# Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion
## Three Dimensional Dynamic Model and Motion Equations of the Robotic Fish
### Three Dimensional Model Equations of Motion  
Based on the ichthyology of carp, a multi-link and autonomous-swimming biomimetic robotic fish prototype is designed. The prototype consists of a rigid body, three active tail links and a passive caudal fin. In order to analyze the guidance states of the robotic fish, 6-DoF body motion equations are presented in this paper. These equalities define the kinematic and dynamic behaviors of the robotic fish in two coordinate frames and also include the hydrodynamic forces and moments acting on the robotic fish body. The moving coordinate frame is fixed to the body and it is expressed as a body-fixed reference frame. The origin of the body-fixed frame is pointed with the center of gravity. Also, the motion of the body-fixed frame depends on the inertial reference frame that is named the earth-fixed frame. The 6-DoF motion of the robotic fish prototype in the Earth-Fixed frame with translation and rotation relations is shown in Figure 9.  
The generalized speed and positions of the robotic fish are completely given by the state variable vector \(s=\left[ \begin{array}{cccccccc}u&v&w&p&q&r&x&y&z&\phi&\theta&\psi\end{array}  \right]^{T}\). The linear speeds \((u,v,w)\) are surge, sway and heave, respectively. The angular speeds \((p,q,r)\) are roll, pitch and yaw speeds, respectively. Surge, sway and heave positions \((x,y,z)\) are determined with the earth-fixed frame. Roll, pitch and yaw angles \((\phi,\theta,\psi)\) are the rigid body orientations with respect to the earth-fixed frame. The kinetic translation between the body-fixed and the earth-fixed frames for the robotic fish motion is expressed as:  
\[\dot{\eta}=\left[ \begin{array}{cc}R_{\Theta}(\eta)&0_{3\times 3}\\ 0_{3\times 3}&T_{\Theta}(\eta)\end{array} \right]v \tag{12}\]  
where, \(\eta=\left[ \begin{array}{cccccccc}x&y&z&\phi&\theta&\psi\end{array} \right]^{T}\) and \(v=\left[ \begin{array}{cccccccc}u&v&w&p&q&r\end{array} \right]^{T}\) are the generalized positions and speeds, respectively. \(\eta\) is relative with the body-fixed frame and speed vector depends on the earth-fixed frame. \(R_{\Theta}(\eta)\) is the rotation matrix and \(T_{\Theta}(\eta)\) is the transformation matrix. In this way, the 3D equation of the rigid body motion is given by:  
\[M\dot{v}+C(v)v+D(v)v+g(\eta)=\tau_{f} \tag{13}\]  
Figure 9: The designed multi-link robotic fish prototype on reference frames. where, \(M=M_{RB}+M_{A}\) is the mass matrix, \(M_{RB}\) is the rigid body mass matrix and \(M_{A}\) is the added mass matrix. \(C(v)=C_{RB}(v)+C_{A}(v)\) is the Coriolis and centripetal matrix, \(C_{RB}(v)\) is the rigid body Coriolis matrix and \(C_{A}(v)\) is the hydrodynamic Coriolis matrix. \(D(v)\) is the hydrodynamic damping matrix including linear and non-linear damping effects. \(\tau_{f}\) is the vector that includes the hydrodynamic forces. Rigid body mass matrix and added mass matrix are defined as:  
\[M_{RB}=\left[ \begin{array}{cccc}mI_{3\times 3}&-mS(r_{G})\\ mS(r_{G})&I_{0}\end{array} \right] \tag{14}\]  
\[M_{A}=\left[ \begin{array}{ccccc}X_{i}&0&0&0&0&0\\ 0&Y_{\dot{v}}&0&0&0&Y_{\dot{r}}\\ 0&0&Z_{\dot{w}}&0&Z_{\dot{q}}&0\\ 0&0&0&K_{\dot{p}}&0&0\\ 0&0&M_{\dot{w}}&0&M_{\dot{q}}&0\\ 0&N_{\dot{v}}&0&0&0&N_{\dot{r}}\end{array} \right] \tag{15}\]  
In these equations, \(m\) is the total body mass. The position of the center of gravity is equal to \(r_{G}=\left[ \begin{array}{cccc}x_{G}&y_{G}&z_{G}\end{array} \right]^{T}\). The moments of inertia are defined as \(I_{0}\). Asymmetric matrix \(S\) is used for easier computation. \(X_{\dot{u}}\) and so forth are the zero-frequency added mass coefficients. \(C_{RB}(v)\) matrix indicates the Coriolis matrix \(\left[ \begin{array}{cccc}p&q&r\end{array} \right]^{T}\times\left[ \begin{array} []{cccc}u&v&w\end{array} \right]^{T}\) and it is expressed by:  
\[C_{RB}(v)=\left[ \begin{array}{cccc}0_{3\times 3}&-S(v_{1}ml_{3\times 3}-v_{2}mS(r _{G}))\\ -S(v_{1}ml_{3\times 3}+v_{2}-mS(r_{G}))&-S(v_{1}mS(r_{G})+v_{2}I_{0})\end{array}