Exploiting ocean energy for improved AUV persistent presence: path planning based on spatiotemporal current forecasts

Abstract

This paper presents a path planner which exploits ocean energy to enable long-range operation of autonomous underwater vehicles (AUVs) in turbulent, cluttered, time varying and uncertain environments. The proposed strategy employs current field forecasts within an evolutionary path planner to find an optimal trajectory for an AUV, with maximum energy utilization of the current motion and minimal time usage to reach its destination. The proposed path planner is simulated to generate an optimal trajectory for an AUV navigating through a spatiotemporal variable ocean environment in the presence of irregularly shaped terrains as well as obstacles whose position coordinates are uncertain. Simulation results show that with integration of an ocean current forecast model, the proposed methodology is able to obtain a more optimized trajectory than one relying on a static current map. Monte Carlo simulations were run to testify the performance of the proposed path planner, and analyse the effects of currents prediction uncertainties on the time optimal path. The results highlight the benefits of including ocean forecasts in the planning, and demonstrate the superiority of the proposed scheme based on a forecasted ocean current map compared with a configured quasi-stationary scenario.

Appendix 1

1.1 Kinematics and dynamics of the vehicle

As mentioned in Sect. 3.1, an AUV simulator (developed using Simulink) is used to test the trajectories generated by the proposed path planners that have been developed in Matlab. Kinematics and dynamics of the vehicle are described here in detail. The AUV is modelled as a free-moving body within a 6 degrees of freedom (6DOF) space as represented in Eqs. 11–17 [60]. The states represent the translation and rotation coordinates of the vehicle $\eta$, in the body frame reference. These states are represented in the North–East–Down (NED) frame (Eq. 13). The first- and second-order derivatives are also presented in Eqs. 11 and 12. The velocities, ${V_{\text{a}}}$, and accelerations, ${\dot {V_{\text{a}}}}$, are defined in the body frame. The velocity, components $u$, $v$, and $w$ represent the surge, sway, and heave linear motions, respectively, while $p$, $q$, and $r$, represent the roll, pitch, and yaw rotational rates, respectively:

$${\dot {V_{\text{a}}}}={\left[ {\begin{array}{*{20}{c}} {\dot {u}}&{\dot {v}}&{\dot {w}}&{\dot {p}}&{\dot {q}}&{\dot {r}} \end{array}} \right]^{\text{T}}},$$

(11)

$${V_{\text{a}}}={\left[ {\begin{array}{*{20}{c}} u&v&w&p&q&r \end{array}} \right]^{\text{T}}},$$

(12)

$$\dot {\eta }=J\left( \eta \right){V_{\text{a}}}.$$

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1.1.1 Euler-based kinematics

Using a Euler representation (14), the relationship between the velocity, ${V_{\text{a}}}$, represented in the body frame, and the NED frame, $\dot {\eta }$, is defined as $J\left( \eta \right)$ in Eq. 15. The linear velocity transformation matrix $R_{{\text{b}}}^{{\text{n}}}\left( \Theta \right)$ and the angular velocity transformation matrix ${T_\Theta }\left( \Theta \right)$ are defined in Eqs. 16 and 17, respectively. Where the integral of which is position in the NED frame, $\eta$, is defined as North, $x$, East, $y$, Down, $z$, and the Euler angles roll, $\phi$, pitch, $\theta$, and yaw, $\psi$:

$$\eta ={\left[ {\begin{array}{*{20}{c}} x&y&z&\phi &\theta &\psi \end{array}} \right]^{\text{T}}},$$

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$$\dot {\eta }=J\left( \eta \right){V_{\text{a}}} \Rightarrow \left[ {~\begin{array}{*{20}{c}} {{{\dot {p}}^n}} \\ {\dot {\Theta }} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {R_{{\text{b}}}^{{\text{n}}}\left( \Theta \right)}&{{0_{3 \times 3}}} \\ {{0_{3 \times 3}}}&{{T_\Theta }\left( \Theta \right)} \end{array}~~~~\begin{array}{*{20}{c}} {v_{{\text{o}}}^{{\text{b}}}} \\ {\omega _{{{\text{nb}}}}^{{\text{b}}}} \end{array}} \right],$$

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$$\begin{aligned} R_{{\text{b}}}^{{\text{n}}}\left( \varvec{\Theta} \right) & =\left[ {\begin{array}{*{20}{c}} {\cos \psi \cos \theta }&{ - \sin \psi \cos \phi +\cos \psi \sin \theta \sin \phi } \\ {\sin \psi \cos \theta }&{\cos \psi \cos \theta +\sin \phi \sin \theta \sin \psi } \\ { - \sin \theta }&{\cos \theta \sin \phi } \end{array}} \right. \\ & \quad \left. {\begin{array}{*{20}{c}} {\sin \psi \sin \phi +\cos \psi \sin \theta \cos \phi } \\ { - \cos \psi \sin \phi +\sin \theta \sin \psi \cos \phi } \\ {\cos \theta \cos \phi } \end{array}} \right], \\ \end{aligned}$$

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$${T_\Theta }\left( \Theta \right)=\left[ {\begin{array}{*{20}{c}} 1&{\sin \phi \tan \theta }&{\cos \phi \tan \theta } \\ 0&{\cos \phi }&{ - \sin \phi } \\ 0&{\frac{{\sin \phi }}{{\cos \theta }}}&{\frac{{\cos \phi }}{{\cos \theta }}} \end{array}} \right].$$

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1.1.2 Vehicle kinetics

This subsection presents the mathematical framework for describing the motion of a torpedo shaped AUV in a kinetic sense.

To summarize the equations presented in the generic sense:

$$M{\dot {V}_a}+{C_{{\text{RB}}}}\left( {{V_{\text{a}}}} \right){V_{\text{a}}}~+{C_{\text{A}}}\left( \mathcal{V} \right)\mathcal{V}+D\left( \mathcal{V} \right)\mathcal{V}+L\left( \mathcal{V} \right)\mathcal{V}+g\left( \eta \right)=Bu+w,$$

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$${V_{\text{a}}}=\mathcal{V}+{V_{\text{c}}},\quad {V_{\text{c}}}=R_{{\text{b}}}^{{\text{n}}}{\left( \varvec{\Theta} \right)^{\text{T}}}{v_{{\text{cn}}}},$$

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where, ${\dot {V}_{\text{a}}}$ is the vehicle acceleration vector, ${V_{\text{a}}}$ is the vehicle velocity vector, $\mathcal{V}$ is the vehicle water-referenced velocity vector, $\eta$ is the vehicle position vector, ${v_{{\text{cn}}}}$ is the water velocity vector NED frame, $u$ is the control vector, $M$ is the mass matrix, ${C_{{\text{RB}}}}\left( \nu \right)$ is the rigid body Coriolis force matrix, ${C_{\text{A}}}\left( {{\nu _{\text{r}}}} \right)$ is the added mass Coriolis force matrix, $D\left( {{\nu _{\text{r}}}} \right)$ is the drag matrix, $L\left( {{\nu _{\text{r}}}} \right)$ is the Lift matrix, $g\left( \eta \right)$ is the Buoyancy and gravitational force vector, $B$ is the actuation dynamic matrix, $w$ is the vector of un-modelled disturbances.

1.2 Spatiotemporal ocean environment

As mentioned in Sect. 3.2.1, it is possible to generate these currents field using analytic equations. Here a procedure is described to generate realistic ocean currents synthetically.

The ocean environment ${V_c}=\left( {{u_c},{v_c}} \right)$ is estimated by:

$${V_{\text{c}}}=f\left( {{{\mathbb{R}}^{\text{o}}},r,\zeta } \right),$$

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$${u_{\text{c}}}\left( {\mathbb{R}} \right)= - r\frac{{y - {y_0}}}{{2\pi {{\left( {{\mathbb{R}} - {{\mathbb{R}}^{\text{o}}}} \right)}^2}}}\left[ {1 - {e^{ - \left( {\frac{{{{\left( {{\mathbb{R}} - {{\mathbb{R}}^{\text{o}}}} \right)}^2}}}{{{\zeta ^2}}}} \right)}}} \right],$$

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$${v_{\text{c}}}\left( {\mathbb{R}} \right)=r\frac{{x - {x_0}}}{{2\pi {{\left( {{\mathbb{R}} - {{\mathbb{R}}^{\text{o}}}} \right)}^2}}}\left[ {1 - {e^{ - \left( {\frac{{{{\left( {{\mathbb{R}} - {{\mathbb{R}}^{\text{o}}}} \right)}^2}}}{{{\zeta ^2}}}} \right)}}} \right],$$

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where ${\mathbb{R}}=\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right]~$ represents the 2-D spatial domain, ${{\mathbb{R}}^{\text{o}}}=\left[ {\begin{array}{*{20}{c}} {{x_{\text{o}}}} \\ {{y_{\text{o}}}} \end{array}} \right]$ is the centre of the vortex and $r$, $\zeta$ are parameters that control the radius and strength of the vortex.

The continuous time varying ocean environment can be estimated by recursive application of Gaussian noise to the three parameters ${{\mathbb{R}}^{\text{o}}},\;r$ and $\zeta$. The random walk dynamic equations of these parameters are therefore:

$${\mathbb{R}}_{i}^{{\text{o}}}=A{\mathbb{R}}_{{i - 1}}^{{\text{o}}}+BX_{{i - 1}}^{{{\mathbb{R}}x}}+CX_{{i - 1}}^{{{\mathbb{R}}y}},$$

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$${r_i}=A{r_{i - 1}}+BX_{{i - 1}}^{r},$$

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$${\zeta _i}=A{\zeta _{i - 1}}+BX_{{i - 1}}^{\zeta },$$

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where $X_{{i - 1}}^{{{\mathbb{R}}x}}\sim N\left( {0,{\sigma _{{\mathbb{R}}x}}} \right)$, $X_{{i - 1}}^{{{\mathbb{R}}y}}\sim N\left( {0,{\sigma _{{\mathbb{R}}y}}} \right)$, $X_{{i - 1}}^{r}\sim N\left( {0,{\sigma _r}} \right)$ and $X_{{i - 1}}^{\zeta }\sim N\left( {0,{\sigma _\zeta }} \right)$ are Gaussian. The current field is changing at every ${\varvec{\Delta}_{{\varvec{t}}\_{\varvec{c}}}}$ value of time. The parameter metrics are given by:

$$A=\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right],\quad B=\left[ {\begin{array}{*{20}{c}} {{\Delta _{t\_c}}} \\ 0 \end{array}} \right],\quad C=\left[ {\begin{array}{*{20}{c}} 0 \\ {{\Delta _{t\_c}}} \end{array}} \right].$$

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