## 5 Experimental Setup and Results Discussion
### Comparison of Algorithm  
After experimental testing, the triangle centroid algorithm was found to produce significant positioning errors. To further evaluate accuracy, we referred to a study published by Nantong University in China [22] and conducted a JAVA-based simulation for comparison. As illustrated in Figure 11, when targeting the coordinate (11, 5), the triangle centroid algorithm yielded a result of (9.73, 2.98), corresponding to a converted positioning error of 145.55 cm and an execution time of 17.49 us. In contrast, the intersection midpoint algorithm proposed in this study produced a positioning result of (11.10, 5.04), with a significantly lower error of 6.93 cm and a shorter execution time of 13.65 microseconds. Figure 12: Test of tag local at (7, 5) using the IMA method. Similarly, the inner triangle centroid algorithm from Nantong University also achieved coordinates of (11.10, 5.04) with the same 6.93 cm error but required 33.28 \(\upmu\)s to execute. To calculate the standard deviation of the TCA, ITCA, and IMA errors from Table 1, the standard deviation (\(\sigma\)) of a dataset is given by the formula:  
\[\sigma=\sqrt{\frac{1}{n}\overline{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}}\,, \tag{13}\]  
where \(x_{i}\) represents each individual data point (i.e., localization error), \(\upmu\) is the average error of these data points, and n is the total number of data points. Based on Equation (13) and the error values provided in Table 1, the calculated standard deviations for the TCA, ITCA, and IMA are 72.91 cm, 3.76 cm, and 5.36 cm, respectively. These results demonstrate that the algorithm proposed in this study offers high positioning accuracy with a simpler computation process, effectively reducing execution time. The experimental results of TCA, ITCA, and IMA positioning are shown in Table 1 below. The comparison error summarized in Table 1 is the distance between the coordinates calculated by the algorithm and the actual position of the target. A total of ten points in the coordinates were taken out for ranging experiments. The test values were simulated on the JAVA program to simulate the positioning accuracy, and the algorithm execution time was calculated. Finally, the average error and standard deviation were calculated. The optimization-based method serves as a benchmark to evaluate the trade-off between computational cost and localization accuracy. In resource-constrained systems, heuristic models such as ITCA and IMA offer a favorable balance between accuracy and computing efficiency. As demonstrated in Table 1, although ITCA achieves the highest accuracy under average error of 11.85 cm, IMA achieves a close accuracy (12.87 cm) with the shortest execution time (13.65 \(\upmu\)s), thus offering superior performance for both computational complexity and algorithm execution time. Although the TCA performs well at certain coordinates, such as an error of 3.31 cm when positioning at (3, 1) and 1.92 cm at (5, 3). It fails to accurately locate the target when the distance from the origin increases, resulting in an average error of 75.36 cm. Based on the algorithm time consumption shown in Table 1, the IMA proposed in this study is notably efficient, requiring only 13.65 \(\upmu\)s to complete a positioning task. In contrast, the ITCA, reported in previous studies, is more complex and takes significantly longer, at 33.28 \(\upmu\)s, to perform the same task. These results demonstrate that the proposed IMA not only simplifies the computational process but also achieves accurate indoor positioning with lower time consumption. In conclusion, the TCA exhibits significant error variation; although it performs accurately at certain coordinates, the positioning error increases markedly with distance from the origin, leading to poor stability. In contrast, the ITCA shows tightly clustered and stable errors, resulting in highly consistent positioning outcomes.