# Smooth 3D Dubins Curves Based Mobile Data Gathering in Sparse Underwater Sensor Networks
## 4 Simulation Results  
A simulation was set up with \(n\) sensor nodes randomly distributed in a cube of \(10\times 10\times 10\) unit3, with the AUV's turning radius of \(r=1\) unit for the Dubins path. The MATLAB 2016b simulator was used as the simulation tool. Figure 8 depicts an initial deployment snapshot of our simulation scenario when \(n=19\), and Figure 9 shows the 3D interpolation results with only \(n=3\) sensor nodes. It is intuitively clear that this simple curve derived by our proposed algorithm is continuous and smooth. Figure 6: Simplified azimuth heading sets. Figure 7: Data gathering at sensor node \(S_{i}\) sphere. With the proposed interpolation method of Section 3, the desired control points and interpolation results for the Bezier curves in the \(Z\)-axis are as illustrated in Figures 10 and 11, respectively. Control points generated with Equation \(\mathcal{M}=\mathcal{A}^{-1}\mathcal{S}\) should satisfy the \(3D-G^{1}\) continuous characteristic in the \(Z\)-axis. Figure 8: Simulation topology snapshot (\(n=19\)). Figure 10: Sensor nodes and control points of Bezier curves. Figure 9: Three-dimensional (3D) Bezier interpolation results with \(n=3\) nodes. Subsequently, following the methods proposed in [22; 23] and Section 3, we obtained comparable results derived from the linear-interpolation-based 3D Dubins curves and Bezier interpolation based 3D Dubins curves, respectively. Figures 12 and 13 illustrate the final trajectory of the linear-interpolation 3D Dubins algorithm and Bezier interpolation 3D Dubins algorithm when \(\phi=\frac{\lambda\pi}{4}\), and their projection curves in the \(X\)-\(Y\) plane are shown in Figures 14 and 15, respectively. It can be shown that the desired 3D curves are both continuous and smooth in the \(X\)-\(Y\) plane. To show this more clearly, Figure 16 gives the \(X\)- and \(Y\)-axis coordinates for each interpolation sequence. Moreover, Figure 17 presents the first-order derivatives of the \(X\)- and \(Y\)-axis coordinates. Clearly, the differential curve is continuous with the interpolation sequences. Figure 16: \(X/Y\)-axis of three-dimensional (3D) Dubins curves (case 1). Figure 14: \(X\)-\(Y\) projection of linear-interpolation three-dimensional (3D) Dubins curves (case 1). Figure 15: \(X\)-\(Y\) projection of Bezier interpolation three-dimensional (3D) Dubins curves (case 1). Moreover, we investigated the smoothness feature in the \(Z\)-axis. As shown in Figures 22 and 23, it is clear that the linear-interpolation 3D Dubins method failed \(G^{1}\) continuity in the \(Z\)-axis, while the Bezier interpolation 3D Dubins method satisfied \(G^{1}\) continuity in both the \(X\)-\(Y\) plane and \(Z\)-axis. Furthermore, we compared the first-order derivatives of the \(Z\)-axis coordinate in Figure 24. It can be seen from the simulation results that the first-order derivative derived from Bezier interpolation was continuous, while that derived from linear interpolation was not continuous, thus proving that the proposed Bezier interpolation 3D Dubins curves belong to 3D smooth curves. Figure 21: \(X\)–\(Y\) projection of Bezier interpolation three-dimensional (3D) Dubins curves (case 2). Figure 7:0 \(X\)–\(Y\) projection of linear-interpolation three-dimensional (3D) Dubins curves (case 2). Figure 7:1 \(Z\)-axis of linear-interpolation three-dimensional (3D) Dubins curves (case 1). Finally, we computed the average total cruising lengths generated by the linear and Bezier interpolation based 3D Dubins algorithms with different numbers of underwater sensors. The average total cruising lengths of the two algorithms were derived from 100 Monte Carlo simulations. The comparison results in Table 1 verify that the Bezier interpolation based 3D Dubins curve planning algorithm had almost the same length compared to the linear-interpolation-based 3D Dubins curve planning algorithm whether for case 1 or 2. This means the Bezier interpolation method only needs a slightly higher path cost than the linear interpolation method.