# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 6 Freeflyer Manipulator Kinematics  
In rendezvous and docking (RVD) manoeuvres, the chaser moves to a proximity position \(\sim\)100-300 m from the target before manoeuvring a series of inspection flyarounds culminating in a V-bar dock [110]. V-bar involves manoeuvring the chaser horizontally along the same orbital velocity vector of the target; R-bar involves manoeuvring the chaser radially along the radial vector from a different orbit than the target. The relative (circular) orbits of the chaser spacecraft and the target spacecraft are described by the Clohessy-Wilshire version of the Hill equations with respect to the target satellite. The Clohessy-Wiltshire equations describe the phasing manoeuvres for a rendezvous between the chaser and the target. In approaching a tumbling satellite, the chaser matches its angular velocity with the line of sight rotation of the target before grappling. Our concern here is primarily with the subsequent phases of a servicing missions. A spacecraft mounting one or more manipulators comprises a chain of multiple bodies of n + 1 rigid bodies connected by n single degree of freedom joints. The first body i = 0 is the spacecraft mount, from which n serial links emanate (plus payload n + 1). The dynamics of this multibody system are based on the conservation of momentum. The barycentric approach to multibody dynamics also locates the system centre of mass. The mounting point of a spacecraft-mounted manipulator is not fixed inertially (unlike in a ground-based manipulator). Furthermore, the system operates under zero-gravity conditions. Inertial coordinates are defined with respect to the system centre of mass which is invariant (i.e., a locally inertial coordinate), and spacecraft base coordinates are defined with respect to the centre of mass (centroid) of the spacecraft base. There are two types of dynamic situation: (i) freefloating systems in which the manipulators are the only actuators, so spacecraft attitude is nominally uncontrolled (with six joint actuators controlling a 12 degree of freedom system, this is an underactuated (redundant) system [111]); (ii) freelying systems in which the manipulators are controlled by manipulator actuators while spacecraft attitude is controlled by attitude actuators. Due to dynamic coupling, motion of the manipulator affects the position and attitude of the spacecraft base in both cases. Manipulator joints incorporate spinning motors which act like gyroscopes on rotating levers with strong Coriolis and centrifugal forces [112]. For example, SRMS had motor gear ratios of 749:1 to 1842:1, so the motors were spinning up to 2000 times the link rotation, enabling high torque amplification at the manipulator joints, but with a significant effect on attitude stabilisation. In freeflyers, the spacecraft base is attitude controlled. While translational control requires the expenditure of propellant, attitude control does not necessarily require fuel expenditure. Although it is possible to stabilise the spacecraft position as well as attitude, this would require the expenditure of propellant thereby limiting the viability of such a spacecraft. Spacecraft position must therefore be uncontrolled in both freeflying and freefloating conditions. Currently, robotic manipulators are operated in free-floating mode with uncontrolled attitude. However, this requires constant adjustment on behalf of the ground operator. For example, the ETS-VII mission employed resolved rate control using the generalised Jacobian computed on the ground, so the attitude control system was switched off [113]. This caused the end effector pitch error to increase rapidly at 0.5-1\({}^{\circ}\) over several minutes until compensated for at the end of the motion by reaction wheels. Hence, the generalised Jacobian did not implement effective attitude control due to deficiencies in the available hardware (not the algorithm itself). Gravity gradient torques acting on the ETS-VII spacecraft over short timescales generated free drift of the spacecraft.