## 3 Results  
### Movement Components of Treadmill Running  
Visualizations of the first five PMs (PM\({}_{\text{1-5}}\)) and their compositions of principal position (PP\({}_{\text{k}}\)-rVAR) and acceleration (PA\({}_{\text{k}}\)-rVAR) are depicted in Figure 1 and movement characteristics are described in Table 2. The first principal movement (PM\({}_{\text{1}}\)), which resembles the swing-phase movement coupled with trunk rotation, shows the highest contribution to the total variance in postural positions and in postural accelerations. Animated stick figures of PM\({}_{\text{2-5}}\) corresponding to phases of the gait cycle are represented in Supplementary Video S2. \begin{table}
\begin{tabular}{l l c c} \hline \hline \(\text{PM}_{\text{k}}\) & **Movement Characteristics** & **PP\({}_{\text{k}}\)-rVAR (\%)** & **PA\({}_{\text{k}}\)-rVAR (\%)** \\ \hline \multirow{2}{*}{k = 1} & Swing-phase movement coupled with trunk rotation: One leg moves & \multirow{2}{*}{72.5 \(\pm\) 7.3} & \multirow{2}{*}{49.0 \(\pm\) 4.2} \\  & forward while the trunk rotates to balance the body’s momentum. & & \\  & Double-leg support coupled with anteroposterior sliding: Both feet are & & \\
2 & on the ground (double-leg support) and involve anteroposterior & & \\  & (forward–backward) movement. & & \\  & Single-leg support coupled with trunk rotation: One leg supports the & & \\
3 & body while the other swings forward. Trunk rotation is also involved & & \\  & here, assisting in the balance during the single-leg support. & & \\  & Double-leg support coupled with hip and knee flexion/extension: Hip & & \\
4 & and knee movements during double-leg support, capturing the flexion & & \\  & and extension of these joints as both feet are in contact with the ground. & & \\
5 & Single-leg support coupled with mediolateral sliding: The mediolateral & & \\  & (side-to-side) movements during the single-leg support phase. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the first five principal movements (PM\({}_{\text{1-5}}\)). Figure 1: Visualizations of the first five principal movements (PM\({}_{1-5}\): left column), shown in sagittal and frontal views, examples of principal position (PP\({}_{\mathbf{k}}\)) and principal acceleration (PA\({}_{\mathbf{k}}\)) over time (middle column), and the space–time representation of the calculated Lyapunov exponent (LyE) for PP\({}_{1-5}\) (right column). The dashed line represents the right limb. Example data were derived from a single male participant. PM\({}_{1}\) reflects swing-phase movement with trunk rotation to balance momentum. PM\({}_{2}\) represents double-leg support with forward–backward sliding. PM\({}_{3}\) illustrates single-leg support with trunk rotation aiding balance. PM\({}_{4}\) captures hip and knee flexion/extension during double-leg support, while PM\({}_{5}\) highlights side-to-side movements during single-leg support. As shown in Figure 1 and described in Table 2, the analysis of movement characteristics revealed distinct patterns in the variance of principal movements (PM\({}_{\text{k}}\)) during running. PM\({}_{1}\), which involves the swing-phase movement coupled with trunk rotation, exhibited the highest variance in both PP1_rVAR (72.5 \(\pm\) 7.3%) and PA1_rVAR (49.0 \(\pm\) 4.2%), indicating that this phase significantly contributes to both postural positions and accelerations. The swing phase plays a critical role in balancing the body's momentum, contributing to high forces acting on the system. In contrast, PM\({}_{2}\), during double-leg support with antero-posterior sliding, showed lower variance, with PP2_rVAR at 12.8 \(\pm\) 6.6% and PA2_rVAR at 2.8 \(\pm\) 0.8%, suggesting that this phase contributes less to both postural positions and accelerations. PM\({}_{3}\), the single-leg support phase coupled with trunk rotation, showed values of 7.3 \(\pm\) 3.2% for PP3_rVAR and 5.3 \(\pm\) 2.1% for PA3_rVAR, suggesting lower forces and greater stability compared to the swing phase.