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Fixed-time Target-guided Coordinate Control of Unmanned Surface Vehicles Based on Dynamic Surface Control
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Chao-yi LI2, Hai-xiang XU1, 2, 3, Wen-zhao YU1, 2, 3, Zhe DU1, 2, Ya-nan DING2
Journal of Ship Mechanics | 2025, 29(6) : 849 - 862
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Journal of Ship Mechanics | 2025, 29(6): 849-862
• Hydrodynamics •
Fixed-time Target-guided Coordinate Control of Unmanned Surface Vehicles Based on Dynamic Surface Control
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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
Published: 2025-06-20 doi: 10.3969/j.issn.1007-7294.2025.06.001
Outline
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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.

unmanned surface vehicle  /  distributed control  /  target-guided coordinate control  /  fixed-time convergence  /  dynamic surface control
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
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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,···,nN} is a node set and ξ={(ni,nj) ∈ {n1,...,nN}×V} is an edge set. The links among N USVs is defined as . If (ni,nj) ∈ ξ, aij=1, otherwise, aij=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, ai0=1, otherwise, ai0=0.
1.2　Kinematic and kinetic models for USVs
The kinematic mathematical model of the ith USV is described as:
where xi, yi, ψbi denote the position in surge, sway and yaw, respectively. ui, vi, ri denote the velocity in surge, sway and yaw, respectively.
Introducing the sideslip angle in Eq. (1), it follows that:
where βi=arctan (vi/ui) is the sideslip angle.
The kinetic mathematical model of the ith USV is described as:
where m11,i, m22,i, m33,i are masses and d11,i, d22,i, d33,i are hydrodynamic damping coefficients for the corresponding freedom respectively. denotes the control force and dwi=[dwui,dwvi,dwri]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 OEXEYE 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 Rt indicates the tracking radius. The target initial heading is defined as ψt 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.
where pit,d is the desired distance between the ith USV and the tracking point.
(2) Make sure the ith USV follows the target velocity vt=[ut,vt,rt]T.
where ut, vt and rt are the surge velocity, sway velocity and yaw velocity, respectively. vi=[ui,vi,ri]T.
(3) Make sure USVs do not collide with each other.
(4) Make sure each USV is within the effective communication distance.
where Rc2 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 βi satisfies |βi| ⩽ 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 ψt(0), the target's initial position pt(0)=[xt(0),yt(0)]T and the radius of the semi-encirclement circle Rt. The tracking points are defined as ptk=[xtk,ytk]T,k=1,2,...,N, the calculation method is as follows:
where ψd=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 dik, which can be further expressed as:
By determining min{dik}, 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:
where epi is the position tracking error, evi is the velocity tracking error, pij,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 Ri 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:
An auxiliary variable is introduced into Eq. (11) to solve the underactuated problem of USVs:
By defining σi=[ui,ri]T and Bi=diag{di,δ0}, substituting the differentiation of Eq. (12) into Eq. (11), it follows that:
where is identity matrix.
The potential function is further introduced in , the total error qti is defined as follows:
where efi 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
where pij=‖pi-pj‖ denotes the distance between the ith and the jth USV, εia, εic, cia, cic are positive constants, Ra2 is the maximum detection region, Ra1 is the minimum detection region, and Rc1 is the minimum connective distance. The details can be seen in Ref. [22].
The estimation errors of and are defined. To estimate βi, a fixed-time ESO is developed as
where kψ and kβ are positive parameters.
Combining Eqs. (13), (14) and (16), the fixed-time kinematic controller is designed as
where udi is the desired surge velocity output and rdi is the desired angular velocity output; kt1 and kt2 are positive gains; 0 < αt < 1, βt > 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.
where Td 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:
where 0 < kvi < 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, kvi 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 epi and evi in the system.
3.2　Fixed-time sliding mode controller
The following kinetic errors are defined:
where uei is the surge velocity tracking error, uei is the yaw velocity tracking error.
Substituting Eq. (3) into the time derivative of Eq. (20) leads to:
The CONN[23] is added to estimate the time-varying disturbance:
where Wi=[Wui Wri]T denotes the weight vector, Pij(ξi) satisfies ‖Pij(ξi)‖ ⩽ ςpi with ςpi > 0, εwi is the approximation error, ξi=[uei,rei]T, , , mn is the number of nodes in the hidden layer of CONN, and .
The weight update law is given as follows:
where is the estimation value of weight vector, is a positive parameter matrix, and kw is a positive parameter.
Assumption 3: The weight vector Wi satisfies , where .
Substituting Eq. (24) into Eq. (21), it follows that:
where , .
The fixed-time sliding surface is given as
where ku1, ku2, kr1, kr2 are positive gains, 0 < αu, αr < 1, βu > 1, βr > 1.
Substituting Eq. (25) into Eq. (26), the fixed-time SMC is given as
4　Stability analysis
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The kinetic error subsystem is given as follows:
Theorem 1: If Assumption 3 holds, the tracking errors of , and in Eq. (28) will converge in a fixed time by choosing ku1,ku2,kr1,kr2,kw appropriately.
Proof: A Lyapunov function is given as follows:
The differentiation of Eq. (29) is as follows:
where , , , , αt1=min{αu,αr,αw}, and βt1=min{βu,βr,βw}.
Eq. (30) can be rewritten as follows:
Applying Lemma 1, the error Et1 will converge in a fixed-time T1 satisfying
is defined as the filter error, its derivative is taken and substituted into Eq. (21), it follows that:
is defined, substituting Eq. (19) into Eq. (14) and combining Eq. (16), the kinematic error subsystem can be given as follows:
where , ke=ke1 + ke2, Aψi is a Hurwitz matrix such that , Pψi is a positive matrix.
Theorem 2: If Assumptions 1-2 and Theorem 1 hold, the tracking errors of , ez and in Eq. (34) will converge in a fixed time, where kt1 and Td need to satisfy and 0 < Td < 2, which are later defined in Eq. (36).
Proof: A Lyapunov function is given as follows:
The substitution of Eq. (34) into the differentiation of Eq. (35) is as follows:
Considering that is a continuous vector, it satisfies , where , and are positive constants.
Defining , , , Eq. (36) can be given as follows:
where , . Bt2=diag[kt2; , αt2=min{αt,αz,αψ}, βt2=min{βt,βz,βψ}, .
According to Theorem 1, can eventually converge to 0. From Eq. (37), when , 0 < Td < 2, it follows that:
Applying Lemma 1, the error Et2 will converge in a fixed-time T2 satisfying
To sum up, the tracking errors of the whole TACC control system converge in a fixed-time T3 satisfying
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 pt=[10 m,10 m,0◦]T. The velocity of the target is set as vt=[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 vi=[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.
Funding
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  • National Natural Science Foundation of China(52201373)
  • Hainan Provincial Joint Project of Sanya Yazhou Bay Science and Technology City(2021CXLH0016)
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Appendix
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Year 2025 volume 29 Issue 6
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doi: 10.3969/j.issn.1007-7294.2025.06.001
  • Receive Date：2024-12-21
  • Online Date：2026-03-24
  • Published：2025-06-20
Article Data
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  • Received：2024-12-21
Funding
National Natural Science Foundation of China(52201373)
Hainan Provincial Joint Project of Sanya Yazhou Bay Science and Technology City(2021CXLH0016)
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

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表12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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