## 2 Modeling of Biomimetic Spherical Robot
### Simplified Modeling of BSR for Trajectory Tracking  
It is necessary to build a robot model to establish a state-space model. To clearly describe the dynamic and kinematic models of the BSR, two reference frames are established: a body reference frame (b-frame) and an inertial reference frame (i-frame). The body reference frame is attached to the vehicle with the origin selected to be the center of gravity. The motion of the BSR is described as the motion of the b-frame with in the inertial reference frame, which fixes the origin at a certain point on Earth. Figure 3 depicts the two coordinate systems. In the i-frame, the BSR position and orientation are represented, as follows:  
Figure 1: The prototype of the biomimetic spherical robot. Figure 2: Detailed structure of the vector propulsion mechanism. \[\mathbf{\eta}=[x,y,z,\varphi,\psi,\theta] \tag{1}\]  
where \((x,y,z)\) is the position and \(\varphi\), \(\psi\), and \(\theta\) are the Euler angle of the roll (along the x-axis), pitch (along the y-axis), and yaw (along the z-axis). The motion states of the BSR with respect to the i-frame are expressed in the following manner:  
\[\mathbf{v}\ =[u,v,w,p,q,r] \tag{2}\]  
where, \(u\), \(v\), and \(w\) are the linear velocity in the x-, y-, and z-direction, respectively, and they are denoted as Surge, Sway, and Heavy. In addition, \(p,q\), and \(r\) are the angular velocity around the x-, y-, and z-axis, respectively, and they are denoted as Roll, Pitch, and Yaw, respectively. The dynamics equation is established according to [PERSON] [PERSON] (1958); [PERSON] (1958):  
\[\mathbf{(M_{RB}+M_{A})\dot{\mathbf{v}}+C(\mathbf{v})\mathbf{v}+\mathbf{(D_{l}+D_{q}( \mathbf{v}))v+g(\mathbf{\eta})}=\tau} \tag{3}\]  
where \(\mathbf{M_{RB}}\) is the inertial matrix and \(\mathbf{M_{A}}\) is the added matrix. In addition, \(\mathbf{C(v)}\) denotes the Coriolis and centripetal matrix, \(\mathbf{D_{l}}\) is the linear damping matrix, \(\mathbf{D_{q}(\mathbf{v})}\) is the nonlinear damping matrix, \(\mathbf{g(\mathbf{\eta})}\) represents the restoring force, and \(\mathbf{\tau}\) is a six-dimensional vector that consists of the driving force and \begin{table}
\begin{tabular}{l l} \hline \hline
**Items** & **Parameters** \\ \hline Dimension(\(Width\times Length\times Height\)) & 30 cm \(\times\) 60 cm \(\times\) 30 cm \\ Mass in air & 6.5 Kg \\ Processors & NVIDIA Jetson Tk1 \\ STM32F407 \\ Max thrust & 3.8 N \\  & Pressure sensor(MS5803-14 BA ) \\ Sensors & IMU(3 DM-GX5-45) \\  & Stereo camera \\  & Acoustic communication module(Micron Sonar) \\ Power & 7.4 V rechargeable Ni-MH batteries (13,200 mAh) \\ Operation time & average 100 min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The Technical Specification of the Robot. Figure 3: The reference coordinate system. moment generated by four thrusters. As the structure of the robot is symmetrical, the inertial matrix can be expressed, as follows:  
\[M_{RB}=\left[ \begin{array}{cccccc}m&0&0&0&0&0\\ 0&m&0&0&0&0\\ 0&0&m&0&0&0\\ 0&0&0&I_{x}&0&0\\ 0&0&0&0&I_{y}&0\\ 0&0&0&0&0&I_{z}\end{array} \right] \tag{4}\]  
where \(m=6.3\) kg is the quality of the BSR. Because the shell of the robot is symmetrical (neglected effects of the thrusters), there is no coupling term. The added matrix can be represented, as follows:  
\[M_{A}=-\left[ \begin{array}{cccccc}X_{\dot{u}}&0&0&0&0&0\\ 0&Y_{\dot{v}}&0&0&0&0\\ 0&0&Z_{\dot{w}}&0&0&0\\ 0&0&0&K_{\dot{p}}&0&0\\ 0&0&0&0&M_{\dot{q}}&0\\ 0&0&0&0&0&N_{r}\end{array} \right] \tag{5}\]  
The Coriolis and centripetal matrix \(\mathbf{C}(\mathbf{v})\) can be ignored, because the speed of the robot is slow. The surrounding flow of the robot can be seen as a Stokes flow because of the low velocity of the robot. The flow viscosity is then \(\mathbf{C}_{l}=1\times 10^{-4}\), and the linear hydrodynamic damping force is expressed as \(\mathbf{D}_{l}=\mathbf{C}_{l}\times diag\left\{1,1,1,1,1,1\right\}\). Because of the symmetrical structure of the robot, the nonlinear hydrodynamic damping force can be represented, as follows:  
\[\mathbf{D}_{q}(v)=-\left[