# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 10 Space Manipulator Force Control  
We must now consider external forces and moment acting at the end effector. The equation of motion for freefloating/freelying robotic servicers including forces exerted by a payload may be given by the Lagrange-Euler formulation:  
\[\left( \begin{array}{cc}D_{s}&D_{c}\\ D_{c}^{T}&D_{m}\end{array} \right)\left( \begin{array}{c}\ddot{q}_{s}\\ \dot{\theta}_{m}\end{array} \right)+\left( \begin{array}{c}C_{s}\\ C_{m}\end{array} \right)=\left( \begin{array}{c}F_{s}\\ \tau_{m}\end{array} \right)+\left( \begin{array}{c}J_{s}^{T}\\ J_{m}^{T}\end{array} \right)F_{ext} \tag{110}\]  
where \(D_{s}\) = spacecraft inertia, \(D_{m}\) = manipulator inertia matrix, \(D_{c}\) = coupling inertia matrix, \(C_{s}\) = spacecraft nonlinear velocity terms, \(C_{m}\) = manipulator Coriolis and centrifugal terms, \(F_{s}\) = spacecraft wrenches (column vector for forces and torques), \(\tau_{m}\) = manipulator torques, \(J_{s}\) = spacecraft Jacobian, \(J_{m}\) = manipulator Jacobian, \(F_{ext}\) = external wrenches. Typical servicing manipulator performance parameters are ~1 mm and 0.1\({}^{\circ}\) positioning accuracy and 150-200 N and 20-30 Nm applied end-effector forces/torques. The spacecraft component is given by:  
\[D_{s}\ddot{q}_{s}+D_{c}\ddot{\dot{\theta}}_{m}+C_{s}=0 \tag{111}\]  
The manipulator dynamics component is given by the Lagrange-Euler formulation (Equation (49)):  
\[\tau_{m}=D_{m}(\theta_{m})\ddot{\theta}_{m}+C_{m}(\theta_{m},\dot{\theta}_{m} )+J_{m}^{T}(\theta_{m})F_{ext} \tag{112}\]  
Integration of the spacecraft component yields total momentum conservation of the system:  
\[\left( \begin{array}{c}P\\ L\end{array} \right)=\int_{0}^{t}F_{s}dt+\int_{0}^{t}J_{s}^{T}F_{ext}dt=D_{s} \dot{q}_{s}+D_{c}\dot{\theta}_{m} \tag{113}\]  
Spaecraft motion is given by:  
\[\dot{q}_{s}=\left( \begin{array}{c}v_{s}\\ w_{s}\end{array} \right)=-D_{s}^{-1}D_{c}\dot{\theta}_{m}=D_{s}^{-1}D_{c}\dot{ \theta}-\int_{0}^{t}F_{s}dt-\int_{0}^{t}J_{s}^{T}F_{ext}dt \tag{114}\]If attitude is controlled, \(\mathrm{w_{s}}=0\). Robotic manipulator regimes of servicing operations may be divided into four phases: (i) free-space manipulator motion to grasp the target using position control; (ii) transitional contact dynamics between manipulator and target; (iii) transitional passivation of target using force control; (iv) servicing operations such as peg-in-hole tasks using hybrid position/force control. The incidence of contact on grasping the target alters the dynamics of the system--prior to contact, the manipulator is in free space mode; on contact and thereafter, the manipulator experiences an external force/moment in acquiring a new payload. This requires re-initialisation of the dynamics of the system. Real-time detection of collision of a manipulator with obstacles may be achieved without explicit force sensors--a collision may treated as a fault in the actuation system [220]. It is dependent on accurate joint torque measurements to generate the residual against the model based on conservation of generalized momentum--a non-zero residual above a threshold indicates collision. However, we assume that force sensors have been adopted. On impact between the end effector and the target, there is an impact transition in which energy is transferred between the chaser and the target spacecraft. We assume that the contact is frictionless (Hertzian) and that contact is made with the target using robotic fingers. The contacting bodies deform over a finite elliptical contact area--Hertzian contact relates the radius of the contact area A to the normal load F [221]:  
\[A=cF^{\gamma} \tag{115}\]  
where \(c=\mathrm{constant}\) depending material properties and fingertip geometry, \(0<\gamma\leq 1/3\) depending on fingertip materials (elastic material has \(\gamma=1/3\)).