# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 10 Space Manipulator Force Control
The vertical depth of depression of a fingertip with a radius of fingertip curvature \(r_{0}\) is given by:  
\[d=\left(\frac{c^{2}}{2r_{0}}\right)F^{2\gamma} \tag{116}\]  
Contact stiffness of a compliant finger is given by:  
\[k=\frac{\partial F}{\partial d}=\frac{c^{2\gamma}}{2\gamma}F^{1-2\gamma} \tag{117}\]  
Hence, contact stiffness is nonlinear and increases quickly with higher contact force. Impact dynamics are determined by the law of conservation of momentum. Impulse is defined as the integral of the impact force between two bodies:  
\[P=\int Fdt=\int m\ddot{q}=m_{1}(v_{1f}-v_{1i})=-m_{2}(v_{2f}-v_{2i}) \tag{118}\]  
Energy loss during impulse is given by:  
\[E=-\int F\dot{q}.dt=-\int Fdq \tag{119}\]  
where \(q=\mathrm{deformation}\). Hertzian contact assumes an elastic indentation on impact at low velocity but some plastic deformation may occur even under nominally Hertzian conditions. There is a loss of energy accompanying any impact due to deformation that is quantified by the coefficient of restitution \(e\)[222; 223]--it relates relative velocities between impactors before and after impact:  
\[e=-\frac{\Delta\dot{q}_{f}}{\Delta\dot{q}_{i}} \tag{120}\]where (perfectly plastic) \(0<e<1\) (perfectly elastic). Impedance control implies end effector behaviour of \(F_{ext}=\overrightarrow{m_{d}}+\tilde{b}\Delta\dot{q}+\tilde{k}\Delta q\) which yields a penetration depth given by:  
\[q(t)=\frac{\Delta\dot{q}_{i}}{w_{d}}e^{-\zeta w_{n}t}sin(w_{d}t) \tag{121}\]  
where \(w_{d}=w_{n}\sqrt{1-\zeta^{2}}\) = damped oscillation frequency, \(\zeta=\frac{\tilde{b}}{2\Delta m_{t}}\), \(w_{n}=\sqrt{\frac{\tilde{k}}{m}}\) = natural frequency of oscillation, \(k\) = stiffness coefficient of impactor, \(b\) = damping coefficient of impactor, \(m_{i}\) = mass of impactor, \(m_{t}\) = mass of target, \(\overline{m}=\frac{m_{t}m_{t}}{m_{i}+m_{t}}\), \(\tilde{b}=\frac{m_{t}}{m_{i}+m_{t}}b\), \(\tilde{k}=\frac{m_{t}}{m_{i}+m_{t}}k\). This may be differentiated thus:  
\[\dot{q}(t)=-e^{-\zeta w_{n}t}\left(\frac{\zeta\Delta\dot{q}_{i}}{\sqrt{1-\zeta ^{2}}}sin(w_{d}t)+\Delta\dot{q}_{i}cos(w_{d}t)\right) \tag{122}\]  
Duration of contact is equated to a half oscillation period:  
\[\Delta t=t_{f}-t_{i}=\frac{\pi}{w_{d}}=\frac{\pi}{w_{n}\sqrt{1-\zeta^{2}}} \tag{123}\]  
\[\text{Therefore, }\Delta\dot{q}(t_{f})=\Delta\dot{q}_{i}exp\left(-\frac{\zeta \pi}{\sqrt{1-\zeta^{2}}}\right) \tag{124}\]  
Hence, the coefficient of restitution may be expressed as:  
\[d=\left(\frac{c^{2}}{2r_{0}}\right)F^{2\gamma} \tag{116}\]0  
When critically damped, \(\zeta\) = 1, e = 0. Hence, coefficient of restitution can be controlled. Rather than the inaccurate Hertzian linear spring-damper model of impact contact, nonlinear viscoelastic force is given by the Hunt-Crossley expression for impact force [224]:  
\[d=\left(\frac{c^{2}}{2r_{0}}\right)F^{2\gamma} \tag{116}\]1  
where \(\gamma\) = constant = 0.25-0.5, \(K\) = stiffness coefficient, \(B=\frac{3K(1-e^{2})}{4\Delta v_{i}}\) = damping coefficient, \(q\) = deformation. The collisional force may be controlled with an integrated force and damping control law of the form [225]:  
\[d=\left(\frac{c^{2}}{2r_{0}}\right)F^{2\gamma} \tag{116}\]2  
where \(f_{imp}\) = impact force. Alternatively, an integral controller on the force error may be implemented which stabilizes a PD controller with high contact stiffness [226]. Once the transient impact phase is completed, physical attachment of the manipulator(s) to the target must be error-tolerant, robust and stable. V-grooves may be employed as physical guides to ensure correct alignment prior to latching. A standardised grappling system comprises a motorised spring-loaded pantograph that mounts a probe which is inserted into a probe receptacle with guide rails mounted on the target [227]. On docking, the pantograph is compressed retracting the docking probe for a secured dock with aligned electrical and fluid connections (the latter are not required for active debris removal).