**6.4 Suboptimal vehicle routing in a time-varying tidal flow**

The last example presented in this chapter is an underwater vehicle routing in Tokyo Bay. In this example, we consider the mission of minimum-time homing to a destination. Due to its narrow entrance and shallow depth, sea currents in Tokyo Bay are hardly affected by the outer ocean currents such as Kuroshio. Instead, like many other littoral zones, currents in Tokyo Bay are dominated by the tidal flow. In this research, we use the time-varying sea current distribution data in Tokyo Bay, generated by a numerical tidal flow simulation model by Kitazawa et al. [19]. As was the case in the previous examples, we first conduct optimal vehicle routing without considering environmental uncertainties. In **Figure 16**, vehicle trajectories generated by the routings of different guidance strategies are shown. As found in the figure, our approach of optimal guidance successfully accomplishes the minimum-time homing mission even the sea current is time varying. Note that the current distribution shown in the figure is the one taken at the traveling time of straight-line tracking, which makes the last arrival at the destination.

**Figure 16.** *Vehicle trajectories in the tidal flow in Tokyo Bay.*

Next, on the basis of our suboptimal strategy, we conduct the vehicle routing simulation, incorporating uncertainties into the time-varying tidal flow in Tokyo Bay. The standard deviation of the uncertainties is set to be *4Ucm*, as was the case in the preceding examples. **Figure 17(a)**–**(f)** shows sequential vehicle trajectories created by our suboptimal vehicle routing. The traveling time of the suboptimal

#### **Figure 17.**

*Sequential vehicle trajectories in Tokyo Bay generated by suboptimal routing (a) t = 4000 s (b) t = 11000 s (c) t = 13000 s (d) t = 14000 s (e) t = 18000 s (f) t =24251 s.*

*Reconfigurable Minimum-Time Autonomous Marine Vehicle Guidance in Variable Sea Currents DOI: http://dx.doi.org/10.5772/intechopen.92013*


#### **Table 2.**

*Traveling times of vehicle routings in Tokyo Bay.*

vehicle routing and those of the optimal, PN, and straight-line tracking obtained in the previous example are summarized in **Table 2**.

In this example, we set *da*, the acceptable limit of deviation distance, to be 1800 m. As seen in the figures, the vehicle has successfully accomplished its homing mission by the suboptimal routing, repeating five revised travels. In view of the results obtained by this example, we find that our suboptimal approach works effectively even in a time-varying environment including uncertainties.

#### **7. Conclusion**

In this chapter, a systematic procedure for obtaining the numerical solution of the optimal guidance law to achieve the minimum-time routing in a region of sea current has been presented. The optimal heading is obtained as the solution of the optimal guidance law, which is fed to the heading control system as the reference.

Reduced computational cost is one of the outstanding features of the proposed procedure. While linearly proportional to the area of a search region in DP, the computational time in our procedure exhibits square root dependence. Moreover, unlike the other path finding algorithms such as DP or GA, when applied to a timevarying environment, our procedure does not increase the search space, resulting in the same computational cost as required in the time-invariant ones.

The performance of the optimal guidance has strong dependency on the current distribution. While an extremely simple configuration, such as uniform flow, hardly allows navigation time reduction by the optimal guidance, a multi-directional complicated flow distribution enhances the potential efficacy of the optimal guidance.

As a fail-safe or fault-tolerable strategy in optimal guidance, the concept of suboptimal guidance has been proposed. The fact that there actually are several possible actions lessening the chance of optimality emphasizes the practical importance of our suboptimal strategy.

We have not considered the problem of unknown or nondeterministic currents. Our approach cannot be applied to an entirely unknown environment. For a sea region with partially or coarsely defined current flow, however, an estimated distribution can be built by means of interpolation and extrapolation. As has already been shown in the optimal and suboptimal vehicle routing examples in actual sea regions, spatiotemporal interpolation of the current velocity successfully derives the converged solution.

#### **Acknowledgements**

The author would like to thank Prof. D. Kitazawa of IIS, the University of Tokyo for providing simulated tidal flow data of Tokyo Bay. Also, the author is particularly grateful to T. Ura, an emeritus professor of the University of Tokyo, for his guidance and support concerning this research work.

*Automation and Control*
