## 2 Small Spherical Robot Set Up and Modeling
### Modeling of The Small Spherical Robot  
Aiming at building the dynamic and kinematic models of the small spherical robot, a body reference frame (\(O-X_{B}Y_{B}Z_{B}\)) and an inertial reference frame (\(O-X_{I}Y_{I}Z_{I}\)) were established, as shown in Figure 5. The position and attitude angle of the robot, denoted as vector \(\mathbf{\eta}\), is described in the inertial coordinate system:  
\[\mathbf{\eta}=[x,y,z,\alpha,\beta,\phi], \tag{1}\]  
where \(x,y,z\) represent the position of the robot. \(\alpha,\beta,\phi\) are the Euler angle of the roll, pitch, and yaw. The motion states of the robot, denoted as vector \(\mathbf{v}\), is represented in the body frame:  
\[\mathbf{v}=[u,v,w,p,q,r], \tag{2}\]  
where, \(u,v,w\) are linear velocities in the x-direction, y-direction, and z-direction, respectively, which are denoted as the surge, sway, and heave, respectively [21]. \(p,q,r\) are the angular velocities around the x-axis, y-axis, z-axis, respectively. As the \"H\"-shaped propulsion strategy was adopted for the trajectory tracking, the driving force could only be provided in the degrees of surge, yaw, and heave. Therefore, the degrees of sway, roll, and pitch will be neglected in this paper. Let \(\mathbf{\eta}=[x,y,z,\phi]_{\mathbf{v}}=[u,w,r]\). The kinematic model of the robot are shown as Equation (3) [20].  
\[\dot{\mathbf{\eta}}=\mathbf{J(\phi)v}, \tag{3}\]  
where  
\[\mathbf{J(\phi)}=\begin{bmatrix}cos\phi&0&0\\ sin\phi&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}. \tag{4}\]  
According to the [PERSON] method, the dynamic model of the small spherical robot was established as follows [20]:  
\[\mathbf{M}\dot{\mathbf{v}}+\mathbf{C(\mathbf{v})v+D(\mathbf{v})v+g(\mathbf{\eta})=\mathbf{\tau}}, \tag{5}\]  
where \(\mathbf{M}\in\mathbf{R}^{3\times 3}\) is the inertia and added mass matrix. \(\mathbf{C(\mathbf{v})\in\mathbf{R}^{3\times 3}}\) is the Coriolis and centripetal matrix. \(\mathbf{D(\mathbf{v})\in\mathbf{R}^{3\times 3}}\) is the damping matrix, which is positive-definite. \(g(\mathbf{\eta})\) is the restoring force and torque vector. \(\mathbf{\tau}\) is the driving force vector. As the structure of the robot was symmetrical, the inertial matrix and added matrix can be expressed by Equation (7) [20]. The Coriolis and centripetal matrix \(\mathbf{C(\mathbf{v})}\) could be ignored because the speed of the robot was less than 0.5 m/s. Moreover, because the center of gravity and the buoyant center were almost at the same position, the restoring force and torque vector \(g(\mathbf{\eta})\) could be set to 0 vector (\(g(\mathbf{\eta})=\mathbf{0}_{4\times 1}\)). The simplified model is shown as follows:  
\[\mathbf{M}\dot{\mathbf{v}}+\mathbf{D(\mathbf{v})v}=\mathbf{\tau}, \tag{6}\]  
where  
\[\mathbf{M}=\begin{bmatrix}m_{11}&0&0\\ 0&m_{22}&0\\ 0&0&m_{33}\end{bmatrix} \tag{7}\]  
\[\mathbf{D(\mathbf{v})}=\begin{bmatrix}d_{11}&0&0\\ 0&d_{22}&0\\ 0&0&d_{33}\end{bmatrix}+\begin{bmatrix}X_{u}|u|&0&0\\ 0&Z_{w}|w|&0\\ 0&0&N_{r}|r|\end{bmatrix} \tag{8}\]  
\[\mathbf{\tau}=[\tau_{u},\tau_{w},\tau_{v}]^{\mathsf{T}}. \tag{9}\]  
Equations (3) and (5) can be written as follows:  
\[\dot{\mathbf{\eta}}=\mathbf{J(\phi)v} \tag{10}\]