## 2 Floating Frame of Reference Formulation  
### Kinematics  
Referring to Figure 1, the pose of the body \(i\) in the global frame is given by \(q^{i}\) which consists of the rigid coordinates \(q_{r}{}^{i}\) and the flexible coordinates \(q_{f}{}^{i}\):  
\[q^{i}=\left[\begin{array}{c}q_{r}{}^{i}\\ q_{f}{}^{i}\end{array}\right]=\left[\begin{array}{c}r_{o}{}^{i}\\ \theta^{i}\\ q_{f}{}^{i}\end{array}\right], \tag{1}\]  
where \(r_{o}{}^{i}\) is the position vector of \(\mathbf{O^{i}}\) with respect to the global frame whereas \(\theta^{i}\) is the orientation vector of the body \(i\) with respect to the global frame.  
Point \(\mathbf{P}\) in Figure 1 denotes the undeformed point whereas point \(\mathbf{P^{\prime}}\) denotes the deformed point. The position of the deformed point \(\mathbf{P^{\prime}}\) in the body \(i\) frame, \(u^{i}\), is the vectorial sum of the position of the undeformed point \(\mathbf{P}\) in the body frame \(i\), \(u_{o}{}^{i}\) and the deformation vector \(u_{f}{}^{i}\):  
\[u^{i}=u_{o}{}^{i}+u_{f}{}^{i}\text{ where }u_{f}{}^{i}=S^{i}q_{f}{}^{i}. \tag{2}\]  
The matrix \(S\) is called the shape matrix whereas \(q_{f}\) is the vector of elastic coordinates which combines the flexible coordinates of all bodies in a multibody system, that is,  
\[q_{f}=\left[\begin{array}{cccc}q_{f}{}^{1}&q_{f}{}^{2}&\cdots&q_{f}{}^{n} \end{array}\right]^{T}, \tag{3}\]  
Figure 1: A deformable body \(i\) in an Euclidean space.  
where \(n\) is the number of bodies in the multibody system. The matrix \(S\) is space dependent whereas the vector \(q_{f}\) is time dependent.  
The position vector of the deformed point \(\mathbf{P^{\prime}}\) in the global frame is given by:  
\[r_{P^{\prime}}^{i}=r_{o}^{\ i}+R^{i}\Big{(}u_{o}^{\ i}+S^{i}q_{f}^{\ i}\Big{)}, \tag{4}\]  
where \(R^{i}\) is the rotation matrix from the body frame \(i\) to the global frame.  
The velocity vector of the deformed point \(\mathbf{P^{\prime}}\) with respect to the global frame is obtained by differentiating (4) with respect to time, that is,  
\[\dot{r}_{P^{\prime}}^{i}=\dot{r}_{o}^{\ i}+\ddot{R}\Big{(}u_{o}^{\ i}+S^{i}q_{ f}^{\ i}\Big{)}+R^{i}S^{i}\dot{q}_{f}^{\ i}, \tag{5}\]  
where a dot above a letter denotes differentiation with respect to time. Using the transport theorem, the time derivative of the rotation matrix \(R^{i}\) can be omitted in the velocity expression. Accordingly, the velocity vector of the deformed point \(\mathbf{P^{\prime}}\) with respect to the global frame can be written in terms of the angular velocity vector \(\alpha^{i}\) as follows:  
\[\dot{r}_{P^{\prime}}^{i}=\dot{r}_{o}^{\ i}+\overline{u^{i}}^{T}\alpha^{i}+R^{ i}S^{i}\dot{q}_{f}^{\ i}, \tag{6}\]  
where \(\overline{u^{i}}\) is the skew-symmetric matrix of the vector \(u^{i}\).  
Upon differentiating Equation (6) with respect to time, the acceleration vector of the deformed point \(\mathbf{P^{\prime}}\) with respect to the global frame can be obtained by utilizing the transport theorem which avoids the time derivative of the rotation matrix \(R^{i}\):  
\[\ddot{r}_{P^{\prime}}^{i}=\ddot{r}_{o}^{\ i}+\Big{(}\alpha^{i}\times\Big{(} \alpha^{i}\times u^{i}\Big{)}\Big{)}+\Big{(}\alpha^{i}\times u^{i}\Big{)}+ \Big{(}2\alpha^{i}\times\Big{(}R^{i}\dot{u}^{i}\Big{)}\Big{)}+R^{i}\dot{u}^{i}, \tag{7}\]  
where a double dot above a letter denotes double differentiation with respect to time and \(\alpha^{i}\) denotes the angular acceleration vector of body \(i\).