# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 8 Spacecraft Attitude Stabilisation
The problem with reaction wheels is exacerbated during grasping when impact occurs at a relative velocity of 0.1 m/s and 1.0 m/s, the latter generating ~1-5 kNm reaction moments on the spacecraft (the former is more reflective of current docking velocities but still yields up to ~500 Nm reaction moments) (Figure 4). We may compare this with the maximum velocity of SPDM of 0.075 m/s and 2.5 \({}^{\circ}\)/s with a payload capability of 500 kg, but we suggest that future servicing missions will proceed at a faster pace under productivity pressures. Clearly, this is far beyond the torqueing capabilities of reaction wheels. Figure 3: Reaction moments generated by manipulator on spacecraft during free space motion N\({}_{\rm rx}\), N\({}_{\rm ry}\) and N\({}_{\rmrz}\)—note that the legend artifactually lists lines 4–6 which are not used in the graph. Control moment gyroscopes (CMG) offer a much higher torque capability as required for force control-based passivation [164]. A control moment gyroscope (CMG) comprises a fixed speed rotor (constant angular momentum) mounted into gimbals so that the angular momentum vector can be altered by driving the gimbals. A small input torque generates a highly amplified output torque on the spacecraft ~100-5000 Nm. In DGCMGs (double gimbal), the rotor is mounted into two gimbals so the rotor angular momentum can be rotated within a sphere (limited only by the gimbal stops). In SGCMGs (single gimbal), the rotor is mounted in a single gimbal so the rotor angular momentum can be rotated in a circle in a plane normal to the gimbal axis. CMGs are driven by a steering law that outputs CMG gimbal rates from spacecraft control torque inputs. DGCMGs have simpler steering laws to SGCMGs but the latter are mechanically much simpler than the former. Any set of n SGCMGs suffers from 2\({}^{\text{n}}\) singular gimbal angles. At singularities, gimbal lock occurs, and control torques cannot be generated to exit the singular condition. We assume that the CMG pyramid lies at the spacecraft centre of mass aligned with the spacecraft principal axes (and so only CMG axial angular momentum is considered). Euler equations for the whole spacecraft with the CMG cluster are given by:  
\[\tau=\dot{H}=\text{I}\ddot{w}+\dot{h}+w\times(Iw+h)\text{ with }\tau_{CMG}=\dot{h}+w\times h \tag{69}\]  
CMG parameters are given by [165]:  
\[\text{h}=\sum_{i=1}^{4}h_{i}(\gamma_{i})=\text{h}\left( \begin{array}{c}-cos \delta sin\gamma_{1}-cos\gamma_{2}+cos\delta sin\gamma_{3}+cos\gamma_{4}\\ cos\gamma_{1}-cos\delta sin\gamma_{2}-cos\gamma_{3}+cos\delta sin\gamma_{4}\\ sin\delta sin\gamma_{1}+sin\delta sin\gamma_{2}+sin\delta sin\gamma_{3}+sin \delta sin\gamma_{4}\end{array} \right) \tag{70}\]  
Figure 4: Reaction moments generated by manipulator on spacecraft during grappling N\({}_{\text{rx}}\), N\({}_{\text{ry}}\) and N\({}_{\text{rz}}\) with N\({}_{\text{ry}}\) at impact velocities 0.1 m/s and 1.0 m/s—note that the legend artifactually shows two additional lines which are not used in the graph. where \(h=\) axial angular momentum, \(\gamma_{i}=\) gimbal angles, \(\delta=\) pyramid skew angle. This may be differentiated to:  
\[\dot{h}=h\sum_{i=1}^{4}\frac{\partial h}{\partial\gamma_{i}}\dot{\gamma}_{i}=hJ( \gamma)\dot{\gamma}=\]  
\[h\left( \begin{array}{ccc}-cos\delta cos\gamma_{1}&sin\gamma_{2}&cos\delta cos \gamma_{3}&-sin\gamma_{4}\\ -sin\gamma_{1}&-cos\delta cos\gamma_{2}&sin\gamma_{3}&cos\delta cos\gamma_{4} \\ sin\delta cos\gamma_{1}&sin\delta cos\gamma_{2}&sin\delta cos\gamma_{3}&sin \delta cos\gamma_{4}\\ \end{array} \right)\left( \begin{array}{c}\dot{\gamma}_{1}\\ \dot{\gamma}_{2}\\ \dot{\gamma}_{3}\\ \dot{\gamma}_{4}\\ \end{array} \right) \tag{71}\]  
The \(3\times n\) Jacobian J(\(\gamma\)) is under-determined. Hence, gimbal rates are given by:  
\[\dot{\gamma}=J^{T}(JJ^{T})^{-1}\tau \tag{72}\]  
This is the Moore-Penrose pseudoinverse--if the rank of the cluster gain JJ^T < 3, the pseudoinverse does not exist (singular state). The pseudoinverse tends to steer the gimbals toward singular states.