# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 4 Space debris Mitigation
ESA's e.Deorbit project before its cancellation was to involve robotic arms to capture its target [48] while the EC's removeDEBRIS project will explore harpoons and nets for de-orbiting at LEO. Tethered harpoons with conical tips have been proposed for attaching a chaser to debris with a safe standoff distance with perforation energy given by [PERSON]'s formula: \(E=kd^{1.5}b^{1.4}\) where \(k=37\times 10^{7}\) for aluminium, \(d=\) cone base diameter, \(b=\) target thickness. Harpoon tests at low temperature indicate brittle fracture in the target but without spallation or fragmentation into the internal space [49]. Tethered nets on the other hand can cause damage to appendages on spacecraft releasing debris. Harpoons and nets are the most likely capture mechanisms to cause breakage and the creation of more debris. Furthermore, both are remote capture mechanisms that are not compatible with tumbling spacecraft targets. Towing with flexible tethers introduces problems with employing thrusters to accelerate/decelerate. A mechanical tether may be employed to re-orbit debris into disposal orbits [50; 51] but they are impractically long. This suggests that the e.Deorbit approach was more versatile than that of removeDEBRIS. Debris mitigation is a role to which robotic on-orbit servicers can naturally adapt themselves. Robotic capture is the most controllable approach in which the debris is captured, manoeuvred to a lower orbit (at LEO) or graveyard orbit (at GEO) and/or a de-orbit device attached (such as propulsive units, tethers, sails or ballutes). Capture requires that the robotic spacecraft control its orbit and attitude to reduce the relative motion between the chaser and the target. A Kalman filter algorithm must estimate the target motion in order to plan the gripper trajectories for grappling with minimum impact [52]. Many of the critical dynamic parameters--specifically kinematic variables--of the target can be obtained by estimation through the Kalman filter based on visual data. CAD models can provide accurate estimates of mass and moment of inertia, but it is feasible to estimate these values if only approximately if shape has been determined visually and assuming a generic spacecraft model with assumptions regarding mass distribution--for a common launch shroud, this should be feasible. Critical to robotic capture is the implementation of force/torque control to acquire and brake the debris target. It has been proposed that electrostatic torques may be employed to detumble the target without physical contact to \(<\)1\({}^{\circ}\)/s from the typical 30\({}^{\circ}\)/s tumbling [53]. This involves controlling the servicer's electrostatic potential using an electron gun to beam charges to the closest feature on the rotating target. However, the feasibility of this approach has yet to demonstrated. An extendible brush of PTFE bristles at the tips of the robotic fingers may be employed to contact and tap the surface of the target to reduce high rotation rates to a manageable level for robotic capture [54]. The frictional force for a bristle model is defined by:  
\[F=k\delta(t) \tag{3}\]  
where \(\delta(t)=\left\{ \begin{array}{cc}\delta(t_{0})+\int_{t_{0}}^{t}\dot{\delta }.dt\;if\;|\delta|<\delta_{max}&\text{where }\delta(t)=\text{bristle displacement, }\delta_{max}=\mu\frac{F_{B}}{k},\,k=\text{bristle }\\ \delta_{max}\;otherwise&\text{where }\delta(t)=\text{bristle displacement, } \delta_{max}=\mu\frac{F_{B}}{k},\,k=\text{bristle }\end{array} \right.\) stiffness, \(\dot{\delta}=\) relative tangential velocity, \(\delta_{max}=\) maximum bristle displacement, \(\mu=\) surface friction. During braking, joint error control may be based on impedance control:  
\[e=\theta_{i}^{d}-\theta_{i}=\frac{J^{T}F_{B}(t)}{k_{i}} \tag{4}\]where \(F_{B}=\) braking force on target, \(\theta_{i}=\) joint angle, \(k_{i}=J^{T}k_{ee}\) = joint stiffness, \(J^{T}=\) Jacobian transpose, which relates stiffness at the end effector to stiffness at joints.