## 3 Virtual Experiment Setup for Human-Robot Collaboration
### Setup for Dynamic Simulations  
For modeling of transient contact/impact in ISO/TS 15066, the maximum permissible energy transfer was estimated as a function of the maximum force or maximum force values found as a result of the quasi-static test described in the previous section. As per ISO/TS 15066, the transient contact was modeled as a fully inelastic contact between a human body region and a robot with effective mass \(m_{R}\), which is given by:  
\[m_{R}=\frac{M}{2}+m_{L} \tag{2}\]  
where \(M\) is the total mass of the moving parts of the robot and \(m_{L}\) is the effective payload of the robot system. This is further used to estimate transient contact speed limits. In this study, the distributed kinetic energy of the links was also considered. The mass of the actuators and the payload of the robot system were assumed to be point masses. The total rotational kinetic energy was calculated for the special case where the robot has its links extended for maximum reach and the shoulder joint, elbow joint, and wrist are operating at an equal angular velocity (\(\omega_{1}=\omega_{2}=\omega_{3}\)), as shown in Figure 2. Typically, robots are not operated in this configuration, with fully extended links, because of singularity concerns where the manipulator loses one or more degrees of freedom [35]. However, this configuration is suitable for the purpose of choosing an appropriate equivalent mass because this represents a harsh operating condition with high effective momentum and tool center point (TCP) velocity. We define the link-length ratios as:  
\[r_{1}=\frac{l_{1}}{L}\ ;r_{2}=\frac{l_{2}}{L}\ ;r_{3}=\frac{l_{3}}{L} \tag{3}\]  
where \(L=l_{1}+l_{2}+l_{3}\). The total kinetic energy of the entire system is equated to the kinetic energy of an equivalent mass \(m_{R}\):  
\[T_{total}=\frac{1}{2}m_{R}(L\omega)^{2} \tag{4}\]  
The resulting equivalent mass of the system is:  
\[m_{R}=m_{a}\Big{(}r_{1}^{2}+2(r_{1}+r_{2})^{2}+1\Big{)}+\frac{m_{d}}{6}\Big{(} r_{1}^{3}+1+3r_{1}^{2}r_{2}+3r_{1}r_{2}^{2}\Big{)}+m_{L} \tag{5}\]  
where \(m_{a}\) is the mass of an actuator and \(m_{d}\) is the total distributed mass of the links, less the actuators. In this study, the effective masses for Universal Robots UR3, UR5, and UR10 were estimated and used in subsequent analysis as examples of Small-, Medium-, and Heavy-Payload-capacity robots, respectively. This approach may be generalized to industrial robots with similar configurations. Dynamic analysis was carried out for the full hand model as well as the index finger under different loading conditions, as summarized in Table 2. In all cases, a rigid cylindrical object, which represents the effective mass of the robot system, was given an initial velocity, which is the velocity of the end-effector of the robot. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Robot Size/Effective Mass** & **Velocity Range** & **Mode of Loading** \\ \hline UR3; Effective Mass: 3.6 kg & & Flexion \\ UR5; Effective Mass: 7.2 kg & 1000 to 2200 mm/s & Hyper-Extension \\ UR10; Effective Mass: 14.4 kg & & Abduction \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design of Virtual Experiments. Figure 2: (**a**) Commercial collaborative robot for accident scenarios [36]. (**b**) Robot Configuration for calculation of equivalent mass. The simulation end time was set at 50 ms. The intent was to capture the stress and strain characteristics as well as the contact forces during the initial moments of the impact. Average human reaction times are around 180 ms; however, unconscious reflex actions are typically much faster at around 50-80 ms [37]. It is assumed that the human subject will not be able to react to the impact stimuli before 50 ms, and the impact event will end as the human subject is expected to move their hand and finger away soon after 50 ms. Collaborative robots, such as the ABB Yumi and Universal Robotics UR10, employ torque sensors and force sensors embedded in joints for collision detection capability [38].