## 2 Dynamic Modeling of Industrial Robot
### Force Analysis of Robot  
In order to achieve optimal dynamic control of an industrial robot in an underwater environment, it is necessary to first establish a dynamic model of the industrial robot. The robot system moving underwater is a nonlinear dynamic system that requires a large number of parameters to be determined. Due to limitations in technology and testing conditions, some parameters cannot be accurately measured or cannot be determined. In order to meet the needs of control, it is necessary to simplify the system and only consider the factors that have a major impact on system performance, such as gravity, buoyancy, thrust, and hydrodynamics. Firstly, the industrial robots are subject to the gravitational force of the Earth, and the resulting forces and moments need to be reflected in the local coordinate system, which can be expressed as  
\[ \begin{pmatrix}F\\ \tau\end{pmatrix} _{G}=G \begin{pmatrix}-\sin\theta&\cos\theta\sin\varphi&\cos \theta\cos\varphi&0&0&0\end{pmatrix} ^{T}, \tag{1}\]  
where due to the coincidence of the robot's center of mass with the origin of the coordinate system, its gravitational moment is 0. At the same time, industrial robots are subjected to buoyancy in the water, and the resulting buoyancy and buoyancy moment can be represented in the local coordinate system as  
\[ \begin{pmatrix}F\\ \tau\end{pmatrix} _{B}=-B \begin{pmatrix}-\sin\theta\\ \cos\theta\sin\varphi\\ \cos\theta\cos\varphi\\ z_{\text{b}}\cos\theta\sin\varphi-y_{\text{b}}\cos\theta\cos\varphi\\ x_{\text{b}}\cos\theta\cos\varphi+z_{\text{b}}\sin\theta\\ -y_{\text{b}}\sin\theta-x_{\text{b}}\cos\theta\sin\varphi\end{pmatrix} , \tag{2}\]  
where \( \begin{pmatrix}x_{\text{b}}&y_{\text{b}}&z_{\text{b}}\end{pmatrix} ^{\text{T}}\) is the coordinate of the center of buoyancy in the local coordinate system. In addition, it is necessary to consider the driving force and torque exerted on industrial robots. As Figure 1 shows, the direction of the thrust force is indicated by arrows, where the thrust forces \(F_{T1}\) and \(F_{T2}\) are located in the oxy plane and symmetrical with respect to the \(x\)-axis, achieving rotation around the \(z\)-axis and translation along the \(x\)-axis. The thrust forces \(F_{T3}\) and \(F_{T4}\) are located in the \(oyz\) plane and are symmetrical with respect to the \(y\)-axis, achieving rotation around the \(x\)-axis and translation along the \(y\)-axis. The thrust forces \(F_{T5}\) and \(F_{T6}\) are located in the \(oxz\) plane and are symmetrical with respect to the \(z\)-axis, achieving rotation around the \(y\)-axis and translation along the z-axis. Figure 1: Dynamic model of industrial robot. Assuming the resultant forces along the \(x\), \(y\), and \(z\) axes are \(T_{x}\), \(T_{y}\), and \(T_{z}\), respectively, and the resultant moments acting on the \(x\), \(y\), and \(z\) axes are \(K_{T}\), \(M_{T}\), and \(N_{T}\), respectively, the resultant forces and resultant moments on each axis can be represented as \(T_{x}=F_{T1}+F_{T2}\), \(T_{x}=F_{T1}+F_{T2}\), \(T_{z}=F_{T5}+F_{T6}\), \(K_{T}=R_{T3}F_{T3}+R_{T4}F_{T4}\), \(M_{T}=R_{T5}F_{T5}+R_{T6}F_{T6}\), \(N_{T}=R_{T1}F_{T1}+R_{T2}F_{T2}\). Robots also experience hydrodynamic forces in water, and the resulting forces and moments need to be reflected in the local coordinate system. Let the velocity of water flow in the geodetic coordinate system be \(u_{f}=\left(u_{f}\quad v_{f}\quad w_{f}\right)^{T}\), and the transformation relationship between the velocity of water flow in the local coordinate system and the geodetic coordinate system is  
\[u_{f}^{\prime}=\left(u_{1}\quad v_{1}\quad w_{1}\right)^{T}=T_{v}^{-1}\Big{(}u _{f}\quad v_{f}\quad w_{f}\Big{)}^{T}. \tag{3}\]  
The velocity of the water flow relative to the robot can be expressed as  
\[u_{r}^{\prime}=\left(u_{r}\quad v_{r}\quad w_{r}\right)^{T}=u^{\prime}-u_{1}^ {\prime}. \tag{4}\]  
The water flow resistance is directly proportional to the square of the relative water flow velocity of the industrial robot.