Approximation-free design for distributed formation tracking of networked uncertain underactuated surface vessels under fully quantized environment

Distributed formation tracking results have been studied in the literature for networked underactuated surface vessels (NUSVs) at the dynamic level. The distributed formation tracker design of NUSVs under a fully quantized environment remains an open problem. This paper provides a solution to this problem in the low-complexity control framework that ensures prescribed performance without the help of any adaptive approximation tools. State variables and underactuated control inputs of followers are quantized and communicated under a directed network. Distributed formation controllers using quantized state feedback signals are constructed for NUSV followers. The design conditions that lead to establishing the prescribed performance of local tracking errors in the presence of state and input quantization are newly derived via distributed quantization errors. It is shown from the recursive analysis of closed-loop quantization errors that although all states and inputs of followers and the communicated position information under a directed network are quantized, the prescribed performance of the presented formation system is ensured.


Introduction
Innovative efforts on cooperative control design and stability analysis of unmanned surface vessels have been motivated by wide applications in marine engineering [1,2]. Based on the control results [3,4] for single surface vessels, several control techniques have been developed for establishing formation control methodologies of multiple surface vessels (MSVs) [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Among them, the prescribed performance control technique [21] has extensively been applied to various formation problems of MSVs because formation control performance can be designed a priori in transient and steady-state responses. In [22][23][24][25], the formation control designs with prescribed performance were developed for the fully actuated models of MSVs. For more practical applications, underactuated surface vessels (USVs) have been considered to address various formation tracker design problems based on prescribed performance. In [26], a distributed prescribed performance formation control strategy was established for networked USVs (NUSVs). The strategy reported in [26] depends on neural-network-based function approximators to approximate unknown nonlinear functions including model uncertainties online.
Thus, computational complexity is increased because the weights and activation functions of the neural networks are computed and updated online. To overcome the design complexity caused by function approximators, low-complexity design strategies, achieved by transforming all error surfaces into performancefunction-based nonlinear errors, have been developed for formation control of uncertain USVs without the help of auxiliary function approximators [27][28][29]. Despite these prescribed performance formation control studies in the low-complexity sense [27][28][29], the state and input quantization problem in the feedback loop is an open problem for distributed formation tracking of NUSVs under a constrained communication bandwidth. For emergency and dangerous cooperative missions in the practical environment, multiple USVs may be required to communicate with control stations located in the land-based center or other ocean vehicles under a network with constrained bandwidth [30,31]. In this case, the network-based communication among USVs also depends on quantization signals. From the need for quantized discrete signals in the feedback and communication loop, two main questions for our study are as follows: (M1) Can we design a distributed low-complexity formation tracking controller for NUSVs by quantized state feedback and communication, and analyze its stability in the presence of quantization errors? (M2) What are the design conditions of local performance functions for ensuring the robustness on quantization errors and nonlinear uncertainties in the fully distributed formation control framework?
With the increase in digital communication channels, quantization has been regarded as an important problem for feedback control under a capacity-limited network. Quantization of state variables only allows the use of discontinuous feedback signals in the control design. Because of this discontinuous state feedback property, the controller design and stability analysis in the presence of state quantization are more difficult than dealing with the input quantization reported in [32][33][34][35][36][37][38][39][40][41]. Although some limited research results have been reported to incorporate the state quantization problem for uncertain nonlinear systems [42][43][44], they cannot be applied to USVs due to the underactuation problem. Thus, some studies [45][46][47] were recently proposed for USVs. However, the design reported in [45] is only applicable to single USVs. In [46,47], the communication signals among vessels were only quantized. The state variables and control inputs need to be quantized for distributed formation tracking of multiple USVs under a capacity-limited network. To the best of our knowledge, the distributed formation tracker design with low complexity, which ensures the prescribed performance of NUSVs with state and input quantization, is still an open problem. The main issues on this problem are as follows: (I1) Since quantized feedback and communication signals are not differentiable, they cannot be directly employed in the Lyapunov-based distributed recursive design for NUSVs at the dynamic level. Thus, the first main issue is to handle discontinuities of quantized states and communication signals in the distributed tracker design of NUSVs with completely unknown nonlinearities. (I2) In this problem, quantized states of NUSVs can be only used to construct nonlinearly transformed distributed errors using performance functions because genuine states are unavailable for feedback. However, the formation control objective can be achieved by analyzing the stability of distributed nonlinear errors using genuine states. That is, the distributed nonlinear errors for the tracker design and those for the prescribed performance analysis are different from each other. Thus, the second main issue is to derive conditions for selecting performance functions that ensure the stability of distributed nonlinear errors using genuine states via the distributed state-quantized formation tracker. (I3) The auxiliary signals can be a solution of the underactuation problem in the kinetics of NUSVs. However, it is difficult to analyze the stability of the control scheme because differentiation of the auxiliary signals is impossible due to the quantized state feedback and communication signals. Thus, the third main issue is to design the auxiliary signals and to analyze their stability by using only quantized state feedback and communication signals.
Based on the above observations, this study aims to give positive solutions to these issues. To design a low-complexity control scheme with prescribed per-formance using quantized states and communication for distributed tracking of NUSVs under a directed network, local state-quantized trackers are constructed by communication of only neighbors' quantized positions in the low-complexity framework. By proving the boundedness of quantization errors of underactuated control inputs recursively, the analysis strategy of the prescribed performance of closed-loop formation systems is provided from the Lyapunov stability theorem. Simulation is presented to verify our theoretical approach.
In sum, these study's unique contributions are as follows: (C1) This study addresses the low-complexity control issue with prescribed performance under a fully quantized environment in the distributed formation control field of NUSVs. The measured state variables and control inputs of the local feedback control designs and the communicated position signals among USVs are all quantized. The local controllers using quantized signals are designed without the employment of any function approximators, low-pass filters for virtual control laws, and adaptive compensation terms. (C2) Conditions for setting the initial parameters of performance functions are derived for ensuring prescribed formation tracking performance, regardless of the unavailability of distributed nonlinear errors using genuine states in the formation tracker. (C3) In [46], the quantized state feedback problem was not considered, and the formation performance cannot be prescribed. Moreover, the fully actuated models of unmanned surface vessels were considered for the location game problem. In contrast, this paper focuses on the underactuated model in the presence of both quantized state feedback and communication. Quantizedstate-based auxiliary signals are designed to deal with the underactuation problem in the kinetics of NUSVs and to analyze the stability of sway dynamics under a fully quantized environment. Furthermore, the prescribed formation tracking performance problem is also addressed in this paper.
The rest of this paper is organized as follows: Section 2 introduces the kinematics and dynamics of NUSVs with state and input quantization and formulates the quantization-based distributed formation control problem of NUSVs. Section 3 elaborates on a quantizationbased distributed low-complexity formation tracker design with prescribed performance and the stability analysis of the closed-loop system considering quantization errors. Simulation results are provided in Sect. 4. Section 5 gives a conclusion of this paper.

Graph theory basis
Graph theory is used to model the network interactions among agents. Considering one leader and N USV followers, a directed communication graph G = {V, E}, where V = {0, 1, 2, . . . , N } is the set of vertices and E ⊆ V × V is the set of edges denoting the topology structure. Node j can obtain information from node i through a directed path (i, j) ∈ E. The neighbors of node i is represented by the set

Models of NUSVs in the presence of state and input quantization
Suppose that N USV followers and a time-varying leader are communicated under a capacity-limited directed network, and each USV and its own local controller are remotely connected through the network. The model of the ith USV follower with asymmetric stern and bow is considered as [50] where i = 1, . . . , N , p i = [x i , y i , ψ i ] with the x-and y-axis position (x i , y i ) and the heading angle ψ i and ν i = [u i , v i , r i ] with the surge, sway, and yaw velocities u i , v i , and r i are the posture and velocity vectors of the ith USV, respectively, 3 ] is the unknown disturbance vector of the ith USV, are the quantized signals of the surge force τ i,u and the yaw moment τ i,r , Q i (·) is the quantization function to be defined later, and Here, m i is the mass of the ith USV, Xu i , Yv i , Yṙ i , and Nṙ i are the added masses, and X u i u i u i denote the hydrodynamic parameters, x i,g denotes the center of gravity in the body-fixed frame, and I i,z is the inertia with respect to the vertical axis.
Owing to the asymmetric stern and bow, there exist off-diagonal terms in M i . Thus, sway dynamics as well as yaw dynamics depends on the yaw moment τ i,r . To deal with this coupling problem, the models (1) and (2) are transformed to [51] 3 ] is defined by

Remark 1
The kinetics (2) of NUSVs contain fewer inputs than degrees of freedom, that is, there is an underactuation problem at the kinetic level. On the other hand, mobile robots are underactuated in the kinematics [52]. Thus, the existing control approaches for mobile robots cannot be directly applied to NUSVs. 3 are unknown and bounded. In addition, the nonlinear functions g i,1 , g i,2 , and g i, 3 in the dynamics (4) are unknown.
In this study, the communication among USVs and between each USV follower and its controller is based on the capacity-limited directed network, as shown in Fig. 1. Thus, the state and input quantization problems in the formation tracker design of multiple USVs are considered for efficient signal transmission under the capacity-limited network. All states and control inputs of USVs before the network transmission are quantized by the uniform-hysteretic quantizer (5), defined at the top of the next page, is the sign function, μ i > 0 denotes the quantization level. Figure 2 shows the map of the uniform-hysteretic quantizer for o i > 0 [53]. [42].

Assumption 2
The quantized positions of the leader (i.e., Q 0 (x 0 ) and Q 0 (ȳ 0 )) are only communicated to a subset of USV followers that has the leader as a neighbor where the function Q 0 (·) is the uniform-hysteretic quantizer with the quantization level μ 0 . Additionally, the leader's velocities u 0 ,v 0 , and r 0 are bounded to provide the stable trajectory of the leader (i.e., for the distributed formation control objective).
The objective of this study is to design the distributed low-complexity formation tracking laws τ i of USV followers under a fully quantized environment so that the USV followers maintain the desired distances to the time-varying leader with the prescribed transient and steady-state performance.
Remark 2 It is noticed that (i) Assumption 1 indicates that the proposed distributed state-quantized formation tracker can be designed without the knowledge of external disturbances and system nonlinearities and their bounds. (ii) Assumption 2 implies that the position information of the leader is also quantized and transmitted to a subset of USV followers through the directed network. That is, the communication between the leader and followers as well as the communica- tracking of USVs aforementioned in Introduction, this paper firstly addresses the fully quantized control problem of NUSVs in the distributed formation tracking framework. In this paper, the prescribed-performance-based distributed formation control strategy and the stability conditions considering quantization errors are derived in the presence of quantized discontinuous state feedback and communication.

Distributed state-quantized low-complexity formation tracker design with prescribed performance
A distributed state-quantized approximation-free design methodology for achieving the prescribed performance formation tracking is investigated in the presence of state and input quantization. The low-complexity control design with prescribed performance for the formation control of USVs basically requires the time derivatives of virtual control laws. However, the quantized state variables cannot be used for the design of the virtual controllers because of the discontinuity of quantized state variables.

Unquantized-states-based distributed intermediate signal design
We define the local error surfaces e i,k , k = 1, . . . , 6, as where i = 1, . . . , N , ς x,i j = ς x,i − ς x, j and ς y,i j = ς y,i − ς y, j are desired positions between agents i and j in the x-and y-axis, respectively; ς x,i , ς x, j , ς y,i , and ς y, j are constants for setting the desired positions of agents i and j, θ i is an approach angle to be designed later, α i,k , k = 1, 2, 3, are intermediate signals,¯ i > 0 is a design constant, and ρ i is an auxiliary signal to deal with the underactuation problem of each USV follower. Then, the normalized error surfaces for ensuring prescribed performance are defined as: where Step 1: Consider the normalized error surfaces η i,1 = e i,1 /γ i,1 and η i,2 = e i,2 /γ i,2 . Then, their time derivatives are represented by: Then, we define the normalized error surface vector (13) and (14), the time derivative of η p is given bẏ where H =L +B, ⊗ denotes the Kronecker prod- , and I 2 is a 2 × 2 identity matrix. Using (9) and (10), the time derivatives of ζ i,1,1 and ζ i,1,2 are obtained as: The intermediate signals α i,1 and α i,2 are constructed as: (17) and (18) into (16) yieldṡ Substituting (19) into (15) yieldṡ Step 2: The normalized error surface η i,3 is considered as where the approach angle θ i is chosen as The intermediate signal α i,3 is designed as where i,2 > 0 is a design constant and φ i, where Step 3: The normalized error surfaces η i,4 , η i, 5 , and η i,6 are considered as η i,4 = e i,4 /γ i,4 , η i,5 = e i,5 /γ i,5 , and η i,6 = e i,6 /γ i,6 , respectively. The time derivatives of the error surfaces η i,4 , η i,5 , and η i,6 are obtained as: where i = 1, . . . , N and α i,u and α i,r are intermediate signals chosen as: , m = 4, 5, 6, and positive constants i,3 , i,4 , i,5 , and ι i .
From (28)(29)(30), (25)(26)(27) becomė where 3.2 Distributed state-quantized formation tracker design and prescribed performance analysis To present the distributed state-quantized formation tracking scheme for USV followers, the normalized error surfacesη i,k using quantized states are defined as: where i = 1, . . . , N and the error surfacesê i,k using quantized state communication are represented by: Here, the proposed state-quantized formation control scheme is designed using the structures of intermediate signals proposed in Sect. 3.1 as follows:

Remark 3
The presented local state-quantized control scheme (44)(45)(46)(47)(48)(49) is constructed without requiring neighbors' orientations and velocities, and any function approximation tools against unknown model nonlinearities g i . That is, the proposed scheme simply consists of normalized error surfacesη i,k and design constants i,k . In this respect, a low-complexity design is achieved, regardless of the state and input quantization of NUSVs.
Remark 4 For achieving prescribed formation tracking performance of NUSVs, the condition |e i,k (0)| < γ i,k (0) related to unquantized states should be satisfied. However, e i,k (0) cannot be calculated because unquantized states are unavailable in this paper. In addition, the proposed control scheme in (44)(45)(46)(47)(48)(49) is implemented via the state-quantized error surfacesη i,k with available errorsê i,k (0). Thus, the design conditions of performance functions γ i,k are presented by considering the quantization effects in the following lemma.
Proof The boundedness of τ i,u − α i,u and τ i,r − α i,r is checked in the recursive manner.
First, we consider the errors e i,k −ê i,k . The errors  (ζ i,1,3 )). 3 is a strictly increasing function and well defined, there exists a constant σ i,3 such that Third, the bounds of the errors |e i,n −ê i,n |, n = 4, 5, 6 are represented by: Because φ i,n , n = 4, 5, 6, are strictly increasing functions and well defined, and η i,n andη i,n are bounded by (65-67), it holds that |φ i,n (η i,n ) − φ i,n (η i,n )| ≤ σ i,n where σ i,n is a constant. Thus, it is ensured that The completes the proof of this lemma.

Remark 6
Compared with the recent formation control approaches [47,56], the proposed design has the following differences.
(i) In [47], the quantization problem of communication signals among NUSVs was only investigated to design a neural network-based adaptive formation controller. That is, the quantization of the full-state feedback information and control inputs was not considered. Besides, the prescribed performance of the formation error cannot be ensured in [47]. The work [47] focuses on the design of the adaptive tuning mechanism using quantized communication signals for weights of neural networks. On the other hand, this study considers a fully quantized environment with the quantization of the full state feedback information, control inputs, and communicated signals for the distributed formation control design of NUSVs. Furthermore, the prescribed formation tracking performance can be guaranteed despite the fully quantized environment. Although all nonlinearities of USV kinetics are unknown, the proposed formation controller does not require any function approximators, filters for virtual controllers, and adaptive compensation terms. Thus, a low-complexity design is achieved. (ii) The key difference between [56] and this study comes from the difference in systems for control design. In [56], multiple nonholonomic mobile robots were treated to design a prescribed performancebased formation tracker in a fully quantized environment. In contrast, this study deals with NUSVs. The NUSVs have fewer inputs than degrees of freedom in kinetics. Namely, there is no control torque for the sway dynamics in (2). On the other hand, the mobile robot is underactuated in kinematics. Therefore, the control method [56] for multiple mobile robots cannot be directly applied to NUSVs. Compared with [56], this study presents the design strategy of quantized-state-based auxiliary signalsρ i to deal with the underactuation problem in the kinetics of NUSVs and to analyze the stability of sway dynamics under a fully quantized environment.

Conclusions
This paper has addressed the distributed formation tracking problem of NUSVs under a fully quantized environment in the framework of low-complexity prescribed performance control. Local controllers of followers using their own quantized states and neighbors' quantized positions have been designed without requiring the knowledge of system nonlinearities in a fully distributed manner. The conditions for designing performance functions have been analytically established to ensure prescribed formation tracking performance with robustness against closed-loop quantization errors. The presented result allows the state and input quantization and quantized communication for distributed prescribed formation tracking of NUSVs under a capacity-limited directed network. However, several study directions remain open, e.g., how to apply the proposed fully quantized control framework to avoid collisions with external obstacles and to preserve network connectivity among NUSVs.

Data Availability Statement
The datasets generated during the current study are available from the corresponding author on reasonable request.

Conflicts of interest
The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.