## 2 Materials and Methods
### Working Principle of V2 SOM  
V2 SOM contains two functional blocks, as depicted in Figure 3a: a nonlinear stiffness generator block (SGB) and a stiffness adjusting block (SAB). To simplify the understanding of the V2 SOM kinematic scheme, Figure 3b represents its cross section, which is symmetrical to the rotation axis \(L_{1}\). The SGB is based on a cam follower mechanism, whereby the cam's rotation \(\gamma\) about the \(L_{5}\) axis, between \(-90^{\circ}\) and \(90^{\circ}\), induces the translation of its follower according to the slider \(L_{6}\). Then the follower extends its attached spring. At this level, a deflection angle \(\gamma\) corresponds to a torque value \(T_{\gamma}\) exerted on the cam. The wide range of this elastic deflection needs to be reduced to a lower range of \(-20\leq\theta\leq 20\), as is widely considered in most VSAs [21; 27]. Figure 4 shows various simplified diagrams necessary to understand the functioning principle of this block. In Figure 4a, the cam follower system is in the resting position, meaning that the springs are relaxed and the cam's applied torque about its rotation axis is \(T_{\gamma}=0\). Applying a torque \(T_{\gamma}\
eq~{}0\) generates a deflection angle \(\gamma\) and the translation of the sliders supporting the followers, which results in the compression of the springs (Figure 3b). The stiffness adjusting block (SAB) acts as a reducer by using a gear ring system, which is considered to be made up of gear mechanisms commonly used to transform the rotary motion into  
Figure 4: Stiffness generator block: (**a**) at rest \(\gamma=0\) (\(T_{\gamma}=0\)); (**b**) at deflection \(\gamma\
eq 0\) (\(T_{\gamma}\)). Figure 3: Block representation of the V2 SOM, (**a**) Two functional blocks of V2 SOM (**b**) V2 SOM kinematic scheme. either rotary or linear motion [28]. Furthermore, the SAB serves as a variable reducer thanks to the linear actuator M, which controls the distance r while driving the gears in a lever-like configuration. The reduction ratio of SAB is continuously tunable, allowing V2 SOM to cope with the external load \(T_{\theta}\), where the link side makes a deflection angle \(\theta\) relative to the actuator side. Figure 5 shows the symmetrical two ring gears (in blue and green in Figure 5a) geared to a central spur gear (in red). These two ring gears are driven by a symmetrical double lever arm system via two rods (in yellow). These rods, which are part of the lever arm \(L_{2}\), as illustrated in Figure 4, slide freely along the pocket of the ring gear, creating the prismatic joint shown in Figure 5b. Adjusting the position \(r\) of the rods along the lever arm via linear actuator M changes the transmission ratio of the lever arm system, and hence the reduction ratio of the SAB. The gears' rotation induces a variation of the rods' position along the pockets, which is characterized by the distance \(x\) in Figure 5b. The V2 SOM CAD (computer aided design) model is presented in Figure 6. More details are given in the V2 SOM patent [29; 30]. Figure 6a shows the two blocks, the stiffness adjustable block and the generator block, with a zoomed-in view. Each block is depicted by additional views of their mechanical parts (Figure 6b-d) with correspondence to the kinematic sketch in Figure 3b. One can identify the linear actuators, the gearing, and cam follower system with its actuation side. The V2 SOM blocks, as shown in Figure 4, are connected rigidly to fulfill separate dedicated tasks:  
* The SGB is characterized by the curve of the torque \(T_{\gamma}\) vs. deflection angle \(\gamma\). This curve is obtained through the cam profile, the followers, and other design parameters. The basic torque curve leading to the torque characteristic of the V2 SOM is depicted in Figure 6a. This basic curve is elaborated with a torque threshold equal to \(T_{max}=2.05\) Nm. * The SAB is considered a quasi-linear continuous reducer (QLCR) and defined by its ratio expression given in Figure 6b.