# Detecting and Controlling Slip through Estimation and Control of the Sliding Velocity
## 5 Sliding Controller  
This section presents a novel slipping control strategy based on the observer presented in the previous section. The basic idea is similar to the one presented in our previous work [16], but with a substantial difference: the control law involves the full state feedback estimated by the observer. The control scheme works in two possible control modes, \"slipping avoidance\" and \"pivoting\". With reference to Figure 6a, in slipping avoidance mode, the objective of the sliding controller is to regulate the estimated slipping velocity \(\widehat{\omega}\) to zero by modulating the grasping force \(f_{n}\). The pivoting mode allows an in-hand manipulation; it consists of letting a manipulated object rotate in hand subject to the gravity so as to change the relative orientation between the gripper and the object. In this modality, the controller allows rotational sliding between the fingertip and the slider. The control strategies for the two control modalities are very similar. We will describe the slipping avoidance mode first, in which all the control components are active. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mu\) & \(\beta_{A}\) & \(\sigma_{0}\) & \(\sigma_{z}\) & \(\delta\) & \(\gamma\) \\ \hline
0.47 & \(4.9\cdot 10^{9}\) Ns/m & \(16.66\) Nm/rad & \(2\) Nms/rad & \(0.0018\) m/N\({}^{\gamma}\) & 0.254 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters estimated on a rigid cardboard slider. Figure 5: Results of the estimation of the slippage velocity after the estimation of parameters with a cardboard slider. Ground-truth robot velocity \(\omega\) (blu), estimated velocity \(\widehat{\omega}\) (red), estimation error (magenta). The inner plots are zooms that show details of the single repetitions; the zoomed areas are highlighted in the black boxes. The control law is composed of two components (Figure 6b):  
\[f_{n}=f_{ns}+f_{nd}. \tag{34}\]  
where \(f_{ns}\) is the static contribution and represents the normal force that is able to avoid the slippage by bringing the maximum torsional friction around the CoR \(g(f_{n},c)\) above the actual dry friction \(\sigma_{0}z\) in (1) and (2). It is computed by exclusively using the estimated dry friction and not the viscous friction, yielding lower grasping forces with respect to the strategy in [16]. The static contribution is able to avoid slippage only in static or quasi-static conditions. In fact, it is well-known that when the load is time-varying, the maximum friction that the contact between the slider and the fingertip can sustain decreases as the rate of variation of the load increases [27]. Thus, the dynamic contribution \(f_{nd}\) is introduced, and it is synthesized by means of the estimated velocity \(\dot{\omega}\) as  
\[f_{nd}=|C_{d}\widehat{\omega}|, \tag{35}\]  
where \(C_{d}\) is a suitable linear differential operator, and the absolute value operator ensures that \(f_{nd}>0\) both for positive and negative slipping velocities. \(C_{d}\) can be represented with a transfer function  
\[C_{d}=k_{cd}\frac{s+z_{cd}}{s+p_{cd}}, \tag{36}\]  
where the real zero and pole (\(-z_{cd}\) and \(-p_{cd}\), respectively) were selected to reduce the high-frequency gain (\(z_{cd}>p_{cd}>0\)) so as to reduce the control sensitivity to the high-frequency noise. The gain \(k_{cd}>0\) was selected to obtain a quick reaction to any relative velocity \(\widehat{\omega}\). The parameters of the controller were tuned to the values \(k_{cd}=10\), \(p_{cd}=7\), \(z_{cd}=250\). The static contribution was designed by resorting to the feedback of the estimated friction state variable \(\hat{z}\). The main idea was the definition of the following slippage indicator  
\[\mathcal{S}=\frac{\sigma_{0}|z|}{g(f_{n},c)}. \tag{37}\]  
It is well-known that the LuGre state \(z\) is bounded by the maximum friction [21], and it always holds that \(\sigma_{0}|z|\leq g(f_{n},c)\). In particular, when \(\sigma_{0}|z|=g(f_{n},c)\), the dry friction has reached its maximum, and slippage can take place.