Adaptive Formation Control for Underactuation multi-USVs with Jointly Connected Switching Typologies

Authors: kunting  yu
yongming  li
kewen  li
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-29 09:46:57

Abstract: This paper investigates the adaptive formation control problem for underactuation Multiply Unmanned Surface Vehicles (multi-USVs) connected by the jointly connected switching typologies. First, the distributed switching observer is devised for each USV to estimate the trajectory signal under the jointly connected switching typologies. Secondly, the underactuation controllers are designed based on the estimated trajectory signal under backstepping structure. In the form of the designed underactuated controller, we have proven through Lyapunov stability theory that each unmanned vessel is able to track the predetermined trajectory. In the end, we validate the effectiveness of the proposed method through numerical simulations.

Main paper: 
 i. introduction: 
 In practical applications, communication networks often change over time for various reasons [1] . Many discussions have been carried out on the control problems of unmanned USVs under switched topologies [2] - [5] . [2] develops a datadriven neural predictor using real-time and historical data, leading to the design of an adaptive kinematic control law ensuring that the tracking error of each unmanned ship is consistently ultimately bounded. [3] designs a switching distributed extended state observer to estimate the unknown dynamics. Building upon this technique, [3] further devises distributed controllers for USVs to cooperatively track the target. The authors in [4] , using orthonormal transformation technique and algebraic graph theory, equivalently formulates the formation control problem as analyzing the strictly dissipative problem of a Markov jump system with model-related time-varying delays. It proves the existence of dissipative martingale solutions. [5] treats Gaussian noise as a Wiener process, transforming the control problem of USV systems into a more specific stochastic problem. By utilizing the average dwell time method and Lyapunov stability theory, it proves that all signals of the USV system remain bounded in mean-square within a specified time. However, the aforementioned control methods for unmanned USVs under switched topologies all necessitate each mode of the switched topology to have a spanning tree, significantly increasing the communication burden of the unmanned USVs under switched topologies.In comparison to the aforementioned switched topologies modes, jointly connected digraphs only require the set of switched topology subsets to have a spanning tree in a certain interval of time. Therefore, jointly connected digraphs do not demand as many communication resources as the switching modes in [2] - [5] . This feature has made jointly connected digraphs a hot research area in the field of switching topologies [6] - [17] . [6] investigates the coordinated output regulation problem of linear multi-agent systems under undirected jointly connected topologies. Building upon this work, [7] and [8] further propose coordinated output regulation control methods for linear multi-agent systems under directed jointly connected topologies. The author in [9] , by avoiding reliance on the leader's state, addresses the robust output regulation control problem for nonlinear models with unknown bounded disturbances under jointly connected topologies.However, currently, there have been no reported results on jointly connected topologies for multi-USVs. This is due to the complex nonlinear dynamic characteristics of USVs, making it challenging to analyze their stability under jointly connected topologies. In order to address this gap, this paper proposes a formation control strategy for underactuated multi-USVs under jointly connected topologies. The innovations of this paper are as follows• By designing a switching trajectory observer, this paper successfully transforms the formation control problem of multi-unmanned ships into a distributed trajectory tracking problem. Building upon the results of [8] , we analyze the stability of trajectory observation errors under jointly connected topologies and provide a stability proof for the trajectory observation errors. • Based on the trajectory estimation vector obtained from the trajectory observer, we design a distributed controller for underactuated unmanned ships, enabling each unmanned ship to track the trajectory based on the relative position in formation, thereby achieving the formation control effect. The rest sections of this paper are organized as follows: Section II shows the problem formulation. In Section III, we give the primary stability results. Section IV depicts the simulation results. And Section V concludes the paper.In this paper, the communication network of the N USVs is modeled as a directed graph P = (G, C), where the edge set is defined as C ⊆ G × G. If j-th USV can transmit information to i-th USV, symbol a ij ∈ V is defined as 1. And if j-th USV can not transmit information to i-th USV, we define a ij = 0. The Laplacian matrix M is designed asM = [m ij ] ∈ R N ×N ,where m ij = -a ij and m ii = N j=1 a ij . Furthermore, symbol b i = 1 is defined while i-th USV can receive the trajectory information. The matrices ∆ and B are defined as ∆ = M+B and B = diag{b i }, respectively.Definition 1 (jointly connected graph) [7] . There exist an infinite sequence of nonempty and non-overlapping time intervals [t l , t l+1 ) such that t l+1 -t l ≤ T for a positive constant T , and a sequence of non-overlapping subintervals[t 0 l , t q l ), • • • , [t q l , t q+1 l) with t 0 l = t l , t q+1 l = t l+1 and T * ≤ t q+1 l -t q l for a positive constant T , respectively. If there exist a time-varying switching signal σ(t) and the union of the subgraphs q+1 p=0 P σ(t p l ,t p+1 l ) contains a spanning tree, then P σ is the jointly connected.Assumption 1 [8] . In this paper, the directed graph P σ is assumed to be jointly connected by setting t q l as the switching time.Assumption 2 [10] . The trajectory signal β 0 (t) = [x 0 , y 0 ] T satisfies β 0 (t) ≤ β * 0 and t t0 β0 (τ ) dτ ≤ β0 such that β * 0 and β0 are positive constants.Lemma 1 [8] . Under Assumption 1, if the system satisfies the following dynamicς(t) = -ε(∆ ⊗ I n )ς(t) then we have ς(t) ≤ -c ς(t) e -c(t-τ ) , ∀t > τwhere ε, c and c are positive constants, ς ∈ R N n is the system state vector.Lemma 2 (Bellman-Gronwall Lemma) [8] . When nonnegative piecewise continuous functions g 1 (t), g 2 (t), g 3 (t) and g 4 (t) satisfyg 1 (t) ≤ g 2 (t) + g 3 (t) t t0 g 4 (τ )g 1 (τ )dτ, t > t 0then the following inequality holdsg 1 (t) ≤ g 2 (t) + g 3 (t) t t0 g 2 (τ )g 4 (τ )e t τ g4(s)g3(τ )dτ dτ, t > t 0In this paper, the kinematics and dynamics of i-th USV is modeled as follows [18] EQUATIONwhere (x i , y i ) is denoted by the positions of each USV, and ψ i represents the yaw angle in relation to the earth-fixed frame. The velocities u i , υ i , and r i correspond to the surge velocity, sway velocity, and yaw rate in the vehicle body-fixed frame, respectively. The disturbances τ i,ωu , τ i,ωυ , and τ i,ωr are factors representing various environmental disturbances like wind, waves, and currents. These disturbances are assumed to be unknown, smooth, and bounded. Control inputs τ i,u and τ i,r are provided by the USV's propeller and rudder, respectively. They influence the motion of the vehicle. The dynamics of the system, encapsulated by f i,1 (•), f i,2 (•), and f i,3 (•), encompass a range of nonlinear effects including hydrodynamic damping forces, Coriolis centripetal forces, and other unmodeled hydrodynamic forces. The form of them can be found in [10] . Control Objective. The control objective of this paper is to design a distributed observer estimating the trajectory signal under jointly connected switching topologies. In addition, underactuated controllers are designed for the multi-USVs to follow the trajectory signal.In this section, we design the distributed observer and analysis the estimation performance. And then we design the underactuated controllers for multi-USVs to track the trajectory estimated signal.In this paper, the distributed observer is designed asEQUATIONwhere κ 1 and κ 2 are designed positive constants, βi,0 = [x i,0 , ŷi,0 ] T and βi,1 = [x i,1 , ŷi,1 ] T .Theorem 1. The states of distributed switching observer can track the trajectory signal within a small neighborhood near the origin.proof. First, we prove the estimation error as βi,0 = β 0βi,1 is bounded under jointly connected switching topologies.By taking the time derivative of the estimation error βi,0 = β 0 -βi,1 , we haveEQUATIONOn the basis of Lemma 1, we can obtainEQUATIONAccording toEQUATIONFurthermore, by employing Lemma 2, we can yield thatβ0 ≤c β0 (t 0 ) e -c(t-t0) + 1 c (1 + √ N )c β0 + c 2 β0 (t 0 ) t t0e -2c(t-τ ) e t τ e -c(t-s)ds dτConsidering the method of integral comparison, the fact e -c(τ -t) dτ = 1 c e t holds. Substituting this fact into (7) , one hasEQUATIONwhere ĉ = min{c, c 2 2c 2 }. Thus, the estimation error βi,0 is bounded fort ∈ [0, ∞).In what follows, we prove the state vector βi,0 can track βi,1 . So that the state vectors βi,0 and βi,1 can estimate the trajectory signal β 0 . Design the error as βi,1 = βi,0 -βi,1 and Lyapunov function asV β = N i=1 1 2βT i,1 βi,1 , respectively. The time derivative of Lyapunov function V β can be calculated asEQUATIONwhereβ1 = [ βT 1,1 , • • • , βT N,1 ] T ∈ R 2N and β1 = [ βT 1,1 , • • • , βT N,1 ] T ∈ R 2N . Under Assumption 1, one obtains Vβ ≤ -(κ 1 -1) βT 1 β1 + (κ 2 ∆ σ(t) + 1) 2 β0 2 + N β * 2 0 (10)From (10) , we can conclude that the estimation error βi,1 is bounded for t ∈ [0, ∞). Therefore, the states of distributed switching observer can track the trajectory signal within a small neighborhood near the origin. The proof of Theorem 1 is completed.In this subsection, the underactuated controllers for multi-USVs are designed to track the trajectory estimated signal βi,0 . Define the position errors of each USV as xi = x i -xi,0 -l i and ỹi = y i -ŷ i,0 -li , the angle error as z i,1 = ψ i -ψ i,d , where (l i , li ) is the relative position of i-th USV in the formation, and ψ i,d = arctan 2( ỹi xi ). The concrete expression of arctan 2(•) can be found in [18] .In what follows, we design the control input τ i,r for underactuated USV model (1).Step 1. The time derivative of angle errorEQUATIONwhere z i,2 = r i -α i,r , α i,r is the virtual control signal for yaw rate r i under backstepping structure. Design the Lyapunov function at this step asV i,1 = 1 2 ln z 2 i,1 ν 2 i,ψ -z 2 i,1with ν i,ψ < π 2 . Its time derivative followsEQUATIONBy applying Young's inequality, one yieldsEQUATIONSubstituting ( 13) into (12), we haveEQUATIONDesign the virtual controller asEQUATIONwhere c i,1 is a designed positive constant. The final version of Lyapunov function V i,1 can be expressed asEQUATIONStep 2. The time derivative of error z i,2 = r i -α i,r can be expressed asEQUATIONBy employing fuzzy logical systems (FLSs) to approximate the unknown dynamic m i,11 -m i,22 u i υ i + m i,33 f i,3 (r i ), (17) can be rewritten asEQUATIONwhere θ * i,1 is the ideal vector of FLSs, ϕ i,1 (u i , υ i , r i ) is fuzzy basis function, and ε i,1 (u i , υ i , r i ) is the bounded approximation error.Design the Lyapunov function at this step asV i,2 = V i,1 + mi,33 2 z 2 i,2 + 1 2 Θ2 i,1 , where Θi,1 = Θ i,1 -Θi,1 , Θ i,1 = θ * T i,1 θ * i,1, and Θi,1 is the estimated variable of unknown product of FLSs vectorEQUATIONInvoking Young's inequality, the following inequalities holdEQUATIONSubstituting ( 20) into (19) , we can obtainEQUATIONIn this step, design the controller τ i,r and adaptive law Θi,1 asEQUATIONwhere c i,2 and γ i,1 are designed positive constants.Substituting ( 22) into ( 21), the final version of Lyapunov function V i,2 can be described asEQUATIONIn what follows, we design the control input τ i,u . To design the surge force, define the global position errors of i-th USV as zi = x2i + ỹ2 i .Step 1. Recalling the definitions of xi , ỹi , zi and ψ i,d , we have xi = zi cos(ψ i,d ) and ỹi = zi sin(ψ i,d ). Therefore, the time derivative of zi satisfiesEQUATIONwhere ψi = z i,1 .Construct the Lyapunov function at this step asV i,3 = 1 2 z2 i .The time derivative of it can be described asEQUATIONDefine the backstepping error as z i,3 = u i -α i,u , where α i,u is the virtual control signal for surge velocity u i . Therefore, (25) can be written asEQUATIONAt this step, the virtual control α i,u is designed asEQUATIONwhere c i,3 is a designed positive constant. Substituting (27) into (26), we get thatEQUATIONNote that cos( ψi ) may be equal to 0, which will result in the singularity problem in (27). To avoid this problem, we construct the Lyapunov function asV i,1 = 1 2 ln z 2 i,1 ν 2 i,ψ -z 2 i,1, which is called barrier Lyapunov function (BLF). If the BLF is bounded and the initial value z i,1 (0)satisfies |z i,1 (0)| ≤ ν i,ψ , it can yield that -π 2 < -ν i,ψ < z i,1 (t) < ν i,ψ < π 2 . From ln ν 2 i,ψ ν 2 i,ψ -z 2 i,1 ≤ z 2 i,1 ν 2 i,ψ -z 2 i,1and (23), choosing the positive constants as ρ= min{c i,1 /2, c i,2 /2, γ i,1 } and ω = N i=1 (κ + 1 2 ε * 2 i,1 + 1 2 τ * 2 i,ωr + γi,1 2 Θ * 2 i,1 ), we have V 2 = N i=1 V i,2 ≤ -ρV 2 + ω.Based on this fact, we can conclude that V 2 is bounded and z i,1 satisfies -π 2 < -ν i,ψ < z i,1 (t) < ν i,ψ < π 2 , ∀t ∈ [0, ∞), respectively. Thus, there is no singularity problem in (27).Step 2. Recalling the definition of backstepping error żi,3 , the time derivative of it follows żi,3 = m i,22 m i,11EQUATIONInvoking FLSs approximating the uncertain dynamic m i,22 υ i r i + m i,11 f i,1 (u i ), we haveEQUATIONwhere θ * i,2 is the ideal vector of FLSs, ϕ i,2 (υ i , r i , u i ) is fuzzy basis function, and ε i,2 (υ i , r i , u i ) is the bounded approximation error.Consider the Lyapunov function at this step asEQUATIONEQUATIONwhere Θi,2 = Θ i,2 -Θi,2 , Θ i,2 = θ * T i,2 θ * i,2 , and Θi,2 is the estimated variable of unknown product of FLSs vectorΘ i,2 = θ * T i,1 θ * i,1Applying Young's inequality, it followsEQUATIONSubstituting (32) into (31), we obtainEQUATIONIn this step, design the controller τ i,u and adaptive law Θi,2 asEQUATIONwhere c i,4 and γ i,2 are designed positive constants.Substituting (34) into (33), the final version of Lyapunov function V i,4 can be described asEQUATIONTheorem 2. The angle tracking error z i,1 and position tracking error zi can converge to a small neighborhood near the origin.Proof. Choosing the global Lyapunov functionEQUATIONΘ2 i,1 + κ + 1 2 ε * 2 i,1 + 1 2 τ * 2 i,ωr + γ i,1 2 Θ * 2 i,1 -c i,3 z2 i -c i,4 z 2 i,3 + κ + 1 2 ε * 2 i,2 + 1 2 τ * 2 i,ωu - γ i,2 2 Θ2 i,2 + γ i,2 2 Θ * 2 i,2 ] ≤ -CV + D (36) where C = min{ ci,1 2 , c i,2 2 , ci,3 2 , ci,4 2 , γ i,1 , γ i,2 } and D = N i=1 {2κ + 1 2 ε * 2 i,1 + 1 2 τ * 2 i,ωr + γi,1 2 Θ * 2 i,1 + 1 2 τ * 2 i,ωu + γi,2 2 Θ * 2 i,2 }.In this section, three USVs are considered with the kinematics and dynamics model (1) to prove the effectiveness of the proposed method. Their parameters are referred [19] , and the switching typologies communication networks are depicted in Fig. 1 , respectively. The switching interval is designed to be 0.3s. From Fig. 1 , we can observe that the considered switching typologies do not need that each switching mode P σ(t) involves a spanning tree, which is different from the existing switching results on USVs [1]- [5] . 2 shows the path trajectories of each USV. It can be seen that each USV can follow the desired path with their own formation position. The formation mission is achieved. Fig. 3 displays the trajectories of angle tracking error z i,1 . From Fig. 3 , we can conclude that each angle tracking error z i,1 are constrained within the designed boundary ν i,ψ , which demonstrates that the virtual controller (27) avoids the singularity problem. V. CONCLUSION This paper proposes a formation control method for Underactuation multi-USVs with jointly connected switching typologies. To provide the trajectory signal for each USV, distributed switching observer is designed. And by employing the signal of distributed switching observer, the underactuated inputs are devised under backstepping structure. In theory, we prove the estimation error and formation error can converge to a small neighborhood near the origin. Simulation results also demonstrate the effectiveness of the proposed method. In practice, finite-time control methods can ensure that unmanned vessels achieve stability within a finite time frame. Therefore, in the next phase of our research, we will extend this method to the field of finite-time control. 
 
 Appendix:  



Reviews: review: title: title: Accept
main_review: main_review: 1.	If lemmas 1 and 2 are not first proposed by the author, the corresponding references need to be added.
2.	In the introduction, the author mentions some comparisons with existing studies (e.g. [2]-[5]) but does not summarize in detail the main contributions of this study and the specific differences with existing studies. It is suggested to add a paragraph to the introduction outlining the innovations of the paper.
3.	It is suggested that the author further provide the analysis of the learning effect of the network under different topology switching modes, especially the influence of the joint-connected topology on the stability and control accuracy of the USV system.
title: title: The topic under consideration is interesting. This paper can be accepted after minor modifications.
main_review: main_review: This paper considers the adaptive formation control problem for underactuation Multiply Unmanned Surface Vehicles connected by the jointly connected switching typologies. The topic under consideration is interesting. Detailed comments and suggestions are listed as follows.
1.	The English writing of the paper needs to be further polished, and some typos should be fixed, such as “the distributed observer is design as…”, “Taking time derivative of it follows”, and “On the basis of the fact …, choosing the positive constants as…, is follows…”.
2.	In Page 5, the derivation of (31) cannot be easily followed. Please give a detailed explanation.
3.	The references should be added for provid ing a more comprehensive summary of the current research status.
title: title: This article is very interesting, but there are still some shortcomings.
main_review: main_review: The conclusion chapter is also too brief. I believe the author can provide relevant opinions on the usefulness and limitations of this method.