## 4 Discussion
This observation suggested that compared with that of the older group, the \(f_{y}\) of the younger group played a more active role in determining \(COP_{y}\). Our second observation is based on [PERSON]'s second law of motion, which states that the acceleration of an object is proportional to the net force acting on the object. The second law implies the following:  
\[COM_{y}\,\propto\,f_{y} \tag{12}\]  
where \(\hat{COM}_{y}\) is the acceleration of the AP component of the COM. If \(f_{y}\) is highly coupled with \(COP_{y}\), \(f_{y}\) loses its ability to independently maneuver the AP component of the COM. This can impair postural stability because the fundamental goal of postural balance control is to maintain the vertical projection of the COM within the safe limits of the support surface. This could partially explain why the values of the proposed feature \(\beta_{AP}\), unlike the values of \(\alpha_{AP}\), were not very close to 1. In addition, the values of \(\beta_{AP}\) summarized in Table 1 demonstrated that the correlation strength between \(COP_{y}\) and \(f_{y}\) for the older group was significantly higher than that for the younger group. Hence, because \(f_{y}\) was more strongly coupled with \(COP_{y}\), the \(f_{y}\) force component of the older group was more constrained to maneuver the AP component of the COM. As a result, postural stability can be undermined. The third observation was related to the effect of \(f_{y}\) on the vertical free moment (FM). As a frictional torque, FM can be computed from the following equation [48; 49]:  
\[FM=M_{z}-COP_{x}f_{y}+COP_{y}f_{x} \tag{13}\]  
where \(M_{z}\) is the vertical moment about the force plate center. Note that including \(M_{z}\), every signal that appeared on the right-hand side of Equation (13) could be directly measured using a force plate. Therefore, the FM is an additional postural variable that can be easily obtained using only a force plate. Similar to Equations (10) and (11), which indicated that moving the COP to the desired position required the coordination of different GRF components, Equation (13) demonstrated that \(M_{z},f_{x}\), and \(f_{y}\) must coordinate with each other to limit the magnitude of FM to prevent the body from rotating excessively in the vertical direction. The capability of \(f_{y}\) in freely attending such a coordination task was weakened when \(f_{y}\) was highly correlated with \(COP_{y}\). As demonstrated by the results of \(\beta_{AP}\) in Table 1, this weakening effect was more prominent in the older group than in the younger group because \(\beta_{AP}\) characterizes the coupling strength between \(COP_{y}\) and \(f_{y}\). In summary, in studying the redundancy problem associated with Equation (11), we found that the coupling strengths between \(COP_{y}\), \(F_{yz}\), and \(f_{y}\) of the younger group were lower than those of the older group. As a result, the younger group exhibited higher flexibility in coordinating \(F_{yz}\) and \(f_{y}\) when forming a given \(COP_{y}\). In addition, by more loosely coupling \(f_{y}\) with \(COP_{y}\), the \(f_{y}\) force component of the younger group could play a more active role than that of the older age group in managing the AP component of the COM and the vertical FM. This relatively more active role of \(f_{y}\) in the younger age group could partially explain why the younger group's posture balance was more stable than that of the older group. In this study, the efficacy of the proposed approach was demonstrated using comparative quiet standing experiments involving participants of two age groups. By quantifying the coupling strengths between COP and its GRF components, the proposed features tried to characterize the complex interplay of biomechanics and neuromuscular control during the balancing process. As discussed in Section 1, conventional COP features were employed to characterize the effects of many health-related conditions on posture balance [13; 16; 17; 18; 19; 20].