A bio-inspired neural network in underwater environments is established (as shown in Fig. 2), in which a neuron represents a grid cell. The information of the grid cell is described by neural activities. Each neuron is connected to adjacent ones to form a network for their transmission of activity (for simplicity, only connections of the central neuron are shown, and those of other neurons are omitted).

The change rule of neuronal activity in the neural network is expressed as [23]:

(1) dukdt=−Auk+(B−uk)([Ik]++∑0<|kl|≤3wkl[ul]+)−(D+uk)[Ik]−

where u_k represents the activity value of the k-th neuron, u_l represents the activity value of other neurons connected to the k-th neuron, and I_k represents the grid cell signal’s input defined as [24]:

(2) Ik={1,ifitisatarget−1,ifitisanobstacle0,otherwise}

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 uk , i.e., uk∈[−D,B] . |kl| is the Euclidean distance between neuron k and its neighbor l on the 3D space,

(3) |kl|=(xk−xl)2+(yk−yl)2+(zk−zl)2

where (x_k, y_k, z_k) and (x_l, y_l, z_l) are the coordinates of the k-th and l-th neurons, respectively, in the 3D coordinate system.

In Eq. (1), w_kl is the connection weights between neuron k and its neighbor l, which can be defined as [25]:

(4) wkl=f(|kl|)={μ/μ|kl|,|kl|,0<|kl|<30,|kl|≥3

where μ is a positive constant, and generally, 0≤wkl≤1 . As the connections between neurons are not directional, the connection weight coefficients are symmetric, i.e., wkl=wlk .

It can be seen from Eq. (1) that the activity values of the target neurons are positive, and those of the obstacle neurons are negative. That means that some excitation signals come from external inputs, and others from internal incremental gains of interconnecting neurons. If Ik≤0 , then ∑0<|kl|≤3wkl[ul]+ , which means there is no external input for neuron k, and all its excitation signals are transmitted through the neuronal network. In contrast, [Ik]− means that all the inhibitory inputs for signals k are external. Therefore, the excitation signals are transmitted between neurons. The value of the 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.

We transform the search path planning problem of an AUV to one of finding its next navigation position. Only by accurately finding the position can an AUV quickly find unknown targets and avoid collisions with obstacles. In this regard, the next navigation position of an AUV must be determined in conjunction with the specific dynamic environment and the previous position and current position of the AUV. Hence, the research of AUV search path planning is transformed to the study of neuron activity output values in the neural network structure. The navigation position of an AUV at the next moment is determined by the distribution of neuron activity output values. In the target search mission, all of an AUVs movements are guided by the dynamic activity landscape of the neural network. The activity of each neuron is obtained by a shunting Eq. (1). The influence of the targets and obstacles, the state workspace varies according to the dynamics of the neural network. The motion of the AUV is determined by the landscape of the dynamic activity of the topologically organized neural network. An AUV's search path selection strategy can be written as [26]:

(5) Pn⇐uPn=max{ul,l=1,2,...,M}

where M represents the number of neurons adjacent to the k-th neuron, P_n represents the AUV’s location at the next moment in the map, and uPn is the highest activity value of the k-th neuron’s neighbors. When an AUV selects a path, it compares the activity value of the neuron of its current location with its neighbors and chooses the one with the largest value as the next step. Repeating this performance, the AUV keeps moving towards the targets.

Fuzzy obstacle-avoidance rules can provide different line speeds and angular velocities for a single AUV when facing obstacles. These rules can improve the smoothness of the AUV trajectory.

The structure of the fuzzy control module embedded in the control system is shown in Fig. 3. It has four main parts [27]:

(1) The fuzzification interface converts input to membership and compares it to rules in the rule library.

(2) The rule base contains rules based on common knowledge and experience.

(3) Decision logic evaluates the current situation, selects the appropriate fuzzy rules, and converts the fuzzy input to fuzzy output.

(4) The defuzzification interface converts the fuzzy output to a non-fuzzy instruction that the AUV recognizes.

The fuzzy controller has the characteristics of intelligence and real-time performance. The output Y(t) of the system is fed back to the fuzzy controller and compared to the reference input R(t) . The input U(t) is extracted by the fuzzy controller to meet the requirements of the underwater robot. The inputs and outputs of the fuzzy controller are shown in Fig. 4.

There are two fuzzy inputs, v and αo in the fuzzy controller, where v is the linear degree of freedom of the AUV, and αo is the angle of attack between the AUV and the obstacle. This design represents line velocity fuzzy terms as VS (velocity small), VM (velocity middle), and VL (velocity large). The angular fuzzy terms are expressed as ANS (angle negative small), ANL (angle negative large), APS (angle positive small), and APL (angle positive large). The membership functions of the above three fuzzy inputs are shown in Fig. 5 [28].

The AUV’s motion is adjusted by the output command of the fuzzy controller, which has two fuzzy outputs:, ωe is the desired curve speed of the AUV, and ve is its desired line speed. To effectively avoid obstacles, ωe has five fuzzy items, represented as NL (negative large), NS (negative small), Z (zero), PS (positive small), and PL (positive large). The fuzzy terms of ve are similarly represented as VS (velocity small), VM (velocity middle), and VL (velocity large), as shown in Fig. 6. For example, if the AUV is required to turn right, then ω would has the fuzzy term PS or PL, depending on the input [29].

The center of gravity algorithm is used for defuzzification, and the fuzzy output is converted to a motion command output to control the motion of the AUV. The fuzzy controller can effectively guide the AUV to avoid obstacles and improve the motion track [30].

The main task of the rule is to make the system have a reasonable output. The rules in Tab. 1 apply to all possible AUV obstacle avoidance scenarios. Each “IF” condition contains sub-conditions of αo and v , and all fuzzy sets basically come from experience and common knowledge.

To test the performance of the proposed algorithm, the search for static targets by multiple AUVs was simulated. The simulation consisted of three targets, three AUVs, and several obstacles in underwater environments. The targets had initial positions (41, 88, 76), (74, 72, 18), and (94, 58, 82), and three AUVs had initial positions (98, 34, 25), (32, 11, 97), and (38, 94, 27), as shown in Fig. 7. At the beginning of the search task, since the target affected the entire search area through neural transmission, the activity of each neuron could be derived from the shunt Eq. (1). When the AUV selected its path, it compared the activity value of the neuron at its current location to its neighbors and selected the neuron with the largest value as the next step. In the proposed algorithm, the target and obstacles respectively are the excitation and suppression of the neural network. Repeating the path selection, the AUV moved toward the target, and could bypass obstacles to avoid collisions. At the same time, through fuzzy obstacle-avoidance, the AUV search path became smoother, and the AUV made no sharp turns when the avoiding obstacles. As shown in Fig. 8, the static targets T1, T2, and T3 were found by AUVs R3, R1, and R2, respectively, which showed that the proposed algorithm could realize the joint search of multiple static targets.

Multiple AUVs searched for multiple dynamic targets in the second simulation. Dynamic targets undoubtedly bring more difficulties to the search task. But the proposed algorithm solved the problem. Due to pre-definition, the activity values of the neurons in the target were positive, and the target globally attracted AUVs through the dynamic neural activity landscape of the model. It can be known from Eq. (1) that when a target’s position changes, the activity values of the adjacent neurons of the AUV will change accordingly. By changing the activity values of neighboring neurons, each AUV can obtain real-time information on the change in the position of a moving target. The algorithm can continuously adjust the search path according to the evolution of adjacent neural activity values.

Figs. 9 and 10 show the process of three AUVs searching for three dynamic targets. A target moves randomly, and an AUV moves according to the proposed algorithm. Fig. 9 shows the initial positions of AUVs, targets, and obstacles. Fig. 10 shows the trajectory of the search targets, where the dotted and solid lines respectively indicate the moving track of the target and the moving trajectory of the AUV. The simulation results showed that the algorithm realized the joint search for multiple dynamic targets.

An AUV may experience mechanical failure when searching in real 3D underwater environments. In the case of partial AUV failure, whether the target search can be completed is an important indicator to measure the fault tolerance of the algorithm. We simulated a target search in the event of an AUV failure. In the simulation environments, four AUVs were to search for three dynamic targets. Initially, the four AUVs searched for targets. After a while, an AUV failed and other AUVs were still working, as shown in Fig. 11a. Although R1 failed, the team continued to search. After target T1 was detected, its activity value was set to 0. Since target T4 was not found, its activity value was 1. R2, R3, and R4 continued to search in its direction. As shown in Fig. 11b, R3 detected target T4. The AUVs found all the targets in the given space, and the search task ended. The simulation showed that the algorithm could complete a search task under the condition of AUV mechanical failure, the algorithm’s good fault-tolerant performance.

Compared to commonly used algorithms such as PSO, the proposed algorithm is expected to improve the efficiency of multi-AUV collaborative search. The PSO algorithm optimizes the solution by improving candidate solutions based on given iterations. The algorithm randomly generates particles, evaluates their fitness, and finds the individual and optimal global positions of the group.

The comparative simulation was of three AUVs searching for three targets in underwater environments with obstacles. The targets had initial positions (41, 88, 76), (78, 78, 6), and (80, 37, 58), and AUVs were randomly distributed, with positions (38, 94, 27), (58, 21, 12) and (93, 55, 83). Figs. 12a, 12b show the search process for two algorithms. Fig. 12 shows that the search path of the algorithm was shorter than that of the PSO algorithm for each AUV member. This is because the solution to the fitness function of PSO is only related to the distance between targets. In complex situations, the PSO algorithm does not adequately guide the AUV to avoid obstacles. The simulation results showed the proposed algorithm’s greater efficiency and adaptability compared to the PSO algorithm.

To compare the trajectory smoothness of the proposed algorithm and the PSO algorithm, Fig. 13 shows the trajectories of the two algorithms to avoid obstacles. Fig. 13a, 13b show the obstacle avoidance result of the proposed algorithm and PSO algorithm, respectively. The fuzzy obstacle avoidance of the proposed algorithm can provide better trajectory smoothness than the PSO control method. An AUV avoids sharp turns when it detects obstacles.