# Detecting and Controlling Slip through Estimation and Control of the Sliding Velocity
## 2 Contact Modeling
Two values are of paramount importance: the point corresponding to a pure translation (\(c\to\infty\)) and a pure rotation (\(c=0\)), which represent the maximum possible values for the translational and torsional friction, respectively [24], i.e.,  
\[f_{\rm max} =\mu f_{n} \tag{3}\] \[\tau_{\rm max} =\mu\xi\delta f_{n}^{\gamma+1}, \tag{4}\]  
where \(\mu\) is the Coulomb friction coefficient, \(\xi\) is a parameter that weakly depends on the particular pressure distribution and varies between \(\frac{3\tau}{16}\) (Hertzian) and \(\frac{2}{3}\) (uniform) [18], and finally, \(\delta\) and \(\gamma\) are additional parameters that relate the radius \(\rho\) of the contact area to the normal force [24] according to the relationship \(\rho=\delta f_{n}^{\gamma}\). Figure 1: Planar slider and the fingertip that exchanges force and torque with it. The CoR axis is highlighted. The point on the LS (\(f_{tLS}\), \(\tau_{LS}\)) can be computed as a function of the CoR position \(c\)[16]  
\[f_{tLS} =-\text{sign}(\omega)\tilde{f}_{tLS}^{*}(\tilde{c})f_{t\max} \tag{5}\] \[\tau_{LS} =-\text{sign}(\omega)\tilde{\tau}_{\text{LS}}^{*}(\tilde{c})\tau_{ \max} \tag{6}\]  
where \(\tilde{c}=\frac{c}{\tilde{\rho}}\) is the normalized CoR position, and \(\tilde{f}_{tLS}^{*}(\tilde{c})\) and \(\tilde{\tau}_{\text{LS}}^{*}(\tilde{c})\) are known functions whose shape weakly depends on the parameter \(\tilde{c}\)[16] and are shown in Figure 3. Given their very weak variation with respect to this parameter, in the following, we will consider the intermediate value \(\tilde{c}=0.6354\) already experimentally identified in [16]. The LS theory was originally conceived as an extension of the Coulomb dry friction for the rototranslational case. During the slippage, viscous friction should also be taken into account to describe the motion. Such a component is modeled by the term \(\sigma_{1}(f_{n},c)\omega\) in (2). In our previous work [16], we extended the limit surface theory to also take into account the viscous friction, which corresponds to points in the force/torque space outside the LS (Figure 2). Thus, the viscous friction as a function of the CoR position can be expressed as  
\[f_{t_{\mathcal{V}}} =\pi\beta_{A}\delta^{2}f_{n}^{2\gamma}\omega c \tag{7}\] \[\tau_{\mathcal{V}} =-\frac{\pi}{2}\beta_{A}\delta^{4}f_{n}^{4\gamma}\omega. \tag{8}\]  
The dynamic model (1) and (2) describes the motion as a pure rotation about the CoR axis, and thus, it is important to estimate the CoR position. The LS theory is typically used to solve the forward problem given the CoR position (i.e., the instantaneous motion)  
\[J\dot{\omega} =-\sigma_{1}(f_{n},c)\omega-\sigma_{0}z-\sigma_{z}\dot{z}+\tau_{ c}, \tag{2}\]0 Graph of the functions \(\tilde{f}_{tLS}^{*}(\tilde{c})\) and \(\tilde{\tau}_{\text{LS}}^{*}(\tilde{c})\) and their weak variation on the pressure distribution type. \[J\dot{\omega} =-\sigma_{1}(f_{n},c)\omega-\sigma_{0}z-\sigma_{z}\dot{z}+\tau_{ c}, \tag{2}\]1 Limit surface represented in the 3-D space. by computing the friction force and torque. We are interested in the inverse problem, i.e., estimating the CoR position \(c\) given the measured friction force and torque. Moreover, when there is no motion (\(\omega=0\)), the CoR is meaningless, but the dynamic model also needs the CoR to be defined in this case. To solve this issue, it is possible to define a virtual CoR (16), i.e., when the velocity is zero, the virtual CoR is the CoR that would result if we lowered the normal force down to the point that a sliding motion occurs. In the following, we will use the acronym CoR to refer to the CoR and the virtual CoR indistinctly. To estimate the CoR from the measurements, it is useful to represent the LS in a space normalized with respect to the maximum friction force and torque (3) and (4).