# Sampling-Based Motion Planning for Free-Floating Space Robot without Inverse Kinematics
## 2 Motion Equations of an FFSR
### Basic Motion Equation and Momentum Conservation Equation in Velocity Form, Jacobian Matrix  
The momentum conservation equation in velocity form is employed to derive the Jacobian matrix related to state transition, the generalized Jacobian matrix, and the Jacobian matrix related to base attitude. In order to obtain the momentum conservation equation in velocity form, the velocity equation of each link must be given. The angular velocity of each link is  
\[\mathbf{\omega}_{i}\ =\ \mathbf{\omega}_{0}+\sum_{j\,=\,1}^{i}\,k_{j}\dot{\rho}_{j},i \ =\ 1,2,\ldots,n \tag{4}\]  
The linear velocity of the mass center of each link is  
\[\mathbf{v}_{i}\ =\ \dot{\mathbf{r}}_{i}\ =\ \mathbf{v}_{0}+\mathbf{\omega}_{0}\times(\mathbf{r}_{i }-\mathbf{r}_{0})+\sum_{j\,=\,1}^{i}\big{[}\mathbf{k}_{j}\times\big{(}\mathbf{r}_{i}-\mathbf{ p}_{j}\big{)}\big{]}\dot{\rho}_{j},i\ =\ 1,2,\ldots,n \tag{5}\]  
In addition, the linear velocity of EE is  
\[\mathbf{v}_{e}\ =\ \dot{\mathbf{p}}_{e}\ =\ \mathbf{v}_{0}+\mathbf{\omega}_{0}\times\big{(}\mathbf{p }_{e}-\mathbf{r}_{0}\big{)}+\sum_{j\,=\,1}^{n}\big{[}\mathbf{k}_{j}\times\big{(}\mathbf{p }_{e}-\mathbf{p}_{j}\big{)}\big{]}\dot{\rho}_{j} \tag{6}\]  
According to Equations (4) and (6), the linear velocity and angular velocity of the EE can be obtained. The matrix form is  
\[\left[ \begin{array}{c}\mathbf{v}_{e}\\ \mathbf{\omega}_{e}\end{array} \right]\ =\ \mathbf{J}_{b}\left[ \begin{array}{c}\mathbf{v}_{0} \\ \mathbf{\omega}_{0}\end{array} \right]+\mathbf{J}_{m}\dot{\Theta} \tag{7}\]  
where \(\mathbf{J}_{b}\) and \(\mathbf{J}_{m}\) are the Jacobian matrixes to the base and the joint angle, respectively. \[\mathbf{J}_{b}\ =\ \left[ \begin{array}{cc}\mathbf{E}_{3\times 3}&-\overline{\mathbf{p}}_{0e} \\ \mathbf{\theta}_{3\times 3}&\mathbf{E}_{3\times 3}\end{array} \right],\mathbf{p}_{0e}\ =\ \mathbf{p}_{e}-\mathbf{r}_{0e},\mathbf{J}_{m}\ =\ \left[ \begin{array}{cc}\mathbf{k}_{1}\times\big{(}\mathbf{p}_{e}-\mathbf{p}_{1}\big{)}&\mathbf{k}_{2 }\times\big{(}\mathbf{p}_{e}-\mathbf{p}_{2}\big{)}&\ldots&\mathbf{k}_{n}\times\big{(}\mathbf{p }_{e}-\mathbf{p}_{n}\big{)}\\ \mathbf{k}_{1}&\mathbf{k}_{2}&\ldots&\mathbf{k}_{1}\end{array} \right]\]  
Given two vectors \(\mathbf{v}\ =\ \left[ \begin{array}{ccc}x&y&z\end{array} \right]^{T}\) and \(\mathbf{w}\), the cross-product operator is defined as follows:  
\[\overline{\mathbf{v}}\ =\ \left[ \begin{array}{ccc}0&-z&y\\ z&0&-x\\ -y&x&0\end{array} \right],\mathbf{v}\times\mathbf{w}\ =\ \overline{\mathbf{v}}\mathbf{w}\]The linear momentum is  
\[\mathbf{P}\ =\ m_{0}\dot{\mathbf{v}}_{0}+\sum_{i\ =\ 1}^{n}\left(m_{i}\dot{\mathbf{v}}_{i} \right)\ =\ m_{0}\dot{\mathbf{v}}_{0}+\sum_{i\ =\ 1}^{n}\left(m_{i}\dot{\mathbf{r}}_{i}\right) \tag{8}\]  
Additionally, the angular momentum is  
\[\mathbf{L}\ =\ \sum_{i\ =\ 0}^{n}\left[I_{i}\mathbf{\omega}_{i}+\mathbf{r}_{i}\times \left(m_{i}\dot{\mathbf{v}}_{i}\right)\right]\ =\ \sum_{i\ =\ 0}^{n}\left[I_{i}\mathbf{\omega}_{i}+\dot{\mathbf{r}}_{i}\times \left(m_{i}\dot{\mathbf{r}}_{i}\right)\right] \tag{9}\]  
Substituting (4) and (5) into the above formulas and sorting them into matrix form, we can get the linear momentum equation:  
\[\mathbf{P}\ =\ \left[ \begin{array}{cc}M\mathbf{E}&M\widetilde{\mathbf{r}}_{og}{}^{T}\end{array}  \right]\ \left[ \begin{array}{c}\dot{\mathbf{v}}_{0}\\ \omega_{0}\end{array} \right]+J_{Tw}\dot{\mathbf{\Theta}} \tag{10}\]  
where  
\[\mathbf{r}_{og}\ =\ \mathbf{r}_{g}-\mathbf{r}_{0},\ \ \mathbf{J}_{Tw}\ =\ \sum_{i\ =\ 1}^{n} \left(m_{i}J_{Ti}\right)\]  
\[\mathbf{J}_{Ti}\ =\ \left[ \begin{array}{cc}\mathbf{k}_{1}\times\left(\mathbf{r}_{i}- \mathbf{p}_{1}\right)&\mathbf{k}_{2}\times\left(\mathbf{r}_{i}-\mathbf{p}_{2}\right)&\ldots& \mathbf{k}_{i}\times\left(\mathbf{r}_{i}-\mathbf{p}_{i}\right)&0&\ldots&0\end{array} \right]\]  
The angular momentum is  
\[\mathbf{L}\ =\ \left[ \begin{array}{cc}M\widetilde{\mathbf{r}}_{og}&\mathbf{H}_{w}\end{array}  \right]\left[ \begin{array}{c}\dot{\mathbf{v}}_{0}\\ \omega_{0}\end{array}