# Event-Based Path-Planning and Path-Following
## 4 Simulation Results  
We conducted numerical simulations using three scenarios in order to validate the performance of the designed control system: with a sharp-edged obstacle, with a U-shaped obstacle, and with densely scattered obstacles. Scenario 1 aims to test the ability of the path planner to cope with obstacles having complex shapes. Scenario 2 demonstrates how the planner can drive the AUV out of a trap made by a U-shaped obstacle. The last scenario shows whether the algorithm is effective in a cluttered environment. In the simulations, we adopted parameters of the model (1) and (2) that correspond to the INFANTE AUV [23]: the nominal mass \(m=2234.5\) kg and moment of inertia \(I_{z}=2000\) N \(\cdot\) m\({}^{2}\). The rest of the model parameters can be found in [23]. The path-following controller was designed with the following parameters: sampling time \(h=0.2\) s, control constraints \(\overline{\mathcal{F}}_{s}=200\) N and \(\overline{\mathcal{G}}_{s}=100\) N m, and maximum path curvature \(c_{\max}=1/R_{\min}=0.05\). Figure 8 shows the feedback gains \(k_{i}\), \(i=\overline{1,5}\) of the path-following controller (7), (8) as functions of the VT's angular speed \(r_{0}\). Path-planning parameters are specified, as follows: \(R_{\min}=20\) m, \(D_{s}=10\) m, \(D_{m}=2\) m, \(\triangle R=5\) m, \(\rho_{vc}=10\) m, \(\rho_{n}=25\) m, \(\rho_{f}=70\) m, \(\rho_{n}^{RP}=30\) m, \(d_{o}=50\) m. Also, for the FLS we specify the number of beams \(N_{b}=60\), the detection range \(\rho_{max}=150\) m, and the field of view \(\psi_{FLS}=120^{\circ}\). The triggering conditions for all events, excluding eEP (the VT has reached the end of the current path) and eES (the VT has reached the end of the current segment), which are monitored with period \(h=0.2\) s, are checked with interval \(h_{e}=2\) s.  
Figures 9-13 show the results of the simulation for Scenario 1. From Figure 1, we can conclude that the AUV successfully avoids collisions with the obstacle by keeping a safe distance from it. In this  
Figure 8: Feedback coefficients \(k_{i}\) (\(i=\overline{1,5}\)) synthesized for the path-following controller (7) as functions of the VT’s angular speed \(r_{0}\). Figure 7: An example of building a Dubins path that connects point \(P_{b}\) with arc \(S\). For this example, \(\mathcal{D}_{s}=\{\text{{\textasci{1}R}}\}\) and \(\mathcal{D}_{f}=\{\text{{\textasci{1}L}}\text{{\textasci{1}R}}\}\). The red line indicates the shortest path. scenario, which lasts approximately 900 s, the AUV rarely corrects the path. The left-hand part of Figure 9 demonstrates that the designed DES changes its state only 14 times and, therefore, the path is corrected the same number of times. Between corrections, the upper control level only checks the event triggering conditions at an interval of 2 s. Obviously, the less the DES changes its state, the less the path is updated, and the less load on the on-board computing device. Note that DES detects not only situations when it is necessary to generate a new path, but also situations when the active waypoint should be changed. For example, during the first 250 s of scenario 1, the active waypoint is changed twice, due to the simultaneous occurrence of elementary events eWPB, eWPR, and eWPDN (rule 1 of the mDOL block in Table 3). As a result of these changes, waypoint 4 of the reference path becomes active before the AUV approaches the line segment between waypoints 3 and 4, and events that triggers the return to the reference path (rule 5 of the mDOL block in Table 3) can be detected. Figure 13 shows that the proposed path-planning algorithms generate paths that have a constrained curvature \(|c_{c}|\leq c_{\rm max}=0.05\), and, consequently, meets the turning radius constraint \(R_{\rm min}=1/c_{\rm max}=20\) m.  
The analysis of Figures 10, 11, and 13 allows for drawing a conclusion that the main source of disturbances causing the growth of path-following errors \(s_{e}\), \(y_{e}\) and velocity discrepancies is jumps in the path curvature.