## 5 Proposed Model
### Fuzzy-Logic Based Movement Decision  
The first step in path planning starts with dividing the area of interest into a set of symmetric virtual hexagons. The MA is supposed to travel from the center of one hexagon to one corner of the same hexagon and the reverse procedure is applied in the next movement. Hexagonal-based movement has been proposed in a number of recent works including [21; 23; 41]. Similar to [23], the hexagons are formed based on the transmission range of \(R_{Tx}\), where the side length of each hexagon is equal to \(R_{Tx}/\sqrt{3}\ m\). Figure 2 shows an example of the virtual hexagons forms in our system. Figure 2: The hexagonal network system model of FLPPL (Fuzzy-Logic based Path Planning for mobility-assisted Localization). The movement of MA from one point (either the center or a corner) to another point will be determined in a real-time approach using an FL system. In such system, three inputs will be used namely the Received Signal Strength Indicator (RSSI) level, the number of neighbours, and the distance to each neighbour. These inputs will be measured from the MA's current location. The Mamdani method was used as the main tool to represent the fuzzy system. Mamdani is considered to be intuitive, widespread accepted and well suited to human input [27]. Table 2 shows the three input functions along with their membership functions. RSSI input measures the strength of the signal of all unknown nodes located within the range of the current location of MA. A scale of \(-100\) dBm to \(0\) dBm is used in this model, where the closer the value to \(0\), the stronger the signal is [42]. The RSSI membership function used in this model is close to the one in [43]. All UNs within the range of the MA will exchange their neighbour table information with the MA. This table includes the node id, the number of neighbours, the neighbour ids, the RSSI strength and the localization status, whether localized (1) or not (0). An example of this table is shown in the following Table 3.  
In the second input, and from its current location, the MA contacts each UN located within its range. Each UN exchanges its neighbours list and the MA evaluates them based on the number of neighbours. A higher chance will be given to the UN with higher number of neighbours. This is meant to reach the maximum number of UNs by considering the ability of each nodes to share its information to as much neighbours as possible. In the third input, although the locations of both the MA and the UN are required to calculate the distance between each of them, it is still possible to estimate using the RSSI. The closer the distance is, the higher chance is given. Considering the module in [44] and assuming \(d_{0}=1\), the distance extracted from the RSSI values is estimated as  
\[d=10^{(-RSSI+P_{L}(d_{0})+N_{a})/(10\times\beta)} \tag{3}\]  
where \(P_{L}(d_{0})\) is the power loss at the reference point \((d_{0})\) in \(dB\), \(N_{a}\) is the zero-mean Gaussian random variable with a standard deviation \(\alpha\) and \(\beta\) is a constant path loss exponent. Two membership functions were chosen to present our parameters, Triangular and Trapezoidal. They are simple, direct and easy to apply in many applications [27; 45]. Simply, triangular membership function is a collection of three points forming a triangle. It is represented using the following equation  
\[\mu_{A1}(x)= \begin{cases}0&x\leq 0\\ \dfrac{x-a1}{b1-a1}&a1\leq x\leq b1\\ \dfrac{c1-x}{c1-b1}&b1\leq x\leq c1\\ 0&c1\leq x\end{cases}  \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input** & \multicolumn{3}{c}{**Membership Function**} \\ \hline RSSI Level & Weak & Medium & Strong \\ Number of Neighbours & Low & Medium & High \\ Distance to each neighbour & Near & Medium & Far \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input functions.