# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 10 Space Manipulator Force Control
Impedance control of the manipulator on contacting the target is given by:  
\[F_{ext}=M_{i}\ddot{\theta}_{i}+B_{i}\dot{e}_{i}+K_{i}\dot{e}_{i} \tag{129}\]  
where \(e_{i}=\theta_{i}^{d}-\theta_{i}\), \(M_{i}\), \(B_{i}\) and \(K_{i}\) are impedance control matrices. This translates to a joint torque given by:  
\[\tau_{m}=D_{m}J_{m}^{-1}[M^{-1}(F_{ext}-B\dot{e}+Ke)+\dot{J}_{m}-\ddot{\theta} _{m}]+C_{m}+J_{m}^{T}F_{ext} \tag{130}\]  
Impedance control during transient impact cannot be employed due to time delays--this requires zero-delay physical compliance. Impedance control with a compliant wrist (such as the remote centre compliance device [233]) is one option [234]. The remote centre compliance is a generalised spring with three linear springs connecting two equilateral triangles. Direct force control requires explicit force feedback from a force/torque sensor typically mounted onto the wrist of the manipulator. Hybrid position/force control partitions the control task into two orthogonal task subspaces [235]: (i) free-space motion of the manipulator under control of position; (ii)constrained interaction of the manipulator with physical surfaces under control of force. A selection matrix \(\mathrm{S=diag(s_{1}\dots s_{n})}\) separates the position-controlled directions from the (I-S) force-controlled directions in cartesian space which may be cast into the equivalent joint space (where \(\mathrm{s_{i}=1}\) for force control and \(\mathrm{s_{i}=0}\) for position control). The S-matrix defines tasks such as the peg-in-hole task--there are two position-controlled degrees of freedom and four force-controlled degrees of freedom, i.e., \(\mathrm{S=diag(110110)}\) for the peg-in-hole task. The hybrid position/force control scheme may be cast into joint coordinates through the compliance matrix \(\mathrm{C=J^{-1}S}\):  
\[\tau=D(\theta)\{\ddot{\theta}+(I-C)[k_{\varphi}(\dot{\theta}^{d}-\dot{\theta}) +k_{p}(\theta^{d}-\theta)]\}+C(\theta,\dot{\theta})+J^{T}F_{ext}^{d}+(C)[k_{f}J ^{T}(F_{ext}^{d}- \tag{131}\]  
A mixed mode of position and force control was implemented in the teleoperated ETS VII space manipulator [236]. The hybrid position/force control scheme based on the geometric orthogonality of free twists and constrained wrenches has been criticized as fallacious [237], but this is unfounded [238]. The chief challenge is that the position-controlled directions require high stiffness but the force-controlled directions require low stiffness--in impedance control, high stiffness implies high impedance and low stiffness implies low impedance. One potential solution is to use explicit force control in the force control directions by converting force into position/velocity using stiffness/damping relationships. Indeed, impedance control does precisely this in indirectly regulating force via position feedback. However, this imposes sluggish behaviour, which favours hybrid position/force control. On the other hand, the selection matrix in hybrid position/force control is unique to a specific task and dynamic task changes require dynamic switching between selection matrices as a task sequence proceeds (this may not be applicable during grappling for active debris removal). It has been proposed to eliminate the selection matrix altogether and resolving conflicts between position and force control through a rule-based priority strategy favouring the force control loop [239]. This is the parallel force/position control scheme, which eliminates the selection matrix and adopts a priority strategy with force control taking priority:  
\[F=m\ddot{q}^{d}+k_{p}e+k_{v}\dot{e}+k_{f}e_{f}+k_{i}\int e_{f}d\tau=0 \tag{132}\]  
where \(\mathrm{e=q^{d}-q}\) and \(\mathrm{e_{f}=F^{d}-F}\). Particularly along a stiff surface, the integral force controller gain should be high. Alternatively, an acceleration-based controller allows implementation of a force constrained position controller without a selection matrix [240]. Control is implemented using position control for small external forces but using force control for large external forces.