## 2 Methods  
### Proposed Features  
Figure 1 illustrates a typical force plate that is used in this study. Force transducers are placed at the four corners of the force plate. In Figure 1, the \(x\)-, \(y\)-, and \(z\)-axes correspond to the medial-lateral (ML) and anterior-posterior (AP), and vertical directions, respectively, and the origin of the coordinate system represents the center of the force plate. Note that the arrows associated with the three axes point to the positive direction of the corresponding axes. With \(\mathbf{F}_{i}\) denoting the GRF component at the \(i\)th corner of the force plate, and \(f_{x}\), \(f_{y}\), and \(f_{z}\) representing the GRF components in the ML, AP, and vertical directions, respectively, the resultant GRF can be expressed by the following vector \(\mathbf{f}\):  
\[\mathbf{f}=f_{x}\mathbf{i}+f_{y}\mathbf{j}+f_{z}\mathbf{k}=\mathbf{F}_{1}+\mathbf{F}_{2}+\mathbf{F}_{3}+ \mathbf{F}_{4} \tag{1}\]  
where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors along the \(x\)-, \(y\)-, and \(z\)-axes, respectively. With \(F_{iz}\) denoting the vertical component of \(\mathbf{F}_{i}\), the vertical component of the resultant GRF \(\mathbf{f}\) can be written as:  
\[f_{z}=\ F_{1z}+F_{2z}+F_{3z}+F_{4z} \tag{2}\]  
Similarly, with \(F_{ix}\) and \(F_{iy}\) denoting the ML and AP components of \(\mathbf{F}_{i}\), respectively, the ML and AP components of the resultant GRF \(\mathbf{f}\) can be represented as:  
\[f_{x}=\ F_{1x}+F_{2x}+F_{3x}+F_{4x} \tag{3}\]  
\[f_{y}=\ F_{1y}+F_{2y}+F_{3y}+F_{4y} \tag{4}\]  
To determine the coordinates of the COP, we must know three geometrical parameters of the force plate: \(d_{x}\), \(d_{y}\), and \(d_{z}\). As shown in Figure 1, \(d_{x}\) and \(d_{y}\) represent the distances along the \(x\)-axis and \(y\)-axis, respectively, from the coordinate axes to the force sensors, and \(d_{z}\) is the distance along the \(z\)-axis from the origin of the coordinate system to the support surface of the force plate. With \(d_{x}\), \(d_{y}\), and \(d_{z}\) known, the following equations can be derived using Newtonian mechanics to determine the coordinates of the COP. \[COP_{x}=\frac{d_{x}F_{xz}+d_{z}f_{x}}{f_{z}} \tag{5}\]  
\[COP_{y}=\frac{d_{y}F_{yz}+d_{z}f_{y}}{f_{z}} \tag{6}\]  
Figure 1: Force plate, its coordinate system, its three geometrical parameters, and locations of the four force sensors. where  
\[F_{xz}=\ F_{1z}+F_{4z}-F_{2z}-F_{3z} \tag{7}\]  
\[F_{yz}=\ F_{1z}+F_{2z}-F_{3z}-F_{4z} \tag{8}\]  
Equation (7) demonstrates how the four vertical GRF components are combined to influence the motion of \(COP_{x}\) which is the ML direction coordinate of COP. Similarly, Equation (8) shows how the four vertical GRF components are combined to change the motion of \(COP_{y}\) which is the AP direction coordinate of the COP. During quiet standing, the vertical direction component of the inertia force is considerably weaker than the weight force, and therefore, we assume that \(f_{z}\) equals the weight force during quiet standing. Consequently, with \(M\) as the mass of the test subject and \(g\) as the acceleration due to gravity, we have:  
\[f_{z}=\ F_{1z}+F_{2z}+F_{3z}+F_{4z}\approx Mg \tag{9}\]  
Under the assumption of a constant \(f_{z}\), Equations (5) and (6) can be rewritten as:  
\[f_{z}=\ F_{1z}+F_{2z}+F_{3z}+F_{4z} \tag{2}\]0  
\[f_{z}=\ F_{1z}+F_{2z}+F_{3z}+F_{4z} \tag{2}\]1  
where \(c_{xz}=d_{x}/Mg\), \(c_{zz}=d_{z}/Mg\), and \(c_{yz}=d_{y}/Mg\). Equation (10) shows that \(COP_{x}\) is determined by its vertical GRF component \(F_{xz}\) and its medial-lateral horizontal GRF component\(f_{x}\). Similarly, Equation (11) indicates that \(COP_{y}\) is determined by its vertical GRF component \(F_{yz}\) and its anterior-posterior horizontal GRF component\(f_{y}\). For a given COP location, Equations (9)-(11) represent an underdetermined system because the number of constraints is lower than the number of variables that can be used to satisfy the constraints.