# Detecting and Controlling Slip through Estimation and Control of the Sliding Velocity
## 5 Sliding Controller
This implies that the slippage indicator \(\mathcal{S}\leq 1\), and, when \(\mathcal{S}=1\), the slider can slip with respect to the finger. Moreover, note that the greater the grasp force \(f_{n}\), the greater the maximum friction \(g(f_{n},c)\); thus, \(\mathcal{S}\) decreases as \(f_{n}\) increases. Given these considerations, the objective of the static contribution \(f_{ns}\) is to regulate the slippage indicator \(\mathcal{S}\) to the value \(\bar{\mathcal{S}}_{a}<1\) designed, taking into account the desired degree of slipping avoidance robustness. The lower the value of \(\bar{\mathcal{S}}_{a}\), the greater the resulting grasp force. In the experiment section, we will select \(\bar{\mathcal{S}}_{a}=0.8\). The regulation of \(\bar{\mathcal{S}}\) is carried out by a classical PI controller with the proportional gain \(k_{p}=4\) and the integral  
Figure 6: Control scheme: (**a**) closed loop system with the interconnections; (**b**) details of the slippage controller. The control mode can be either slipping avoidance or pivoting. gain \(k_{i}=10\) tuned so as to obtain the fastest possible response time with the hardware used in the experiments (Section 6). In pivoting mode, which is the controlled sliding maneuver considered here, the control scheme is almost the same as the slipping avoidance mode, as depicted in Figure 6b. This time, the objective is to allow slippage; \(\mathcal{S}\) has to be regulated to 1, and the grasping force has to be continuously decreased as long as the sliding motion is complete. To this aim, it is sufficient to activate the switches in Figure 6b. The dynamic contribution is deactivated (\(f_{nd}=0\)) because the sliding velocity does not have to be controlled to zero, and the reference slippage indicator is set to the value \(\mathcal{S}=\mathcal{S}_{p}>1\). Additionally, the static control law is switched to a first-order filter with a time constant \(\tau_{p}\). This way, the grasp force will decrease down to the point for which \(\mathcal{S}=1\) reaches its maximum; then, the constant error \(\mathcal{S}_{p}-1\) will cause an exponential decrease in the static contribution \(f_{ns}\) governed by the time constant \(\tau_{p}\). As soon as the pivoting motion has been completed, the slipping avoidance mode is reactivated. At the beginning of the pivoting, the object is grasped far from its center of gravity (CoG), and the friction sustains the gravitational torque. Decreasing the grasp force would cause the object to rotate about the grasp axis in a pendulum-like motion until its equilibrium point, where the gravitational torque is zero and the object is in a \"vertical\" configuration (with the CoG just below the grasp axis). In the beginning, the object is more prone to rotationally slide subject to the torque; at the end, the object is more prone to translate. These two phases can be discriminated by the estimated CoR position (13); low values of \(\tilde{c}\) mean that the object will rotate, and high values mean that the object will translate. This can be graphically visualized in Figure 7. The figure shows the normalized LS parametrized with respect to \(\tilde{c}\). Low values of \(\tilde{c}\) correspond to high-friction torque values and vice versa. Thus, \(\tilde{c}\) can be used in the pivoting mode to detect the end of the pivoting maneuver and stop the decreasing of the grasping force. In the experiments, as soon as the normalized CoR position is such that \(\tilde{c}>1\), the pivoting maneuver is considered completed.