Self-triggered Distributed Formation Control of Under-actuated Unmanned Surface Vehicles in GPS-denied Environments

Authors: xintong  he
lu  liu
haoliang  wang
zhouhua  peng
dan  wang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-30 16:30:20

Abstract: This paper presents a self-triggered distributed formation control method for multiple unmanned surface vehicles (USVs) with limited communication resources in GPS-denied environments. Initially, we configure a velocity quantization-based self-triggered mechanism to reduce the data exchange rate through the wireless network infrastructure. As information transmitted between vehicles is indexed rather than raw data, this approach enhances security and makes data transmission more efficient and stable. Subsequently, we develop a self-triggered distributed guidance law that functions independently of the position information of one's own vehicle as well as that of neighboring vessels. Finally, simulation results validate the effectiveness of the proposed method and highlight its potential applications in environments with weak or nonexistent GPS signals.

Main paper: 
 i. introduction: 
 In recent years, unmanned surface vehicles (USVs) have witnessed a notable surge in various fields from industry to military in recent years [1] - [4] . The distributed formation control of USVs has been getting a lot of attention owing to its recognized potential to significantly enhance operational efficiency when compared to the performance of individual USVs [5] - [8] . The fundamental concept of distributed formation control involves multiple vehicles to maintain a predesigned geometric formation while following a predetermined target.Over the past two decades, an array of strategies for formation control has been developed, encompass-Identify applicable funding agency here. If none, delete this.ing approaches like the leader-follower technique, virtual structural methods, and behavioral-based strategies [9] - [11] . Among these strategies, the leader-follower technique for managing formations has seen extensive application in USVs, noted for its straightforwardness and expansibility, as mentioned in references [11] - [13] . Reference [11] introduces a sophisticated scheme for leader-follower formation control that is predicated on the positional information of the leading unit and a prearranged formation pattern. The work in [12] details the introduction of an observer-driven control technique designed to establish a containment formation swiftly. Concurrently, [13] tackles the combined issue of steady formation and trajectory tracking with a streamlined, adaptive control structure. In the literature above, actual velocity information needs to be transmitted, and quantization problems are not mentioned. In [14] , to save the restricted communication bandwidth, hysteretic quantization has been utilized. However, its controller needs continuous information from neighbors.To alleviate the communication load on vehicular wireless networks, some event-triggered and self-triggered control methods have been proposed to reduce data transmission frequency. In [15] , for the problem of path following for USVs, a dynamic controller is further improved by considering the event-triggered condition. There have also been some achievements in applying event-triggered control to distributed control for multiple USVs [16] , [17] .In conventional operating systems, event-triggered mechanisms are typically adequate; however, under certain circumstances, such as in large-scale or highly dynamic environments, their utilization may lead to resource inef-ficiencies. In contrast, self-triggering represents an active communication paradigm where the next triggering instance can be forecasted based on data from the preceding event. Due to its reliance solely on the local information state of individual agents to ascertain the subsequent triggering event, self-triggering finds favorable applications in distributed control scenarios. In [18] , a self-triggered cooperative vector field is developed to avoid continuous listening. In [19] , a self-triggered path updating law is designed to reduce the sampling times. The self-triggering mechanisms in [18] and [19] need to estimate the states of neighbors, which increases the computational burden on vehicles. Since one's own and neighbors' position information are all needed to compute the next triggering time, these mechanisms cannot be applied in GPS-denied environments.Inspired by the preceding discussions, this article is motivated by three key factors. Firstly, although eventtriggering is extensively utilized in distributed formation control for multiple USVs, self-triggering is a more suitable approach for such systems and practical operational environments, yet it remains relatively unexplored. Secondly, the majority of self-triggering mechanisms require the reception of neighboring state information at sampling intervals, which may hinder their effectiveness in environments with weak GPS signals. Therefore, it is imperative to devise a novel triggering strategy capable of predicting the next triggering time without relying on absolute states. Thirdly, given the practical limitations of communication bandwidth, quantization cannot be overlooked. Additionally, transmitting the index of quantized information, rather than actual data, to each neighbor significantly mitigates the risk of data leakage.The contributions of this method can be summarized as follows:1) In contrast to the event-triggered distributed methods presented in [16] , [17] , a self-triggered distributed guidance law is developed, eliminating the need for continuous listening. Each vehicle calculates its next triggering time and communicates it to neighboring USVs during the current triggering event. To the best of our knowledge, this is the first attempt to consider the self-triggering approach in the design of distributed formation controllers for USVs.2) In contrast to previous self-triggering mechanisms proposed for the formation of USVs, such as those in [18] , [19] , we propose a self-triggering method that focuses on relative states, rather than absolute states. Each vehicle is equipped with distance sensors to measure the relative states between its own ship and its neighbors. Therefore, we do not need to estimate the states of all neighbors, which reduces the computational burden on the vehicles. Combined with the above design that eliminates the need to transmit absolute states, the method proposed in this paper can be applied to challenging environments.3) In contrast to the triggering methods described in [15] - [19] , each vehicle needs to transmit the index of its own quantized velocity to neighbors and decode the indices received from others. The information transmitted between vessels is the index, not the actual data, which can reduce the risk of data leakage. Furthermore, since the information is already quantized at the sensing stage, it can be transmitted without any errors, even through channels with finite capacity.The structure of this paper is organized as following. In Section II, key background concepts are introduced, and the addressed problem is formally defined. The proposed strategy is then presented in Section III, with details provided on the design of a novel self-triggered mechanism and algorithm. Simulations discussed in Section IV aid in evaluating the method's performance, accompanied by results and analysis. Lastly, Section V draws conclusions for this paper.In this document, we employ the following symbols: R N stands for the n-dimensional Euclidean space; the term diag(•) is used to indicate a diagonal matrix. Each vehicle is assigned a label i ∈ N := {1, 2, ..., N }. We denote by N i the set of all neighbors of vehicle i. The adjacency matrix is defined asA = [a ij ] ∈ R N ×N with a ij = 1, if i,j are neighors; and a ij = 0, otherwise. The Laplacian matrix is defined as L = D -A with D = diag{d 1 , ..., d N }. Meanwhile, we define d i = N j=1 a ij , i = 1, ..., N . For a symmetric matrix L ∈ R N ×N with eigenvalues λ 1 ≥ λ 2 ≥ ... ≥ λ N arranged such that λ 1 ≥ λ 2 ≥ ... ≥ λ N , we define λ 2 (L) := λ 2 .The Lambert W -function, expressed as W (y) for y ≥ 0, specifically corresponds to the non-negative solution x which satisfies the transcendental equation xe x = y. In this paper, the subsequent fact would be consistently used without comment. By defining a, c > 0, the solution x = x * of the transcendental equation a(x -b) = e -cx can be written asEQUATIONConsider a network of underactuated USVs labeled from 1 to N, the motion of the ith USV is represented by a kinematic modelEQUATIONwhere η i = [x i , y i ] T represents the position in an earthfixed inertial frame; ψ i denotes the heading angle; u i , υ i , and r i denote the velocities in the surge, sway, and yaw directions in a body-fixed frame (as illustrated in Fig. 1 ).A graphical representation depicting the distributed formation control for USVs is presented in Fig. 1 EQUATION)whereη i = [x i , y i ] T ; η 0 = [x 0 , y 0 ] T ∈ R 2 , the path of the leader; η id (t) = [x id , y id ] T ∈ R 2 , a positional divergence from the leader; l 1 ∈ R is a positive constant.To move on, the following assumptions are needed. Assumption 1: The velocities of the USV are bounded, that is, Given that the USV is a mechanical system, it adheres to the principles of Newton's second law. This adherence inherently limits the possible velocity and acceleration profiles of the USV. Therefore, the conditions set out in Assumption 1 are well justified. The literature, specifically Guo et al. [20] , provides established constraints on velocities and accelerations that align with these principles.|u i | ≤ u * , |v i | ≤ v * , and |r i | ≤ r * . u * , v * ,In this section, a joint design method combining quantization and self-triggered sampling is proposed for multiple USVs, enabling asymptotic consensus while avoiding Zeno behaviors. The architecture of the strategy presented in this study is depicted in Fig. 2 . The distributed kinematic control law is designed to generate the control inputs for each USV based on self-triggered sampling of relative states and the estimated information obtained by an extended state observer.A. Triggering mechanism 1) Quantization scheme: Let P ∈ N be the number of quantization levels, which is an odd number, i.e., P = 2P 0 + 1 for some P 0 ∈ N 0 and V 0 ∈ N be a quantization range. We apply uniform quantization to the interval [-V 0 , V 0 ]. More precisely, a quantization function Q V0,P is defined byQ V0,P [m] :=      2sV0 P if (2s-1)V0 P < m ≤ (2s+1)V0 P 0 if -V0 P ≤ m ≤ V0 P -Q V0,P [-m] if m < -V0 P , (4) for -V 0 ≤ m ≤ V 0 , where s = 1, 2, ..., P 0 .2) Computation of inter-event times: To study the consensus of multi-USV systems, the following assumption is needed.Assumption 3: There is a boundary G 0 > 0 satisfyingη i (t 0 ) - 1 N j∈N η j (t 0 ) ≤ [G 0 , G 0 ] T , i ∈ N ,is known by all USVs, where t 0 is the initial time.A range G is defined by the following functionEQUATIONwhere M ≥ max{u * , v * , r * }, γ ≤ λ 2 (L); ω > 0, a given decay paramater . We denote the sampling times of vehicle i by {t i n } n∈N with t i 0 := 0 and inf n∈N0 (t i n+1 -t i n ) > 0. According to [21] , the following unsaturation condition ( 6) is satisfied at time t = t i n for some n ∈ N under Assumption 3:EQUATION)whereη ijd (t i n ) = η id (t i n )-η jd (t i n ), η ij (t i n ) = η i (t i n )-η j (t i n ), for all i ∈ N and j ∈ N i . Therefore, the multi-USV system reaches a consensus at an exponential pace characterized by a decay rate.Let {t i n } n∈N be the sampling times of vehicle i with t i 0 := 0. We defineEQUATION)Using the range G(t), we computate inter-event timesEQUATIONwhere τ i xn , τ i yn ∈ R, δ i >0 is a given threshold and τ i min , τ i max > 0 are the lower and upper bounds of inter-event times, respectively, i, e., τ i min ≤ τ i n ≤ τ i max . Thus, we define the (n + 1)th triggering time t i n+1 of vehicle i ∈ N byEQUATION)Due to the need for vehicles to communicate velocity information with each other, the quantization function Q V0,P is used to quantize velocities at triggering time. We defineEQUATION)where V 0 ≥ max{u * + v * }. In fact, h i (t i n ) belongs to the finite set2sV 0 P : s ∈ Z, -P 0 ≤ s ≤ P 0 .An encoder of vehicle i assigns an index to each 2sV 0 /P and transmits to a decoder of each neighbor the index corresponding to the h i (t i n ) at triggering time. Since vehicles share V 0 , P , the decoder can generate h i (t i n ) from the received index without any errors even through finite capacity, which is then used by neighbors to compute the next triggering time.We design the set {t ℓ } ℓ∈N0 , which consists of sampling times for all the vehicles without duplication, while n i (ℓ) denotes the count of sampling instances of vehicle i within the interval (0, t ℓ ]. Next, we describe how vehicles compute sampling times in a self-triggered fashion.Step 1. Let i ∈ N and n ∈ N. At time t = t i n := t ℓ0 , vehicle i executes the subsequent set of actions.i) Upon receiving an index from a neighboring vehicle at the instant t = t ℓ0 , the index is decoded, and subsequently, the velocity data of the respective neighboring vehicles is updated accordingly.ii) Compute τ i n,0 as follows. According to the triggering condition (8), we defineτ i xn,0 τ i yn,0 = inf    τ > 0 : τ d i h i (t i n ) -τ j∈Ni h j (t j nj (ℓ0) )> δ i e -ωτ ℓ0 ).(11) By the Lambert W -function W (1), we define a function ϕ as ϕ(a, b, c) :=EQUATIONfor a, c ∈ R and b ≥ 0.Then ( 11) can be rewritten asEQUATIONwhereEQUATION)and we can obtain the τ i n,0 byEQUATION)iii) Set k = 0.iiii) Send information to neighbors. The quantized velocity information h i (t i n ) of the own vehicle is encoded into an index belonging to a finite set of cardinality 2P 0 +1, and this index is then communicated to each neighboring vehicle.Step 2. Vehicle i plans to activate the sensor at timet = t i n + τ i n,k .Step 3-a. If vehicle i receives information from some neighbor in the interval (t ℓ k , t i n +τ i n,k ), i needs to compute a candidate of the next triggering time again. In this situation, let k ∈ N and i receive new signals from its neighbors at times t = t ℓ1 , ..., t ℓ k . At time t = t ℓ k , vehicle i carries out the subsequent actions i)-iii). Then go back to Step 2. i) Set k to k + 1 and record the time t ℓ k at which the information is received from neighors.ii) Update the velocity information h j t j nj (ℓ k ) of the neighbor.iii) Compute τ i n,k by followings. The defination is similar as (11) , and is omitted here. Vehicle i computesτ i xn,k τ i yn,k = ϕ(a i n,k , b i n,k , c i n,k ),whereEQUATIONand we can obtain new inter-event time,EQUATION)Then t i n + τ i n,k is the candidate of the next triggering time of i.Step 3-b. If vehicle i does not receive any information in the interval(t ℓ k , t i n + τ i n,k ), then it sets t i n+1 = t i n + τ i n,k .Step 4. Vehicle i sets n to n + 1. Then go back to Step 1.In this subsection, we devise a self-triggered, distributed guidance strategy, grounded on relative positions acquired through sensors. To attain the desired formation configuration, we define a formation control error specific to the ith vehicle, as elaborated below:EQUATIONwhere η ij = η i -η j , η ijd = η id -η jd , the rotation matrix J i (ψ i ) is expressed asJ i (ψ i ) = cosψ i -sinψ i sinψ i cosψ i .The self-triggered information of the ith USV is defined by the following aperiodic scheme[η e ij (t), η e i0 (t)] = [η ij (t i n ), η i0 (t i n )], t ∈ [t i n , t i n+1), where t i n is the triggering time with n ∈ N, t i n+1 is determined by the scheme in III-A 3).According to (18) , we define the selft-triggered formation control error asEQUATIONwhere the relative state η e ij , η e i0 can be sampled by the sensors at triggering time.To mitigate the challenges posed by under-actuation, we introduce an error transformation as followsEQUATION)where ∆ 0 ∈ R is a positive constant. Taking the time derivative of (19) yieldsEQUATION)whereC i = diag{d i , ∆ 0 }, B = 0 -1 1 0 , σ i = -Σ j∈Ni a ij J T i (ψ i ) ηj -Σ j∈Ni a ij J T i (ψ i ) ηijd + [0, d i v i ] T + Σ j∈Ni a ij J T i (ψ i ) ηijd -a i0 J T i (ψ i ) η0 -a i0 J T i (ψ i ) ηi0 , ηij = η ij -η e ij, and ∥ σi ∥ ≤ σ * with σ * being a positive constant. Note that σ i is totally unknown due to the unavailable continuous neighboring velocites ηj , η0 and self-triggered error ηij . To estimate it, an extended state observer is proposed asEQUATION)where ze i = ẑe i -ze i is the the estimated triggering error, ϑ i = C i [u i , r i ] T and p i1 , p i2 are positive constants. Based on the extended state observer, we formulate a selftriggered, distributed kinematic guidance law asEQUATIONwhereO i = diag{o ix , o iy } ∈ R 2×2 with o ix ∈ R and o iy ∈ R being positive constants; ϶ i ∈ R is a positive constant.In this section, simulation results are provided to illustrate the self-triggered distributed formation controllers for multiple underactuated USVs with limited communication resources in GPS-denied environments. Consider a networked system consisting of five USVs. Since there is no thrust in the direction of underactuated ship roll, a small transverse speed will be generated under the influence of wind and wave currents, resulting in sideslip angle. To compensate for this, v is set to 0.2 for all USVs.The initial states of five USVs are given as η 1 = [4.8, 5.8] T , η 2 = [-1.1, 6] T , η 3 = [5, 0] T , η 4 = [-4, 6] T , η 5 = [K i = diag{2, 2}, ∆ 0 = 0.1, ϵ i = 0.01, ω = 0.1046, δ 1 = δ 2 = δ 3 = 0.01, δ 4 = δ 5 = 0.03, τ 1 max = τ 2 max = τ 3 max = 3s, τ 4 max = τ 5 max = 5s.To avoid Zeno behaviors, we set τ 1 min = τ 2 min = τ 3 min = 12.483 × 10 -3 s, τ 4 min = τ 5 min = 55.857 × 10 -3 s. Vehicles 4 and 5 have larger thresholds and lower bounds for inter-event times due to their smaller number of neighbors compared to the other USVs.The communication topology among the fleet is depicted in Fig. 3 . The simulation results, as shown in Fig. 4 -Fig. 7 , confirm that the proposed self-triggered distributed controller successfully manages a time-varying formation involving five USVs. Fig. 5 displays the trajectory of selftriggered coordinated errors, and it is shown that these errors converge to a boundary at steady state. The relative straight-line distances between each USV and its neighbors are illustrated in Fig. 6 , where it can be observed that the inter-vehicle distances stabilize within a narrow band around the value 7.07. Additionally, Fig. 7 shows that triggerring occurs frequently in the interval [0, 3] but the frequency of triggering decreases after 3s. In this paper, we consider a self-triggered distributed formation control method for multiple underactuated USVs with limited communication, which uses measurements of relative states and quantized velocity information. A distributed guidance law is developed for each vehicle based on a self-triggered consensus approach and an extended state observer. Compared with previous self-triggered studies, we use relative states instead of absolute ones. Additionally, the estimation of the states of neighbors is not needed, reducing the computational burden on USVs. Therefore, the proposed algorithm can be applied to GPS-denied environments. Meanwhile, since the velocity information is already quantized at the sensing stage, it can be conveyed without any errors even through finite capacity channels. The information transmitted between vessels is the index rather than the actual data, which can prevent information leakage. Simulation results substantiate the effectiveness of the proposed integrated guidance and self-triggered method by relative states for underactuated USVs. In the future, it will be interesting to implement the proposed selftriggered distributed formation control algorithm in a realworld multi-USV system. 
 
 Appendix:  



Reviews: review: title: title: The article proposes a self triggering communication method that uses relative position information to predict the next triggering time for unmanned ship formation control in environments with weak GPS signals. The selected topic is in line with the research background in the field of control and has good research significance. In order to better publish this paper, the following suggestions are proposed:
main_review: main_review: 1) There are multiple missing spaces and incorrect use of singular and plural words in the article.
2) There are some symbol font issues in the article, where the N-symbol font representing the set is inconsistent before and after.
title: title: Accepted, but some recommendations remain
main_review: main_review: This paper designs a self-triggered distributed formation control method for multiple unmanned surface vehicles. The self-triggering mechanisms proposed in this paper are effected by comparing the relative states between vehicles, which reduces the computational burden and eliminates the need for absolute states. Meanwhile, secure transmission tasks achieved by the fact that the information transmitted between vessels is not actual data. There are still some  suggestions:
1. The comparative simulation of the event triggering methods presented in [16], [17] could be added.
2. This paper mentions that this is the first attempt to consider the self-triggering approach in the design of distributed formation controllers for USVs. Hopefully, the author will conduct more research to determine if this fact holds true.
title: title: Self-triggered Distributed Formation Control of Under-actuated Unmanned Surface Vehicles in GPS-denied Environments
main_review: main_review: This paper presents a self-triggered distributed formation control method for multiple unmanned surface vehicles (USVs) with limited communication resources in GPS-denied environments. This work is well organized. Below are some comments.
(1)	The contributions should be illustrated in a clearer manner. For example, what is the main improvement of the paper compared to the existing results. The authors should explain the unique contributions of this paper.
(2)	It is better to give a guideline of selection for all control parameters.
(3)	The simulation results should be explained more carefully.
(4)    The self-triggered method  should be compared with other methods to highlight its advantage.
(5)	The paper is well presented, spelled correctly. I recommend authors to carefully read the entire paper to find possible misspellings.
title: title: Accept
main_review: main_review: This paper proposes a self-triggered formation control method for situations with missing GPS signals. The velocity quantization-based self-triggered mechanism reduces data exchange frequency through wireless network infrastructure, and the effectiveness of the proposed method is validated through simulation results. However, to further enhance the quality of the paper, the following suggestions should be considered: 1. The self-triggered method emphasized in the main contributions should be compared with similar methods to highlight its effectiveness; 2. Some formulas in the paper, such as Formula 3, exhibit non-standard forms, and some parameters provided lack specified ranges