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

#### **Figure 5.**

*Schematic of the numerical solution procedure AREN.*

the Euler-Lagrange equation, which is a typical example of the two-point boundary value problem, characterized by split boundary conditions in states and costates [8, 11]. To obtain the solution of a two-point boundary value problem, an iterative solution procedure is usually required. The most famous and commonly used numerical procedures for such purpose are the shooting and relaxation methods [17]. However, direct applications of these methods to our problem have significant difficulties. In applying shooting method to a two-point boundary problem in time domain, governing ODEs with proper initial guesses should be integrated until reaching the upper limit of the boundary. However, as noticeable from its name, i.e., the minimum-time guidance, our problem is a so-called free boundary one, having unspecified upper limit in time domain. In treating a free boundary problem by relaxation method, on the other hand, the independent variable should be transformed into a new one defined between 0 and 1. Here, we can anticipate an intrinsic serious difficulty in determining the stepsize in free boundary problems. Properness of temporal grid distribution ensuring convergence is initially unknown, and to know, it is extremely difficult before the end time-marching computation. Moreover, strong initial guess dependency of the solution is another serious concern in applying the relaxation method to our problem. Inappropriate initial guess possibly leads to local optimality or divergence [17].

As a new approach for deriving the numerical solution of the optimal guidance law Eq. (4), we presented a search procedure, which determines correct initial heading. Being named AREN (Arbitrary REference Navigation), our procedure works globally on the basis of the minimum principle. **Figure 5** summarizes the algorithmic scheme of our solution procedure.

Note that in **Figure 5** and hereafter, an asterisked variable denotes the one corresponding to the optimal solution. In applying AREN, we first have to make a vehicle routing simulation in which the vehicle travels to the destination following an arbitrary guidance. It is noted here that the traveling time must be registered at the final stage of this simulation. We call the traveling time the reference final time and use it as the key criterion in seeking optimal initial heading. The navigation applied to the simulation is called reference navigation, which is arbitrary if only the vehicle's arrival at the destination is assured. Therefore, simple one such as proportional navigation (PN) based on the line-of-sight (LOS) guidance is frequently used as the reference navigation. To find the correct initial heading, the interval of *0*–*2π* is divided by equally spaced *N-1* subintervals, as represented by:

$$
\Delta\varphi\_0^{(i)} = i\Delta\psi \quad \text{for} \quad i = 0, 1, \ldots N \text{-} I \tag{5}
$$

where is *(i)*th initial heading guess, and is the increment of the guess. Next, by applying an initial heading guess to Eq. (4), we solve Eq. (4) in time domain. This produces a simulated vehicle routing starting from . The routing having been produced here is called the *(i)*th trial adjoining to . Once the vehicle passes through the destination by the *(i)*th trial, it is regarded as a possible optimal routing since the correct initial heading incorporated into the optimal guidance law lets a vehicle reach the destination. Therefore, *N* trials are the candidates for the simulated minimum-time routing. In practice, however, discretization error in the optimal initial heading causes the residual in the optimal trajectory, making the optimal solution identified in an approximate manner. For the vehicle trajectory generated by a trial, we define the "minimum distance" as the shortest distance between the destination and the trajectory. In **Figure 6**, , , and are the minimum distances corresponding to *(k-1)*th*, (k)*th, and *(k + 1)*th trials, respectively.

When the minimum distance of *(k)*th trial is smaller than any other one, satisfying:

$$l\_{\\_min}^{(k)} \le l\_{\\_min}^{(i)} \text{ for } \ i = 0, 1, \dots \\ N \text{-} I \tag{6}$$

we choose the *(k)*th trial as the optimal routing because the vehicle approaches the destination marking the smallest deviation. In determining the optimal routing among the trials, however, there still remains a serious drawback. We have no idea how long we have to continue a trial not to miss the true minimum distance of the trial. We settle this problem by exploiting the result of reference navigation. The reference navigation is apparently a nonoptimal one based on an arbitrary guidance only assuring the arrival at the destination. Therefore, the reference final time must be larger or equal to that of the optimal routing as follows:

$$0 < t\_f^{\top} \le t\_{f\_{\underline{r}}r\underline{r}f} \tag{7}$$

where represents the traveling time of the optimal routing. It should be noted here that by the minimum principle [8, 11], we can set up a sufficient condition for

**Figure 6.** *Minimum distances of trials.*

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

#### **Figure 7.**

*Determining optimal routing among trials.*

seeking the optimal solution. By the minimum principle, once a trial has started with an initial heading sufficiently close to the optimal value, the vehicle obviously passes by the vicinity of the destination at the traveling time smaller than . In other words, the reference final time qualifies as the upper limit of the necessary simulation time of any trial, which assures the convergence to the vicinity of the destination in case the trial is near optimal. In **Figure 7**, *(k)*th trial is selected as the optimal routing among all trials terminated at since is the smallest minimum distance.

The minimum distance of the optimal routing is to be interpreted as the residual error in the converged solution. Therefore, it can be said that the smaller the minimum distance is, the better the convergence is. When is still unacceptably large though the *(k)*th trial has been accepted as the optimal routing, the initial heading interval of is subdivided, and the trials are repeated starting from these subdivisions pursuing finer convergence.
