# Smooth 3D Dubins Curves Based Mobile Data Gathering in Sparse Underwater Sensor Networks
## 3 Smooth 3D Dubins Curves Planning-Based Mobile Data Gathering Algorithm
### 3D Dubins Curves with Bezier Interpolation  
Dubins curves of the two cases in Figure 2 have been proven to realize smooth paths in 2D planes [17]. The Dubins curves satisfy the motion constraints by a combination of maximum curvature arcs (\(R\) or \(L\)) or a straight line segment (\(S\)). The presence of kinematic constraints of AUVs implies that the distance between pairs of underwater nodes depends on the incoming and outgoing directions of sensor nodes. The incoming and outgoing heading angles are selected on the basis of the optimal type of Dubins curve, which is one of \(\{RSL,RSR,LSL,LSR,RLR,LRL\}\). Our former works [22; 23] extend 2D Dubins curves to 3D Dubins curves from sensor node \(S_{i}(x_{i},y_{i},z_{i},\Phi_{i})\) to sensor node \(S_{j}(x_{j},y_{j},z_{j},\Phi_{j})\) (as in Figure 3) by assigning each \((x,y,\phi)\in\mathbb{R}^{2}\) and \((x,y,z,\Phi)\in\mathbb{R}^{3}\) with  
\[z=z_{i}+\frac{\mathcal{L}_{(x_{i},y_{i})}^{(x,y)}}{\mathcal{L}_{(x_{i},y_{i}) }^{(x_{i},y_{i})}}\times(z_{j}-z_{i}), \tag{6}\]  
where \(\mathcal{L}_{(x_{i},y_{i})}^{(x,y)}\) and \(\mathcal{L}_{(x_{i},y_{i})}^{(x_{i},y_{i})}\) denote lengths of 2D Dubins curves from \((x_{i},y_{i},\phi_{i})\) to \((x,y,\phi)\) and \((x_{j},y_{j},\phi_{j})\), respectively.  
Thus, the length of a 3D Dubins curve with linear interpolation from underwater sensor \(S_{i}(x_{i},y_{i},z_{i},\Phi_{i})\) to \(S_{j}(x_{i},y_{j},z_{j},\Phi_{j})\) is demonstrated as  
Figure 3: Principle of three-dimensional (3D) Dubins curves.  
Figure 2: Two cases of Dubins curves.  
\[\mathcal{L}_{Linear-3D}=\sqrt{(\mathcal{L}_{(x_{i},y_{i})}^{(x_{i},y_{i})})^{2}+(z_ {j}-z_{i})^{2}}. \tag{7}\]  
However, the above 3D linear interpolation fails to maintain the \(G^{1}\) continuity in the \(Z\)-axis. Therefore, we propose Bezier interpolation based smooth 3D Dubins path planning for MDG in USNs. In the upper part of Figure 4, we use a traditional Dubins curve to design the path in the \(X\)-\(Y\) plane. In the lower part of Figure 4, we extend the 2D Dubins curve to a 3D Dubins curve from sensor node \(S_{i}(x_{i},y_{i},z_{i},\Phi_{i})\) to \(S_{j}(x_{j},y_{j},z_{j},\Phi_{j})\) by assigning \(z\) with a quadratic Bezier curve \(C_{i}(u)\)[24]:  
\[z=(1-u)^{2}z_{i}+2u(1-u)M_{i}+u^{2}z_{j}, \tag{8}\]  
where \(M_{i}\) is the \(Z\)-axis coordinate value of a control point in the quadratic Bezier curve.  
The traditional Bezier curve is defined as  
\[C(u)=\sum_{t=0}^{m}B_{m,t}(u)D_{t}, \tag{9}\]  
where \(D_{t}\) denotes end points and \(B_{m,t}(u)=\frac{m}{t(m-t)}u^{t}(1-u)^{m-t}\).  
For simpler quadratic Bezier curves, we have  
\[C(u)=(1-u)^{2}D_{0}+2u(1-u)D_{1}+u^{2}D_{2}, \tag{10}\]  
where \(D_{0}\) and \(D_{2}\) denote end points and \(D_{1}\) denotes a control point. It is noted that the first-order derivative of a quadratic Bezier curve is continuous [24].  
Figure 4: Two-dimensional (2D) and three-dimensional (3D) Dubins curves.  
The approximate length of a quadratic Bezier curve can be calculated as follows [24]:  
\[\begin{split}\mathcal{L}_{z_{i}z_{j}}&\approx\frac{2}{3 }\times\left|z_{i},z_{j}\right|+\frac{1}{3}\times(\left|z_{i},M_{i}\right|+ \left|M_{i},z_{j}\right|)\\ &\geq\frac{2}{3}\times\left|z_{i},z_{j}\right|+\frac{1}{3}\times \left|z_{i},z_{j}\right|\\ &\geq\left|z_{i},z_{j}\right|.\end{split} \tag{11}\]  
It can be proved from the above equation that the Bezier curve interpolation method will lead to a slightly longer length than linear interpolation in the \(Z\)-axis. As a result, the length of a 3D Dubins curve with Bezier interpolation is  
\[\mathcal{L}_{Bezier-3D}=\sqrt{(\mathcal{L}_{(x_{i},y_{i})}^{(x_{i},y_{j})})^{ 2}+(\mathcal{L}_{z_{i}z_{j}})^{2}}. \tag{12}\]