## 4 Analysis and Results
### Dynamic Analysis
#### 4.2.1 Case I--Hyperextension  
For this series of virtual experiments, hyperextension of the index finger was studied under different angles and velocities of impact. Figure 4 shows the hyperextension of the index finger under impact at 1.8 m/s velocity, at an angle of 30\({}^{\circ}\) from the vertical, by a robot with an effective mass of 3.6 kg (UR3 or similar robot). The shear strain exceeded the tissue damage threshold limit of 5.54 MPa at 30 ms. The maximum principal stresses developed in the bone exceeded the fracture threshold at 35 ms and reached a peak value of 171 MPa in the proximal phalange. The strain energy of the index finger reached a  
Figure 3: Quasi-Static test. (**a**) Maximum Principal Stress (**b**) Maximum Shear Elastic Strain (**c**) Maximum Shear Stress.  
peak value of 4.0 Joules, which far exceeds the energy limit value of 0.49 J for the hands and fingers specified in ISO/TS 15066. The maximum contact force developed for this simulation was 273.3 N. Based on the total contact area, this corresponded to an average pressure of 127 N/cm\({}^{2}\). This is lower than the maximum permissible pressures specified in ISO/TS 15066. However, the contact area of the cylindrical end effector was much higher than that of the plate used in the quasi-static case described earlier. Additionally, localized pressure at certain points may be higher than the permissible value even if the average pressure is lower.  
Figure 4: (**a**) Results for hyperextension of index finger; (**b**) Maximum Shear Elastic Strain; (**c**) Maximum Principal Stress.  
As the end effector was not a sharp object and the motion of the finger was not severely constrained, it is understandable that the principal stress in the bone might have exceeded fracture limits close to the same time as the shear stress in the soft tissue exceeded permissible values.  
Virtual experiments were run for velocities between 1 m/s and 2.2 m/s. Figures 5 and 6 show the maximum shear stress and maximum principal stresses developed in the tissue and bone, respectively. The results of the quasi-static benchmark test were used to highlight the pain onset threshold.  
Another set of experiments with hyperextension was carried out, where the cylindrical end effector was given a velocity at different angles, with an initial velocity of 1.2 m/s and an end effector with an effective mass of 7.2 kg and 14.4 kg (corresponding to UR5 and UR10 or equivalent robots). It was observed that shear stress in the tissue as well as principal stress in the bones initially increased with an increase in the moment arm (Figure 7). When the angle of the initial velocity was directed at an angle of greater than 90 degrees, the end effector was effectively free to move past the finger without transferring much of its initial kinetic energy to the finger. As shown in Figures 8 and 9, the shear strain and principal stress did not change markedly with an increase in robot effective mass under these conditions.  
Figure 6: Maximum Principal Stress as a function of end effector velocity and effective mass, with contour results at the indicated point.  
Figure 7: Maximum Principal Stress vs. Angle of Impact, UR5 and UR10, with contour results at the points shown.