# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 7 Freeflyer Manipulator Dynamics  
The dynamics formulation of robotic manipulators exploits the manipulator kinematics to determine the joint torques. The Lagrange-Euler formulation of manipulator dynamics is given by:  
\[\tau=D_{m}(\theta)\ddot{\theta}+C_{m}(\theta,\dot{\theta})+J(\theta)^{T}F_{ext} +F(\theta,\dot{\theta}) \tag{49}\]  
where \(D_{m}\) = manipulator inertia, \(C_{m}\) = manipulator coriolis/centrifugal forces, \(F_{ext}\) = external forces, \(F\) = friction. Of particular note, the frictional term \(F(\theta,\dot{\theta})\) in the Lagrange-Euler formulation hides a high degree of complexity. Effective friction compensation requires accurate friction models. The simplest models of friction characterise slip as a combination of static friction, Coulomb (dynamic) friction and viscous friction which varies in proportion to load and opposes motion independent of contact area. Static friction can be reduced below the level of Coulomb friction through the use of lubricants so that stick-slip will be eliminated, e.g., static friction is typically \(\sim\)0.02 Nm compared with viscous friction of \(\sim\)0.1 Nm (except in ageing motors). Motor friction is commonly modelled as a nonlinear Stribeck curve, which lumps the effects of backlash, deadzone and velocity-dependent friction together, but it does not model static friction, which yields stick-slip overshoot. Neural networks may be employed to learn manipulator inverse dynamics by storing association patterns which potentially resolve friction difficulties [147]. Although there exist sophisticated friction models such as [148] which has seven parameters, for simplicity, most friction models consider Coulomb friction and viscous friction only additively:  
\[\tau=D_{m}(\theta)\ddot{\theta}+C_{m}(\theta,\dot{\theta})+J(\theta)^{T}F_{ ext}+b\dot{\theta}+ksgn(\dot{\theta}) \tag{50}\]  
where \(b\) = viscous friction coefficient (assumed zero), \(k\) = Coulomb friction parameter (assumed zero in most cases). Joint friction is the primary limitation on precision and can lead to stick-slip behaviour or oscillations, especially for low-speed, low-amplitude movements. Torque sensors in the joints can effectively eliminate joint friction (as implemented in DLR manipulator joints). Active force control compensates for unknown friction effects through direct measurement of joint acceleration and forces exploiting a growing/pruning neural network approach [149]. However, self-sensing without the use of specific joint sensors can be achieved by monitoring back EMF generated within the motor (assuming constant field strength) since the back EMF is linearly related to the motor velocity (which may be integrated to give displacement) [150]. Freeflyer dynamics bears some resemblance to the dynamics of mobile manipulators mounted onto rover vehicles [151; 152; 153]. There are some resemblances of space manipulator dynamics to underwater robotic vehicle dynamics but with added mass forces and moments (additional inertial resistance imposed by fluid), drag and lift forces, and buoyancy forces [154], an example of which may be exploited for the exploration of Europa's subglacial oceans [155]. However, the space manipulator application is what concerns us here. The Lagrange-Euler formulation of manipulator dynamics is defined by \(\tau=\frac{d}{dt}\frac{d\tau}{\partial q}-\frac{dL}{\partial q}\) where \(L=T-U=\) Lagrangian, \(T=\) kinetic energy, \(U=\) potential energy = 0 in space.