# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 9 Freeflyer Manipulator Control Systems
The Lagrangian dynamics may be partitioned into rigid body and flexural components by [192]:  
\[\left( \begin{array}{c}\tau\\ 0\end{array} \right)=\left( \begin{array}{cc}D_{11}&D_{12}\\ D_{21}&D_{22}\end{array} \right)\left( \begin{array}{c}\ddot{\theta}\\ \ddot{\zeta}\end{array} \right)+\left( \begin{array}{c}0\\ k_{w}\zeta\end{array} \right)+\left( \begin{array}{c}C_{r}\\ C_{f}\end{array} \right) \tag{87}\]  
The model-based computed torque technique is reasonably effective in controlling tip motion by treating flexible link manipulators as perturbations to the rigid manipulator despite there being fewer control actions than degrees of freedom [193]. However, to reduce settling time, active vibration control by feedforward preshaping of the motion is generally required--this involves using two impulses separated by T/2 where T = vibration period [194]. Normal rigid body robotic control systems can be readily adapted to modal control to suppress vibrations by using a \"virtual rigid manipulator\" approach--this is achieved by replacing actual endpoint kinematic variables by those of a \"virtual rigid manipulator\" [195]. The virtual rigid manipulator is an assumed rigid manipulator with the same joint angles as the real deformed link manipulator but with coincident end effector positions to the undeformed link manipulator. A modal feedback controller based on LQC observer can estimate modal coordinates:  
\[D_{\zeta}\ddot{\zeta}+k_{w}\zeta=B_{\zeta}\tau-C_{\zeta} \tag{88}\]where \(D_{\zeta}=(D_{22}-D_{21}D_{11}^{-1}D_{12})\), \(B_{\zeta}=-D_{21}D_{11}^{-1}\), \(C_{\zeta}=C_{f}-D_{21}D_{11}^{-1}C_{r}\approx 0\). Hence, elimination of the C terms linearises the dynamics permitting the adoption of state space representation:  
\[\dot{\zeta}=A\zeta+B\tau \tag{89}\]  
Measurement of link strains give estimates on modal amplitude coordinates. Bending strain energy measure along n + 1 flexible links may be given by:  
\[U=\frac{1}{2}\sum_{i=1}^{n+1}EI\int_{0}^{i}\left(\frac{d\alpha_{ij}}{d\theta_{ ij}}\right)^{2}d\theta \tag{90}\]  
This allows computation of the additional joint torque components to dampen the modal vibrations. The use of adaptive control approaches with high controller gains can stiffen the joints. However, control of flexible appendages is difficult as sensors and actuators at the joints are not co-located with the beam shape unless mode shapes between tip and actuators are detected using embedded strain gauges. Hence, vibration suppression of flexible multi-link manipulators may be implemented through distributed arrays of piezoelectric sensor/actuators which alter the stiffness of the structure [196]. Similar approaches have been proposed for vibration suppression in large space structures [197] which can be induced by flexible space manipulators which typically have fundamental vibration eigenfrequencies of around 1 Hz depending on the payload. Flexible joints are the dominant flexible element of space manipulators due to the high ratio gearboxes and/or flexsplines of harmonic drives [198]. Joint elasticity is generally modelled as torsional springs with a characteristic stiffness \(\sim\)10\({}^{4}\)-10\({}^{5}\) Nm/rad which is dominant over link elasticity. This may be approximated as a linear gearbox spring by:  
\[\tau=J_{eff}\ddot{\theta}+b_{eff}\dot{\theta}+nk\Delta\theta \tag{91}\]  
where \(k\) = joint stiffness, \(\Delta\theta\) = difference between motor and link angles. However, the gearbox stiffness may not be linear--the flexspline stiffness is typically sigmoidal. Simple PD controllers may be applied to elastic joints inherent in harmonic drive systems to provide stability to fast dynamics [199]. The harmonic drive is characterized by nonlinear compliance with hysteresis and nonlinear damping. The addition of an integral controller to form a PID controller that follows a rigid model may be employed to ensure robustness.