## 3 Safety Criteria: V2 SOM vs. Constant Stiffness
### Human-Robot Collision Model  
The human head is the most critical body region when dealing with the safety problems of pHRI. Indeed, some previous works [13; 14] have investigated this issue. Furthermore, they proceeded with theoretical modeling of dummy head hardware in crash tests. The resulting model (see Figure 11) has been well-tuned experimentally and validated.  
The model in Figure 11 is parameterized according to [PERSON] et al. [14], with:  
* Neck viscoelastic parameters \(d_{N}=12\) (N-s/m), \(k_{N}=3300\) [N/m].
* Head's mass \(M_{head}=5.09\) (Kg) and linear displacement x.
* The contact surface viscoelastic parameters \(d_{c}=10\) (N-s/m), \(k_{c}=1500\) (N/m).
* Robot arm contact position l and inertia \(I_{arm}\).
* Rotor inertia \(I_{rotor}\), torque \(\tau_{rotor}\), and angular position \(\theta_{1}\).
* VSM's stiffness \(K\) and angular deflection \(\theta=\theta_{1}-\theta_{2}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Body Region**} & \multicolumn{2}{c}{**ImpF (N) for AIS \(\leq\) 1**} & \multicolumn{2}{c}{**CC (N/mm)**} & \multicolumn{1}{c}{**HIC\({}_{15 ms}\) for AIS = 0**} \\ \hline \multirow{2}{*}{Head/Neck} & Face & 90 & 75 & 150 \\ \cline{2-5}  & Neck/sides & 190 & 50 & \\ \hline Trunk & Belly & 160 & 10 & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Safety criteria thresholds for most critical body regions from ISO/TS15066 and Payne [31]. ImpF = impact force; AIS = abbreviated injury scale; CC = compression criterion; HIC = head injury criterion.  
The collision model allows us to evaluate both ImpF and HIC. The first criterion is directly deduced from the simulation data as the maximum value of the contact force. HIC is evaluated by solving the following optimization problem:  
\[\begin{split} HIC_{15}=\max_{t_{1},t_{2}}\left[\left(\frac{1}{(t_{2 }-t_{1})}\right)\frac{t_{2}^{2}}{\tilde{x}(t)dt}\right]^{2.5}(t_{2}-t_{1}) \right],\\ \text{Subject to }t_{2}-t_{1}\leq 15 ms,\end{split} \tag{3}\]  
where \(\tilde{x}(t)\) is the head acceleration value at instant \(t\).  
In the next sections, to compare between V2 SOM and constant stiffness (CS), a VSM is carried out via a simulation of the HR collision model in the Matlab/Simulink platform. Here, the CS value is set so that the two profiles match at a torque value of 0.8 \(T_{max}\), as shown in Figure 8. This torque value defines the deflection range of the normal operational mode for the V2 SOM, after which the shock-absorbing mode is considered to be triggered.  
The following simulations are meant to highlight V2 SOM's inertia and torque decoupling capacities in comparison to a statistically equivalent CS-based VSM. These capacities represent the robust passive tackling of a HR collision in the fast and critical phase before the collision detection and reaction take place (e.g., within a range of 15 ms), as quantified via the HIC. These simulations were carried out using parametrization of Table 4 to evaluate both safety criteria (HIC and ImpF).  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Figure 13** & **Figure 14** \\ \hline \((I_{rotor},I_{arm})\) (kgm\({}^{2}\)) & (0.0875\(\rightarrow\)0.525, 0.14) & (0.175, 0.14) \\ \hline \((\tau_{rotor},T_{max},T_{1})\) (Nm) & \((10,15,12)\) & (7.5\(\rightarrow\)30,15,12) \\ \hline \(c\) & 37 & 37 \\ \hline \(\hat{\partial}_{1}\left(\text{rad}\cdot\text{s}^{-1}\right)\) & \(\pi\) & \(\pi\) \\ \hline \(k_{c}\) (N/m) & 1500 & 1500 \\ \hline \(l\) (m) & 0.6 & 0.6 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation parameters.  
Figure 11: Mechanical model of dummy head hardware collision against a robot arm. Simulation results of human-robot collision.