船舶力学

Kinematic control parameters	SigAPF parameters	Kinetic control parameters
N	4	δ₀	0.15	R_a1/m	9	R_a2/m	5	k_u1	1.2	k_u2	1.2
k_t1	diag[0.4; 0.15]	k_t2	diag[0.4; 0.15]	R_c1/m	100	R_c2/m	70	k_r1	0.3	k_r2	0.3
α_t	0.8	β_t	1.2	ε_ia	0.005	c_ia	0.1	α_u	0.7	α_r	0.7
k_i1	10	k_i2	20	ε_ic	0.001	c_ic	0.5	β_u	1.5	β_r	1.5
α_ψ	0.8	β_ψ	1.2					γ_i	0.1×diag[1; 1; 1; 1; 1]	k_w	10⁻⁴
T_d	0.5	k_vi	0.1					W_i

Kinematic control parameters	SigAPF parameters	Kinetic control parameters
N	4	δ₀	0.15	R_a1/m	9	R_a2/m	5	k_u1	1.2	k_u2	1.2
k_t1	diag[0.4; 0.15]	k_t2	diag[0.4; 0.15]	R_c1/m	100	R_c2/m	70	k_r1	0.3	k_r2	0.3
α_t	0.8	β_t	1.2	ε_ia	0.005	c_ia	0.1	α_u	0.7	α_r	0.7
k_i1	10	k_i2	20	ε_ic	0.001	c_ic	0.5	β_u	1.5	β_r	1.5
α_ψ	0.8	β_ψ	1.2					γ_i	0.1×diag[1; 1; 1; 1; 1]	k_w	10⁻⁴
T_d	0.5	k_vi	0.1					W_i

基于动态面技术的固定时间多无人艇分布式目标跟踪控制

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李超逸 ² , 徐海祥 ¹^,²^,³ , 余文曌 ¹^,²^,³ , 杜哲 ¹^,² , 丁亚楠 ²

船舶力学 | 流体力学 2025,29(6): 849-862

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船舶力学 | 流体力学 2025, 29(6): 849-862

基于动态面技术的固定时间多无人艇分布式目标跟踪控制

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李超逸², 徐海祥¹^,²^,³, 余文曌¹^,²^,³, 杜哲¹^,², 丁亚楠²

作者信息

^1.高性能船舶技术教育部重点实验室（武汉理工大学），武汉　430062

^2.武汉理工大学　船海与能源动力工程学院，武汉　430062

^3.武汉理工大学　三亚科教创新园，海南　三亚　572024

李超逸（1996-），男，博士研究生

徐海祥（1975-），男，博士，武汉理工大学教授

余文曌（1989-），男，博士，武汉理工大学副研究员

杜哲（1993-），男，博士，武汉理工大学讲师

丁亚楠（1998-），女，硕士研究生。

通讯作者:

corresponding author, E-mail：wzyu@whut.edu.cn

Fixed-time Target-guided Coordinate Control of Unmanned Surface Vehicles Based on Dynamic Surface Control

Chao-yi LI², Hai-xiang XU¹^,²^,³, Wen-zhao YU¹^,²^,³, Zhe DU¹^,², Ya-nan DING²

Affiliations

^1.Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430062, China

^2.School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430062, China

^3.Sanya Science and Education Innovation Park of Wuhan University of Technology, Sanya 572024, China

出版时间: 2025-06-20 doi: 10.3969/j.issn.1007-7294.2025.06.001

文章导航

摘要

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本文针对多无人艇分布式目标跟踪控制问题进行研究。在跟踪非合作目标的场景下，只有部分个体能够获取目标的状态信息。为了实现对非合作目标的快速安全跟踪，本文提出一种基于动态面控制的固定时间跟踪控制方法。首先，设计一种运动学和动力学解耦的分布式目标跟踪控制结构，以降低控制系统设计的复杂度。其次，提出一种基于动态面技术的运动学控制律，其中包含跟踪点预分配策略和避碰势场函数，可在跟踪过程中避免碰撞并优化运动学控制输出。最后，提出一种固定时间分布式目标跟踪控制系统，实现运动学和动力学误差的快速收敛。仿真对比结果验证了本文所提方法在提高目标跟踪安全性和减少控制输出抖振方面的有效性。

关键词

多无人艇 / 分布式控制 / 目标跟踪控制 / 固定时间收敛 / 动态面控制

Abstract

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This paper presents an investigation on the target-guided coordinated control (TACC) of unmanned surface vehicles (USVs). In the scenario of tracking non-cooperative targets, the status information of the target can only be obtained by some USVs. In order to achieve semi-encirclement tracking of non-cooperative targets under maritime security conditions, a fixed-time tracking control method based on dynamic surface control (DSC) is proposed in this paper. Firstly, a novel TACC architecture with decoupled kinematic control law and decoupled kinetic control law was designed to reduce the complexity of control system design. Secondly, the proposed DSC-based target-guided kinematic control law including tracking points pre-allocation strategy and sigmoid artificial potential functions (SigAPFs) can avoid collisions during tracking process and optimize kinematic control output. Finally, a fixed-time TACC system was proposed to achieve fast convergence of kinematic and kinetics errors. The effectiveness of the proposed TACC approach in improving target tracking safety and reducing control output chattering was verified by simulation comparison results.

Key words

unmanned surface vehicle / distributed control / target-guided coordinate control / fixed-time convergence / dynamic surface control

引用本文

李超逸, 徐海祥, 余文曌, 杜哲, 丁亚楠. 基于动态面技术的固定时间多无人艇分布式目标跟踪控制. 船舶力学, 2025 , 29 (6) : 849 -862 . DOI: 10.3969/j.issn.1007-7294.2025.06.001

Chao-yi LI, Hai-xiang XU, Wen-zhao YU, Zhe DU, Ya-nan DING. Fixed-time Target-guided Coordinate Control of Unmanned Surface Vehicles Based on Dynamic Surface Control[J]. Journal of Ship Mechanics, 2025 , 29 (6) : 849 -862 . DOI: 10.3969/j.issn.1007-7294.2025.06.001

正文

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0　Introduction

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The utilization and exploration of ocean resources and the escort of important shipping lanes have attracted full attention around the world^[1]. In order to effectively protect maritime territory and cooperative targets from infringement, path-guided coordinate control^[2-3], trajectory-guided coordinate control^[4-5], target-guided coordinate control^[6] and collaborative target surrounding^[7-8] have become important working modes for multiple USVs. In particular, collaborative target tracking control technology of USVs has gained broad applications in formation escort, maritime patrol and intrusion target expulsion.

During the past few years, research on USVs TACC has made great progress. Fahimi^[9] transformed the formation tracking problem into the virtual target tracking problem. By designing a sliding mode controller to track the position and heading of a virtual target, formation maintenance was achieved during the movement of USVs. Works on decentralized control for tracking a cooperative target was discussed in Refs. [10-13]. In Ref. [10], NN was developed to estimate model uncertainties and external environmental disturbance. Meanwhile, dynamic surface control technology was utilized in the kinematic control layer to smooth the kinetic control output chattering. To avoid actuator saturation, Khoshnam^[11] introduced a generalized saturation function into the kinetic error and further used RBFNN based on Ref. [10] to accurately estimate internal and external disturbances. To address the problems of angle constraints and actuator faults, barrier Lyapunov functions (BLFs) were designed in Ref. [12], and the finite time convergence of the control system was achieved. Sun et al^[13] developed a continuous function to replace the signum function in the sliding mode control method, which effectively reduced the chattering based on Ref. [9]. Different from those in Refs. [10-13], Liu et al^[14] studied the problem of unavailable non-cooperative target velocity. An extended state observer (ESO) was designed in the kinematic control law to estimate the uncertain target dynamics due to the unavailable velocity.

However, the above studies are all based on decentralized control structures. Since there is no information exchange between USVs, the tracking synchronization is not optimal. Based on the above issues, the control structure of USVs TACC gradually transitions from decentralized to distributed^[15-20]. On the basis of Ref. [14], Gao et al^[15] introduced an event triggering mechanism in the distributed ESO (DESO) to simultaneously deal with the problem of target state unavailable and communication delay. Considering the unknown obstacles during target tracking, Gao et al^[16] added a control barrier function (CBF) as a solution into the kinematic control law in Ref. [15]. In Ref. [17], a potential function was designed to avoid collisions between USVs and targets. Meanwhile, the event-trigger scheme was introduced into the kinematic layer, which could reduce the actuator activity. Considering that only some USVs could obtain position information of the target under different scenarios, Ma et al^[18] further proposed a switched topology DESO based on the works in Ref. [14] and Ref. [15]. Different from Ref. [18], Zhu et al^[19] proposed three substitution strategies for the substitution problem of damaged USV individuals to ensure the continuation of the tracking task. Considering the convergence speed of the tracking system, Zhu et al^[20] designed three new initial USVs selection strategies to achieve fast implementation of tracking tasks in the case of USVs redundancy. On the other hand, finite-time convergence was also achieved in the proposed control system.

Motivated by the mentioned works, this paper is aimed to propose a novel TACC method for USVs. The highlights of the study cover: (i) division of the TACC architecture into two parts, including decoupled kinematic control layer and kinetic control layer, designed to reduce the complexity of control system design, (ii) proposal of a DSC-based target-guided kinematic control law including tracking points pre-allocation strategy and sigmoid-APFs to avoid collisions during tracking process and optimize kinematic control output chattering, (iii) proposal of a fixed-time TACC system to make errors converge fast, and estimation of time-varying unknown sideslip angle and external environmental disturbance by utilizing fixed-time ESO and Chebyshev orthogonal neural network (CONN) estimator, respectively.

Compared with the existing research, the salient features of the proposed method are as follows: (i) different from Refs. [15-20], not only the kinematic control layer is decoupled from the kinetic control layer, but also the velocity synchronization control objective is considered, which reduces the complexity of the system design and makes it more suitable for engineering practice, (ii) different from the one-line formation in Refs. [14, 18] and the triangle formation in Refs. [9-11, 13, 17] and the quadrangle formation in Refs. [15, 16], a semi-encirclement tracking formation for the patrol and eviction conditions of suspicious targets at sea is proposed; different from Refs. [16] and [17], the collision avoidance constraints and connectivity maintenance constraints between individuals are considered, and a monotonic and bounded APFs is utilized based on sigmoid function, which effectively ensures the safety of the whole tracking process, (iii) based on the finite-time form in Refs. [12] and [20], the fixed-time TACC system is further designed and the stability is analyzed through the Lyapunov theory.

1　Preliminaries and problem formulation

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In this paper, the following notations are used. Given any

, it follows that

denotes the maximum value in the array.

1.1　Graph theory

The communication topology is described by a graph G={V,ξ}, where V={n₀,···,n_N} is a node set and ξ={(n_i,n_j) ∈ {n₁,...,n_N}×V} is an edge set. The links among N USVs is defined as

. If (n_i,n_j) ∈ ξ, a_ij=1, otherwise, a_ij=0. The virtual links between USVs and the non-cooperative target is defined as

. If the ith USV can obtain the status information of the non-cooperative target, a_i0=1, otherwise, a_i0=0.

1.2　Kinematic and kinetic models for USVs

The kinematic mathematical model of the ith USV is described as:

(1)

where x_i, y_i, ψ_bi denote the position in surge, sway and yaw, respectively. u_i, v_i, r_i denote the velocity in surge, sway and yaw, respectively.

Introducing the sideslip angle in Eq. (1), it follows that:

(2)

where β_i=arctan (v_i/u_i) is the sideslip angle.

The kinetic mathematical model of the ith USV is described as:

(3)

where m_11,i, m_22,i, m_33,i are masses and d_11,i, d_22,i, d_33,i are hydrodynamic damping coefficients for the corresponding freedom respectively.

denotes the control force and d_wi=[d_wui,d_wvi,d_wri]^T is the time-varying disturbance.

1.3　Problem formulation

The semi-encirclement target tracking scene is shown in Fig. 1. OXY is the body-fixed coordinate and O_EX_EY_E is the north-east-down coordinate. The ith USV position is defined as

. The black circle outside the target indicates the desired tracking circle, and R_t indicates the tracking radius. The target initial heading is defined as ψ_t and position as

. Four control objectives need to be met.

(1) Make sure the ith USV follows the target and stays at the tracking point.

(4)

where p_it,d is the desired distance between the ith USV and the tracking point.

(2) Make sure the ith USV follows the target velocity v_t=[u_t,v_t,r_t]^T.

(5)

where u_t, v_t and r_t are the surge velocity, sway velocity and yaw velocity, respectively. v_i=[u_i,v_i,r_i]^T.

(3) Make sure USVs do not collide with each other.

(6)

(4) Make sure each USV is within the effective communication distance.

(7)

where R_c2 is the maximum connective distance.

1.4　Lemmas and assumptions

Lemma 1^[21]: Consider the nonlinear function

. If there exists a continuous radially unbounded function V(x) such that V(x)=0 ⇔ x=0 and satisfies

for some κ₁,κ₂,ζ > 0, ε₁ ∈ (0,1) and ε₂ ∈ (1,∞), then the origin of the system is globally actual fixed-time stable. Furthermore, given a scalar quantity ε₃ ∈ (0,1), the convergence time T satisfies T ⩽ 1/(κ₁ε₃(1-ε₁)) + 1/(κ₂ε₃(ε₂−1)).

Assumption 1: The target velocity is lower than USVs. The target initial heading can be obtained.

Assumption 2: The sideslip angle β_i satisfies |β_i| ⩽ β^∗ with β^∗ > 0.

2　Target tracking points pre-allocation

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To avoid conflicts, tracking points need to be evenly distributed over the semi-encirclement. The specific scenario is shown in Fig. 2. The position of the tracking points is determined by the target's initial heading ψ_t(0), the target's initial position p_t(0)=[x_t(0),y_t(0)]^T and the radius of the semi-encirclement circle R_t. The tracking points are defined as p_tk=[x_tk,y_tk]^T,k=1,2,...,N, the calculation method is as follows:

(8)

where ψ_d=π/(N + 1).

After the tracking points are determined, the negotiation method is needed to assign the tracking points to USVs. The distance is defined between the ith USV and the kth tracking point as d_ik, which can be further expressed as:

(9)

By determining min{d_ik}, the kth tracking point is allocated to the ith USV, and the allocation process is repeated for the remaining tracking points and USVs until all allocations are completed. It is worth noting that when a tracking point is allocated to multiple USVs at the same time, it should be allocated to the farthest USV before continuing with subsequent allocation tasks.

Based on the above strategy, the matching matrix between USV individuals and tracking points can be defined as

, which is further used in the design of the kinematic control law.

3　Distributed fixed-time collaborative controller design

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3.1　Fixed-time kinematic control law design

Combining the graph theory, the kinematic control error is given as follows:

(10)

where e_pi is the position tracking error, e_vi is the velocity tracking error, p_ij,d is the desired distance between USVs,

is the desired distance among USVs and target,

is the kth tracking point position corresponding to the ith USV, and R_i is the coordinate rotation matrix.

Remark 1: Considering that the velocity of the non-cooperative target is unavailable, the velocity synchronization error is calculated by using the change rate of the target position in the north-east-down coordinate, and then converted to the body-fixed coordinate for kinematic control error calculation.

The time derivative of Eq. (10) is given as follows:

(11)

An auxiliary variable

is introduced into Eq. (11) to solve the underactuated problem of USVs:

(12)

By defining σ_i=[u_i,r_i]^T and B_i=diag{d_i,δ₀}, substituting the differentiation of Eq. (12) into Eq. (11), it follows that:

(13)

where

is identity matrix.

The potential function is further introduced in

, the total error q_ti is defined as follows:

(14)

where e_fi is the sum of the potential function gradients,

is the inter-USVs collision avoidance potential function,

is the connectivity constraint potential function. The latter two can be further described as

(15)

where p_ij=‖p_i-p_j‖ denotes the distance between the ith and the jth USV, ε_ia, ε_ic, c_ia, c_ic are positive constants, R_a2 is the maximum detection region, R_a1 is the minimum detection region, and R_c1 is the minimum connective distance. The details can be seen in Ref. [22].

The estimation errors of

and

are defined. To estimate β_i, a fixed-time ESO is developed as

(16)

where k_ψ and k_β are positive parameters.

Combining Eqs. (13), (14) and (16), the fixed-time kinematic controller is designed as

(17)

where u_di is the desired surge velocity output and r_di is the desired angular velocity output; k_t1 and k_t2 are positive gains; 0 < α_t < 1, β_t > 1.

In order to smooth the kinematic output chattering caused by the acceleration signals of

and

, dynamic surface control technology is introduced in Eq. (17). The first-order low-pass filter is introduced.

(18)

where T_d is the time constant,

is the filtered speed value of USVs and the target.

Substituting the differentiation of Eq. (18) into Eq. (17), it follows that:

(19)

where 0 < k_vi < 1.

Remark 2: Based on the principle of PID controller, Eq. (19) can be regarded as a P, I, D type controller. The sideslip angle estimation can be regarded as the integral term, meanwhile the acceleration signals of

and

in Eq. (19) can be regarded as differential terms. Filtering the acceleration signal also draws on incomplete differential control technology, which can theoretically prove the rationality of the proposed method for smoothing the kinematic control output.

Remark 3: In order to further smooth the chattering signal brought by the acceleration signal to the kinematic control law, k_vi is set to reduce the chattering amplitude of the desired signal without sacrificing the convergence speed. It can also be regarded as a weight parameter to adjust the proportion of e_pi and e_vi in the system.

3.2　Fixed-time sliding mode controller

The following kinetic errors are defined:

(20)

where u_ei is the surge velocity tracking error, u_ei is the yaw velocity tracking error.

Substituting Eq. (3) into the time derivative of Eq. (20) leads to:

(21)

The CONN^[23] is added to estimate the time-varying disturbance:

(22)

(23)

where W_i=[W_ui W_ri]^T denotes the weight vector, P_ij(ξ_i) satisfies ‖P_ij(ξ_i)‖ ⩽ ς_pi with ς_pi > 0, ε_wi is the approximation error, ξ_i=[u_ei,r_ei]^T,

, m_n is the number of nodes in the hidden layer of CONN, and

The weight update law is given as follows:

(24)

where

is the estimation value of weight vector,

is a positive parameter matrix, and k_w is a positive parameter.

Assumption 3: The weight vector W_i satisfies

, where

Substituting Eq. (24) into Eq. (21), it follows that:

(25)

where

The fixed-time sliding surface is given as

(26)

where k_u1, k_u2, k_r1, k_r2 are positive gains, 0 < α_u, α_r < 1, β_u > 1, β_r > 1.

Substituting Eq. (25) into Eq. (26), the fixed-time SMC is given as

(27)

4　Stability analysis

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The kinetic error subsystem is given as follows:

(28)

Theorem 1: If Assumption 3 holds, the tracking errors of

and

in Eq. (28) will converge in a fixed time by choosing k_u1,k_u2,k_r1,k_r2,k_w appropriately.

Proof: A Lyapunov function is given as follows:

(29)

The differentiation of Eq. (29) is as follows:

(30)

where

, α_t1=min{α_u,α_r,α_w}, and β_t1=min{β_u,β_r,β_w}.

Eq. (30) can be rewritten as follows:

(31)

Applying Lemma 1, the error E_t1 will converge in a fixed-time T₁ satisfying

(32)

is defined as the filter error, its derivative is taken and substituted into Eq. (21), it follows that:

(33)

is defined, substituting Eq. (19) into Eq. (14) and combining Eq. (16), the kinematic error subsystem can be given as follows:

(34)

where

, k_e=k_e1 + k_e2, A_ψi is a Hurwitz matrix such that

, P_ψi is a positive matrix.

Theorem 2: If Assumptions 1-2 and Theorem 1 hold, the tracking errors of

, e_z and

in Eq. (34) will converge in a fixed time, where k_t1 and T_d need to satisfy

and 0 < T_d < 2, which are later defined in Eq. (36).

Proof: A Lyapunov function is given as follows:

(35)

The substitution of Eq. (34) into the differentiation of Eq. (35) is as follows:

(36)

Considering that

is a continuous vector, it satisfies

, where

and

are positive constants.

Defining

, Eq. (36) can be given as follows:

(37)

where

. B_t2=diag[k_t2;

, α_t2=min{α_t,α_z,α_ψ}, β_t2=min{β_t,β_z,β_ψ},

According to Theorem 1,

can eventually converge to 0. From Eq. (37), when

, 0 < T_d < 2, it follows that:

(38)

Applying Lemma 1, the error E_t2 will converge in a fixed-time T₂ satisfying

(39)

To sum up, the tracking errors of the whole TACC control system converge in a fixed-time T₃ satisfying

(40)

5　Simulation results

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In this chapter, four USVs and one moving target are considered, where status information of the non-cooperative target can be directly obtained by a USV shown in Fig. 3. The effects of the two cooperative kinematics control laws of Eqs. (17) and (19) on the proposed TACC control system are compared, respectively. The initial position of the target is set as p_t=[10 m,10 m,0^◦]^T. The velocity of the target is set as v_t=[1 m/s,0 m/s,0.0045×sin(0.001×T) rad/s]^T, where T is the number of control cycles. The initial position of USVs are given as p₁=[−10 m,−20 m,0^◦]^T, p₂=[−10 m,0 m,0^◦]^T, p₃=[−10 m,20 m,0^◦]^T, and p₄=[−20 m,0 m,0^◦]^T. The initial velocity of USVs are all set as v_i=[0.2 m/s,0 m/s,0 rad/s]^T. The mathematical model of USVs are the same as in Ref. [24]. The main simulation parameters are shown in Tab. 1.

Fig. 4 shows the trajectories of the target and 4 USVs. The four USVs are conducting coordinated patrol missions in a certain formation and initial velocity at the initial moment, and immediately carry out target tracking missions after discovering the target, and each USV is allocated to its own tracking point. After a period of time, USVs quickly change from a triangle formation to a semi-enclosed formation, and the tracking formation is always maintained during subsequent movements. Fig. 5 presents the position tracking errors between USVs and tracking points. After 50 s, the USVs position tracking errors achieve convergence. And the post-convergence mean square errors (MSE) are 0.94 m, 0.83 m, 1.03 m and 1.09 m respectively. The velocity tracking errors between USVs and the target is shown in Fig. 6. The USVs velocity tracking errors achieve convergence like the position tracking errors, and the MSE are 0.14 m/s, 0.0325 m/s, 0.0329 m/s and 0.0077 m/s respectively. The result in Fig. 6 further proves the rationality of Remarks 3-4.

Fig. 7 and Fig. 8 show the compared results of the control method with and without DSC in desired velocity tracking and kinetic control output respectively. In Fig. 7, the control method with DSC and the control method without DSC can both converge the actual velocity to the desired kinematic control output. However, the control method with DSC smoothes the chattering of the desired velocity signal, so that the actual velocity signal is also optimized. The result in Fig. 7 is further reflected in Fig. 8. The control method without DSC has a maximum amplitude of 20 N on the surge control output and a trend of gradually increasing and diverging, and a maximum amplitude of 0.6 N·m on the yaw control output. In comparison, the output of the control method with DSC is greatly optimized, which is more conducive to practical engineering applications.

Fig. 9 shows the results of APFs. In the first 50 s, there is a risk of collision when the USVs move towards their respective tracking points. Under the inter-USVs collision avoidance APF, the navigation safety of the USVs is guaranteed. Fig. 10 and Fig. 11 show the verification of the effectiveness of the fixed-time ESO and CONN estimator in estimating time-varying sideslip angle and external environmental disturbance respectively.

6　Concluding remarks

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In this paper, semi-encirclement TACC of USVs under maritime security conditions is studied, and a novel fixed-time TACC approach is proposed. Firstly, A decoupled TACC control structure is designed, and the independent debugging of kinematic control law and kinetic control law is more suitable for practical engineering applications and reduces the complexity of system design. Secondly, a DSC-based target-guided kinematic control law including tracking points pre-allocation strategy and sigmoid-APFs is proposed, where collision avoidance during target tracking is effectively achieved by using sigmoid-APFs. Simultaneously, dynamic surface control technology greatly optimizes the subsequent control output. Finally, a fixed-time TACC system is proposed for the first time. In the kinetic layer, the designed fixed-time sliding mode controller combined with CONN can achieve fast convergence of control errors and achieve accurate compensation of time-varying sideslip angle and external environmental disturbance simultaneously. The effectiveness of the proposed TACC approach in improving target tracking safety and reducing control output chattering is verified by the simulation comparison results.

Currently, the situation where tracking points are dynamically allocated due to non-cooperative target movement or changes in the navigation environment is not considered in this paper. In addition, it is ideal to assume that the target state can be obtained by some USVs. Combined with the real detection range of the sensors mounted on USVs, state estimation of the target under switching topology is another research content in the future.

基金

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国家自然科学基金资助项目(52201373)
海南省科技计划三亚崖州湾科技城科技创新联合项目(2021CXLH0016)

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2025年第29卷第6期

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doi: 10.3969/j.issn.1007-7294.2025.06.001

接收时间：2024-12-21
首发时间：2026-03-24
出版时间：2025-06-20

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收稿日期：2024-12-21

基金

National Natural Science Foundation of China(52201373)

国家自然科学基金资助项目(52201373)

Hainan Provincial Joint Project of Sanya Yazhou Bay Science and Technology City(2021CXLH0016)

海南省科技计划三亚崖州湾科技城科技创新联合项目(2021CXLH0016)

作者信息

^1.高性能船舶技术教育部重点实验室（武汉理工大学），武汉　430062

^2.武汉理工大学　船海与能源动力工程学院，武汉　430062

^3.武汉理工大学　三亚科教创新园，海南　三亚　572024

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corresponding author, E-mail：wzyu@whut.edu.cn

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2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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Kinematic control parameters				SigAPF parameters				Kinetic control parameters
Symbol	Value	Symbol	Value	Symbol	Value	Symbol	Value	Symbol	Value	Symbol	Value
N	4	δ₀	0.15	R_a1/m	9	R_a2/m	5	k_u1	1.2	k_u2	1.2
k_t1	diag[0.4; 0.15]	k_t2	diag[0.4; 0.15]	R_c1/m	100	R_c2/m	70	k_r1	0.3	k_r2	0.3
α_t	0.8	β_t	1.2	ε_ia	0.005	c_ia	0.1	α_u	0.7	α_r	0.7
k_i1	10	k_i2	20	ε_ic	0.001	c_ic	0.5	β_u	1.5	β_r	1.5
α_ψ	0.8	β_ψ	1.2					γ_i	0.1×diag[1; 1; 1; 1; 1]	k_w	10⁻⁴
T_d	0.5	k_vi	0.1					W_i