Adaptive Formation Learning Control for Cooperative AUVs under Complete Uncertainty
Emadodin Jandaghi Mingxi Zhou Paolo Stegagno Chengzhi Yuan

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Running Title]Adaptive Formation Learning Control for Cooperative AUVs under Complete Uncertainty 


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    Received 2024; accepted 2024
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§ ABSTRACT 




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This paper introduces an innovative two-layer control framework for Autonomous Underwater Vehicles (AUVs), tailored to master completely uncertain nonlinear system dynamics in the presence of challenging hydrodynamic forces and torques. Distinguishing itself from prior studies, which only regarded certain matrix coefficients as unknown while assuming the mass matrix to be known, this research significantly progresses by treating all system dynamics, including the mass matrix, as unknown. This advancement renders the controller fully independent of the robot’s configuration and the diverse environmental conditions encountered. The proposed framework is universally applicable across varied environmental conditions that impact AUVs, featuring a first-layer cooperative estimator and a second-layer decentralized deterministic learning (DDL) controller. This architecture not only supports robust operation under diverse underwater scenarios but also adeptly manages environmental effects such as changes in water viscosity and flow, which influence the AUV's effective mass and damping dynamics. The first-layer cooperative estimator fosters seamless inter-agent communication by sharing crucial system estimates without the reliance on global information, while the second-layer DDL controller utilizes local feedback to precisely adjust each AUV’s trajectory, ensuring accurate formation control and dynamic adaptability. Furthermore, the capability for local learning and knowledge storage through radial basis function neural networks (RBF NNs) is pivotal, allowing AUVs to efficiently reapply learned dynamics following system restarts. Comprehensive simulations validate the effectiveness of our groundbreaking framework, highlighting its role as a major enhancement in the field of distributed adaptive control systems for AUVs. This approach not only bolsters operational flexibility and resilience but is also essential for navigating the unpredictable dynamics of marine environments.
  

§ KEYWORDS:
 Environment-independent Controller, Autonomous Underwater Vehicles (AUV),  Dynamic learning, formation learning control, Multi-agent systems, Neural network control, Adaptive control, Robotics. 




§ INTRODUCTION


Robotics and autonomous systems have a wide range of applications, spanning from manufacturing and surgical procedures to exploration in challenging environments <cit.>. However, controlling robots in challenging environments, such as space and underwater, presents significant difficulties due to the unpredictable dynamics in these settings. In the context of underwater exploration, Autonomous Underwater Vehicles (AUVs) have emerged as indispensable tools. These vehicles are not only cost-effective and reliable but also versatile in adapting to the dynamic and often harsh conditions of underwater environments <cit.>. Unlocking the mysteries of marine environments hinges on the effective use of AUVs, making advancements in their control and operation critical.

As the complexity of tasks assigned to AUVs increases, there is a growing need to enhance their operational capabilities. This includes developing sophisticated formation control strategies that allow multiple AUVs to operate in coordination, mimicking collaborative behaviors seen in natural swarms such as fish schools or bird flocks. These strategies are essential for ensuring efficient, stable, and precise operations in varied underwater tasks, ranging from pipeline inspections to seafloor mapping <cit.>. Multi-AUV formation control presents significant challenges due to the intricate nonlinear dynamics and complex interaction behaviors among the AUVs, as well as the uncertain and dynamic nature of underwater environments. Despite these difficulties, the problem has garnered substantial interest from the marine technology and control engineering communities, driven by the extensive applications of AUVs in modern ocean industries, making effective multi-AUV formation control increasingly critical <cit.>.



Historically, formation control research has predominantly utilized the behavioral approach <cit.>, which segmented the overall formation control design problem into multiple subproblems. Each vehicle's control action is then determined by a weighted average of solutions to these subproblems, facilitating decentralized implementation. However, defining the appropriate weighting parameters often presents challenges. Another method, the leader-following approach <cit.>, selected one vehicle as the leader while the rest function as followers. The leader pursues a specified path, and the followers maintain a pre-defined geometric relationship with the leader, enabling control over the formation behavior by designing specific motion behaviors for the leader. Additionally, the virtual structure approach <cit.> modified this concept by not assigning a physical leader but instead creating a virtual structure to set the desired formation pattern. This method combines the benefits of the leader-following approach with greater flexibility, as the dynamics of the virtual leader can be tailored for more adaptable formation control designs.

In the field of formation control, several key methodologies have emerged as significant contributors to research. The behavioral approach <cit.> addresses the formation control challenge by partitioning the overall design problem into distinct subproblems, where the control solution for each vehicle is computed as a weighted average of these smaller solutions. This technique supports decentralized control but often complicates the calibration of weighting parameters. Conversely, the leader-following strategy <cit.> assigns a primary vehicle as the leader, guiding a formation where subordinate vehicles adhere to a specified geometric arrangement relative to this leader. This approach effectively governs the collective behavior by configuring the leader’s trajectory. Furthermore, the virtual structure method <cit.> builds upon the leader-following paradigm by substituting the leader with a virtual structure, thereby enhancing the system’s adaptability. This advanced method allows for the flexible specification of virtual leader dynamics, which can be finely tuned to meet diverse operational needs, thus inheriting and enhancing the leader-following model's advantages.

Despite extensive literature in the field, much existing research assumes homogeneous dynamics and certain system parameters for all AUV agents, which is unrealistic in unpredictable underwater environments. Factors such as buoyancy, drag, and varying water viscosity significantly alter system dynamics and behavior. Additionally, AUVs may change shape during tasks like underwater sampling or when equipped with robotic arms, further complicating control. Typically, designing multi-AUV formation control involves planning desired formation paths and developing tracking controllers for each AUV. However, accurately tracking these paths is challenging due to the complex nonlinear dynamics of AUVs, especially when precise models are unavailable. Implementing a fully distributed and decentralized formation control system is also difficult, as centralized control designs become exceedingly complex with larger AUV groups.

In previous work, such as <cit.>, controllers relied on the assumption of a known mass matrix, which is not practical in real-world applications. Controllers’ dependent on known system parameters can fail due to varying internal forces caused by varying external environmental conditions. The solution lies in developing environment-independent controllers that do not rely on any specific system dynamical parameters, allowing the robot to follow desired trajectories based on desirable signals without any knowledge on system dynamical parameters.


This paper achieves a significant advancement by considering all system dynamics parameters as unknown, enabling a universal application across all AUVs, regardless of their operating environments. This universality is crucial for adapting to environmental variations such as water flow, which increases the AUV’s effective mass via the added mass phenomenon and affects the vehicle's inertia. Additionally, buoyancy forces that vary with depth, along with hydrodynamic forces and torques—stemming from water flow variations, the AUV’s unique shape, its appendages, and drag forces due to water viscosity—significantly impact the damping matrix in the AUV's dynamics. By addressing these challenges, the proposed framework ensures robust and reliable operation across a spectrum of challenging scenarios.

The framework's control architecture is ingeniously divided into a first-layer Cooperative Estimator Observer and a lower-layer Decentralized Deterministic Learning (DDL) Controller. The first-layer observer is pivotal in enhancing inter-agent communication by sharing crucial system estimates, operating independently of any global information. Concurrently, the second-layer DDL controller utilizes local feedback to finely adjust each AUV’s trajectory, ensuring resilient operation under dynamic conditions heavily influenced by hydrodynamic forces and torques by considering system uncertainty completely unknown. This dual-layer setup not only facilitates acute adaptation to uncertain AUV dynamics but also leverages radial basis function neural networks (RBF NNs) for precise local learning and effective knowledge storage. Such capabilities enable AUVs to efficiently reapply previously learned dynamics following any system restarts. This framework not only drastically improves operational efficiency but also significantly advances the field of autonomous underwater vehicle control by laying a robust foundation for future enhancements in distributed adaptive control systems and fostering enhanced collaborative intelligence among multi-agent networks in marine environments. Extensive simulations have underscored the effectiveness of our pioneering framework, demonstrating its substantial potential to elevate the adaptability and resilience of AUV systems under the most demanding conditions.

The rest of the paper is organized as follows: Section <ref> provides an initial overview of graph theory, Radial Basis Function (RBF) Neural Networks, and the problem statement. The design of the distributed cooperative estimator and the decentralized deterministic learning controller are discussed in Section <ref>. The formation adaptive control and formation control using pre-learned dynamics are explored in Section <ref> and Section <ref>, respectively. Simulation studies are presented in Section <ref>, and Section <ref> concludes the paper.




§ PRELIMINARIES AND PROBLEM STATEMENT




 §.§ Notation and Graph Theory

Denoting the set of real numbers as ℝ, we define ℝ^m × n as the set of m × n real matrices, and ℝ^n as the set of n × 1 real vectors.
The identity matrix is symbolized as 𝐈. 
The vector with all elements being 1 in an n-dimensional space is represented as 1_n. The sets 𝕊_+^n and 𝕊_-^n+ stand for real symmetric n × n and positive definite matrices, respectively. A block diagonal matrix with matrices X_1, X_2, …, X_p on its main diagonal is denoted by diag{X_1, X_2, …, X_p}. 
A ⊗ B signifies the Kronecker product of matrices A and B. 
For a matrix A, A⃗ is the vectorization of A by stacking its columns on top of each other. For a series of column vectors x_1, …, x_n, col{x_1, …, x_n} represents a column vector formed by stacking them together.
Given two integers k_1 and k_2 with k_1 < k_2, 𝐈[k_1, k_2] = {k_1, k_1 + 1, …, k_2}. For a vector x ∈ℝ^n, its norm is defined as |x| := (x^T x)^1/2. For a square matrix A, λ_i(A) denotes its i-th eigenvalue, while λ_min(A) and λ_max(A) represent its minimum and maximum eigenvalues, respectively.

A directed graph G = (V, E) comprises nodes in the set V = {1, 2, …, N} and edges in E ⊆ V × V. An edge from node i to node j is represented as (i, j), with i as the parent node and j as the child node. Node i is also termed a neighbor of node j. N_i is considered as the subset of V consisting of the neighbors of node i. A sequence of edges in G, (i_1, i_2), (i_2, i_3), …, (i_k, i_k+1), is called a path from node i_1 to node i_k+1. Node i_k+1 is reachable from node i_1. A directed tree is a graph where each node, except for a root node, has exactly one parent. The root node is reachable from all other nodes. A directed graph G contains a directed spanning tree if at least one node can reach all other nodes. The weighted adjacency matrix of G is a non-negative matrix A = [a_ij] ∈ℝ^N × N, where a_ii = 0 and a_ij > 0  (j, i) ∈ E. The Laplacian of G is denoted as L = [l_ij] ∈ℝ^N × N, where l_ii = ∑_j=1^N a_ij and l_ij = -a_ij if i ≠ j. 
It is established that L has at least one eigenvalue at the origin, and all nonzero eigenvalues of L have positive real parts. 
From <cit.>, L has one zero eigenvalue and remaining eigenvalues with positive real parts if and only if G has a directed spanning tree.


 §.§ Radial Basis Function NNs
 
The RBF Neural Networks can be described as f_nn(Z)=∑_i = 1^Nw_i s_i(Z)=W^TS(Z), where Z∈Ω_Z⊆ℝ^q and W=w_1,...,w_N^T∈ℝ^N as input and weight vectors respectively <cit.>. N indicates the number of NN nodes, S(Z)=[s_1(||Z-μ_i||),...,s_N(||Z - μ_i||)]^T with s_i(·)  is a RBF, and μ_i(i = 1,...,N) is distinct points in the state space.
The Gaussian function s_i (||Z - μ_i||)=exp[-(Z-μ_i)^T(Z-μ_i)/η_i^2] is generally used for RBF, where μ_i=[μ_i1,μ_i2,...,μ_iN]^T is the center and η_i is the width of the receptive field. The Gaussian function categorized by localized radial basis function s in the sense that s_i (||Z - μ_i||)→ 0 as ||Z||→∞.
Moreover, for any bounded trajectory Z(t) within the compact set Ω_Z, f(Z) can be approximated using a limited number of neurons located in a local region along the trajectoryf(Z) = W^*_ζ S_ζ(Z) + ϵ_ζ, where ϵ_ζ is the approximation error, with ϵ_ζ = O(ϵ) = O(ϵ^*), S_ζ(Z) = [s_j_1(Z), …, s_j_ζ(Z)]^T ∈ℝ^N_ζ, W^*_ζ = [w^*_j_1, …, w^*_j_ζ]^T ∈ℝ^N_ζ, N_ζ < N_n, and the integers j_i = j_1, …, j_ζ are defined by |s_j_i(Z_p)| > ι (ι > 0 is a small positive constant) for some Z_p ∈ Z(t). This holds if Z(t) - ξ_j_i < ϵ for t > 0. The following lemma regarding the persistent excitation (PE) condition for RBF NNs is recalled from <cit.>.

    
    Consider any continuous recurrent trajectory[A recurrent trajectory represents a large set of periodic and periodic-like trajectories generated from linear/nonlinear dynamical systems. A detailed characterization of recurrent trajectories can be found in <cit.>.] Z(t) : [0, ∞) →ℝ^q. Z(t) remains in a bounded compact set Ω_Z ⊂ℝ^q. Then for an RBF Neural Network (NN) W^T S(Z) with centers placed on a regular lattice (large enough to cover the compact set Ω_Z), the regressor subvector S_ζ(Z) consisting of RBFs with centers located in a small neighborhood of Z(t) is persistently exciting.



 §.§ Problem Statement

A multi-agent system comprising N AUVs with heterogeneous nonlinear uncertain dynamics is considered. The dynamics of each AUV can be expressed as <cit.>:

    η̇_i = J_i(η_i)ν_i, 
       M_iν̇_i + C_i(ν_i)ν_i + D_i(ν_i)ν_i + g_i(η_i) + Δ_i(χ_i) = τ_i.

In this study, the subscript i ∈ I[1, N] identifies each AUV within the multi-agent system. For every i ∈ I[1, N], the vector η_i = [x_i, y_i, ψ_i]^T ∈ℝ^3 represents the i-th AUV's position coordinates and heading in the global coordinate frame, while ν_i = [u_i, v_i, r_i]^T ∈ℝ^3 denotes its linear velocities and angular rate of heading relative to a body-fixed frame. The positive definite inertial matrix M_i = M_i^T ∈𝕊_3^+, Coriolis and centripetal matrix C_i(ν_i)∈ℝ^3 × 3, and damping matrix D_i(ν_i)∈ℝ^3 × 3 characterize the AUV's dynamic response to motion. The vector g_i(η_i)∈ℝ^3 × 1 accounts for the restoring forces and moments due to gravity and buoyancy. The term Δ_i(χ_i) ∈ℝ^3 × 1, with χ_i := col{η_i, ν_i}, describes the vector of generalized deterministic unmodeled uncertain dynamics for each AUV.

The vector τ_i ∈ℝ^3 represents the control inputs for each AUV. The associated rotation matrix J_i(η_i) is given by:
J_i(η_i) = [  cos(ψ_i)  sin(ψ_i)         0; -sin(ψ_i)  cos(ψ_i)         0;         0         0         1 ],
Unlike previous work <cit.> which assumed known values for the AUV's inertia matrix and rotation matrix, this study considers not only the matrix coefficients C_i(ν_i), D_i(ν_i), g_i(η_i), and Δ_i(χ_i) but also treats the inertia matrix and rotation matrix as completely unknown and uncertain. When all system parameters are considered unknown, external forces do not influence the controller, making it universally applicable to any AUV regardless of shape, weight, and environmental conditions, including variations in water flow or depth that typically affect damping and inertia forces during operation.

In the context of leader-following formation tracking control, the following virtual leader dynamics generates the tracking reference signals:

    χ̇_0 = A_0χ_0


With “0” marking the leader node, the leader state χ_0 := col{η_0, ν_0} with η_0 ∈ℝ^3 and ν_0 ∈ℝ^3, A_0 ∈ℝ^3 × 3 is a constant matrix available only to the leader’s neighboring AUV agents.

Considering the system dynamics of multiple AUVs (<ref>) along with the leader dynamics (<ref>), we establish a non-negative matrix A = [a_ij], where i, j ∈ I[0, N] such that for each i ∈ I[1, N], a_i0 > 0 if and only if agent i has access to the reference signals η_0 and ν_0. All remaining elements of A are arbitrary non-negative values, such that a_ii = 0 for all i. Correspondingly, we establish G = (V, E) as a directed graph derived from A, where V = {0, 1, …, N} designates node 0 as the leader, and the remaining nodes correspond to the N AUV agents. We proceed under the following assumptions:


All the eigenvalues of A_0 in the leader’s dynamics (<ref>) are located on the imaginary axis.


The directed graph G contains a directed spanning tree with the node 0 as its root.














Assumption <ref> is crucial for ensuring that the leader dynamics produce stable, meaningful reference trajectories for formation control. It ensures that all states of the leader, represented by χ_0 = col{η_0, ν_0}, remain within the bounds of a compact set Ω_0 ⊂ℝ^6 for all t ≥ 0. The trajectory of the system, starting from χ_0(0) and denoted by ϕ_0(χ_0(0)), generates periodic signal. This periodicity is essential for maintaining the Persistent Excitation (PE) condition, which is pivotal for achieving parameter convergence in Distributed Adaptive (DA) control systems. Modifications to the eigenvalue constraints on A_0 mentioned in Assumption <ref> may be considered when focusing primarily on formation tracking control performance, as discussed later.

Additionally, Assumption <ref> reveals key insights into the structure of the Laplacian matrix L of the network graph G. Let Ψ be an N × N non-negative diagonal matrix where each i-th diagonal element is a_i0 for i ∈ I[1, N]. The Laplacian L is formulated as:

    L = [     ∑_j=1^N a_0j -[a_01, …, a_0N];            -Ψ1_N                H ],

where a_0j > 0 if (j, 0) ∈ E and a_0j = 0 otherwise. This results in 𝐇1_N = Ψ1_N since L1_N+1 = 0. As cited in <cit.>, all nonzero eigenvalues of 𝐇, if present, exhibit positive real parts, confirming 𝐇 as nonsingular under Assumption <ref>.

    
    In the context of a multi-AUV system (<ref>) integrated with virtual leader dynamics (<ref>) and operating within a directed network topology G, the aim is to develop a distributed NN learning control protocol that leverages only local information. The specific goals are twofold:
    
1) Formation Control: Each of the N AUV agents will adhere to a predetermined formation pattern relative to the leader, maintaining a specified distance from the leader's position η_0.

2) Decentralized Learning: The nonlinear uncertain dynamics of each AUV will be identified and learned autonomously during the formation control process. The insights gained from this learning process will be utilized to enhance the stability and performance of the formation control system.

remarkRemark
 
    
The leader dynamics described in Equation (<ref>) are designed as a neutrally stable LTI system. This design choice facilitates the generation of sinusoidal reference trajectories at various frequencies which is essential for effective formation tracking control. This approach to leader dynamics is prevalent in the literature on multiagent leader-following distributed control systems like <cit.> and <cit.>.

 
    
    It is important to emphasize that the formulation assumes formation control is required only within the horizontal plane, suitable for AUVs operating at a constant depth, and that the vertical dynamics of the 6-DOF AUV system, as detailed in <cit.>, are entirely decoupled from the horizontal dynamics.

As shown in Fig <ref>, a two-layer hierarchical design approach is proposed to address the aforementioned challenges. The first layer,the Cooperative Estimator, enables information exchange among neighboring agents. The second layer, known as the Decentralized Deterministic Learning (DDL) controller, processes only local data from each individual AUV. The subsequent two sections will discuss the development of the DA observer within the first layer and the formulation of the DDL control strategy, respectively.



§ TWO-LAYER DISTRIBUTED CONTROLLER ARCHITECTURE




 §.§ First Layer: Cooperative Estimator

In the context of leader-following formation control, not all AUV agents may have direct access to the leader’s information, including tracking reference signals (χ_0) and the system matrix (A_0). This necessitates collaborative interactions among the AUV agents to estimate the leader’s information effectively. Drawing on principles from multiagent consensus and graph theories <cit.>, we propose to develop a distributed adaptive observer for the AUV systems as:





    χ̇̂̇_i(t) = A_i(t)χ̂_i(t) + β_i1∑_j=0^N a_ij (χ̂_j(t) - χ̂_i(t)),   ∀ i ∈ I[1, N]

The observer states for each i-th AUV, denoted by χ̂_i = [η̂_i, ν̂_i]^T ∈ℝ^6, aim to estimate the leader's state, χ_0 = [η_0, ν_0]^T ∈ℝ^6. As t →∞, these estimates are expected to converge, such that η̂_i approaches η_0 and ν̂_i approaches ν_0, representing the leader's position and velocity, respectively. The time-varying parameters A_0(t) for each observer are dynamically adjusted using a cooperative adaptation law:

    Â̇_i(t) = β_i2∑_j=0^N a_ij (Â_j(t) - Â_i(t)),   ∀ i ∈ I[1, N]

Â_i are used to estimate the leader's system matrix A_0 while both are ∈ℝ^6*6. The constants β_i1 and β_i2 are all positive numbers and are subject to design.
 
    
    Each AUV agent in the group is equipped with an observer configured as specified in equations (<ref>) and (<ref>), comprising two state variables, χ_i and A_i. For each i ∈ I[1, N], χ_i estimates the virtual leader’s state χ_0, while A_i estimates the leader’s system matrix A_0. The real-time data necessary for operating the i-th observer includes: (1) the estimated state χ̂_j and matrix Â_̂ĵ, obtained from the j-th AUV itself, and (2) the estimated states χ_j and matrices A_j for all j ∈ N_i, obtains from the j-th AUV’s neighbors. Note that in equations (<ref>) and (<ref>), if j ∉ N_i, then a_ij = 0, indicating that the i-th observer does not utilize information from the j-th AUV agent. This configuration ensures that the proposed distributed observer can be implemented in each local AUV agent using only locally estimated data from the agent itself and its immediate neighbors, without the need for global information such as the size of the AUV group or the network interconnection topology.

By defining the estimation error for the state and the system matrix for agent i as χ̃_i = χ̂_i - χ_0 and Ã_i = Â_i - A_0, respectively, we derive the error dynamics:

    χ̇̃̇_i(t)    = Â_i(t)χ̂_i(t) + A_0χ_0(t) + β_i1∑_j=0^N a_ij (χ̂_j(t) - χ̂_i(t)) 
       = Â_i(t) χ̂_i (t) - A_0χ̂_i (t) + A_0χ̂_i (t ) - A_0χ _0 (t ) + β _1∑ _j=0^Na_ij (χ̂_j (t ) - χ _0 (t ) + χ _0 (t ) - χ̂_i (t ) ) 
       = A_0χ̃_i (t ) + Ã_i (t )χ̃_i (t ) + Ã_i (t )χ _0 (t ) + β _1∑ _j=0^Na_ij (χ̃_j (t ) - χ̃_i (t ) )
    
    Ã̇_i(t)    = β_i2∑_j=0^N a_ij - (Ã_j(t) - Ã_i(t)) ,   ∀ i ∈ I[1, N].

Define the collective error states and adaptation matrices: χ̃ = col{χ̃_1, …, χ̃_N} for the state errors,
Ã = col{Ã_1, …, Ã_N} for the adaptive parameter errors,
Ã_b = diag{Ã_1, …, Ã_N} representing the block diagonal of adaptive parameters,
B_β_1 = diag{β_11, …, β_N1} and
B_β_2 = diag{β_12, …, β_N2} for the diagonal matrices of design constants. 
With these definitions, the network-wide error dynamics can be expressed as:

    χ̇̃̇(t)    = ((I_N ⊗ A_0) - B_β_1(H ⊗ I_2n)) χ̃(t) + (Ã_b(t) ⊗ I_2n) χ̃(t) + Ã_b(t) (1_N ⊗χ_0(t)), 
    Ã̇(t)    = -B_β_2(H ⊗ I_n) Ã(t).

theoremTheorem

 Consider the error system equations (<ref>). Under Assumptions <ref> and  <ref>, and given that β_1, β_2 > 0, it follows that for all i ∈ I[1, N] and for any initial conditions χ_0(0), χ_i(0), A_i(0), the error dynamics of the adaptive parameters and the states will converge to zero exponentially. Specifically, lim_t →∞Ã_i(t) = 0 and lim_t →∞χ̃_i(t) = 0.

This convergence is facilitated by the independent adaptation of each agent's parameters within their respective error dynamics, represented by the block diagonal structure of Ã_b and control gains B_β_1 and B_β_2. These matrices ensure that each agent's parameter updates are governed by local interactions and error feedback, consistent with the decentralized control framework.


Proof: We begin by examining the estimation error dynamics for Ã as presented in equation <ref>. This can be rewritten in the vector form:

    Ã̇⃗̇_0(t) = -β_2(I_6⊗ (ℋ⊗ I_3)) Ã⃗_0(t)

Under assumption <ref>, all eigenvalues of ℋ possess positive real parts according to <cit.>. Consequently, for any positive β_2 > 0, the matrix -β_2(I_6 ⊗ (H ⊗ I_3)) is guaranteed to be Hurwitz, Which implies the exponential stability of system (<ref>). Hence, it follows that lim_t →∞Ã⃗_0(t) = 0 exponentially, leading to lim_t →∞Ã_i0(t) = 0 exponentially for all i ∈ I[1, N].

Next, we analyze the error dynamics for χ_0 in equation (<ref>). Based on the previous discussions, We have lim_t →∞Ã_b(t) = 0 exponentially, and the term Ã_b(t)(1_N ⊗χ_0(t)) will similarly decay to zero exponentially. Based on <cit.>, if the system defined by

    χ̇̃̇_0(t) = ((I_N⊗ A_0) - β_1(ℋ⊗ I_6)) χ̃_0(t)

is exponentially stable, then lim_t →∞χ_0(t) = 0 exponentially. With assumption <ref>, knowing that all eigenvalues of A_0 have zero real parts, and since ℋ as nonsingular with all eigenvalues in the right-half plane, system (<ref>) is exponentially stable for any positive β_1 > 0. Consequently, this ensures that lim_t →∞χ_0(t) = 0, i.e., lim_t →∞χ_i0(t) = 0 exponentially for all i ∈ I[1, N].

Now, each individual agent can accurately estimate both the state and the system matrix of the leader through cooperative observer estimation (<ref>) and (<ref>). This information will be utilized in the DDL controller design for each agent's second layer, which will be discussed in the following subsection.



 §.§ Second Layer: Decentralized Deterministic Learning Controller


To fulfill the overall formation learning control objectives, in this section, we develop the DDL control law for the multi-AUV system defined in (<ref>). We use d^*_i to denote the desired distance between the position of the i-th AUV agent η_i and the virtual leader's position η_0. Then, the formation control problem is framed as a position tracking control task, where each local AUV agent's position η_i is required to track the reference signal η_d,i := η_0 + d^*_i.

Besides, due to the inaccessibility of the leader’s state information χ_0 for all AUV agents, the tracking reference signal η̂_d,i := η̂_0,i + d^*_i is employed instead of the reference signal η̂_d,i. As established in theorem <ref>, η̂_d,i is autonomously generated by each local agent and will exponentially converge to η_d,i. This ensures that the DDL controller is feasible and the formation control objectives are achievable for all i ∈ I[1, N] using η̂_d,i.

To design the DDL control law that addresses the formation tracking control and the precise learning of the AUVs' complete nonlinear uncertain dynamics at the same time, we will integrate renowned backstepping adaptive control design method outlined in <cit.> along with techniques from <cit.> and <cit.> for deterministic learning using RBF NN. Specifically, for the i-th AUV agent described in system (<ref>), we define the position tracking error as z_1,i = η_i - η̂_d,i for all i ∈ I[1, N]. Considering J_i(η_i) J_i^T(η_i) = I for all i ∈ I[1, N], we proceed to:

    ż_1,i = J_i(η_i)ν_i - η̇̂̇_i,   ∀ i∈𝐈[1,N].

To frame the problem in a more tractable way, we assume ν_i as a virtual control input and α_i as a desired virtual control input in our control strategy design, and by implementing them in the above system we have:

    z_2,i   = ν_i - α_i, 
    α_i    = J_i^T(η_i) (-K_1,iz_1,i + η̇̂̇_i),   ∀ i ∈𝐈[1, N].

A positive definite gain matrix K_1,i∈𝕊_3^+ is used for tuning the performance. Substituting ν_i = z_2,i + α_i into ((<ref>)) yields:

    ż_1,i   = J_i(η_i)z_2,i - K_1,iz_1,i,   ∀ i ∈𝐈[1, N].

Now we derive the first derivatives of the virtual control input and the desired control input as follows:

    ż_2,i   = ν̇_i - α̇_i 
       = M_i^-1(-C_i(ν_i)ν_i - D_i(ν_i)ν_i - g_i(η_i) - Δ_i(χ_i) + τ_i) - α̇_i,


    α̇_i    = J̇_i^T(η_i)(-K_1,iz_1,i + η̇̂̇_i) + J_i^T(η_i)(K_1,iη̇̂̇_i - K_1,iJ_i(η_i)ν_i + η̈̂̈_i).   ∀ i ∈𝐈[1, N],

As previously discussed, unlike earlier research that only identified the matrix coefficients C_i(ν_i), D_i(ν_i), g_i(η_i), and Δ_i(χ_i) as unknown system nonlinearities while assuming the mass matrix M_i to be known, this work advances significantly by also considering M_i as unknown. Consequently, all system dynamic parameters are treated as completely unknown, making the controller fully independent of the robot's configuration—such as its dimensions, mass, or any appendages—and the uncertain environmental conditions it encounters, like depth, water flow, and viscosity. This independence is critical as it ensures that the controller does not rely on predefined assumptions about the dynamics, aligning with the main goal of this research. To address these challenges, we define a unique nonlinear function F_i(Z_i) that encapsulates all nonlinear uncertainties as follows:

    F_i(Z_i) = M_iα̇_i + C_i(ν_i)ν_i + D_i(ν_i)ν_i + g_i(η_i) + Δ_i(χ_i),

where F_i(Z_i) = [f_1,i(Z_i), f_2,i(Z_i), f_3,i(Z_i)]^T and Z_i = col{η_i, ν_i}∈Ω_Z_i⊂ℝ^6, with Ω_Z_i being a bounded compact set.
We then employ the following RBF NNs to approximate f_k,i for all i ∈𝐈[1, N] and k ∈𝐈[1, 3]:

    f_k,i(Z_i) = W_k,i^*TS_ki(Z_i) + ϵ _k,i(Z_i),

where W_k,i^* is the ideal constant NN weights, and ϵ_k,i(Z_i) is the approximation error ϵ^*_k,i > 0 for all i ∈𝐈[1, N] and k ∈𝐈[1, 3], which satisfies |ϵ_k,i(Z_i)| ≤ϵ^*_k,i. This error can be made arbitrarily small given a sufficient number of neurons in the network.
A self-adaptation law is designed to estimate the unknown W_k,i^* online. We aim to estimate W_k,i^* with Ŵ_k,i by constructing the DDL feedback control law as follows:

    τ _i = -J_i^T(η _i)z_1,i - K_2,iz_2,i + Ŵ_k,i^TS_k,i(Z_i)

To approximate the unknown nonlinear function vector F_i(Z_i) in (<ref>) along the trajectory Z_i within the compact set Ω_Z_i, we use:

    Ŵ_i^T S_i^F(Z_i) = 
    [ Ŵ_1,i^T S_1,i(Z_i); Ŵ_2,i^T S_2,i(Z_i); Ŵ_3,i^T S_3,i(Z_i) ].

Also, K_2,i∈𝕊_3^+ is a feedback gain matrix that can be tuned to achieve the desired performance. From (<ref>) and (<ref>) we have:

    M_iν̇_i + C_i(ν_i)ν_i + D_i(ν_i)ν_i + g_i(η_i) + Δ_i(χ_i) = τ _i = -J_i^T(η _i)z_1,i - K_2,iz_2,i + Ŵ_k,i^TS_k,i(Z_i)

By subtracting W_k,i^*TS_k,i(Z_i) + ϵ_k,i(Z_i) from both sides and considering Equations (<ref>) and (<ref>), we define W̃_k,i := Ŵ_k,i - W_k,i^*, leading to:

    ż_2,i = M_i^-1 (-J_i^T(η _i)z_1,i - K_2,iz_2,i + Ŵ_k,i^TS_k,i(Z_i) - ϵ _k,i(Z_i))

For updating Ŵ_k,i online, a robust self-adaptation law is constructed using the σ-modification technique <cit.> as follows:

    Ŵ̇_k,i = -Γ_k,i (S_k,i(Z_i)z_2k,i + σ _kiŴ_k,i)

where z_2,i = [z_21,i, z_22,i, z_23,i]^T, Γ_k,i = Γ_k,i^T > 0, and σ_k,i > 0 are free parameters to be designed for all i ∈𝐈[1, N] and k ∈𝐈[1, 3].
Integrating equations (<ref>), (<ref>) and (<ref>) yields the following closed-loop system:

    {ż_1,i   = -K_1,iz_1,i + J_i(η_i)z_2,i, 
    ż_2,i   = M_i^-1(-J_i^T(η_i)z_1,i - K_2,iz_2,i + Ŵ_ki^TS_ki(Z_i) - ϵ_k,i(Z_i)), 
    Ẇ̃̇_k,i   = -Γ_k,i(S_k,i(Z_i)z_2,k,i + σ_k,iŴ_k,i),
    .

where, for all i ∈𝐈[1, N] and k ∈𝐈[1, 3], W̃_i^T S_i(Z_i) = [W̃_1,i^T S_1,i(Z_i), W̃_2,i^T S_2,i(Z_i), W̃_3,i^T S_3,i(Z_i)]^T, and ϵ_i(Z_i) = [ϵ_1,i(Z_i), ϵ_2,i(Z_i), ϵ_3,i(Z_i)]^T.
 
    
Unlike the first-layer DA observer design, the second-layer control law is fully decentralized for each local agent. It utilizes only the local agent's information for feedback control, including χ_i, χ̂_i, and W_k,i, without involving any information exchange among neighboring AUVs.

The following theorem summarizes the stability and tracking control performance results of the overall system:


Consider the local closed-loop system (<ref>). For each i ∈𝐈[1, N], if there exists a sufficiently large compact set Ω_Z_i such that Z_i ∈Ω_Z_i for all t ≥ 0, then for any bounded initial conditions, we have:
1) All signals in the closed-loop system remain uniformly ultimately bounded (UUB).
2) The position tracking error η_i - η_d,i converges exponentially to a small neighborhood around zero in finite time T_i > 0 by choosing the design parameters with sufficiently large λ(K_1,i) > 0 and λ(K_2,i) > 2λ(K_1,i) > 0, and sufficiently small σ_k,i > 0 for all i ∈𝐈[1, N] and k ∈𝐈[1, 3].

Proof:
1) Consider the following Lyapunov function candidate for the closed-loop system (<ref>):

    V_i    = 1/2z_1,i^T z_1,i + 1/2z_2,i^T M_i z_2,i + 1/2∑_k=1^3W̃_k,i^T Γ_k,i^-1W̃_k,i.

Evaluating the derivative of V_i along the trajectory of (<ref>) for all i ∈𝐈[1, N] yields:

    V̇_i    = z_1,i^T(-K_1,iz_1,i + J_i(η_i)z_2,i) 
          + z_2,i^T(-J_i^T(η_i)z_1,i - K_2,iz_2,i + W̃_k,i^TS_k,i(Z_i) - ϵ_k,i(Z_i))) 
          - ∑_k=1^3W̃_k,i^T (S_k,i(Z_i)z_2k,i + σ_k,iŴ_k,i) 
       = -z_1,i^T K_1,i z_1,i - z_2,i^T K_2,i z_2,i - z_2,i^T ϵ_k,i(Z_i) 
          - ∑_k=1^3σ_k,iW̃_k,i^T Ŵ_k,i,   ∀ i ∈𝐈[1, N].

Choose K_2,i = K_1,i + K_22,i such that K_1,i, K_22,i∈𝕊_3^+. Using the completion of squares, we have:

    -σ_k,iW̃_k,i^T Ŵ_k,i   ≤ -σ_k,iW̃_k,i^2/2 + σ_k,iW_k,i^*^2/2, 
    
        -z_2,i^T K_22,i z_2,i - z_2,i^T ϵ_i(Z_i)    ≤ϵ_i^T(Z_i) ϵ_i(Z_i)/4 λ(K_22,i)≤ϵ_i^*^2/4 λ(K_22,i).

where ϵ_i^* = [ϵ_1,i^*, ϵ_2,i^*, ϵ_3,i^*]^T. Then, we obtain:

    V̇_i ≤   -z_1,i^T K_1,i z_1,i - z_2,i^T K_1,i z_2,i + ϵ_i^*^2/4 λ(K_22,i)
       + ∑_k=1^3 ( -σ_k,iW̃_k,i^2/2 + σ_k,iW_k,i^*^2/2 .

It follows that V̇_i is negative definite whenever:

    z_1,i   > ϵ_i^*/2√(λ(K_1,i) λ(K_22,i)) + ∑_k=1^3( √(σ_k,i/2 λ(K_1,i))W_k,i^*), 
    z_2,i   > ϵ_i^*/2√(λ(K_1,i) λ(K_22,i)) + ∑_k=1^3( √(σ_k,i/2 λ(K_1,i))W_k,i^*), 
    W̃_k,i   > ϵ_i^*/2√(σ_k,iλ(K_22,i)) + ∑_k=1^3W_k,i^*:= W̃_k,i^*.

For all i ∈𝐈[1, N], ∃ k ∈𝐈[1, 3]. This leads to the Uniformly Ultimately Bounded (UUF) behavior of the signals z_1,i, z_2,i, and W̃_k,i for all i ∈𝐈[1, N] and k ∈𝐈[1, 3]. As a result, it can be easily verified that since η_di = η_i + d^*_i with η_i bounded (according totheorem<ref> and assumption<ref>), η_i = z_1,i + η_i is bounded for all i ∈𝐈[1, N]. Similarly, the boundedness of ν_i = z_2,i + α_i can be confirmed by the fact that α_i in (<ref>) is bounded.
In addition, W_k,i = W̃_k,i + W_k,i^* is also bounded for all i ∈𝐈[1, N] and k ∈𝐈[1, 3] because of the boundedness of W̃_k,i and W_k,i^*. Moreover, in light of (<ref>), α̇_i is bounded as all the terms on the right-hand side of (<ref>) are bounded. This leads to the boundedness of the control signal τ_i in (<ref>) since the Gaussian function vector S_i^F(Z_i) is guaranteed to be bounded for any Z_i. As such, all the signals in the closed-loop system remain UUB, which completes the proof of the first part.

2) For the second part, it will be shown that η_i will converge arbitrarily close to η_di in some finite time T_i > 0 for all i ∈𝐈[1, N]. To this end, we consider the following Lyapunov function candidate for the dynamics of z_1,i and z_2,i in (<ref>):

    V_z,i   = 1/2z_1,i^T z_1,i + 1/2z_2,i^T M_i z_2,i,   ∀ i ∈𝐈[1, N].

The derivative of V_z,i is:

    V̇_z,i   = z_1,i^T (-K_1,iz_1,i + J_i(η_i)z_2,i) + z_2,i^T (-J_i^T(η_i)z_1,i - K_2,iz_2,i + W̃_k,i^T S_k,i(Z_i) - ϵ_i(Z_i)) 
       = -z_1,i^T K_1,i z_1,i - z_2,i^T K_2,i z_2,i + z_2,i^T W̃_k,i^T S_k,i(Z_i) - z_2,i^T ϵ_k,i(Z_i),   ∀ i ∈𝐈[1, N].

Similar to the proof of part one, we let K_2,i = K_1,i + 2K_22,i with K_1,i, K_22,i∈𝕊_3^+. According to <cit.>, the Gaussian RBF NN regressor S_i^F(Z_i) is bounded by S_i^F(Z_i)≤ s_i^* for any Z_i and for all i ∈𝐈[1, N] with some positive number s_i^* > 0. Through completion of squares, we have:

    -z_2,i^T K_22,i z_2,i + z_2,i^T W̃_i^T S_i^F(Z_i)    ≤W̃_i^*^2 s_i^*2/4 λ(K_22,i), 
    
        -z_2,i^T K_22,i z_2,i - z_2,i^T ϵ_i(Z_i)    ≤ϵ_i^*^2/4 λ(K_22,i).

where W̃_i^* = [W̃_1,i^*, W̃_2,i^*, W̃_3,i^*]^T. This leads to:

    V̇_z,i   ≤ -z_1,i^T K_1,i z_1,i - z_2,i^T K_1,i z_2,i + δ_i 
       ≤ -2λ(K_1,i)(1/2z_1,i^T z_1,i + 1/2λ(M_i)z_2,i^T M_i z_2,i) + δ_i 
       ≤ -ρ_i V_z,i + δ_i,   ∀ i ∈𝐈[1, N],

where ρ_i = min{2λ(K_1,i), 2λ(K_1,i)/λ(M_i)} and δ_i = (W̃_i^*^2 s_i^*2/4 λ(K_22,i)) + (ϵ_i^*^2/4 λ(K_22,i)), ∀ i ∈𝐈[1, N]. Solving the inequality (<ref>) yields:

    0 ≤ V_z,i(t) ≤ V_z,i(0)exp(-ρ_i t) + δ_i/ρ_i

which together with (<ref>) implies that:

    min{1, λ(M_i)}1/2(z_1,i^2 + z_2,i^2) ≤ V_z,i(0)exp(-ρ_i t) + δ_i/ρ_i,
             ∀ t ≥ 0,   i ∈𝐈[1, N].

Also

    z_1,i^2 + z_2,i^2 ≤2/min{1, λ(M_i)} V_z,i(0) exp(-ρ_i t) 
        + 2 δ_i/ρ_i min{1, λ(M_i)}.

Consequently, it is straightforward that given δ̅_i > √(2δ_i/ρ_i min{1, λ(M_i)}), there exists a finite time T_i > 0 for all i ∈𝐈[1, N] such that for all t ≥ T_i, both z_1,i and z_2,i satisfy z_1,i(t)≤δ̅_i and z_2,i(t)≤δ̅_i ∀ i ∈𝐈[1, N], where δ̅_i can be made arbitrarily small by choosing sufficiently large λ(K_1,i) > 0 and λ(K_2,i) > 2 λ(K_1,i) > 0 for all i ∈𝐈[1, N]. This ends the proof.

By integrating the outcomes of theorems 1 and 2, the following theorem is established, which can be presented without additional proof:

    
 By Considering the multi-AUV system (<ref>) and the virtual leader dynamics (<ref>) with the network communication topology G and under assumptions <ref> and <ref>, the objective 1) of Problem 1 (i.e., η_i converges to η_0 + d_i^* exponentially for all i ∈𝐈[1, N]) can be achieved by using the cooperative observer (<ref>), (<ref>) and the DDL control law (<ref>) and  (<ref>) with all the design parameters satisfying the requirements in theorems <ref> and <ref>, respectively.
 
    

 
    
    With the proposed two-layer formation learning control architecture, inter-agent information exchange occurs solely in the first-layer DA observation. Only the observer's estimated information, and not the physical plant state information, needs to be shared among neighboring agents. Additionally, since no global information is required for the design of each local AUV control system, the proposed formation learning control protocol can be designed and implemented in a fully distributed manner.

 
    
    It is important to note that the eigenvalue constraints on A_0 in <ref> are not needed for cooperative observer estimation (as detailed in the section (<ref>) or for achieving formation tracking control performance (as discussed in this section). This indicates that formation tracking control can be attained for general reference trajectories, including both periodic paths and straight lines, provided they are bounded. However, these constraints will become necessary in the next section to ensure the accurate learning capability of the proposed method.



§ ACCURATE LEARNING FROM FORMATION CONTROL


It is necessary to demonstrate the convergence of the RBF NN weights in Equations (<ref>) and (<ref>) to their optimal values for accurate learning and identification. The main result of this section is summarized in the following theorem.



 Consider the local closed-loop system (<ref>) with assumptions <ref> and <ref>. For each i ∈𝐈[1, N], if there exists a sufficiently large compact set Ω_Z_i such that Z_i ∈Ω_Z_i for all t ≥ 0, then for any bounded initial conditions and W_k,i(0) = 0 ∀ i ∈𝐈[1, N], k ∈𝐈[1, 3], the local estimated neural weights W_ζ,k,i converge to small neighborhoods of their optimal values W_ζ,k,i^* along the periodic reference tracking orbit ϕ_ζ,i(Z_i(t))|_t ≥ T_i (denoting the orbit of the NN input signal Z_i(t) starting from time T_i). This leads to locally accurate approximations of the nonlinear uncertain dynamics f_k,i(Z_i) ∀ k ∈𝐈[1, 3] in (<ref>) being obtained by W_k,i^T S_k,i(Z_i), as well as by W̅_k,i^T S_k,i(Z_i), where ∀ i ∈𝐈[1, N], k ∈𝐈[1, 3].


    W̅_k,i = mean_t∈ [t_a,i, t_b,i]Ŵ_k,i(t)

Where [t_a,i, t_b,i] (t_b,i > t_a,i > T_i) represents a time segment after the transient process.

Proof: From theorem <ref>, we have shown that for all i ∈𝐈[1, N], η_i will closely track the periodic signal η_d,i = η_0 + d_i^* in finite time T_i. In addition, (<ref>) implies that ν_i will also closely track the signal J_i^T(η_i) η̇_0^i since both z_1,i and z_2,i will converge to a small neighborhood around zero according to <ref>. Moreover, since η̇_0^i will converge to η̇_0 according to <ref>, and <ref> J_i(η_i) is a bounded rotation matrix, ν_i will also be a periodic signal after finite time T_i, because η̇_0 is periodic under <ref>.
Consequently, since the RBF NN input Z_i(t) = col{η_i, ν_i} becomes a periodic signal for all t ≤ T_i, the PE condition of some internal closed-loop signals, i.e., the RBF NN regression subvector S_ζ,k,i(Z_i) (∀ t ≥ T_i), is satisfied according to Lemma <ref>. It should be noted that the periodicity of Z_i(t) leads to the PE of the regression subvector S_ζ,k,i(Z_i), but not necessarily the PE of the whole regression vector S_k,i(Z_i). Thus, we term this as a partial PE condition, and we will show the convergence of the associated local estimated neural weights W_ζ,k,i→ W_ζ,k,i^*, rather than W_k,i→ W_k,i^*.

Thus, to prove accurate convergence of local neural weights W_ζ,k,i associated with the regression subvector S_ζ,k,i(Z_i) under the satisfaction of the partial PE condition, we first rewrite the closed-loop dynamics of z_1,i and z_2,i along the periodic tracking orbit ϕ_ζ,i(Z_i(t))|_t ≥ T_i by using the localization property of the Gaussian RBF NN:

    ż_1,i   = -K_1,i z_1,i + J_i(η_i) z_2,i, 
    ż_2,i   = M_i^-1( -W_ζ,i^*T S_ζ,i^F(Z_i) - ϵ_ζ,i - J_i^T(η_i) z_1,i - K_2,i z_2,i + Ŵ_ζ,i^T S_ζ,i^F(Z_i) + Ŵ_ζ̅,i^T S_ζ̅,i^F(Z_i) ) 
       = M_i^-1 ( -J_i^T(η_i) z_1,i - K_2,i z_2,i + W̃_ζ,i^T S_ζ,i^F(Z_i) - ϵ'_ζ,i ).

where F_i(Z_i) = W_ζ,i^*T S_ζ,i^F(Z_i) + ϵ_ζ,i with W_ζ,i^*T S_ζ,i^F(Z_i) = [W_ζ,1,i^*T S_ζ,1,i(Z_i), W_ζ,2,i^*T S_ζ,2,i(Z_i), W_ζ,3,i^*T S_ζ,3,i(Z_i)]^T and ϵ_ζ,i = [ϵ_ζ,1,i, ϵ_ζ,2,i, ϵ_ζ,3,i]^T being the approximation error. Additionally, W_ζ,i^T S_ζ,i^F(Z_i) + W_ζ̅,i^T S_ζ̅,i^F(Z_i) = W_i^T S_i^F(Z_i) with subscripts ζ and ζ̅ denoting the regions close to and far away from the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, respectively. According to <cit.>, W_ζ̅,i^T S_ζ̅,i^F(Z_i) is small, and the NN local approximation error ϵ'_ζ,i = ϵ_ζ,i - W_ζ̅,i^T S_ζ̅,i^F(Z_i) with ϵ'_ζ,i = O(ϵ_ζ,i) is also a small number. Thus, the overall closed-loop adaptive learning system can be described by:


    [ [     ż_1,i;     ż_2,i; Ẇ̃̇_ζ,1,i; Ẇ̃̇_ζ,2,i; Ẇ̃̇_ζ,3,i ]]    = [ [                                                          [             -K_1,i           J_i(η_i); -M_i^-1 J_i^T(η_i)      -M_i^-1 K_2,i ]                                                                                                                                       Ξ_i; [                     0 -Γ_ζ,1,i S_ζ,1,i(Z_i);                     0 -Γ_ζ,2,i S_ζ,2,i(Z_i);                     0 -Γ_ζ,3,i S_ζ,3,i(Z_i) ]                                                                                                                                         0 ]] ×[ [    z_1,i;    z_2,i; W̃_ζ,1,i; W̃_ζ,2,i; W̃_ζ,3,i ]] + [ [                    0;              -ϵ'_ζ,i; -σ_i,1Γ_ζ,1,iŴ_ζ,1,i; -σ_i,2Γ_ζ,2,iŴ_ζ,2,i; -σ_i,3Γ_ζ,3,iŴ_ζ,3,i ]]


and

    [ Ẇ̃̇_ζ̅,i,1; Ẇ̃̇_ζ̅,i,2; Ẇ̃̇_ζ̅,i,3 ]   = [ -Γ_ζ̅,1,i( S_ζ̅,1,i(Z_i) z_2,i + σ_i,1Ŵ_ζ̅,1,i); -Γ_ζ̅,2,i( S_ζ̅,2,i(Z_i) z_2,i + σ_i,2Ŵ_ζ̅,2,i); -Γ_ζ̅,3,i( S_ζ̅,3,i(Z_i) z_2,i + σ_i,3Ŵ_ζ̅,3,i) ].

where

    Ξ_i    = [                                                                                                                                                  0; M_i^-1[ S_ζ,1,i^T(Z_i)              0              0;              0 S_ζ,2,i^T(Z_i)              0;              0              0 S_ζ,3,i^T(Z_i) ] ].

for all i ∈𝐈[1, N]. The exponential stability property of the nominal part of subsystem (<ref>) has been well-studied in <cit.>, <cit.>, and <cit.>, where it is stated that PE of S_ζ,k,i(Z_i) will guarantee exponential convergence of (z_1,i, z_2,i, W̃_ζ,k,i) = 0 for all i ∈𝐈[1, N] and k ∈𝐈[1, 3]. Based on this, since ϵ'_ζ,i = O(ϵ_ζ,i) = O(ϵ_i), and σ_k,iΓ_ζ,k,iŴ_ζ,k,i can be made small by choosing sufficiently small σ_k,i for all i ∈𝐈[1, N], k ∈𝐈[1, 3], both the state error signals (z_1,i, z_2,i) and the local parameter error signals W̃_ζ,k,i (∀ i ∈𝐈[1, N], k ∈𝐈[1, 3]) will converge exponentially to small neighborhoods of zero, with the sizes of the neighborhoods determined by the RBF NN ideal approximation error ϵ_i as in (<ref>) and σ_k,iΓ_ζ,k,iŴ_ζ,k,i.
The convergence of W_ζ,k,i→ W_ζ,k,i^* implies that along the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, we have

    f_k,i(Z_i)    = W_ζ,k,i^*T S_ζ,k,i(Z_i) + ϵ_ζ,k,i
       = Ŵ_ζ,k,i^T S_ζ,k,i(Z_i) - W̃_ζ,k,i^T S_ζ,k,i(Z_i) + ϵ_ζ,k,i
       = Ŵ_ζ,k,i^T S_ζ,k,i(Z_i) + ϵ_ζ_1,k,i
       = W̅_ζ,k,i^T S_ζ,k,i(Z_i) + ϵ_ζ_2,k,i,

where for all i ∈𝐈[1, N], k ∈𝐈[1, 3], ϵ_ζ_1,k,i = ϵ_ζ,k,i - W̃_ζ,k,i^T S_ζ,k,i(Z_i) = O(ϵ_ζ,i) due to the convergence of W̃_ζ,k,i→ 0. The last equality is obtained according to the definition of (<ref>) with W̅_ζ,k,i being the corresponding subvector of W̅_k,i along the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, and ϵ_ζ_2,k,i being an approximation error using W̅_ζ,k,i^T S_ζ,k,i(Z_i). Apparently, after the transient process, we will have ϵ_ζ_2,k,i = O(ϵ_ζ_1,k,i), ∀ i ∈𝐈[1, N], k ∈𝐈[1, 3].
Conversely, for the neurons whose centers are distant from the trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, the values of S_ζ̅,k,i(Z_i) will be very small due to the localization property of Gaussian RBF NNs. From the adaptation law (17) with W^k_i(0) = 0, it can be observed that these small values of S_ζ̅,k,i(Z_i) will only minimally activate the adaptation of the associated neural weights W_ζ̅,k,i. As a result, both W_ζ̅,k,i and W_ζ̅,k,i^T S_ζ̅,k,i(Z_i), as well as W̅_ζ̅,k,i and W̅_ζ̅,k,i^T S_ζ̅,k,i(Z_i), will remain very small for all i ∈𝐈[1, N], k ∈𝐈[1, 3] along the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i.
This indicates that the entire RBF NN W_k,i^T S_k,i(Z_i) and W̅_k,i^T S_k,i(Z_i) can be used to accurately approximate the unknown function f_k,i(Z_i) locally along the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, meaning that

    f_k,i(Z_i)    = Ŵ_ζ,k,i^T S_ζ,k,i(Z_i) + ϵ_ζ_1,k,i = Ŵ_k,i^T S_k,i(Z_i) + ϵ_1,k,i
       = W̅_ζ,k,i^T S_ζ,k,i(Z_i) + ϵ_ζ_2,k,i = W̅_k,i^T S_k,i(Z_i) + ϵ_2,k,i.

with the approximation accuracy level of ϵ_1,k,i = ϵ_ζ_1,k,i - W_ζ̅,k,i^T S_ζ̅,k,i(Z_i) = O(ϵ_ζ_1,k,i) = O(ϵ_k,i) and ϵ_2,k,i = ϵ_ζ_2,k,i - W̅_ζ̅,k,i^T S_ζ̅,k,i(Z_i) = O(ϵ_ζ_2,k,i) = O(ϵ_k,i) for all i ∈𝐈[1, N], k ∈𝐈[1, 3]. This ends the proof.


    
    The key idea in the proof of theorem <ref> is inspired by <cit.>. For more detailed analysis on the learning performance, including quantitative analysis on the learning accuracy levels ϵ_1,i,k and ϵ_2,i,k as well as the learning speed, please refer to <cit.>. Furthermore, the AUV nonlinear dynamics (<ref>) to be identified do not contain any time-varying random disturbances. This is important to ensure accurate identification/learning performance under the deterministic learning framework. To understand the effects of time-varying external disturbances on deterministic learning performance, interested readers are referred to <cit.> for more details.


    
    Based on (<ref>), to obtain the constant RBF NN weights W̅_k,i for all i ∈𝐈[1, N], k ∈𝐈[1, 3], one needs to implement the formation learning control law (<ref>), (<ref>) first. Then, according to Theorem 4, after a finite-time transient process, the RBF NN weights W_k,i will converge to constant steady-state values. Thus, one can select a time segment [t_a,i, t_b,i] with t_b,i > t_a,i > T_i for all i ∈𝐈[1, N] to record and store the RBF NN weights W_k,i(t) for t ∈ [t_a,i, t_b,i]. Finally, based on these recorded data, W̅_k,i can be calculated off-line using (<ref>).


    
    It is shown in theorem <ref> that locally accurate learning of each individual AUV's nonlinear uncertain dynamics can be achieved using localized RBF NNs along the periodic trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i. The learned knowledge can be further represented and stored in a time-invariant fashion using constant RBF NNs, i.e., W̅_k,i^T S_k,i(Z_i) for all i ∈𝐈[1, N], k ∈𝐈[1, 3]. In contrast to many existing techniques (e.g., <cit.> and <cit.>), this is the first time, to the authors’ best knowledge, that locally accurate identification and knowledge representation using constant RBF NNs are accomplished and rigorously analyzed for multi-AUV formation control under complete uncertain dynamics.



§ FORMATION CONTROL WITH PRE-LEARNED DYNAMICS


In this section, we will further address objective 2 of problem <ref>, which involves achieving formation control without readapting to the AUV's nonlinear uncertain dynamics. To this end, consider the multiple AUV systems (<ref>) and the virtual leader dynamics (<ref>). We employ the estimator observer (<ref>), (<ref>) to cooperatively estimate the leader's state information. Instead of using the DDL feedback control law (<ref>), and self-adaptation law (<ref>), we introduce the following constant RBF NN controller, which does not require online adaptation of the NN weights:

    τ_i = -J_i^T(η_i)z_1,i - K_2,iz_2,i + W̅_i^TS_i^F(Z_i)

where W̅_ki^TS_ki^F(Z_i) = [W̅_1,i^TS_1,i(Z_i), W̅_2,i^TS_2,i(Z_i), W̅_3,i^TS_3,i(Z_i)]^T is ubtained from (<ref>). The term W̅_k,i^T S_k,i(Z_i) represents the locally accurate RBF NN approximation of the nonlinear uncertain function f_k,i(Z_i) along the trajectory ϕ_ζ,i(Z_i(t))|_t ≥ T_i, and the associated constant neural weights W̅_k,i are obtained from the formation learning control process as discussed in remark <ref>.

    
    Consider the multi-AUV system (<ref>) and the virtual leader dynamics (<ref>) with the network communication topology G. Under assumptions <ref> and <ref>, the formation control performance (i.e., η_i converges to η_0 + d^*_i exponentially with the same η_0 and d^*_i defined in theorem <ref> for all i ∈𝐈[1, N]) can be achieved by using the DA observer (<ref>), (<ref>) and the constant RBF NN control law (<ref>) with the constant NN weights obtained from (<ref>).

Proof: The closed-loop system for each local AUV agent can be established by integrating the controller (<ref>) with the AUV dynamics (<ref>). 

    ż_1,i=   -K_1,iz_1,i + J_i(η _i)z_2,i
    ż_2,i=   M_i^-1(-J_i^T(η _i)z_1,i - K_2,iz_2,i + W̅_i^TS_i^F(Z_i) - F_i(Z_i))
    
           =   M_i^-1(-J_i^T(η _i)z_1,i - K_2,iz_2,i - ϵ _2,i),   ∀ i∈𝐈 [1,N ]

where ϵ_2,i = [ϵ_21,i, ϵ_22,i, ϵ_23,i]^T. Consider the Lyapunov function candidate V_z,i = 1/2 z_1,i^T z_1,i + 1/2 z_2,i^T M_i z_2,i, whose derivative along the closed-loop system described is given by:

    V̇_z,i =    z_1,i^T (-K_1,i z_1,i + J_i(η_i) z_2,i) + z_2,i^T (-J_i^T(η_i) z_1,i - K_2,i z_2,i - ϵ_2,i) 
    
            =    -z_1,i^T K_1,i z_1,i - z_2,i^T K_2,i z_2,i - z_2,i^T ϵ_2,i.

Selecting K_2,i = K_1,i + K_22,i where K_1,i, K_22,i∈𝕊_3^+, we can utilize the method of completing squares to obtain:

    -z_2,i^T K_22,i z_2,i - z_2,i^T ϵ_2,i≤(ϵ_2,i^2/4 λ(K_22,i)) ≤(ϵ^*_2,i^2/4 λ(K_22,i)),

which implies that:

    V̇_z,i≤ -z_1,i^T K_1,i z_1,i - z_2,i^T K_1,i z_2,i + (ϵ^*_2,i^2/4 λ(K_22,i)) ≤ -ρ_i V_z,i + δ_i,   ∀ i∈𝐈 [1,N ]

where ρ_i = min{2 λ(K_1,i), (2 λ(K_1,i) / λ(M_i))} and δ_i = (ϵ^*_2,i^2 / 4 λ(K_22,i)).
Using similar reasoning to that in the proof of theorem <ref>, it is evident from the derived inequality that all signals within the closed-loop system remain bounded. Additionally, η_i - η^d_i will converge to a small neighborhood around zero within a finite period. The magnitude of this neighborhood can be minimized by appropriately choosing large values for λ(K_1,i) > 0 and λ(K_2,i) > λ(K_1,i) across all i ∈𝐈[1, N]. In line with theorem <ref>, under assumptions <ref> and <ref>, the implementation of the DA observer  DA observer (<ref>), (<ref>) facilitates the exponential convergence of η_i towards η_0. This conjunction of factors assures that η_i rapidly aligns with η_d,i = η_0 + d^*_i, achieving the objectives set out for formation control.

    
    Building on the locally accurate learning outcomes discussed in Section <ref>, the newly developed distributed control protocol comprising (<ref>), (<ref>), and  (<ref>) facilitates stable formation control across a repeated formation pattern. Unlike the formation learning control approach outlined in Section <ref>, which involves (<ref>), (<ref>) coupled with (<ref>) and (<ref>), the current method eliminates the need for online RBF NN adaptation for all AUV agents. This significantly reduces the computational demands, thereby enhancing the practicality of implementing the proposed distributed RBF NN formation control protocol. This innovation marks a significant advancement over many existing techniques in the field.



§ SIMULATION


We consider a multi-AUV heterogeneous system composed of 5 AUVs for the simulation. The dynamics of these AUVs are described in THE system (<ref>). The system parameters for each AUV are specified as follows:

    M_i   = [ m_11,i      0      0;      0 m_22,i m_23,i;      0 m_23,i m_33,i ],
    
            C_i   = [                        0                        0 -m_22,i v_i - m_23,i r_i;                        0                        0              -m_11,i u_i;  m_22,i v_i + m_23,i r_i              -m_11,i u_i                        0 ],
    
            D_i   = [ d_11,i (ν_i)            0            0;            0 d_22,i (ν_i) d_23,i (ν_i);            0 d_32,i (ν_i) d_33,i (ν_i) ],    g_i = 0,
    Δ_i   = [ Δ_1,i (χ_i); Δ_2,i (χ_i); Δ_3,i (χ_i) ],   ∀ i ∈𝐈 [1, 5]

where the mass and damping matrix components for each AUV i are defined as:

    m_11,i   = m_i - X_u̇,i,     m_22,i   = m_i - Y_v̇,i, 
    
        m_23,i   = m_i x_g,i - Y_ṙ,i,     m_33,i   = I_z,i - N_ṙ,i, 
    
        d_11,i   = -(X_u,i + X_uu,iu_i),     d_22,i   = -(Y_v,i + Y_vv,iv_i + Y_rv,ir_i), 
    
        d_23,i   = -(Y_r,i + Y_vr,iv_i + Y_rr,ir_i),     d_32,i   = -(N_v,i + N_vv,iv_i + N_rv,ir_i), 
    
        d_33,i   = -(N_r,i + N_vr,iv_i + N_rr,ir_i).

 according to the notations in <cit.> and <cit.> the coefficients {X(·), Y(·), N(·)} are hydrodynamic parameters. For the associated system parameters are borrowed from <cit.> (with slight modifications for different AUV agents) and simulation purposes and listed in table <ref>. For all i ∈𝐈[1,5], we set x_g,i = 0.05 and Y_ṙ,i = Y_rv,i = Y_vr,i = Y_rr,i = N_rv,i = N_rr,i = N_vv,i = N_vr,i = N_r,i = 0.
Model uncertainties are given by:


    Δ _1=   0,   Δ _2 =
        [ 0.2u_2^2+0.3v_2           -0.95         0.33r_2 ]^T
    Δ _3=   [ -0.58+cos (v_3 )        0.23r_3^3        0.74u_3^2 ]^T
    Δ _4=   [             -0.31                 0 0.38u_4^2 + v_4^3 ]^T
    Δ _5=   [       sin (v_5 ) cos (u_5 + r_5 )            -0.65 ]^T.

Fig. <ref> illustrates the communication topology and the spanning tree where agent 0 is the virtual leader and is considered as the root, in accordance with assumption <ref>. The desired formation pattern requires each AUV, η_i, to track a periodic signal generated by the virtual leader η_0. The dynamics of the leader are defined as follows:


    [ η̇_0; ν̇_0 ] =    [                                0 [  1  0  0;  0 -1  0;  0  0  1 ]; [ -1  0  0;  0  1  0;  0  0 -1 ]                                0 ][ η_0; ν_0 ], 
    [ η_0(0); ν_0(0) ] =    [  0 80  0 80  0 80 ]^T.

The initial conditions and system matrix are structured to ensure all eigenvalues of A_0 lie on the imaginary axis, thus satisfying Assumption <ref>. The reference trajectory for η_0 is defined as [80sin(t), 80cos(t), 80sin(t)]^T. The predefined offsets d^*_i, which determine the relative positions of the AUVs to the leader, are specified as follows:

    d^*_1 = [0, 0, 0]^T,   d^*_4 = [-10, 10, 0]^T,   d^*_2 = [10, -10, 0]^T,   d^*_5 = [-10, -10, 0]^T,   d^*_3 = [10, 10, 0]^T

Each AUV tracks its respective position in the formation by adjusting its location to η_i = η_0 + d^*_i.


 §.§ DDL Formation Learning Control Simulation


The estimated virtual leader's state, derived from the cooperative estimator in the first layer (see equation (<ref>) and (<ref>)), is utilized to estimate each agent's complete uncertain dynamics within the DDL controller (second layer) using equations (<ref>) and (<ref>). The uncertain nonlinear functions F_i(Z_i) for each agent are approximated using RBF NNs, as described in equation (<ref>). Specifically, for each agent i ∈{1, …, 5}, the nonlinear uncertain functions F_i(Z_i), dependent on ν_i, are modeled. The input to the NN, Z_i = [u_i, v_i, r_i]^T, allows the construction of Gaussian RBF NNs, represented by W^T_k,i S_k,i(Z_i), utilizing 4096 neurons arranged in an 16 × 16 × 16 grid. The centers of these neurons are evenly distributed over the state space [-100, 100] × [-100, 100] × [-100, 100], and each has a width γ_k,i = 60, for all i ∈{1, …, 5} and k ∈{1, 2, 3}.

The observer and controller parameters are chosen as β_1 = β_2 = 5, and the diagonal matrices K_1,i = 800 * diag{1.2, 1, 1} and K_2,i = 1200 * diag{1.2, 1, 1}, with Γ_k,i = 10 and σ_k,i = 0.0001 for all i ∈{1, …, 5} and k ∈{1, 2, 3}. The initial conditions for the agents are set as η_1(0) = [30, 60, 0]^T, η_2(0) = [40, 70, 0]^T, η_3(0) = [50, 80, 0]^T, η_4(0) = [10, 70, 0]^T, and η_5(0) = [10, 50, 0]^T. Zero initial conditions are assumed for all the distributed observer states (χ^i_0, A^i_0) and the DDL controller states W^k_i for all i ∈{1, …, 5} and k ∈{1, 2, 3}. Time-domain simulation is carried out using the DDL formation learning control laws as specified in equations (<ref>) and (<ref>), along with equations (<ref>) and (<ref>).

Figure <ref> displays the simulation results of the cooperative estimator (first layer) for all five agents. It illustrates how each agent's estimated states,η̂_i, converge perfectly to the leader's states η̂_0 thorough (<ref>) and (<ref>).





    [b]0.5
        
    < g r a p h i c s >

        x̂_i → x_0  (m).
        
      
   


    [b]0.5
        
    < g r a p h i c s >

        ŷ_i → y_0  (m).
        
    



    [b]0.5
        
    < g r a p h i c s >

        ψ̂_i →ψ_0  (deg).
        
    
    


    Distributed observer for all  three states of each AUVs
    

Fig <ref> presents the position tracking control responses of all agents. Fig <ref>a, <ref>b, and <ref>c illustrate the tracking performance of AUVs along the x-axis, y-axis, and vehicle heading, respectively, demonstrating effective tracking of the leader's position signal. While the first AUV exactly tracks the leader's states, agents 2 through 5 are shown to successfully follow agent 1, maintaining prescribed distances and alignment along the x and y axes, and matching the same heading angle. These results underscore the robustness of the real-time tracking control system, which enforces a predefined formation pattern, initially depicted in Fig <ref>. Additionally, Fig <ref> highlights the real-time control performance for all agents, showcasing the effectiveness of the tracking strategy in maintaining the formation pattern.





    [b]0.5
        
    < g r a p h i c s >

        x_i → x_0  (m).
        
      
   


    [b]0.5
        
    < g r a p h i c s >

        y_i → y_0  (m).
        
    



    [b]0.5
        
    < g r a p h i c s >

        ψ_i →ψ_0  (deg).
        
    
    


    Postition Tracking Control for all three states of each AUVs
    



 
The sum of the absolute values of the neural network (NN) weights in Fig. <ref>, along with the NN weights depicted in Fig. <ref>, demonstrates that the function approximation from the second layer is achieved accurately. This is evidenced by the convergence of all neural network weights to their optimal values during the learning process, which is in alignment with Theorem <ref>.
 
Fig. <ref> presents the neural network approximation results for the unknown system dynamics F_3(Z_3) (as defined in (<ref>) for the third AUV, using a Radial Basis Function (RBF) Neural Network. The approximations are plotted for both W^T_k,3 S_k,3(Z_3) and W̅^T_k,3 S_k,3(Z_3) for all k ∈ I[1,3]. The results confirm that locally accurate approximations of the AUV's nonlinear dynamics are successfully achieved. Moreover, this learned knowledge about the dynamics is effectively stored and represented using localized constant RBF NNs.










    [b]0.5
        
    < g r a p h i c s >

        Ŵ_1,3.
        
      
   


    [b]0.5
        
    < g r a p h i c s >

        Ŵ_2,3.
        
    



    [b]0.5
        
    < g r a p h i c s >

        Ŵ_3,3.
        
    
    


    Neural Network weights convergence for the third AUV (i = 3).
    







    [b]0.5
        
    < g r a p h i c s >

        k=1.
        
      
   


    [b]0.5
        
    < g r a p h i c s >

        k=2.
        
    



    [b]0.5
        
    < g r a p h i c s >

        k=3.
        
    
    


    Function approximation for k = 1, 2, 3 for the 3^rd AUV (i = 3): Comparison of W̅_k,3^TS_k,3(Z_3), W_k,3^TS_k,3(Z_3), and f_k,3(Z_3).
    




 §.§ Simulation for Formation Control with Pre-learned Dynamics


To evaluate the distributed control performance of the multi-AUV system, we implemented the pre-learned distributed formation control law. This strategy integrates the estimator observer (<ref>) and (<ref>), this time coupled with the constant RBF NN controller (<ref>). We employed the virtual leader dynamics described in (<ref>) to generate consistent position tracking reference signals, as previously discussed in Section <ref>. To ensure a fair comparison, identical initial conditions and control gains and inputs were used across all simulations. Fig <ref> illustrates the comparison of the tracking control results from (<ref>) and (<ref>) with the results using pre-trained weights w̅ in (<ref>).






    [b]0.5
        
    < g r a p h i c s >

        x_i → x_0  (m).
        
      
   


    [b]0.5
        
    < g r a p h i c s >

        y_i → y_0  (m).
        
    



    [b]0.5
        
    < g r a p h i c s >

        ψ_i →ψ_0  (deg).
        
    
    


    Postition Tracking Control using pretrained weights (w̅)
    


The control experiments and simulation results presented demonstrate that the constant RBF NN control law (<ref>) can achieve satisfactory tracking control performance comparable to that of the adaptive control laws (<ref>) and (<ref>), but with notably less computational demand. The elimination of online recalculations or readaptations of the NN weights under this control strategy significantly reduces the computational load. This reduction is particularly advantageous in scenarios involving extensive neural networks with a large number of neurons, thereby conserving system energy and enhancing operational efficiency in real-time applications.

For the design of this framework, all system dynamics are assumed to be unknown, which allows its application to any AUV, regardless of environmental conditions. This universality includes adaptation to variations such as water flow, which can increase the AUV's effective mass through the added mass phenomenon, as well as influence the AUV's inertia. Additionally, buoyancy force, which varies with depth, impacts the inertia. The lift force, dependent on varying water flows or currents and the AUV's shape or appendages, along with the drag force from water viscosity, significantly influences the damping matrix in the AUV's dynamics. These factors ensure the framework's robustness across diverse operational scenarios, allowing effective operation across a range of underwater conditions including those affected by hydrodynamic forces and torques.

The controller's design enables it to maintain stability and performance in dynamic and often unpredictable underwater environments. The stored weights from the training process enable the AUVs to recall and apply learned dynamics efficiently, even after the systems are shut down and restarted. This feature ensures that AUVs equipped with our control system can quickly resume operations with optimal control settings, regardless of environmental changes or previous operational states. Our approach not only demonstrates a significant advancement in the field of autonomous underwater vehicle control but also establishes a foundation for future enhancements that could further minimize energy consumption and maximize the adaptability and resilience of AUV systems in challenging marine environments



§ CONCLUSION


In conclusion, this paper has introduced a novel two-layer control framework designed for Autonomous Underwater Vehicles (AUVs), aimed at universal applicability across various AUV configurations and environmental conditions. This framework assumes all system dynamics to be unknown, thereby enabling the controller to operate independently of specific dynamic parameters and effectively handle any environmental challenges, including hydrodynamic forces and torques. The framework consists of a first-layer distributed observer estimator that captures the leader's dynamics using information from adjacent agents, and a second-layer decentralized deterministic learning controller. Each AUV utilizes the estimated signals from the first layer to determine the desired trajectory, simultaneously training its own dynamics using Radial Basis Function Neural Networks (RBF NNs). This innovative approach not only sustains stability and performance in dynamic and unpredictable environments but also allows AUVs to efficiently utilize previously learned dynamics after system restarts, facilitating rapid resumption of optimal operations. The robustness and versatility of this framework have been rigorously confirmed through comprehensive simulations, demonstrating its potential to significantly enhance the adaptability and resilience of AUV systems. By embracing total uncertainty in system dynamics, this framework establishes a new benchmark in autonomous underwater vehicle control and lays a solid groundwork for future developments aimed at minimizing energy use and maximizing system flexibility.









The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


§ AUTHOR CONTRIBUTIONS

Emadodin Jandaghi and Chengzhi Yuan led the research and conceptualized the study. Emadodin Jandaghi developed the methodology, conducted the derivation, design, implementation of the method, and simulations, and wrote the manuscript. Chengzhi Yuan, Mingxi Zhou, and Paolo Stegagno contributed to data analysis and provided insights on the control and oceanographic aspects of the research. Chengzhi Yuan supervised the project, assisted with securing resources, and reviewed the manuscript. All authors reviewed and approved the final version of the manuscript.


§ FUNDING

This work is supported in part by the National Science Foundation under Grant CMMI-1952862 and CMMI-2154901.









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