# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 9 Freeflyer Manipulator Control Systems
The gains can be adapted using the MIT rule so that the motor output torque tracks the desired behaviour of the linear mass-damper reference model parameters given by:  
\[\dot{k}_{p}=-\frac{\dot{j}_{eff}k_{p}}{J_{eff}} \tag{94}\]  
\[\dot{k}_{v}=k_{p}\dot{b  
Most adaptive controllers however are based on segmenting all uncertain parameters into a regressor matrix of the form [209; 210]:  
\[D(\theta)\ddot{\theta}+C(\theta,\dot{\theta})=Y(\theta,\dot{\theta},\ddot{\theta })a \tag{100}\]  
where Y = regressor matrix of known parameters, \(a=-\)PY\({}^{T}\)_e_ = vector of unknown physical parameters, P = predicted estimate gain, \(\dot{P}=\lambda(P-PK^{-1}P)-PY^{T}\)_YP_, K = upper bound on P, \(\lambda\) = forgetting factor. Passivity through the Jacobian transpose takes the following form:  
\[\tau=J^{T}k_{f}(\Delta\dot{q}+\alpha\Delta q)=D(\theta)\ddot{\theta}+C(\theta, \dot{\theta}) \tag{101}\]  
where \(s=\Delta\dot{q}+\alpha\Delta q\) = sliding surface. Adaptive controllers have been employed in space manipulators to deal with parameter variations such as inertial or payload parameters [211; 212]. An adaptive control model with a regressor matrix was investigated for a free-floating space-based manipulator [213]. The problem of formulating the regressor matrix lies in linearising the dynamics in inertial space when dealing with an unknown payload which cannot be decoupled dynamically from known manipulator dynamic parameters for a free-floating manipulator [214]. The primary problem with adaptive computed torque controllers is the requirement for measurements of joint velocity and acceleration which are difficult and/or corrupted by noise. A robust adaptive computed torque controller uses joint position measurements and input torques in an extended Kalman filter to estimate joint position, velocities and accelerations [215]. The Kalman filter is a Bayesian technique where the posterior probability p(x|z) of state x is given by the prior probability p(x) of the state x determined by the model combined with the measurement z of state x and conditional probability p(z|x) through Bayes' rule. The state vector \(x=(\theta^{T}\ddot{\theta}^{T}\phi^{T})\) and input u(t) = \(\tau\)+w(t) yields a nonlinear model with process noise w given by:  
\[\dot{x}=f(x,u)+g(x)w=\left( \begin{array}{c}\dot{\theta}\\ D^{-1}(\theta,\phi)[u(t)-C(\theta,\dot{\theta},\phi)]\\ 0\end{array} \right)+\left( \begin{array}{c}0\\ D^{-1}(\theta,\phi)&0\\ 0\end{array} \right)w(t) \tag{102}\]  
To apply the extended Kalman filter, the system must be linearised through a first order Taylor expansion with respect to a reference trajectory:  
\[f(x(t))\approx f(\dot{x}(t))+\frac{\partial f}{\partial\dot{x}}[x(t)-\dot{x}( t)] \tag{103}\]  
Hence, the EKF process model is given by:  
\[\dot{x}(t)=F(t)x(t)+G(t)w(t) \tag{104}\]  
\[\text{where }F(t)=\left( \begin{array}{ccc}0&I&0\\ -D^{-1}\Big{(}\frac{\partial D}{\partial\dot{\theta}}\ddot{\theta}+\frac{ \partial C}{\partial\dot{\theta}}\Big{)}&-D^{-1}\Big{(}\frac{\partial C}{ \partial\dot{\theta}}\Big{)}&-D^{-1}\Big{(}\frac{\partial D}{\partial\dot{ \phi}}\ddot{\theta}+\frac{\partial C}{\partial\dot{\phi}}\Big{)}\\ 0&0&0\end{array} \right) \tag{105}\]  
The measurement model with measurement noise \(v(t)\) is given by:  
\[z(t)=H(t)x(t)+v(t) \tag{106}\]  
\(H(t)\) models the fact that a sensor measures the target state x indirectly and incompletely. Both process and measurement noise are independent Gaussian functions with zero mean and covariancesgiven by: Q(t) = cov(w(t),w(\(\tau\))) and R = cov(v(t),v(\(\tau\))) respectively.