# Detecting and Controlling Slip through Estimation and Control of the Sliding Velocity
## 2 Contact Modeling  
In this paper, the manipulated object is modeled as a planar slider. With reference to Figure 1, a planar slider is a rigid body that can rotate and translate in a 2-D space. The robotic finger is modeled as a hemispherical soft pad that makes contact with the slider with a non-zero contact area. This way, the fingertip exchanges both friction forces and torques with the slider. The friction can be indirectly controlled by acting on the normal force that the fingertip exerts on the body, which in turn changes the maximum friction force and torque that the contact can sustain. Any 2-D roto-translational motion can be instantaneously described as a pure rotation about the CoR. Based on our previous friction modeling [19] and inspired by the revisited LuGre friction model [21], in this paper, we propose a novel planar slider dynamic model that considers the micro-damping between the micro-asperities in contact [21]. The slider motion about the CoR is modeled as  
\[\dot{z} =\omega-\frac{\sigma_{0}}{g(f_{n},c)}z|w| \tag{1}\] \[J\dot{\omega} =-\sigma_{1}(f_{n},c)\omega-\sigma_{0}z-\sigma_{z}\dot{z}+\tau_{ c}, \tag{2}\]  
where \(\tau_{c}\) is the external torque acting on the slider, \(\omega\) is the slider's rotational velocity about the CoR, \(z\) is the friction state variable, which represents the deflection of the micro asperities, \(\sigma_{0}\) and \(\sigma_{z}\) are the so-called stiffness and damping of the micro asperities, respectively, \(J\) is the inertia moment of the slider about the CoR; \(g(f_{n},c)\) is the maximum rotational friction torque about the CoR depending on the normal force \(f_{n}\) and the CoR position \(c\), and finally, \(\sigma_{0}z\) and \(\sigma_{1}(f_{n},c)\omega\) are the dry and viscous friction torques, respectively. The relevant difference of the model in (1) and (2) with respect to the classical revisited LuGre friction model is the extension to the roto-translational case. This generalization is achieved by incorporating the LS method into the model and, specifically, in the computation of the maximum friction torque \(g(f_{n},c)\), which requires the estimation of the instantaneous CoR. The only available control input is the normal force \(f_{n}\) that is nonlinearly related to the maximum dry friction and the viscous one. The greater the value of \(f_{n}\), the greater the fingertip deformation (and the contact area), and the greater the available friction. This relationship can be found by resorting to the limit surface concept (LS) [18]. The LS is a closed surface defined in the space of the friction force and torque (Figure 2) and represents the maximum dry friction that a contact can sustain without sliding. With reference to Figure 1, we define a contact frame located in the center of pressure (CoP) of the contact area, with the \(y\)-axis along the direction of the tangential force \(f_{t}\) and the \(z\)-axis normal to the contact surface and aligned to the normal force \(f_{n}\). It is well known that, assuming an axisymmetric pressure distribution [24], the limit surface is axisymmetric as well, and we can represent it using any radial cross-section. Moreover, the CoR position is always orthogonal to the tangential force, and, thus, it lies on the \(x\)-axis [18]. Consequently, it is possible to represent the CoR position with the scalar \(c\). Finally, taking into account the load-motion inequality [12], the friction torque \(\tau\) and \(c\) have opposite signs. Given one particular CoR position, it is possible to generate a point on the LS.