# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 4 Effects of Tool Compliance
### Simulated Results  
The effects of the stiffness and damping of the decoupling system on impact mechanics were analyzed by carrying out parametric simulations with \(m_{t}=0.125\) kg, \(m_{p}=0.25\) kg, \(v_{t}=1\) m/s and \(v_{p}=0\) m/s. The robot was assumed to start braking after the collision. The damping coefficient was defined by means of the viscous damping ratio [21]\(\zeta=\frac{c_{t}}{2\sqrt{k_{t}m_{t}}}\). Figure 16 represents the final velocity of the object for different values of contact stiffness \(k_{c}\), as a function of the stiffness \(k_{t}\) and considering a constant damping ratio, which is equal to the critical value \(\zeta=1\). A perfect elastic impact (\(e=1\)) is assumed. The comparison between Figures 13a and 16, shows that the decoupling system leads to a significant reduction in the velocity of the object after the collision. The increase in stiffness \(k_{t}\) and in damping \(c_{t}\) of the decoupling system increases the final velocity of the object, in agreement with Equations (20) and (22). Moreover, Figure 16 shows that an increase in \(k_{c}\) leads to a decrease in the final velocity of the object. This phenomenon is related to the fact that an increase in \(k_{c}\) produces an increase in the contact force which, in turn, generates a large compression of the decoupling system, which moves backwards. Since \(e=1\), the relative velocity \(V_{rel}\) before and after the impact are equal and, when there is a backwards motion of the tool, the absolute final velocity of the object \(V_{p}=V_{t}+V_{rel}\) is reduced. When \(k_{t}\) is about 90 \(\frac{N}{m}\), the curve corresponding to the largest contact stiffness suddenly changes its slope. After the collision, the elastic energy stored inside the spring of the decoupling system is released and the tool is subjected to a rebound. If the impact duration is short, the restitution of the elastic energy starts very quickly. In this condition, multiple impacts can occur. Therefore, the sudden increment in slope highlights the presence of multiple collisions, which leads to an increment in the final velocity. Figure 17a,b represents the robot, tool and object velocities with \(k_{t}=150\)\(\frac{N}{m}\), \(\zeta=1\), \(e=1\). In Figure 17a, the contact stiffness is set to \(k_{c}=10^{5}\)\(\frac{N}{m^{1.5}}\), whereas in Figure 17b, the contact stiffness is set to \(k_{c}=10^{4}\)\(\frac{N}{m^{1.5}}\). The contact force applied to the object and the force generated by the decoupling system on the tool are depicted in Figure 17c,d, respectively. Figure 16: Effects of the stiffness and of the damping of the decoupling system on the final velocity of the object with \(e=1\). The end-effector is compressed by the inertia force during the acceleration of the robot. Therefore, the velocity of the tool is slightly higher than the robot velocity when the latter moves at a constant velocity due to the restitution of the elastic energy absorbed during the acceleration. The increase in contact stiffness increases the deceleration of the tool. Figure 17c highlights two clear peaks in the contact force profile and confirms the presence of multiple impacts. When inelastic impacts are considered, the results show that the final velocity of the object decreases (see Figure 2). Furthermore, the decrease in contact stiffness \(k_{c}\) leads to a reduction in the final velocity of the object; this behavior is opposite to that found with \(e=1\). Parametric simulations were carried out to analyze this aspect considering \(m_{t}=0.125\) kg, \(m_{p}=0.25\) kg, \(v_{t}=1\) m/s and \(v_{p}=0\) m/s. The robot was assumed to start braking after the collision. Figure 18 represents the final velocity of the object for different values of contact stiffness \(k_{c}\) as a function of the stiffness \(k_{t}\) considering a constant damping ratio equal to the critical value (\(\zeta=1\)). A nearly completely inelastic impact (\(e=0.1\)) was simulated.