# AUV Trajectory Planning for Optimized Sensor Data Collection in Internet of Underwater Things
## 3 Results and Discussion
These sensor nodes are clustered using the k-medoids algorithm, and a CH is selected for each cluster, as shown in Figure 4. Figure 4 depicts the communication sphere of radius \(R_{i}\) for each CH, along with the horizontal plane at depth \(z_{d}\), which defines the AUV's operating level for data collection. In Figure 4, each \(\mathcal{C}_{i}\) is shown as a dashed circle representing the communication boundary of CH \(i\) on the horizontal plane, centered at \(\left(x_{CH_{i}},y_{CH_{i}},z_{d}\right)\) with radius \(\rho_{i}\). The AUV can collect data from CH \(i\) by reaching any point along the circumference of \(\mathcal{C}_{i}\). The task is to determine the AUV's start and end points, along with a trajectory that visits exactly one point on the circumference of each \(\mathcal{C}_{i}\), ensuring complete data collection  
Figure 4: System overview showing: (**a**) sensor node distribution (black dots); (**b**) CH selection (white dots); (**c**) CH communication spheres intersecting the AUV’s horizontal operation plane; (**d**) the horizontal communication regions of each CH (dashed black circles). while maximizing the total residual VoI upon returning to the USV on the surface. With all parameters defined, the proposed forward DP algorithm is applied to identify the optimal set of waypoints. A Dubins-like trajectory is then constructed to connect these waypoints. The first step involves distributing a finite set of \(L=30\) candidate waypoints uniformly along the circumference of each \(\mathcal{C}_{i}\) (see Figure 5a). Any waypoints that fall outside the area of interest are excluded. The remaining waypoints constitute the state space for each CH, and the forward DP algorithm outlined in Algorithm 1 is employed to efficiently determine the optimal visiting sequence and associated waypoint selection, one point per \(\mathcal{C}_{i}\), denoted by \(\left(\Phi_{\text{opt}},\mathcal{P}_{\text{opt}}\right)\), as illustrated in Figure 5b. The \(x\) and \(y\) coordinates of the first point in \(\mathcal{P}_{\text{opt}}\) are assigned to the USV's starting point \(S_{s}\), with its altitude set to \(z=0\) to ensure placement on the sea surface. Likewise, the \(x\) and \(y\) coordinates of the final point in \(\mathcal{P}_{\text{opt}}\) define the USV's endpoint \(S_{f}\), with its altitude also set to \(z=0\). Figure 5c illustrates the complete AUV trajectory, starting from \(S_{s}\), marked with a blue dot, continuing through the selected waypoints in the optimal order, and ending at \(S_{f}\), marked with a red dot, following the Dubins-like trajectory design described in Section 2.3.2. Figure 5d shows the 2D view of the final trajectory of the AUV. The proposed method demonstrated effectiveness by preserving approximately 66% of the total residual VoI across all CHs. The results further indicate that high-importance data decays more rapidly than medium-importance data. Nevertheless, the method retained around 64% of the high-importance information, which is considered satisfactory (see Table 1). This level of preservation supports the timely detection of abnormal events with minimal information loss, enabling rapid intervention in hazardous situations. However,  
Figure 5: Implementation of the proposed method showing: (**a**) candidate waypoints distributed on each CH’s horizontal circle (red dots); (**b**) optimal waypoints selected by the forward DP algorithm (red dots); (**c**) final 3D trajectories of the AUV (green solid line) and the USV (magenta solid line) from the start point \(S_{s}\) (blue dot) to the end point \(S_{f}\) (yellow dot); (**d**) the final 2D trajectory of the AUV. the performance declines when the AUV operates over a larger area with widely dispersed CHs and a deeper surveillance plane, indicating that further improvements are required to maintain performance in more demanding environments.