Multiple Autonomous Underwater Vehicle Formation Obstacle Avoidance Control Using Event-Triggered Model Predictive Control

Abstract

In this paper, multiple autonomous underwater vehicle (multi-AUV) formation control with obstacle avoidance ability in 3D complex underwater environments based on an event-triggered model predictive control (EMPC) is proposed. Firstly, multi-AUV motion model systems are developed. The navigation reference trajectory of the follower AUVs can be obtained using a multi-AUV relative motion model. Secondly, in order to overcome the speed jump and obstacle avoidance problem in multi-AUV systems, compatibility constraints are presented in MPC that limit the uncertainty deviation of each AUV. The event-triggered mechanism (ET) is designed to decrease the computational load, which is based on the error between the optimal predicted and current state of the AUV. Finally, the effectiveness and superiority of the proposed algorithm are confirmed via simulation and compared with those of other algorithms.

1. Introduction

In the complex and changeable marine environment, how to use the autonomous underwater vehicle (AUV) to carry out maritime search and rescue, environmental investigation, surveillance and reconnaissance, anti-submarine, environmental monitoring, and other search behaviors is a subject worthy of being studied [1]. However, when faced with high-efficiency, large-scale task requirements, a single AUV cannot meet the requirements. A new approach for resolving intricate marine tasks is offered using multi-AUV systems [2]. Some scholars have recently shown interest in consensus control [3], formation control [4,5], flocking control [6], and containment control [7] of multi-agent systems (MASs). The primary algorithms utilized for multi-AUV formation control are the leader-follower [8], behavior-based [9], and virtual structure [10]. The behavior-based approach allows for the functional diversification of multi-AUV formation, but with poor formation stability [9]. The virtual structure approach offers good stability but imposes a heavy burden on computation and communication [10]. The leader-follower approach operates on the control principle of designating a single AUV as the leader AUV, while assigning the remaining AUVs as the follower AUVs [8]. The motion and attitude of the leader AUV are referenced to calculate the speed and direction of the follower AUVs so that multiple AUVs can sail in formation [8]. The structure of the leader-follower formation algorithm is straightforward and effortless to execute [1].

The AUV relies heavily on the technical foundation provided by the controller’s design. In the past decades, a variety of control methods have been presented by scholars, such as PID [11], backstepping (BS) [12], sliding mode control (SMC) [13], and model predictive control (MPC) [14]. PID is a simple, low-complexity control method; however, accurate control can only be provided in the case of linear models and constant disturbances [11]. Backstepping control is an effective method for controlling nonlinear systems; however, as the order of a system increases, so does its computational complexity [12]. SMC is robust to uncertain models and time-varying parameters. However, SMC suffers from inaccurate control with a low degree of precision, excessive energy usage, and a high level of power consumption [13]. In addition, the above control methods are challenging to manage input and state constraints. In the controller design phase, the constraints of the system are ignored in the above control methods, which leads to poor control performance to a certain extent [14].

MPC uses the current state information of an agent to obtain its predictive state information, which is a closed-loop optimal control strategy. Multiple uncertainties and state constraints in multi-AUV systems can be handled using MPC [15]. MPC aims to tackle constrained optimization problems based on rolling optimization and feedback correction. Moreover, regardless of whether the controlled object is a linear model or a nonlinear model, the predictive model can be combined with the controlled object to achieve a more accurate control. MPC was applied to track the trajectory of unmanned vehicles by Yu et al. [16]. An MPC strategy was used to obtain the current control action online through solving predictive horizon open-loop optimal control [17]. A multi-objective MPC was developed to address the problem of path-following for an AUV [18]. An MPC strategy was presented to address the problem of trajectory tracking [19]. In [20], a Laguerre function-based adaptive predictive control was developed to address the challenge of trajectory tracking for AUVs operating in intricate underwater environments. Simultaneously, the real-time performance and tracking accuracy of the AUV system were improved. A target-tracking approach for AUVs was presented in [21], which utilized an MPC algorithm to account for dynamic behavior, allowing the accurate tracking of dynamic targets and trajectories. Additionally, MPC was employed in [22] to guarantee the control efficacy for linear systems that experience additive perturbations as well as multiple constraints.

The multi-AUV formation encounters various obstacles, such as buoys, reefs, and piers, while sailing. To address the challenge of obstacle avoidance, artificial potential field (APF) [23], MPC [24], and online optimal control method [25] have been developed by researchers. APF suffers from two key issues: goal non-reachable with obstacle nearby (GNWON) and jitter. Online optimal control requires a large amount of computation when the formation size is substantial, leading to a reduction in real-time performance [26]. In [27], the assumed prediction information of a neighbor agent was used to avoid collision. Under the constraints of robot dynamics and obstacle avoidance, a planner based on MPC was designed in [28]. A compatibility constraint was redefined in [29], which limits the deviation between the assumed and actual predicted states of local optimization. The model predictive control of AUV was studied in [30], and an integrated tracking control algorithm that included AUV dynamic path planning was developed. In [31], a dynamic path planning and trajectory tracking algorithm for autonomous satellites was proposed. Furthermore, a finite-horizon MPC strategy was presented. In [32], a framework of path planning was proposed for autonomous vehicles. In [33], an improved particle swarm optimization (IPSO)-based MPC was proposed to address the challenge of path planning.

The significant computational requirement is a disadvantage of MPC. The ET mechanism can effectively decrease computational load [34,35]. Therefore, it is advantageous to integrate the ET mechanism with MPC. It can lower the computational cost while retaining the features of MPC because the controller only updates when events are triggered. The EMPC for multi-vehicle systems was proposed in [36], and the ET mechanism was used to lower the computational load of MPC.

The majority of prior research on multi-agent systems predominantly focused on linear systems [37], as evident from the aforementioned studies, whereas an AUV is a typical nonlinear system. The primary focus of many studies have been on multi-agent systems like UAVs and unmanned vehicles. However, research on the AUV system has been limited due to the heightened complexity of the underwater environment in comparison to air and land environments.

Motivated by these existing research results and issues, a model predictive control method based on an event-triggered mechanism (EMPC) is proposed for multi-AUV formation in multi-obstacle underwater environments. The relative motion model of the multi-AUV formation is established according to the leader-follower formation control method. A compatible constrained MPC is proposed to plan the obstacle avoidance path of the multi-AUV formation so that the formation can avoid obstacles and reach the target point smoothly. The control problem of the multi-AUV formation is transformed into an optimization problem with input constraints and state constraints through MPC. The next optimal input is calculated according to the current state of the AUV and the desired trajectory, enabling the AUV to track the desired reference path accurately. In addition, an event-triggered mechanism based on the predicted and actual state values of the AUV is established, which can greatly reduce the calculation amount for solving optimization problems. The contributions of this paper are described as follows:

(1)

The norm-form compatibility constraint is proposed to address the issue of convergence for multi-AUV formation, that is, the uncertainty deviation between the actual and assumed trajectories of the AUV.

(2)

Since MPC has multiple constraints, the constraints of the AUV are embedded into the optimization of MPC, and the speed jump of multi-AUV formation systems can be avoided.

(3)

Combining event-triggered mechanisms and MPC, an EMPC strategy is designed to handle multiple constraints and reduce the computational burden of the multi-AUV system.

The remainder of this paper is organized as follows: The preliminaries are introduced in Section 2. In Section 3, the EMPC for multi-AUV systems considering obstacle avoidance is developed. Section 4 presents the main results to demonstrate the efficacy of the proposed approach. In Section 5, a simulation is conducted to further confirm the efficacy of the proposed algorithm. Section 6 concludes this paper and outlines future research.

2. Preliminaries

In this paper, an EMPC for multi-AUV systems obstacle avoidance is designed. The structure flow framework of EMPC is shown in Figure 1, where N represents the quantity of AUVs. The inputs of EMPC are derived from the information detected by the sensor. Then, multi-AUV formation generation, obstacle avoidance, and formation maintenance are realized using actuators. The notations and problem statement are presented in this section.

2.1. Notations

The following notations are adopted in this paper. The Euclidean norm is represented by $\left|\right|\cdot \left|\right|$. $\left|\right|\eta (t)|{|}_B$ represents the norm of $\eta (t)$, weighted by $B$, where $B$ is a positive definite matrix. The real space of $n\times n$ dimensional matrix is denoted as ${\Re }^{n\times n}$. ${\lambda }_{\min }(\cdot )$ and ${\lambda }_{\max }(\cdot )$ represent the minimum and the maximum of matrix eigenvalues, respectively.

2.2. The Model of Multi-AUV System

The establishment of the inertial coordinate $(X-Y-Z)$ and the body-fixed coordinate $(x_0-y_0-z_0)$ are illustrated in Figure 2. ${\eta }_i(t)={[p_i(t),{\phi }_i(t)]}^T$ is the state of the $AUVi$, where $i\in N$ is the number of AUVs, $p_i(t)=[x_i(t),y_i(t),z_i(t)]$, ${\phi }_i(t)$ is the position and orientation of the $AUVi$. The kinematic equation of the AUV is as follows [38].

$$\dot{\eta }=f(\eta (t),u(t))=\left[\begin{array}{cccc}\cos \phi & -\sin \phi & 0& 0\\ \sin \phi & \cos \phi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]u(t)$$

(1)

where $f(\cdot )$ is the continuously nonlinear function, and the control input is denoted as $u(t)={[v, r]}^T$, where $v={[u,v,w]}^T$. The rate of surge, sway, heave, and yaw are respectively represented by $u$, $v$, $w$, and $r$.

Let ${\eta }_i^d(t)$ be the desired state, ${\phi }_i^d(t)$ be the desired heading angle, ${\eta }_i^e(t)={\eta }_i-{\eta }_i^d$ be the state error, and ${\phi }_i^e(t)={\phi }_i(t)-{\phi }_i^d(t)$ be the angle error of the $AUVi$. Note that the multi-AUV system is usually subjected to external disturbances. The multi-AUV system with external interference is represented as

$$\dot{\eta }=f(\eta (t),u(t))+C(t), t\ge t_0$$

(2)

where the initial instant is denoted as $t_0$, $C(t)\in {\Re }^n$ is a bounded external disturbance, and $\left|\right|C(t)\left|\right|\le C_0$, $C_0$ is a positive constant.

A typical relative motion model for multiple AUV systems is shown in Figure 3. In this scenario, all AUVs possess identical structural and kinematic models. According to the length $l_{iL}^d$ and azimuth ${\phi }_{iL}^d$ of the intended arrangement of multi-AUV systems, the intended location $(x_i^d,y_i^d,z_i^d)$ and orientation ${\phi }_i^d$ of the follower AUV can be determined as (3).

$$\{\begin{array}{c}{x}_i^d(t)=x_L(t)-l_{iL}^d\cos ({\phi }_L(t)+{\phi }_{iL}^d)\\ y_i^d(t)=y_L(t)-l_{iL}^d\sin ({\phi }_L(t)+{\phi }_{iL}^d)\\ z_i^d(t)=z_L(t)-z_i(t)\\ {\phi }_i^d={\phi }_L\end{array}$$

(3)

where $p_L=[x_L,y_L,z_L]$ is the location of the leader AUV, and ${\phi }_L$ is the heading angle of the leader AUV.

2.3. Problem Statement

In order to clearly illustrate the multi-AUV formation obstacle avoidance control, the following assumptions are made.

(1)

The AUV can be perfectly located, and the map environment is known.

(2)

The starting point, target point, and obstacle position are known before the formation operation.

(3)

The multi-AUV formation is achieved without communication delay, packet loss, mechanical failure, or sensor noise.

To ensure the generation, collision, and obstacle avoidance of multi-AUV systems, control objectives are established as follows:

Formation generation:

$$\begin{gathered} \underset{t\to \infty }{\lim }(p_L(t)-p_i(t))=l_{iL}^d, {\phi }_i(t)={\phi }_L(t) \\ \underset{t\to \infty }{\lim }(p_j(t)-p_i(t))=d_{ij}, j\in N_i \end{gathered}$$

(4)

Collision avoidance of multi-AUV systems:

$$\Vert p_j(t)-p_i(t)\Vert \ge R_o, j\in N_v\backslash i$$

(5)

Obstacle avoidance of multi-AUV systems:

$$\Vert p_i(t)-p_o\Vert \ge R_o, o\in N_o$$

(6)

where $d_{ij}$ is the desired distance between the follower $AUVi$ and $AUVj$, $p_o$ denotes the position of an obstacle, $R_o$ is the safe distance between the AUV and an obstacle, $N_o$ is the number of obstacles, and $N_i$ is the neighbor set of the $AUVi$. In addition, for the multi-AUV system, only obstacles with distances less than $h$ away from the $AUVi$ are considered in this paper. Because if an obstacle is too far away from the AUV, the operation of the multi-AUV formation will not be affected. There is

$$\Vert {\overline{p}}_{io}(s|t_{k+1})\Vert =\Vert {\overline{p}}_i(s|t_{k+1})-p_o\Vert \le h, h>0$$

(7)

where $t_{k+1}\in [t_k, t_k+T]$ is the next update time. The relationship between all the time intervals is shown in Figure 4. At each update time, $T$ is the prediction horizon and $t_k$ is the current instant. The shortest duration between two events is $\inf \{t_{k+1}-t_k\}=\sigma$. $\sup \{t_{k+1}-t_k\}=T$ is the longest period of time that can elapse between two events.

Remark 1.

Other neighboring AUVs are regarded as obstacles to avoid collision. Therefore, the safe distances for collision and obstacle avoidance are identical.

As shown in Figure 5, ${\Omega }_i$ and ${\Omega }_{\epsilon i}$ are designed as concentric. The invariant set is denoted by the larger ellipse, and the terminal set is denoted by the smaller ellipse.

$${\Omega }_i\triangleq \{p_i^e(t_k):{\Vert {\widehat{p}}_i^e(s)\Vert }_Z^2\le {\delta }^2\}, {\Omega }_{\epsilon i}\triangleq \{p_i^e(t_k):{\Vert {\widehat{p}}_i^e(t_k+T)\Vert }_Z^2\le {\epsilon }^2\}$$

(8)

where $s\in [t_k,t_k+T]$, ${\widehat{p}}_i^e(s|t_k)$ represents the assumed predictive tracking error state trajectory, generated using the assumed predictive input ${\widehat{u}}_i(s|t_k)$ of the $AUVi$. $p_i^{e*}(s|t_k)$ is the optimal predictive tracking error state trajectory, and ${\overline{u}}_i(s|t_k)$ is the feasible predictive input. $Z$ is the positive definite symmetric matrix, and $\delta$ and $\epsilon$ are constants related to the invariant set and the terminal set, respectively. The designed controller is established as follows:

$${\widehat{u}}_i(s|t_{k+1})=\{\begin{array}{l}{u}_i^{*}(s|t_k), \begin{array}{ccc}& & \end{array} s\in [t_{k+1},t_k+T],\\ -K_i({\widehat{p}}_i^{e*}(s|t_k)), s\in [t_k+T,t_{k+1}+T].\end{array}$$

(9)

If the actual tracking error state trajectory of the follower $AUVi$ is outside the invariant set ${\Omega }_i$, that is, $P_i^e(t_k)\notin {\Omega }_i$, the control input is $u_i^{*}(s|t_k)$. If the actual tracking error state trajectory of the follower $AUVi$ is inside the invariant set ${\Omega }_i$, that is, $P_i^e(t_k)\in {\Omega }_i$, the control input is $-K_i({\widehat{p}}_i^{e*}(s|t_k))$.

Assumption 1.

$v_i(t)$ is bounded by $a$, that is, $|v_i(t)|\le a$, where$a$ is the upper bound of the linear velocity of the $AUVi$. $r$ is bounded by $\overline{r}$, and $\overline{r}$ is the upper bound of the angular velocity of the AUV.

Assumption 2.

According to [36], the nonlinear function $f({\eta }_i(t),u_i(t))$ is for the $AUVi$. There is the following inequality when the Lipschitz constant $L_f>0$.

$${\Vert f({\eta }_1(t),u_1(t))-f({\eta }_2(t),u_2(t))\Vert }_{p_i}\le L_f{\Vert {\eta }_1(t)-{\eta }_2(t)\Vert }_{p_i}$$

(10)

Assumption 3.

According to [39], the local convergence condition for the $AUVi$ is

$${\displaystyle \sum _{i\in N}({\dot{g}}_i(p_i^e(s|t_k))+L_i}(p_i^e(t_k),u_i^e(t_k)))\le 0$$

(11)

Lemma 1.

According to[[40]], if there is a Lipschitz constant $L_f=a$, the system (1) is locally Lipschitz in ${\eta }_i(t)$with $u_i(t)\in U$.

Lemma 2.

The deviation of the AUV’s trajectory between the actual and optimal states is limited by

$$\Vert {\eta }_i(s)-{\eta }_i^{*}(s|t_k)\Vert \le \rho (s-t_k)$$

(12)

where $s\in [t_k,t_k+T]$, $\rho (\tau )=C_{0}s{e}^{a\tau }$.

Lemma 3.

According to [41], if $\Vert u_L(t)\Vert <\alpha /\beta$holds, the ideal control input of the AUV is constructed using $\left|{\tilde{v}}^d(t)\right|/\tilde{a}+\left|{\tilde{r}}^d(t)\right|/\tilde{\overline{r}}\le \gamma$, $\gamma \in (0,1)$ is a constant. The minimum value of $\gamma$ is

$${\gamma }_{\min }=\max \left\{{\Vert u_L(t)\Vert }_2\right\}\frac{\beta }{\alpha }.$$

(13)

where $\alpha =\frac{\mu a\overline{r}}{\sqrt{a^2+{(\mu \overline{r})}^2}}$, $\beta ={\Vert M\Vert }_2$, the control input of the leader AUV is denoted as $u_L(t)$, $\mu$ is the offset, and $M$ is a matrix.

3. Design of EMPC for Multi-AUV Systems

There are multiple uncertainties and state constraints in multi-AUV systems. Uncertainties refers to system uncertainty due to the limitations and imperfect models of the AUV, making it impossible for the AUV to fully and accurately predict its own behavior and respond to environmental changes during task execution [42]. In this section, EMPC is designed to control the multi-AUV system. Simultaneously, an ET mechanism is designed to lower the computational burden caused by using MPC. The design of EMPC is mainly presented in this section.

3.1. Methodology

A multi-AUV formation obstacle avoidance control based on event-triggered model predictive control (EMPC) is proposed in this paper. The control structure diagram of the multi-AUV system is shown in Figure 6. Under the control of EMPC, each AUV in the formation converges to the desired state. The key steps are as follows:

(1)

A mathematical model of the multi-AUV system is established.

(2)

A norm-form compatibility constraint is proposed to address the issues of convergence and obstacle avoidance for multi-AUV systems.

(3)

Multiple constraints are embedded in MPC to avoid the speed jump in multi-AUV systems.

(4)

An EMPC is designed to handle multiple constraints and minimize the computational workload of multi-AUV systems.

(5)

Finally, the feasibility and stability of the system are analyzed.

3.2. Optimization Problem

The optimal cost should be considered in MPC, so it is necessary to construct the cost and optimization function of MPC. The cost function of the $AUVi$ is

$$J_i\left(p_i^e(t_k),u_i^e(t_k)\right)={\displaystyle {\int }_{t_k}^{t_k+T}{L}_i}\left(p_i^e(t_k),u_i^e(t_k)\right)ds+g_i\left(p_i^e(t_k+T)\right),$$

(14)

where

$$L_i(p_i^e(t_k),u_i^e(t_k))={\Vert p_i^e(s|t_k)\Vert }_P^2+{\displaystyle \sum _{j\in N_i}{\Vert p_{ij}(s|t_k)\Vert }_Q^2}+{\Vert p_{io}(s|t_k)\Vert }_R^2+{\Vert u_i^e(s|t_k)\Vert }_S^2,$$

(15)

$$g_i\left(p_i^e(t_k+T)\right)={\Vert p_i^e(t_k+T|t_k)\Vert }_Z^2+{\Vert p_{ij}(t_k+T|t_k)\Vert }_Z^2+{\Vert p_{io}(t_k+T|t_k)\Vert }_Z^2,$$

(16)

where $L_i\left(p_i^e(t_k),u_i^e(t_k)\right)$ is the stage cost function, ${\Vert p_i^e(s|t_k)\Vert }_P^2$ is the position tracking penalty term, ${\Vert p_{ij}(s|t_k)\Vert }_Q^2$ is the position formation penalty term, ${\Vert p_{io}(s|t_k)\Vert }_R^2$ is the obstacle avoidance penalty term, and ${\Vert u_i^e(s|t_k)\Vert }_S^2$ is the velocity tracking penalty term. The terminal cost function is denoted as $g_i\left(p_i^e(t_k+T)\right)$. $P$, $Q$, $R$, $S$, and $Z$ are positive definite symmetric matrices.

The optimization of MPC for the $AUVi$ is constructed as follows:

$$\underset{u_i(s|t_k)}{\min } J_i\left(p_i^e(t_k),u_i^e(t_k)\right)$$

(17)

$$s.t. {\eta }_i(t_k|t_k)={\eta }_i(t_k)$$

(18)

$${\dot{\eta }}_i(s|t_k)=f\left({\eta }_i(s|t_k),u_i(s|t_k)\right)$$

(19)

$$u_i(s|t_k)\in U_i$$

(20)

$$\Vert p_i^e(s|t_k)\Vert \le \frac{T\epsilon }{s-t_k}$$

(21)

$$p_i^e(t_k+T|t_k)\in {\Omega }_{\epsilon i}$$

(22)

$$C_1\left({\pi }_i(s|t_k),{\varphi }_i(t_k),{\psi }_i(t_k)\right)\le 0$$

(23)

$$C_2\left(p_i(s|t_k),{\widehat{p}}_j(s|t_k)\right)\le 0$$

(24)

$$C_3\left(p_i(s|t_k),p_o\right)\le 0$$

(25)

where $t_k$ is an update instant, $s\in [t_k,t_k+T]$. The optimal cost function is expressed as follows:

$$J_i\left(p_i^{e*}(t_k),u_i^{e*}(t_k)\right)={\displaystyle {\int }_{t_k}^{t_k+T}{L}_i}(p_i^{e*}(t_k),u_i^{e*}(t_k))ds+g_i(p_i^{e*}(t_k+T)).$$

(26)

3.3. The Event-Triggered Mechanism

To lower the computational load, the ET mechanism is established. The condition for ET is expressed as

$$\Vert {\eta }_i(s)-{\eta }_i^{*}(s|t_k)\Vert \ge \rho (\sigma ),$$

(27)

where $\sigma \in (0,T)$. When Equation (27) is not satisfied, a lower bound of triggering time is obtained as follows:

$${\tilde{t}}_{k+1}={\inf }_{s\in [t_k,t_k+T]}s:\Vert {\eta }_i(s)-{\eta }_i^{*}(s|t_k)\Vert \ge \rho (\sigma )$$

(28)

Note that at both $t_0$ and $t_k+T$, the multi-AUV system is obliged to be triggered at least once. The designed triggering time $t_{k+1}$ is:

$$t_{k+1}=\min \{{\tilde{t}}_{k+1},t_k+T\}.$$

(29)

Remark 2.

The disparity between adjacent update times in two events is $t_{k+1}-t_k$. Zeno behavior usually exists in the ET mechanism, indicating repeated triggering within a finite time limit. The minimum time between two events is $\inf \{t_{k+1}-t_k\}=\sigma$, and the max time between two events is $\sup \{t_{k+1}-t_k\}=T$. Hence, the Zeno behavior is naturally avoided in this paper.

3.4. The Constraints Design

The constraints of multi-AUV systems are designed in this section.

Firstly, let

$${\varphi }_{ij}(t_k)={\min }_{j\in N\backslash i}\left\{\frac{\Vert {\widehat{p}}_j(s|t_k)-{\widehat{p}}_i(s|t_k)\Vert -R_o}{2}\right\}, s\in [t_k,t_k+T].$$

(30)

The limit on collision avoidance ${\varphi }_i(t_k)$ is denoted as

$${\varphi }_i(t_k)=\underset{j\in N\backslash i}{\min }\left\{{\varphi }_{ij}(t_k)\right\}.$$

(31)

The bound of tracking and formation ${\psi }_i(t_k)$ is [36]:

$${\psi }_i(t_k)=\frac{\vartheta \left({\Vert {\eta }_i^e(t_k)\Vert }_P^2+{\displaystyle {\sum }_{j\in N_i}{\Vert p_{ij}(t_k)\Vert }_Q^2}\right)}{(T-T_p){\displaystyle {\sum }_{j\in N_i}{\lambda }_{\max }(Q)(2e_{ij}(k)+3{\varphi }_{ij}(k))}},$$

(32)

where

$$e_{ij}(t_k)=\underset{s\in [t_{k+1},t_k+T]}{\max }\Vert {\widehat{p}}_j(s|t_k)-{\widehat{p}}_i(s|t_k)+d_{ji}\Vert ,$$

(33)

where the sampling period is denoted as $T_p$, and $\vartheta \in [0,1)$ is a constant. The uncertain deviation between the assumed and actual trajectories of the $AUVi$ ${\pi }_i(s|t_k)$ is introduced.

$$\Vert {\pi }_i(s|t_k)\Vert \le \min \left\{{\varphi }_i(t_k),{\psi }_i(t_k)\right\},$$

(34)

where ${\pi }_i(s|t_k)={\widehat{p}}_i(s|t_k)-p_i(s|t_k)$.

The compatibility constraint is constructed as

$$C_1\left({\pi }_i(s|t_k),{\varphi }_i(t_k),{\psi }_i(t_k)\right)=\Vert {\pi }_i(s|t_k)\Vert -\min \{{\varphi }_i(t_k),{\psi }_i(t_k)\},$$

(35)

and the collision avoidance constraint for the $AUVi$ is

$$\Vert p_j(s|t_k)-p_i(s|t_k)\Vert \ge R_o.$$

(36)

Considering the constraint (34), there is

$$\begin{array}{cc}\Vert p_j(s|t_k)-p_i(s|t_k)\Vert & =\Vert {\widehat{p}}_j(s|t_k)-p_i(s|t_k)-{\pi }_j(s|t_k)\Vert \\ & \ge \Vert p_j(s|t_k)-p_i(s|t_k)\Vert -{\varphi }_{ij}(t_k)\end{array}.$$

(37)

According to inequalities (36) and (37), we have

$$\Vert {\widehat{p}}_j(s|t_k)-p_i(s|t_k)\Vert \ge R_o+{\varphi }_{ij}(t_k), j\in N\backslash i$$

(38)

The designed constraint of collision avoidance is

$$C_2(p_i(s|t_k),{\widehat{p}}_j(s|t_k))=R_o+{\varphi }_{ij}(t_k)-\Vert {\widehat{p}}_j(s|t_k)-p_i(s|t_k)\Vert .$$

(39)

According to (6), we have

$$\Vert p_i(s|t_k)-p_o\Vert \ge R_o.$$

(40)

The obstacle avoidance constraint is

$$C_3(p_i(s|t_k),p_o)=R_o-\Vert p_i(s|t_k)-p_o\Vert .$$

(41)

3.5. EMPC Design

The EMPC strategy is proposed in this section to accomplish control objectives (4)–(6). To reduce the computational burden, the ET mechanism is integrated into MPC, which is shown in Algorithm 1. The first step is to choose the parameters. Each AUV maintains its communication. The optimal control sequence and the trajectory of the tracking error state are calculated prior to the entry of the tracking error state into ${\Omega }_i$. The terminal controller is activated to drive the tracking error state to enter ${\Omega }_{\epsilon i}$. The ET strategy is developed based on the variation between the actual trajectory and the optimal prediction. The ET condition determines the next update time.

Algorithm 1. Algorithm to design EPMC scheme.

Input: The initial and target state of AUVs; the parameters $P$, $Q$, $R$, $S$, $Z$, $\vartheta$, ${\widehat{k}}_1$ and ${\widehat{k}}_2$. Output: Begin: Step 1: At $t_0$, according to ${\widehat{p}}_i(s|t_0)=p_i(t_0)$, the $AUVi$ sends ${\widehat{p}}_i(s|t_0)$ to the $AUVj$, and receives ${\widehat{p}}_j(s|t_0)$ from the $AUVj$. Step 2: At sample instant $t_p$, compute ${\varphi }_{ij}(t_k)$, ${\varphi }_i(t_k)$, ${\psi }_i(t_k)$. Step 3: At $t_k$, if $p_i^e(t_k)\notin {\Omega }_i$, the optimization problem (13a)-(13i) is solved to obtain $u_i^{*}(s)$ and ${\widehat{p}}_i^{e*}(s)$, $s\in [t_k,t_k+T]$. If $p_i^e(t_k)\in {\Omega }_i$, switch to the terminal controller, ${\widehat{u}}_i(s|t_k)=-K_i({\widehat{p}}_i^e(t_k))$. Step 4: $t_{k+1}=\min \{{\widehat{t}}_{k+1},t_k+T\}$ is given as: (a)    If (27) is satisfied, then $t_{k+1}={\widehat{t}}_{k+1}={\inf }_{s\in [t_k,t_k+T]}s:\left|\right|{\eta }_i(s)-{\eta }_i^{*}(s|t_k)||\ge \rho (\sigma )$. (b)    Or else, $t_{k+1}=t_k+T$. Step 5: Set $t_k=t_{k+1}$, and proceed to Step 1. End

Remark 3.

In Algorithm 1, the selection of $T$ is necessary. Whereas the computational load will be heavy if $T$ is too long, the control effect will be worse if $T$ is too short. Selecting a suitable sampling period $T_p$ is crucial. If $T_p$ is too large, it will lead to a suboptimal control effect. If $T_p$ is too small, the computational burden will be heavy.

4. Main Results

In this section, we present the proposed analysis of feasibility and stability.

4.1. Feasibility Analysis

The feasibility analysis is proposed in this part.

Theorem 1.

Set

$${\Omega }_{\epsilon i}=\left\{p_i^{e*}(t_k+T|t_k):{\Vert p_i^{e*}(t_k+T|t_k)\Vert }^2\le {\epsilon }^2\right\}$$

(42)

as the terminal region.

If Equations (17)–(25) are solved at $t_k$, they will be feasible at $t_{k+1}$. For all $t_k>0$, the $AUVi$ neither collides with $AUVj$ nor with obstacles, and

$$C_0\le \frac{e^{-aT}}{\sigma }(\delta -\epsilon ), \sigma \ge \frac{1}{\widehat{k}}\ln \frac{\delta }{\epsilon }, \varpi \delta \le \epsilon <\delta ,$$

(43)

where $\delta =\frac{a(1-\gamma )}{\sqrt{{\widehat{k}}_1^2+{\widehat{k}}_2^2}}$, ${\widehat{k}}_1$ and ${\widehat{k}}_2$ are the parameters for feedback gain values, $\widehat{k}=\min \{{\widehat{k}}_1,{\widehat{k}}_2\}$, $\varpi =\max \left\{\frac{T-\sigma }{T},\frac{1}{T\widehat{k}}\right\}$ with $T\widehat{k}>1$.

The detailed proof is reported in Appendix A.

4.2. Stability Analysis

Theorem 2.

In the multi-AUV system, the EMPC strategy with function (14) is considered. If the following inequality holds, the multi-AUV system is stable.

$$\begin{array}{l}\sigma ({\lambda }_{\min }(P){\epsilon }^2+R_o^2)-\frac{{\lambda }_{\max }(P)C_0^2{\sigma }^2}{2a}(e^{2aT}-e^{2a\sigma })-2R{\lambda }_{\max }(Z)C_0\sigma e^{aT}-h^2({\lambda }_{\max }(R)T+{\lambda }_{\max }(Z))\\ -\frac{2{\lambda }_{\max }(P)C_0\sigma \epsilon }{\sqrt{2a}}{(\frac{T^2}{\sigma }-T)}^{\frac{1}{2}}{(e^{2aT}-e^{2a\sigma })}^{\frac{1}{2}}>0\end{array}$$

(44)

The detailed proof is reported in Appendix A.

5. Simulation

In this section, the multi-AUV formation control in complex obstacle environments is simulated directly using a MATLAB R2019a software platform, without using Simulink or other toolboxes.

The following aspects are verified through simulation.

(1)

Whether the multi-AUV formation can be generated, and its desired reference path can be accurately tracked, under the control of EMPC. Compared with MPC and the backstepping control method, the advantages of the proposed algorithm are verified.

(2)

Can obstacle avoidance and collision avoidance be satisfied by the proposed algorithm?

(3)

Can the ET mechanism effectively reduce the computational load?

5.1. Simulation Results in 3D Obstacle-Free Environments

Firstly, the efficacy of the suggested algorithm is validated in a 3D obstacle-free environment with a formation of three AUVs, that is, $N=3$. The multi-AUV formation forms an equilateral triangle that has a side length of $4$ m, that is, the length $l_{iL}^d=4$ m, and azimuth ${\phi }_{iL}^d=\pi /6$. AUV0 is the leader AUV, and AUV1,2 are follower AUVs. Let $a=2$ m/s, $\overline{r}=1$ rad/s, $T=2$s, and $T_p=0.1$ s. The parameters of BS are set as $k_{xy}=3$ and $k_z=k_{\phi }=6$.

Table 1. Simulation parameters.

Parameters	Value	Parameters	Value
$\vartheta$	0.99	$\epsilon$	0.35
${\widehat{k}}_1$	0.88	$\mu$	$0.12$
${\widehat{k}}_2$	1.08	$P$	$diag\{0.5,0.5,0.5\}$
$C_O$	$0.02$	$Q$	$diag\{0.04,0.04,0.04\}$
$\sigma$	$2\times {10}^{-3}$	$R$	$diag\{0.02,0.02,0.02\}$
$R_o$	2	$S$	$diag\{0.7,0.7,0.7\}$
$\gamma$	0.89	$Z$	$diag\{0.5,0.5,0.5\}$
$\delta$	0.75

displays the simulation parameters.

The starting condition of each AUV is as follows: ${\eta }_0(t_0)={\left[-2,5,2,0\right]}^T$, ${\eta }_1(t_0)={\left[-3,4,1,0\right]}^T$, ${\eta }_2(t_0)={\left[2,3,0,0\right]}^T$. The target state of AUV0 is set as $(20,20,20,0)$. The desired trajectory of AUV0 is as follows: $x_d(t)=t$, $y_d(t)=t$, $z_d(t)=t$, ${\phi }_d(t)=\pi /4$. The reference speed is as follows: $u_d(t)=\sqrt{2}$ m/s, $v_d(t)=0$ m/s, $w_d(t)=1$ m/s, $r_d(t)=0$ rad/s.

The trajectory of each AUV using the MPC, EMPC, and backstepping methods is shown in Figure 7. Figure 7a illustrates that under the control of EMPC, every AUV has the capability to track the desired trajectory in a stable manner and effectively reach the intended target location. According to Figure 7b, MPC requires a significant amount of time for each AUV to follow the desired trajectory, about 4 s. According to Figure 8, Figure 9 and Figure 10, under the control of EMPC and MPC, the speed of each AUV does not exceed the limit.

According to Figure 8, Figure 9 and Figure 10, under backstepping control, both the linear speed and the angular speed exceed the limit (the speed limit is indicated by the black dotted line). However, in practical applications, each AUV is given a corresponding speed limit. Therefore, when the speed of the AUV exceeds the limit, the maximum speed limit is used as the control input to obtain the track trajectory results, as depicted in Figure 7c. It is evident from Figure 7c that under backstepping control, each AUV fails to track the desired path and does not reach the target point.

Considering the large amount of computation required when MPC solves optimization problems, an event-triggered mechanism is proposed in this paper. Traditional MPC controllers are time-triggered, and the problem of optimization should be solved at each time step, i.e., each sampling will trigger a controller update. The triggering instants of each follower AUV in obstacle-free environments is shown in Figure 11. The number of triggers rises notably at the beginning, due to state errors in the initial formation generation stage.

The complete simulation duration for implementing the MPC approach is 156.58 s, which is significantly longer than the 34.69 s required for utilizing the suggested EMPC method. The comparison outcomes for the overall distance for each AUV under the control of the EMPC and the MPC strategy are shown in Figure 12. According to Figure 12, the total distance of each AUV with the EMPC strategy is less than that with the MPC strategy.

5.2. Simulation Results in 3D Obstacle Environments

Simulation experiments are adopted to validate obstacle avoidance as well as collision ability. Note that the backstepping method has no corresponding constraints, so it does not have the ability to avoid obstacles and collisions. The initial condition of each AUV is ${\eta }_0(t_0)={\left[1,1,5,0\right]}^T$, ${\eta }_1(t_0)={\left[1,10,-7,0\right]}^T$, ${\eta }_2(t_0)={\left[1,-7,-2,0\right]}^T$. The location parameters of each obstacle are shown in

Table 2. The location parameters of each obstacle.

Obstacles	Position (m)	Diameter/Side Length (m)
1	(10, 10, 19)	D = 6
2	(20, 23, 25)	D = 5
3	(40, 46, −85)	D = 3, h = 85
4	(40,60, −40)	D = 6, h = 85
5	(37,22, −40)	L = 12.5, w = 12.5, h = 45

, where $D$ represents the diameter, $L$ is the length, $W$ is the width, and $h$ is the height of the cylinder.

It can be seen from Figure 13 and Figure 14 that under the proposed constraints, the five obstacles in the figure can be successfully avoided under the EMPC and MPC strategies, and the multi-AUV formation operation can be maintained. The speed of each AUV under the control of EMPC/MPC in complex obstacle environments is shown in Figure 15, Figure 16 and Figure 17. Compared with the MPC strategy, the formation convergence rate is faster when the EMPC strategy is adopted. In addition, each AUV can quickly track the reference trajectory under EMPC control after avoiding obstacles. Figure 15, Figure 16 and Figure 17 illustrate that the speed changes in the formation generation phase as well as the obstacle avoidance phase, but the speed of each AUV is within the constraint range.

The triggering moments of each follower AUV are displayed in Figure 18. There are state errors in the initial formation generation phase and the obstacle avoidance phase. Therefore, the number of triggers increases significantly at the initial time and when obstacles are encountered. The entire simulation duration for implementing the MPC approach is 156.58 s, and the duration for using the EMPC strategy is 45.32 s. The comparison results of the total distance for each AUV under the control of the EMPC and the MPC strategies are shown in Figure 19. According to Figure 19, the total distance of each AUV with the EMPC strategy is less than that with the MPC strategy.

Evidently, when the ET mechanism is adopted, the number of each follower AUV triggering instant is significantly reduced. When the EMPC strategy is adopted, the time consumed by the system and the total distance of each AUV are less than those of the MPC strategy, which indicates that the event-triggered mechanism proposed in this paper can effectively save computational resources.

6. Conclusions

An EMPC strategy for tracking and obstacle avoidance in a multi-AUV formation system with bounded interference is presented in this paper. The uncertain deviation of each AUV is bounded by a compatibility constraint, which guarantees the convergence of the multi-AUV formation. In addition, avoidance constraints are designed between each AUV and obstacles as well as between each AUV and other AUVs, to ensure that each AUV can avoid obstacles and collisions. The feasibility of EMPC and the stability of the system are guaranteed. Since MPC has multiple constraints, the problem of speed jump can be avoided. The proposed EMPC strategy has been validated through simulation examples, demonstrating its effectiveness and advantages. In addition, MPC presents a heavy computational burden. An event-triggered control mechanism that reduces the communication and computational burden is proposed.

However, most existing studies have focused on static obstacles, and future related work should be aimed at obstacle avoidance control in dynamic obstacle environments as well as the control of heterogeneous systems. In addition, the ocean current environment is not considered in this paper. In the future, obstacle avoidance in the ocean current environment will be further studied. In addition, we will focus on comparing our work with state-of-the-art solutions in subsequent research to increase the quality of this paper.