**Abstract**

In this chapter, we present an approach of reconfigurable minimum-time guidance of autonomous marine vehicles moving in variable sea currents. Our approach aims at suboptimality in the minimum-time travel between two points within a sea area, compensating for environmental uncertainties. Real-time reactive revisions of ongoing guidance followed by tracking controls are the key features of our reconfigurable approach. By its reconfigurable nature, our approach achieves suboptimality rather than optimality. As the basic tool for achieving minimum-time travel, a globally working numerical procedure deriving the solution of an optimal heading guidance law is presented. The developed solution procedure derives optimal reference headings that achieve minimum-time travel of a marine vehicle in any deterministic sea currents including uncertainties, whether stationary or time varying. Pursuing suboptimality, our approach is robust to environmental uncertainties compared to others seeking rigorous optimality. As well as minimizes the traveling time, our suboptimal approach works as a fail-safe or fault-tolerable strategy for its optimal counterpart, under the condition of environmental uncertainties. The efficacy of our approach is validated by simulated vehicle routings in variable sea currents.

**Keywords:** guidance, minimum-time, marine vehicle, sea current, suboptimal, environmental uncertainty

#### **1. Introduction**

It is well known that the sea environment contains several kinds of flows, which possibly interacts with the motion of surface or submerged vehicles. Among these, sea or ocean currents are the most significant ones, directly affecting traveling speed, power consumption, and thus the endurance and range of a vehicle. Suppose that a marine vehicle is to travel to a given destination starting from a point in the region of flow disturbance. Then, it is quite natural that the traveling time of the vehicle should change according to the selection of a specific path. In case the power consumption of a vehicle is controlled to be constant throughout the travel, the total energy consumption is directly proportional to the traveling time.

Recently, autonomous marine vehicles (AMVs) are playing important roles in diverse applications, such as oceanographic survey, marine patrol, undersea oil/gas production, and various military applications [1]. Relying on onboard energy

storage as the main energy source, the endurance and moving range of an AMV are limited by its power consumption and its capacity of energy storage. Therefore, it can be said that the reduced traveling time of an AMV enhances vehicle safety and mission effectiveness [2].

Considerable research works have been done on the guidance or path planning for a mobile vehicle through varied fluid environments. Though aiming at the same objectives, the most notable difference between the guidance and the path planning of a vehicle is the consideration of its dynamical constraints. While, in general, dynamical constraints in vehicle motion are incorporated into the formulation of vehicle guidance problems [3, 4], they are ignored in most path planning problems [5, 6]. This allows great flexibility in the target path generation, enabling the use of combinatorial optimization techniques in path planning approaches. Dynamic programming (DP) might be one of the most classical and popular techniques for combinatorial optimization. In [6], the problem of minimal-time vessel routing in a region of deterministic wave environment is treated on the basis of the dynamic programming approach. In this problem, sea region is subdivided into several subregions of different sea states. The optimal path is derived by determining the sequence of subregions to be visited, which minimizes the traveling time to a given destination. Aside from the difficulty in establishing a practically available numerical procedure adjoining the formulation, the significant solution dependency on the regional subdivision is a critical issue in this approach. Some recent researches reported the application of a generic algorithm (GA) to path planning for an underwater vehicle in a variable ocean [5]. Major advantages of the GA over dynamic programming are reduced computational complexity and time, although it is susceptible to local minima, however. Also, one of its significant drawbacks is a strong constraint in generating the optimal path. In a path planning application on the basis of GA, a user-defined primary coordinate should strictly maintain a monotonic increase in the optimal path [5]. This is such a strong constraint that makes it impossible to generate the optimal path containing interim backward intervals. Minimum-time guidance of a mobile vehicle in a fluid environment of arbitrary flow field is a strongly nonlinear optimization problem, quite difficult to solve numerically as well as analytically. One of the recent approaches to treating this sort of problems is cell mapping [3]. Though it is known to be especially adequate for strongly nonlinear problems, computational demand of cell mapping for obtaining a stable solution is enormous.

Path finding or guidance algorithms can be classified into two categories according to the instant when its solution is generated. While a pregenerative one derives an unchangeable solution prior to a mission, a reactive algorithm allows revised solution during the mission [5, 7]. In this research, as a reactive strategy for achieving minimum-time travel in varied sea current environments, we propose an approach of suboptimal guidance. In our problem, minimum-time travel of a vehicle is attempted on the basis of the optimal guidance law presented by Bryson and Ho [8]. The solution of this guidance law is a time sequence of the optimal headings. In an actual field application for the minimum-time travel, obtained optimal headings are tracked by a vehicle as the reference in its heading control. Compact as it is, the optimal guidance law is derived without considering any specific dynamic constraint, like many other path planning approaches. In our suboptimal strategy, we compensate for this drawback by incorporating reactive revisions of optimal reference heading. Once there happens a failure in tracking current optimal reference attributed to the ignorance of limitations in vehicle dynamics, onboard autopilot reroutes the vehicle by reapplying the optimal guidance law.

In addition to the dynamic constraints, there are several unfavorable environmental factors that might be fatal in achieving the proposed optimal vehicle routing.

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

Examples of such factors are uncertainties in sea environments, severe sensor noises, or temporally faulty actuators [9, 10]. As a fail-safe or fault-tolerable strategy, our suboptimal approach can compensate for the failure in ongoing minimumtime travel due to any of the abovementioned factors. The suboptimal guidance does not achieve rigorous optimality. However, it achieves a near-optimality realized by the utmost in-situ actions as possible, which is useful and important in a practical sense.

Though provides superior adaptiveness, robustness, and more flexibility, the reactive approach in marine vehicle guidance incurs a heavy computational cost in its onboard implementation [3, 5, 10]. In this research, we present a practical solution procedure of highly reduced computational cost required for implementing our minimum-time guidance in a suboptimal manner. This is a simple procedure applicable to any sea current whether stationary or time-varying, provided that its distribution at a specified instant is deterministic. Robust global convergence is another advantage of our procedure. On the basis of the minimum principle [8, 11], our procedure realizes an efficient search space reduction, enabling optimal solution search in a global manner. Due to this algorithmic nature, our numerical procedure bears crucially lower possibility of taking local minima than other search algorithms primarily relying on initial guesses.

As mentioned previously, deterministic sea current is the prerequisite for implementing our suboptimal and optimal strategies. It is noted that, however, in many cases, it is not easy to obtain a predescribed current distribution of the sea region of interest. One of the simplest ways to build up the database of sea current distribution is direct measurement. Many governmental, public, or private institutions related to maritime affairs provide tabulated surface current distributions which are obtained by field measurements [12, 13]. The availability of these data is more or less restrictive because there are many sea regions for which the current distribution data are not built up or treated as confidential. As another source of sea current information, numerical estimation models are playing an important role. By assimilating the field measurement into them, some recent numerical models provide both forecasts and nowcasts of ocean fields with sufficiently accurate mesoscale resolution [14]. In this research, we employ two kinds of sea current data generated in totally different ways; the measurement-based stationary current distribution in Northwest Pacific, near Japan, and the sequential tidal current distribution in Tokyo Bay obtained by a numerical forecasting model.

Unlike path-planning approaches, our approach leads to simulation-based resultant optimal trajectory rather than optimal reference path. In our optimal guidance problem, not the position but the heading of a vehicle is employed as the design variable to be optimized. And, since we consider the dynamics of a specific vehicle as a constraint, optimal trajectory is to be generated by the simulated vehicle routing following the optimal reference heading. It is noted here that while the term of path is used as a route or track between one place and another, the trajectory means a curved path that an object describes on the basis of its kinematic scheme in this research [15].

#### **2. Minimum-time guidance**

#### **2.1 Problem statement**

The problem of minimum-time guidance for AMVs moving in flow fields is described in this section. Consider a marine vehicle traveling through a sea region of flows such as sea currents, whose properties are the function of space, or both space and time. The vehicle is to travel to a predetermined destination starting from the initial position at the initial time *t0*. Then, it is easily anticipated that the traveling time of the vehicle varies depending on its traveling path (**Figure 1**). Furthermore, it is also anticipated that an ingenious traveling path can minimize the traveling time to the destination. In this research, we refer to the optimal path as the path of minimum traveling time to the destination. As a minimum-time problem, our problem merely takes the traveling time as the performance index. In other words, in the general form used in optimal control, the integrand of performance index takes the value of 1.

$$J = \int\_{t\_0}^{t\_f} L d\tau = \int\_{t\_0}^{t\_f} d\tau = t\_f \cdot t\_0 \tag{1}$$

In Eq. (1), *J* is the performance index, and *t0* and *tf* are the initial and final time of the travel.
