# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 2 Impact between the Tool and the Object
### Instantaneous Impact  
If the duration of the impact is very short and the impact force is so large that other forces can be neglected, the impact between the tool of the moving robot and the object can be studied according to the classical impact theory [10]. Two limit cases are represented in Figure 1.  
Figure 1: Instantaneous impacts. (**a**) A tool rigidly fixed to the robot; (**b**) a tool decoupled from the robot.  
In the first limit case, the tool, which has a plane face, is rigidly connected to the robot, and the object is a spherical point mass. Since the mass of the robot with the tool is much larger than the mass of the object, its velocity change is negligible. In other words, the velocity of the tool after the impact (\(V_{t}\)) is equal to the velocity before the impact (\(v_{t}\)) and is imposed by the control system of the robot. The final velocity of the object can be calculated using the restitution coefficient (\(e\)), which is the ratio of the relative velocity of the two masses after the impact to that before; \(e=1\) means that the contact is elastic (no kinetic energy is lost), and \(e=0\) means that the contact is completely inelastic.The following equation holds:  
\[V_{p}=V_{t}+e(v_{t}-v_{p}) \tag{1}\]  
in which \(v_{p}\) and \(V_{p}\) are the initial and final velocities of the object, respectively. It is worth noting that the mass of the object has no effect on its final velocity. If the tool is moving with velocity \(v_{t}\) = 1 m/s and the object is initially stationary, the final velocity of the object is \(V_{p}\) = 2 m/s with \(e=1\) (pure elastic impact) and \(V_{p}\) = 1 m/s with \(e=0\) (completely inelastic impact).  
In the second limit case, the tool is completely decoupled from the robot and is represented by a simple point mass \(m_{t}\) with plane faces. The initial velocity \(v_{t}\) of the tool is imposed by the motion of the robot, but it can change after the impact and the tool can move backwards. The conservation of momentum of the mechanical system depicted in Figure 1b leads to the following:  
\[m_{t}v_{t}+m_{p}v_{p}=m_{t}V_{t}+m_{p}V_{p} \tag{2}\]  
The final velocities of the two masses can be found by coupling Equation (2) with Equation (1):  
\[V_{t}=v_{t}+\frac{m_{p}}{m_{t}+m_{p}}(1+e)(v_{p}-v_{t}) \tag{3}\]  
\[V_{p}=v_{p}-\frac{m_{t}}{m_{t}+m_{p}}(1+e)(v_{p}-v_{t}) \tag{4}\]  
Figure 2 shows the effect of \(e\) on the final velocities. A tool with \(m_{t}\) = 0.125 kg and \(v_{t}\) = 1 m/s and an object with \(m_{p}\) = 0.25 kg and \(v_{p}\) = 0 m/s are considered. The increase in the restitution coefficient leads to increases in the final velocity of the object and in the velocity variation of the tool.  
The effect of tool mass is depicted in Figure 3. A decrease in the mass of the tool causes a decrease in the final velocity of the object for every value of the restitution coefficient.  
Figure 2: Influence of \(e\) on the end velocities of the tool and of the object.  
The comparison between the final velocities of the object calculated in the two limit cases shows that the variation in the velocity of the tool leads to a large reduction in the final velocity of the object. In fact, the tool is neither rigidly fixed to a massive and stiff robot nor completely free to move. Nevertheless, this simple analysis shows that the decrease in the restitution coefficient and the mechanical decoupling between the tool and the robot are needed to minimize the object's final velocity.