# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 12 The Nature of Dexterity and Tactility  
We suggest that the technologies in the preceding sections are all essential components of a robust approach to the robotics capabilities necessary for active debris removal. We consider in this last section a few more sophisticated capabilities related to dexterity and tactility. Compliance is an essential feature of fine manipulation inherent in hand grasping [265]. Compliant fingertips offer damping and can be a partially effective method for suppressing vibrations [271]. There are several dexterity measures based on the grasp Jacobian matrix G that defines the kinematic geometry of grasping at contact i with respect to object coordinates [272]:  
\[f_{obj}=\left( \begin{array}{ccc}G_{1}&\ldots&G_{n}\end{array} \right)\left( \begin{array}{c}f_{1}\\ \vdots\\ f_{n}\end{array} \right) \tag{140}\]  
where \(G=\left( \begin{array}{ccc}n&s&a&0\\ p\times n&p\times s&p\times a&n\end{array} \right)\), \(p=\)position vector contact point i, \(\left( \begin{array}{ccc}n&s&a\end{array} \right)=\)orientation of contact point i, f\({}_{\rm i}=\)wrenches at contact point i, f\({}_{\rm obj}=\)(f\({}^{\rm T}\) n\({}^{\rm T}\))\({}^{\rm T}=\)wrench on object, F\({}_{\rm ext}=\)(\(f_{1}^{\rm T}\ldots f_{n}^{\rm T}\))\({}^{\rm T}\). To control grasping forces, the inversion of the grasp matrix is required which invokes the Moore-Penrose pseudo-inverse of G as it is non-square:  
\[\left( \begin{array}{c}f_{1}\\ \vdots\\ f_{n}\end{array} \right)=G^{+}f_{obj}+(I-G^{+}G)\lambda \tag{141}\]  
where \(G^{+}=\)Moore-Penrose pseudo-inverse, \(F_{int}=(I-G^{+}G)\lambda=\)internal forces, \(\lambda=\)Lagrange multiplier. The internal forces do not contribute to the motion of the object so should be close to zero. The squeeze forces within the nullspace of the grasp Jacobian allow minimization of slippage to zero, i.e., force closure. The stability of the object is determined by the shape of the potential well about the equilibrium position generated by the grasp configuration. The grasp manipulability index quantifies how far the grasp configuration is from a singular configuration: \(\sigma=\sqrt{det(GG^{T})}\geq 0.5\) while the condition number n\({}_{\rm c}\) quantifies the distribution of effort between fingers \(n_{\rm c}=\left|G^{T}\right|\)\(\left|G^{-T}\right|\)\(\rightarrow\) 1 for optimal distribution. Distribution of forces between fingers is given by: \(f_{i}=G^{T}f_{ext}\) subject to the force ellipsoid \(f_{ext}^{T}(GG^{T})f_{ext}\leq 1\) and \(\tau=\)\(\int^{T}f_{ext}\) where J=hand Jacobian for torque minimisation. To accommodate grasp dynamics, the grasp manipulability index may be adapted to \(\sigma=\sqrt{det(GD^{-1}G^{T}D^{-T})}\) where D(\(\theta\)) = grasp inertia matrix. Grasp equilibrium requires that the sum of all applied wrenches on the object are zero, \(\left(\sum f^{T},\sum n^{T}\right)^{T}=0\). Minimisation of the squared wrench to zero is given by [273]:  
\[\varepsilon=\varpi^{T}\varpi \tag{142}\]  
where \(\varpi=\sum_{i=1}^{m}\left(f_{x}^{i}f_{y}^{i}f_{z}^{i}n_{x}^{i}n_{y}^{i}n_{z}^ {i}\right)^{T}=\)wrench. This is the force closure condition in which any arbitrary external force can be exerted on an object with frictional contacts while internal forces are generated in the nullspace of the grasp matrix G. Form closure is force closure under frictionless contacts. Dextrous manipulation requires pure rolling in the contact plane with maintenance of contact (v\({}_{\rm z}=\) 0), no sliding (v\({}_{\rm x}=\) v\({}_{\rm y}=\) 0) and no twisting around the contact normal (w\({}_{\rm z}=\) 0) [274]. To firmly grasp an object, normal contact force must be positive and resolution of tangential and normal contact forces \(f_{\rm c}\) must lie within the friction cone. When the tangential force/moment lie within the friction limit surface, there is no sliding between the fingertip and object. While slippage on Earth is usually due to gravitational effects, slippage can occur in space due to inertial effects.