# A Brunnstrom Stage Evaluation System with Fuzzy-Accelerometer
## 2 System Development
### Features Analysis  
Figure 3 shows that, during the Flexion action, the acceleration along the \(y\)-axis is about \(-1\) g during preparation, and the acceleration along the \(y\)-axis increases as the arm begins to rise and then decreases as the arm begins to fall. Figure 3 shows that each action has a maximum (peak) value, and this peak is used in Equation (2) to calculate the angle, which is then averaged as shown in Figure 4. As shown in the diagram of Equation (2), rather than taking the maximum or minimum angle of the user's motion, averaging the angles provides a more objective representation of the angle of user motion, because there may be discrepancies between the angle of each action, thus the average angle better matches the value of a specific level. \[\overline{y}_{ANG}=\frac{\sum_{i=1}^{n}(y_{i}+1)\times 90}{n}, \tag{2}\]  
Figure 3: Flexion motion’s \(y\)-axis’s filtered data from top to bottom as ST2, ST3, ST4, ST5, ST6, Healthy. where \(\overline{y}_{ANG}\) is the average angle, \(n\) is the total number of peaks, and \(y_{i}\) is the _ith_ peak. Figure 5a shows the results of the peaks for the ST4 acceleration data in Figure 3. Since ST4 falls between ST3 and ST5, the signal easily falls below or above the stage, thus ST4 is used as an example. Although the average angle more easily fits a specific level, the average angle disregards the peak levels for each action, and thus the precise angle of each action is unknown. It can be seen that the five peaks in Figure 5a are small and the motion is relatively stable, but the motion angle of 62-85.3\({}^{\circ}\) in Figure 5b represents a significant drop from the 71.1-79.2\({}^{\circ}\) angle in Figure 5a. The \(\overline{y}_{ANG}\) of both are 73.62\({}^{\circ}\), thus the average angle alone cannot objectively represent the user's movement, and the peak is incorporated into Equation (3) to calculate the standard deviation of the angle, where a smaller value indicates a small degree of variation. \[y_{ASD}=\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}-\overline{y}_{ANG}\right)^{2}}{ n}}, \tag{3}\]  
where \(y_{ASD}\) is the angular standard deviation, \(\overline{y}_{ANG}\) is the average angle, \(n\) is the total number of peaks, and \(y_{i}\) is the _ith_ peak. Figure 4: Schematic diagram of Equation (2). Figure 5: Flexion average angle comparison (**a**) Stability peak value and (**b**) Non stability peak value. Figure 6 shows the \(x\)-axis acceleration data for the Flexion motion. Normally this data will remain relatively static during the motion. Figure 6a shows that \(x\)-axis acceleration data was both above and below 0 g during the motion. If the arm does not bend during the motion, the data will produce a line close to zero with minimal variation, as shown in Figure 6b. Figure 6b also shows that, during Flexion, the \(x\)-axis acceleration data varies greatly with respect to Figure 6a. Since the arm is also bent during the action, the value is not as smooth as in Figure 6a. Therefore, the system calculates the standard deviation using Equation (4) from the \(x\)-axis acceleration data, and uses this parameter to assess the degree of horizontal stability with which the subject performs the Flexion action, with a smaller value indicating higher stability. \[x_{ASD}=\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}{n}}, \tag{4}\]  
where \(x_{ASD}\) is the standard deviation of the \(x\)-axis, \(\overline{x}\) is the average amplitude of the \(x\)-axis, \(n\) is the total number of data points on the \(x\)-axis, and \(x_{i}\) is the \(i\)th data. The Pronation action uses the same features as the Flexion action. Pronation stretches and rotates the hand 180 degrees, and is used to determine the degree of write joint rotation. This action is calculated using acceleration data along the \(z\)-axis.