Formation Control of Multiple Autonomous Underwater Vehicles under Communication Delay, Packet Discreteness and Dropout
Abstract
:1. Introduction
 (1)
 The communication delay is estimated based on the kernel density estimation method. Kernel density estimation is a nonparametric estimation method that does not need prior knowledge and an accurate mathematical model of communication delay. Instead, we can obtain an accurate distribution of delay with this method according to the characteristics and properties of delay values based on underwater experiments. Compared with other methods, this method has more extensive applications.
 (2)
 The packet discreteness and dropout problems are solved by information prediction based on the curve fitting method. This method can be used to predict the key states of the leader AUV to generate a continuous and precise trajectory for the follower AUVs.
 (3)
 We derive a kinematic model for the error of the formation control system. The follower controller is designed using the input–output feedback linearization method and the stability of this method is proved by Lyapunov stability theory.
 (4)
 Both the simulation in MATLAB and the field tests on the lake are carried out to verify the feasibility of the scheme presented in this paper.
2. Problem Formulation and Preliminaries
2.1. AUV Model
2.2. Description of Formation Control
3. Solutions of Communication Delay, Packet Discreteness and Dropout
3.1. Estimation of Communication Delay
3.2. Prediction of Leader States
4. Formation Control Scheme
4.1. Generate Continuous Trajectories of the Leader
Algorithm 1. Continuous Trajectories of the Leader 
1: The followers receive packets from the leader $\left({x}_{1}\left({t}_{i}\right),{y}_{1}\left({t}_{i}\right),{u}_{1}\left({t}_{i}\right),{\Psi}_{1}\left({t}_{i}\right)\right)$ 
2: if $K<k$ 3: if$t={t}_{i}$, where $t$ denotes the time 4: then we obtain the communication delay using Equation (17) and the states of the leader are corrected 5:
$$\widehat{{x}_{1}}\left({t}_{i}\right)={x}_{1}\left({t}_{i}\right)+\tau {u}_{1}\left({t}_{i}\right)\mathrm{sin}\left(\Psi \left({t}_{i}\right)\right)$$
$$\widehat{{y}_{1}}\left({t}_{i}\right)={y}_{1}\left({t}_{i}\right)+\tau {u}_{1}\left({t}_{i}\right)\mathrm{cos}\left(\Psi \left({t}_{i}\right)\right)$$
7: then continuous trajectories of the leader are predicted 8:
$$\widehat{{x}_{1}}\left(t\right)={x}_{1}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)+\left(t{t}_{i}\right){u}_{1}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)\mathrm{sin}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)$$
$$\widehat{{y}_{1}}\left(t\right)={y}_{1}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)+\left(t{t}_{i}\right){u}_{1}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)\mathrm{cos}\left(\mathrm{arg}\underset{{t}_{i}}{\mathrm{min}}(t{t}_{i})\right)$$

9: else$K\ge k$ 10: if$t={t}_{i}$ 11: then we obtain the communication delay using Equation (17) and the states of the leader are corrected 12:
$$\widehat{{x}_{1}}\left({t}_{i}\right)=x\left({t}_{i}+\tau \right)=P({t}_{i}+\tau )M$$
$$\widehat{{y}_{1}}\left({t}_{i}\right)=y\left({t}_{i}+\tau \right)=P({t}_{i}+\tau )N$$
14: then $M$ and $N$ are updated using Equation (20) and the continuous trajectories of the leader are predicted according to Equations (21) and (22) 15: end 
4.2. Design the Follower Controller
5. Simulation Results
5.1. Simulation Environment
 Experimental methods
 Experimental scenario
 Communication environments
5.2. Simulation Results and Discussion
6. Field Tests
6.1. Vehicle Characteristics
6.2. Preparation and Scenario
6.3. Test Results and Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Parameter  Leader AUV  Follower AUV_{1}  Follower AUV_{2}  Follower AUV_{3} 

Starting point (x, y)  (0,0)  (0,0)  (0,0)  (0,0) 
Yaw (rad)  $\pi /2$  $\pi /2$  $\pi /2$  $\pi /2$ 
Linear velocity (m/s)  1.0289  none *  none  none 
Yaw velocity (rad/s)  −0.0175  none  none  none 
$\mathrm{Formation}\mathrm{structures}({L}^{d},{\Phi}^{d}$)  none  $\left(10,\pi \right)$  $\left(5\sqrt{2},\frac{5\pi}{4}\right)$  $\left(5\sqrt{2},\frac{3\pi}{4}\right)$ 
Maximum linear velocity (m/s)  1.5433  1.5433  1.5433  1.5433 
Maximum yaw velocity (rad/s)  0.1  0.1  0.1  0.1 
Parameter  Environment1  Environment2  Environment3  Environment4 

Communication delay (s)  $N\left(3,0.1\right)$ *  $N\left(3,0.1\right)$  $N\left(5,0.1\right)$  $N\left(3,0.1\right)$ 
Interval of sent packets (s)  1  2  1  1 
Packet dropout rate (%)  20  20  20  40 
Parameter  Environment1  Environment2  Environment3  Environment4  Average 

Errors of distance (m)  3.902  4.988  5.754  4.658  4.826 
Errors of angle (rad)  0.226  0.261  0.284  0.253  0.256 
Parameter  Environment1  Environment2  Environment3  Environment4  Average 

Errors of estimated communication delay (s)  0.092  0.095  0.087  0.098  0.093 
Errors of the prediction of the leader states (m)  0.239  0.236  0.277  0.232  0.246 
Errors of distance (m)  0.242  0.234  0.283  0.251  0.216 
Errors of angle (rad)  0.021  0.022  0.026  0.021  0.023 
Parameter  Method1  Method2  Method3  Method 4 

Errors of distance (m)  4.826  0.216  2.137  2.695 
Errors of angle (rad)  0.256  0.023  0.152  0.169 
Parameter  Test1  Test2  Test3  Test4  Average 

Packet dropout rate (%)  31.3  32.4  31.8  31.1  31.5 
Errors of distance (m)  12.204  11.989  12.221  12.187  12.150 
Errors of angle (rad)  0.408  0.419  0.412  0.399  0.410 
Parameter  Test1  Test2  Test3  Test4  Average 

Packet dropout rate (%)  33.1  31.2  30.5  32.3  31.8 
Errors of estimated communication delay (s)  0.103  0.092  0.096  0.108  0.998 
Errors of the prediction of the leader states (m)  0.833  1.142  0.986  1.048  1.002 
Errors of distance (m)  5.453  5.398  5.429  5.364  5.411 
Errors of angle (rad)  0.218  0.205  0.211  0.196  0.208 
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Li, L.; Li, Y.; Zhang, Y.; Xu, G.; Zeng, J.; Feng, X. Formation Control of Multiple Autonomous Underwater Vehicles under Communication Delay, Packet Discreteness and Dropout. J. Mar. Sci. Eng. 2022, 10, 920. https://doi.org/10.3390/jmse10070920
Li L, Li Y, Zhang Y, Xu G, Zeng J, Feng X. Formation Control of Multiple Autonomous Underwater Vehicles under Communication Delay, Packet Discreteness and Dropout. Journal of Marine Science and Engineering. 2022; 10(7):920. https://doi.org/10.3390/jmse10070920
Chicago/Turabian StyleLi, Liang, Yiping Li, Yuexing Zhang, Gaopeng Xu, Junbao Zeng, and Xisheng Feng. 2022. "Formation Control of Multiple Autonomous Underwater Vehicles under Communication Delay, Packet Discreteness and Dropout" Journal of Marine Science and Engineering 10, no. 7: 920. https://doi.org/10.3390/jmse10070920