## 4 Discussion
The gait tube approach extends these traditional methods by offering a continuous, temporally resolved representation of COM variability across the gait cycle. Unlike eventbased or stride-averaged methods such as MOS, Lyapunov exponents [16; 24; 34], or Floquet analysis [18; 30], gait tubes allow for detailed visualization and interpretation of when and where variability increases. The multidimensional gait tubes in Figure 1 demonstrate these patterns visually: powered conditions resulted in wider, more dispersed tubes during stance transitions, while the NoExo condition exhibited narrow, stable trajectories throughout. Group-level comparisons showed that HighLater, which involved high-magnitude torque with a late end timing, produced the largest mean ellipsoid volume (99,937 mm\({}^{3}\)/s\({}^{3}\); z = 2.3), confirming this condition had the greatest destabilizing effect (Figure 4A; Table 1). This finding supports our hypothesis regarding the destabilizing role of prolonged assistance but also shows that the most sensitive period was earlier in the gait cycle than initially predicted. In direction-specific analyses, AP variability remained stable in most conditions, with only HighLater showing a significant increase (\(p\) = 0.0371; Figure 4C; Table 1). This supports the understanding that forward motion in walking is inherently momentum-driven and robust against moderate perturbations [3]. In contrast, ML variability responded more noticeably to changes in condition. Surprisingly, the PowerOff condition (i.e., passive exoskeleton [35; 36; 37]) led to a significant reduction in ML variability (\(p\) = 0.0488; Figure 4B), suggesting that passive structure alone may restrict lateral sway and contribute to stability. These findings offer design insights, indicating that passive and active features influence dynamic balance [35; 36; 37; 38; 39]. To explore how ellipsoid volume reflects overall variability, we compared it with total variability (sum of ML, AP, and VT directions). We found consistently strong correlations across powered conditions (r \(\geq\) 0.65), with the strongest relationship seen in HighLatest (r = 0.91, \(p\) = 0.0002; Figure 5C). This suggests that ellipsoid volume is a robust surrogate for multidimensional variability and supports its use in future studies or clinical applications. The z-score comparison between ellipsoid volume and total variability further supports their alignment as complementary metrics (Figure 5B and Table 1). In most conditions, both scores shifted in the same direction relative to NoExo, confirming that multidimensional variability captured by the gait tube method reflects the magnitude of direction-specific fluctuations in COM velocity. The comparison of z-scores for ellipsoid volume and total variability (Table 1; Figure 5B) revealed consistent directional trends across most powered conditions, supporting their alignment as complementary metrics. While ellipsoid volume captures the multidimensional structure of COM fluctuations, total variability provides a unidimensional aggregation. The similarity in z-score patterns across both metrics strengthens the validity of gait tube analysis as a summary method for quantifying stability under varying assistance profiles. The inclusion of metabolic cost in Table 1 reveals intriguing relationships with gait variability metrics, shedding light on the interplay between energy expenditure and stability under hip exoskeleton assistance. Notably, conditions with high-magnitude torque, such as HighLater and HighLatest, which exhibited the most enormous mean ellipsoid volumes (99,937 mm\({}^{3}\)/s\({}^{3}\) and 92,314 mm\({}^{3}\)/s\({}^{3}\), respectively) and significant VT (\(p\) \(\leq\) 0.0059), were associated with reduced metabolic cost compared to the NoExo condition, consistent with prior findings on exoskeleton efficiency [4; 12].