## 2 Materials and Methods
### Hill-Type-Based Muscular Model  
Calf muscle is activated by activation stimulation and contracts to produce tension or muscle force (\(F_{M}\)). The ankle joint torque is predicted by the summation of tensions of the calf muscle tendon units (MTUs). The Hill-type muscular model is implemented to calculate \(F_{m}\). The conventional form described in [22] is adopted. There are two main parts, namely, the passive serial element (SE) and the contractile element (CE) parts. The SE part represents the elastic property of the muscle tendon. The CE part is composed of a contractile element (CE) and a passive element (PE). PE represents the muscle fiber passive force property. As a passive stiffness part is added in SE, PE is ignored in the calculation. The muscle fore, \(F_{M}\), can be calculated as follows:  
\[F_{M}=f(a,l,v)=F_{a}=\left(F_{CE}+F_{PE}\right)\cos\theta_{p}F_{CE}\cos\theta _{p} \tag{7}\]  
\[\left\{\begin{array}{l}F_{CE}=F_{\max}a(u)\cdot f_{l}\cdot f_{p}\\ f_{l}=\exp\left(-0.5\left(\left(L_{CE}/L_{CE_{0}}-1.05\right)/0.19\right)^{2} \right)\\ f_{v}=0.1433\left(0.1074+\exp\left(-1.3\sinh\left(2.8\frac{V_{CE}}{V_{CE_{0}} }+1.64\right)\right)\right)^{-1}\\ \theta(t)=\arcsin(\frac{L_{CE_{0}}\sin\theta_{p0}}{L})\end{array}\right. \tag{8}\]  
where \(F_{PE}\) and \(F_{CE}\) are the passive and active force generated by muscle fiber, respectively. \(F_{\max}\) is the maximum muscle fiber force exerted at the optimal fiber length \(L_{CE0}\). \(V_{CE0}\) is the maximum rate of change of the muscle fiber length. \(\theta_{p}\) and \(\theta_{p0}\) are, respectively, current and initial pennation angles. The equation for calculating \(\theta_{p}\) is derived based on the assumption that the thickness and the volume of the muscle are constant. Due to the small change of ankle joint angle in this study, \(\theta_{p}\) can be regarded as a constant. The initial parameter values of \(F_{\max}\), \(L_{CE0}\), \(V_{CE0}\) and \(\theta_{p0}\) are based on those reported in [40]. Prior to using these parameters, they are scaled by the weight and height of the subject. In this study, it is assumed that that the fiber length is located at its optimum during quiet standing. Based on the assumption mentioned here, \(F_{m}\) is the maximum value at the same muscle activation level. Under such a condition, the human metabolic energy consumption can be minimized in order to maintain upright balance [41]. This means that the initial value of \(L_{CE}\) is assumed to bethe optimal length \(L_{CE0}\) in the calibration. Moment arm (\(r\)) is gained from a general model depicted in [40] and linearly scaled by the subject's height. As the ankle joint angle changes within a small range, \(r\) is set to be a constant value. The Covariance Matrix Adaptation-Evolution Strategy (CMA-ES) optimization method adopted in [42] is implemented to find the proper parameters of the muscular model. The measured ankle joint torque, \(\tau_{ankle}\), is calculated by the data recorded from a force plate as follows:  
\[\tau_{ankle}=F_{z}\cdot(d_{COP}+d_{ankle})=F_{z}\cdot(\sqrt{x_{COP}{}^{2}+y_{ COP}{}^{2}}+d_{ankle}) \tag{9}\]  
where \(d_{COP}\) is the distance between GRF point and ankle joint ground projection point. \(x_{COP}\) and \(y_{COP}\) are calculated from force plate data as follows:  
\[x_{COP}=-\frac{M_{y}+Z_{off}\times F_{x}}{F_{z}} \tag{10}\]  
\[y_{COP}=\frac{M_{x}-Z_{off}\times F_{y}}{F_{z}} \tag{11}\]  
where \(F_{x}\), \(F_{y}\), \(F_{z}\), \(M_{x}\), \(M_{y}\), Mz are forces and moments in the x, y and z directions, respectively. Zoff is the height of the force plate sensor and is provided by the manufacturer. \(d_{ankle}\) is the distance in the ground projection between the original point of the coordinate on the force plate and the center of the ankle joint of the subject.