# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 10 Space Manipulator Force Control
This is the basis of the ESA mechanical latch which requires a target capture pin [228]. However, these types of docking mechanism require that the target spacecraft has been designed for servicing and provides a pre-existing interface mechanism for debris removal. This will not, in general, be the case. The issue of a secure latch on an uncooperative target has yet to be solved (this is the rationale behind harpoons and nets), as most latch designs assume a benign target. This suggests that the versatility of grippers renders them the most likely of solutions. Once contact has occurred, there are interaction forces between the manipulator and the target that must be controlled. The environment stiffness is typically high \(k_{e}\)\(\sim\)10\({}^{5}\)-10\({}^{6}\) N/m and produces little damping so high natural frequency responses are generated with frequency:  
\[d=\left(\frac{c^{2}}{2r_{0}}\right)F^{2\gamma} \tag{116}\]3  
where \(k_{f}=\frac{\partial F}{\partial\dot{\eta}}=J^{T}k_{p}J=\mathrm{manipulator compliance}\), \(\mathrm{m}_{\mathrm{w}}=\mathrm{wrist mass}\). This oscillation will be persistent and will generate contact-make/contact-break cycles. If \(k_{f}k_{e}\delta t<1\), the system is stable, so stiff environments must be met with low stiffness manipulation. Hence, force sensing requires high sample rates \(\sim\)100+ Hz. On grasping, hybrid position/force control and impedance control are the most popular methods of force control. Hybrid position/force control is based on separating movement from force exertion; impedance control is based on the interaction between the manipulator and the environment in which force equates to both stiffness and viscosity. Twist is defined as the \(6\times 1\) generalised velocity vector indicating free space motion, \(\dot{q}=\left( \begin{array}{c}v\\ w\end{array} \right)=J(\dot{\theta})\dot{\theta}\) while wrench is defined as the \(6\times 1\) generalised force vector indicating constrained motion, \(\omega=\left( \begin{array}{c}f\\ n\end{array} \right)\). Impedance control (\(F=\ddot{q}^{d}+k_{m}^{-1}(k_{p}\Delta\dot{q}+k_{p}\Delta q+F_{ext})\)) is a generalisation of compliance control (\(F=k_{p}\Delta q\) where \(k_{p}=\mathrm{stiffness gain}\)) and admittance control (\(F=k_{p}\Delta\dot{q}\) where \(k_{p}=\mathrm{damping gain}\)) that controls the mechanical impedance at the end effector in contact with a rigid environment [229; 230; 231]. Admittance control incorporates an outer force feedback controller that generates position commands to an inner position controller. Unfortunately, it is stable only in non-rigid environments. During manipulator-environment interaction, one side must behave as an impedance and the other as an admittance--since the environment generally presents admittances, the manipulator must act as an impedance through impedance control. Indirect force control such as impedance control does not require explicit measurement of force rather it employs motion control. Impedance of the manipulator involves controlling contact forces through stiffness and damping control, i.e., a PD control scheme. Impedance control relates force to motion through mechanical impedance--the manipulator is modelled as a mass-springer-damper. Impedance control converts position error into force and eliminates the problem of instabilities in direct force feedback. Impedance control lacks positional accuracy compared with hybrid position/force control but requires no switching on contact. Physical contact between the manipulator and the target during grappling introduces external forces and torques to the manipulator(s). Initial impact generates transient contact dynamics which in the simplest analysis may be modelled as pre- and post-impact momentum depending on the coefficient of restitution. Contact forces are typically computed from a Hertz model. Impedance control may be employed to provide mass-spring-damper behaviour at the contact between the manipulator and the target to absorb impact energy [232].