodynamic drag during the above-mentioned braking phase of undulation (Figure 3c). Also, because the underactuated joint is an environment-dependent one, its implementation may prevent the tail fin from being opposed to the direction of water flow. However, even if complete looseness might result in lower hydrodynamic drag during the braking phase of undulation (Figure 3c), it would also result in lower thrust in the initial phase of undulation (Figure 3a).  
From this standpoint, we hypothesize that a joint with limited freeness may allow both (i) a reduction in drag during braking and (ii) the production of enough thrust during thrusting to possibly favorably shift the balance of these two phases, yielding an overall increase in the propulsion force. Therefore, we have designed a \"limited underactuated\" tail joint that is allowed to move freely but only within a limited angular range. The angular limiter defines the maximum angle of underactuation \(\delta\) (Figures 4 and 5) that could be achieved by the joint while moving depending on the environmental conditions. Moreover, the angle \(\delta\) could be manually adjusted for the purpose of investigating its optimal value from the viewpoint of achieving a maximum speed or maximum energy efficiency of locomotion. However, it is fixed during undulation.  
We can compare the behavior of the tail fin featuring the proposed limited underactuated joint (Figure 4) with that of the canonical fully actuated one (Figure 3). The main concern that we intend to address is the unwanted hydrodynamic drag during the braking phase (Figures 3c and 4c) caused by the stall of the fin due to the higher angles of attack of the latter. Regarding the expected behavior of a single undulation with a limited underactuated joint (Figure 4c), the braking phase is associated with an orientation of the tail fin mainly along the water flow (and, consequently, with an overall reduction in the angle of attack), which we expect to result in a reduction in the hydrodynamic pressure drag. As undulation has the same period for both analyses, we can conclude that the lower angle of attack would result in both (i) a delayed (during the thrust phase of the undulation as illustrated in Figure 4a,b) and (ii) shorter and less severe (during the braking phase, as shown in Figure 4c) stall of the tail fin. However, because, compared to the fully actuated joint, the reduced angle of attack due to the limited underactuation--according  
Figure 4: Phases of undulation of fishbot featuring a limitedly underactuated posterior joint. \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{a}\), and \(\delta\) denote the angular deviation of the two joints, the angle of attack of the tail fin, and the maximum angle of the limited underactuation, respectively. The phases of undulation feature reduced values of the angle of attack compared to the analogical phases of undulation of the fully actuated fishbot (Figure 3). Step **d** represent the gesture and position of the fishbot body when a new phase of undulation is starting.  
to Equation (4)--might also reduce the effectiveness of the thrusting phase of undulation (Figures 3a and 4a), we are unsure about whether (depending on the angle of underactuation \(\delta\)) the limited actuation would result in increased net propulsion.