# Development of Portable Magnetic Adsorption Amphibious Robot
## 5 Dynamic Modeling and Motion Simulation of Amphibious Robot  
### Robot Motion Model Building  
The robot coordinate system diagram is shown in Figure 18. In order to better control the underwater motion of the robot, it was necessary to establish the underwater motion equation for the amphibious robot. In general, the center of gravity (\(G(x_{g},y_{g},z_{g})\)) of the robot and the origin (\(O\)) of the robot coordinate system are not concentric, and the resultant motion equation of the amphibious robot with six degrees of freedom can be derived when the robot is regarded as a rigid body. \[\left\{ \begin{array}{l}X=m\left[\dot{u}-vr+wq-x_{g}(q^{2}+r^{2})+y_{g}(pq- \dot{r})+z_{g}(pr+\dot{q})\right]\\ Y=m\left[\dot{v}-wp+ur-y_{g}(p^{2}+r^{2})+z_{g}(qr-\dot{p})+x_{g}(qp+\dot{r}) \right]\\ Z=m\left[\dot{w}-uq+vp-z_{g}(q^{2}+p^{2})+z_{g}(rp-\dot{q})+y_{g}(rq+\dot{p}) \right]\\ K=l_{x}\dot{p}+(l_{z}-I_{y})qr+m\left[y_{g}(\dot{w}+vp-uq)-z_{g}(\dot{v}+ur-wp) \right]\\ M=I_{y}\dot{q}+(I_{x}-I_{z})rp+m\left[z_{g}(\dot{u}+wq-vr)-x_{g}(\dot{w}+vp-uq) \right]\\ N=I_{z}\dot{r}+(I_{y}-I_{x})pq+m\left[x_{g}(\dot{v}+ur-wp)-y_{g}(\dot{u}+wq-vr) \right]\end{array} \right. \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Remanent Magnetic** & **Coercivity** & **Maximum Magnetic Energy** & **Operating Temperature** \\
**Induction** & **(H\({}_{c}\)/(kA\(\times\)m\({}^{-1}\)))** & **Product** & **(T\({}_{w}\)/\({}^{g}\) C)** \\
**(Br/mT)** & **(H\({}_{c}\)/(kA\(\times\)m\({}^{-1}\)))** & **(T\({}_{w}\)/\({}^{g}\) C)** \\ \hline
1430–1480 & 828–907 & 398–422 & \(<\)60 \\ \hline \end{tabular}
\end{table} 
Table 4: Performance parameters of NdFeB N52. Figure 17: Magnetic inductance line of permanent magnet adsorption unit. where \(v=[u\,v\,w\,p\,q\,r]^{T}\) is the velocity vector defined in the moving coordinate system and the components correspond to longitudinal velocity, transverse velocity, float velocity, roll angular velocity, pitch velocity, and course angular velocity, respectively; \(m\) stands for the amphibious robot's mass; \(I_{x},I_{y},I_{z}\), is the moment of inertia of the amphibious robot on the three coordinate axes; and \(K,M,N\) is the moment of the amphibious robot's force. The amphibious robot is a nonlinear system, and its movement in water is mainly affected by hydrodynamic force (\(F_{w}\)), floatage (\(B\)), gravity (\(G\)), thruster thrust (\(T_{i}\)), and cable force (\(F_{i}\)). The external force on an amphibious robot in water can be expressed as follows:  
\[F=F_{w}+B+G+\sum_{i=1}^{n}T_{i}+F_{i} \tag{10}\]  
The amphibious robot uses a zero-buoyancy cable, and the cable force can be ignored. The calculation of thruster thrust needs to consider the arrangement of the thruster. As shown in Figure 19, there are four vertical thrusters distributed with symmetrical origins. They are respectively represented by \(T_{1},T_{2},T_{3},T_{4}\), and the two horizontal thrusters that are symmetrical to each other are respectively represented by \(T_{5},T_{6}\). The thruster thrust formula is as follows:  
\[T=K_{T}\rho D^{4}n^{2} \tag{11}\]  
The thrust torque formula is as follows:  
\[M_{T}=K_{0}\rho D^{5}n^{2} \tag{12}\]  
Figure 19: Layout of push machine. Figure 18: Schematic diagram of robot coordinate system. where \(\rho\) is the water density, \(n\) is the propeller speed, \(D\) is the airscrew diameter, \(K_{T}\) is the thrust coefficient, and \(K_{Q}\) is the moment coefficient. Vertical motion is powered by four vertical thrusters, which provide opposite thrust by controlling the direction of rotation of the propeller. The thrust and torque formulas for vertical motion are as follows:  
\[\left[ \begin{array}{c}T\\ M\end{array} \right]=\left[ \begin{array}{c}\sum\limits_{i=1}^{4}T_{i}\\ \sum\limits_{i=1}^{4}M_{i}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ T_{1}+T_{2}+T_{3}+T_{4}\\ I_{1}(T_{1}-T_{2}+T_{3}-T_{4})\\ I_{1}(T_{1}+T_{2}-T_{3}-T_{4})\\ 0\end{array}