# AUV Trajectory Planning for Optimized Sensor Data Collection in Internet of Underwater Things
## 2 Materials and Methods
### System Model
#### 2.1.3 Communication Region Model  
Let the coordinates of the CHs be denoted as \(\big{(}x_{CH_{i}},\ y_{CH_{i}},\ z_{CH_{i}}\big{)}\), for \(i=1,2,\ldots,M\). These CHs are stationary; however, their depth values \(z_{CH_{i}}\) vary due to deployment over an uneven seafloor. Each CH communicates using acoustic signals and can connect with the AUV within a communication range \(R_{i}>0\)[11]. Within this communication region, data transmission between the AUV and the corresponding CH takes place.  
Although transmission time may vary due to factors such as distance, data size, and channel quality, this study simplifies the model by assuming a fixed transmission duration for all CHs [6; 7]. Let \(\tau>0\) denote the minimum time the AUV must remain within the communication range of any CH \(i\) to successfully complete data transmission. A successful connection with CH \(i\) is said to be established at time \(t_{i}\) if, during the interval \([t_{i}-\tau,\ t_{i}]\), the AUV remains within the communication region \(R_{i}\). This means that the distance \(d_{i}(t)\) between the AUV and CH \(i\) must satisfy \(d_{i}(t)\leq R_{i}\) for all \(t\) in that period.  
Given that the maximum surge speed of the AUV is \(u_{\max}\), it cannot move more than \(\tau u_{\max}\) meters in \(\tau\) seconds. Therefore, if the AUV is within a distance of \(R_{i}-\tau u_{\max}\) from CH \(i\) at time \(t_{i}\), it is guaranteed to remain within the full communication sphere of radius \(R_{i}\) during the interval \([t_{i}-\tau,t_{i}]\). The corresponding horizontal constraint on the AUV's position, projected onto the plane \(z(t)=z_{d}\), is given by:  
\[\rho_{i}=\sqrt{\left(R_{i}-\tau u_{\max}\right)^{2}-\big{(}z_{d}-z_{CH_{i}} \big{)}^{2}}. \tag{5}\]  
If the AUV lies within a circle of radius \(\rho_{i}\), centered at \(\big{(}x_{CH_{i}},y_{CH_{i}},z_{d}\big{)}\) at time \(t_{i}\), its 3D distance to node \(i\) is less than \(R_{i}-\tau u_{\max}\). This ensures that the AUV remains within the communication range for the entire duration \(\tau\), enabling a successful data transmission (refer to Figure 2). This condition holds under the assumption:  
\[R_{i}>\big{|}z_{d}-z_{CH_{i}}\big{|}+\tau u_{\max}\quad\forall i, \tag{6}\]  
which guarantees that the square root is real, and the projected region is nonempty.  
It is important to note that Equation (5) shows that as the vertical distance between the CH's altitude \(z_{CH_{i}}\) and the AUV's altitude \(z_{d}\) decreases, the radius \(\rho_{i}\) of the horizontal communication circle increases. This increase in \(\rho_{i}\) allows the AUV to maintain a greater horizontal distance from the CH while still successfully completing the data transmission. In the context of an uneven seafloor, this is beneficial because a larger \(\rho_{i}\) reduces the need for the AUV to approach the uneven terrain of the seafloor where the CH is located, hence lowering the risk of collision. Thus, in addition to satisfying constraint (2), Equation (5) implicitly contributes to collision avoidance by enabling communication at a safer distance from CHs located on elevated regions of the seafloor.