# An Overview of Indoor Localization System for Human Activity Recognition (HAR) in Healthcare
## 4 Localization Methods
### Trilateration or True-Range Multilateration  
Lateration, also called range measurement, computes the position of an object by measuring its distance from multiple reference positions. It is a technique that calculates the physical position of the target node by knowing the positions of the three fixed non-collinear referent nodes in the 2D space. Finding a location in two dimensions requires distance measurements from three non-collinear points. In three dimensions, distance measurements from four non-coplanar points are required. Using the geometry of the circles, it is possible to determine the positioning of the moving node [31]. Circles are drawn with the coordinates of the reference node as their center and radii equal to the estimated distance, for which, ideally, the target node is located at the point of intersection of the three circles (Figure 8). When more reference points are used than necessary, then we speak of multilateration or true-range multilateration. The circumference can be described through the following equation:  
\[\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}=r_{i}^{2}, \tag{8}\]  
where \(\left(x_{i},y_{i}\right)\) are land coordinates of the various centers, and \(r_{i}\) is the radius of the \(i\)-th circle. Accordingly, the coordinates of the target node can be obtained by solving the following system:  
\[\begin{array}{l}\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=r_{1}^{2} \\ \left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}=r_{2}^{2}\\ \left(x-x_{3}\right)^{2}+\left(y-y_{3}\right)^{2}=r_{3}^{2}\end{array}. \tag{9}\]  
In practice, since the distance estimated is never perfect, the coordinates of the target node are rarely found at the point of intersection of the three circles; hence, it is likely that it is located within the area of intersection between the circles or even that the circles do not touch. To overcome this drawback, the AML (adapted multilateration) method developed by [PERSON] et al. can be adopted [32] based on an iterative process applied to the three beacon nodes. Two circles are drawn around two randomly chosen beacon nodes. If the intersection point is unique, the coordinates of this point are acquired. If the circles do not touch, the spokes are increased proportionally to touch each other in one place. If, on the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Technique** & **Accuracy** & **Cost** & **Advantages** & **Disadvantages** \\ \hline TOA & High & High & Scalability, does not require any fingerprint & Needs time synchronization, difficult to implement, produces multipath effects \\ \hline TOP & High & High & No need time synchronization among devices and received nodes, does not require any fingerprint & Requires time synchronization between the received nodes, difficult to implement in narrow bandwidth, multipath effects \\ \hline RTT & High & High & Does not require time synchronization, low complexity & Affected by multipath effects and noise, different processing time delays \\ \hline AOA & Medium & High & No need for any fingerprint, no need for time synchronization, low number of APs & Requires additional directional antennas, decreases in accuracy as distance from source increases \\ \hline RSS & Low & Medium & No need for synchronization, can be used with different technologies, easy to implement & Suffers from multipath effect, noise, can require fingerprint \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of signal measurement techniques.  
other hand, the two circles touch each other at two points, the one that has a distance from the third beacon closest to the estimate of the distance calculated for this node is chosen. At the end of the iterative process, the average of the coordinates that will determine the position of the target node is calculated.