## 2 Materials and Methods
### Identification of a Suitable Assistance Delay Time  
Figure 3 depicts ankle angle in a gait cycle with an assumed case of insufficient ankle dorsiflexion on the paralyzed side of a stroke survivor. The ankle dorsiflexion angle in the swing phase on the paralyzed side is typically smaller than that on the healthy side. However, the difference between the paralyzed and healthy sides varies between participants, possibly due to recovery status. An ideal assistance timing that ensures voluntary movement before assistance and makes up for an insufficient dorsiflexion angle is labeled in the figure. An excessively early assistance could hinder voluntary effort, while an excessively late assistance could decrease MTC or force compensatory movements. From the previous assumption, we aim to derive an equation to identify an ideal assistance timing that simultaneously ensures voluntary dorsiflexion movement and prevents compensatory movements. Ankle angle data to derive an identification equation were obtained from medical records from 2017 to 2020 in our previous studies on a vibrotactile system [30; 31; 32; 33]. The data were collected with a portable motion gait analysis system (RehaGait, HASOMED GmbH, Magdeburg, Saxony-Anhalt, Germany) [34]. In some of these studies, a vast with vibrators was used to provide biofeedback on toe-off and heel contact events to stroke survivors who can walk individually with or without canes [30; 31]. Improvements in swing-phase dorsiflexion and push-off angles were observed during and after intervention. In the other studies [32; 33], a waist belt with vibrators was used to provide cadence cueing. With implicitly increasing cadence cued to the stroke survivors, their walking speed increased without worsening of gait kinematics. Ankle angle data  
Figure 3: Assumed figure of ankle angle data for a stroke survivor with insufficient swing-phase dorsiflexion. Figure 2: (**a**) Control diagram; (**b**) system control in a gait cycle. before intervention from 16 of the stroke participants ([PERSON]'s stages V-VI) whose paralyzed sides suffer insufficient dorsiflexion in the swing phase were selected. Data from 11 of these participants were used for the multiple linear regression analysis, and the others for verification and validation. As Figure 4 indicates, the ideal assistance timing was set when the angular acceleration equaled zero (i.e., the second derivative of angle data equaled zero) after toe-off, and the assistance delay \(y\) was the time from toe-off to the assistance timing activation. With the ankle angle data, multiple regression was used to test whether the explanatory variables in Figure 4 (\(x_{1},x_{2},x_{3},x_{4}\)) could significantly predict the suitable assistant delay time \(y\) in milliseconds. The regression equation for assistance delay is as follows:  
\[y=Ax_{1}+Bx_{2}+Cx_{3}+Dx_{4}+E \tag{1}\]  
The results of the multiple linear regression are shown in Table 1. According to the results, gait cycle ratios of duration from toe-off to the maximum dorsiflexion angle at the healthy side (\(x_{4}\)) did not significantly predict the delay time for activating dorsiflexion assistance after toe-off (\(y\)). The multiple linear regression analysis was repeated using only the remaining three explanatory variables, and the results are shown in Table 2. The identification equation for delay time for optimal dorsiflexion assistance was derived as follows:  
\[y=-0.089x_{1}+58.87x_{2}+711.811x_{3}+42.031\left(p<0.05,\ R^{2}=0.74\right) \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Explanatory Variable** & **Coefficient** &
\begin{tabular}{c} **Significance** \\ **Judgment 1** \\ \end{tabular} & **VIF 2** \\ \hline \(x_{1}\) & \(-0.0925\) & \(*\) & 1.087 \\ \(x_{2}\) & \(57.407\) & \(*\) & 1.062 \\ \(x_{3}\) & \(874.31\) & \(*\) & 2.728 \\ \(x_{4}\) & \(-173.38\) & & 2.755 \\ constant & \(56.643\) & - & - \\ \hline \hline \end{tabular}  
* \(p<0.05\); 2 VIF, variance inflation factor.  
\end{table}