# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 6 Freeflyer Manipulator Kinematics
Given that accuracies of 0.01-0.1\({}^{\circ}\) are required for grasping during the final target approach, attitude should be controlled. The control of attitude is also essential to maintain pointing requirements of solar panels to the sun and antennae to the Earth. Active attitude control is desirable particularly under conditions of higher levels of autonomy. A dedicated attitude control system will be required when the manipulators are not in operation, and indeed, all spacecraft possess attitude control systems to compensate for attitude perturbations. Both the SRMS and SPDM had limited angular position accuracy of 1\({}^{\circ}\) compared with \(<\)0.1\({}^{\circ}\) for reaction wheels. The spacecraft base attitude may be controlled using internal angular momentum management devices such as reaction wheels, momentum wheels and/or control moment gyroscopes. The wheels or gimbals are mounted in orthogonal directions often with a fourth skewed wheel for redundancy. The strawman robotic servicer that we have adopted is the ATLAS (advanced telerobotic actuation system) concept for LEO servicing with a wet mass of 1425 kg (of which 476 kg is propellant) (Figure 1). It is configured with two identical manipulators with a mass of 45 kg each (with a similar kinematic structure as the PUMA 560/600 but with inline outboard joints from the shoulder), including grappling end effector but excluding payload (assumed to be a 225 kg ORU typical of MMS design). We consider initially that one manipulator remains stowed to emulate single-manipulator operation and it employs control moment gyroscopes for three-axis stabilisation. Lie groups is an alternative representation of kinematics that uses the product of exponentials formulation [114; 115], but we have adopted the more familiar and more intuitive Denavit-Hartenburg matrix representation. Manipulator kinematics define the relationship between the manipulator joint angles, link geometry and the end effector position and orientation. For ground-mounted manipulators, they are given by:  
\[q=f(\theta) \tag{17}\]  
\[\dot{q}=\left( \begin{array}{c}v\\ w\end{array} \right)=J(\theta)\dot{\theta} \tag{18}\]  
\[\ddot{q}=\left( \begin{array}{c}\dot{v}\\ \dot{w}\end{array} \right)=J\ddot{\theta+}\dot{j}\dot{\theta} \tag{19}\]  
where \(q=\) cartesian generalised position/orientation of end effector, \(\dot{q}=\) cartesian generalised velocities of end effector, \(\ddot{q}=\) cartesian generalised accelerations of end effector, \(v=\) translational cartesian end effector velocity, \(w=\) angular cartesian end effector velocity, \(\theta=\) manipulator joint angles, \(\dot{\theta}=\) manipulator joint velocities, \(\ddot{\theta}=\) manipulator joint accelerations, \(f(\theta)=\) cumulative Denavit-Hartenburg transform, \(J=\) manipulator Jacobian, \(\dot{j}=\) differential of Jacobian. For control purposes, these need to be inverted to yield joint kinematic parameters in terms of cartesian kinematic parameters. For a space-based manipulator, the reference body is typically selected to be the mounting spacecraft though it can be any body of the kinematic chain [116] (Figure 2). The situation for space-based manipulators requires the imposition of constraints--conservation of momentum--to yield solutions. Holonomic systems can be integrated from velocity relations to position relations whereas nonholonomic systems cannot be so integrated. Conservation of linear and angular momentum states:  
\[P=\sum_{i=0}^{n+1}m_{i}\dot{r}_{i}=m_{T}\dot{r}_{cm}=const \tag{20}\]  
\[L=\sum_{i=0}^{n+1}I_{i}w_{i}+m_{i}r_{i}\times\dot{r}_{i}=I_{cm}w_{cm}+m_{T}r_{ cm}\times\dot{r}_{cm}=const \tag{21}\]  
Conservation of linear momentum as a holonomic system can be integrated into the equilibrium of moments principle:  
\[P=\sum_{i=0}^{n+1}m_{i}r_{i}=m_{T}r_{cm}=const \tag{22}\]  
Conservation of angular momentum is not integrable due to the non-commutativity of rotation and so is a non-holonomic constraint.