# Smooth 3D Dubins Curves Based Mobile Data Gathering in Sparse Underwater Sensor Networks
## 2 Preliminaries and Problem Statement  
As depicted in Figure 1, there are \(n\) sensor nodes \((S_{i},i=1,2,\cdots,n)\) with coordinates \((x_{i},y_{i},z_{i})\in\mathbb{R}^{3}\) and one mobile AUV deployed randomly in a \(W\times W\times W\) cubic underwater space. Each underwater sensor node has the ability of monitoring parameters of the surroundings and transmitting data via acoustic communication. Compared with the vast monitoring region of USNs, the drifting movement of underwater sensors by oceanic current can be ignored. Additionally, it is assumed that the battery energy of mobile AUVs is enough to support the complete traversal of a scheduled trajectory. As USNs might be deployed sparsely, sensory data gathering in the manner of traditional multi-hop transmission between underwater sensors is not applicable. The investigated USN model considers the larger-range 3D topological property but ignores such factors as aquatic medium and acoustic communication. The nonholonomic motion constraint requires that AUVs move along a continuous and smooth path to satisfy kinematic constraints. As shown in Figure 1, the kinematics of AUVs with spatial position \([x_{A},y_{A},z_{A},\psi_{A}]\) are defined as  
\[\left\{ \begin{array}{l}x_{A}(\tau+1)=x_{A}(\tau)+\rho\times(\vec{g}cos\psi_{A }-\vec{v}sin\psi_{A}),\\ y_{A}(\tau+1)=y_{A}(\tau)+\rho\times(\vec{g}sin\psi_{A}+\vec{v}cos\psi_{A}),\\ z_{A}(\tau+1)=z_{A}(\tau)+\rho\times\vec{\omega},\\ \psi_{A}(\tau+1)=\psi_{A}(\tau)+\rho\times\vec{\gamma},\end{array} \right. \tag{1}\]  
where \(x_{A}\), \(y_{A}\), and \(z_{A}\) are the positions in the north, east, and downward axes, respectively; \(\psi_{A}\) denotes the rotation angle around the downward axes; and \(\rho\) is a scalar governing the trajectory updating of AUVs, that is, the time interval for updating the kinematics model. Additionally, \(\vec{g}\), \(\vec{v}\), and \(\vec{\omega}\) are the linear velocities in surge, sway, and heave and \(\vec{p}\), \(\vec{q}\), and \(\vec{\gamma}\) are the angle velocities in roll, pitch and yaw, respectively [21]. The smoothness of curves and junctions can be measured by geometric and parametric continuity. First of all, we propose the definition of \(G^{1}\) continuous curves:  
\(G^{1}\)_continuous curves_: Let \(P(s_{0},s_{1}:s)\) and \(Q(t_{0},t_{1}:t)\) be two parametric curves \(P(s_{1})=Q(t_{0})=J\) and \(\dot{P}(s)|_{s_{1}}=\dot{Q}(t)|_{t_{0}}\); then the two parametric curves meet at joint \(J\) with \(G^{1}\) continuity. If \(\dot{P}(s)|_{s}\) and \(\dot{Q}(t)|_{t}\) are both continuous, the connecting curves by \(P(s)\) and \(Q(t)\) belong to \(G^{1}\) continuous curves. Figure 1: Network topology and dynamics of autonomous underwater vehicles (AUVs) (\(n=10\) sensor nodes and one AUV deployed randomly in a three-dimensional underwater region). With this, we can define and design smooth curves in 3D space. The investigated 3D smooth curves are defined as follows:  
_3D smooth curves_: Let \(P(s)=[x(s),y(s),z(s)]\) and \(Q(t)=[x(t),y(t),z(t)]\) be two regular parametric 3D curves in a 3D region, where \(s\in[0,1]\) and \(t\in[0,1]\) are the parameters with 0 and 1 denoting the starting and ending points, respectively. Moreover, \(P(s)\) and \(Q(t)\) are \(G^{1}\) continuous curves in both the \(X\)-\(Y\) plane and \(Z\)-axis. If \(P(1)=Q(0)=J\) and \(\hat{P}(s)|_{s=1}=\hat{Q}(t)|_{t=0}\), then the two parametric curves meet at joint \(J\) with \(3D-G^{1}\) continuity; that is, these curves belong to 3D smooth curves. In other words, 3D smooth (\(3D-G^{1}\) continuous) curves refer to such curves satisfying \(G^{1}\) continuity in the \(X\)-, \(Y\)-, and \(Z\)-axes. In this paper, the sensory data gathering process is involved in the cruising process of AUVs; that is to say, the AUV does not have to stop at a certain collection position to receive data. From this perspective, 3D path planning for MDG aims to determine a smooth path from an initial node to a final node through a complicated 3D underwater space.