# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 3 Effects of Robot Compliance
### Numerical Results  
In the simulations, the tool approached an object along a linear trajectory. Moreover, it was assumed that the tool moved between the starting and final point following a trapezoidal velocity profile, and a collision occurred when the robot moved at a constant velocity. The position and velocity are referred to the fixed reference frame of the first joint of the robot (see Figure 9). Figure 11 represents the imposed position (\(x_{r}\)) and the imposed velocity (\(v_{r}=\dot{x}_{r}\)) of the robot flange before the collision. In order to evaluate the behavior of the robot during the interaction with an object, the robot was assumed to be equipped with a control system that was capable of detecting the contact force and ordering the robot to start braking. The force detection threshold was set to 1 N. The reaction to the impact was not instantaneous, but it was assumed that the robot began to perform braking after 16 ms in order to take into account the delay in processing the external input signal [20]. The circular marker (Figure 11) indicates when the impact happens. Figure 11: Position and velocity profiles imposed on the tool by the robot control system. The circular dot indicates the impact time. Figure 10: Flow chart of the steps taken to evaluate the terms \(M_{X,11}\), \(D_{X,11}\) and \(K_{X,11}\) along the trajectory of the tool. DOF: degree of freedom. The trends of the lumped element parameters of the robot \(m_{r}\), \(c_{r}\), \(k_{r}\) as functions of the imposed position \(x_{r}\) are depicted in Figure 12. The circular marker highlights the values of these parameters when the impact occurs. Several scenarios were simulated, with the aim of evaluating the effects of robot properties during the collision. The simulation results depicted in Figure 13a,b represent the velocities of the tool and of the object when the robot had rigid and compliant joints, respectively. A perfectly elastic collision was considered (\(e=1\)) with \(k_{c}=10^{5}\)\(\frac{N}{m^{13}}\); the masses of the tool and of the object were \(m_{t}=0.125\) kg and \(m_{p}=0.25\) kg, respectively. In Figure 13a, the object velocity is exactly twice the approach velocity of the robot and the collision is so short that the robot actuators start braking after the impact. In this condition, as the impact occurred in a period of time in which the mass \((m_{r}+m_{t})\) maintained the same velocity, the robot with rigid joints was equivalent to a slider moving at constant velocity. The final velocity of the object confirmed this assumption (see Section 2.1). In the case of compliant joints (Figure 13b), the impact velocity is about 0.98 m/s due to the vibrations of the tool (the impact takes place when there is a local minimum of robot velocity), and the  
Figure 12: Lumped components \(m_{r}\), \(c_{r}\) and \(k_{r}\) as a function of the imposed position \(x_{r}\). The circular dot indicates the impact time. Figure 13: Time history of tool and object velocity. (**a**) Robot with rigid joints; (**b**) robot with compliant joints. velocity of the object is about 1.94 m/s, which is almost twice the approach velocity of the robot. Although the total colliding mass of the robot \((m_{r}+m_{t})\) is not rigidly coupled with the imposed motion, due to the compliances represented by the elements \(k_{r}\) and \(c_{r}\) of the model in Figure 8, the actual behavior is nearly the same as the robot with rigid joints. Two factors explain these results: firstly, the colliding masses are very different, since the term \((m_{r}+m_{t})\) is equal to 5.35 kg just before the impact, and Equation (4) shows that object velocity increases asymptotically towards the limit case, when the tool mass becomes large; secondly, since the contact force is small, joint compliances have a small influence. In Figure 14a, a heavy object with a mass of 2.5 kg is considered. This mass roughly corresponds to the rated payload of the robot and is about 48% of the term \((m_{r}+m_{t})\), which is equal to 5.35 kg just before the impact.