## 2 Materials and Methods
### Data Analysis  
The positions of the center of pressure (COP) of the GRF were calculated using forces and moments measured by the force plates [40]. For standing balance tests, the total distance traveled by the COP over the course of the trial duration (DCOP) was calculated [25]. The COP sway area (ACOP) was calculated as that of an equivalent ellipse covering 95% of the samples, and the major and minor axes of the ellipse were determined using principal component analysis [41].  
For gait balance analysis, all body segments, namely the head/neck, trunk, pelvis, arms, forearms, thighs, shanks, and feet, were modeled as rigid bodies. Using the measured marker data, each body segment was embedded with an orthogonal coordinate system with the origin at the segment's COM and the positive x-axis directed anteriorly, the positive y-axis superiorly, and the positive z-axis to the right. The 13-body segment model of the body was then used to calculate the position of the body's COM (C) as the weighted sum of the segmental COMs of all body segments as follows [29]:  
\[\overset{\rightharpoonup}{C}=\frac{\sum_{i=1}^{13}m_{i}\overset{\rightharpoonup}{C}_{i}}{BM} \tag{1}\]where \(m_{i}\) and \(\overset{\rightharpoonup}{c}_{i}\) were the mass and position of the COM of the \(i\)th body segment calculated using the marker data and an optimization-based method [17; 42]. \(BM\) was the total body mass of the subject. During the double stance phase, a resultant COP was calculated from the COP and GRF of each foot [40]. A global optimization method was used to reduce the effects of soft tissue artifacts of the skin markers on the pelvis-leg apparatus by minimizing the weighted sum of squared distances between measured and calculated marker positions [43].  
For the description of the motions of the COM relative to the COP, the line of progression was defined as that which bisected the mediolateral range of motion of the COM during bilateral single-limb support (SLS) and double-limb support (DLS), a zero value being on the bisection line and a positive value being on the right side of the bisection line. The IAs in the sagittal plane and frontal plane were calculated as follows [44]:  
\[\overset{\rightharpoonup}{t}=\left(\overset{\rightharpoonup}{Z}\times\frac{ \overset{\rightharpoonup}{P}_{COM-COP}}{\left|\overset{\rightharpoonup}{P}_{ COM-COP}\right|}\right) \tag{2}\]  
\[\text{Sagittal IA}=\sin^{-1}(t_{Y}) \tag{3}\]  
\[\text{Frontal IA}=\left\{\begin{aligned} &-\sin^{-1}(t_{X}), \quad for\ the\ right\ limb\\ &\sin^{-1}(t_{X}),\quad\quad for\ the\ left\ limb\end{aligned}\right. \tag{4}\]  
where \(\overset{\rightharpoonup}{P}_{COM-COP}\) was the vector pointing from the COP to the COM, \(\overset{\rightharpoonup}{Z}\) was the unit vector of the vertical axis of the global coordinate system, and \(\overset{\rightharpoonup}{X}\) was the unit vector pointing in the direction of progression. The RCIA was also calculated by smoothing and differentiating the trajectories of sagittal and frontal IAs using the generalized cross-validatory spline method [45]. Positive values of the sagittal and frontal IAs indicated that the COM was anterior to and away from the COP toward the contralateral limb, respectively (Figure 1). The force plate data were also used to determine the gait events of heel-strike and toe-off [46]. The values of the IA and RCIA at heel-strike and toe-off were obtained from the gait cycles of both sides. The ranges and time-averaged values of IA and RCIA over DLS were also obtained. Spatiotemporal parameters of gait were also obtained, namely stride length, step length, step width, cadence, gait speed, and stride time.  
Figure 1: The sagittal and frontal COM–COP inclination angles (IA, denoted \(\alpha\) and \(\beta\), respectively) during level walking (**a**) at the end of single-limb support at contralateral heel-strike; and (**b**) during terminal double-limb support. The trajectories of the COM and COP are also shown.