**2.2 Formulation**

As mentioned previously, we employ the vehicle heading as the output in our optimal guidance problem. Here, it is noted that we adopt the so-called GNC (Guidance, Navigation, and Control) system based on the hierarchical control architecture consisting of two control layers. That is, the high-level control for guidance and navigation, and the low-level control for pure tracking purpose (**Figure 2**). The optimal heading derived by solving the optimal guidance law is used as the reference output for low-level heading tracking control. In this research, we use the optimal heading guidance law presented by Bryson and Ho [8]. In deriving the optimal guidance law, two sets of coordinate systems are used: the inertial (earth-fixed) coordinate system *o-xy* and the body fixed coordinate system *o'-x'y'* (**Figure 3**).

As the marine vehicle used in our problem, we employ an autonomous underwater vehicle (AUV) "r2D4" as described by Kim and Ura [10]. In **Figure 3**, actuator inputs and kinematic variables are described. *ψ* is the yaw displacement of the vehicle. While *δpr* denotes the main thruster axis deflection, *δel* and *δer* are the deflections of elevators on left and right sides, respectively.

**Figure 1.** *Dependence of traveling time on traveling path.*

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

**Figure 2.** *Two-layer hierarchical control architecture for an AMV.*

**Figure 3.** *Coordinate systems for optimal guidance problem formulation.*

In this research, we approximate that the direction of the vehicle's advance velocity coincides with the *x'*-axis. It is arguable in the rigorous definition since there certainly occurs sideslip during a turning motion of an underactuated vehicle. However, as mentioned by Lewis et al. [16], the hydrodynamic sideslip induced by a low-speed, slender vehicle is bounded within a sufficiently small range, justifying our approximation. Since the distribution of a sea current is considered to be deterministic in our research, current velocity is described as a function of the position and time. Therefore, on the assumption that the advance velocity of a vehicle and the current velocity are superimposable, the resultant vehicle velocity is expressed as follows:

$$
\mu = \dot{\mathbf{x}} = U\_o \cos \psi + \mu\_c(\mathbf{x}, \mathbf{y}, t) \tag{2}
$$

$$\mathbf{v} = \dot{\mathbf{y}} = U\_o \sin \boldsymbol{\varphi} + \mathbf{v}\_c(\mathbf{x}, \mathbf{y}, t) \tag{3}$$

#### *Automation and Control*

where *u* and *v* are the components of the vehicle velocity relative to the inertial frame, *U0* is the advance speed of the vehicle in still water, and *uc* and *vc* are the components of current velocity at a given position and time. It is noted that we assume *U0* is the constant throughout a travel, which implies the operating condition of steady cruise.

Eq. (4) shows the minimum-time guidance law originally presented by Bryson and Ho [8]. Detailed procedures deriving Eq. (4) are well explained by Kim and Ura [10]. It is noted here that if only deterministic, there is no restriction on the type of the sea current in Eq. (4). That is, not only stationary but also time-varying sea current can be applied to Eq. (4). This leads to one of the most powerful aspects of our approach over many other path planning approaches based on combinatorial optimization.

$$\dot{\boldsymbol{\nu}} = \sin^2 \boldsymbol{\psi} \frac{\partial \boldsymbol{\nu}\_c}{\partial \mathbf{x}} + \left(\frac{\partial \boldsymbol{u}\_c}{\partial \mathbf{x}} - \frac{\partial \boldsymbol{\nu}\_c}{\partial \mathbf{y}}\right) \sin \boldsymbol{\psi} \cos \boldsymbol{\nu} \boldsymbol{\nu} \cdot \cos^2 \boldsymbol{\psi} \frac{\partial \boldsymbol{u}\_c}{\partial \mathbf{y}} \tag{4}$$

The optimal guidance law shown above is a nonlinear ordinary differential equation of unknown vehicle heading. The solution of optimal guidance law is used as the optimal reference heading, by tracking which a vehicle achieves the minimum-time travel to the destination, leaving the trail of optimal trajectory.

### **3. Numerical solution procedure**

Eq. (4) is a nonlinear ordinary differential equation (ODE) for an unspecified vehicle heading *ψ(t)*. If the functions *uc(x,y,t)* and *vc(x,y,t)* describing current velocity distribution are differentiable and deterministic, the solution of Eq. (4) seems to be attainable with an initial value of *ψ(t)*, in terms of an appropriate numerical solution algorithm such as Runge-Kutta. However, in practice, with an arbitrary initial heading a vehicle following the guidance law Eq. (4) does not reach the destination, as depicted in **Figure 4**.

More precisely, consisting of a part of the solution, the initial vehicle heading is not arbitrary but is to be assigned correctly. This is because Eq. (4) is derived from

**Figure 4.** *Solution convergence affected by initial heading.*
