# AUV Trajectory Planning for Optimized Sensor Data Collection in Internet of Underwater Things
## 2 Materials and Methods
### Proposed Solution
#### 2.3.2 Path Design  
After determining the optimal sequence of waypoints that maximizes the residual VoI from all \(M\) CHs, the next step is to develop a trajectory design method based on the AUV's kinematic model. In this section, we design the path of the AUV to navigate through the selected waypoints, starting from the initial USV position \(S_{\text{s}}\) and ending at the final USV position \(S_{f}\) and determine the control input values required for its motion. Knowing these values enables the calculation of the travel time between waypoints, which is essential for implementing the forward DP algorithm. Let the optimal waypoint set be denoted as \(\mathcal{P}_{\text{opt}}=\left\{q_{l_{\text{s}\text{opt}}(1)}^{\text{opt}}, \ldots,\,q_{l_{\text{s}\text{opt}}(M)}^{\text{opt}}\right\}\in\mathcal{Q}_{ \Phi_{\text{opt}}}\), where \(q_{l_{\text{s}\text{opt}}(i)}^{\text{opt}}\in\mathcal{Q}_{\phi\text{s}\text{ opt}(i)}\) represents the selected waypoint from the \(i\)-th CH in the optimal permutation \(\Phi_{\text{opt}}=\left\{s^{\text{opt}}(1),\ldots,s^{\text{opt}}(M)\right\}\), and each waypoint is defined as \(q_{l_{\text{s}\text{opt}}(i)}^{\text{opt}}=\left(x_{q_{l_{\text{s}\text{opt} }(i)}^{\text{opt}}}^{\text{opt}},y_{q_{l_{\text{s}\text{opt}}(i)}^{\text{opt }}}^{\text{opt}},z_{d}\right)\). Accordingly, the AUV starts from the surface location \(S_{\text{s}}=\left(x_{q_{l_{\text{s}\text{opt}}(1)}^{\text{opt}}}^{\text{opt }},y_{q_{l_{\text{s}\text{opt}}(1)}^{\text{opt}}}^{\text{opt}},z_{d}\right)\), descends vertically to the first waypoint \(q_{l_{\text{s}\text{opt}}(1)}^{\text{opt}}\), and then follows a horizontal trajectory through all waypoints in \(\mathcal{P}_{\text{opt}}\). Once the final waypoint \(q_{l_{\text{s}\text{opt}}(M)}^{\text{opt}}\) is reached, the AUV ascends vertically to the surface location \(S_{f}=\left(x_{q_{l_{\text{s}\text{opt}}(M)}^{\text{opt}}}^{\text{opt}},y_{q_ {l_{\text{s}\text{opt}}(M)}^{\text{opt}}}^{\text{opt}},z_{d}\right)\). For simplicity, we define a new set \(P_{final}=\left\{p_{1},\ldots,p_{F}\right\}\), representing the complete sequence of points along the AUV's path, where the indices are labeled by \(\gamma=1,\ldots,F\) and \(F=M+2\). This set includes the starting point \(S_{s}\), all waypoints in \(\mathcal{P}_{\text{opt}}\), and the final point \(S_{f}\). Since we are navigating between discrete waypoints, control inputs are determined for each subtrajectory connecting two consecutive points. Specifically, to move from point \(p_{\gamma}\) to point \(p_{\gamma+1}\), the control inputs \(u_{\gamma}\), \(\omega_{\gamma}\), and \(r_{\gamma}\) are determined, enabling calculation of the corresponding travel time and transition dynamics required for the forward DP algorithm. The AUV follows a trajectory composed of three phases: a vertical descent from \(p_{1}=S_{\text{s}}\) to \(p_{2}\); horizontal movement through the waypoints \(p_{2},\ldots,p_{F-1}\) along the horizontal plane at depth \(z_{d}\), and a vertical ascent from \(p_{F-1}\) to \(p_{F}=S_{f}\). During the vertical phases, the control inputs \(u_{\gamma}\) and \(r_{\gamma}\) are set to zero, while \(\omega_{\gamma}\) is set to \(-\omega_{\text{max}}\) for descent and \(\omega_{\text{max}}\) for ascent. For the horizontal movement, the AUV follows a Dubins-like trajectory design adapted from [24], combining a single arc turn with a straight segment to meet its kinematic constraints. The AUV rotates from its current waypoint \(p_{\gamma}\) at a constant yaw rate \(r_{\gamma}\), selecting either \(r_{\text{max}}\) or \(-r_{\text{max}}\) for \(\gamma=2,\ldots,F-1\). Rotation continues until the heading \(\psi_{\gamma}\) aligns with the vector leading to the subsequent waypoint \(p_{\gamma+1}\), reaching the tangent point \(\xi_{\gamma}\) (see Figure 3). At \(\xi_{\gamma}\), the yaw rate \(r_{\gamma}\) is set to zero, and the AUV proceeds straight toward \(p_{\gamma+1}\)[24].