# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 6 Freeflyer Manipulator Kinematics
Dynamic singularities are configuration and path dependent making path planning challenging. One approach to combat this is to introduce singular value decomposition to obtain a pseudoinverse of the generalised Jacobian [134; 135]. The pseudoinverse has a severe computational cost in making the already complex generalised Jacobian even more complex. We can simplify the generalised Jacobian formulation by adopting the freeflyer approach, which employs attitude control so that \(\dot{R}_{0}=0\), J\({}_{s}=0\) and w\({}_{0}=0\) yielding the extended Jacobian:  
\[v^{*}=\ \dot{p}_{cm}^{*}+\left(\frac{m_{0}}{m_{T}}\right)\dot{\wp}_{0}+\sum_{i= 1}^{n}\dot{R}_{i}\lambda_{i}-\left(\frac{m_{n+1}}{m_{T}}\right)\dot{R}_{n+1}r _{n+1} \tag{40}\]  
Since \(\dot{p}_{cm}^{*}=0\) by definition, \(\dot{\wp}_{0}=0\) since s\({}_{0}\) is fixed and \(\dot{R}_{n+1}=0\), because any grasp is invariant:  
\[v*=\sum_{i=1}^{n}\dot{R}_{i}\lambda_{i}=\left(\sum_{i=1}^{n}\sum_{k=1}^{i} \frac{\partial R_{i}}{\partial\theta_{k}}\lambda_{i}\right)\dot{\theta}_{i} \tag{41}\]  
\[\text{where }J_{m}=\sum_{i=1}^{n}\sum_{k=1}^{i}\frac{\partial R_{i}}{\partial \theta_{k}}\lambda_{i}=\text{extended Jacobian} \tag{42}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No. of Manipulators** & & **Conventional Jacobian** & **Generalised Jacobian** \\ \hline
1 & Mult & 40n \(-\) 24 = 216 for n = 6 & 162n + 87 = 1059 for n = 6 \\  & Add & 24n \(-\) 18 = 126 for n = 6 & 132n + 59 = 869 for n = 6 \\
2 & Mult & 432 & 168n + 87 = 2103 for n = 6 \\  & Add & 252 & 138n + 65 = 1721 for n = 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational complexity of Jacobians. This extended Jacobian \(J_{m}\) is different to the manipulator Jacobian \(J_{m^{\prime}}\) earlier in Equation (35) in that it incorporates parameters related to the spacecraft base. The cross-product formulation of the Jacobian can be derived similarly:  
\[v*=\sum_{i=1}^{n}\left(1-\frac{1}{m_{T}}\sum_{j=i+1}^{n+1}m_{j}\right)\left(z_{ i-1}\times l_{i}\right)\dot{\theta}_{i}-\frac{1}{m_{T}}\sum_{i=1}^{n}m_{i}(z_{i-1} \times r_{i})\dot{\theta}_{i}=\sum_{i=1}^{n}\left(z_{i-1}\times\lambda_{i} \right)\dot{\theta}_{i} \tag{43}\]  
Resolved acceleration follows directly by differentiation:  
\[\left( \begin{array}{c}\dot{v}^{*}\\ \dot{w}\end{array} \right)=J_{m}\ddot{\theta}_{i}+\dot{J}_{m}\dot{\theta}_{m} \tag{44}\]  
Whereas the generalised Jacobian suffers from unpredictable dynamic singularities, the extended Jacobian (equivalent to the \"dynamic\" Jacobian [136]) is dependent on kinematic parameters and masses which are fixed, and so singularities are kinematic in nature and predictable. Hence, the implementation of attitude measurement and control solves the problem of unpredictability of dynamic singularities [119]. Nevertheless, the spacecraft base does translate, but does not rotate:  
\[v_{0}=-\frac{1}{m_{T}}\sum_{j=1}^{n}\sum_{i=1}^{n}m_{j}v_{ci} \tag{45}\]  
Due to the desirability of maintaining a stable spacecraft attitude in freefloaters, many efforts have focussed on minimising reaction disturbances from the manipulator to the mounting base, e.g., visual servoing requires stabilisation of spacecraft attitude [137]. The implementation of cycling motion superimposed over the desired trajectory acts equivalently to a set of reaction wheels [138], but this introduces the potential for collision in grasping a target. The reaction null space approach exploited null space to simultaneously control spacecraft attitude and manipulator joint angles by applying a cost function to the pseudoinverse to minimise reaction torques on the spacecraft. [139; 140]:  
\[\ddot{\theta}=(J^{*})^{+}(\ddot{q}-\dot{J}^{*}\theta)+\dot{(}I-(J^{*})^{+}J^{*} )\ddot{\zeta} \tag{46}\]  
where \(\zeta\) = null space vector of dimension \(n-3\). The reactionless approach to satellite capture by a manipulator exploits reaction null space to eliminate attitude movements of the mounting base [141].