**4. Dynamic constraint**

As mentioned previously, we adopt GNC system based on the hierarchical control architecture consisting of two control layers. In the high-level control layer,

**Figure 8.** *Graphical description of the equation of motion for lateral dynamics.*

i.e., the optimal guidance, the guidance law derived irrespective of specific vehicle dynamics is used. In the low-level control layer, however, vehicle dynamics is implemented as an implicit constraint. When conducting the trials explained in previous section, closed-loop dynamics of a specific vehicle is used. Therefore, vehicle trajectories generated by the trials are feasible ones subject to the dynamic constraint of a specific vehicle. Eq. (8) is the state-space model of the lateral dynamics of r2D4 describing its sway, roll, and yaw responses. In **Figure 8**, kinematic variables and actuations appearing in Eqs. (8)–(10) are described graphically. By solving Eq. (8) in time domain, velocities and attitudes of the vehicle are obtained.

$$
\begin{bmatrix}
\dot{\boldsymbol{\nu}} \\
\dot{\boldsymbol{p}} \\
\dot{\boldsymbol{r}} \\
\dot{\boldsymbol{\phi}}
\end{bmatrix} = \boldsymbol{A}\_{lat} \begin{bmatrix}
\boldsymbol{\nu} \\
\boldsymbol{p} \\
\boldsymbol{r} \\
\boldsymbol{\phi}
\end{bmatrix} + \boldsymbol{B}\_{lat} \begin{bmatrix}
\boldsymbol{\delta}\_{pr} \\
\boldsymbol{\delta}\_{sl} \\
\boldsymbol{\delta}\_{sr}
\end{bmatrix} \tag{8}
$$

where

$$A\_{tot} = \begin{bmatrix} -0.5266 & 0.0029 & -0.7754 & -0.0052 \\ -3.5308 & -4.7898 & 7.9422 & -10.5336 \\ -0.1472 & -0.0334 & -0.5742 & -0.0722 \\ 0.0 & 1.0 & -0.0129 & 0.0 \end{bmatrix} \tag{9}$$

$$B\_{lat} = \begin{bmatrix} -0.0335 & -0.0011 & 0.0011 \\ 0.0244 & -2.1704 & 2.1704 \\ 0.0486 & -0.0149 & 0.0149 \\ 0.0 & 0.0 & 0.0 \end{bmatrix} \tag{10}$$

### **5. Minimum-time guidance in stationary current flows**

#### **5.1 Reference navigation**

As mentioned previously, in order to practice the numerical procedure AREN, a vehicle routing simulation by the reference navigation has to be conducted beforehand. Among several strategies for mobile vehicle navigation, the simplest one ensuring arrival at the destination might be the PN based on LOS guidance. In all of our optimal guidance examples presented in this paper, we employ PN as the reference navigation.

#### **5.2 Linear shearing flow**

The first example of the optimal guidance in this paper is the minimum-time routing in a current disturbance of linear shearing flow, taken from Bryson and Ho [8]. The current velocity in this problem is described by:

$$u\_{\varepsilon}(\mathbf{x}, \boldsymbol{\wp}) = 0 \tag{11}$$

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

$$\mathbf{v}\_c(\mathbf{x}, \mathbf{y}) = \mathbf{-}U\_c \mathbf{x} \mathcal{H} \tag{12}$$

In Eq. (12), *Uc* and *h* are constants whose numerical values are set to be 1.54 m/s and 100 m, respectively. In this example, the vehicle is to travel to the destination located at the origin, starting from the initial position at (�186 m, 366 m). As an operating condition, the vehicle is assumed to maintain its thrust power constant throughout its travel, producing a constant advance speed of 1.54 m/s. This is an important operating condition applied to all examples presented thereafter. With the current distribution given as Eqs. (11) and (12), we can derive the analytic optimal guidance law in a closed form, shown as follows:

$$\frac{\mathcal{X}}{h} = \csc\psi \mathbf{\cdot} \cdot \csc\psi \mathbf{\cdot}\_{\mathcal{f}} \tag{13}$$

$$\frac{\mathbf{y}}{\hbar} = \frac{1}{2} \left[ \mathbf{c} \mathbf{c} \mathbf{c} \boldsymbol{\psi}\_f (\mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu} - \mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu}\_f) + \mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu} (\mathbf{c} \mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu}\_f - \mathbf{c} \mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu}) + \log \frac{\mathbf{c} \mathbf{c} \mathbf{c} \boldsymbol{\psi}\_f \cdot \mathbf{c} \boldsymbol{\nu} \boldsymbol{\nu}\_f}{\mathbf{c} \mathbf{c} \boldsymbol{\psi} \cdot \mathbf{c} \boldsymbol{\psi} \boldsymbol{\nu}} \right] (\mathbf{1} \mathbf{4})$$

where *ψ<sup>f</sup>* represents the final vehicle heading taken at the destination. The analytic optimal guidance law shown above is similar to that found by Bryson and Ho [8] but has some differences due to the switched *x* and *y*.

Vehicle trajectories are shown in **Figure 9**. In **Figure 10**, vehicle headings obtained by the routings conducted by PN, that is, the reference navigation, and by optimal guidance are shown.

During the vehicle routing by PN, significant adverse drift happens at the initial stage. This is because the speed of current flow exceeds the advance speed of the vehicle in the region |*x*| > 100 m, as noticeable from Eqs. (11) and (12). On the other hand, optimal guidance detours the vehicle across the upper half plane of the flow region on purpose, taking advantage of the strong current flowing to favorable direction. This leads to the dramatic decrease in traveling time. The traveling times by PN and optimal guidance are 353.8 and 739.2 s, respectively, implying a 52% reduction in the optimal guidance. As seen in **Figure 9**, the optimal trajectory

**Figure 9.** *Vehicle trajectories in a stationary linear shearing flow.*

**Figure 10.** *Vehicle headings during travels in a stationary linear shearing flow.*

obtained by the numerical solution shows extremely good agreement with the analytic one.

As a criterion for evaluating the reference tracking performance in our two-layer control architecture, we employ Normalized Root-Mean Square Error (NRMSE) fit defined as follows:

$$
\hat{\xi}\_{\hat{\epsilon}}f t = I \frac{\left\| \tilde{\xi}\_{re^{\circ}} - \tilde{\xi} \right\|}{\left\| \tilde{\xi}\_{re^{\circ}} - \overline{\tilde{\xi}} \right\|}\tag{15}$$

where *ξref* and *ξ* are the vectors of output reference and output, and represents the mean value of *ξ*. The value of NRMSE fit varies between �∞ to 1, implying full decorrelation to perfect fit between the output reference and the output. In this simulation, NMRES fit between the optimal reference heading and the actually tracked one has marked 0.993. This means highly good heading tracking result, which is also found in **Figure 10**. The numerical solution approximates the analytic solution with extremely high accuracy (**Figure 9**), validating AREN as an effective numerical procedure for the optimal guidance law Eq. (4). As shown in this example, our approach based on the numerical procedure AREN and two-layer control architecture works properly achieving minimum-time AMV routing in a given sea current field.

#### **5.3 Sea current in Northwest Pacific near Japan**

In response to the successful result obtained from the benchmark example shown in previous section, we apply our optimal guidance to the minimum-time routing problem in an actual sea region of stationary sea current. The sea region selected is located in the Northwest Pacific Ocean near Japan. The daily updated sea current data of this region are available from [18] presented by the Japan Meteorological Agency. The most notable environmental characteristic in this sea region is the current field dominated by Kuroshio. The Kuroshio is a strong western

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

#### **Figure 11.**

*Vehicle trajectories in a sea current in Northwest Pacific near Japan.*

boundary current flowing northeastward along the coast of Japan. In the sea current data from [18], current velocity is defined only on the grid nodes covering the region. As noticeable from Eq. (4), however, in order to derive the optimal heading reference, current velocity and its gradient at every vehicle position have to be available. In the previous example, their exact values are easily obtained by analytic formulae. In this example, however, since they are defined only on the grid nodes, current velocity and its gradient are estimated by interpolating the predefined values on grid nodes surrounding the present vehicle position. In applying the current velocity interpolation, the grid node nearest the present vehicle position is identified first. Then, the current velocity at the present vehicle position is estimated by 2-D bi-quadratic interpolation utilizing the values on the nearest node and

**Figure 12.** *Time sequences of vehicle headings during the travel.*

eight nodes surrounding current vehicle position. Gradients of current velocities are obtained by the same manner. Since the velocity gradients are not provided from the database, however, prior to the interpolation, we calculate their nodal values by finite difference approximation.

**Figure 11** shows the vehicle trajectories obtained by the guidance of three different objectives already explained in the previous example. Time sequences of the vehicle headings corresponding to the vehicle trajectories shown in **Figure 11** are depicted in **Figure 12**.

As shown in the figure, like the preceding examples in which the current velocities and their gradients are analytically available anywhere in the region, optimal reference trajectory has successfully been derived by interpolation-based current velocities and gradients. Moreover, subject to its dynamic constraint, the vehicle tracks the optimal reference trajectory with a negligibly small deviation, resulting in the NMRES fit to be 0.986. This demonstrates the validity of our optimal guidance strategy in any actual sea currents, if only their distribution is deterministic.

#### **6. Suboptimal strategy**

#### **6.1 Environmental uncertainty**

In the following example, we apply our optimal guidance strategy to a vehicle routing in the same sea region shown in the preceding example. The only thing different from the preceding example is that we consider uncertainty in our sea current data in this example. The uncertainty components in sea current velocities are expressed as additive white Gaussian noise (AWGN). Taking the sea current velocities in the Northwest Pacific Ocean used beforehand as the mean values, the on-site current velocity including uncertainty is given by:

$$
\mu\_{cs}(\mathbf{x}, \mathbf{y}) = \mu\_c(\mathbf{x}, \mathbf{y}) + e\_u(\sigma) \tag{16}
$$

$$\mathbf{v}\_{cs}(\mathbf{x}, \mathbf{y}) = \mathbf{v}\_c(\mathbf{x}, \mathbf{y}) + \mathbf{e}\_\mathbf{v}(\boldsymbol{\sigma}) \tag{17}$$

where *ucs* and *vcs* are the components of onsite current velocity, *uc* and *vc* are the components of deterministic current velocity taken from the database, and *eu(σ)* and *ev(σ)* are the AWGNs with standard deviation *σ*. As the parameter for specifying the value of *σ*, we introduce the regional mean current speed *Ucm* defined as follows:

$$\begin{aligned} \sum\_{cm}^{N} \sqrt{\mathbf{u}\_{cl}^{2} + \mathbf{v}\_{cl}^{2}} \\ \hline N \end{aligned} \tag{18}$$

where *i* represents the index covering all grid nodes on which the database-based current velocities are defined.

Vehicle trajectories by optimal vehicle routings conducted on two different level uncertainties are shown in **Figure 13**. When the level uncertainty is such that *σ* = *2Ucm*, optimal heading reference derived without considering any uncertainty still seems acceptable. As a result, though slightly deviating from the destination, the final position of the vehicle remains in the vicinity of the destination. When the level of uncertainty increases up to *σ* = *4Ucm*, however, the vehicle following the

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

#### **Figure 13.**

*Vehicle trajectories in Northwest Pacific near Japan. The sea current velocities in this example include uncertainties modeled by AWGN.*

optimal reference heading can no longer approach the destination, which means the failure in accomplishing the minimum-time travel to the destination.

#### **6.2 Suboptimal guidance**

The suboptimal guidance proposed in this research is a fail-safe or fault-tolerable strategy toward robust field implementation of our optimal guidance strategy. The optimal reference heading obtained by our approach is the one derived without considering the dynamics of a specific vehicle. This means the optimal trajectory may not be realized by a specific vehicle. Hence, we note that the dynamic constraint is one possible source of the failure in putting our approach into practice for an actual field application. Another significant source of the failure is the environmental uncertainty, as already shown above. It is easily expected that as a vehicle progresses following the optimal heading in the sea region of environmental uncertainty, due to the interaction with the current flow different from that was used in deriving the optimal heading, its actual trajectory deviates away from the optimal reference trajectory, and eventually, it might fail in reaching the destination. The basic idea of our suboptimal approach is rather simple. Let *d1* denotes the deviation distance between the present vehicle position and the preassigned one on the optimal reference trajectory obtained by AREN. When *d1* exceeds a prescribed acceptable limit *da*, the high-level controller in autopilot is activated to reroute the vehicle by reapplying AREN. This rerouting is repeated whenever *d1* exceeds a predefined acceptable limit. The resulting vehicle routing is not rigorously optimal, since it includes past nonoptimal travels. However, Bellman's principle of optimality [8, 11] states that it is evidently the best strategy we can take under the condition we are faced with. We, therefore, call this approach the suboptimal guidance. **Figure 14** depicts the schematic of our suboptimal guidance explained thus far.

**Figure 14.** *Schematic of the vehicle routing by suboptimal guidance.*
