# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 2 Impact between the Tool and the Object
### Non-Instantaneous Impact  
The theoretical analysis presented in the previous sub-section in many practical cases gives a good prediction of final velocities, but it does not give information about the contact force and duration, which are very important for robot control. In recent years, many contact force models have been presented [11]. The impact between the two bodies can be schematized as in Figure 4, in which there an elastic element and a damping element in parallel are present. The elastic element can be linear or non-linear; the damping element is usually non-linear and the damping force depends both on the deformation velocity \(\dot{\delta}\) and on deformation \(\delta\). Most of the proposed contact models [11; 12; 13] can be formulated according to this equation:  
\[F=k_{c}\delta^{m}+\chi\delta^{n}\dot{\delta} \tag{5}\]  
in which \(k_{c}\) is stiffness, \(\chi\) is a constant and \(m\) and \(n\) are two exponents. In the present case, \(F\) is the force that a tool exerts on the object, and \(\delta=x_{t}-x_{p}\), with \(x_{t}-x_{p}>0\) to guarantee the contact. Figure 4: Non-instantaneous impacts. (**a**) A tool rigidly fixed to the robot; (**b**) a tool decoupled from the robot. Figure 3: Influence of moving mass on end velocities. The motion of the two masses after the impact can be calculated by solving the two differential equations:  
\[m_{t}\dot{x}_{t}=-F \tag{6}\] \[m_{p}\dot{x}_{p}=F\]  
If the tool is rigidly connected to the robot, only the second equation has to be solved, because \(x_{t}\) is assigned. Since in general \(F\) is non-linear, a numerical integration was carried out by means of Simulink. It is worth noting that Equation (5) with \(m=1\) and \(n=0\) represents a simple linear spring-damper contact model. The simple linear contact model makes it possible to predict the contact forces and impact duration and leads to a simple relationship between the damping ratio of the relative motion and the coefficient of restitution [10], but it has some drawbacks. The coefficient of restitution does not depend on the relative velocity before the impact (as experimentally assessed [12]). The viscous component of the contact force is discontinuous, since it suddenly increases at the beginning of the contact, is not zero at the end of the contact and is opposed to the separation between the object and the tool. This behavior is not consistent with physical intuition, since at the beginning and at the end of the contact there is no penetration and the force should be zero [12]. A non-linear contact model overcomes most of the above-mentioned limits. In the framework of this research, a contact model with \(m=\frac{3}{2}\) and \(n=\frac{3}{2}\) was considered. Since the tool and the object can be schematized as a plane surface and a sphere, respectively [14], the exponent of the elastic force comes from the well-known Hertz contact law [12]. The calculation of the energy lost in the hysteresis loop of the damping force shows that exponent \(n\) has to be set equal to \(m\)[12]. The damping constant \(\chi\) is related to the contact stiffness and to the restitution coefficient by this equation, as derived in [15]:  
\[\chi=\frac{3}{2}\frac{(1-e)}{e}k_{c}\frac{1}{\dot{\delta}_{in}} \tag{7}\]  
in which \(\dot{\delta}_{in}\) is the initial relative velocity. It is worth noting that Equation (7) comes from an improvement of the impact model presented in [12], which extends the validity of the model in the range of small restitution coefficients. Figure 5 shows the total contact force and its elastic and damping components calculated for the two limit cases of Figure 4. In the simulations, the following parameters were adopted: \(m_{t}=0.125\) kg, \(m_{p}=0.25\) kg, \(k_{c}=10^{5}\)\(\frac{N}{m^{1.5}}\), \(e=0.5\), \(v_{t}=1\) m/s and \(v_{p}=0\) m/s. The non-linear model leads to a realistic shape of the damping force (without discontinuities) and of the contact force.