# 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
\right] \tag{16}\]  
\(C_{A}(v)\) matrix contains Munk moments and it is given by:  
\[C_{A}(v)=\left[ \begin{array}{ccccc}0&0&0&0&-Z_{\dot{w}}w&Y_{\dot{v}}v\\ 0&0&0&Z_{\dot{w}}w&0&-X_{\dot{u}}u\\ 0&0&0&-Y_{\dot{v}}v&X_{\dot{u}}u&0\\ 0&-Z_{\dot{w}}w&Y_{\dot{v}}v&0&-N_{\dot{r}}r&M_{\dot{q}}q\\ Z_{\dot{w}}w&0&-X_{\dot{u}}u&N_{\dot{r}}r&0&-K_{\dot{p}}p\\ -Y_{\dot{v}}v&X_{\dot{u}}u&0&-M_{\dot{q}}q&K_{\dot{p}}p&0\end{array} \right] \tag{17}\]  
Hydrodynamic damping effects of the robotic fish are separated into linear and non-linear terms. These matrixes can be defined as below equations:  
\[D_{L}=\left[ \begin{array}{ccccc}X_{u}&0&0&0&0&0\\ 0&Y_{v}&0&0&0&-Y_{r}\\ 0&0&Z_{w}&0&Z_{q}&0\\ 0&0&0&K_{p}&0&0\\ 0&0&M_{w}&0&M_{q}&0\\ 0&-N_{\dot{v}}&0&0&0&N_{r}\end{array} \right] \tag{18}\]  
\[D_{NL}=\left[ \begin{array}{ccccc}X_{|u|u}\Big{|}u\Big{|}&0&0&0&0&0\\ 0&Y_{|v|v}\Big{|}v&0&0&0&-Y_{|v|r}\Big{|}v\Big{|}\\ 0&0&Z_{|w|w}\Big{|}w\Big{|}&0&Y_{|w|q}\Big{|}w\Big{|}&0\\ 0&0&0&K_{|p|p}\Big{|}p&0&0\\ 0&0&M_{|q|w}\Big{|}q\Big{|}&0&M_{|q|q}\Big{|}q\Big{|}&0\\ 0&-N_{|r|v}\Big{|}r&0&0&0&N_{|r|r}\Big{|}r\Big{|}\end{array} \right] \tag{19}\]\(D_{L}\) and \(D_{NL}\) are the diagonal matrixes and these matrixes depend on the body shape and speed of the robotic fish, respectively. \(X_{u}\), \(X_{|u|u}\) and so forth terms are linear and non-linear positive damping coefficients. In marine applications, underwater vehicles move at low speed and adding mass terms can be neglected. However, these effects are used to obtain complete motion model. \(W=mg\) is the gravity force and \(B=\rho g\Delta\) is the buoyancy force. Here, \(\Delta\) is the submerged volume. Thus, hydrostatic effect \(g(\eta)\) is calculated as:  
\[g(\eta)=\left[ \begin{array}{c}(W-B)s(\theta)\\ -(W-B)c(\theta)s(\phi)\\ -(W-B)c(\theta)c(\phi)\\ -(y_{g}W-y_{b}B)c(\theta)c(\phi)+(z_{g}W-z_{b}B)c(\theta)s(\phi)\\ -(z_{g}W-z_{b}B)s(\theta)+(x_{g}W-x_{b}B)c(\theta)c(\phi)\\ -(x_{g}W-x_{b}B)c(\theta)s(\phi)-(y_{g}W-y_{b}B)s(\theta)\end{array} \right] \tag{20}\]  
Note that (\(s\)), \(c\)), \(t\)) are equal to (\(sin()\), \(cos()\), \(tan()\)). \(r_{B}=\left[ \begin{array}{cc}x_{B}&y_{B}&z_{B}\end{array} \right]^{T}\) is the position vector from the body origin to center of buoyancy. The force vector \(\tau_{f}\) includes the hydrodynamic forces and moments acting on the fish body and it is expressed as:  
\[\tau_{f}=\left[ \begin{array}{cccc}F_{x}&F_{y}&F_{z}&\tau_{x}&\tau_{y}&\tau_{ z}\end{array} \right]^{T} \tag{21}\]  
where, \((F_{x},F_{y},F_{z})\) represent the surge, sway and heave forces. \((\tau_{x},\tau_{y},\tau_{z})\) are the yaw, pitch and roll moments, respectively. It is considered that \(F_{x}=F_{F}\cos(\theta_{2}+\theta_{3}+\theta_{4})\). \(F_{y}\) and \(F_{z}\) are chosen as zero, and \(\tau_{z}=((l_{1}+l_{2})\cos(\theta_{2})+l_{3}\cos(\theta_{2}+\theta_{3})+l_{4 }\cos(\theta_{2}+\theta_{3}+\theta_{4}))F_{F}\sin(\theta_{2}+\theta_{3}+\theta)\). Finally, 6-DoF motion block diagram is given in Figure 10 that includes the fish body and fish tail equations both body-fixed frame and earth-fixed frame. By solving these equations, whole 3D non-linear dynamic model of the robotic fish is obtained.