## Materials and Methods
### Model and Control System Design  
The system's control logic was developed by the finite state machine (FSM) criterium as in the previous prototypes [28; 29; 30]. The FSM involved the following phases:  
* Idle: the actuator's rod remains still in the retracted position;
* Approach: the piston nears the target body following a constant speed reference signal. It is triggered by the occurrence of the start command delivered by the operator;
* Strike: the actuator meets the target body and force reference control is issued. It is triggered by a 2.5 N threshold on the load cell signal;
* Retraction: the actuator's rod returns to the retracted position with constant speed after having completed or failed to enact the strike phase. The rod's idle, approach, and retraction phases, concerning only the actuator part, are managed by the driver's embedded PI speed controller. The strike phase, of higher importance due to the phenomena involving the human-machine interaction, is controlled by the real-time machine through an MPC-based force control design. The driver's embedded PLC is responsible for the toggling between the two types of control. System modeling has thus exclusively involved the strike phase. The human body recovers low balance disturbances in the orthostatic position configuration mainly by ap \begin{table}
\begin{tabular}{c c} \hline \hline
**Number** & **Component** \\ \hline
1 & GD16Q motor (NiLAB GmbH, Klagenfurt am Worthersee, Austria) \\ \hline
2 & SLVD1N driver (Parker Hannifin Corp., Cleveland, USA) \\ \hline
3 & Q4 XTULAF400-Q8 optical sensor (Banner Engineering Corp., Plymouth, MN, USA) \\ \hline
4 & UMM 50 kgf-ranged load cell (Dacell Co. Ltd., Cheongju, Korea) \\ \hline
5 & Expanded polyethylene interface \\ \hline
6 & Baseline Real-Time target machine (Speed goat Inc., Natick, MA, USA) \\ \hline
7 & Aluminum sliding plates \\ \hline
8 & Target body weight \\ \hline
9 & Custom made viscoelastic dampers \\ \hline
10 & C-SHR28-1000-B4 linear guides (MISUMI, Europa GmbH, Frankfurt, Germany) \\ \hline
11 & PZ-34-A-100 displacement transducers (GEFRAN, Provaglio d’Iseo, Italy) \\ \hline
12 & DEWE-RACK-4 (Dewetron GmbH, Grambach, Austria) \\ \hline
13 & Connection box for Real-Time target system \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of test bench key components for HIL testing. Numbers refer to labels in Figure 1.  
Figure 1: (**a**) Test bench model in SolidWorks environment; (**b**) test bench specimen; (**c**) example of experimental trial. plying a correcting torque at the ankle (Tc). Dynamic equilibrium is reached once the body's CoM is vertically aligned with the CoP, i.e., the resultant of the gravity forces (\(-m_{tot}g\)) lies on the same (vertical) axis of the ground reaction force (R\({}_{\text{y}}\)). This approximation allows studying human balance control in the sagittal plane through a simple inverse pendulum model, as shown in Figure 2a. The same approach can be used to describe the oscillations of the operator about the ankle joint while maneuvering the automatic perturbator. As for the nonlinearities involved in the strike phenomenon, the literature offers many solutions in order to efficiently model the contact force, mainly depending on the properties of the materials involved and on the definition of boundary conditions. Nonlinearities generally refer to polynomial forms in the bodies' relative penetration \(\delta\), its respective speed \(\dot{\delta}\), and a hysteresis damping factor \(\chi\)[44]. Due to the difficult measurement of these parameters, said aforementioned nonlinear relations were approximated by adopting a lumped parameter system that models the actuator's and target's bodies as directly connected, therefore modeling the contact force as a stress. This approximation, in addition to linearizing the angular configuration of the phenomenon, as in Figure 2a, through Taylor's series, leads to the linearized lumped parameter model shown in Figure 2b.