## 5 Conclusions  
This study demonstrates that gait tube analysis is a powerful method for assessing phase-dependent gait stability under hip exoskeleton assistance. The results show that powered conditions significantly increase multidimensional variability, particularly during early-to-mid stance phases where torque overlaps with transitions in support. VT was the most sensitive direction to assistance, while AP variability remained unchanged. The PowerOff condition uniquely reduced ML variability, highlighting the influence of passive structural elements. Ellipsoid volume was strongly correlated with total variability across conditions, validating its use as a summary metric. To support safer and more effective exoskeleton use, future device designs should consider implementing phase-specific control strategies that reduce assistance during early-to-mid stance--when users appear most sensitive to stability disruptions--and increase assistance during late stance with caution. Adaptive control systems that monitor individual stability responses in real time could further personalize support to optimize both energy efficiency and balance. Clinically, the ability to detect phase-dependent instability offers significant potential for fall-risk assessment and rehabilitation planning in older adults or individuals with neurological impairments. Incorporating gait tube analysis into clinical gait evaluations could help identify specific instability-prone phases and guide targeted interventions. Together, these findings suggest that gait tube analysis offers a comprehensive, interpretable framework for evaluating the effects of wearable assistive devices. This approach can inform the development of phase-specific controllers and support personalized rehabilitation strategies. Future studies should focus on translating these insights to clinical populations, optimizing torque delivery, and enhancing the real-world applicability of exoskeleton technologies. These findings confirm our hypothesis that the timing of assistance plays a critical role in modulating gait stability and suggest that exoskeleton controllers should implement phase-specific torque strategies to reduce variability during early stance. Clinically, this method could be used to identify instability-prone gait phases and guide individualized rehabilitation protocols that address balance control in users with neurological or age-related gait impairments.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15137530/s1](https://www.mdpi.com/article/10.3390/app15137530/s1)]([https://www.mdpi.com/article/10.3390/app15137530/s1](https://www.mdpi.com/article/10.3390/app15137530/s1)), S1: Supplementary Material: Gait Tube Stability method via the MATLAB function (_GaitTubePlot_Supplementary_MATLAB_Code.m_); S2: Gait Tube Analysis Mathematical Model explanations (_Gait Tube Analysis Mathematical Model.doc_).  
Conceptualization A.M.G.; Methodology A.M.G. and F.F.; Software A.M.G.; Validation A.M.G.; Formal analysis A.M.G.; Investigation A.M.G. and F.F.; Resources A.M.G. and F.F.; Data curation A.M.G. and F.F.; Writing--original draft preparation A.M.G. and F.F.; Writing--review and editing A.M.G. and F.F.; Visualization A.M.G. and F.F.; Supervision A.M.G.; Project administration A.M.G.; Funding acquisition A.M.G. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Institutes of Health (P20 GM109090 and R01 HD090333), the National Science Foundation (2203143), and the U.S. Department of Veterans Affairs Rehabilitation Research and Development Service (101 RX000604 and 101 RX003266). The content is solely the authors' responsibility and does not represent the official views of the funding agencies.  
Not applicable.  
Data Availability Statement: The original contributions presented in this study are included in the article/Supplementary Materials. Further inquiries can be directed to the corresponding author.