# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 6 Freeflyer Manipulator Kinematics
)\) = R = \(3\times 3\) rotation matrix identical to terrestrial manipulators, \(n=3\times 1\) normal vector, \(s=3\times 1\) slide vector, \(a=3\times 1\) approach vector  
\[p*=r_{c0}+R_{0}s_{0}+\sum_{i=1}^{n}R_{i}l_{i}=p_{cm}^{*}+\left(\frac{m_{0}}{m_ {T}}\right)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
The virtual manipulator (VM) approach assumes a serial link chain of zero mass with its base fixed in inertial space at the spacecraft-manipulator system centre of mass (virtual ground) and whose end effector coincides with that of the real manipulator [125; 126]. The dynamically equivalent manipulator (DEM) is similar to the VM approach in that it is a virtual manipulator with its base fixed to the centre of mass of the spacecraft-manipulator system. Although the spacecraft base is fixed, the first joint at the manipulator mount is spherical and passive with the end effector coinciding with the real one [127]. According to the dynamically equivalent manipulator model (DEM), the position of the spacecraft mount is given by:  
\[r_{cm}=r_{0}-\frac{(\sum_{i=1}^{n}m_{i})(s_{0}+r_{1})}{m_{T}}-\frac{(\sum_{i= 2}^{n}m_{i})(s_{1}+r_{2})}{m_{T}}-\ldots-\frac{m_{n}(s_{n-1}+r_{n})}{m_{T}} \tag{29}\]  
\[\text{Therefore, }p*=r_{cm}+s_{0}+\sum_{i=1}^{n}\left(r_{i}+s_{i}\right) \tag{30}\]  
where \(s_{i}=\frac{1}{m_{T}}\sum_{j=0}^{i}m_{j}s_{i}\) and \(r_{i}=\frac{1}{m_{T}}\sum_{j=0}^{i}m_{j}r_{i}\)  
The difference between our formulation and the others is that our formulation maps from joint to joint rather than link centroid to link centroid--this is essential as a step towards driving the joint motors. This permits our formulation to be used directly into the standard terrestrial robotic manipulator algorithms including the Denavit-Hartenburg matrix and Newton-Euler dynamics with only minor changes, namely the link length parameters. Joint trajectories for a space manipulator may be parametrised by polynomials similar to a terrestrial manipulator [128]. The inverse solutions follow from the relative position of the end effector with respect to the manipulator mounting point:  
\[p_{er}^{\star}=\sum_{i=1}^{n}R_{i}\lambda_{i}=p*-p_{cm}^{\star}-\left(\frac{m _{0}}{m_{T}}\right)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!  
Hence,  
\[\dot{q}=v*=J_{s}(-D_{s}^{-1}D_{m}\dot{\theta}_{m})+J_{m^{\prime}}\dot{\theta}_{m} =(J_{m^{\prime}}-J_{s}D_{s}^{-1}D_{m})\dot{\theta}_{m}=J^{\prime}\dot{\theta} \tag{38}\]  
\[\text{where }J^{\star}=J_{m^{\prime}}-J_{s}D_{s}^{-1}D_{m}=\text{generalised Jacobian} \tag{39}\]  
The generalised Jacobian thus incorporates dynamic as well as kinematic parameters. Its computational complexity can be compared with a traditional Jacobian (Table 4) [131]. For N manipulator arms with n DOF each, the generalised Jacobian has a complexity of (156 + 6N)n + 87 Multiplications and (126 + 6N)n + 6N + 53 additions. The issue of computational complexity becomes acute in the inversion of the Jacobian required for the inverse kinematics. The importance of algorithm development may be illustrated by the replacement of Gaussian elimination (used in Jacobian inversion) with a complexity of O(n\({}^{3}\)/3) with Levinson's recursion with a reduced complexity of O(n\({}^{2}\) + 2n). The higher computational complexity of the generalised Jacobian favours splitting the computational load between the manipulator control system and a dedicated attitude control system. Singularities in terrestrial manipulators are purely kinematic because the Jacobian is kinematic. Singular joint configurations in freefloaters cannot be mapped into unique points in the cartesian workspace because the generalised Jacobian is a dynamic function rather than a purely kinematic function like the traditional Jacobian. These singularities cannot be predicted from the kinematic configuration alone because they are functions of the history of the end effector path (as a consequence of the non-commutativity of rotations) [132]. The generalised Jacobian suffers from dynamic singularities that are unpredictable [133].