# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 12 The Nature of Dexterity and Tactility
To prevent slippage, the contact force must remain within the friction cone, i.e., tangential contact forces do not exceed the normal friction force: \(\sqrt{f_{x}^{2}+f_{z}^{2}}\leq\mu f_{z}\) where \(\mu=\)coulomb friction coefficient = 0.4-0.8 typically (0.65 for aluminium), \(f_{z}\) = normal contact force \(>0\)[275; 276]. From this, a critical finger measure may be defined as a grasp stability metric based on the fingertip \"distance\" from its friction limit [277]:  
\[\alpha=\frac{\sqrt{\left(f_{extx}+\delta f_{extx}\right)^{2}+\left(f_{exty}+ \delta f_{exty}\right)^{2}}}{\mu\big{|}f_{extz}+\delta f_{extz}\big{|}}\text{ with slip occurring when }\alpha>1 \tag{143}\]  
Similarly, a similar potential function-based progression-to-slip measure may be defined [278]:  
\[\Delta U=\frac{f_{extx}\delta f_{extx}+f_{exty}\delta f_{exty}}{\sqrt{\left(f_ {extx}\right)^{2}+\left(f_{exty}\right)^{2}}}-\mu\delta f_{extz}sgn(f_{extz}) \tag{144}\]  
Slip prevention may require that internal forces to be generated in the null space of the grasp matrix. A significant facet of grasping is that it involves planning with small mid-range movements over extreme angle movements [279] Regrasping, finger gaiting (sequence of regrasps) and finger sliding/rolling provide sophisticated mechanisms for altering grasp configuration [280]. Grasp stability can be maintained if fingertip impedances can be controlled with positive stiffness gains on frictionless contacts [281]. There are several approaches to learning grasp configurations, including modelling mirror neurons acting as hand state-object state association schemas and learning grasp-object shape pattern matching [282]. Grasped object shapes may be modelled as superquadratics:  
\[s(\theta,\phi)=\left( \begin{array}{cc}a_{1}c^{\epsilon_{1}}\theta c^{\epsilon _{2}}\phi\\ a_{2}c^{\epsilon_{1}}\theta c^{\epsilon_{2}}&\phi\\ a_{3}s^{\epsilon_{1}}\theta\end{array} \right) \tag{145}\]  
where \(\epsilon_{1}\) = shape parameters, and \(\text{a}_{\text{i}}\) = scaling factors. The superquadratic is fitted to the grasp data. For rectangular objects, \(\epsilon_{1}\) and \(\epsilon_{2}\)\(\approx\) 0.1 while for cylindrical objects, \(\epsilon_{1}\)\(\approx\) 1 and \(\epsilon_{2}\)\(\approx\) 0.1. These superquadratic models can be used to estimate orientation and axes and guide further contour following exploration to generate further grasp data. Complex objects may be constructed from generalised cylinders generated through superquadratics. Clearly, although fine manipulation may not be required during grappling for active debris removal, the problem of grasping without slippage is highly relevant to this application--it may be also be considered that regrasping and finger gaiting during transport may also be advantageous but we consider that secure latching would be a simpler approach. Tactile sensing is more complex than force/torque sensing--although dominated by static and dynamic force/pressure measurement, it is supplemented by temperature, vibration, stretch and texture measurements. Vibration sensing exploits the fact that slip generates low frequency \(\sim\)10-100 Hz vibrations. Haptic exploration involves rolling/sliding fingers to determine surface features such as texture [283]. Texture comprises two major dimensions--roughness and softness--which can be detected by tactile micropressure sensor arrays [284]. Capacitative, piezoresistive, piezoelectric, inductive and optical measurements are superior to thermoresistive, inductive and magnetic measurements [285]. Example tactile sensors include [286; 287; 289]. PZT tactile sensors in conjunction with pressure sensors with a feedback circuit can provide sensitive tactile measurements including differentiating between hard and soft objects [290]. The theory of elasticity provides the basis for determining physical contact parameters from tactile sensor measurements [291]. The four basic human skin sensations of pressure, texture, temperature and pain have been replicated for teleoperation [292].