## 2 The Configuration of SFRM
### Time-Varying Mass Model  
The total mass and weight of the SFRM can be time-varied based on the following equations:  
\[W(t)=m(t)g \tag{1}\]  
\[m(t)=m_{0}+\sum_{i=1}^{2}m_{i}(t) \tag{2}\]  
\[m_{i}(t)=\int\dot{m_{i}}(t)dt=\rho Q_{i}(t),\;i=1,2 \tag{3}\]  
where \(t\) is defined as time; \(W(t)\) and \(m(t)\) are the total weight and mass of the SFRM; \(m_{0}\) is the initial mass of the SFRM; \(m_{1}(t)\) and \(m_{2}(t)\) represent the masses of forward ballast tank and aft ballast tank; \(\dot{m_{1}}(t)\) and \(\dot{m_{2}}(t)\) denote the rates of masses of forward ballast tank and aft ballast tank; \(Q_{1}(t)\) and \(Q_{2}(t)\) denote the flow rates of forward ballast tank and aft ballast tank; \(\rho\) is the water density; \(t\) is defined as time.  
Since the attitude of the SFRM is determined by the mass variation of the ballast tank, the coordinate of CG can be determined as below:  
\[x_{g}(t)=\frac{m_{0}x_{go}+x_{b1}(t)m_{1}(t)+x_{b2}(t)m_{2}(t)}{m_{0}+m_{1}(t) +m_{2}(t)} \tag{4}\]  
\[y_{g}(t)=\frac{m_{0}y_{go}+y_{b1}(t)m_{1}(t)+y_{b2}(t)m_{2}(t)}{m_{0}+m_{1}(t) +m_{2}(t)} \tag{5}\]  
\[z_{g}(t)=\frac{m_{0}z_{go}+z_{b1}(t)m_{1}(t)+z_{b2}(t)m_{2}(t)}{m_{0}+m_{1}(t) +m_{2}(t)} \tag{6}\]  
where (\(x_{go}\), \(y_{go}\), \(z_{go}\)) is the initial coordinate of CG; (\(x_{b1}\), \(y_{b1}\), \(z_{b1}\)) is the coordinate of CG for the forward ballast tank, and (\(x_{b2}\), \(y_{b2}\), \(z_{b2}\)) is the coordinate of CG for the aft ballast tank.  
Subsequently, the inertia moments of the SFRM are defined by:  
Figure 2: The configuration of ballast tanks and the attitude adjusting system.  
\[I_{xx}(t) = I_{x0}+[I_{xb1}(t)+(y_{b1}(t)^{2}+z_{b1}(t)^{2})\times m_{1}(t)]+[I_{xb2 }(t)+(y_{b2}(t)^{2}+\,z_{b2}(t)^{2})\times m_{2}(t)] \tag{7}\] \[I_{yy}(t) = I_{y0}+[I_{y01}(t)+(x_{b1}(t)^{2}+z_{b1}(t)^{2})\times m_{1}(t)]+[ I_{yb2}(t)+(x_{b2}(t)^{2}+z_{b2}(t)^{2})\times m_{2}(t)]\] (8) \[I_{zz}(t) = I_{z0}+[I_{zb1}(t)+(x_{b1}(t)^{2}+y_{b1}(t)^{2})\times m_{1}(t)]+[ I_{zb2}(t)+(x_{b2}(t)^{2}+y_{b2}(t)^{2})\times m_{2}(t)]\] (9) \[I_{xy}(t) = I_{xy0}+\Big{[}I_{xyb1}(t)+x_{b1}(t)y_{b1}(t)m_{1}(t)\Big{]}+ \Big{[}I_{xyb2}(t)+x_{b2}(t)y_{b2}(t)m_{2}(t)\Big{]}\] (10) \[I_{xz}(t) = I_{xz0}+[I_{xzb1}(t)+x_{b1}(t)z_{b1}(t)m_{1}(t)]+[I_{xzb2}(t)+x_ {b2}(t)z_{b2}(t)m_{2}(t)]\] (11) \[I_{yz}(t) = I_{yz0}+\Big{[}I_{yzb1}(t)+y_{b1}(t)z_{b1}(t)m_{1}(t)\Big{]}+ \Big{[}I_{yzb2}(t)+y_{b2}(t)z_{b2}(t)m_{2}(t)\Big{]} \tag{12}\]  
where \(I_{x0}\), \(I_{y0}\), \(I_{z0}\), \(I_{xy0}\), \(I_{xz0}\), and \(I_{yz0}\) are the inertia moments of the SFRM corresponding to the origin of the initial coordinate; \(I_{xb1}\), \(I_{yb1}\), \(I_{zb1}\), \(I_{xyb1}\), \(I_{xzb1}\), and \(I_{yzb1}\) are the inertia moments due to mass variation corresponding to the CG of the forward ballast tank; \(I_{yb2}\), \(I_{zb2}\), \(I_{xyb2}\), \(I_{xzb2}\), and \(I_{yzb2}\) are the inertia moments due to mass variation corresponding to the CG of the aft ballast tank.  
In conditions of full ballast tanks, the weight of either the forward or aft water tank is about 600 g. Since the mass variation of ballast tanks causes the change of the pitch angle, the time-varying mass model would be adopted to understand the time series of the SFRM's attitude in the powerless state. Due to asymmetrical positions of ballast tanks on both sides of the SFRM, the time-varying model of each individual ballast tank was used to discuss the effect of ballast tanks on the inertia moments in this study.