## 3 System Model
### Trajectories of AUV Mobility
#### 3.2.3 Elliptical Trajectory  
The path of the AUV plays a vital role in efficient data gathering. A number of trajectories, like linear trajectory, circular trajectory, square-shaped trajectory and random way-points trajectory, exist in the literature. However, these trajectories have three major drawbacks: (i) high end-to-end delay; (ii) low packet delivery ratio; and (iii) high energy consumption. We, therefore, focus on the sub-optimized elliptical trajectory of the AUV, rather than the actual setting of the nodes for the network lifetime  
Figure 4: Trajectory of the AUV along the HC. prolongation, throughput maximization and end-to-end delay minimization. For example, in the case of the MST, there is no uniform movement of the AUV, which results in data loss (the packet dropped ratio increase). Nodes lying in the corner of the network area are at farther distances from the MST and, thus, need surplus energy to transmit data packets to the GNs. This type of movement of the AUV on the MST leads to non-uniform consumption of energy. Therefore, we convert the random trajectory (_i.e._, the actual setting of the nodes or the MST) into an elliptical trajectory to minimize the non-uniform consumption of energy. Moreover, the pattern of the GNs is also disturbed due to the random path followed by the AUV on the MST. The major and minor axis of the ellipse are _2a_ and _2b_, respectively, as shown in Figure 5. The major and minor axis are important factors to calculate the area of an ellipse. In other words, we can say that if the optimal area of an ellipse is calculated, then the network throughput is increased. If the area of the first quadrant is determined, then it is easier to compute the area of the remaining three quadrants. Therefore, we focus on finding the area of ellipse (\(A_{ellipse}\)) by using the general equation as follows:  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1 \tag{8}\]  
The detailed steps to compute the area of the first quadrant of ellipse (\(A_{FQ-ellipse}\)) is given in Appendix B. After further simplification, we have calculated:  
\[A_{FQ-ellipse}=\frac{ab\pi}{4} \tag{9}\]  
Let \(b=r\); then, we can write:  
\[b=\frac{\sum_{\forall\ e(i,j)\in EMST}d\left(i,j\right)+\alpha}{\pi} \tag{10}\]  
The relation between the semi-major and semi-minor axes in terms of \(\beta\) is given. The sub-optimized value of \(\beta\) is calculated in the next section. \[a=b+b\beta\hskip 36.135 pt0<\beta,<1,\hskip 5.690551 pta>b \tag{11}\]  
Figure 5: Elliptical trajectory of the AUV. Figure 6 shows that we have simulated different trajectories of the AUV by varying the value of \(\beta\). When the value of \(\beta\) is equal to zero, the trajectory of the AUV is circular. In the circular trajectory, the distance between end nodes and the AUV increases, and hence, the number of transmission(s) and reception(s) also increases. This leads to increased energy consumption in transmitting and receiving the sensed data. As we increase the value of \(\beta\) from zero, the distance between end nodes and the AUV decreases, which leads to a decreased number of hops. A reduced number of hops results in low transmissions and receptions, which leads to a decrease in the energy consumption. Therefore, the maximum number of nodes are alive for a longer duration by increasing the value of \(\beta\), as shown in Figure 7a,b. Simulation results with varying values of \(\beta\) depict that the majority of the nodes are alive for a longer duration when the value of \(\beta\) is equal to one. Hence, the best results for \(\beta=1\) prove the consideration to prefer the elliptical trajectory. If the nodes are located on a straight line, then going on the straight line along the location of nodes seems to be more logical. However, as per our considerations, the network area is a 3D cube in which nodes are randomly deployed. If the AUV moves as per the actual settings of nodes, then random movements at surplus positions would lead to very high end-to-end delay.