Dual-Loop Prescribed-Time 3D Formation Control of Underactuated Multi-AUVs

doi:10.12382/bgxb.2025.0046

Abstract

Abstract:

The cooperative operation of multiple autonomous underwater vehicles (AUVs) shows broad application prospects in underwater combat scenarios.A dual-loop prescribed-time formation control method is proposed for the control of underactuated multi-AUV three-dimensional formation under model uncertainties and unknown environmental disturbances.A mathematical model of AUV motion is established and an underactuated coordinate transformation is introduced to solve the underactuation problem of AUV.To mitigate the adverse effects of model uncertainty and unknown environmental disturbances on the system,a fixed-time extended state observer is designed to estimate and compensate for model uncertainty and unknown environmental disturbances.In order to achieve the prescribed-time convergence in multi-AUV formation,a dual-loop prescribed-time controller,consisting of an outer-loop control law and an inner-loop control law,is designed.The outer-loop control law is used to adjust the desired velocity of AUV to achieve the formation objective,while the inner-loop control law is responsible for tracking the desired velocity.Theoretical analysis demonstrates that the proposed method enable the system to converge within a prescribed time.+++Simulated results show that,compared to the finite-time and fixed-time control methods,the proposed method can achieve prescribed-time convergence and further enhances the system's convergence speed,providing a new approach for multi-AUV formation control in tasks with high coordination requirements.

Key words: autonomous underwater vehicle, formation control, prescribed-time control, dual-loop control, extended state observer

CLC Number:

TP273

WANG Zeqing, XU Haixiang, YU Wenzhao, DU Zhe, WANG Hongmei. Dual-Loop Prescribed-Time 3D Formation Control of Underactuated Multi-AUVs[J]. Acta Armamentarii, 2025, 46(8): 250046-.

Figures/Tables 13

Fig.1 Earth-fixed coordinate system and body- fixed coordinate system

Fig.2 Schematic diagram of AUV dual-loop prescribed time formation control

Fig.3 Multi-AUV 3D formation control system

Fig.4 Communication topology of AUV formation

Table 1 Initial states and desired formation of AUV

参数	数值	参数	数值
η₁(0)	[-6m,-20m,3m,0°,0°]^T	$δ ¯ 1$	[0m,-20m,0m]^T
η₂(0)	[-4m,-8m,-2m,0°,0°]^T	$δ ¯ 2$	[0m,-10m,0m]^T
η₃(0)	[-6m,0m,3m,0°,0°]^T	$δ ¯ 3$	[0m,0m,0m]^T
η₄(0)	[-4m,7m,-1m,0°,0°]^T	$δ ¯ 4$	[0m,10m,0m]^T
η₅(0)	[-5m,19m,-2m,0°,0°]^T	$δ ¯ 5$	[0m,20m,0m]^T
υ_i(0)	[0m/s,0m/s,0m/s, 0°/s,0°/s]^T	$η ¯ 0$	[0.4tm,0.1tm, 2sin(0.1t) m]^T

Table 1 Initial states and desired formation of AUV

参数	数值	参数	数值
η₁(0)	[-6m,-20m,3m,0°,0°]^T	$δ ¯ 1$	[0m,-20m,0m]^T
η₂(0)	[-4m,-8m,-2m,0°,0°]^T	$δ ¯ 2$	[0m,-10m,0m]^T
η₃(0)	[-6m,0m,3m,0°,0°]^T	$δ ¯ 3$	[0m,0m,0m]^T
η₄(0)	[-4m,7m,-1m,0°,0°]^T	$δ ¯ 4$	[0m,10m,0m]^T
η₅(0)	[-5m,19m,-2m,0°,0°]^T	$δ ¯ 5$	[0m,20m,0m]^T
υ_i(0)	[0m/s,0m/s,0m/s, 0°/s,0°/s]^T	$η ¯ 0$	[0.4tm,0.1tm, 2sin(0.1t) m]^T

Table 2 Controller parameters

控制环节	参数	数值	参数	数值
外环控制律	K_η	diag{0.4,0.4,0.4}	T_η	20
内环控制律	K_υ	diag{5,5,5}	T_υ	20
固定时间 ESO	F₁	diag{5,5,5}	H₁	diag{6,6,6}
	F₂	diag{6,6,6}	H₂	diag{10,10,10}
	F₃	diag{12,12,12}	H₃	diag{12,12,12}
	f	0.9	h	1.11

Fig.5 Motion trajectories of multi-AUV formation

Fig.6 Outer-loop control tracking errors

Fig.7 Inner-loop control tracking errors

Fig.8 Lumped disturbance estimation errors

Fig.9 Control forces

Table 3 Integral absolute formation error of each AUV

控制方法	P₁	P₂	P₃	P₄	P₅
指定时间控制	8.63	10.66	8.16	9.77	5.74
固定时间控制	16.57	17.54	16.49	15.99	8.42
有限时间控制	25.28	27.42	26.60	24.50	11.78

Fig.10 Total formation error of multi-AUV

References 25

[1]	黄琰, 李岩, 俞建成, 等. AUV智能化现状与发展趋势[J]. 机器人, 2020, 42(2):215-231. doi: 10.13973/j.cnki.robot.190392
	HUANG Y, LI Y, YU J C, et al. State-of-the-art and development trends of AUV intelligence[J]. Robot, 2020, 42(2):215-231. (in Chinese) doi: 10.13973/j.cnki.robot.190392
[2]	智达. 国外海军无人潜航器发展综述[J]. 船舶工程, 2022, 44(8):170-172.
	ZHI D. Review of the development of foreign naval unmanned underwater vehicles[J]. Ship Engineering, 2022, 44(8):170-172. (in Chinese)
[3]	王童豪, 彭星光, 胡浩, 等. 海上有人/无人协同系统及其关键技术综述[J]. 兵工学报, 2024, 45(10):3317-3340. doi: 10.12382/bgxb.2024.0327
	WANG T H, PENG X G, HU H, et al. Maritime manned/unmanned collaborative systems and key technologies:a survey[J]. Acta Armamentarii, 2024, 45(10):3317-3340. (in Chinese)
[4]	张鑫明, 韩明磊, 余益锐, 等. 潜艇与UUV协同作战发展现状及关键技术[J]. 水下无人系统学报, 2021, 29(5):497-508.
	ZHANG X M, HAN M L, YU Y R, et al. Development and key technologies of submarine-UUV cooperative operation[J]. Journal of Unmanned Undersea Systems, 2021, 29(5):497-508. (in Chinese)
[5]	崔云菲. 多无人水下航行器自抗扰编队包含控制方法研究[D]. 哈尔滨: 哈尔滨工程大学, 2024:3-4.
	CUI Y F. Research on Anti-disturbance formation containment control for multiple unmanned underwater vehicles[D]. Harbin: Harbin Engineering University, 2024:3-4. (in Chinese)
[6]	ER M J, GONG H B, LIU Y, et al. Intelligent trajectory tracking and formation control of underactuated autonomous underwater vehicles:a critical review[J]. IEEE Transactions on Systems Man Cybernetics-Systems, 2024, 54(1):543-555.
[7]	DU H B, WEN G H, WU D, et al. Distributed fixed-time consensus for nonlinear heterogeneous multi-agent systems[J]. Automatica, 2020, 113:108797.
[8]	NGO A T, PHAM N N T, HO P H A. Adaptive finite-time leader-follower formation control for multiple AUVs regarding uncertain dynamics and disturbances[J]. Ocean Engineering, 2023, 269:113503.
[9]	LI J, TIAN Z Y, ZHANG H H, et al. Robust finite-time control of a multi-AUV formation based on prescribed performance[J]. Journal of Marine Science and Engineering, 2023, 11(5):897.
[10]	LI L N, LIU Z X, GUO S F, et al. Adaptive practical predefined-time control for uncertain teleoperation systems with input saturation and output error constraints[J]. IEEE Transactions on Industrial Electronics, 2024, 71(2):1842-1852.
[11]	POLYAKOV A. Nonlinear feedback design for fixed-time stabilization of linear control systems[J]. IEEE Transactions on Automatic Control, 2012, 57(8):2106-2110.
[12]	LI X J, QIN H D, LI L Y. Fixed-time formation control for AUVs with unknown actuator faults based on lumped disturbance observer[J]. Ocean Engineering, 2023, 269:113495.
[13]	王静耀, 杜佳璐. 欠驱动AUV分布式事件触发固定时间三维编队控制[J]. 控制与决策, 2025, 40(1):231-241.
	WANG J Y, DU J L. Distributed event-triggered fixed-time 3-D formation control of underactuated AUVs[J]. Control and Decision, 2025, 40(1):231-241. (in Chinese)
[14]	HOLLOWAY J, KRSTIC M. Prescribed-time output feedback for linear systems in controllable canonical form[J]. Automatica, 2019, 107:77-85.
[15]	黄秀颖, 刘海涛, 田雪虹. 基于输入饱和的欠驱动水面舰艇预定义时间跟踪控制[J]. 中国舰船研究, 2024, 19(1):98-110.
	HUANG X Y, LIU H T, TIAN X H. Predefined time tracking control of underactuated surface vessel with input saturation[J]. Chinese Journal of Ship Research, 2024, 19(1):98-110. (in Chinese)
[16]	YANG T T, ZHANG P F, CHEN H Y. Distributed prescribed-time leader-follower formation control of surface vehicles with unknowns and input saturation[J]. ISA Transactions, 2023, 124:16-27.
[17]	YAN Z P, GONG P, ZHANG W, et al. Model predictive control of autonomous underwater vehicles for trajectory tracking with external disturbances[J]. Ocean Engineering, 2020, 217:107884.
[18]	XIA G Q, ZHANG Y, ZHANG W, et al. Dual closed-loop robust adaptive fast integral terminal sliding mode formation finite-time control for multi-underactuated AUV system in three dimensional space[J]. Ocean Engineering, 2021, 233:108903.
[19]	CHEN H X, TANG G Y, WANG S F, et al. Adaptive fixed-time backstepping control for three-dimensional trajectory tracking of underactuated autonomous underwater vehicles[J]. Ocean Engineering, 2023, 275:114109.
[20]	HUA C C, NING P J, LI K. Adaptive prescribed-time control for a class of uncertain nonlinear systems[J]. IEEE Transactions on Automatic Control, 2022, 67(11):6159-6166.
[21]	YANG Y N, LIU X S, ZENG X D. Dynamic event-triggered prescribed-time zero-error tracking control for non-strict feedback nonlinear systems[J]. Nonlinear Dynamics, 2025, 113(1):567-581.
[22]	WANG T Q, LIU Y T, ZHANG X F. Extended state observer-based fixed-time trajectory tracking control of autonomous surface vessels with uncertainties and output constraints[J]. ISA Transactions, 2022, 128:174-183.
[23]	LI J H. 3D trajectory tracking of underactuated non-minimum phase underwater vehicles[J]. Automatica, 2023, 155:111149.
[24]	SHOJAEI K. Neural network formation control of underactuated autonomous underwater vehicles with saturating actuators[J]. Neurocomputing, 2016, 194(19):372-384.
[25]	KHAC D D, JIE P. Control of ships and underwater vehicles:design for underactuated and nonlinear marine systems[M]. Berlin,Germany: Springer, 2009.