# Location-Based Lattice Mobility Model for Wireless Sensor Networks
## 3 Problem Statement and Lattice Mobility Model Formulation
In the case of the rectangular Bravais lattice, the new location of the mobile sensor node is depicted in Figure 3a and obtained by multiplying both sides by \(\exp\left(2\pi ciC\times L/M\right)\) and adding over \(L\):  
\[\psi(L,\ T_{n})-T_{n}\
abla(C)\psi(C,T_{n})=1 \tag{12}\]  
where \(C\): Lattice points on the cell corners; \(M\): The number of dimensions  
\[\text{with}\ \psi(L,\ T_{n})=\sum_{L}\exp[2\pi iC\times\text{L}]\psi(L,T_{n}) \tag{13}\]  
Figure 3: _Cont._  
In the rectangular lattice, the node moves to the new location and creates a \(90^{\lx@math@degree}\) angle from the distance of lattice node, but does not meet the \(x\)-axis and \(y\)-axis coordinate points. The oblique is another type of Bravavais lattice that helps determine the location of mobile sensor node when it creates a trajectory smaller than \(90^{\lx@math@degree}\). Accordingly, the new location of the moving sensor node is obtained slightly differently. Figure 3: (**a**) Discovery of the new location of a moving sensor node using the rectangular Bravais lattice. (**b**) Discovery of the new location of a moving sensor node using the obliqueBravais lattice. (**c**) Discovery of new location of moving sensor node using hexagonalBravais lattice. (**d**) Discovery of the new location of a moving sensor node using the rhombicBravais lattice. (**e**) Discovery of the new location of a moving sensor node using the squareBravais lattice. Its curve is depicted in Figure 3b and obtained by:  
\[\psi(L,\ T_{n})=\sum\limits_{L}T(L)\exp[(\frac{2\pi iC\times L}{M})] \tag{14}\]  
The hexagonal Bravais lattice is used to determine the location of a mobile sensor node when the trajectory at the new location is exactly \(120^{\lx@math@degree}\). Thus, the new location of a moving sensor node is obtained by applying the Fourier inverse of \(\psi(C,\ T_{n})\). The location of a mobile sensor node is depicted in Figure 3c. \[\psi(L,\ T_{n})=\frac{1}{L^{2}}\sum\limits_{C}\frac{\exp\left(-2\pi iC\times \frac{L}{M}\right)}{1-T_{n}\
abla(\theta)} \tag{15}\]  
The rhombic is the type of Bravais lattice that can be used to find the location of a mobile sensor node when its trajectory is between \(30^{\lx@math@degree}\) and \(60^{\lx@math@degree}\). Hence, the new location of mobile sensor node is obtained by using the following integrands, and the trajectory of the mobile sensor node is depicted in Figure 3d:  
\[\psi(L,T_{n})=\frac{1}{(2\pi)^{2}}\iint\limits_{\pi}^{-\pi}\frac{\exp\left(-i \theta\times L\right)}{1-T_{n}\
abla(\theta)}\ d\theta \tag{16}\]  
The last type of two dimensional Bravais lattices is the square, which can be used to locate a mobile sensor node when its trajectory is larger than \(90^{\lx@math@degree}\) and it satisfies the \(x\)-axis and \(y\)-axis coordinate points. Therefore, the trajectory of the mobile sensor node is depicted in Figure 3e and its location is obtained as:  
\[\psi(L,T_{n})=\frac{1}{(2\pi)^{2}}\iint\limits_{\pi}^{-\pi}\left[\
abla(\theta )\right]^{K}\exp\left(-i\theta\times L\right)d\theta \tag{17}\]  
The transition probability '\(T(L)\)' is used to determine the distance from the  
Thus, the total energy '\(E\)' consumed in updating the locations \(L_{u}\) of the sink nodes and the cost for each complete cycle (Even-monitoring process) \(C_{c}\) used in [23] can be given by:  
\[L_{ue}=\sum_{i=1}^{N}e(K_{n})\times C_{m} \tag{26}\]  
Equation (26) denotes the energy consumed for the location update:  
\[E_{s}=\frac{n\times(\Delta C_{p}\times R_{e})+n\times(A_{e}\times C_{m})}{2M_{ e}}+r^{2}\;(K_{n}-1) \tag{27}\]  
Equation (27) shows the consumed energy for synchronization:  
\[\text{with}\ \psi(L,\ T_{n})=\sum_{L}\exp[2\pi iC\times\text{L}]\psi(L,T_{n}) \tag{13}\]0  
Equation (28) shows the consumed energy for receiving and forwarding the data to the next node:  
\[