# 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
\right]+\mathbf{H}_{w0}\dot{\mathbf{\Theta}} \tag{11}\]  
where  
\[\mathbf{H}_{w}\ =\ \sum_{i\ =\ 1}^{n}\left(I_{i}+m_{i}\widetilde{\mathbf{r}}_{0i}{}^{T }\widetilde{\mathbf{r}}_{0i}\right)+I_{0},\ \ \ \mathbf{H}_{w0}\ =\ \sum_{i\ =\ 1}^{n}\left(I_{i}\mathbf{J}_{Ri}+m_{i}\widetilde{\mathbf{r}}_{0i}\mathbf{J}_{Ti} \right)\mathbf{r}_{0i}\ =\ \mathbf{r}_{i}-\mathbf{r}_{0}\]  
\[\mathbf{J}_{Ri}\ =\ \left[ \begin{array}{cccccccc}\mathbf{k}_{1}&\mathbf{k}_{2}&\ldots& \mathbf{k}_{i}&0&\ldots&0\end{array} \right]\]  
Put linear momentum and angular momentum equations together, and we can get  
\[\left[ \begin{array}{c}\mathbf{P}\\ \mathbf{L}\end{array} \right]\ =\ \left[ \begin{array}{cc}M\mathbf{E}&M\widetilde{\mathbf{r}}_{og}{}^{T} \\ M\widetilde{\mathbf{r}}_{og}&\mathbf{H}_{w}\end{array} \right]\left[ \begin{array}{c} \dot{\mathbf{v}}_{0}\\ \mathbf{\omega}_{0}\end{array} \right]+\left[ \begin{array}{c}\mathbf{J}_{Tw}\\ \mathbf{H}_{w0}\end{array} \right]\dot{\mathbf{\Theta}} \tag{12}\]  
Let  
\[\mathbf{H}_{b}\ =\ \left[ \begin{array}{cc}M\mathbf{E}&M\widetilde{\mathbf{r}}_{og}{}^{T} \\ M\widetilde{\mathbf{r}}_{og}&\mathbf{H}_{w}\end{array} \right],\mathbf{H}_{m}\ =\ \left[ \begin{array}{c}\mathbf{J}_{Tw}\\ \mathbf{H}_{w0}\end{array} \right]\]  
where \(\mathbf{H}_{b}\) is always invertible. The linear momentum and angular momentum of the FFSR are conserved. The initial momentum of the system is assumed to be 0, and the base velocity can be expressed as a linear function of the joint angular velocity:  
\[\left[ \begin{array}{c}\dot{\mathbf{v}}_{0}\\ \mathbf{\omega}_{0}\end{array} \right]\ =\ -\mathbf{H}_{b}{}^{-1}\mathbf{H}_{m}\dot{\mathbf{\Theta}}\ =\ \mathbf{H}_{b0}\dot{\mathbf{\Theta}}\ =\ \left[ \begin{array}{c}\mathbf{H}_{v}\\ \mathbf{H}_{w}\end{array} \right]\dot{\mathbf{\Theta}} \tag{13}\]  
Combining Equation (7), we can get  
\[\left[ \begin{array}{c}\dot{\mathbf{v}}_{e}\\ \mathbf{\omega}_{e}\end{array} \right]\ =\ \mathbf{I}_{g}\dot{\mathbf{\Theta}} \tag{14}\]where \(I_{g}\) is the generalized Jacobian matrix  
\[I_{g}\ =\ -I_{b}H_{b\Theta}+I_{m}\]  
According to (13), we can also get  
\[\omega_{0}\ =\ H_{\omega}\dot{\Theta} \tag{15}\]  
where \(H_{\omega}\) is called the Jacobian matrix related to base attitude. For the rigid body, the relationship between the first derivative of attitude angle and angular velocity is  
\[\omega\ =\ N\left[ \begin{array}{ccc}\dot{\alpha}&\dot{\beta}&\dot{\gamma}\end{array}  \right]^{T} \tag{16}\]  
where  
\[N\ =\ \left[ \begin{array}{ccc}1&\tan(\beta)*\sin(\alpha)&\tan(\beta)*\cos( \alpha)\\ 0&\cos(\alpha)&-\sin(\beta)\\ 0&\sin(\alpha)/\cos(\beta)&\cos(\alpha)/\cos(\beta)\end{array} \right]\]  
\(\omega\) is expressed in the coordinate system of the base in Equation (16). Considering the Formulas (15) and (16), the state transition equation of the FPSR can be gotten  
\[\left[ \begin{array}{c}\dot{\Theta}\\ \dot{\Psi}_{b}\end{array} \right]\ =\ \left[ \begin{array}{c}E_{m\times n}\\ N^{-1}\Big{(}I_{0}\Big{)}^{T}\end{array} \right]\dot{\Theta}\ =\ J_{x}\dot{\Theta} \tag{17}\]  
where \(J_{x}\) is the Jacobian matrix related to state transition.