# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 8 Spacecraft Attitude Stabilisation  
The feedforward signal from the robot manipulator to the spacecraft attitude control system is computed as a byproduct of the Newton-Euler manipulator dynamics:  
\[N_{r}=N_{0}+s_{0}\times F_{0}=N_{T}+(p_{cm}^{*}-r_{c0}-s_{0})\times F_{T} \tag{66}\]  
This feedforward signal is input to a three-axis active attitude control system which is torqued to nullify the reaction forces on the spacecraft imposed by the manipulator to maintain a fixed attitude. This approach avoids the dynamic singularities of uncontrolled attitude. The robotic arm transfers angular momentum to the mounting platform where it may be compensated with torqueing wheels within the constraints of these actuators. The torque required by three perpendicular reaction wheels is given by:  
\[\tau_{w}=\dot{h}_{w}+w_{s}\times h_{w} \tag{67}\]  
where \(h_{w}=I_{1}\Omega_{1}i+I_{2}\Omega_{2}j+I_{3}\Omega_{3}k\), \(\dot{h}_{w}=I_{w}\dot{w}_{w}\), \(w_{s}=\) satellite angular velocity, \(I_{i}=\) reaction wheel moment of inertia, \(\Omega_{1}\) = reaction wheel speed. A simple PID controllers may be applied to three-axis spacecraft attitude control [159]:  
\[\tau_{w}=K_{p}(\theta_{s}^{d}-\theta_{s})+K_{\circ}(w_{s}^{d}-w_{s}) \tag{68}\]  
Optimal control of 6 DOF spacecraft is commonly achieved through LQR-based control algorithms of both position and attitude by virtue of its superiority to PD controllers [160]. There have been explorations in determining optimal PID control gains by minimising augmented integral square error [161], but parameter variations suggest adaptive and/or robust controllers. Adaptive control is concerned with tracking environmental variations while robust control is concerned with disturbance rejection. Nonlinear adaptive control has also been applied to spacecraft attitude control to accommodate parameter uncertainties [162]. Typically, a fourth reaction wheel skewed at angle of \(tan^{-1}\sqrt{2}\) rad = 54.7\({}^{\circ}\) typically from the other three is employed for redundancy. This requires employing an algorithm that minimises the maximum wheel torque using the pseudo-inverse matrix that transforms the four wheel torques to the three-axis spacecraft torques. This cannot be integrated to yield attitude uniquely because angular momentum is path-dependent. However, minimisation of both maximum wheel torque (angular acceleration) and maximum angular momentum (angular velocity) maximises agility by imposing further constraints to eliminate path-dependence while preventing uneven wheel angular momentum distribution [163]. A simulation of the reaction moments exerted on a spacecraft due to the mounted manipulator are shown in Figures 3 and 4. Figure 3 shows a freespace manipulator movement reacting on the spacecraft mount prior to grasping; Figure 4 shows a force-controlled passivation of the target from the initial grasp. In Figure 3, the six knot points define the free-space manoeuvre from the initial manipulator configuration to its grasp of the target--an acceleration of the end effector from rest over two knot points, a constant velocity coast over two knot points and a deceleration to rest over two knot points. The knot points are defined at cartesian level so are equidistant in time. The reaction moments generated on the spacecraft due to the freespace manipulator motions using ATLAS parameters were ~1-5 Nm (Figure 3). This is consistent with Ranger NBV system simulations which yield typical reaction moments of ~0.25 Nm for freespace position control. However, reaction wheels offer limited torque capability at ~0.1-1 Nm, which is insufficient for the agile attitude capability necessary for feedforward compensation. Figure 4 shows the reaction moment trajectory from the instant of target capture of a 225 kg payload (with initial impact force determined by the end effector velocity along the y direction) and then a slightly under-critically damped force controller passivating the target. A PI-based force controller passivated the spacecraft attitude.