Research ArticleA Coordination Law for Multiple Air Vehicles in DistributedCommunication Scenarios

Zhongtao Cheng Mao Su Lei Liu Bo Wang and Yongji Wang

National Key Laboratory of Science and Technology on Multispectral Information ProcessingSchool of Artificial Intelligence and Automation Huazhong University of Science and Technology Wuhan China

Correspondence should be addressed to Lei Liu liuleihusteducn

Received 21 November 2019 Revised 20 May 2020 Accepted 19 June 2020 Published 4 July 2020

Academic Editor Hocine Imine

Copyright copy 2020 Zhongtao Cheng et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

(is paper proposes a consensus-based guidance methodology for multiple air vehicles to arrive at the same spot cooperativelyFirst based on the Lyapunov stability theory a guidance law with only one control parameter is proposed and the exactexpression of total flight time can be obtained with a control parameter equal to one(en a two-step guidance scheme which canachieve a finite-time consensus of the flight time is built upon the Lyapunov-based guidance law In the first step on account ofthe information exchange between the air vehicles through an undirected and connected communication topology a time-varyingcontrol parameter is designed to reduce the disparities of the flight time After the consensus of the flight time the controlparameter will remain constant at one and simultaneous arrival can be achieved Besides the guidance strategy is applied in aleader-follower case that one of the vehicles cannot receive information from the others and acts as the leader(e effectiveness ofthe proposed method is demonstrated with simulations

1 Introduction

(e consensus of multiple dynamic agents has a long historyand can be found in many applications such as unmannedunderwater vehicles [1] mobile robots [2] sensor networks[3] and other areas With the vigorous development ofunmanned air vehicles in recent years the simultaneousarrival of multiple air vehicles has attracted much attentionBased on information exchange between the agents thecontrol strategy is designed to make sure that the agentsreach an agreement on the state or output values It isobvious that simultaneous arrival can be achieved if all theagents synchronize the arrival time

A widely used strategy to achieve simultaneous arrival isindividual homing Studies in this direction usually set acommon arrival time as the desired time for each vehiclebefore homing and then each vehicle tries to arrive at thetarget with the same specific time independently In this wayany guidance law with the ability to control flight time can beapplied to this kind of simultaneous arrival (e earliest

appearance in this direction was proposed in [4] a guidancelaw was proposed as a combination of two terms with theability to control the flight time One term was the pro-portion navigation guidance (PNG) law and the other wasthe flight time error(en it was applied to the simultaneousarrival case (e work in [4] was extended in [5] by con-trolling both arrival time and angle (e design frameworkin [4] was further enhanced by taking the field of viewconstraint into consideration in [6]

Some other studies solve the flight time control problem forsimultaneous arrival with nonlinear control theory Two-di-mensional and three-dimensional guidance laws with arrivaltime constraint were derived using the Lyapunov stabilitytheory in [7] where the Lyapunov candidate function con-tained the flight time error directly While the work in [8] dealtwith the problem indirectly a Lyapunov candidate functionconcerning the heading angle error was proposed and the exactexpression of the flight time was derived However the in-complete beta function was used and the flight time controlcovered only a small-time interval

HindawiJournal of Advanced TransportationVolume 2020 Article ID 1810962 10 pageshttpsdoiorg10115520201810962

Other studies on individual homing take advantage ofthe polynomial function [9ndash12] In [9] the guidance com-mand was proposed as a polynomial function with threeunknown coefficients one of which was determined tosatisfy the flight time constraint (e guidance law in [10]solved the flight time control problem by following the lookangle profile that was polynomial in time both second-orderand third-order polynomials were considered (e designframework in [10] was extended in [11] by considering anadaptive guidance scheme (e work in [12] generalizedsome preliminary solutions in this direction and extendedthe polynomial function to any order through mathematicalinduction On the other hand a very recent work presented aguidance scheme that can extend a certain class of existingguidance laws to satisfy the flight time constraints (e onlyrequirement was that the time-to-go prediction was pro-vided by the existing laws [13]

It should be noted that individual homing strategyusually requires the calculation of the achievable range offlight time for each air vehicle before homing (en theintersection is determined from each vehiclersquos achievablerange A suitable common time is chosen from the inter-section It means that the cooperative arrival may fail if theintersection between each vehicle is null

(e disadvantage of the individual homing can beavoided by cooperative homing which requires no pre-determined suitable time selected from the intersection (econsensus of the flight time is achieved through commu-nications among themselves To the best of the authorrsquosknowledge more interest has been directed to individualhoming in the guidance literature and rare instances ofcooperative homing can be found except in [14ndash17] wherethe consensus of the vehiclesrsquo time-to-go estimates wasaddressed to synchronize the arrival time

(e work in [14] proposed a centralized cooperativeproportion navigation (CPN) guidance law to achieve theconsensus of time-to-go through a time-varying navigationgain the calculation of which required the instantaneoustime-to-go information of the air vehicle and that of all theothers (e navigation gain would be updated at each timestep until the instantaneous time-to-go variances went tozero As an improvement of the work in [14] a distributedguidance law was proposed in [15] where each air vehiclemerely exchanged information through an undirected andconnected communication topology with its neighborsrather than all the other vehicles Inspired by the work in[15] the research in [16] introduced a more practical case byconsidering both the perpendicular acceleration and thetangential acceleration and the results were further ex-tended to communication failure case where one of thegroup vehicles cannot get information from the others (erobust guidance law proposed in [17] could still achievesimultaneous arrival without the information of faulty ve-hicles which was different from the unidirectional com-munication error in [16] It should be noted that theaforementioned centralized law or distributed laws achievedthe cooperative arrival by the consensus of the time-to-goestimates It should be noted that the aforementionedguidance laws are based on the consensus of the time-to-go

estimation which is accurate only toward the end of thehoming process

Motivated by the previous work this paper proposes aconsensus-based guidance law for multiple vehicles arrivingat a target cooperatively Specifically the exact expression fortotal flight time can be obtained from the Lyapunov-basedguidance law with control parameter equal to one At eachtime step we assume the control parameters are initializedwith one and the total flight time for each vehicle can becalculated (en by exchanging the total flight time betweenthe vehicle and its neighbors under an undirected andconnected communication topology the control parameterwill be adjusted to reduce the disparities of the arrival timeAfter the consensus of the flight time the control parameterswill remain constant at one Furthermore the guidance lawis applied in a leader-follower case that one of the vehiclescannot receive information from the others and acts as theleader (e effectiveness of the proposed method is dem-onstrated with simulations (e main contributions of thispaper are stated as follows

(1) (e cooperative guidance law is distributed andrequires only neighboring information rather thanglobal information which reduces the communi-cation burden

(2) (e previous work needs the information of thetime-to-go estimates However the proposed guid-ance law deals with the consensus of the real flighttime directly rather than the estimation of the time-to-go (is improves the accuracy of the guidancelaw

(3) To ensure the scalability of this coordinationmethod the number of vehicles is not specific in themodeling and design process Furthermore thevalidity of the law is then examined in the single nodefailure case

(e paper is structured as follows Lyapunov-basedguidance law is introduced in Section 2 Coordination lawfor multiple air vehicles is offered in Section 3 (en theguidance law is extended to a communication failure case inSection 4 Simulations are carried out in Section 5 to showthe effectiveness of the proposed law Finally the conclusionof the work is proposed in Section 6

2 Lyapunov-Based Guidance Law Design

In this paper a scenario where multiple air vehicles arrive ata common target is considered in two-dimensional space(e planar multiple agents system is profiled in Figure 1 inwhich X minus O minus Y is an inertial reference frame denoting thevertical plane To ensure the scalability of this coordinationmethod the number of vehicles which can be hundreds ormore is not specific in the modeling and design process

Considering that the communication scenarios can bevery complex for a group of vehicles communication statebetween different vehicles is defined as a binary variable andcommunication topology between the air vehicles is denotedby G(E A) Before moving on it is necessary to present

2 Journal of Advanced Transportation

some basic fundamental facts E stands for the set of edgesedge (i j) means that ith vehicle and jth vehicle areneighbors and jth vehicle can receive information from ith

vehicle A graph is called undirected if for any (i j) isin E(j i) isin E An undirected graph is called connected if there isan undirected path between any two different vehicles A

[aij]NtimesN is the adjacency matrix aij 1 if the ith vehicle canget information from the jth vehicle and aij 0 if it cannotBesides the Laplacian matrix L [lij]NtimesN of G associatedwith adjacency matrix A is defined as lij minus aij ine j andlij 1113936

Nj1 aij i j

(e following assumptions are claimed before deriving thekinematic equations First the target is assumed to be sta-tionary Second the speed of each air vehicle remains constantduring the process but may not be the same as that of othervehicles (ird the communication topology G of the multi-agents system is assumed to be undirected and connected (eaforementioned assumptions can lead to the following lemmas

Lemma 1 (see [18]) One eigenvalue of L is zero with 1 beingthe right eigenvector It can be expressed mathematically asL1 01 where 1 denotes a column vector with all entriesequal to one Moreover all nonzero eigenvalues have positivereal parts

Lemma 2 (see [19]) xTLxge λxTx if x satisfies 1Tx 0where x refers to any x isin Rn and λ denotes the smallestnonzero value of the Laplacian matrix L

In Figure 1 a subscript i is added to demonstrate variablesassociated with the ith vehicle θ denotes the heading anglewhich is the angle between the velocity vector and the fixedreference axis q denotes the line of sight (LOS) angle (eheading error angle σ is the angle between the velocity vectorand LOS vector All the angles are measured counterclock-wise (e relationship between the aforementioned angles is

σi θi + minus qi( 1113857 θi minus qi i 1 n (1)

We can obtain the two-dimensional kinematic equationsfrom the engagement geometry as

_xi Vi cos θi i 1 n (2)

_yi Vi sin θi i 1 n (3)

where x and y denote the instantaneous positions of the airvehicle (e heading angle turning rate is connected with thelateral acceleration a by

_θi ai

Vi

i 1 n (4)

and R denotes the relative range between target and vehicle(e differential equations for the relative range and LOSangle are

_Ri minus Vi cos σi i 1 n (5)

_qi minus Vi sin σi

Ri

i 1 n (6)

In order to arrive at the target with zero miss distancethe velocity vector should aim directly at the target whichmeans the heading error angle should reach zero before or atthe instant of arrival Considering this the following Lya-punov candidate function is proposed

Wi 2sin2σi

2 i 1 n (7)

(e derivative of W with respect to time is_Wi sin σi middot _σi i 1 n (8)

To make each vehicle satisfy the Lyapunov asymptoticstability condition the heading error rate is proposed as

_σi minusciVi

Ri

sin σi ci gt 0 i 1 n (9)

where c is the control parameter for each vehicleSubstituting equation (9) into equation (8) we have

_Wi minusciVi

Ri

sin2σi i 1 n (10)

It is obvious that _W will be negative definite if cgt 0Besides equation (7) implies that W is positive definiteHence the Lyapunov asymptotic stability condition can bemet under the proposed law

Dividing equations (9) and (5) side by side yieldsdσi

tan σi

ci

Ri

dRi i 1 n (11)

Integrating both sides of equation (11) we have

sin σi Ri

R0i

1113888 1113889

ci

sin σ0i i 1 n (12)

Equation (12) illustrates that the heading error is con-nected with the relative range the value of which will declineto zero as engagement proceeds In the meantime equation(5) indicates that the relative range will decrease monoto-nously (is signifies that the value of the heading error willalso converge to zero at the end of the flight

X

Y

O

Ri

Vi

ai

Mi

qiθi

R1

V1

a1

q1

θ1

M1

Rn

Mn

Vn

anqn

θn

T

Figure 1 Engagement geometry

Journal of Advanced Transportation 3

Combining equations (12) and (9) yields

_σi minusciVi

Rci

0i

Ri( 1113857ci minus 1 sin σ0i i 1 n (13)

According to equation (13) control value clt 1 will in-evitably lead to an undesirable situation as the relative rangegoes to zero in the terminal guidance situation(is valuableinformation indicates that it is necessary to require cge 1 inthe terminal guidance situation

Since the proposed guidance law can also be used in amidcourse guidance situation the relative range of which willnot go near zero there is no need to worry about the un-desired situation caused by zero relative range(en the valueof the control parameter just needs tomeet the requirement ofthe Lyapunov asymptotic stability condition which is cgt 0 Sothe reasonable range for the control parameter is

ci gt 0midcourse guidance i 1 n

ci ge 1 terminal course guidance i 1 n1113896 (14)

Differentiating equation (1) with respect to time results in

_σi _θi minus _qi i 1 n (15)

Substituting equations (6) and (9) into equation (15) weget

_θi minus ci + 1( 1113857Vi

Ri

sin σi i 1 n (16)

(e following guidance command can be obtained fromequations (4) and (16)

ai minus ci + 1( 1113857V2

1Ri

sin σi i 1 n (17)

Substituting equation (12) into equation (17) yields

ai minus ci + 1( 1113857V2

i

Rci

0i

Rci minus 1 sin σ0i i 1 n (18)

Suppose that the control parameter c for the vehicles isfixed at one and we are going to see the flight time cal-culation under this specific circumstance

Substituting c 1 into equation (13) yields

_σi minusVi

R0i

sin σ0i i 1 n (19)

From equation (19) we know that the heading error rateremains negative meaning that heading error will decreasemonotonously From equation (12) we know that the headingerror will go to zero with relative range As a result the headingerror will decrease from the initial value all along to zero at theend of the flight Furthermore equation (19) also implies thatthe heading error rate is constant Hence dividing the totalvariation of the heading error by its change rate the analyticalform of the total flight time can be acquired as

ti 0 minus σ0i

_σi

σ0iR0i

Vi sin σ0i

i 1 n (20)

If every vehiclersquos total flight time calculated fromequation (20) is equal to the others simultaneous arrival can

be achieved(en the main objective of this paper is to find aguidance law to reduce the flight time disparities betweendifferent vehicles

Remark 1 (e proposed Lyapunov-based guidance law canachieve the basic objective of reducing the relative distanceto an acceptable order of magnitude Utilizing of the Lya-punov stability condition can make sure that the system isstable

Remark 2 Equation (20) gives the mathematical expressionof the flight time with the vehiclersquos initial condition (eexact mathematical expression of the vehiclesrsquo total flighttime can be derived if the control parameter equals one Noestimation or linearization is used in the process

3 Coordination Law for Multiple Air Vehicles

31 Design Strategy Enlightened by the mathematical ac-quisition of the total flight time in equation (20) a two-stepcontrol strategy is proposed here to achieve the cooperativeguidance law

First assume that all the vehicles are under the proposedLyapunov-based guidance law with control parameter equalto one such that equation (20) can be used to calculate thetotal flight time once the initial conditions are given (eneach time step is viewed as the initial time and the in-stantaneous states are treated as the initial states useequation (20) to recalculate the flight time (en equation(20) should be updated accordingly

tiprime

σiRi

Vi sin σi

i 1 n (21)

It is obvious that the flight time calculated from equation(21) can also be viewed as the real time-to-go(e total flighttime can be written as

ti t + tiprime i 1 n (22)

where t is the instantaneous flight time for the vehicles Wechoose the vehiclersquos total flight time as the consensus var-iable (e consensus error of the vehiclesrsquo total flight timeunder the undirected and connected communication to-pology is defined as

εi 1113944n

j1aij tj minus ti1113872 1113873 i 1 n (23)

Second adjust the control parameter to make the totalflight time reach an agreement In the previous discussionwe know that the consensus of the total flight time for thevehicles can lead to a simultaneous arrival Once the con-sensus error calculated from equation (23) is zero thecontrol parameter for all the vehicles will change to one andremain there

32 Coordination Law Substituting equation (22) intoequation (23) yields

4 Journal of Advanced Transportation

εi 1113944n

j1aij tjprime minus tiprime1113872 1113873 i 1 n (24)

(e analytical form of the flight time in equation (20) isderived from dividing the total variation of the heading errorby its changing rate In order to achieve the consensus of theflight time the vehicles with larger flight time should in-crease their heading error changing rate while the otherswith smaller flight time decrease their changing rate to delaythe flight time Based on the information exchange betweenvehicles via the communication network the heading errorin the first step is proposed as

_σi minus 1 minus ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 11138571113872 1113873

Vi sin σi

Ri

i 1 n (25)

where u is a constant that satisfies 0lt ult 1 It is obvious thatthe proposed heading error rate is under the Lyapunov-based guidance law structure where the control parameterfor each vehicle is ci 1 minus |εi|

usgn(εi) Once the total flighttime arrives at a consensus the control parameter for eachvehicle will be fixed at one

Before moving on two other lemmas are introduced inadvance

Lemma 3 (see [20]) For xi isin R i 1 n 0lt ale 1 then

1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868a ge 1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

a

(26)

Lemma 4 (see [21]) If there exists a Lyapunov function V(x)

such that

_V(x)le minus aVm

(x) (27)

where agt 0 and 0ltmlt 1 then V(x) will converge to zero ora small neighborhood of zero before the final time6e settlingtime T depending on initial condition state x0 is given by

TleV x0( 1113857

1minus m

a(1 minus λ) (28)

Theorem 1 6e proposed heading error rate in equation (25)can make εi i 1 n converge to zero in finite time andthe simultaneous arrival problem for the multivehicles systemin Section 2 can be solved

Proof Differentiating equation (21) with respect to timeyields

_tiprime

_σiRi

V sin σi

+σi

_Ri

Vi sin σi

minus_σiσiRi cos σi

Vi sin σi

i 1 n

(29)

Substituting equation (25) into (29) we have

_tiprime ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873 minus

σi cos σi

sin σi

minuski εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873σi cos σi

sin σi

i 1 n

(30)

where σi are usually small angles then sin σi asymp σi andcos σi asymp 1 minus σ2i 2 Hence equation (30) can be rewritten as

_tiprime 1 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 i 1 n (31)

(e following Lyapunov candidate function isconsidered

V1 12

1113944ijisin

aij tjprime minus tiprime1113872 1113873

212tTLt (32)

where t [t1 tn] (e derivative of V1 with respect totime is given by

_V1 _tTLt minus 1113944

n

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u (33)

Note that the last equality in equation (33) is obtained byusing the fact that L1 0 and ε minus Lt Define

k min 1 minus cos σi( 1113857ki i 1 n (34)

(en we have

_V1 le minus k 1113944

n

i1εi

11138681113868111386811138681113868111386811138681113868u+1 le minus k εTε1113872 1113873

u+12 (35)

As 1TL1 0 (L121)T(L121) we can get L121 0(en we have 1TL12t 0 According to Lemma 1 we can gettTLLtge λtTLt which can be written as εTεge 2λV On ac-count of these analyses the following equation can be drivenfrom equation (35)

_V1 le minus k(2λ)1+u2

V1+u21 (36)

According to finite-time convergence theory fromLemma 4 V1 will converge to zero or a small neighbor ofzero in finite time (e convergence of V1 also means thatthe consensus error εi will converge to zero Once theconsensus error reaches zero the simultaneous arrival canbe achieved In addition the consensus time is given by

Tle2V

(1minus u2)1

k(1 minus u)(2λ)1minus u2 (37)

which completes the proof of (eorem 1

Remark 3 Different from previous works [14ndash17] where theconsensus of the time-to-go estimations is considered thispaper deals with the consensus of the flight time directlyMoreover the assumption that ri gt 0 and σi ne 0 before theconsensus is not necessary (us the guidance law is moreoperationally effective Compared with [14 15] only theneighboring information is required rather than the globalinformation in this method Hence the guidance law isdistributed

Journal of Advanced Transportation 5

4 Extension to a Communication Failure Case

In this subsection the communication faults scenario thatone of the group vehicles cannot receive information fromother vehicles is considered As a result the flight time forthis fault vehicle cannot be adjusted (e only way to makecooperative arrival possible in this case is that all the othervehicles coordinate their flight time with the fault one whichwill be viewed as the leader

(e communication topology is viewed as a leader-follower graph Gprime with the fault vehicle as the root whichwill be denoted as the nth one In this case the controlparameter for the nth vehicle will remain constant at one

With the assumption in this section the Laplacianmatrix of Gprime can be denoted as

L L1 L2

01times(nminus 1) 0⎡⎣ ⎤⎦ (38)

where L1 isin R(nminus 1)times(nminus 1) is symmetric and L2 isin Rnminus 1 It isobvious that

L11 minus L2 (39)

Theorem 2 6e proposed heading error rate in equation (25)can solve the simultaneous arrival problem for the multi-vehicles system when the communication topology is Gprime

Proof Let 1113957t [t1 tnminus 1]T the Lyapunov candidate

function is proposed as

V2 12

1113957t minus 1113957tn1( 1113857TL1 1113957t minus 1113957tn1( 1113857 (40)

It can be concluded from Lemma 1 that L1 is positivedefinite Let 1113957ε [ε1 εnminus 1]

T we have

1113957ε minus L1 L21113858 1113859 1113957t tn1113858 1113859T (41)

Combining equations (39) and (41) yields1113957ε minus L1 1113957t minus tn1( 1113857 (42)

Differentiating equation (40) with respect to time wehave

_V2 1113957t minus 1113957tn1( 1113857TL1 _1113957t minus _1113957tn11113872 1113873

minus 1113944nminus 1

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u

(43)

Similar to the proof (eorem 1 the following equationcan be driven

_V2 le minus 1113957k 1113957εT1113957ε1113872 1113873

u+12 (44)

where1113957k min 1 minus cos σi( 1113857ki i 1 n minus 1 (45)

Note that

L121 1113957t minus 1113957tn1( 11138571113872 1113873TL1 L121 1113957t minus 1113957tn1( 11138571113872 1113873ge λ L121 1113957t minus 1113957tn1( 11138571113872 1113873

TL121 1113957t minus 1113957tn1( 11138571113872 1113873

(46)

which means that 1113957εT1113957εge 2λV2 On account of these analysesthe following equation can be driven from equation (35)

_V2 le minus 1113957k(2λ)1+u2

V1+u22 (47)

According to finite-time convergence theory fromLemma 4 V2 will converge to zero or a small neighbor ofzero in finite time (e convergence of V2 also means thatthe consensus error εi will converge to zero and the si-multaneous arrival can be achieved Hence (eorem 2 hasbeen proven

5 Simulations

In this section numerical simulations are carried out toshow the effectiveness of the proposed strategies (esimulation step is 001 s All the simulations are terminatedwhen the sign of the relative velocity becomes positive or therelative range is less than 001m We consider four vehiclesarriving at a common target from different directions andthe target is fixed at (8000 0)m Detailed simulation pa-rameters for the vehicles are tabulated in Table 1

51 Case 1 Undirected and Connected In this subsectionsimulations are carried out to show the effectiveness of theproposed law under undirected and connected communi-cation topology which is demonstrated in Figure 2 (edetailed simulation parameters are tabulated in Table 1

An undirected path exists between any two differentvehicles Hence all the vehicles can receive informationfrom their neighbors (e Laplacian matrix of the com-munication topology can be acquired as

L

3 minus 1 minus 1 minus 1

minus 1 2 minus 1 0

minus 1 minus 1 3 minus 1

minus 1 0 minus 1 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Simulation results are demonstrated in Figure 3 Solidline dashed line dash-dotted line and dotted line stand forthe results of vehicles 1 2 3 and 4 respectively Combiningthe vehicle trajectories in Figure 3(a) and range variation inFigure 3(d) we can see that simultaneous arrival can beachieved under the proposed guidance law (e variance ofthe heading error angles is in Figure 3(b) all of which declineto zero at the end of the engagement which verifies theanalysis in equation (12) (e consensus error of the flighttime is demonstrated in Figure 3(e) It is obvious that theflight time of each vehicle can reach an agreement in finitetime under the proposed law Once the consensus of flighttime is achieved the control parameter will remain constantat 1 We know that the acceleration will remain constant ifthe control parameter remains at 1 which is consistent withthe simulation results in Figure 3(c) (is simulation proves

6 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

some basic fundamental facts E stands for the set of edgesedge (i j) means that ith vehicle and jth vehicle areneighbors and jth vehicle can receive information from ith

vehicle A graph is called undirected if for any (i j) isin E(j i) isin E An undirected graph is called connected if there isan undirected path between any two different vehicles A

[aij]NtimesN is the adjacency matrix aij 1 if the ith vehicle canget information from the jth vehicle and aij 0 if it cannotBesides the Laplacian matrix L [lij]NtimesN of G associatedwith adjacency matrix A is defined as lij minus aij ine j andlij 1113936

Nj1 aij i j

(e following assumptions are claimed before deriving thekinematic equations First the target is assumed to be sta-tionary Second the speed of each air vehicle remains constantduring the process but may not be the same as that of othervehicles (ird the communication topology G of the multi-agents system is assumed to be undirected and connected (eaforementioned assumptions can lead to the following lemmas

Lemma 1 (see [18]) One eigenvalue of L is zero with 1 beingthe right eigenvector It can be expressed mathematically asL1 01 where 1 denotes a column vector with all entriesequal to one Moreover all nonzero eigenvalues have positivereal parts

Lemma 2 (see [19]) xTLxge λxTx if x satisfies 1Tx 0where x refers to any x isin Rn and λ denotes the smallestnonzero value of the Laplacian matrix L

In Figure 1 a subscript i is added to demonstrate variablesassociated with the ith vehicle θ denotes the heading anglewhich is the angle between the velocity vector and the fixedreference axis q denotes the line of sight (LOS) angle (eheading error angle σ is the angle between the velocity vectorand LOS vector All the angles are measured counterclock-wise (e relationship between the aforementioned angles is

σi θi + minus qi( 1113857 θi minus qi i 1 n (1)

We can obtain the two-dimensional kinematic equationsfrom the engagement geometry as

_xi Vi cos θi i 1 n (2)

_yi Vi sin θi i 1 n (3)

where x and y denote the instantaneous positions of the airvehicle (e heading angle turning rate is connected with thelateral acceleration a by

_θi ai

Vi

i 1 n (4)

and R denotes the relative range between target and vehicle(e differential equations for the relative range and LOSangle are

_Ri minus Vi cos σi i 1 n (5)

_qi minus Vi sin σi

Ri

i 1 n (6)

In order to arrive at the target with zero miss distancethe velocity vector should aim directly at the target whichmeans the heading error angle should reach zero before or atthe instant of arrival Considering this the following Lya-punov candidate function is proposed

Wi 2sin2σi

2 i 1 n (7)

(e derivative of W with respect to time is_Wi sin σi middot _σi i 1 n (8)

To make each vehicle satisfy the Lyapunov asymptoticstability condition the heading error rate is proposed as

_σi minusciVi

Ri

sin σi ci gt 0 i 1 n (9)

where c is the control parameter for each vehicleSubstituting equation (9) into equation (8) we have

_Wi minusciVi

Ri

sin2σi i 1 n (10)

It is obvious that _W will be negative definite if cgt 0Besides equation (7) implies that W is positive definiteHence the Lyapunov asymptotic stability condition can bemet under the proposed law

Dividing equations (9) and (5) side by side yieldsdσi

tan σi

ci

Ri

dRi i 1 n (11)

Integrating both sides of equation (11) we have

sin σi Ri

R0i

1113888 1113889

ci

sin σ0i i 1 n (12)

Equation (12) illustrates that the heading error is con-nected with the relative range the value of which will declineto zero as engagement proceeds In the meantime equation(5) indicates that the relative range will decrease monoto-nously (is signifies that the value of the heading error willalso converge to zero at the end of the flight

X

Y

O

Ri

Vi

ai

Mi

qiθi

R1

V1

a1

q1

θ1

M1

Rn

Mn

Vn

anqn

θn

T

Figure 1 Engagement geometry

Journal of Advanced Transportation 3

Combining equations (12) and (9) yields

_σi minusciVi

Rci

0i

Ri( 1113857ci minus 1 sin σ0i i 1 n (13)

According to equation (13) control value clt 1 will in-evitably lead to an undesirable situation as the relative rangegoes to zero in the terminal guidance situation(is valuableinformation indicates that it is necessary to require cge 1 inthe terminal guidance situation

Since the proposed guidance law can also be used in amidcourse guidance situation the relative range of which willnot go near zero there is no need to worry about the un-desired situation caused by zero relative range(en the valueof the control parameter just needs tomeet the requirement ofthe Lyapunov asymptotic stability condition which is cgt 0 Sothe reasonable range for the control parameter is

ci gt 0midcourse guidance i 1 n

ci ge 1 terminal course guidance i 1 n1113896 (14)

Differentiating equation (1) with respect to time results in

_σi _θi minus _qi i 1 n (15)

Substituting equations (6) and (9) into equation (15) weget

_θi minus ci + 1( 1113857Vi

Ri

sin σi i 1 n (16)

(e following guidance command can be obtained fromequations (4) and (16)

ai minus ci + 1( 1113857V2

1Ri

sin σi i 1 n (17)

Substituting equation (12) into equation (17) yields

ai minus ci + 1( 1113857V2

i

Rci

0i

Rci minus 1 sin σ0i i 1 n (18)

Suppose that the control parameter c for the vehicles isfixed at one and we are going to see the flight time cal-culation under this specific circumstance

Substituting c 1 into equation (13) yields

_σi minusVi

R0i

sin σ0i i 1 n (19)

From equation (19) we know that the heading error rateremains negative meaning that heading error will decreasemonotonously From equation (12) we know that the headingerror will go to zero with relative range As a result the headingerror will decrease from the initial value all along to zero at theend of the flight Furthermore equation (19) also implies thatthe heading error rate is constant Hence dividing the totalvariation of the heading error by its change rate the analyticalform of the total flight time can be acquired as

ti 0 minus σ0i

_σi

σ0iR0i

Vi sin σ0i

i 1 n (20)

If every vehiclersquos total flight time calculated fromequation (20) is equal to the others simultaneous arrival can

be achieved(en the main objective of this paper is to find aguidance law to reduce the flight time disparities betweendifferent vehicles

Remark 1 (e proposed Lyapunov-based guidance law canachieve the basic objective of reducing the relative distanceto an acceptable order of magnitude Utilizing of the Lya-punov stability condition can make sure that the system isstable

Remark 2 Equation (20) gives the mathematical expressionof the flight time with the vehiclersquos initial condition (eexact mathematical expression of the vehiclesrsquo total flighttime can be derived if the control parameter equals one Noestimation or linearization is used in the process

3 Coordination Law for Multiple Air Vehicles

31 Design Strategy Enlightened by the mathematical ac-quisition of the total flight time in equation (20) a two-stepcontrol strategy is proposed here to achieve the cooperativeguidance law

First assume that all the vehicles are under the proposedLyapunov-based guidance law with control parameter equalto one such that equation (20) can be used to calculate thetotal flight time once the initial conditions are given (eneach time step is viewed as the initial time and the in-stantaneous states are treated as the initial states useequation (20) to recalculate the flight time (en equation(20) should be updated accordingly

tiprime

σiRi

Vi sin σi

i 1 n (21)

It is obvious that the flight time calculated from equation(21) can also be viewed as the real time-to-go(e total flighttime can be written as

ti t + tiprime i 1 n (22)

where t is the instantaneous flight time for the vehicles Wechoose the vehiclersquos total flight time as the consensus var-iable (e consensus error of the vehiclesrsquo total flight timeunder the undirected and connected communication to-pology is defined as

εi 1113944n

j1aij tj minus ti1113872 1113873 i 1 n (23)

Second adjust the control parameter to make the totalflight time reach an agreement In the previous discussionwe know that the consensus of the total flight time for thevehicles can lead to a simultaneous arrival Once the con-sensus error calculated from equation (23) is zero thecontrol parameter for all the vehicles will change to one andremain there

32 Coordination Law Substituting equation (22) intoequation (23) yields

4 Journal of Advanced Transportation

εi 1113944n

j1aij tjprime minus tiprime1113872 1113873 i 1 n (24)

(e analytical form of the flight time in equation (20) isderived from dividing the total variation of the heading errorby its changing rate In order to achieve the consensus of theflight time the vehicles with larger flight time should in-crease their heading error changing rate while the otherswith smaller flight time decrease their changing rate to delaythe flight time Based on the information exchange betweenvehicles via the communication network the heading errorin the first step is proposed as

_σi minus 1 minus ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 11138571113872 1113873

Vi sin σi

Ri

i 1 n (25)

where u is a constant that satisfies 0lt ult 1 It is obvious thatthe proposed heading error rate is under the Lyapunov-based guidance law structure where the control parameterfor each vehicle is ci 1 minus |εi|

usgn(εi) Once the total flighttime arrives at a consensus the control parameter for eachvehicle will be fixed at one

Before moving on two other lemmas are introduced inadvance

Lemma 3 (see [20]) For xi isin R i 1 n 0lt ale 1 then

1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868a ge 1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

a

(26)

Lemma 4 (see [21]) If there exists a Lyapunov function V(x)

such that

_V(x)le minus aVm

(x) (27)

where agt 0 and 0ltmlt 1 then V(x) will converge to zero ora small neighborhood of zero before the final time6e settlingtime T depending on initial condition state x0 is given by

TleV x0( 1113857

1minus m

a(1 minus λ) (28)

Theorem 1 6e proposed heading error rate in equation (25)can make εi i 1 n converge to zero in finite time andthe simultaneous arrival problem for the multivehicles systemin Section 2 can be solved

Proof Differentiating equation (21) with respect to timeyields

_tiprime

_σiRi

V sin σi

+σi

_Ri

Vi sin σi

minus_σiσiRi cos σi

Vi sin σi

i 1 n

(29)

Substituting equation (25) into (29) we have

_tiprime ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873 minus

σi cos σi

sin σi

minuski εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873σi cos σi

sin σi

i 1 n

(30)

where σi are usually small angles then sin σi asymp σi andcos σi asymp 1 minus σ2i 2 Hence equation (30) can be rewritten as

_tiprime 1 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 i 1 n (31)

(e following Lyapunov candidate function isconsidered

V1 12

1113944ijisin

aij tjprime minus tiprime1113872 1113873

212tTLt (32)

where t [t1 tn] (e derivative of V1 with respect totime is given by

_V1 _tTLt minus 1113944

n

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u (33)

Note that the last equality in equation (33) is obtained byusing the fact that L1 0 and ε minus Lt Define

k min 1 minus cos σi( 1113857ki i 1 n (34)

(en we have

_V1 le minus k 1113944

n

i1εi

11138681113868111386811138681113868111386811138681113868u+1 le minus k εTε1113872 1113873

u+12 (35)

As 1TL1 0 (L121)T(L121) we can get L121 0(en we have 1TL12t 0 According to Lemma 1 we can gettTLLtge λtTLt which can be written as εTεge 2λV On ac-count of these analyses the following equation can be drivenfrom equation (35)

_V1 le minus k(2λ)1+u2

V1+u21 (36)

According to finite-time convergence theory fromLemma 4 V1 will converge to zero or a small neighbor ofzero in finite time (e convergence of V1 also means thatthe consensus error εi will converge to zero Once theconsensus error reaches zero the simultaneous arrival canbe achieved In addition the consensus time is given by

Tle2V

(1minus u2)1

k(1 minus u)(2λ)1minus u2 (37)

which completes the proof of (eorem 1

Remark 3 Different from previous works [14ndash17] where theconsensus of the time-to-go estimations is considered thispaper deals with the consensus of the flight time directlyMoreover the assumption that ri gt 0 and σi ne 0 before theconsensus is not necessary (us the guidance law is moreoperationally effective Compared with [14 15] only theneighboring information is required rather than the globalinformation in this method Hence the guidance law isdistributed

Journal of Advanced Transportation 5

4 Extension to a Communication Failure Case

In this subsection the communication faults scenario thatone of the group vehicles cannot receive information fromother vehicles is considered As a result the flight time forthis fault vehicle cannot be adjusted (e only way to makecooperative arrival possible in this case is that all the othervehicles coordinate their flight time with the fault one whichwill be viewed as the leader

(e communication topology is viewed as a leader-follower graph Gprime with the fault vehicle as the root whichwill be denoted as the nth one In this case the controlparameter for the nth vehicle will remain constant at one

With the assumption in this section the Laplacianmatrix of Gprime can be denoted as

L L1 L2

01times(nminus 1) 0⎡⎣ ⎤⎦ (38)

where L1 isin R(nminus 1)times(nminus 1) is symmetric and L2 isin Rnminus 1 It isobvious that

L11 minus L2 (39)

Theorem 2 6e proposed heading error rate in equation (25)can solve the simultaneous arrival problem for the multi-vehicles system when the communication topology is Gprime

Proof Let 1113957t [t1 tnminus 1]T the Lyapunov candidate

function is proposed as

V2 12

1113957t minus 1113957tn1( 1113857TL1 1113957t minus 1113957tn1( 1113857 (40)

It can be concluded from Lemma 1 that L1 is positivedefinite Let 1113957ε [ε1 εnminus 1]

T we have

1113957ε minus L1 L21113858 1113859 1113957t tn1113858 1113859T (41)

Combining equations (39) and (41) yields1113957ε minus L1 1113957t minus tn1( 1113857 (42)

Differentiating equation (40) with respect to time wehave

_V2 1113957t minus 1113957tn1( 1113857TL1 _1113957t minus _1113957tn11113872 1113873

minus 1113944nminus 1

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u

(43)

Similar to the proof (eorem 1 the following equationcan be driven

_V2 le minus 1113957k 1113957εT1113957ε1113872 1113873

u+12 (44)

where1113957k min 1 minus cos σi( 1113857ki i 1 n minus 1 (45)

Note that

L121 1113957t minus 1113957tn1( 11138571113872 1113873TL1 L121 1113957t minus 1113957tn1( 11138571113872 1113873ge λ L121 1113957t minus 1113957tn1( 11138571113872 1113873

TL121 1113957t minus 1113957tn1( 11138571113872 1113873

(46)

which means that 1113957εT1113957εge 2λV2 On account of these analysesthe following equation can be driven from equation (35)

_V2 le minus 1113957k(2λ)1+u2

V1+u22 (47)

According to finite-time convergence theory fromLemma 4 V2 will converge to zero or a small neighbor ofzero in finite time (e convergence of V2 also means thatthe consensus error εi will converge to zero and the si-multaneous arrival can be achieved Hence (eorem 2 hasbeen proven

5 Simulations

In this section numerical simulations are carried out toshow the effectiveness of the proposed strategies (esimulation step is 001 s All the simulations are terminatedwhen the sign of the relative velocity becomes positive or therelative range is less than 001m We consider four vehiclesarriving at a common target from different directions andthe target is fixed at (8000 0)m Detailed simulation pa-rameters for the vehicles are tabulated in Table 1

51 Case 1 Undirected and Connected In this subsectionsimulations are carried out to show the effectiveness of theproposed law under undirected and connected communi-cation topology which is demonstrated in Figure 2 (edetailed simulation parameters are tabulated in Table 1

An undirected path exists between any two differentvehicles Hence all the vehicles can receive informationfrom their neighbors (e Laplacian matrix of the com-munication topology can be acquired as

L

3 minus 1 minus 1 minus 1

minus 1 2 minus 1 0

minus 1 minus 1 3 minus 1

minus 1 0 minus 1 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Simulation results are demonstrated in Figure 3 Solidline dashed line dash-dotted line and dotted line stand forthe results of vehicles 1 2 3 and 4 respectively Combiningthe vehicle trajectories in Figure 3(a) and range variation inFigure 3(d) we can see that simultaneous arrival can beachieved under the proposed guidance law (e variance ofthe heading error angles is in Figure 3(b) all of which declineto zero at the end of the engagement which verifies theanalysis in equation (12) (e consensus error of the flighttime is demonstrated in Figure 3(e) It is obvious that theflight time of each vehicle can reach an agreement in finitetime under the proposed law Once the consensus of flighttime is achieved the control parameter will remain constantat 1 We know that the acceleration will remain constant ifthe control parameter remains at 1 which is consistent withthe simulation results in Figure 3(c) (is simulation proves

6 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

Combining equations (12) and (9) yields

_σi minusciVi

Rci

0i

Ri( 1113857ci minus 1 sin σ0i i 1 n (13)

According to equation (13) control value clt 1 will in-evitably lead to an undesirable situation as the relative rangegoes to zero in the terminal guidance situation(is valuableinformation indicates that it is necessary to require cge 1 inthe terminal guidance situation

Since the proposed guidance law can also be used in amidcourse guidance situation the relative range of which willnot go near zero there is no need to worry about the un-desired situation caused by zero relative range(en the valueof the control parameter just needs tomeet the requirement ofthe Lyapunov asymptotic stability condition which is cgt 0 Sothe reasonable range for the control parameter is

ci gt 0midcourse guidance i 1 n

ci ge 1 terminal course guidance i 1 n1113896 (14)

Differentiating equation (1) with respect to time results in

_σi _θi minus _qi i 1 n (15)

Substituting equations (6) and (9) into equation (15) weget

_θi minus ci + 1( 1113857Vi

Ri

sin σi i 1 n (16)

(e following guidance command can be obtained fromequations (4) and (16)

ai minus ci + 1( 1113857V2

1Ri

sin σi i 1 n (17)

Substituting equation (12) into equation (17) yields

ai minus ci + 1( 1113857V2

i

Rci

0i

Rci minus 1 sin σ0i i 1 n (18)

Suppose that the control parameter c for the vehicles isfixed at one and we are going to see the flight time cal-culation under this specific circumstance

Substituting c 1 into equation (13) yields

_σi minusVi

R0i

sin σ0i i 1 n (19)

From equation (19) we know that the heading error rateremains negative meaning that heading error will decreasemonotonously From equation (12) we know that the headingerror will go to zero with relative range As a result the headingerror will decrease from the initial value all along to zero at theend of the flight Furthermore equation (19) also implies thatthe heading error rate is constant Hence dividing the totalvariation of the heading error by its change rate the analyticalform of the total flight time can be acquired as

ti 0 minus σ0i

_σi

σ0iR0i

Vi sin σ0i

i 1 n (20)

If every vehiclersquos total flight time calculated fromequation (20) is equal to the others simultaneous arrival can

be achieved(en the main objective of this paper is to find aguidance law to reduce the flight time disparities betweendifferent vehicles

Remark 1 (e proposed Lyapunov-based guidance law canachieve the basic objective of reducing the relative distanceto an acceptable order of magnitude Utilizing of the Lya-punov stability condition can make sure that the system isstable

Remark 2 Equation (20) gives the mathematical expressionof the flight time with the vehiclersquos initial condition (eexact mathematical expression of the vehiclesrsquo total flighttime can be derived if the control parameter equals one Noestimation or linearization is used in the process

3 Coordination Law for Multiple Air Vehicles

31 Design Strategy Enlightened by the mathematical ac-quisition of the total flight time in equation (20) a two-stepcontrol strategy is proposed here to achieve the cooperativeguidance law

First assume that all the vehicles are under the proposedLyapunov-based guidance law with control parameter equalto one such that equation (20) can be used to calculate thetotal flight time once the initial conditions are given (eneach time step is viewed as the initial time and the in-stantaneous states are treated as the initial states useequation (20) to recalculate the flight time (en equation(20) should be updated accordingly

tiprime

σiRi

Vi sin σi

i 1 n (21)

It is obvious that the flight time calculated from equation(21) can also be viewed as the real time-to-go(e total flighttime can be written as

ti t + tiprime i 1 n (22)

where t is the instantaneous flight time for the vehicles Wechoose the vehiclersquos total flight time as the consensus var-iable (e consensus error of the vehiclesrsquo total flight timeunder the undirected and connected communication to-pology is defined as

εi 1113944n

j1aij tj minus ti1113872 1113873 i 1 n (23)

Second adjust the control parameter to make the totalflight time reach an agreement In the previous discussionwe know that the consensus of the total flight time for thevehicles can lead to a simultaneous arrival Once the con-sensus error calculated from equation (23) is zero thecontrol parameter for all the vehicles will change to one andremain there

32 Coordination Law Substituting equation (22) intoequation (23) yields

4 Journal of Advanced Transportation

εi 1113944n

j1aij tjprime minus tiprime1113872 1113873 i 1 n (24)

(e analytical form of the flight time in equation (20) isderived from dividing the total variation of the heading errorby its changing rate In order to achieve the consensus of theflight time the vehicles with larger flight time should in-crease their heading error changing rate while the otherswith smaller flight time decrease their changing rate to delaythe flight time Based on the information exchange betweenvehicles via the communication network the heading errorin the first step is proposed as

_σi minus 1 minus ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 11138571113872 1113873

Vi sin σi

Ri

i 1 n (25)

where u is a constant that satisfies 0lt ult 1 It is obvious thatthe proposed heading error rate is under the Lyapunov-based guidance law structure where the control parameterfor each vehicle is ci 1 minus |εi|

usgn(εi) Once the total flighttime arrives at a consensus the control parameter for eachvehicle will be fixed at one

Before moving on two other lemmas are introduced inadvance

Lemma 3 (see [20]) For xi isin R i 1 n 0lt ale 1 then

1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868a ge 1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

a

(26)

Lemma 4 (see [21]) If there exists a Lyapunov function V(x)

such that

_V(x)le minus aVm

(x) (27)

where agt 0 and 0ltmlt 1 then V(x) will converge to zero ora small neighborhood of zero before the final time6e settlingtime T depending on initial condition state x0 is given by

TleV x0( 1113857

1minus m

a(1 minus λ) (28)

Theorem 1 6e proposed heading error rate in equation (25)can make εi i 1 n converge to zero in finite time andthe simultaneous arrival problem for the multivehicles systemin Section 2 can be solved

Proof Differentiating equation (21) with respect to timeyields

_tiprime

_σiRi

V sin σi

+σi

_Ri

Vi sin σi

minus_σiσiRi cos σi

Vi sin σi

i 1 n

(29)

Substituting equation (25) into (29) we have

_tiprime ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873 minus

σi cos σi

sin σi

minuski εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873σi cos σi

sin σi

i 1 n

(30)

where σi are usually small angles then sin σi asymp σi andcos σi asymp 1 minus σ2i 2 Hence equation (30) can be rewritten as

_tiprime 1 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 i 1 n (31)

(e following Lyapunov candidate function isconsidered

V1 12

1113944ijisin

aij tjprime minus tiprime1113872 1113873

212tTLt (32)

where t [t1 tn] (e derivative of V1 with respect totime is given by

_V1 _tTLt minus 1113944

n

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u (33)

Note that the last equality in equation (33) is obtained byusing the fact that L1 0 and ε minus Lt Define

k min 1 minus cos σi( 1113857ki i 1 n (34)

(en we have

_V1 le minus k 1113944

n

i1εi

11138681113868111386811138681113868111386811138681113868u+1 le minus k εTε1113872 1113873

u+12 (35)

As 1TL1 0 (L121)T(L121) we can get L121 0(en we have 1TL12t 0 According to Lemma 1 we can gettTLLtge λtTLt which can be written as εTεge 2λV On ac-count of these analyses the following equation can be drivenfrom equation (35)

_V1 le minus k(2λ)1+u2

V1+u21 (36)

According to finite-time convergence theory fromLemma 4 V1 will converge to zero or a small neighbor ofzero in finite time (e convergence of V1 also means thatthe consensus error εi will converge to zero Once theconsensus error reaches zero the simultaneous arrival canbe achieved In addition the consensus time is given by

Tle2V

(1minus u2)1

k(1 minus u)(2λ)1minus u2 (37)

which completes the proof of (eorem 1

Remark 3 Different from previous works [14ndash17] where theconsensus of the time-to-go estimations is considered thispaper deals with the consensus of the flight time directlyMoreover the assumption that ri gt 0 and σi ne 0 before theconsensus is not necessary (us the guidance law is moreoperationally effective Compared with [14 15] only theneighboring information is required rather than the globalinformation in this method Hence the guidance law isdistributed

Journal of Advanced Transportation 5

4 Extension to a Communication Failure Case

In this subsection the communication faults scenario thatone of the group vehicles cannot receive information fromother vehicles is considered As a result the flight time forthis fault vehicle cannot be adjusted (e only way to makecooperative arrival possible in this case is that all the othervehicles coordinate their flight time with the fault one whichwill be viewed as the leader

(e communication topology is viewed as a leader-follower graph Gprime with the fault vehicle as the root whichwill be denoted as the nth one In this case the controlparameter for the nth vehicle will remain constant at one

With the assumption in this section the Laplacianmatrix of Gprime can be denoted as

L L1 L2

01times(nminus 1) 0⎡⎣ ⎤⎦ (38)

where L1 isin R(nminus 1)times(nminus 1) is symmetric and L2 isin Rnminus 1 It isobvious that

L11 minus L2 (39)

Theorem 2 6e proposed heading error rate in equation (25)can solve the simultaneous arrival problem for the multi-vehicles system when the communication topology is Gprime

Proof Let 1113957t [t1 tnminus 1]T the Lyapunov candidate

function is proposed as

V2 12

1113957t minus 1113957tn1( 1113857TL1 1113957t minus 1113957tn1( 1113857 (40)

It can be concluded from Lemma 1 that L1 is positivedefinite Let 1113957ε [ε1 εnminus 1]

T we have

1113957ε minus L1 L21113858 1113859 1113957t tn1113858 1113859T (41)

Combining equations (39) and (41) yields1113957ε minus L1 1113957t minus tn1( 1113857 (42)

Differentiating equation (40) with respect to time wehave

_V2 1113957t minus 1113957tn1( 1113857TL1 _1113957t minus _1113957tn11113872 1113873

minus 1113944nminus 1

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u

(43)

Similar to the proof (eorem 1 the following equationcan be driven

_V2 le minus 1113957k 1113957εT1113957ε1113872 1113873

u+12 (44)

where1113957k min 1 minus cos σi( 1113857ki i 1 n minus 1 (45)

Note that

L121 1113957t minus 1113957tn1( 11138571113872 1113873TL1 L121 1113957t minus 1113957tn1( 11138571113872 1113873ge λ L121 1113957t minus 1113957tn1( 11138571113872 1113873

TL121 1113957t minus 1113957tn1( 11138571113872 1113873

(46)

which means that 1113957εT1113957εge 2λV2 On account of these analysesthe following equation can be driven from equation (35)

_V2 le minus 1113957k(2λ)1+u2

V1+u22 (47)

According to finite-time convergence theory fromLemma 4 V2 will converge to zero or a small neighbor ofzero in finite time (e convergence of V2 also means thatthe consensus error εi will converge to zero and the si-multaneous arrival can be achieved Hence (eorem 2 hasbeen proven

5 Simulations

In this section numerical simulations are carried out toshow the effectiveness of the proposed strategies (esimulation step is 001 s All the simulations are terminatedwhen the sign of the relative velocity becomes positive or therelative range is less than 001m We consider four vehiclesarriving at a common target from different directions andthe target is fixed at (8000 0)m Detailed simulation pa-rameters for the vehicles are tabulated in Table 1

51 Case 1 Undirected and Connected In this subsectionsimulations are carried out to show the effectiveness of theproposed law under undirected and connected communi-cation topology which is demonstrated in Figure 2 (edetailed simulation parameters are tabulated in Table 1

An undirected path exists between any two differentvehicles Hence all the vehicles can receive informationfrom their neighbors (e Laplacian matrix of the com-munication topology can be acquired as

L

3 minus 1 minus 1 minus 1

minus 1 2 minus 1 0

minus 1 minus 1 3 minus 1

minus 1 0 minus 1 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Simulation results are demonstrated in Figure 3 Solidline dashed line dash-dotted line and dotted line stand forthe results of vehicles 1 2 3 and 4 respectively Combiningthe vehicle trajectories in Figure 3(a) and range variation inFigure 3(d) we can see that simultaneous arrival can beachieved under the proposed guidance law (e variance ofthe heading error angles is in Figure 3(b) all of which declineto zero at the end of the engagement which verifies theanalysis in equation (12) (e consensus error of the flighttime is demonstrated in Figure 3(e) It is obvious that theflight time of each vehicle can reach an agreement in finitetime under the proposed law Once the consensus of flighttime is achieved the control parameter will remain constantat 1 We know that the acceleration will remain constant ifthe control parameter remains at 1 which is consistent withthe simulation results in Figure 3(c) (is simulation proves

6 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

εi 1113944n

j1aij tjprime minus tiprime1113872 1113873 i 1 n (24)

(e analytical form of the flight time in equation (20) isderived from dividing the total variation of the heading errorby its changing rate In order to achieve the consensus of theflight time the vehicles with larger flight time should in-crease their heading error changing rate while the otherswith smaller flight time decrease their changing rate to delaythe flight time Based on the information exchange betweenvehicles via the communication network the heading errorin the first step is proposed as

_σi minus 1 minus ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 11138571113872 1113873

Vi sin σi

Ri

i 1 n (25)

where u is a constant that satisfies 0lt ult 1 It is obvious thatthe proposed heading error rate is under the Lyapunov-based guidance law structure where the control parameterfor each vehicle is ci 1 minus |εi|

usgn(εi) Once the total flighttime arrives at a consensus the control parameter for eachvehicle will be fixed at one

Before moving on two other lemmas are introduced inadvance

Lemma 3 (see [20]) For xi isin R i 1 n 0lt ale 1 then

1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868a ge 1113944

n

i1xi

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

a

(26)

Lemma 4 (see [21]) If there exists a Lyapunov function V(x)

such that

_V(x)le minus aVm

(x) (27)

where agt 0 and 0ltmlt 1 then V(x) will converge to zero ora small neighborhood of zero before the final time6e settlingtime T depending on initial condition state x0 is given by

TleV x0( 1113857

1minus m

a(1 minus λ) (28)

Theorem 1 6e proposed heading error rate in equation (25)can make εi i 1 n converge to zero in finite time andthe simultaneous arrival problem for the multivehicles systemin Section 2 can be solved

Proof Differentiating equation (21) with respect to timeyields

_tiprime

_σiRi

V sin σi

+σi

_Ri

Vi sin σi

minus_σiσiRi cos σi

Vi sin σi

i 1 n

(29)

Substituting equation (25) into (29) we have

_tiprime ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873 minus

σi cos σi

sin σi

minuski εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 minus 11113872 1113873σi cos σi

sin σi

i 1 n

(30)

where σi are usually small angles then sin σi asymp σi andcos σi asymp 1 minus σ2i 2 Hence equation (30) can be rewritten as

_tiprime 1 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868usgn εi( 1113857 i 1 n (31)

(e following Lyapunov candidate function isconsidered

V1 12

1113944ijisin

aij tjprime minus tiprime1113872 1113873

212tTLt (32)

where t [t1 tn] (e derivative of V1 with respect totime is given by

_V1 _tTLt minus 1113944

n

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u (33)

Note that the last equality in equation (33) is obtained byusing the fact that L1 0 and ε minus Lt Define

k min 1 minus cos σi( 1113857ki i 1 n (34)

(en we have

_V1 le minus k 1113944

n

i1εi

11138681113868111386811138681113868111386811138681113868u+1 le minus k εTε1113872 1113873

u+12 (35)

As 1TL1 0 (L121)T(L121) we can get L121 0(en we have 1TL12t 0 According to Lemma 1 we can gettTLLtge λtTLt which can be written as εTεge 2λV On ac-count of these analyses the following equation can be drivenfrom equation (35)

_V1 le minus k(2λ)1+u2

V1+u21 (36)

According to finite-time convergence theory fromLemma 4 V1 will converge to zero or a small neighbor ofzero in finite time (e convergence of V1 also means thatthe consensus error εi will converge to zero Once theconsensus error reaches zero the simultaneous arrival canbe achieved In addition the consensus time is given by

Tle2V

(1minus u2)1

k(1 minus u)(2λ)1minus u2 (37)

which completes the proof of (eorem 1

Remark 3 Different from previous works [14ndash17] where theconsensus of the time-to-go estimations is considered thispaper deals with the consensus of the flight time directlyMoreover the assumption that ri gt 0 and σi ne 0 before theconsensus is not necessary (us the guidance law is moreoperationally effective Compared with [14 15] only theneighboring information is required rather than the globalinformation in this method Hence the guidance law isdistributed

Journal of Advanced Transportation 5

4 Extension to a Communication Failure Case

In this subsection the communication faults scenario thatone of the group vehicles cannot receive information fromother vehicles is considered As a result the flight time forthis fault vehicle cannot be adjusted (e only way to makecooperative arrival possible in this case is that all the othervehicles coordinate their flight time with the fault one whichwill be viewed as the leader

(e communication topology is viewed as a leader-follower graph Gprime with the fault vehicle as the root whichwill be denoted as the nth one In this case the controlparameter for the nth vehicle will remain constant at one

With the assumption in this section the Laplacianmatrix of Gprime can be denoted as

L L1 L2

01times(nminus 1) 0⎡⎣ ⎤⎦ (38)

where L1 isin R(nminus 1)times(nminus 1) is symmetric and L2 isin Rnminus 1 It isobvious that

L11 minus L2 (39)

Theorem 2 6e proposed heading error rate in equation (25)can solve the simultaneous arrival problem for the multi-vehicles system when the communication topology is Gprime

Proof Let 1113957t [t1 tnminus 1]T the Lyapunov candidate

function is proposed as

V2 12

1113957t minus 1113957tn1( 1113857TL1 1113957t minus 1113957tn1( 1113857 (40)

It can be concluded from Lemma 1 that L1 is positivedefinite Let 1113957ε [ε1 εnminus 1]

T we have

1113957ε minus L1 L21113858 1113859 1113957t tn1113858 1113859T (41)

Combining equations (39) and (41) yields1113957ε minus L1 1113957t minus tn1( 1113857 (42)

Differentiating equation (40) with respect to time wehave

_V2 1113957t minus 1113957tn1( 1113857TL1 _1113957t minus _1113957tn11113872 1113873

minus 1113944nminus 1

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u

(43)

Similar to the proof (eorem 1 the following equationcan be driven

_V2 le minus 1113957k 1113957εT1113957ε1113872 1113873

u+12 (44)

where1113957k min 1 minus cos σi( 1113857ki i 1 n minus 1 (45)

Note that

L121 1113957t minus 1113957tn1( 11138571113872 1113873TL1 L121 1113957t minus 1113957tn1( 11138571113872 1113873ge λ L121 1113957t minus 1113957tn1( 11138571113872 1113873

TL121 1113957t minus 1113957tn1( 11138571113872 1113873

(46)

which means that 1113957εT1113957εge 2λV2 On account of these analysesthe following equation can be driven from equation (35)

_V2 le minus 1113957k(2λ)1+u2

V1+u22 (47)

According to finite-time convergence theory fromLemma 4 V2 will converge to zero or a small neighbor ofzero in finite time (e convergence of V2 also means thatthe consensus error εi will converge to zero and the si-multaneous arrival can be achieved Hence (eorem 2 hasbeen proven

5 Simulations

In this section numerical simulations are carried out toshow the effectiveness of the proposed strategies (esimulation step is 001 s All the simulations are terminatedwhen the sign of the relative velocity becomes positive or therelative range is less than 001m We consider four vehiclesarriving at a common target from different directions andthe target is fixed at (8000 0)m Detailed simulation pa-rameters for the vehicles are tabulated in Table 1

51 Case 1 Undirected and Connected In this subsectionsimulations are carried out to show the effectiveness of theproposed law under undirected and connected communi-cation topology which is demonstrated in Figure 2 (edetailed simulation parameters are tabulated in Table 1

An undirected path exists between any two differentvehicles Hence all the vehicles can receive informationfrom their neighbors (e Laplacian matrix of the com-munication topology can be acquired as

L

3 minus 1 minus 1 minus 1

minus 1 2 minus 1 0

minus 1 minus 1 3 minus 1

minus 1 0 minus 1 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Simulation results are demonstrated in Figure 3 Solidline dashed line dash-dotted line and dotted line stand forthe results of vehicles 1 2 3 and 4 respectively Combiningthe vehicle trajectories in Figure 3(a) and range variation inFigure 3(d) we can see that simultaneous arrival can beachieved under the proposed guidance law (e variance ofthe heading error angles is in Figure 3(b) all of which declineto zero at the end of the engagement which verifies theanalysis in equation (12) (e consensus error of the flighttime is demonstrated in Figure 3(e) It is obvious that theflight time of each vehicle can reach an agreement in finitetime under the proposed law Once the consensus of flighttime is achieved the control parameter will remain constantat 1 We know that the acceleration will remain constant ifthe control parameter remains at 1 which is consistent withthe simulation results in Figure 3(c) (is simulation proves

6 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

4 Extension to a Communication Failure Case

In this subsection the communication faults scenario thatone of the group vehicles cannot receive information fromother vehicles is considered As a result the flight time forthis fault vehicle cannot be adjusted (e only way to makecooperative arrival possible in this case is that all the othervehicles coordinate their flight time with the fault one whichwill be viewed as the leader

(e communication topology is viewed as a leader-follower graph Gprime with the fault vehicle as the root whichwill be denoted as the nth one In this case the controlparameter for the nth vehicle will remain constant at one

With the assumption in this section the Laplacianmatrix of Gprime can be denoted as

L L1 L2

01times(nminus 1) 0⎡⎣ ⎤⎦ (38)

where L1 isin R(nminus 1)times(nminus 1) is symmetric and L2 isin Rnminus 1 It isobvious that

L11 minus L2 (39)

Theorem 2 6e proposed heading error rate in equation (25)can solve the simultaneous arrival problem for the multi-vehicles system when the communication topology is Gprime

Proof Let 1113957t [t1 tnminus 1]T the Lyapunov candidate

function is proposed as

V2 12

1113957t minus 1113957tn1( 1113857TL1 1113957t minus 1113957tn1( 1113857 (40)

It can be concluded from Lemma 1 that L1 is positivedefinite Let 1113957ε [ε1 εnminus 1]

T we have

1113957ε minus L1 L21113858 1113859 1113957t tn1113858 1113859T (41)

Combining equations (39) and (41) yields1113957ε minus L1 1113957t minus tn1( 1113857 (42)

Differentiating equation (40) with respect to time wehave

_V2 1113957t minus 1113957tn1( 1113857TL1 _1113957t minus _1113957tn11113872 1113873

minus 1113944nminus 1

i11 minus cos σi( 1113857ki εi

11138681113868111386811138681113868111386811138681113868u

(43)

Similar to the proof (eorem 1 the following equationcan be driven

_V2 le minus 1113957k 1113957εT1113957ε1113872 1113873

u+12 (44)

where1113957k min 1 minus cos σi( 1113857ki i 1 n minus 1 (45)

Note that

L121 1113957t minus 1113957tn1( 11138571113872 1113873TL1 L121 1113957t minus 1113957tn1( 11138571113872 1113873ge λ L121 1113957t minus 1113957tn1( 11138571113872 1113873

TL121 1113957t minus 1113957tn1( 11138571113872 1113873

(46)

which means that 1113957εT1113957εge 2λV2 On account of these analysesthe following equation can be driven from equation (35)

_V2 le minus 1113957k(2λ)1+u2

V1+u22 (47)

According to finite-time convergence theory fromLemma 4 V2 will converge to zero or a small neighbor ofzero in finite time (e convergence of V2 also means thatthe consensus error εi will converge to zero and the si-multaneous arrival can be achieved Hence (eorem 2 hasbeen proven

5 Simulations

In this section numerical simulations are carried out toshow the effectiveness of the proposed strategies (esimulation step is 001 s All the simulations are terminatedwhen the sign of the relative velocity becomes positive or therelative range is less than 001m We consider four vehiclesarriving at a common target from different directions andthe target is fixed at (8000 0)m Detailed simulation pa-rameters for the vehicles are tabulated in Table 1

51 Case 1 Undirected and Connected In this subsectionsimulations are carried out to show the effectiveness of theproposed law under undirected and connected communi-cation topology which is demonstrated in Figure 2 (edetailed simulation parameters are tabulated in Table 1

An undirected path exists between any two differentvehicles Hence all the vehicles can receive informationfrom their neighbors (e Laplacian matrix of the com-munication topology can be acquired as

L

3 minus 1 minus 1 minus 1

minus 1 2 minus 1 0

minus 1 minus 1 3 minus 1

minus 1 0 minus 1 3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Simulation results are demonstrated in Figure 3 Solidline dashed line dash-dotted line and dotted line stand forthe results of vehicles 1 2 3 and 4 respectively Combiningthe vehicle trajectories in Figure 3(a) and range variation inFigure 3(d) we can see that simultaneous arrival can beachieved under the proposed guidance law (e variance ofthe heading error angles is in Figure 3(b) all of which declineto zero at the end of the engagement which verifies theanalysis in equation (12) (e consensus error of the flighttime is demonstrated in Figure 3(e) It is obvious that theflight time of each vehicle can reach an agreement in finitetime under the proposed law Once the consensus of flighttime is achieved the control parameter will remain constantat 1 We know that the acceleration will remain constant ifthe control parameter remains at 1 which is consistent withthe simulation results in Figure 3(c) (is simulation proves

6 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

Table 1 Initial parameters for the four vehicles

Vehicle Initial relative range (m) Velocity (ms) Initial heading angle (deg) Initial LOS angle (deg)1 8000 270 60 02 7500 250 30 03 7700 220 45 04 7000 200 30 0

1 2

34

Figure 2 Undirected and connected communication topology among vehicles

0 2000 4000 6000 8000 10000Downrange (m)

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

ndash100

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

Figure 3 Continued

Journal of Advanced Transportation 7

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

that the proposed guidance law can be applied in cooperativearrival for multiple vehicles

52 Case 2 Leader-Follower In this subsection the leader-follower communication topology between the vehiclesis demonstrated in Figure 4 We consider that four ve-hicles arrive at a target (e detailed simulation pa-rameters are the same as those of case 1 which aretabulated in Table 1 Vehicle 3 acts as the leader whichmeans that vehicle 3 cannot receive information from theother vehicles

(e Laplacian matrix of the communication topologycan be acquired as

L

2 minus 1 0 minus 1

minus 1 3 minus 1 minus 1

0 minus 1 2 minus 1

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(49)

Simulation results are demonstrated in Figure 5Dashed line dotted line solid line and dash-dotted linestand for vehicles 1 2 3 and 4 respectively It can beconcluded from the vehicle trajectories in Figure 5(a)that all four vehicles can arrive at the target Further-more the range variation in Figure 5(d) means that allthe vehiclesrsquo relative ranges converge to zero at the sametime implying that a successful simultaneous arrival isachieved under the proposed law (e variance of theheading error angles is depicted in Figure 5(b) and all ofthem decline to zero at the end of the engagement whichis in line with the analysis in equation (12) Vehicle 3 actsas the leader which means its control parameter willremain constant at 1 during the homing process (eother vehicles will adjust their control parametersaccording to vehicle 3 After the follower vehicles reach

an agreement with the leader in flight time all the ve-hiclesrsquo control parameters will be 1 (is is consistentwith the simulation results in Figures 5(c) and 5(e) It isobvious that the flight time of each vehicle can reach anagreement in finite time under the proposed law (issimulation proves the proposed guidance law can also beapplied in cooperative arrival even if communicationfailures exist

6 Conclusion

(is paper proposes a guidance law for multiple vehiclesarriving at a target cooperatively (e Lyapunov-basedguidance law is proposed and the flight time can be cal-culated with control parameter equal to one Specifically weassume that the control parameters are initialized with one ateach time step (en by exchanging the total flight timebetween the vehicle and its neighbors under an undirectedand connected communication topology the control pa-rameter will be adjusted to reduce the disparities of the flighttime After the consensus of the flight time the controlparameters will remain constant at one (e effectiveness of

0 5 10 15 20 25 30 35 40Time (s)

15

10

5

0

ndash5

ndash10

ndash15

ndash20

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(e)

Figure 3 Simulation results under undirected and connected communication topology (a) Vehicle trajectory (b) Heading error (c) Lateralacceleration (d) Range variation (e) Consensus error of flight time

1 2

34

Figure 4 Leader-follower communication topology amongvehicles

8 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

0 2000 4000 6000 8000 10000Downrange (m)

3000

2500

2000

1500

1000

500

ndash500

0

Alti

tude

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(a)

0 5 10 15 20 25 30 35 40Time (s)

70

60

50

40

30

20

10

0

Hea

ding

angl

e err

or (d

eg)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(b)

0 5 10 15 20 25 30 35 40Time (s)

60

40

20

0

ndash20

ndash40

ndash60

ndash80

Acce

lera

tion

(ms

2 )

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(c)

0 5 10 15 20 25 30 35 40Time (s)

8000

7000

6000

5000

4000

3000

2000

0

1000

Relat

ive r

ange

(m)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

(d)

0 5 10 15 20 25 30 35 40Time (s)

Con

sens

us er

ror o

f im

pact

tim

e (s)

Vehicle 1Vehicle 2

Vehicle 3Vehicle 4

12

10

8

6

4

2

0

ndash2

ndash4

(e)

Figure 5 Simulation results under leader-follower communication topology (a) Vehicle trajectory (b) Heading error (c) Lateral ac-celeration (d) Range variation (e) Consensus error of flight time

Journal of Advanced Transportation 9

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation

the proposed method is demonstrated with simulationsCompared with previous work this paper deals with theconsensus of the flight time directly rather than the esti-mation of time-to-go In future related work the tangentialacceleration should be considered in the design of guidancelaw

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was cosupported in part by the National NaturalScience Foundation of China (nos 61903146 61873319 and61803162)

References

[1] S Ruiz L Guichard N Pilon and K Delcourte ldquoA new airtraffic flowmanagement user-driven prioritisation process forlow volume operator in constraint simulations and resultsrdquoJournal of Advanced Transportation vol 2019 Article ID1208279 21 pages 2019

[2] K Raghuwaiya B Sharma and J Vanualailai ldquoLeader-fol-lower based locally rigid formation controlrdquo Journal of Ad-vanced Transportation vol 2018 Article ID 527856514 pages 2018

[3] S Hao L Yang L Ding and Y Guo ldquoDistributed cooperativebackpressure-based traffic light control methodrdquo Journal ofAdvanced Transportation vol 2019 Article ID 748148914 pages 2019

[4] I S Jeon J I Lee and M J Tahk ldquoImpact-time-controlguidance law for anti-ship missilesrdquo IEEE Transactions onControl Systems Technology vol 14 no 2 pp 260ndash266 2006

[5] I S Jeon J I Lee and M J Tahk ldquoGuidance law to controlimpact time and anglerdquo in Proceedings of the InternationalConference on Control and Automation pp 852ndash857 HongKong China March 2007

[6] H-G Kim J-Y Lee H J Kim H-H Kwon and J-S ParkldquoLook-angle-shaping guidance law for impact angle and timecontrol with field-of-view constraintrdquo IEEE Transactions onAerospace and Electronic Systems vol 56 no 2 pp 1602ndash1612 2020

[7] M Kim B Jung B Han S Lee and Y Kim ldquoLyapunov-basedimpact time control guidance laws against stationary targetsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 51 no 2 pp 1111ndash1122 2015

[8] Z Cheng B Wang L Liu and Y Wang ldquoA compositeimpact-time-control guidance law and simultaneous arrivalrdquoAerospace Science and Technology vol 80 pp 403ndash412 2018

[9] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 49 no 4 pp 2806ndash2817 2013

[10] R Tekin K S Erer and F Holzapfel ldquoPolynomial shaping ofthe look angle for impact-time controlrdquo Journal of GuidanceControl and Dynamics vol 40 no 10 pp 2668ndash2673 2017

[11] R Tekin K S Erer and F Holzapfel ldquoAdaptive impact timecontrol via look-angle shaping under varying velocityrdquoJournal of Guidance Control and Dynamics vol 40 no 12pp 3247ndash3255 2017

[12] R Tekin and K S Erer ldquoImpact time and angle control againstmoving targets with look angle shapingrdquo Journal of GuidanceControl and Dynamics vol 43 no 5 pp 1020ndash1025 2020

[13] M-J Tahk S-W Shim S-M Hong H-L Choi andC-H Lee ldquoImpact time control based on time-to-go pre-diction for sea-skimming antiship missilesrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol 54 no 4pp 2043ndash2052 2018

[14] I-S Jeon J-I Lee and M-J Tahk ldquoHoming guidance law forcooperative attack of multiple missilesrdquo Journal of GuidanceControl and Dynamics vol 33 no 1 pp 275ndash280 2010

[15] J Zhou J Yang and Z Li ldquoSimultaneous attack of a sta-tionary target using multiple missiles a consensus-basedapproachrdquo Science China Information Sciences vol 60 no 7Article ID 070205 2017

[16] Z Hou L Liu Y Wang J Huang and H Fan ldquoTerminalimpact angle constraint guidance with dual sliding surfacesand model-free target acceleration estimatorrdquo IEEE Trans-actions on Control Systems Technology vol 25 no 1pp 85ndash100 2017

[17] S Wang Y Guo S Wang Z Liu and S Zhang ldquoCooperativeguidance considering detection configuration against targetwith a decoyrdquo IEEE Access vol 8 pp 66291ndash66303 2020

[18] W Ren R W Beard and E M Atkins ldquoInformation con-sensus in multivehicle cooperative controlrdquo IEEE ControlSystems vol 27 no 2 pp 71ndash82 2007

[19] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEE Transactions on Automatic Control vol 49 no 9pp 1520ndash1533 2004

[20] N Zoghlami L Beji and RMlayeh ldquoFinite-time consensus ofnetworked nonlinear systems under directed graphrdquo inProceedings of the European Control Conference pp 546ndash551Strasbourg France June 2014

[21] D Zhou S Sun and K L Teo ldquoGuidance laws with finite timeconvergencerdquo Journal of Guidance Control and Dynamicsvol 32 no 6 pp 1838ndash1846 2009

10 Journal of Advanced Transportation