# AUV Trajectory Planning for Optimized Sensor Data Collection in Internet of Underwater Things
## 2 Materials and Methods
### Problem Formulation
#### 2.2.1 Objective Function  
This work investigates the problem of planning an optimal trajectory for a single AUV assigned to collect data from \(M\) CHs deployed across the UASN. The AUV begins and ends its mission at two distinct surface locations, \(S_{s}\) and \(S_{f}\), both associated with a USV. The objective is to determine the optimal starting and ending locations, along with the most efficient visiting sequence of CHs and corresponding waypoints, such that the total residual VoI of the collected data is maximized. This enables the timely delivery of critical information to the USV for rapid response to abnormal events. Several studies [6; 7; 8; 9; 11] have proposed different VoI definitions for underwater data, considering importance, timeliness, and rarity. In this paper, we adopt a model similar to that in [1], which classifies data as emergency or routine and utilizes an adaptive decay function that links value loss to data significance. It also computes the initial VoI based on historical data, offering a practical and realistic approach [1]. To illustrate the adopted VoI concept, consider a typical underwater pipeline surveillance scenario. Emergency events such as leaks are often indicated by sudden pressure drops or abnormal chemical readings [6; 9]. In contrast to routine measurements, these events are rare and carry high importance, requiring urgent data transmission to prevent serious consequences. In such cases, a normal distribution can be applied to historical data to capture the rarity and significance of these events when calculating the VoI [1]. Let \(C\) denote the set of measured data, where \(C_{kj}\) represents the \(k\)-th reading obtained by the \(j\)-th sensor node in cluster \(i\). Here \(k=1,2,\ldots\), and \(j=1,2,\ldots,U\). Each sensor node \(j\) maintains historical data characterized by a mean \(\mu_{j}\) and standard deviation \(\sigma_{j}\). The abnormality of a reading is quantified using an importance score \(\mathcal{I}\left(C_{kj}\right)\), defined as [1]:  
\[\mathcal{I}\left(C_{kj}\right)=\frac{1}{\sigma_{j}\sqrt{2\pi}}\int_{-\infty}^{ \mu_{j}+|C_{kj}-\mu_{j}|}e^{\frac{-(c-\mu_{j})^{2}}{2\sigma_{j}^{2}}}\,dc, \tag{7}\]  
where \(c\) is the integral variable. Equation (7) captures the degree of deviation of a reading \(C_{kj}\) from the historical norm. The most abnormal reading from sensor node \(j\) is determined as \(\mathcal{I}\left(C_{j}\right)=\max_{k}\left\{\mathcal{I}\left(C_{kj}\right)\right\}\)[1]. The CH then computes the cluster's data importance as \(\mathcal{I}_{i}=\max_{j}\left\{\mathcal{I}\left(C_{j}\right)\right\}\) based on the data received from all sensor nodes, to identify the most important data in cluster \(i\)[1]. Figure 2: The design of the horizontal communication region for sensor node \(i\). At this point, the initial VoI corresponding to the data from cluster \(i\), denoted by \(\mathcal{V}_{int_{i}}\), is calculated using [1]:  
\[\mathcal{V}_{int_{i}}=2\;\mathcal{I}_{i}-1. \tag{8}\]  
Equation (8) highlights the relationship between data abnormality and its importance, where a higher value of \(\mathcal{V}_{int_{i}}\) indicates higher urgency of the data [1]. To reflect how urgency decreases over time, a decay-based VoI model is constructed. When the AUV collects data with initial VoI \(\mathcal{V}_{int_{i}}\) from CH \(i\) and delivers it to the USV at time \(t_{c}\), the residual VoI is computed as [1]:  
\[V_{r_{i}}(t_{c})=\mathcal{V}_{int_{i}}(1-\beta\;\mathcal{I}_{i})^{t_{c}}, \tag{9}\]  
where \(\beta\) is the decay coefficient and \(t_{c}\) is the time duration from the start of data collection to its delivery to the USV. Equation (9) reflects that data with higher abnormality, i.e., greater deviation from historical norms \(\left|C_{bj}-\mu_{j}\right|\), undergoes more rapid value decay [1]. Hence, the objective is to plan a trajectory that preserves the maximum possible residual VoI.