Dynamic Optimal Obstacle Avoidance Control of AUV Formation Based on MLoTFWA Algorithm

Abstract

In addressing the optimal formation obstacle avoidance control problem for Autonomous Underwater Vehicles (AUVs) in environments with unknown and moving obstacles, this paper employs the Modified Fireworks Algorithm based on a Loser Elimination Mechanism (MLoTFWA) and constructs a Distributed Model Predictive Control (DMPC) framework to achieve obstacle avoidance for AUV formations. Initially, a prediction model is established, followed by feedback compensation to mitigate the effects of unknown perturbations. An appropriate fitness function is then formulated, and enhancements such as the loser elimination rule are introduced to optimize the fireworks algorithm. Additionally, the concept of an adaptive DMPC prediction window is proposed to conserve resources. The local and global stability of the DMPC formation control framework is theoretically proven. Simulations verify that the control system based on the DMPC framework ensures safe obstacle avoidance for the formation, maintains formation consistency, and achieves the shortest and smoothest path. The improved fireworks algorithm demonstrates superior performance compared with the original fireworks algorithm and other optimization algorithms. In testing, the improved fireworks algorithm exhibits better adaptability, higher average fitness, and best fitness, along with a significantly faster convergence speed. Compared with the ordinary fireworks algorithm, the convergence speed is reduced by 30%.

1. Introduction

Over the past few decades, significant advancements in science and technology have propelled the field of Autonomous Underwater Vehicles (AUVs) both domestically and internationally. These advancements have yielded substantial positive outcomes across various marine activities.

Autonomous Underwater Vehicles (AUVs) have a wide range of applications in both military and civilian fields. In the military domain, these applications include anti-submarine warfare, mine warfare, intelligence reconnaissance, patrol and surveillance, logistical support, terrain mapping, and underwater construction [1,2]. In the civilian sector, AUVs are primarily used for marine environmental surveys, exploration of seabed mineral and biological resources, pipeline inspection, seawater quality inspection, marine geological exploration, and submarine cable detection [3,4]. China and Russia jointly developed the CR-01 and CR-02 (6000 m) AUVs, capable of performing acoustic, optical, and hydrographic surveying tasks in polymetallic nodule mining areas with flat topography on the deep seafloor. In 2011, the Monterey Bay Oceanographic Institute (MBOI) in the United States deployed the “Dorado deep-sea AUV”, which has a minimum navigational altitude of 50 m above the seafloor and can accurately map the seafloor topography around craters [5,6].

Given the critical importance of Autonomous Underwater Vehicles (AUVs) in both civil and military domains, significant advancements have been made in AUV technology over the past few decades. These advancements have led to the development of multi-autonomous underwater vehicle (MAUV) formation systems tailored to meet specific mission requirements.

In real marine environments, multi-AUV formations inevitably encounter various dynamic obstacles [7,8,9]. The process of avoiding these obstacles is referred to as obstacle avoidance. Additionally, collision avoidance is necessary to prevent collisions between AUV formation members. Online methods for collision and obstacle avoidance are primarily categorized into three types: the artificial potential field method, reactive methods, and optimization-based methods.

(1)

Artificial Potential Field Method: This method achieves obstacle avoidance by applying forces and moments to the AUV controller through the generation of virtual gravitational and repulsive forces in regions such as target points and obstacles. However, the artificial potential field method has several limitations. For instance, it can lead to deadlock situations when the gravitational and repulsive forces are equal, causing the AUV to halt at a specific point without reaching its destination. Additionally, in environments with dense obstacles, the superposition of multiple gravitational and repulsive forces can cause the AUV to deviate from its intended target. Furthermore, in narrow and elongated channels, this method can induce oscillations.

(2)

Reactive methods: These include the velocity obstacle (VO) method [10] and the buffered Voronoi cell method. The velocity obstacle method ensures that AUVs do not collide by calculating a set of expected velocity directions [11]. The buffered Voronoi cell method ensures that cells do not overlap and that obstacles do not enter the cells by calculating Voronoi cells for each AUV [12]. While these methods are practical and efficient, they do not account for system constraints and optimality, potentially leading to resource wastage.

(3)

Optimization-based methods: These include mixed integer linear programming (MILP) [13], sequential convex programming (SCP) [14], and model predictive control (MPC) [15,16,17]. Among these, MPC stands out for its ability to explicitly handle constraints, predict, and correct future trajectories through rolling optimization and feedback correction. Compared with MILP and SCP, MPC is highly resistant to interference and offers greater flexibility. Mingxing Qin and his team [15] have proposed a novel distributed model predictive control (DMPC) algorithm, which achieves distributed synchronous execution by introducing assumed trajectories and low-conservative compatibility constraints. Chao Lun Zhao et al. [16] employed the pilot-follower method combined with distributed model predictive control technology to address the formation and maintenance challenges of multi-quadrotor UAV formations. Zhang [17] optimized MPC using a differential evolution algorithm, introducing adaptive adjustments to the prediction range of the model predictive controller and analyzing the asymptotic convergence of rolling optimization. Liu [18] proposed an MPC method based on Gaussian processes, which combines the strengths of Gaussian process regression and MPC. Song, Y. [19] developed an MPC method based on radial basis function neural networks (RBF-MPC), which compensates for model uncertainty through the development of an RBF neural network approximator and the real-time construction of a feedback state training dataset to achieve obstacle avoidance.Despite its advantages, MPC has the drawback of requiring substantial computational resources and imposing high communication demands, which remains a critical challenge to be addressed in MPC control [20].

The Fireworks Algorithm (FWA) [21,22,23] is a global optimization algorithm proposed by Tan and Zhu [21] in 2010. It performs a global search by mimicking the phenomenon of fireworks exploding. Due to its unique explosive search method, this algorithm has garnered extensive research interest since its inception and has been widely applied to practical optimization problems. Li [24] proposed a new information interaction mechanism to enable parallel computation of the fireworks algorithm on modern hardware, addressing its difficulty in parallel computation. Yu [25] designed a novel Tent-Levy FWA based on a discrete update process by introducing integer coding, Tent chaos mapping, and Levy variation. Experimental results indicate that the average cost of TLFWA is 8.17% and 13.73% lower than that of FWA and PSO, respectively. Fan [26] utilized orthogonal arrays to control the distribution of sparks and employed reinforcement learning to calculate algorithm parameters. Zhang [27] introduced the explosion operation of the fireworks algorithm when the artificial fish swarm algorithm fell into regional convergence, enhancing variability and thereby improving the optimization speed and capability of the algorithm. Wu G. [28] proposed a CCPP method based on an improved genetic algorithm, incorporating a stretching fitness function, adaptive mutation operator, and crossover operator. This method combined key operators from the fireworks algorithm and optimized turning and obstacle avoidance during the operation of unmanned boats. Specific methods are shown in the

Table 1. Formation obstacle avoidance method.

Method Category	Representative Methods	Advantages	Disadvantages
Artificial Potential Field Method	Artificial Potential Field Method	Achieves obstacle avoidance by applying virtual gravitational and repulsive forces	Can lead to deadlock; may cause deviation from the target in dense obstacle environments; prone to oscillations in narrow channels
Reactive Methods	Velocity Obstacle (VO) Method, Buffered Voronoi Cell Method	Practical and efficient, ensures no collisions	Does not account for system constraints and optimality, may result in resource wastage
Optimization-Based Methods	Mixed Integer Linear Programming (MILP), Sequential Convex Programming (SCP), Model Predictive Control (MPC)	MPC can explicitly handle constraints, highly resistant to interference, flexible	Requires substantial computational resources, high communication demands

and

Table 2. “Fireworks Algorithms and their Improvements” category.

Method Class	Representative Method	Key Improvement Point	Advantages	Disdvantages
Fireworks Algorithm (FWA)	Tan and Zhu [21]	-	The unique explosive search method is widely used in practical optimization problems	It is difficult to compute in parallel on modern hardware
Improved FWA	Parallel Computation [24]	New information interaction mechanism	Enhances parallel computation capability	Increased complexity
Improved FWA	Tent-Levy FWA [25]	Introduction of integer coding, Tent chaos mapping, and Levy variation	Reduces computational cost, significant optimization effect	Requires complex parameter adjustments
Improved FWA	Orthogonal Array and Reinforcement Learning [26]	Utilization of orthogonal arrays to control spark distribution, reinforcement learning for parameter calculation	Improves optimization speed and accuracy	Complex implementation
Improved FWA	Artificial Fish Swarm Algorithm with FWA [27]	Introduction of explosion operation in artificial fish swarm algorithm to enhance variability	Enhances optimization speed and capability	Limited applicability
Improved FWA	Improved Genetic Algorithm with FWA [28]	Incorporation of stretching fitness function, adaptive mutation operator, and crossover operator	Optimizes turning and obstacle avoidance in unmanned boat operations	High algorithm complexity

:

Building on the aforementioned research results, this paper proposes a dynamic optimal obstacle avoidance control for AUV formations based on the MLoTFWA algorithm. In this control method, the Fireworks Algorithm (FWA) is employed to address the dynamic optimization problem while considering the influence of the sliding window on DMPC control. The specific contributions of this paper are summarized as follows:

(1)

The explosion operation of the fireworks algorithm incorporates Levy flight search, which focuses on the area near the explosion site. Additionally, an adaptive Gaussian variation constant is proposed to maintain the search proximity to the existing solution.

(2)

The loser elimination operation is incorporated into the fireworks algorithm to facilitate information exchange between fireworks. This mechanism allows locally optimal fireworks to escape from suboptimal regions, thereby mitigating initialization issues.

(3)

A DMPC control method based on an adaptive prediction window is proposed, wherein adaptive values are defined to balance control performance and computational efficiency.

2. Description of the Problem

This paper addresses the problem of moving an AUV formation in an unknown dynamic environment from one point to another with minimum cost while maintaining the formation’s formation and without collision. For each AUV in the formation, the objectives are shown in the following equation:

The conditions of formation maintenance are

$$li{m}_{t\to \infty }∥p_j\left(t\right)-p_i\left(t\right)∥=d_{ij,\mathrm{ref}},\forall j\in F_i.$$

(1)

The collision avoidance conditions between formations are

$$li{m}_{t\to \infty }∥p_j\left(t\right)-p_i\left(t\right)∥>d_{\mathrm{sp}},\forall j\in F_i.$$

(2)

Formation obstacle avoidance conditions are

$$li{m}_{t\to \infty }∥p_o\left(t\right)-p_i\left(t\right)∥>d_{\mathrm{so}},\forall o\in O_i.$$

(3)

The optimal objective of the formation is

$$\mathrm{Minimize}J\left(x_i\right)\in \mathbb{R}$$

(4)

where $p\left(t\right)$ denotes the position information of AUVs. $d_{ij,\mathrm{ref}}$ is the expected relative distance between AUVs j and i in the formation, $d_{sp}$ denotes the minimum distance between AUVs, and $d_{so}$ denotes the minimum distance between AUVs and obstacles. $F_i$ denotes the set of formations, and $O_i$ denotes the set of obstacles. $J\left(x_i\right)$ is the objective optimization function, including the costs associated with formation maintenance, speed consistency, obstacle avoidance, path smoothness, and energy consumption. Due to the complexity of the unknown dynamic environment, this paper does not impose consistency constraints on the speed in the problem description but makes the formation speed converge as much as possible in the optimization function.

It is assumed that the communication situation is completely ideal without any time delay. Each AUV has exactly the same parameters and structure, and the obstacle is simplified to a sphere to save computation. The position information of the obstacle is unknown outside the detection domain of the AUVs, and the absolute velocity is 0–0.3 m/s. There exists an unknown current with a velocity range of 0.3–0.6 m/s and an acceleration of no more than 0.1 m/s², and a rate of change of direction of no more than 5°/s.

In order to simplify the processing of the sonar model, assume that the radius of sonar detection is $R_{\mathrm{v}}=100$ m, the current coordinates of the AUV are $\left(x_0,y_0,z_0\right)$, the co-ordinates of the obstacle are $\left(x_1,y_1,z_1\right)$, and set the relative position of the two $\left(x_m,y_m,z_m\right)=\left(x_1,y_1,z_1\right)-\left(x_0,y_0,z_0\right)$. The angles that can be detected in the $XOY$ plane and the $XOZ$ plane are $\alpha$ and $\beta$, respectively, and the conditions set in this paper for the obstacle to be detected by the AUV are

$$\left\{\begin{array}{c}{\displaystyle \frac{\left|y_m\right|}{\sqrt{x_m^2+y_m^2}}}\le sin{\displaystyle \frac{\alpha }{2}}\hfill \\ \sqrt{x_m^2+y_m^2+z_m^2}\le R_v.\hfill \\ {\displaystyle \frac{\left|z_m\right|}{\sqrt{x_m^2+y_m^2}}}\le sin{\displaystyle \frac{\beta }{2}}\hfill \end{array}\right.$$

(5)

The sonar detection model of the AUV is shown in Figure 1. In order to better describe the problem, the detection region of AUV is simplified. The detection area of the AUV is a cone with itself as the apex, and the distance from the apex to the bottom edge along the side of the cone is 100 m. Obstacles located within the cone can be detected by all of the AUVs. In order to save the communication cost of the formation, the positions of the detected obstacles are not shared among all AUVs in the formation.

Once an obstacle is detected, the area is called the caution area with a radius of $R_{warning}+R_l$, where $R_l$ is the radius of the obstacle. In addition, AUVs are required to maintain a certain distance from the obstacle, and a certain-sized area near the obstacle is a hazardous area that AUVs are not allowed to enter, called $R_{dangerous}+R_l$. Other AUVs in the team also belong to a kind of obstacle, but the radius of their warning area is ${R_{warning}}^{\prime }$, and the radius of their danger area is ${R_{dangerous}}^{\prime }$, which is different from ordinary obstacles. A diagram of the warning and danger zones is shown in Figure 2.

3. Adaptive DMPC Formation Control Based on Improved Fireworks Algorithm

The marine environment is complex and variable, and there are more moving and unknown obstacles in the ocean. If the mission is carried out in the unknown sea, the general offline path planning algorithms can no longer adapt to the needs in that kind of environment, so the AUV formation needs to be online for collision avoidance and obstacle avoidance.

In order to balance the optimality of formation with the safety of collision and obstacle avoidance, as well as to explicitly deal with the constraints of AUV formation, Model Predictive Control [29] is widely used, which predicts the future behavior of the system and optimizes the current control inputs in order to achieve the optimal control effect.

The advantages of MPC control include the following [30,31]:

• Ability to handle multi-variable, non-linear, and constrained systems: MPC can effectively handle systems with multiple inputs and outputs, non-linear dynamics, and various constraints. By considering the constraints of the system, it is possible to ensure that the control inputs are within the permissible range and satisfy the requirements of the system.
• Consideration of future information: by predicting future system behavior, MPC is able to better cope with the dynamic characteristics of the system and adopt appropriate control strategies in advance. This makes MPC robust to dealing with time-varying characteristics, uncertainties, and perturbations of the system.
• Performance indicators can be optimized: MPC can be optimized for specific performance indicators, such as minimizing error and maximizing efficiency. With the online optimization algorithm, the MPC is able to calculate the optimal control inputs in each control cycle to achieve the desired control effect.
• Easy to integrate constraints and restrictions: MPC can easily integrate various constraints and restrictions, such as physical constraints, safety constraints, etc. This makes MPC highly adaptable in dealing with complex systems and practical engineering problems.

MPC control is an online optimization strategy based on prediction windows, and its basic framework is shown in Figure 3. At each sampling time, the MPC method optimizes the control problem for the following P-step sampling time and applies only the first element of the optimal control sequence to the prediction model, where the control domain is m. The prediction output $y_{mi}$ is then obtained. The modified prediction output $y_{pi}$ is determined by feedback compensation. The control input $u_i$ is generated by backward optimization of the optimization index $J_i$. The system output $y_i$ is then obtained. The whole process will be iterated online to achieve distributed MPC control for each AUV. Since the control structure in this MPC control mode is distributed, it is called the DMPC (Distributed Model Predictive Control) control algorithm.

3.1. Predictive Modelling

In Figure 4, $E-\xi \eta \zeta$ is the north–east coordinate system, i.e., the fixed coordinate system, and $O-xyz$ is the hull coordinate system, i.e., the inertial coordinate system.

The AUV’s executive body is as follows: main thrusters are arranged at the transom of the AUV and are responsible for the longitudinal motion control of the AUV; the auxiliary thrusters are arranged symmetrically at the two sides and the top of the AUV and are responsible for the transverse and pendulous motion control of the AUV; the vertical rudder is responsible for the bow-turning motion control of the AUV; and the horizontal rudder is responsible for the longitudinal pitching motion control of the AUV. The space motion of this AUV has the characteristic of full drive, and its space motion coordinate system is shown in Figure 4. The mathematical model established under the hull coordinate system:

$$\left\{\begin{array}{c}\dot{\eta }=J\left(\eta \right)\nu \hfill \\ M\dot{\nu }=g^{\prime }\tau -C\left(\nu \right)\nu -D\left(\nu \right)\nu -g\left(\eta \right)\hfill \end{array}\right.$$

(6)

where: $\eta ={(x,y,z,\theta ,\psi )}^T\in R^5$ denotes the position vector of the AUV in the fixed coordinate system; the velocity vector of the AUV in the hull coordinate system is: $v={(u,v,w,q,r)}^T\in R^5$; M is the inertia matrix quantity; $J\left(\eta \right)$ is the transformation matrix; $C\left(v\right)$ is the Coe’s force and centripetal force matrix; $D\left(v\right)$ is the lifting moment and hydrodynamic drag; $g\left(\eta \right)$ is the restoring force and torque vector; $\tau ={({\tau }_u,{\tau }_v,{\tau }_w,{\tau }_q,{\tau }_r)}^T\in R^5$ is the input vector of the AUV actuator; and $g^{\prime }\in R^{5\times 5}$ is the actuator’s parameter matrix. The higher-order damping terms and the effect of transverse hull rocking on the AUV motion are not considered, where the detailed mathematical model of the AUV kinematics is established as follows:

Kinematic Modeling:

$$\left\{\begin{array}{c}\dot{x}=ucos\psi cos\theta -vsin\psi +wcos\psi sin\theta \hfill \\ \dot{y}=usin\psi cos\theta +vcos\psi +wsin\psi sin\theta \hfill \\ \dot{z}=-usin\theta +wcos\theta \hfill \\ \dot{\theta }=q\hfill \\ \dot{\psi }=r/cos\theta .\hfill \end{array}\right.$$

(7)

Predictive model: At each discrete moment, the predictive control inputs $x_i\left(k\right)\in [x_{min},x_{max}]$ are taken as variables, and the kinematic model of the system is used to recursively predict the future state of the system $y_{mi}\left(k\right)\in [y_{min},y_{max}]$, and then the optimal control problem is solved to obtain the optimal control input sequence.

Let the sampling time length period be T. Define the velocity state quantity of the ith member at the moment of k to be $x_i\left(k\right)\in [x_{min},x_{max}]$, the input quantity to be $u_i\left(k\right)\in [u_{min},u_{max}]$, and $z_{1i}\left(k\right),u_i\left(k\right)\in R^5$, then its output is the bit position state quantity to satisfy $y_{mi}\left(k\right)\in [y_{min},y_{max}]$. At this time, the MPC control prediction model is shown in the following equation:

$$\left\{\begin{array}{c}{x}_i(k+1)=x_i\left(k\right)+\underset{kT}{\overset{(k+1)T}{\int }}f(x_i\left(t\right),u_i\left(k\right))dt\hfill \\ y_{mi}\left(k\right)=T\ast \left[C{x}_i\left(k\right)\right].\hfill \end{array}\right.$$

(8)

The matrix of Equation (8) is the state transition coefficient matrix. From Equation (7), the following can be deduced:

$$C=\left[\begin{array}{ccccc}cos\psi cos\theta & -sin\psi & cos\psi sin\theta & 0& 0\\ sin\psi cos\theta & cos\psi & sin\psi sin\theta & 0& 0\\ -sin\theta & 0& cos\theta & 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1/cos\theta \end{array}\right]$$

(9)

f is the kinetic function to transform the input and output.

3.2. Feedback Compensation

In the P step-window online optimization of MPC, the predicted control sequences are computed by $U_i\left(k|k\right)$ and $U_i\left(k+1|k\right)$, …, $U_i\left(k+P-1|k\right)$, and the predicted outputs are obtained by $y_{mi}(k+1\mid k)$, $y_{mi}(k+2\mid k)$, … $y_{mi}(k+P\mid k)$. However, due to possible external disturbances, the predicted output $y_{mi}(k+P\mid k)$ is difficult to match with the actual output $y_i(k+j)$. Therefore, the predicted output should be based on the error between the actual output value and the predicted output value, which should be further revised as follows with reference to the literature [32]:

$$y_{pi}(k+j|k)=y_{mi}(k+j|k)+k_{1}H{\dot{e}}_i\left(k\right)+k_{2}H{\left|s_i\left(k\right)\right|}^{1-\alpha }\mathrm{sgn}\left(s_i\left(k\right)\right)+k_{3}Hswi\left(s_i\left(k\right)\right)$$

(10)

where $H=I_{3\times 3}$ is the error coefficient matrix and $e_i\left(k\right)$ is defined as

$$e_i\left(k\right)=y_i\left(k\right)-y_{mi}\left(k\right|k-1)$$

(11)

$s_i\left(k\right)$ is defined as

$$s_i\left(k\right)=k_1{e}_i\left(k\right)+k_2{\dot{e}}_i\left(k\right).$$

(12)

Among them, the parameters and functions such as $k_1,k_2,k_3$, and $swi\left(s_i\left(k\right)\right)$ are the same as those in the literature [6], and the stability has been proved before, so we will not repeat it here.

3.3. Optimization Functions

The optimization function is the core part of MPC. Only after defining the optimization function can the relative optimal solution be found through the continuous interaction between the intelligent body and the environment, trial and error, and according to the optimization strategy.

In this paper, for the ith AUV in the formation at the moment k, the optimization function $J_i\left(k\right)$ is given by

$$min{J}_i\left(k\right)={\lambda }_1{J}_i^1\left(k\right)+{\lambda }_2{J}_i^2\left(k\right)+{\lambda }_3{J}_i^3\left(k\right)+{\lambda }_4{J}_i^4\left(k\right)+{\lambda }_5{J}_i^5\left(k\right).$$

(13)

$J_i^1\left(k\right)$, $J_i^2\left(k\right)$, $J_i^3\left(k\right)$, $J_i^4\left(k\right)$, $J_i^5\left(k\right)$, denote the formation formation keeping cost, speed consistency cost, obstacle avoidance cost, path smoothness cost, and energy cost, respectively. ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, and ${\lambda }_5$ are weighting factors that satisfy ${\lambda }_1+{\lambda }_2+{\lambda }_3+{\lambda }_4+{\lambda }_5=1$.

The formation formation cost function $J_i^1\left(k\right)$ is defined by the following equation:

$$J_i^1\left(k\right)=\sum _{j=1}^P{∥{\mathbf{y}}_{pi}(k+j|k)-y_{di}(k+j)∥}_P^2$$

(14)

and satisfies $y_{min}\le y_{pi}(k+j|k)\le y_{max}$, where P is the size of the time prediction range and ${∥·∥}_P^2$ is the square 2-parameter weighted by the positive definite matrix P. If the ith AUV is the leader, $y_{di}\left(k\right)$ denotes the given predetermined trajectory or end position. Conversely, if the i th AUV is a follower, $y_{di}\left(k\right)$ denotes the desired position computed by the follower based on the pilot’s predicted position information $y_{dl}(k+j|k)$ with formation constraints. The speed consistency cost function $J_i^2\left(k\right)$ is defined by the following equation:

$$J_i^2\left(k\right)=\sum _{j=1}^P{∥x_i(k+j|k)-x_l(k+j|k)∥}_Q^2$$

(15)

and satisfies $x_{min}\le x_{pi}(k+j|k)\le x_{max}$, where ${∥·∥}_Q^2$ is the square 2-paradigm weighted by the positive definite matrix Q.

$x_i(k+j|k)$ indicates the speed information of the ith AUV in the formation at the time of k predicted at the time of $k+j$, and $x_l(k+j|k)$ indicates the speed information of the pilot in the formation at the time of k predicted at the time of $k+j$.

The formation collision avoidance and obstacle avoidance cost function $J_i^3\left(k\right)$ is defined by the following equation:

$$\begin{array}{cc}\hfill J^3\left(k\right)& =\sum _{l=1}^O\sum _{j=1}^p{\kappa }_l(k+j\mid k),{\kappa }_l(k+j\mid k)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{∥{\mathbf{y}}_{pl}(k+j\mid k)-{\eta }_l∥}_{0_{2_1}}^2+\epsilon }},\hfill & ∥{\mathbf{y}}_{pl}\left(k+j\mid k\right)-{\eta }_l∥<R_{dangsronoo}\hfill \\ {\displaystyle \frac{1}{{∥{\mathbf{y}}_{pl}(k+j\mid k)-{\eta }_l∥}_{0_{2_2}}^2+\epsilon }},\hfill & R_{dangsrono}+R_l<∥{\mathbf{y}}_{pl}\left(k+j\mid k\right)-{\eta }_l∥<R_{dannog}+R_l\hfill \\ 0,\hfill & ∥{\mathbf{y}}_{pl}\left(k+j\mid k\right)-{\eta }_l∥\ge R_{dannog}+R_l\hfill \end{array}\right.\hfill \end{array}$$

(16)

where O is the summed number of obstacles and other AUVs in the detection area, ${\eta }_l=(x_l,y_l,z_l)$, is the position of the obstacle or AUV center of mass. $R_{obstacle}$ is the obstacle or AUV radius; G1 and G2 are positive definite matrices. representing the fitness coefficients in the hazardous and alert regions, respectively, and $\epsilon$ is a very small constant to prevent the denominator of ${\kappa }_l(k+j|k)$ from being zero.

According to Figure 1, obstacle avoidance behavior is closely related to the measured distance of the detected obstacle. Specifically, the obstacle avoidance cost will be taken into account when the AUV enters the warning area. Based on the measured distance of obstacles, the obstacle avoidance cost of each AUV at each time point k is calculated to decide the obstacle avoidance action. And when entering the danger zone, the obstacle avoidance cost function will become extra high to try to avoid the AUV entering the danger zone.

The path smoothness cost function $J_i^4\left(k\right)$ is defined by the following equation:

$$J_i^4\left(k\right)={\displaystyle \frac{1}{P}}\sum _{j=1}^P\left(\right|{\psi }_i(k+j|k)-{\psi }_i(k+j-1|k)|+|{\gamma }_i(k+j|k)-{\gamma }_i(k+j-1|k)\left|\right)$$

(17)

and satisfies ${\psi }_{min}\le {\psi }_i(k+j|k)\le {\psi }_{max},{\gamma }_{min}\le {\gamma }_i(k+j|k)\le {\gamma }_{max}$.

$J_i^4\left(k\right)$ represents the average change in path angle, reflecting the smoothness of the AUV navigation path. This introduction penalizes large corners during obstacle avoidance and helps to keep the path smooth.

The energy cost function $J_i^5\left(k\right)$ is defined by the following equation:

$$J_i^5\left(k\right)=\sum _{i=1}^m{\Vert u_i(k+j-1\mid k)\Vert }_R^2$$

(18)

and satisfy $u_{min}\le u_i(k+j|k)\le u_{max}$ as the control domain. The smaller the input force and moment, the lower the energy cost.

3.4. Scrolling Optimization

Since the optimization function is extremely complex and non-trivial, it is difficult to find the optimal value through traditional optimization methods or purely mathematical methods such as gradient search, Hessians search, and linear search to determine the next sampling point based on the characteristics of the problem. Hidden optimization using population intelligence algorithms is a simple and effective method. The DMPC algorithm itself is computationally intensive, so it needs to reduce the amount of computation as much as possible and strike a good balance between the quality of optimization and real-time performance.

The fireworks algorithm has the advantages of high diversity, both global and local search ability, being easy to implement and adjust, and a strong effect of low-dimensional operations, which makes it very suitable as an algorithm for hidden optimization of optimization functions. Therefore, in this paper, the fireworks algorithm is chosen to improve it in order to carry out an optimization algorithm for DMPC control.

DMPC control requires high dynamic performance of the algorithm due to the large amount of computation caused by the computational approach. In 2018, Tan [33], the founder of Fireworks Algorithm, proposed a fireworks algorithm based on loser-elimination mechanism (LoTFWA), introducing the concept of predictive fitness. If the predicted fitness is worse than the best fireworks in the current generation, the fireworks will be reinitialized to save the algorithm running time and avoid searching in inefficient places. However, the LoTFWA algorithm still has drawbacks, i.e., the communication between fireworks is not sufficient and the cost of reinitialization is too high. In this paper, its loser elimination mechanism is improved and combined with the improved fireworks algorithm to design the following improved fireworks algorithm based on the loser elimination mechanism (MLoTFWA).

Let the MLoTFWA algorithm optimize the following objective function for the current AUV at k time:

$$\mathrm{Minimize} J\left(u_i^k\right)\in \mathbb{R}, u_{\min }⩽u_i^k⩽u_{\max }.$$

(19)

In the above equation, the target $J\left(u_i^k\right)$ is Equation (13). In the above equation, the target $u_i^k={[u_i\left(k\right|k),u_i(k+1|k),\dots ,u_i(k+P-1|k)]}^T$ is the set vector of inputs in all windows, P is the current prediction window size of the DMPC algorithm, and $u_{min}$ and $u_{max}$ refer to the upper and lower bounds of the inputs, respectively.

The specific steps of the MLoTFWA algorithm are as follows:

• Initialisation

First, establish locations to lay the Firework, and for each $u_i^k,i=1,2,\dots ,n$, make sure $u_{min}⩽u_i^k⩽u_{max}$ is in place.

2.

Explosive operations

After the initialization operation is performed, the explosion operation is performed. The explosion operation is designed to change the position of the sparks to generate Sparks (sub-sparks), which in turn are searched in the neighborhood of the sparks, and is divided into two elements: explosion intensity and explosion magnitude. Each Firework generates a different number of Sparks according to the explosion intensity and searches in different neighborhoods according to the explosion amplitude.

Explosion intensity is the number of Sparks generated per Firework. A well-adapted Firework should narrow the explosion and search carefully, while a poorly adapted Firework should expand the search. The explosion intensity formula is defined as follows:

$$S_i=m{\displaystyle \frac{y_{max}-J\left(u_i^k\right)+\epsilon }{{\displaystyle \sum _{i=1}^N}(y_{max}-J(u_i^k\left)\right)+\epsilon }}$$

(20)

where $S_i$ refers to the number of Sparks generated by the ith Firework, refers to the total number of Sparks in each generation, and defines $y_{max}=maxJ\left(u_i^k\right),i=1,2,\dots ,n,y_{min}=minJ\left(u_i^k\right),i=1,2,\dots ,n$. $\epsilon$ is a very small constant to avoid the denominator being zero. In order to make the number of Sparks generated by each Firework explosion an integer, define it as in the following equation:

$${\widehat{s}}_i=\left\{\begin{array}{cc}round(a^{\prime }·m)\hfill & \mathrm{if} s_i<a^{\prime }m\hfill \\ round(b^{\prime }·m)\hfill & \mathrm{if} s_i>b^{\prime }m, a<b<1\hfill \\ round\left(s_i\right)\hfill & otherwise.\hfill \end{array}\right.$$

(21)

Explosion magnitude refers to the search range of each Firework. The explosion range formula is defined as follows:

$$A_i=A·{\displaystyle \frac{J\left(u_i^k\right)-y_{min}+\epsilon }{{\sum }_{i=1}^n(J\left(u_i^k\right)-y_{min})+\epsilon }}.$$

(22)

For each Firework $u_i^k$, a Spark ${\tilde{u}}_i^{kl}$ is copied, and any $z=round(d\ast rand(0,1\left)\right)$ dimension is selected for explosion operation to spread the Spark, where d is the dimension of. $u_i^k$. For each selected dimension ${\tilde{u}}_i^{kl}$ of each Firework, there are the following explosion operations:

$${\tilde{u}}_i^{kl}={\tilde{u}}_i^{kl}+A_i\ast Levy\left(u^k\right)$$

(23)

where $\mathrm{Levy}\left(u^k\right)=0.01\times {\displaystyle \frac{r_a\times \sigma }{{\left|r_b\right|}^{\frac{1}{\chi }}}}$, where $r_a,r_b$ is [0, 1] two normally distributed random numbers with constant $\chi$ = 1.5, $\sigma ={\left({\displaystyle \frac{\Gamma (1+\chi )\times sin\left(\frac{\pi \chi }{2}\right)}{\Gamma \left(\frac{1+\chi }{2}\right)\times \chi \times 2^{\left(\frac{\chi -1}{2}\right)}}}\right)}^{1/\chi }$, $\Gamma \left(x\right)=(\mathrm{x}-1)!$.

If ${\tilde{u}}_i^{kl}<u_{min}^k$ or ${\tilde{u}}_i^{kl}>u_{max}^k$, a mapping operation is performed on it:

$${\tilde{u}}_i^{kl}=\left\{\begin{array}{c}{u}_{max}^k-\left|{\tilde{u}}_i^{kl}-u_{max}^k\right|,{\tilde{u}}_i^{kl}>u_{max}^k\hfill \\ u_{min}^k+\left|{\tilde{u}}_i^{kl}-u_{min}^k\right|,{\tilde{u}}_i^{kl}<u_{min}^k\hfill \end{array}\right..$$

(24)

After the blast operation, Firework has a total of $n+m$ with Spark.

3.

Gaussian variation operation

First, randomly select a dimension of a Firework or Spark ${\tilde{u}}_i^{km}$. Then, make a copy of the Firework or Spark and perform the following Gaussian mutation operation:

$${\tilde{u}}_i^{km}={\tilde{u}}_i^{km}+N(-1,1).$$

(25)

$N(-1,1)$ is a Gaussian distribution. If ${\tilde{u}}_i^{km}<u_{min}^k$ or ${\tilde{u}}_i^{km}>u_{max}^k$, then it is performed on Equation (24) the mapping operation.

Define the jth generation Gaussian constant of variation $z_j$ as

$$z_j=\left\{\begin{array}{c}{z}_{init},j=0\hfill \\ z_{fin}+{\displaystyle \frac{(z_{init}-z_{fin})}{g_{max}}}\ast (g_{max}-j),0<j<g_{max}\hfill \\ z_{fin},j=g_{max}\hfill \end{array}\right.$$

(26)

where $g_{max}$ is the number of generations to terminate the iteration, $z_{init}$ is the Gaussian variation constant of the starting state, and $z_{fin}$ is the Gaussian variation constant at the termination of the iteration.

Repeat this operation $G={\displaystyle \sum _{j=1}^{g_{max}}}{z}_j$ times, and G is the preset number of Gaussian mutations. After the Gaussian mutation operation, Firework and Spark have a total of $n+m+G$.

4.

Selection strategy

After the Gaussian mutation operation, n is selected as the initial Firework of the next generation from the set K consisting of $n+m+G$ positions. Since a loser elimination operation is also required after the selection, n spark with the highest fitness is selected to save computational resources.

5.

Loser-elimination operations

The loser-elimination operation is designed to exchange information between fireworks and provide an opportunity for those stuck in the best local location to jump out.

Define the fitness difference between the ith firework in the gth generation and the previous generation as ${\delta }_i^g$:

$${\delta }_i^g=J\left(u_i^{k(g-1)}\right)-J\left(u_i^{kg}\right)\ge 0.$$

(27)

Then, the predicted fitness of this firework in the last generation $g_{max}$ was calculated as

$$J\left({\widehat{u}}_i^{k{g}_{max}}\right)=J\left(u_i^{kg}\right)-(g_{\max }-g){\delta }_i^g.$$

(28)

For each firework, if its predicted fitness is worse than the best firework in the current generation, the firework is reinitialized. This is for the original loser elimination operation. This method greatly accelerates the convergence efficiency, but sometimes the best individual fireworks perform too well, which will lead to other fireworks keep repeating the initialization, wasting resources. Therefore, this paper follows Equation (27). After the calculation of predictive fitness, the expected fitness of each firework is divided into four grades, following different mechanisms to change its search pattern. Firstly, define the operator

$${\Delta }_g=J\left({\widehat{u}}_i^{k{g}_{max}}\right)-\underset{j}{min}\left(J\left(u_j^{kg}\right)\right)$$

(29)

(1)

Reservations to ensure stability

If a firework is predicted to be better adapted than the optimal firework in the current generation, i.e., $J\left({\widehat{u}}_i^{k{g}_{max}}\right)<{min}_j\left(J\left(u_j^{kg}\right)\right)$, it should be kept unchanged into the next generation.

(2)

Interactive learning to enhance exploration

If the predicted fitness of a firework $u_i^{kg}$ is inferior to the optimal firework in the current generation but not much different from the optimal firework in the current generation, it should be learned by leading it to the optimal firework. Define the interactive learning operation to be performed when ${min}_j\left(J\left(u_j^{kg}\right)\right)<J\left({\widehat{u}}_i^{k{g}_{max}}\right)<{min}_j\left(J\left(u_j^{kg}\right)\right)+{\displaystyle \frac{1}{2}}{\Delta }_g$ is as follows.

$$u_i^{kg}=min((1-\gamma )u_i^{kg}+\gamma u_b^{kg},u_i^{kg})$$

(30)

where $\gamma$ is the learning factor and $0<\gamma <1$, $u_b^{kg}$ are the best-adapted individuals in the g generation.

(3)

Mutation within a range to enhance exploration

If the predicted fitness of a firework $u_i^{kg}$ is inferior to the optimal firework in the current generation and the gap is large, the usefulness of interaction learning will be greatly diminished due to the distance from the optimal firework. When ${min}_j\left(J\left(u_j^{kg}\right)\right)+{\displaystyle \frac{1}{2}}{\Delta }_g<J\left({\widehat{u}}_i^{k{g}_{max}}\right)<{min}_j\left(J\left(u_j^{kg}\right)\right)+{\displaystyle \frac{3}{4}}{\Delta }_g$ should be made to mutate again, all dimensions of that firework should be selected for the Gaussian mutation operation in step 3.

(4)

Reinitialization

If the fitness of a firework $u_i^{kg}$ is really too poor, it is reinitialized to avoid wasting computational resources by searching too much in useless regions.

(5)

Repeat iterations

If a predetermined number of iteration steps is reached or a termination condition is reached, the algorithm is terminated; otherwise, go back to step 2.

Above are all the steps of the MLoTFWA algorithm, while the DMPC algorithm performs rolling optimization based on the MLoTFWA algorithm, and rolling optimization of the ith AUV at the moment can be summarized as the following steps:

(1)

Load the initial state of the AUV, initialize the formation communication topology, and reference position requirements. Initialize the population set of the fireworks algorithm, where each individual is a candidate solution for $U_i\left(k\right)$, where $U_i\left(k\right)$ contains multiple values from $U_i\left(k|k\right)$ to $U_i\left(k+P-1|k\right)$.

(2)

Execute the MLoTFWA algorithm for predictive modeling and feedback compensation for each individual to obtain the predicted output $y_{pi}\left(k+j|k\right)$.

(3)

Based on the predicted output $y_{pi}\left(k+j|k\right)$, calculate the cost of each individual and record the individual with the lowest cost.

(4)

Screening the next generation of new individuals based on a loser culling mechanism to ensure that better individuals remain in the population.

(5)

Repeat steps 2 to 4 until the control algorithm converges or the number of iterations reaches an upper limit.

(6)

The best individual ${U^{\ast }}_i\left(k\right)$ was identified, and its first element ${U^{\ast }}_i\left(k\right|k)$ was applied to the prediction model and the AUV formation. The corresponding outputs were obtained.

(7)

Go to step 1 to perform the operation for the next time window.

The specific process is shown in Figure 5.

3.5. Adaptive Prediction Window Size

The size of the prediction window interval P in the DMPC control framework is a key determinant of optimization performance. If P is too small, the optimization effectiveness of the DMPC control framework will be limited because the dynamic changes in the future environment are not adequately taken into account. Although increasing will significantly improve the optimization effect of DMPC control, too large a dynamic window will significantly increase the computational cost of DMPC control, and the mechanism of DMPC control determines that it must have good real-time performance, which is very difficult to ensure with the excessive computational volume of MPC control in practical applications.

Therefore, a careful trade-off between performance and cost is needed when choosing the size of P, especially when multiple AUVs are in formation in an obstacle-filled environment. Depending on the number of obstacles in the detection region, we can define the adaptive P value to achieve a balance between optimal control performance and computational efficiency. The size of the prediction window interval P for the DMPC control framework in this paper is described in the following equation:

$$P={\displaystyle \frac{{\displaystyle \sum _{i\in D_w}}max(\left|d_i\right|,R_{dangerous})}{R_{warning}}}+1$$

(31)

where $D_w$ is the set of obstacles in the alert area of this AUV, and $d_i$ is the set of the AUV with the ith obstacle in the set. If there are few or even no obstacles in the detection area, the AUV formation can be done with relative ease, and thus the value of P can be lowered to reduce the computational cost. On the contrary, if there are a large number of obstacles in the detection area, the safe flight of the AUV formation will face great challenges. In this case, increasing the value of P can improve the prediction performance, taking into account the multi-step motion trend.

The specific process of this section is shown in the Figure 6 below:

4. Proof of Stability of DMPC Formation Control

It is known that the DMPC method divides the control problem into a series of local optimizations, each of which is solved by the self-MLOTFWA algorithm. In this section, the asymptotic convergence of the local optimizations is analyzed.

Assuming the moment k, the optimal solution set of the cost function $J\left(u_k\right)$ can be defined as follows:

$$S^{\ast }\left(k\right)=\left\{u^{\ast }:\overline{J}\left(u^{\ast }\right|k)=min\left\{\overline{J}\left(u\right|k):u\in S\left(k\right)\right\}\right\}$$

(32)

where $S\left(k\right)$ is the measurable search space and $u^{\ast }$ denotes the optimal solution individual.

However, it is difficult to fulfil the conditions in the actual control Equation (32). Therefore, an extended set of optimal solutions is considered as follows:

$$S_{\epsilon }^{\ast }\left(k\right)=\{\tilde{u}\in S\left(k\right):|\overline{J}\left(\tilde{u}\right|k)-\overline{J}\left(u^{\ast }\right|k)|<\epsilon \}$$

(33)

where $\epsilon$ > 0, is a small positive real number. $\mu$ denotes the Leberger measure, we assume that for each $\epsilon$, there is $\mu \left({S^{\ast }}_{\epsilon }\left(k\right)\right)>0$. Extending the optimal Equation (33) can be approximated as an optimal solution.

Before analyzing the convergence of local optimization, we need to introduce the concept of probabilistic convergence as follows:

Define $\{u_G,G=1,2,\dots \}$ as the sequence of populations generated by the MLoTFWA algorithm for the local optimization problem where $u_G=\{u_i^G,G=1,2,\dots ,n\}$. It is required that the algorithm can converge probabilistically to the optimal solution set if and only if the following equation holds.

$$\underset{G\to \infty }{lim}P\{u_G\cap S_{\epsilon }^{\ast }\left(k\right)\ne \varnothing \}=1$$

(34)

where P denotes the probability of an event. The algorithm is stable when it must converge to the optimal solution set.

The following proof of Equation (34) must hold, and the DMPC control is locally stable at moment k:

Based on the above definition, it is assumed that for each population $u_G$ of the MLotFWA method, there exists at least one individual $u^G$ such that:

$$P\{u^G\in S_{\epsilon }^{\ast }\left(k\right)\}\ge \alpha >0.$$

(35)

$\alpha$ is a small positive value. Suppose a single $u_{rand}^G$ is randomly generated in each population $u_G$. The probability of $u_{rand}^G$ belonging to the optimal solution set is given by the following equation:

$$P\{u_{\mathrm{rand}}^G\in S_{\epsilon }^{\ast }\left(k\right)\}={\displaystyle \frac{\mu \left(S_{\epsilon }^{\ast }\left(k\right)\right)}{\mu \left(S\right(k\left)\right)}}=\alpha >0.$$

(36)

That is $P\{u_{\mathrm{rand}}^G\notin S_{\epsilon }^{\ast }\left(k\right)\}=1-\alpha >0$, so the probability that an individual in the first d population is not included in the set of optimal solutions is

$$\prod _{G=1}^{d}P\left\{u_G\cap S_{\epsilon }^{\ast }\left(k\right)=\varnothing \right\}\le {(1-\alpha )}^d.$$

(37)

Based on the basic idea of the MLoTFWA algorithm, the best individual in the population $u_G$ has the same or better fitness value as the best of all previous populations, which means that:

$$\underset{G\to \infty }{lim}P\left\{u_G\cap S_{ϵ}^{\ast }\left(k\right)=\varnothing \right\}=\underset{d\to \infty }{lim}\prod _{G=1}^{d}P\left\{u_G\cap S_{\epsilon }^{\ast }\left(k\right)=\varnothing \right\}\le \underset{d\to \infty }{lim}{(1-\alpha )}^d=0.$$

(38)

In other words:

$$\underset{G\to \infty }{lim}P\left\{u_G\cap S_{\epsilon }^{\ast }\left(k\right)\ne \varnothing \right\}=1-\underset{d\to \infty }{lim}\prod _{\sigma }^{d}P\{u_G\cap S_{\epsilon }^{\ast }\left(k\right)=\varnothing \}=1-0=1.$$

(39)

Equation (34) is established to be proved. At this point, the proof of local stability of DMPC is complete.

The basic idea of the DMPC control framework is that each AUV employs the same control algorithm, i.e., the Improved Fireworks Algorithm, in each step of the local optimization. Therefore, the local convergence analysis for a certain predicted moment k is fully applicable to the next predicted moment $k+j$, $j=1,2,\dots$. In addition, the local optimization of the predicted moment $k+j$ is executed on the basis of the optimization result of the predicted moment $k+j-1$, i.e., the optimization of the current moment is necessarily better than the previous one. As long as the control convergence can be guaranteed for each predicted moment, the convergence of the whole control process can be guaranteed.

5. Simulation

5.1. Simulation of Three Position Obstacle Avoidance Motion of AUV Formation

To evaluate the obstacle avoidance performance and compare the performance differences between various algorithms and the improved fireworks algorithm designed in this paper, which utilizes the MLoTFWA-based algorithm under simulated real-world conditions, the following simulation experiments are designed.

In this section of the simulation, the AUV formation adopts a pilot-follower strategy with one pilot and four followers in the formation. It sails from one point to another point. The start position of the navigator is (0, 0, 100) and the end point is (250, 0, 100). The desired trajectories are listed as follows:

$$\left\{\begin{array}{c}x=250t\hfill \\ y=0\hfill \\ z=-100\hfill \end{array}\right.$$

(40)

The placement positions of each follower in the AUV formation and the relative positions of the follower and the navigator are shown in

Table 3. Follower’s initial position and relative position to the leader’s expectations.

Follower	Initial Deployment Position/m	Desired Relative Position to Pilot/m
Follower 1	[−20, 20, −100]	[−20, 20, −100]
Follower 2	[−10, 10, −100]	[−10, 10, −100]
Follower 3	[−10, −10, −100]	[−10, −10, −100]
Follower 4	[−20, −20, −100]	[−20, −20, −100]

.

AUV Detection Distance $R_v=100$ m, Detection Angle $\alpha ={30}^{\circ }$, $\beta ={60}^{\circ }$, Warning Zone Radius $R_{warning}=50$ m, $R_{warning}^{\prime }=10$ m. Danger zone radius $R_{dangerous}=20$ m, $R_{dangerous}^{\prime }=5$ m. Obstacles are simplified to be regarded as spheres with different positions and radii, which move irregularly with a speed of 0–0.3 m/s in the plane and 0–0.03 m/s in the vertical direction, and their initial positions and radii are shown in the following

Table 4. Initial position and radius of obstacles.

Hindrance	Initial Deployment Position/m	Radius/m
Obstacle 1	[80, 0, −100]	15
Obstacle 2	[180, 0, −75]	22
Obstacle 3	[220, 40, −100]	20
Obstacle 4	[120, 60, −80]	30

.

The parameters of the loser elimination operation in the MLoTFWA algorithm have been mentioned in the above section, while the other parameters are $m=50$, $A=20$, $a^{\prime }=0.2$, $b^{\prime }=1,z_{init}=140$, $z_{fin}=70$, $\gamma =0.4$, and the parameters of the basic fireworks algorithm are $m=50$, $a^{\prime }=0.04$, $b^{\prime }=1$, $A=20$, $z=100$. The parameters of the particle swarm algorithm are inertia weights $w=0.707$, individual learning factor $c_1=2$, and social learning factor $c_2=2$. The parameters of the improved Gray Wolf algorithm are balance factor B = 0.8, convergence factor a = 2, and r1 and r2 are two random numbers. The number of particles and the maximum number of generations of each algorithm are 50. The parameters of the feedback compensation link of DMPC are $k_1=0.8$, $k_2=1$, $k_3=0.8$, $k_4=0.5$, $\alpha =0.7$, $a=5$, $b=10$. None of them will be repeated here. The other parameters in the simulation of this section are as follows: ${\lambda }_1=0.368$, ${\lambda }_2=0.01$, ${\lambda }_3=0.4$, ${\lambda }_4=0.02$, ${\lambda }_5=0.001$, $G_1=0.0005I_5$, $G_2=0.005I_5$, $R=I_5$. The velocity/angular velocity and acceleration/angular acceleration limits are shown in

Table 5. Status quantities and control input limits for each AUV.

Parametric	Limit Value
Pilot longitudinal speed (m/s)	[−1.5, 1,5]
Follower longitudinal velocity (m/s)	[−2.5, 2.5]
Pilot lateral speed (m/s)	[−0.8, 0.8]
Follower lateral velocity (m/s)	[−1, 0.8]
Pilot vertical speed (m/s)	[−0.8, 0.8]
Follower vertical velocity (m/s)	[−1, 1]
Full AUV longitudinal control input ($\mathrm{m}/{\mathrm{s}}^2$)	[−0.3, 0.3]
Full AUV lateral control input ($\mathrm{m}/{\mathrm{s}}^2$)	[−0.3, 0.3]
Full AUV droop control input ($\mathrm{m}/{\mathrm{s}}^2$)	[−0.3, 0.3]
Full AUV longitudinal angular velocity($\mathrm{rad}/{\mathrm{s}}^2$)	[−$\pi$/36, $\pi$/36 ]
Full AUV longitudinal angular velocity input ($\mathrm{rad}/{\mathrm{s}}^2$)	[−$\pi$/36, $\pi$/36]
Full AUV bow angular velocity ($\mathrm{rad}/{\mathrm{s}}^2$)	[−$\pi$/9, $\pi$/9]
Full AUV bow angular velocity input ($\mathrm{rad}/{\mathrm{s}}^2$)	[−2$\pi$/9, 2$\pi$/9]

below.

In order to ensure the formation, the linear speed limit of the leader will be more strict so that the follower can catch up when it falls out of the formation.The model of the AUV is still the same as in Section 2, and the simulation step size is $T=0.5$ s. During the simulation, there are still unknown currents of size 0–0.3 m/s with a direction of 45°, but the size and direction of the currents are unknown for the AUV formation.

Simulate the real marine environment by placing obstacles. Multiple AUVs form a formation to reach the target point by avoiding obstacles through path planning. The simulation results are shown below.

The follower and the navigator all broadcast their own position and location information and prediction window information to the formation. Figure 7 and Figure 8 show the side and top views of the 3D motion curves of the AUV formation, respectively, while the obstacles in Figure 9 and Figure 10 are actually counted as the danger zone of the obstacles. It can be seen that the AUV formation does not collide with obstacles and does not enter the danger zone while trying to maintain the formation formation as much as possible.

Figure 11, Figure 12 and Figure 13 show the trajectories of the AUV formation in the lateral, longitudinal, and vertical directions, and it can be seen that the AUV formation has more constant spacing in the x-axis and y-axis directions, while the z-axis direction has a large adjustment to complete the obstacle avoidance. Figure 14 and Figure 15 show the attitude angle information of the AUVs, and it can be seen that the longitudinal inclination and bow angle of the follower and the navigator maintain good consistency.

Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 shows the state information of the AUV formation’s linear and angular velocities in each direction. It can be seen that the angular velocities converge fast and keep good consistency; the longitudinal inclination angular velocity fluctuates slightly, but within 5, the influence is not big; the lateral velocity and the longitudinal velocity fluctuate not much, and the overall consistency is better, while the vertical velocity of each follower and the leader is not quite consistent because each AUV has to avoid touching and avoiding obstacles.

The three figures in Figure 21 show the longitudinal, transverse, and vertical thrust of the AUV formation, and the two figures in Figure 22 show the horizontal and vertical rudder angles of the AUV formation, which can be seen to be in the range of not exceeding the maximum tolerance range of the AUV and can be applied to the actual AUV system.The spacing between the various members of the AUV formation in the simulation is shown in Figure 23, and it can be very clearly seen that the AUVs can form the formation quickly and can be stably maintained when there is no obstacle, while 1–2 of the followers can avoid obstacles within a small range while trying to keep the formation not scattered and can ensure collision avoidance with each other. Figure 24 represents the distance curve of each AUV from the danger zone of the current nearest obstacle. When the value is greater than 0, the AUV formation can ensure safety, and the AUV formation has been maintaining a distance close to 0 but not exceeding 0, indicating that the AUVs traveling along the tangent of the obstacle can maintain the formation’s safety while shortening the path, promoting the optimality of the AUV formation. This experiment demonstrates that the AUV formation can safely travel from the initial location to the target location within a dynamic environment under the adaptive DMPC formation control based on the improved fireworks algorithm.

5.2. Improved Fireworks Algorithm Performance Verification

To validate the performance improvements of the enhanced Fireworks Algorithm (FWA) over the original FWA, a comparative simulation was conducted. Additionally, to demonstrate the advantages of the improved FWA over widely-used algorithms such as Particle Swarm Optimization (PSO) [34] and the improved Grey Wolf Optimizer (GWO) [35], the simulation included these algorithms as well. This simulation utilized the improved FWA, standard FWA, PSO, and improved GWO to solve several benchmark test functions. Through this simulation, the performance of the improved FWA was compared against the standard FWA and other algorithms to highlight its effectiveness and superiority. This approach ensures a comprehensive evaluation of the enhanced algorithm’s capabilities in various optimization scenarios.

The test function is as follows:

(1)

Sphere Function: $f_1\left(x\right)={\displaystyle \sum _{i=1}^{30}}{x}_i^2,min\left(f_1\right)=f_1(0,\dots ,0)=0$

(2)

Schwefel’s Problem 2.22 Function: $f_2\left(x\right)={\displaystyle \sum _{i=1}^{30}}|x_i|+{\displaystyle \prod _{i=1}^{30}}\left|x_i\right|min\left(f_2\right)=f_2(0,\dots ,0)=0$

(3)

Rosenbrock Function: $f_3\left(x\right)={\displaystyle \sum _{i=1}^{29}}[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2]$, $min\left(f_3\right)$ = $f_3(1,\dots ,1)=0$.

The minimum value of these test functions is 0. Therefore, the closer the absolute value of the adaptation in the test function is to 0, the better the performance.

The test range for each function, i.e., the range of values for (x), is presented in the

Table 6. Test functions and ranges.

Test Function	Sphere	Schwefel’s Problem 2.22	Rosenbrock
Test Range	[−5.12, 5.12]	[−10, 10]	[−30, 30]

below.

Since path planning is a low-dimensional vector short iteration problem, it is uniformly set to 100 rounds of 30-dimensional vector iterations, the number of individual particles is all 50, and 50 Monte Carlo experiments are conducted to take the average of the results.

The specific parameters of the algorithm are consistent with the simulations described above.

Below is a

Table 7. Test function adaptation experimental results.

Test Function	Test Metrics	MFWA	FWA	PSO	GWO
Sphere	Average adaptation	$6.77\times {10}^{-6}$	$1.22\times {10}^{-4}$	0.65	$4.81\times {10}^{-5}$
	optimal adaptation	$2.47\times {10}^{-8}$	$7.47\times {10}^{-6}$	0.34	$1.04\times {10}^{-7}$
	worst-case scenario	$1.09\times {10}^{-5}$	$7.67\times {10}^{-3}$	0.98	$1.72\times {10}^{-4}$
Schwefel’s Problem 2.2	Average adaptation	$8.86\times {10}^{-6}$	0.78	6.25	$5.23\times {10}^{-4}$
	optimal adaptation	$7.08\times {10}^{-23}$	$3.46\times {10}^{-11}$	4.39	$1.24\times {10}^{-17}$
	worst-case scenario	$2.40\times {10}^{-5}$	2.21	11.8	$9.93\times {10}^{-5}$
Rosenbrock	Average adaptation	28.5	28.91	$1.10\times {10}^4$	29.4
	optimal adaptation	27.7	28.6	$3.42\times {10}^3$	28.1
	worst-case scenario	29.3	30.2	$2.58\times {10}^4$	31.2

of the fitness of each algorithm for each test function:

It is evident that both the average fitness and the best fitness of the improved fireworks algorithm surpass those of the standard fireworks algorithm and other conventional algorithms.

5.3. Adaptive Prediction Window Simulation

To further validate the performance of the adaptive prediction window and the MLoTFWA algorithm, the integration of absolute error (IAE, hereafter referred to as IAE), global smoothness (hereafter referred to as GS), and velocity consistency (hereafter referred to as VC) are introduced here. abbreviated as VC).

The expression for IAE is shown in the following equation:

$$\mathrm{IAE}={\displaystyle \frac{1}{N·H_{\mathrm{T}}}}\sum _{i=1}^N\sum _{k=1}^{H_{\mathrm{T}}}{∥y_i\left(k\right)-r_i\left(k\right)∥}_{I_{3\times 3}}^2$$

(41)

where $r_i\left(k\right)$ represents the desired trajectory of each AUV, N represents the number of AUVs in the formation, and $H_{\mathrm{T}}$ represents the total number of steps in the simulation. IAE mainly represents the deviation value of each AUV from the intended trajectory, which can represent both the goodness of formation consistency and the length and reasonableness of the path.

The expression for GS is shown in the following equation:

$$\mathrm{GS}={\displaystyle \frac{1}{N·H_{\mathrm{T}}}}·\sum _{i=1}^N\sum _{k=1}^{H_{\mathrm{T}}}\left(\left|{\psi }_i(k+1)-{\psi }_i\left(k\right)\right|+\left|{\gamma }_i(k+1)-{\gamma }_i\left(k\right)\right|\right).$$

(42)

GS mainly represents the smoothness of the path.

The VC expression is shown in the following equation:

$$\mathrm{VC}={\displaystyle \frac{1}{N·H_{\mathrm{T}}}}\sum _{i=1}^N\sum _{k=1}^{H_{\mathrm{T}}}{∥x_i\left(k\right)-x_l\left(k\right)∥}_{I_{3\times 3}}^2$$

(43)

where $x_l\left(k\right)$ represents the speed of the AUV’s navigator, and VC mainly represents the formation consistency of the AUV’s speed, but sometimes the consistency of the speed will not be too high in formation avoidance, and the comparison of VC is only meaningful when it is necessary to ensure that the formation safety and IAE values are both small.

The results of the comparison between the prediction window P constant and the adaptive prediction window are shown in

Table 8. Comparison of the performance of different prediction window strategy formations.

Betactful	IAE/m	GS/°	VC
P = 1	2.125	1.419	0.215
P = 2	1.635	1.025	0.187
adaptive P	0.925	0.607	0.164

.

It can be seen that the adaptive prediction window provides a more satisfactory performance of the formation. The window length selected for the adaptive prediction window in the formation is shown in Figure 25.

In order to better illustrate the computational effort of the MLoTFWA algorithm proposed in this chapter, a comparative simulation is introduced here, where the basic fireworks algorithm (FWA) is compared with the particle swarm algorithm (PSO). The corresponding results are shown in

Table 9. Comparison of formation performance of different algorithms.

Arithmetic	IAE/m	GS/°	VC
MLoTFWA	0.925	0.607	0.164
FWA	1.193	0.741	0.220
PSO	1.965	1.819	0.272

. All the algorithms in Table 8 and Table 9 are performed under the adaptive prediction window strategy.

It can be seen that the MLoTFWA algorithm provides more satisfactory formation performance compared with both the FWA algorithm and the PSO algorithm. In addition to the formation performance, due to the computationally intensive nature of DMPC control itself, the amount of computation is also a major indicator of the superiority of DMPC control. The computational duty cycle is defined below $T_c$:

$$T_c={\displaystyle \frac{T_{CPU}}{T}}$$

(44)

where $T_{CPU}$ is the algorithm running time and T is the sampling time.

The simulation results are shown in

Table 10. Comparison of running time of different algorithms.

Arithmetic	${\mathit{T}}_{\mathit{c}}/\mathit{P}=1$	${\mathit{T}}_{\mathit{c}}/\mathit{P}=2$	${\mathit{T}}_{\mathit{c}}/\mathit{Adaptive}\mathit{P}$
MLoTFWA	0.522	0.649	0.581
FWA	0.508	0.626	0.552
PSO	0.857	1.015	0.954

.

From the definition, it is easy to know that the computational duty cycle $T_c$ needs to be less than 1, otherwise the simulation time is larger than the actual time, and the predictive control will lose its original meaning. The fireworks algorithm proposed in this chapter greatly improves the performance of the algorithm without increasing the amount of computation, while the adaptive windowing strategy avoids the case of computational duty cycles greater than 1 and makes the best use of computational resources.

6. Conclusions

In this paper, we address the optimal formation obstacle avoidance control problem for Autonomous Underwater Vehicles (AUVs) in environments with unknown obstacles and dynamic conditions. We enhance the fireworks algorithm and develop a Dynamic Model Predictive Control (DMPC) framework based on this improved algorithm to achieve effective obstacle avoidance for AUV formations. Key improvements to the fireworks algorithm include the loser elimination rule and rolling optimization. Additionally, to optimize resource utilization, we propose the theory of an adaptive DMPC prediction window. Simulation results demonstrate that the control system based on the DMPC framework ensures safe obstacle avoidance while maintaining formation consistency, achieving the shortest and smoothest paths. The improved fireworks algorithm outperforms the original fireworks algorithm and other conventional algorithms in terms of both average fitness and best fitness.

Looking ahead, future work will focus on further optimizing the obstacle avoidance control method for AUV formations based on the MLoTFWA algorithm. Specifically, we will explore more complex environments and various obstacle types to validate the robustness and adaptability of the algorithm. Furthermore, we plan to integrate other advanced optimization algorithms with the improved fireworks algorithm to enhance the obstacle avoidance performance and path planning efficiency of AUV formations. Through these studies, we aim to provide more reliable and efficient solutions for AUV formations in practical applications.