# A Low-Cost IoT-Based Bidirectional Torque Measurement System with Strain Gauge Technology
## 4 Results
A fifth-order polynomial regression was applied to the data, and the resulting fit is shown as a dotted line in Figure 4. The equation of the polynomial and the \(\mathrm{R^{2}}\) value are displayed on the graph. The high \(\mathrm{R^{2}}\) value confirms the suitability of the polynomial model for accurate calibration across the full range of applied torque. The mean absolute error (MAE) is the standard way to express the average magnitude of errors in a set of predictions, without considering their direction, as calculated by Equation (4) [47]:  
\[\mathrm{MAE}=\frac{1}{n}\sum_{i=1}^{n}\left|T_{i}-\hat{T}_{i}\right| \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline
**Applied Torque** & **Averaged Digital Value** & **Averaged Digital Value** \\
**[N\(\cdot\)m]** & **(Clockwise)** & **(Counterclockwise)** \\ \hline
5 & 416,014.40 & 526,014.00 \\
10 & 379,772.40 & 607,232.00 \\
15 & 342,533.00 & 702,498.00 \\
20 & 279,097.20 & 813,371.00 \\
25 & 238,616.20 & 941,406.00 \\
30 & 225,340.20 & 1,088,161.00 \\
35 & 195,485.80 & 1,255,192.00 \\
40 & 174,029.80 & 1,444,057.00 \\
45 & 137,405.80 & 1,656,311.00 \\
50 & 117,235.80 & 1,893,513.00 \\ \hline \end{tabular}
\end{table} 
Table 1: Correlation between applied torque and sensor averaged digital value for clockwise and counterclockwise directions. The MAE provides a single value that summarizes the average magnitude of the errors between the indicated and actual torque values, regardless of direction. Intuitively, the MAE tells us how far, on average, the measurement system's readings deviate from the true applied torque. Thus, a lower MAE means that the indicated torque values are, on average, closer to the actual torque, directly reflecting higher measurement accuracy. Table 2 presents the absolute error for each calibration point across the tested torque range. The mean absolute error (MAE) calculated over all measurements was 0.559 N\(\cdot\)m. This value represents the average deviation between the sensor output and the reference torque, providing a quantitative measure of the system's overall accuracy. Torque was applied in both clockwise (positive values) and counterclockwise (negative values) directions to evaluate the sensor's bidirectional accuracy. Reporting MAE for both directions allows for the assessment of any asymmetry or hysteresis in the sensor's response, which is a common characteristic in strain-gauge-based torque transducers. The observed MAE of 0.559 N\(\cdot\)m reflects average accuracy, while the maximum precision for 95% probability reaches \(\pm\)2.5 N\(\cdot\)m across the tested range. This performance is consistent with the requirements for laboratory and industrial torque measurements using strain-gauge-based sensors. The detailed reporting of absolute errors at each calibration point, alongside the overall MAE, provides transparency and allows for the identification of any systematic trends or direction-dependent effects. Repeatability was assessed by calculating the standard deviation (\(\sigma\)) of repeated measurements at each torque level, as defined in Equation (5) [48]:  
\[\sigma=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\bigl{(}T_{i}-\overline{T}\bigr{)}^{2}} \tag{5}\]  
where \(x_{i}\) represents repeated measurements and \(\overline{x}\) is the mean of the repeated measurements. This metric quantifies the precision of the system by indicating how closely grouped  
Figure 4: Calibration curve showing applied torque vs. indicated digital value using fifthorder polynomial regression. the measurements are for the same applied torque. However, it does not reflect closeness to the true value (accuracy), which is instead evaluated through the MAE. Table 3 presents the statistical analysis of the calibration data for the torque measurement system. For each applied torque value, the observed average torque was calculated from multiple repeated measurements.