## 2 Floating Frame of Reference Formulation
### Inertia  
The inertia matrix of body \(i\) can be derived from its kinetic energy expression:  
\[T^{i}=\frac{1}{2}\!\!\int\limits_{\
u^{i}}\rho^{i}\dot{r}^{i}\dot{r}^{i}\dot{ r}^{i}dV^{i}, \tag{8}\]  
where \(\rho^{i}\) is the mass density of body \(i\) and \(\dot{r}^{i}\) denotes \(\dot{r}_{P^{\prime}}^{i}\) by omitting the subscript \(\mathbf{P^{\prime}}\) for brevity.  
The inertia matrix of the body \(i\) is then given by the following symmetric matrix:  
\[M^{i}=\left[\begin{array}{cc}m_{rr}^{\ i}&m_{rf}^{\ i}\\ m_{ff}^{\ i}&m_{ff}^{\ i}\end{array}\right]=\left[\begin{array}{cc}m_{r_{o} \alpha_{o}^{\ i}}&m_{r_{o}\alpha_{o}^{\ i}}&m_{r_{o}f_{i}^{\ i}}\\ &m_{\theta\theta}^{\ i}&m_{\theta f}^{\ i}\\ sym&m_{ff}^{\ i}\end{array}\right] \tag{9}\]  
where the elements of the inertia matrix are defined by the following volume integrals:  
\[m_{r_{o}\alpha_{o}}^{\ \ i}=\int\limits_{\
u^{i}}\rho^{i}Id\
u^{i} \tag{10}\]  
\[m_{r_{o}\alpha}^{\ \ i}=\int\limits_{\
u^{i}}\rho^{i}\dot{B}^{i}dV^{i};\quad\text{ where}\quad B^{i}=-R^{i}\overline{u^{i}}G^{i}\quad\text{and}\quad\alpha^{i}=G^{i} \dot{\theta}^{\ i} \tag{11}\]\[m_{r_{\theta}f}{}^{i}=R^{i}\int\limits_{\mathcal{V}^{i}}\rho^{i}s^{i}dV^{i} \tag{12}\]  
\[m_{\theta\theta}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}B^{iT}R^{i}dV^{i} \tag{13}\]  
\[m_{\theta f}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}B^{iT}R^{i}S^{i}dV^{i} \tag{14}\]  
\[m_{ff}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}S^{iT}S^{i}dV^{i}. \tag{15}\]  
For a planar case, which is the case in this work, the integrals Equations (10)-(15) are given by the following:  
\[m_{r_{\theta}r_{\theta}}{}^{i}=\left[\begin{array}{cc}m^{i}&0\\ 0&m^{i}\end{array}\right]\text{ where }m^{i}\text{ is the mass of the body }i \tag{16}\]  
\[m_{r_{\theta}o}{}^{i}=R_{\theta}{}^{i}\!\left(\!I_{1}{}^{i}+\overline{S}^{i}q _{f}{}^{i}\!\right)\text{ where }R_{\theta}{}^{i}=\frac{\partial R^{i}}{\partial\theta ^{i}} \tag{17}\]  
\[I_{1}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}u_{o}{}^{i}dV^{i} \tag{18}\]  
\[\overline{S}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}S^{i}dV^{i} \tag{19}\]  
\[m_{r_{\theta}}{}^{i}=R^{i}\overline{S}^{i} \tag{20}\]  
\[m_{\theta\theta}{}^{i}=\left(m_{\theta\theta}{}^{i}\right)_{r_{\theta}r_{ \theta}}+\left(m_{\theta\theta}{}^{i}\right)_{r_{\theta}f}+\left(m_{\theta \theta}{}^{i}\right)_{ff}\text{, where} \tag{21}\]  
\[\left(m_{\theta\theta}{}^{i}\right)_{r_{\theta}r_{\theta}}=\int\limits_{Vi} \rho^{i}{u_{o}}^{iT}u_{o}{}^{i}dV^{i} \tag{22}\]  
\[\left(m_{\theta\theta}{}^{i}\right)_{r_{\theta}f}=2\left(\int\limits_{ \mathcal{V}^{i}}{\rho^{i}{u_{o}}^{iT}S^{i}dV^{i}}\right)q_{f}{}^{i} \tag{23}\]  
\[\left(m_{\theta\theta}{}^{i}\right)_{ff}=q_{f}{}^{iT}m_{ff}q_{f}{}^{i} \tag{24}\]  
\[m_{ff}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}S^{iT}S^{i}dV^{i} \tag{25}\]  
\[m_{\theta f}{}^{i}=\int\limits_{\mathcal{V}^{i}}\rho^{i}{u_{o}}^{iT}\overline {S}^{i}dV^{i}+q_{f}{}^{i}\!\left(\overline{S}_{12}-\overline{S}_{21}\right) \text{, where} \tag{26}\]  
\[\overline{I}=\left[\begin{array}{cc}0&1\\ -1&0\end{array}\right] \tag{27}\]  
\[m_{ff}{}^{i}=\overline{S}_{11}{}^{i}+\overline{S}_{22}{}^{i}. \tag{28}\]  
Notice that and \(\overline{S}_{kl}\) in Equations (26) and (28) is given by:  
\[\overline{S}_{kl}=\int\limits_{\mathcal{V}^{i}}\rho^{i}{S_{k}}^{iT}S_{l}dV^{i} ;\quad k,l=1,2. \tag{29}\]