# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 6 Freeflyer Manipulator Kinematics
The non-holonomic character of the freefloating system arises due to the non-integrability of angular momentum. Different end effector paths to the same cartesian end effector position will result in different spacecraft attitudes. Solutions can be obtained through the introduction of manipulator \"coning\" motion onto the desired manipulator trajectory to maintain attitude stabilisation, but this is a rather ungainly approach, as well as having a high risk of collision. With two arms, one can perform the required task while the other undergoes coning motion to stabilise the base \"bucking bronco\" style, but the ungainliness, as well as collision hazard, remains. The holonomy/non-holonomy division suggests that separate treatment of linear kinematics and angular kinematics may be warranted. While it is possible to separate the motion of the spacecraft base and the mounted manipulator [117], a more natural separation is between the linear and angular aspects of the system dynamics. We have adopted a 6-degrees-of-freedom manipulator, but such manipulators can suffer from singular configurations. A singularity occurs when the manipulator Jacobian loses rank, i.e., a degree of freedom. Singular configurations may occur when \(|\mathbb{J}^{\ast}\mathbb{J}|=0\), but they are predictable from the kinematics. From this, manipulability is defined by \(\sigma=\sqrt{det|\mathbb{J}^{\ast}\mathbb{J}|}\). To eliminate singularities, redundant joints may be added. It is often assumed that the spacecraft-mounted manipulator possesses 7 degrees of  
Figure 2: Kinematic configuration of a single-manipulator mounted spacecraft servicer. freedom for kinematic redundancy for greater manipulability, e.g., SSRMS. Kinematic redundancy offers great flexibility of postures for any given task but it requires the Moore-Penrose pseudoinverse of the Jacobian matrix for solution, \(\dot{\theta}=J^{+}\dot{q}\) where \(J^{+}=J^{T}(JJ^{T})^{-1}\) = Moore-Penrose pseudoinverse. Several performance conditions can be incorporated:  
\[\dot{\theta}=J^{+}\dot{q}+(I-J^{+}J)z \tag{23}\]  
where \((I-J^{+}J)\) represents a projection onto the nullspace of J while z is the nullspace vector. For example, the damped least square inverse Jacobian includes a damping factor for greater smoothness in the neighbourhood of singularities [118; 119]: \(J^{+}=J^{T}(JJ^{T}+\lambda^{2}I)^{-1}\) where \(\lambda\) = damping factor. The additional redundant degree of freedom comes at the cost of additional mass imposed by the extra joint and the Moore-Penrose pseudoinverse is more complex to compute compared to the standard Jacobian. Since a freelying servicing spacecraft provides a re-orientable spacecraft mount for the manipulator, a 6-degrees-of-freedom manipulator may be more appropriate (though the formulation we present is applicable to any kinematic configuration). We have generalised the approach used for a shuttle-mounted manipulator employing attitude control [120; 121] to any kinematic configuration [122; 123]. This is the freeflyer approach. For a freeflying manipulator with n links plus a payload n + 1 mounted on a spacecraft of mass m\({}_{0}\) employing dedicated attitude control of the spacecraft mount, the Denavit-Hartenburg matrix may be represented as:  
\[q=\left( \begin{array}{cccc}n&s&a&p*\\ 0&0&0&1\end{array} \right) \tag{24}\]  
where \(( \begin{array}{cccc}n&s&a\end{array}