## Materials and Methods
### Model and Control System Design
The system includes the appropriate transducers required to monitor relevant state variables, as shown in the test-bench design (see Figure 1). The modeled system neglects the CoP displacement signal, which is the target of biomedical analysis only and therefore is not used in the control loop computations. All the system parameters are listed in Table 2.  
The Lagrange approach, adopting the displacements displayed in Table 2 as degrees of freedom q(t), leads to the following dynamic equations:  
\[[\text{M}]\ddot{\text{q}}(\text{t})+[\text{C}]\dot{\text{q}}(\text{t})+[\text{ K}]\text{q}(\text{t})\ =\ \{\text{F}(\text{t})\},\ \ \text{q}(\text{t})\ =\ \left\{ \begin{array}{c}\text{x}_{ \text{a}}(\text{t})\\ \text{x}(\text{t})\\ \text{x}_{\text{b}}(\text{t})\end{array} \right\} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Description** \\ \hline m\({}_{\text{a}}\), x\({}_{\text{a}}(\text{t})\), F\({}_{\text{a}}(\text{t})\) & Actuator’s mass, displacement, and issued force \\ c\({}_{\text{a}}\), k\({}_{\text{a}}\) & Operator response’s damping and stiffness \\ m\({}_{\text{b}}\), c\({}_{\text{b}}\), k\({}_{\text{b}}\), x\({}_{\text{b}}(\text{t})\) & Target body’s mass, linear damping, stiffness, and displacement \\ m\({}_{\text{tr}}\), x\({}_{\text{0}}\) & Rod’s mass, elongation at impact \\ c\({}_{\text{i}}\), k\({}_{\text{i}}\), F\({}_{\text{i}}(\text{t})\), x\({}_{\text{t}}(\text{t})\) & Interface’s damping, stiffness, contact force, and displacement \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Modeled system key parameters list. Symbols refer to Figure 2b. All displacements are referred to the respective equilibrium positions. Figure 2: (a) Human body’s reaction to low balance disturbances; (b) model of the system during strike phase for control application: actuator (in blue), interface (in green), target body (in red), state and measured variables (in black). \[[\mathrm{M}]\;=\;\left[ \begin{array}{ccc}\mathrm{m_{a}}&0&0\\ 0&\mathrm{m_{r}}&0\\ 0&0&\mathrm{m_{b}}\end{array} \right],\;[\mathrm{C}]\;=\;\left[ \begin{array}{ cccc}\mathrm{c_{a}}&0&0\\ 0&\mathrm{c_{i}}&-\mathrm{c_{i}}\\ 0&-\mathrm{c_{i}}&\mathrm{c_{i}}+\mathrm{c_{b}}\end{array} \right],\;[\mathrm{K }]\;=\;\left[ \begin{array}{ccc}\mathrm{k_{a}}&0&0\\ 0&\mathrm{k_{i}}&-\mathrm{k_{i}}\\ 0&-\mathrm{k_{i}}&\mathrm{k_{i}}+\mathrm{k_{b}}\end{array} \right] \tag{2}\]  
\[\{\mathrm{F(t)}\}\;=\;\left\{ \begin{array}{c}-\mathrm{F_{a}(t)}\\ \mathrm{F_{a}(t)}\\ 0\end{array} \right\} \tag{3}\]  
Standard dynamic to state-space (SS) representation transformation is performed through the reduction of order technique by the following equations:  
\[\left\{\begin{array}{c}[\mathrm{M}]\dot{\mathrm{v}}+[\mathrm{C}]\mathrm{v}+[ \mathrm{K}]\mathrm{q}\;=\;\{\mathrm{F(t)}\}\stackrel{{\mathrm{def }}}{{=}}\{\mathrm{T}\}\mathrm{u}(\mathrm{t})\;\;,\;\mathrm{z}(\mathrm{t})\;= \;\left\{ \begin{array}{c}\mathrm{v}(\mathrm{t})\\ \mathrm{q}(\mathrm{t})\end{array} \right\}\!,\;\mathrm{T}\;=\;\left\{ \begin{array} []{c}-1\\ 1\\ 0\end{array} \right\}\right\} \tag{4}\]  
\[\dot{\mathrm{z}}(\mathrm{t})\;=\;\left[ \begin{array}{cc}-[\mathrm{M}]^{-1} [\mathrm{C}]&-[\mathrm{M}]^{-1}[\mathrm{K}]\\ \mathrm{I_{3\times 3}}&\mathrm{O_{3\times 3}}\end{array} \right]\mathrm{z}( \mathrm{t})+\left[ \begin{array}{c}[\mathrm{M}]^{-1}\{\mathrm{T}\}\\ 0_{3\times 1}\end{array} \right]\mathrm{u}(\mathrm{t})\stackrel{{ \mathrm{def}}}{{=}}[\mathrm{A}]\mathrm{z}(\mathrm{t})+[\mathrm{B}]\mathrm{u}( \mathrm{t}) \tag{5}\]  
\[\mathrm{y}(\mathrm{t})\;=\;\left\{ \begin{array}{c}\mathrm{F_{i}(t)}\\ \mathrm{x}(\mathrm{t})-\mathrm{x_{a}(t)}\end{array} \right\}\;=\;\left[ \begin{array} []{cccc}0&-\mathrm{c_{i}}&\mathrm{c_{i}}&0&-\mathrm{k_{i}}&\mathrm{k_{i}}\\ 0&0&0&-1&1&0\end{array}