# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 9 Freeflyer Manipulator Control Systems
The Shuttle RMS' high inertia with low viscous damping generated a lightly damped mode at the resonant frequency of 70 rad/s when unloaded which required slow speed movement, typically 0.004 rad/s. Robust adaptive control used for rigid manipulators may be generalized to apply to flexible joint manipulators using a reduced order dynamic model by adding an extra corrected control signal to compensate for the effects of flexibility to force the flexible manipulator to behave similarly to a rigid manipulator [200; 201], such as global linearisation into passivity through structural damping [202]. Electric motors require a variety of techniques for feedback control to cope with nonlinear dynamics, unobserved states and parameter uncertainties [203]. Suitable techniques include exact linearization using computed torque methods, state estimation using Kalman filtering, error tolerance through robust control and parameter identification through adaptive control techniques. Robust controllers however adopt fixed parameters such as high gain that are tolerant of error to maintain consistent behaviour. Robust controllers include Ho control [204] and sliding mode (variable structure) controller, the latter being common for underwater robotic vehicles [154] and robotic manipulators [205]. The sliding mode controller drives the system state onto a pre-designed switching surface in state space--the sliding surface \(s=\dot{e}+\lambda_{1}e+\lambda_{2}\int_{0}^{t}e.dt=0\) when \(e\to 0\) as \(t\rightarrow\infty\) and \(\dot{s}=\ddot{\theta}^{d}+\lambda_{1}\dot{e}+\lambda_{2}e\) where \(e=\theta^{d}-\theta\) and \(\ddot{\theta}^{d}=D(\theta)^{-1}[\tau-C(\theta,\dot{\theta})]\). The sliding mode controller then maintains the system in the vicinity of the sliding surface through the control law--this renders it robust. The control input is given by:  
\[\tau=\lambda_{1}\dot{e}+\lambda_{2}e+k.sgn(s)\text{ where }k.sgn(s)=\left\{ \begin{array}{l}1\text{ }for\text{ }s>0\\ -1\text{ }for\text{ }s<0\\ 0\text{ }for\text{ }s=0\end{array} \right. \tag{92}\]  
Unfortunately, the discontinuous sliding mode control law generates high-frequency switching through the sliding surface--chattering--due to finite time delays. This can excite unmodelled high-frequency dynamics into mechanical oscillation--this is particularly problematic for space manipulators where there is no damping medium unlike in submersible-based manipulators. One means to reduce chattering is to replace the discontinuous controller with a continuous saturation function:  
\[u=K.sat(s/\phi)=\left\{ \begin{array}{l}1\text{ }for\text{ }s/\phi>1\\ -1\text{ }for\text{ }s/\phi<1\\ 0\text{ }for\text{ }s/\phi=1\end{array} \right.\text{ where }\phi=\text{boundary constraint} \tag{93}\]  
Sliding mode controllers have also been married to fuzzy controllers and neural network controllers to compensate for their noise sensitivity close to the sliding surface introduced by the saturation function. Adaptive controllers adjust unknown parameters online on the basis of performance. For example, the SSRMS shoulder yaw joint experienced the significant moment of inertia variations of up to 3.0 kgm\({}^{2}\), which can be accommodated using adaptive control. There are several adaptive control schemes applied to manipulators--examples include model predictive control [206], passivity-based adaptive control [207] such as using Jacobian transpose [208] and neural networks. The model reference adaptive control is based on the MIT rule of the form \(\frac{d\theta}{dt}=-\gamma e\frac{\partial\epsilon}{\partial\theta}\). For example, the computed torque controller decouples the manipulator into a set of linear PD control laws for each motor of the form: \(\tau=\int_{eff}^{actual}\ddot{\theta}+\int_{eff}^{actual}\dot{\theta}\) where \(\ddot{\theta}=\ddot{\theta}^{d}+k_{v}\dot{e}+k_{p}e\), \(\int_{eff}^{actual}=\) effective motor and gear inertia, \(\int_{eff}^{actual}=\) effective motor and gear viscous damping, \(e=\theta^{d}-\theta,\ k_{p}=\) proportional gain, \(k_{v}=\) damping gain.