Autonomous Underwater Vehicle Motion Planning in Realistic Ocean Environments Using Penalty Function-Particle Swarm Optimization Technique

Abstract

In this work, the penalty functions technique together with the particle swarm optimization (PSO) algorithm are employed to develop a motion planning algorithm for autonomous underwater vehicles under real environment conditions. The objective functions include the traveling time, consumed energy and a combination of these quantities. The optimization procedure is conducted based on the optimal control using the trigonometric swarm. The constraints are implemented through the penalty functions approach by introducing the velocity approximation strategy to reduce the optimization process runtime. The method is verified by solving the problem under investigation in a simulated environment and performing comparison studies with the results of the other related methods. After that, its robustness and efficiency for real conditions are demonstrated by performing the motion planning of AUVs under disturbed conditions. The effects of real currents of an area in Persian Gulf (PG) on the proposed motion planning algorithm are investigated. Finally, it is shown that the use of the trigonometric swarm and also the velocity approximation strategy improve the motion planning results. In addition, it is shown that the velocity approximation strategy decreases the process runtime by up to almost 30%, and the trigonometric swarms instead of common swarms (spline) improves the objective function minimization by up to almost 31%.

Appendices

Appendix A. Realistic Ocean Currents Values of Persian Gulf

To calculate system input at real ocean currents, it needs the current velocity and direction. The velocity directions of the area shown in Fig. 1 are presented in Fig. 

Fig. 20

The direction and magnitude of currents in the selected PG region during 12 h in the x–y–t space

20 in the x–y-t space. Because of the dynamic changes and discrete information, the direction of the currents is shown at each node in the x–y plane in Fig. 

Fig. 21

The direction of real currents during 12 h in the x–y plane

21.

Figure 

Fig. 22

The currents values at nodes since 5:00 am to 17:00 pm, (Blue line) $\dot{x}_c$, (red line) $\dot{y}_c$

22 shows the time histories of $\dot{x}_c$ and $\dot{y}_c$ of each node during 12 h (since 5:00 am to 17:00 pm) that considered for path planning. The values and directions are used as disturbance for path planning of AUVs. There are generally $6 \times 8 = 48$ nodes that the currents information is available. As mentioned in the main text, the three-dimensional interpolations are used to derive the information at the other nodes.

Appendix B. The Vehicle Parameters

The vehicle parameters defined in Eqs. (1)–(3) is shown in Table 9 (Prestero 2001). As mentioned before, the vehicle dynamics is important to calculate VOF and saturations of inputs as constraints at 3rd example.

Table 9 The definitions and values of the parameters of the sample AUV equations of motion (1)–(3) (Prestero 2001)

Appendix C. The Formula and Procedure for Trigonometric Swarm Generation as Initial Guess (X _i,j) for PFPSO Process

There is a combined transfer function for swarm generation. The swarm generation contains four steps. At first step, 71 elements are considered for basic swarm generation. Then, because $\left| {{\varvec{\eta }}_{\bf{i}} - {\varvec{\eta }}_{\bf{f}} } \right| =$ 30,079, a contract factor (a = 429.44) is used (dilation mapping). It makes swarm the same size as the length of the path. At 3^rd step, a rotation in conformal mapping occurs to set the direction of swarms between the initial and final points because the angle of direction is $\alpha = 0.3979\pi$ rad (rotation mapping). The initial point is ${\varvec{\eta }}_{\bf{i}} = \left[ { - 6700,7590} \right]$. At final step, there is a single transformation to set the initial and final points of swarm on ${\varvec{\eta }}_{\bf{i}}$ and ${\varvec{\eta }}_{\bf{f}}$, respectively. The conformal mapping process of swarm generation is shown in Fig. 23. The generated trigonometric swarm in the x–y plane are as follows

$$ y_{1,1} = 0 $$

(C.1)

$$ \left\{ {\begin{array}{*{20}l} {y_{1,n} = ( - 1)^n \times 2n\left( {\sin \left( {\frac{\pi k}{{25}} - \frac{9\pi }{{10}}} \right) + 1} \right)} \hfill & {k > 10,k < 61} \hfill \\ {y_{1,n} = 0} \hfill & {k \le 10,k \ge 61} \hfill \\ \end{array} } \right.\quad n = 2 - 5 $$

(C.2)

$$ \begin{gathered} y_{1,6} = ( - 1)^2 \times 8 \times \sin \left( {\frac{k\pi }{{71}}} \right), \hfill \\ y_{1,7} = ( - 1)^3 \times 10 \times \sin \left( {\frac{k\pi }{{71}}} \right), \hfill \\ y_{1,8} = ( - 1)^2 \times 8 \times \sin \left( {\frac{2k\pi }{{71}}} \right) \hfill \\ \end{gathered} $$

(C.3)

$$ \begin{gathered} y_{1,9} = ( - 1) \times 10 \times \sin \left( {\frac{2k\pi }{{71}}} \right), \hfill \\ y_{1,10} = ( - 1)^2 \times 8 \times \sin \left( {\frac{3k\pi }{{71}}} \right), \hfill \\ y_{1,11} = ( - 1) \times 10 \times \sin \left( {\frac{3k\pi }{{71}}} \right) \hfill \\ \end{gathered} $$

(C.4)

$$ x_{1,n} = 50\left[ {\sin \left( {\frac{(k - 1)\pi }{{70}} - \frac{\pi }{2}} \right) + 1} \right],k = 1,2, \ldots ,71,\ n = 1,2, \ldots ,11 $$

(C.5)

Dilation mapping:

$$ \begin{gathered} x_{2,n} = 429.44x_{1,n} , \hfill \\ y_{2,n} = 429.44y_{1,n} , \hfill \\ z_{2,n} = x_{2,n} + iy_{2,n} \hfill \\ \end{gathered} $$

(C.6)

Rotation mapping:

$$ z_{3,n} = e^{i\alpha } \times z_{2,n} ,\alpha = 0.3616\pi ; $$

(C.7)

Translation mapping

$$ \begin{aligned} z_n = &\, z_{3,n} + z_0 , \\ x_n + iy_n = &\, z_n , \\ z_0 = &\, - 6700 + 7590i \\ \end{aligned} $$

(C.8)

In addition, the time swarm is easily generated as:

$$ t_k = k - 1 \to \left\{ \begin{gathered} t_{1,k} = 200t_k \hfill \\ t_{2,k} = 400t_k \hfill \\ t_{3,k} = 600t_k \hfill \\ \end{gathered} \right.\quad k = 1,2, \ldots ,71 $$

(C.9)

The general swarms are a combination of time (t) and position (x–y), which are shown in Fig. 24. Each line in this figure represents one particle (X_i,j). The equivalent spline particles are also used for comparative study in cases 1–5 (Tables 4, 5, 6, 7, 8). It has to be mentioned that increase in particle numbers will improve the result (Pitrowski et al. 2020), anyhow, to increase computation speed, there are generally 33 particles considered (i_max $\times$ j_max = 33).

Fig. 23

The process of generating x–y swarm

Fig. 24

The generated swarms: a x–y swarm, b t swarm, c three-dimensional x–y–t swarm