# Proximity-Based Adaptive Indoor Positioning Method Using Received Signal Strength Indicator
## 3 Proposed Indoor Positioning Method
### Adaptive Location Estimation  
For a general state which is not in proximity to any AP, the trilateration technique [14; 15] is employed for position estimation. Conversely, the proposed algorithm adopts the MP technique for position estimation if the receiver has a large proximity to a specific AP. The trilateration technique involves estimating the position using the straight-line distances from three points. Indoor positioning using the trilateration technique converts the RSSI collected from three APs into distances and can thereby estimate the location of the receiver. Figure 5 shows the determination of a receiver location through three AP distances. Three circles in Figure 5 are formed with the distance as the radius based on the locations of AP1, AP2, and AP3 and the distance from each AP to the receiver. The receiver location is determined as a point where the three circles commonly intersect. Figure 4: RSSI correction through Kalman filter. The distance \(r_{i}\) between \(i\)th AP and the receiver is calculated as follows:  
\[r_{1}^{2} =(x-x_{1})^{2}+(y-y_{1})^{2} \tag{8}\] \[r_{2}^{2} =(x-x_{2})^{2}+(y-y_{2})^{2}\] \[r_{3}^{2} =(x-x_{3})^{2}+(y-y_{3})^{2}\]  
where \((x,y)\) represents the coordinates of the receiver, and \((x_{i},y_{i})\) (\(i=1,2,3\)) are the coordinates of \(i\)th AP. The coordinates of the receiver are calculated by rearranging Equation (8) with \(x\) and \(y\) as follows:  
\[x =\frac{(r_{2}^{2}-r_{3}^{2}-r_{2}^{2}+x_{3}^{2}-y_{2}^{2}+y_{3}^{2 })(y_{2}-y_{1})-(y_{3}-y_{2})(r_{1}^{2}-r_{2}^{2}-x_{1}^{2}+x_{2}^{2}-y_{1}^{2 }+y_{2}^{2})}{((y_{2}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{2}-x_{1}))} \tag{9}\] \[y =\frac{(r_{2}^{2}-r_{3}^{2}-x_{2}^{2}+x_{3}^{2}-y_{2}^{2}+y_{3}^{2 })(x_{2}-x_{1})-(x_{3}-x_{2})(r_{1}^{2}-r_{2}^{2}-x_{1}^{2}+x_{2}^{2}-y_{1}^{2 }+y_{2}^{2})}{((x_{2}-x_{1})(y_{3}-y_{2})-(x_{3}-x_{2})(y_{2}-y_{1}))}\]  
A previous approach for positioning [34] primarily identifies which AP is closest to the receiver to estimate the approximate region. However, this technique has proven difficult for indoor localization due to significant errors in the physical location of the actual receiver. Although this problem can be alleviated by increasing the number of AP installations to refine the region and identify the locations, it is not an efficient positioning technique because the cost increases in proportion to the number of APs. The proposed MP technique can solve the limitations of the previous proximity technique. Since the MP technique is selectively applied only in cases of extremely close proximity to a specific AP, it does not need additional AP installations in other regions. Unlike the existing proximity technique, the MP technique considers the distance to the identified nearby AP based on the received RSSI and the AP locations. Therefore, the MP technique is a more accurate positioning method compared to the conventional proximity technique. The MP technique internally varies its positioning technique depending on the identified locations of nearby APs. Virtual numbers are sequentially mapped to the installed APs, as shown in Figure 6. If the AP number is not first or last, the AP location is estimated by vertically moving the position a distance \(r\) from the nearby AP to the inner region, as shown in Figure 7. The method of position estimation through vertical shifting differs based on whether the identified AP number is odd or even. If the number identified as a nearby AP is odd, such as AP3 in Figure 7, the distance calculated by calculating the RSSI collected from the nearby  
Figure 5: Trilateration example. AP is defined as \(r\). Afterward, the coordinates obtained by adding the distance \(r\) to the \(y\)-coordinate of the identified nearby AP location coordinates are taken as the location estimation result. If the identified nearby AP number is even, the AP location is estimated by subtracting distance \(r\) from the \(y\)-coordinate of the respective AP's location coordinates.