# Robust Adaptive Neural Cooperative Control for the USV-UAV Based on the LVS-LVA Guidance Principle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem for Formulation and Preliminaries

#### 2.1. Mathematical Model of USV-UAV with Euler–Lagrange Form

**Remark**

**1.**

#### 2.2. RBF-NNs

**Lemma**

**1**

**.**For any given nonlinear continuous function $f\left(\mathit{x}\right)$ with $f\left(0\right)=0$ in the compact set $\mathsf{\Omega}\in \Re $, one can approximate the nonlinear function by the RBF-NNs.

#### 2.3. LVS-LVA Guidance for the USV-UAV System

**Remark**

**2.**

## 3. Robust Adaptive Neural Cooperative Controller

#### 3.1. Control Design

**Step 1**: One defines the position error ${x}_{je},{y}_{je},{z}_{ae},j=s,a$ for the USV-UAV system.

**Case**

**1.**

**Case**

**2.**

**Step 2**: According to the relationship between ${\beta}_{u},{\beta}_{x},{\beta}_{y},{\beta}_{z}$ and the attitudes information of the UAV, the reference roll angle and the pitch angle can be deduced as Equation (22) by using the nonlinear decoupling technique.

**Remark**

**3.**

#### 3.2. Stability Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**4.**

## 4. Numerical Simulation

**Remark**

**5.**

**Remark**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**The position and heading errors for the USV-UAV ((

**a**) is heading errors for USV-UAV. (

**b**) is position errors for USV-UAV).

**Figure 9.**The adaptive parameters for the USV-UAV ((

**a**) is adaptive parameters for USV. (

**b**) is adaptive parameters for UAV).

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**MDPI and ACS Style**

Li, J.; Zhang, G.; Li, B.
Robust Adaptive Neural Cooperative Control for the USV-UAV Based on the LVS-LVA Guidance Principle. *J. Mar. Sci. Eng.* **2022**, *10*, 51.
https://doi.org/10.3390/jmse10010051

**AMA Style**

Li J, Zhang G, Li B.
Robust Adaptive Neural Cooperative Control for the USV-UAV Based on the LVS-LVA Guidance Principle. *Journal of Marine Science and Engineering*. 2022; 10(1):51.
https://doi.org/10.3390/jmse10010051

**Chicago/Turabian Style**

Li, Jiqiang, Guoqing Zhang, and Bo Li.
2022. "Robust Adaptive Neural Cooperative Control for the USV-UAV Based on the LVS-LVA Guidance Principle" *Journal of Marine Science and Engineering* 10, no. 1: 51.
https://doi.org/10.3390/jmse10010051