# A Novel Integrated Gliding and Flapping Propulsion Biomimetic Mana-Ray Robot
## 2 The Design of the Manta Ray Robot
### The Flapping Propulsion System Design  
The gliding propulsion has the advantages of high efficiency, energy-saving, and environmental friendliness. However, the maneuverability and stability of the gliding propulsion are not good. When the robot needs high maneuverability and stability, the flapping propulsion method makes up for it. In this section, we design the pectoral fin structure and analyze the flapping propulsion method.  
The mechanism schematic is described in Figure 5a, and the three-dimensional model of the pectoral fin is shown in Figure 2d. The flexible pectoral fin comprises driving steering gears 1 and 2, a secondary link mechanism, flexible carbon fiber fin rays, and an elastic skin [36]. When the pectoral fin flaps, the motors drive the steering gear to make the driving rod swing back and forth along the spanwise direction. The central pattern generators (CPGs) generate the motors' motion law. Controlled by the CPGs, the flexible flapping wing has chord-wise wave transmission and spanwise swing, which imitate the movement of the real manta ray. Symmetrical pectoral fin flapping enables the robotic fish to generate chord-wise power. When the pectoral fins flap asymmetrically, causing a yaw force, this allows the robotic fish to complete heading control (Figure 5b).  
The pectoral fins are driven by CPGs, which are implemented as a network of coupled nonlinear oscillators [37,38]. Figure 5a shows the topology of the CPG network and the behavior of the output of each oscillator during flapping propulsion. There is a single oscillator for each motor. The first oscillator (CPG 1) is the one that drives motor 1, oscillators 2-4 drive motors 2-4 (Figure 5a). Each oscillator is implemented as a phase oscillator denoted by three equations as  
\[\ddot{a}_{i}=k_{i}\left[\frac{k_{i}}{4}(A_{i}-a_{i})-\dot{a}_{i}\right] \tag{7}\]  
\[\dot{\phi}_{i}=2\pi f_{i}+\sum_{j}k_{ij}\sin\left(\phi_{j}-\phi_{i}-\Delta\phi _{ij}\right) \tag{8}\]  
\[\dot{\phi}_{i}=a_{i}\cos\phi_{i}+b_{i} \tag{9}\]  
where \(\phi_{i}\) and \(a_{i}\) are the state variables representing the phase and the amplitude of oscillator \(i\), respectively; \(f_{i}\) and \(A_{i}\) determine its intrinsic frequency and amplitude and \(k_{i}\) is a positive  
Figure 5: Analysis of the flapping propulsion method. The motors 1, 2, 3, and 4 are controlled by CPG 1, 2, 3, and 4, respectively. (**a**) Schematic diagram of the pectoral fins and CPG network topology; (**b**) the four CPG outputs when the robot goes straight and turns. When the robot goes straight, the CPG 1 and CPG 3 output the same signals, and the CPG 2 and the CPG 4 output the same lines. When the robot turns, the CPGs will output different drive signals. The red, blue, green, and brown lines denote the CPG1, CPG2, CPG3, and CPG4, respectively.  
constant. Couplings between oscillators are defined by the weights \(k_{ij}\) and phase biases \(\Delta\phi_{ij}\). \(\vartheta_{i}\) represents the output of the oscillator after the application of an offset \(b_{i}\), which is used to steer the robot. Each output \(\vartheta_{i}\) is sent as a set point to the controllers for each motor.