## 2 Materials and Methods
### Different VSMs' Basic Stiffness Curves  
Previous works on VSAs resulted in design concepts that differ in several aspects, such as mass to volume ratio, elastic energy to mass or volume ratios, working principal, etc. (see [13; 17; 18; 19; 20] for more details). Herein, we focus on the VSA's stiffness profile. The VSAs mentioned in other studies (see [14; 17; 18; 19; 20], representative of the current state-of-the-art) can be classified into one of the three categories illustrated in Table 1.  
The stiffness profiles of VSAs might be viewed singularly as a tunable basic stiffness profile. These basic stiffness profiles are shown in Figure 2:  
* Constant stiffness: this basic stiffness profile is shown in Figure 2a and exemplified in the second column of Table 1 by an actuator with adjustable stiffness (AwAS-II) [19]. At each curve of the AwAS-II characteristic, the stiffness remains practically constant. To adapt this mechanism to a variable load, the torque's slope is tuned via a small motor. * Biomimetically inspired stiffness, shown in Figure 2b, represents different VSAs, such as the ones developed at the German Aerospace Center (DLR) Institute of Robotics and Mechatronics, the floating spring joint (FS) [20], QA-joint (Quasi-Antagonistic Joint) [21], FAS (Flexible Antagonistic Spring) [22] or the one presented by [PERSON] et al. [23]. The third column of Table 1 shows the floating spring joint (FJS) prototype with a stiffness that increases along with the deflection, the same as in biological muscles [23]. * Torque limiter: [PERSON] introduced [24; 25] the safe joint mechanism (SJM) (see fourth column of Table 1), which is based on a slider-crank mechanism. Its basic stiffness curve corresponds to the one in Figure 2c. This mechanism is supposed to remain stiff under a certain torque level \(T_{1}\). Upon exceeding it, the stiffness rapidly drops to maintain the torque at \(T_{max}\) level, which represents the safety threshold. As the SJM is a slider-crank-based mechanism that is well known for its sensitivity to friction in the vicinity of zero deflection, the safety threshold may be easily exceeded in the case of blunt shocks, as shown by [PERSON] and colleagues [26]. For this reason, we introduced the V2 SOM basic curve (see Figure 2c,d). In the next section, the full V2 SOM characteristic, which allows for coping with load variation, unlike the SJM, is shown. The latter curve presents a finite value stiffness near zero deflection, which continuously drops to display a constant torque threshold \(T_{max}\). The following equation depicts the desired behavior: \[T_{\gamma}(\gamma)=T_{max}(1-e^{-s\gamma}),\] (1) where \(s\) is a positive constant and the \(\gamma\) elastic deflection angle is in the range of \(0\) to \(\pi/2\). Figure 1: Variable stiffness actuator (VSA) including the variable stiffness safety-oriented mechanism (V2 SOM). To compare the three stiffness profiles, we adopted the following factors:  
* Factor 1. The maximal stored elastic energy in \"normal working conditions,\" (i.e., not during collision scenarios). This quantifies how much elastic energy is stored in the VSM that can be unleashed as collision kinetic energy, hence increasing the damaging effect of a potential collision. * Factor 2. Passive torque limitation: in the case of a fast HR collision, it is more convenient to instantaneously contain the exerted torque with the VSM rather than as a control-based reaction [5, 6]. * Factor 3. Gravity-induced elastic deflection: this criterion quantifies the VSM's ability to passively limit this elastic deflection, which reduces the controller's compensation action. Based on the three factors, we ranked the stiffness profiles in Figure 2 accordingly, where the number of '+' reflects the factor's qualitative value. The results are given in Table 2, which shows that the V2 SOM basic curve displays better features in normal working conditions (i.e., factors 1 and 3), and in a collision scenario based on factor 2.