# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 5 End-Effector Design  
In order to optimize the the force profile generated by the decoupling system, non-linear solutions were analyzed to meet the following requirements:  
1. The system has to be able to exert a minimum force on the tool to keep it in the desired position during the acceleration of the robot;
2. The system has to prevent rebounds of the tool in order to avoid multiple collisions and the restitution of the energy absorbed in the impact;
3. The impulse transmitted to the object should be as small as possible, according to Equation (22). The bi-stable mechanism shown in Figure 21 is considered. This system is characterized by two stable equilibrium positions and, in general, consists of a spring connected to a sliding element, which is constrained to move along the \(x\)-axis. The spring  
Figure 21: Schematic representation of a bi-stable mechanism with a single spring. Figure 20: Impulse generated by the decoupling system. generates an elastic force on the sliding element which has two components: the first is parallel to the \(x\)-axis (\(F_{T}\)) and influences the dynamics of the slider; the second is orthogonal to the sliding path and does not have important effects on the movement of the slider. The component \(F_{T}\) generated by the spring is given by the following equations:  
\[F_{T} =k(L_{0}-L)\frac{x}{L} \tag{23}\] \[L =\sqrt{H^{2}+x^{2}} \tag{24}\]  
where \(k\) is the stiffness of the spring, \(L\) is its actual length, \(L_{0}\) is the spring free length, \(x\) is the position of the slider relative to the vertical alignment position of the spring and \(H\) is the distance between the upper revolute joint of the spring and the sliding direction. In real applications, the slider could be constrained to move between two mechanical constraints that introduce a limit to its movement [22; 23]. The trends of the elastic force \(F_{T}\) as a function of the distance from the vertical alignment position are reported in Figure 22a in the case with \(L_{0}=0.07\) m and \(k=50\)\(\frac{N}{m}\). In the same figure, the effect on the force profiles of two mechanical constraints are schematized through the two orange areas, which identify the off-limit zones. The scheme depicted in Figure 22b shows a physical representation of the mechanical constraints. Figure 22a shows for \(x>0\) that the force is positive, whereas for \(x<0\), the force is negative. The limitation of the movement of the slider between the two mechanical constraints leads to a system that is characterized by a specific portion of the force profile. If the tool is leaning against the constraint placed on the right of the \(x=0\) position and an external force pushes it back towards the negative direction of the \(x\)-axis, the system reacts with a positive elastic force that tends to counterbalance the external force. However, when the vertical alignment position of the spring is crossed, the tool is pulled back against the left constraint and is kept there by the elastic force. The figure shows that it is possible to adjust the maximum positive and negative values of the force profiles modifying the  
Figure 22: (**a**) Tangential force profiles for different values of the parameter H; (**b**) physical representation of mechanical constraints. geometrical parameter \(H\), but also to vary the vertical alignment position of the spring in relation to the mechanical constraints. The maximum positive value of the elastic force should be large enough to keep the tool in position against the right constraint before the impact, in order to avoid oscillations, but, at the same time, it should be as small as possible to minimize the impulse transmitted to the object. The maximum negative value of the elastic force should be large enough to pull the tool back and keep it against the robot flange (left constraint). This bi-stable mechanism was integrated in the numerical model of the end-effector, and a force profile corresponding to \(H=0.028\) m was implemented.