3.1. Biological inspired neurodynamics model

In the search process, search paths are determined by BINM. Firstly, a biological inspired neural network is built according to the underwater environment (as shown in Figure 3).

Figure 3.

Schematic diagram of BINM.

Firstly, an AUV two-dimensional (2-D) grid coordinate map of working environment is represented by a 2-D Biological Inspired Neural Network (BINN). There is one-to-one correspondence between each neuron in the neural network and the position of the grid map. Secondly, the path strategy of AUV is set on the basis of the distribution of neurons' active output value in the BINM. In Figure 3, the receptive field R is chosen as 2. The receptive field of the l-th neuron is represented by a circle with a radius of R. w _kl is the connection weights between neuron k and its neighbour l. Thus, it has lateral connections only to its eight neighbouring neurons within its receptive filed. The AUV may move to one of eight corresponding positions of the grid map next in Figure 3. The real next position of AUV motion is determined by the neurons' active output value of the eight neighbouring neurons within its receptive field. In this model, each individual neuron is connected with the adjacent ones to form a network for their transmission of activity. The dynamic characteristics of the neurons' activity value x _k can be denoted as (Ögmen and Gagné, Reference Ögmen and Gagné1990):

(1)$$\displaystyle{{d{x_k}} \over {dt}} = - A{x_k} + (B - {x_k})S_k^e (t) - (D + {x_k})S_k^i (t)$$

setting $S_k^e = {[{I_k}]^ +} + \sum\limits_{l = 1}^M {{w_{kl}}{{[{x_l}]}^ +}} $, S _kⁱ = [I _k]⁻, then Equation (1) can be converted to (Li et al., Reference Li, Yang and Seto2009):

(2)$$\eqalign{\displaystyle{{d{x_k}} \over {dt}} & = - A{x_k} + \left( {B - {x_k}} \right)\left( {{{[{I_k}]}^ +} + \sum\limits_{l = 1}^M {{w_{kl}}{{[{x_l}]}^ +}}} \right) \cr & \quad - \left( {D + {x_k}} \right){[{I_k}]^ -}} $$

where x _k represents the activity value of the k-th neuron, x _l represents the activity value of those connected to k, M the numbers of the neighbouring neurons, and I _k the external signals input in k. The external input I _k to the k-th neuron is defined as

(3)$${I_k} = \left\{ \matrix{E, & {if \, it \, is \, a \, target} \hfill \cr - E, & {if \, it \, is \, an \, obstacle} \hfill \cr 0, & otherwise \hfill } \right.$$

A, B and D are positive constants, -A reflects the passive decay rate of neuron k's activity, B and D are upper and lower limits of x _k, i.e. x _k ∈ [−D,B].

In Equation (2), w _kl is the connection weights between neuron k and its neighbour l, which can be defined as (Yang and Luo, Reference Yang and Luo2004):

(4)$${w_{kl}} = f\,(\left\vert {{q_k} - {q_l}} \right\vert) = \left\{ {\matrix{ {{\mu / {\left\vert {{q_k} - {q_l}} \right\vert,}}} & {0 \lt \left\vert {q_k - {q_l}} \right\vert \lt R} \cr {0,} & {\left\vert {{q_k} - {q_l}} \right\vert \ge R} \cr } } \right.$$

where |q _k − q _l| is the Euclidean distance between vector q _k and q _l on the state space, μ and R are both positive constants, and generally, 0 ≤ w _kl ≤ 1. As the connections between neurons are not directional, the connection weight coefficients are symmetrical, that is w _kl = w _lk.

From Equation (2), the neuronal excitatory input signal S _k^e includes [I _k]⁺ and $\sum\limits_{l = 1}^M {{w_{kl}}{{[{x_l}]}^ +}} $. That means one part of the excitation signals comes from external input while the other part comes from internal incremental gains of interconnecting neurons. If I _k ≤ 0, then $S_k^e = \sum\limits_{l = 1}^M {{w_{kl}}{{[{x_l}]}^ + }} $, which means there is not any external input for neuron k and all its excitation signals are transmitted through the neuronal network. In contrast, S _kⁱ = [I _k]⁻ means all the inhibitory input for signals k are external. Therefore, the excitation signals transmit between neurons, the value of positive activity of neurons in a neural network has a global effect while the inhibitory signals do not transmit and the negative activity of neurons has only a local effect.

In the search task, the AUV keeps moving toward the location with maximum neural activity in the search area. The strategies of AUVs' paths selection can be denoted as follows:

(5)$${P_n} \Leftarrow {x_{{P_n}}} = \max \{ {x_l},l = 1,2,...,M\} $$

where P _n represents AUVs' location at the next moment in the map.

Figure 4 shows the process of path selection. When an AUV makes a selection of its path, it compares the activity value of the neuron of its current location with its neighbours and chooses the one with the highest value as the next step. The target and obstacles remain at the peak and valley of the activity landscape of the neural network, respectively. Repeating this process, the AUV keeps moving towards the targets. Thus, the AUV should be able to search for the target efficiently with obstacle avoidance until the search task ends. By this way, the AUVs can realise cooperative searching efficiently.

Figure 4.

Process of path selection. (a) Process of path selection in the map (b) Process of path selection in the neural network

In the BINM-based path planning approach above, BINM requires no specimen learning information and parameter adjustment, thus is relatively simple to apply in practice. In addition, the BINM algorithm is able to plan collision-free search paths. It does not offset for the effects of ocean current so the next step is to apply the velocity synthesis algorithm to the optimal search path for each AUV in the ocean current environment.

3.2. Velocity synthesis algorithm

Taking into consideration the effect of ocean currents in a real underwater environment, AUVs may deviate from their search paths planned by BINM, resulting in a failure of the search task, particularly in cases where the current is opposite to AUV's moving direction (Soulignac, Reference Soulignac2011; Jan et al., Reference Jan, Chang and Parberry2008). To solve this problem, the VS algorithm, simple but effective, is included (Sahbani et al., Reference Sahbani, Sahar and Philippe2012; Yang et al., Reference Yang, Chao and Cheng-Kuo2011). Figure 5 shows how VS is applied to modify the search paths.

Figure 5.

Velocity synthesis algorithm.

In Figure 5, vector L represents the planned direction by BINM. The angle between L and the x-axis is ai. Vector V _c represents the ocean current, and vector V denotes the AUV's velocity whose magnitude is given and can be adjusted according to different requirements. ai2 represents the angle between V and the x-axis. The angle between V _c and the x-axis is ai1. The goal is to control the direction of vector V to make sure the resultant vector directly points to the desired direction. The implementation of the velocity synthesis algorithm is based on the precondition that |V _c| < |V|. Let |V _cn| = |V _c|cos(ai − ai1) be the ocean current component assisting motion along vector L and |V _cd| = |V _c|sin(ai − ai1) be the ocean current component that is perpendicular to the vector L. Similarly, we make |V _n| = |V|cos(ai2 − ai) the AUV's velocity component along the vector L and |V _d| = |V|sin(ai2 − ai) is the AUV's velocity perpendicular component. Staying on the planned direction requires V _d to cancel V _cd, which can be interpreted by the following expression (Zhu et al., Reference Zhu, Huan and Yang2013):

(6)$$\left\vert V\right\vert \sin (ai2 - ai) = \left\vert V_c\right\vert \sin (ai - ai1)$$

In accordance with vector inside accumulate theorem, another equation is given as follows:

(7)$$ai2 = \arcsin (\left\vert {{V_c}} \right\vert\sin (ai - ai1)/\left\vert V \right\vert) + ai$$

Summarising what we have discussed earlier and combining Equations (6) and (7), we can easily calculate ai2.

4.1. Static target in constant ocean current

The first case deals with a search for a static target in constant ocean current with fixed speed and direction. In this simulation, we set the speed of the ocean current to 0·2 and an angle of ai1 = 135°. According to the principle, the effect of ocean current is equal to neurons activity value increased by 0·2. Since BINM follows the principle of moving towards neurons with a higher activity value for search paths, when the ocean current occurs, AUVs will change their direction towards the target.

Figure 6(a) clearly shows this process. At the beginning of the search task, four AUV members all move with the ocean current at an angle of ai1 = 135°, after a while, R2 and R4 are more affected by neural activity of the target rather than the ocean current and thus gradually turn to the target. On the other hand, because R1 and R3 are much farther from the target than R2 and R4, the target could not overcome the effect with enough activity. Even when R2 has finally finished the search task, R1 and R3 are still fighting against the ocean current.

Figure 6.

Search for static target in constant ocean current. (a) Search process by BINM. (b) Search process by BINM & VS.

In Figure 6(b), however, the assistance of the VS algorithm leads to a quite different result. According to Equation (7), VS re-adjusts the AUV's direction by making the AUV's vertical velocity component equal to the currents. In this way, the current's effect is counteracted. As shown in Figure 6(b), four AUVs all move smoothly towards the target during the searching process before R4 finally finds it. Making a comparison between Figure 6(a) and 6(b), the total length of the search paths, either covered by each single AUV or the work team as a whole, are much shorter based on the improved BINM algorithm. It thus realises the goal of improving work efficiency and saving energy.

4.2. Static target in variable ocean current

Variable ocean current refers to current whose speed and direction will change. In this simulation, the variable current is first set a speed of 0·2 and an angle of ai1 = 45°, then 20 seconds later, it changes to a speed of 0·3 and an angle of ai1 = −45°. During the search process, the working principle is similar to that in constant ocean current. The only difference is that when there is a change in the ocean current speed and direction, the neurons that are affected by it will also correspond. In this case, it is initially the neuron whose θ = 45° is affected but then the one whose θ = −45°.

In Figure 7(a), it is clearly shown that AUV's search direction swings from ai1 = 45°to ai1 = −45° at Location 1 where the ocean current's direction changes. However, in Figure 7 (b), with the integration of BINM and VS, AUVs are rarely affected no matter how the current changes but move straight ahead for their target.

Figure 7.

Search for static target in variable ocean current. (a) Search process by BINM. (b) Search process by BINM & VS.

4.3. Mobile target in constant ocean current

In a real maritime environment, targets may not necessarily remain stationary. This will undoubtedly cause more difficulties for the search task if ordinary algorithms are adopted. But the biological inspired neural network proposed in this paper avoids this problem. This is because according to the pre-definition, the neural activity value of the target is set as the highest, and AUVs must keep tracking for locations of the highest activity value in the whole dynamic search process. Real-time information about the ever-changing locations of the mobile target can be obtained immediately by each AUV. BINM can constantly re-plan new search paths according to changes of the target's neural activity value and re-assign tasks for each multi-AUV team member no matter how the target swings around.

We set the speed of the constant ocean current to 0·3, the angle ai1 = −45°, and the target moves along a straight line southward at a speed of 0·1. The simulation results are presented in Figure 8. Comparing Figure 8(a) and 8(b), it can be found that due to the effect of ocean current, BINM takes much longer search paths than the proposed algorithm. Particularly, R2 appears confined in a very small search area and zigzags at random with repetitive coverage of the same area.

Figure 8.

Search for mobile target in constant ocean current. (a) Search process by BINM. (b) Search process by BINM & VS.

4.4. Comparison

In order to make a comparison between simulation cases in dynamic underwater environments with ocean currents, Table 2 lists the search path length for each AUV as well as for each multi-AUV team as a whole. By analysing these figures, we can make the following conclusions:

1. 1) In all three cases, BINM takes a much longer search path compared with BINM and VS, which proves to be inferior in time and energy saving.

2. 2) As to BINM, the total path lengthens from 100, 105, to 124 when searching for static target in constant current, static target in variable current, or mobile target in constant current. Also the increase rate widens from cases for static targets to mobile ones. This means that BINM is susceptible to the dynamic environment and the more complicated the ocean current is, the lower work efficiency it presents.

3. 3) By BINM and VS, the total path length varies little from 73, 70, to 77, which shows that velocity synthesis algorithm could effectively offset the effect of ocean current and is more adaptable to the dynamic underwater environment.

Table 2.

Path length for four cases by the two algorithms.

To better illustrate this, see Figure 9. It is easy to find that the proposed BINM and VS algorithm not only takes a shorter search path in all cases but also shows little distinction between them. Compared with BINM that varies greatly in amplitude, it turns out to be more stable.

Figure 9.

Comparison of total path length by the two algorithms.