areal-timeobstacleavoidancemethodfor multi-auvclusterbasedon...

CHINESE JOURNAL OF SHIP RESEARCHVOL13NO6DEC 2018

Received2018 - 06 - 20Supported by Natural Science Foundation of Heilongjiang Province (F2018009) Key Laboratory Fund for Equipment Pre-re

search (614221801050717) Open Project of State Key Laboratory of Ocean Engineering Shanghai Jiao Tong University (1616)

Author(s) XU Bo (Corresponding author) male born in 1982 PhD associate professor Research interests inertial navigationintegrated navigation initial alignment and information fusion theoretical algorithm simulation andexperiment of transfer alignment for multi-platform cooperative navigation E-mail xubocartersinacomZHANG Jiao female born in 1992 master degree candidate Research interest inertial navigationintegrated navigation and information fusion E-mail 1814708268qqcom

To cite this articleXU B ZHANG J WANG C A real-time obstacle avoidance method for multi-AUV cluster based on artificialpotential field[JOL] Chinese Journal of Ship Research 2018 13(6) httpwwwship-researchcomENY2018V13I666

DOI1019693jissn1673-3185 01326

A real-time obstacle avoidance method formulti-AUV cluster based on

artificial potential field

XU BoZHANG JiaoWANG ChaoCollege of AutomationHarbin Engineering UniversityHarbin 150001China

Abstract［Objectives］This paper proposes a real-time obstacle avoidance method based on artificial potentialfieldin order to cope with the many potential obstacles in complex underwater environment which affects themovement plan for multi-autonomous underwater vehicle（AUV） cluster［Methods］Firstlya formation methodbased on dynamic network topology is adoptedand the AUV is regarded as a node in the network The potential fieldfunction is set to meet the formation requirements Thenbased on the artificial potential field methodthe potentialfield function is established for the region where both targets and obstacles exist simultaneously Afterwardsthepotential field function is upgraded to an exponential functionsuch that the AUVs can be planned online for real-timeaccomplishment of the mission of obstacle avoidance for multi-AUV cluster Finally10 AUVs and 6 obstacles aresimulated with Matlab software［Results］ The simulation results show that with this methodeach AUV cansuccessfully avoid obstacles and reach the safe area at the target point［Conclusions］The artificial potential fieldfunction method may enable the multi-AUV to accurately avoid obstacle in real time The advancement of thistechnology has important and positive significance for improving military operational capabilityKey wordsAutonomous Underwater Vehicle（AUV）artificial potential fieldmovement in a clusterformationcontrolobstacle avoidanceCLC number U66482

0 Introduction

As Unmanned Aerial Vehicles (UAVs) UnmannedGround Vehicles (UAVs) and robot soldiers gradually show more and more power on the battlefield thedevelopment of unmanned battle has shown a relatively clear blueprint Intelligent unmanned vehiclesin ocean space such as Unmanned Surface Vehicle(USV) Autonomous Underwater Vehicle (AUV) andUnmanned Underwater Vehicle (UUV) have also developed rapidly which have begun to play an increasingly important role in the future marine territorial security and marine development Great impor

tance has been paid to the research in this field inChina and abroad and many encouraging resultshave been achieved which has been gradually applied in military and other fields [1]

Movement in a cluster is a very common phenomenon in nature which is typically represented by thecollective flight of birds the collective migration ofinsects and even the collective movement of proteins and other substances in life bodies [2-5] In thesimilar movements of these groups how to form a coordinated and orderly collective movement mode andhow to quickly change the current movement statehas been a hot issue in the research of cluster move

46

downloaded from wwwship-researchcom

ment control In the movement formation method ofmulti-AUV cluster there are many problems to beconsidered such as stability and controllability Tosolve these problems many scholars have proposedtheir own ideas such as Leader-follower method behavior-based method graph theory method and artificial potential field method [6]

Balch et al [7] proposed a behavior-based formation method which decomposes formation control into a series of basic behaviors and realizes motion control through the integration of behaviors This method has explicit formation feedback and achieves distributed control but it has no clear definition ofgroup behavior which is difficult to carry out mathematical analysis and cannot guarantee the stabilityof formation

Pan et al [8] proposed a virtual structure methodnamely that the AUV formation as a whole is regarded as a virtual structure of a rigid body and eachAUV is a fixed point on the rigid body with fixed relative position This method can control the motion ofAUV by defining the behavior of rigid body but itcannot change the formation according to the changeof environment so its application scope is limited

Yu et al [9] studied a Leader-follower methodwhich divides AUVs into groups of two namely oneleader AUV with one follower AUV The formationcontrol can be achieved by keeping certain angleand distance between followers and leaders According to the relative position relationship between leaders and followers different network topologies can beformed When they encounter obstacles in the environment the obstacles can be avoided by changingthe formation However the self-adaptability and robustness of this master-slave strategy are not strongenough to fully reflect the ability of individuals in nature to select or track their own targets

Khatib[10] first proposed the concept of artificial potential field which assumes that the individual inthe field moves in the direction of the minimum potential energy function when subjected to the force ofthe target and obstacle In the early stage this method was only used in static environment but in dynamic environment because many dynamic factorswere neglected only relative position was chosen asinput To solve these problems Khatib improved themethod to combine it with other methods which ismore suitable for multi-AUV formation control

Liu et al[11] proposed that clustering of clustermovement should be studied in the field of formationcontrol because clustering emphasizes the move

ment differentiation in the cluster and different target tasks can be achieved through different groupssuch as obstacle avoidance and tracking Howeverthey only built theoretical models for this problemand experimental verification lacked The number ofAUVs has an important influence on the navigationand control algorithm of multi-AUV cluster motionThe larger number of AUVs leads to more difficultcontrol

Combining the above problems a cluster motioncontrol method based on dynamic network topologyand an obstacle avoidance method based on artificialpotential field method were proposed in this paperwhich improves the artificial potential field functionand verifies the feasibility of the algorithm throughsimulation experiments with Matlab software1 Basic model of cluster motion

It is assumed that the control law of the ith AUV isui and

ui = uαi + uo

i + uγ

i （1）where uα

i is the formation control law uoi is the ob

stacle avoidance control law and uγ

i is the controllaw of moving toward the target point According tothese three sub-control laws AUV is controlled [12]

Considering the cluster consisting of N AUVs thedynamic equation is

qi = pi

pi = ui

iIcirc Z qipiuiIcirc R2 （2）where qi = [xiyi]

T which is the individual positionpi = [xi yi]

T is the velocity vector ui = [uxiuyi

]T isthe control input Z is the positive integer set R2

is the two-dimensional real number set qij = qi - q j which is the relative distance vector between the ith

and the jth AUVsEach AUV in a cluster is considered as a node in

a network oriented digraph The network topology ofAUV will change dynamically in the course of itsmovement Using the dynamic network method thegroup motion behavior of AUV can be modeled sothat the network node can maintain the balance distance between AUV and its neighbors

Definition 1 (adjacency graph) Let G = (vε) be aweighted oriented digraph of n nodes wherev = 12n is a set of vertices and ε is a set ofedges A = [aij] which is the weighted adjacency matrix where for i jIcirc I = 12n i sup1 jaij 0 for iIcirc I aij = 0 Let r be the distance between any

XU B et al A real-time obstacle avoidance method for multi-AUV cluster based on artificial potential field 47


CHINESE JOURNAL OF SHIP RESEARCHVOL13NO6DEC 2018two AUV nodes and the adjacent set of node vi isNi which is defined as

Ni = jIcirc v qi - q j lt r （3）where times is Euclidean norm under Rm ( Rm is aset of m-dimensional real numbers) and for equilibrium distance r gt 0 Adjacent network G(q) =

(vε(q)) can be determined by point set v and edgeset ε(q) where the edge set is

ε(q) = (i j)Icirc v acute v qi - q j lt r i sup1 j （4）Obviously the edge set ε(q) is determined by q ( q

is the individual position) and (G(q)q) is the adjacent structure2 Research on formation control

based on dynamic network to-pology

Each AUV in the cluster is regarded as a node inthe network and AUV forms a dynamic network inmotion In order to capture obvious spatial order in areal cluster the geometric structure of AUV clusternodes is simulated by using grids We found out a series of q and n nodes and let these nodes and adjacent nodes maintain the same distance described byFormula (5) as

q j - qi = d jIcirc Ni(q) （5）In the formula the selection of q affects the ex

pected formation of AUV cluster to a great extent soit is defined as a lattice object

Definition 2 The q geometric configuration under the restriction of Formula (5) is regarded as anα lattice with size d and the AUV individual iscalled an α individual Several adjacent individualsare connected to form α lattices The edge lengthsof adjacent networks formed by α lattices are equal

In order to construct a smooth integrated potentialfield and a corresponding spatial adjacency matrix anon-negative map of σ -norm is defined as follows

zσ= 1ε

[ 1 + z2- 1] （6）

In this paper the value of ε remains unchangedThe new norm is established here because z

σis

differentiable at any time and z is not differentiable at z = 0 ( z denotes function variable)

Impulse function ρh(z) is a scalar function and issmooth between 0 and 1 It is used to constructsmooth potential field function and adjacency matrix The impulse function is selected as follows

ρh(z) =

igrave

iacute

icirc

iumliuml

iumliuml

1 z Icirc[0h)

12

[1 + cos(π(z - h)(1 - h)

] z Icirc[h1]

0 The others

（7）

where h Icirc(01) impulse function ρh(z) is in[1yen) ρh(z) = 0 || ρh(z) tends to z uniformly Using this impulse function the spatial adjacency matrix A(q) can be defined

aij(q) = ρh( q j - qiσrα)Icirc[01] j sup1 i （8）

where rα = rσ and for any i and q there is

aij(q) = 0 When h = 1 ρh(z) is constant 1 in [01)

and 0 in other intervals Here the significance of selecting impulse function is to build a matrix depending on the adjacent network with only 0 and 1 matrixelements

In order to construct a smooth paired potentialfield we also introduced the behavioral functionϕα(z)

ϕα(z) = ρh(zrα)ϕ(z - dα)Icirc[01] （9）ϕ(z) = 1

2[(a + b)σ1(z + c) + (a - b)] （10）

where σ1(z) = z 1 + z2 ϕ(z) is a non-uniformS-shaped function dα is the edge length of the connection network composed of α lattice a b c are arbitrary real numbers and 0 lt a b c = ||a - b 4ab

to ensure ϕ(0) = 0 Paired gravitational or repulsionpotential field functions are defined as follows

ψα(z) = dα

z

ϕα(s)ds （11）Finally a distributed control algorithm was pro

posed to study formation control in multi-AUV cluster motion

uαi = cα1 aring

jIcirc N αi

ϕα( q j - qiσ)ni j +cα2 aring

jIcirc N αi

aij(q)( p j - pi)

（12）The first half of the equation is the distance gradientand the second half is the velocity uniformity termni j = σε(q j - qi) =

q j - q

1 + ε q j - qi

2is a vector con

necting qi and q j where εIcirc(01) is a fixed parameter of the σ -norm The algorithm describes the motion rules between any two AUVs and satisfies threerules proposed by Reynolds about cluster motion3 Multi-AUV cluster motion plan-

ning based on artificial potentialfield

The basic idea of artificial potential field methodis to construct gravitational field Uatt and repulsive

48


field U rep at the locations of the target point and obstacle respectively The corresponding potential fieldforce will attract AUV to move near the target pointand prevent AUV from moving towards the obstacleUnder the ultimate effect of resultant force AUVwill be guided to move towards the target point

The design of repulsive field function is related toobstacles and when the function is closer to the obstacles the repulsion force is larger the gravitationalfield function is similar to it In order to carry outnecessary theoretical study on the method a two-dimensional planar artificial potential field model withstatic threat was constructed on the basis of simplifying the problem without losing generality [13]31 Gravitational field modeling

The function of distance between AUV and targetpoint in two-dimensional plane is defined as

ρ(rg) = r(x1y1) - g(x2y2) （13）The gravitational field generated is

Uatt (ρ(rg)) = ξ ρ(rg)m

（14）where r(x1y1) refers to the position of AUVg(x2y2) refers to the position of target point m is apositive constant and the value of m is the factordetermining the shape of potential field functioncurve and ξ is the target potential field coefficientHere we take m = 2 because when m = 1 there willbe unbounded gravitational field when the gravitational field derivation is conducted which will causethe AUV to jitter near the target point Therefore m

takes the minimum value of 2The attraction of the target point to AUV is as fol

lowsFatt = -NtildeUatt (ρ(rg)) = mξ r(x1y1) - g(x2y2)

m - 1nRG

（15）where nRG is the unit vector from AUV to the targetpoint

In this paper because the target of AUV is fixedthe relative motion between AUV and target is notconsidered when designing gravitational potentialfield (Fig 1)32 Establishment of repulsive field

model

The position of obstacle is assumed to be o(x3y3)

and F rep1 is the repulsion between AUV and the obstacle which makes AUV move away from the obstacle Similar to the above analysis it is assumed thatthe obstacle is static regardless of the effect of relative velocity between AUV and the obstacle on the

AUV motion F rep is a repulsion generated by AUVspeed which is proportional to the velocity component on the connection line between AUV and the obstacle and the direction is along the AUV-obstacleline and away from the obstacle (Fig 2)

The repulsive potential field isU reps = aring

i = 1

M

U reps(ρi(ro)) （16）where M is the number of obstacles ρi(ro) =

r(x1y1) - oi(x3y3) is the distance function between AUV and the ith obstacle Therefore Formula (15)can be written as

U reps(ρi(ro)) =

igraveiacuteicirc

iuml

iuml

12η( 1ρi(ro)

- 1ρ(rg)

)2 ρi(ro) lt ρ(rg)

0 ρi(ro) gt ρ(rg)（17）

where η is the coefficient of repulsive potential fieldThe repulsion corresponding to the repulsive po

tential field of the ith obstacle in two-dimensionalspace S is

F reps(ρi(ro)) = -NtildeU reps(ρi(ro)) （18）Therefore the total repulsion is

F reps = aringi = 1

N

F reps(ρi(ro)) （19）33 Spatial composite potential field

modeling

The gravitational potential field and repulsive potential field in space are synthesized and the synthe

Fig1 Gravity in two-dimensional space S

Gravitation

y

xo

AUV

Target

Fig2 Repulsion in two-dimensional space S

Repulsion

y

xo

AUV

Obstacle



CHINESE JOURNAL OF SHIP RESEARCHVOL13NO6DEC 2018sized potential field is obtained as follows

U = Uatt (rg) + U reps(ρ(ro)) （20）The spatial composite potential field force isF = Fatt + F rep = -NtildeUatt (rg) - Ntildearing

i = 1

N

U reps(ρi(ro))

（21）The independent variable of the conventional arti

ficial potential field function is the square of distance The curve of this form of function rises andfalls too fast which also makes the gravitation and repulsion change too fast Therefore parameter selection is more stringent and it is likely that the pathplanning cannot be completed due to the problem ofparameter selection Therefore the potential fieldfunctions adopted here are all improved and the exponential function is used as the potential field function whose structure is as follows

U (ρ(rg)) = α exp(-X ) （22）For gravitational field the expression of exponen

tial function independent variable X isX = Xrg = β ρ(rg)

m（23）

For the repulsive field the expression of X isX = Xro = γ ρ(ro)

n（24）

When the exponential function is selected the potential field function is still a function of distancebut the function curve is no longer a quadratic curvebut an exponential curve According to the characteristics of exponential function the values of m andn are greater than 1 and the coefficients of gravitational potential field β and repulsive potential fieldγ are positive constants so X gt 0 And according tothe characteristics of exponential function curve theU (ρ(rg)) image takes the interval part [0yen) which changes slowly and is easy to control thusachieving a relatively stable effect4 Simulation

In order to validate the multi-AUV path planningalgorithm based on artificial potential field methodthe following simulation validation was carried outTarget potential field coefficient ξ was 5 for gravitational field the number of adjacent individuals α

was 5 and the gravitational field coefficient β was05 for repulsive field the number of adjacent individuals α was 1 and the repulsive field coefficientγ was 2 Ten AUVs were set up and the obstacleavoidance motion was completed under the combined action of the gravitation of the target point andthe repulsion of the obstacle The initial velocity of10 AUVs (unit ms) is set to

Vx0 =［1650980651303230496798038615366915808711991］

Vy0 =［9138121764935859085990980809895651315614539］

In Fig 3 the x sign in the lower left corner represents 10 AUVs and the X sign in the upperright corner represents the position of the targetpoint with the unchanged coordinates the circles inthe plane represent obstacles with coordinates of(80 m 100 m) (100 m 100 m) (120 m 100 m)(180 m 200 m) (200 m 150 m) and (240 m 200 m)respectively the dashed line is the movement trackof 10 AUVs Fig 3(b) shows the track of 10 AUVsafter a 08 ms deceleration Fig 4 shows the movement track of multi-AUV cluster in x and y directions before and after deceleration

From the simulation results of Figs 3 and 4 it canbe seen that all AUVs can avoid obstacles smoothlyand reach the safe area near the target point accurately using the obstacle avoidance method based onthe artificial potential field In the case of slowerspeed the AUV is farther from the obstacle and thecollision avoidance effect is better It can be seenthat the artificial potential field method based on exponential function can accurately achieve AUV ob

（a）Movement track before deceleration

（b）Movement track after decelerationFig3 Multi-AUV cluster obstacle avoidance movement track

-5 0 5 10 15 20 25 30 35xm

35302520151050

-5

ym

35302520151050

-5

ym

-5 0 5 10 15 20 25 30 35xm

50


stacle avoidance5 Conclusion

Aiming at the problem of keeping clustering andavoiding collision among individuals in multi-AUVcooperative positioning process in this paper we adopted a formation method based on dynamic networktopology which regards AUV as a node in the network and satisfies the formation requirements by setting potential field function In addition aiming atthe possible obstacles in navigation artificial potential field method was used for on-line planning andthe potential field function was simulated as an exponential function The simulation results show that inthe multi-AUV collision avoidance method based onartificial potential field function the exponentialfunction method can achieve real-time obstacleavoidance for multi-AUVReferences［1］ XU Y RSU Y MPANG Y J Expectation of the devel

opment in the technology on ocean space intelligent unmanned vehicles［J］ Chinese Journal of Ship Research20061（3）1-4（in Chinese）

［2］ MELANCHENKO A G Coordinated spacecrafts translational motion control within linear architecture cluster［J］ Journal of Automation and Information Sciences201446（9）49-67

［3］ KHALIL I S MABELMANN LMISRA S Magnetic-based motion control of paramagnetic microparticleswith disturbance compensation［J］ IEEE Transactionson Magnetics201450（10）1-10

［4］ YOUAKIM KEHAB MHATEM Oet al Paramagnetic microparticles sliding on a surfacecharacterizationand closed-loop motion control［C］Proceedings of2015 IEEE International Conference on Robotics andAutomation Seattle WA USA IEEE 20154068-4073

［5］ KIM WSUNG M Scalable motion control system using EtherCAT-based shared variables［C］Proceedings of the 2015 IEEE 20th Conference on EmergingTechnologies amp Factory Automation LuxembourgLuxembourgIEEE2015

［6］ JIANG D P Research on coordinated control technology for multiple autonomous underwater vehicles［D］HarbinHarbin Engineering University2011（in Chinese）

［7］ BALCH TARKIN R C Behavior-based formation control for multirobot teams［J］ IEEE Transactions on Robotics and Automation199814（6）926-939

［8］ PAN W WJIANG D PPANG Y Jet al Amulti-AUV formation algorithm combining artificial potential field and virtual structure［J］ Acta Armamentarii201738（2）326-334（in Chinese）

［9］ YU HWANG Y JLIU L Control of flocking motion ofthe flock with a leader based on dynamic topology［J］Systems Engineering and Electronics200628（11）

1721-1724（in Chinese）［10］ KHATIB O Real-time obstacle avoidance for manipu

lator and mobile robots［J］ IJRR19865（1）90-98［11］ LIU M YYANG P PLEI X Ket al ICD（informa

tion coupling degree） based control algorithm forbreaking up of swarm of AUVs（autonomous underwater vehicles） into subswarms［J］ Journal of Northwestern Polytechnical University201432（4）

581-585（in Chinese）［12］ ZHANG X Study on the control of trajectory tracking

and flocking motion for the autonomous navigation ofmobile robots［D］ XianChangan University2012（in Chinese）

［13］ ZHANG D FLIU F Research and development trendof path planning based on artificial potential fieldmethod［J］ Computer Engineering and Science201335（6）88-95（in Chinese）

（a）AUV movement track in x and y directions before deceleration

（b）AUV movement track in x and y directions after decelerationFig4 Multi-AUV cluster motion trajectories

in x and y directions

0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm




一种基于人工势场多AUV集群的实时避障方法

徐博张娇王超哈尔滨工程大学自动化学院黑龙江哈尔滨 150001

摘要［目的目的］针对复杂水下环境中可能存在的多种障碍物影响多自主式水下机器人（AUV）集群运动规划

问题提出一种基于人工势场的多AUV集群实时避障方法［方法方法］首先采用一种基于动态网络拓扑的编队方

法将AUV看作网络中的节点通过设置势场函数来满足编队要求然后基于人工势场法对同时存在目标和障

碍的区域建立势场函数并将势场函数改进为指数函数对AUV进行在线规划实时完成多AUV集群避障的任

务最后在Matlab软件中仿真设置 10台AUV和 6个障碍物进行仿真验证［结果结果］仿真结果表明采用该方法

AUV可以全部顺利地避开障碍物准确到达目标点的安全区域［结论结论］人工势场函数法可准确实现多 AUV的

实时避障该技术的进步对提高军事作战能力具有重要的意义

关键词自主式水下机器人人工势场集群运动编队控制避障

自主式水下机器人水下对接技术综述

郑荣 1宋涛 12孙庆刚 13国婧倩 12

1 中国科学院沈阳自动化研究所机器人学国家重点实验室辽宁沈阳 1100162 东北大学机械工程与自动化学院辽宁沈阳 110819

3 中国科学院大学北京 100049

摘要自主式水下机器人（AUV）作为水面支持平台和海底空间站以及深海长期观测系统之间的重要纽带其

水下对接技术一直以来都是国内外的研究热点在归纳分析国内外AUV水下对接技术如水下箱（笼）式对接

机械手或载体辅助式对接杆类引导对接平台阻拦索式对接和喇叭口式引导对接技术的基础上介绍各种

AUV对接技术的实现方法和结构原理以及对接技术的发展现状与趋势并针对目前应用较为广泛的喇叭口

式引导对接方式详细介绍一种针对重型AUV的水下对接系统经试验验证该系统模块化强对横滚姿态要

求低适用于多种尺寸AUV对接系统的对接成功率高所做工作可为今后AUV水下对接技术的发展提供参考

关键词自主式水下机器人水下对接结构原理喇叭口式引导对接综述

[Continued from page 37]

10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52


ment control In the movement formation method ofmulti-AUV cluster there are many problems to beconsidered such as stability and controllability Tosolve these problems many scholars have proposedtheir own ideas such as Leader-follower method behavior-based method graph theory method and artificial potential field method [6]

Balch et al [7] proposed a behavior-based formation method which decomposes formation control into a series of basic behaviors and realizes motion control through the integration of behaviors This method has explicit formation feedback and achieves distributed control but it has no clear definition ofgroup behavior which is difficult to carry out mathematical analysis and cannot guarantee the stabilityof formation

Pan et al [8] proposed a virtual structure methodnamely that the AUV formation as a whole is regarded as a virtual structure of a rigid body and eachAUV is a fixed point on the rigid body with fixed relative position This method can control the motion ofAUV by defining the behavior of rigid body but itcannot change the formation according to the changeof environment so its application scope is limited

Yu et al [9] studied a Leader-follower methodwhich divides AUVs into groups of two namely oneleader AUV with one follower AUV The formationcontrol can be achieved by keeping certain angleand distance between followers and leaders According to the relative position relationship between leaders and followers different network topologies can beformed When they encounter obstacles in the environment the obstacles can be avoided by changingthe formation However the self-adaptability and robustness of this master-slave strategy are not strongenough to fully reflect the ability of individuals in nature to select or track their own targets

Khatib[10] first proposed the concept of artificial potential field which assumes that the individual inthe field moves in the direction of the minimum potential energy function when subjected to the force ofthe target and obstacle In the early stage this method was only used in static environment but in dynamic environment because many dynamic factorswere neglected only relative position was chosen asinput To solve these problems Khatib improved themethod to combine it with other methods which ismore suitable for multi-AUV formation control

Liu et al[11] proposed that clustering of clustermovement should be studied in the field of formationcontrol because clustering emphasizes the move

ment differentiation in the cluster and different target tasks can be achieved through different groupssuch as obstacle avoidance and tracking Howeverthey only built theoretical models for this problemand experimental verification lacked The number ofAUVs has an important influence on the navigationand control algorithm of multi-AUV cluster motionThe larger number of AUVs leads to more difficultcontrol

Combining the above problems a cluster motioncontrol method based on dynamic network topologyand an obstacle avoidance method based on artificialpotential field method were proposed in this paperwhich improves the artificial potential field functionand verifies the feasibility of the algorithm throughsimulation experiments with Matlab software1 Basic model of cluster motion

It is assumed that the control law of the ith AUV isui and

ui = uαi + uo

i + uγ

i （1）where uα

i is the formation control law uoi is the ob

stacle avoidance control law and uγ

i is the controllaw of moving toward the target point According tothese three sub-control laws AUV is controlled [12]

Considering the cluster consisting of N AUVs thedynamic equation is

qi = pi

pi = ui

iIcirc Z qipiuiIcirc R2 （2）where qi = [xiyi]

T which is the individual positionpi = [xi yi]

T is the velocity vector ui = [uxiuyi

]T isthe control input Z is the positive integer set R2

is the two-dimensional real number set qij = qi - q j which is the relative distance vector between the ith

and the jth AUVsEach AUV in a cluster is considered as a node in

a network oriented digraph The network topology ofAUV will change dynamically in the course of itsmovement Using the dynamic network method thegroup motion behavior of AUV can be modeled sothat the network node can maintain the balance distance between AUV and its neighbors

Definition 1 (adjacency graph) Let G = (vε) be aweighted oriented digraph of n nodes wherev = 12n is a set of vertices and ε is a set ofedges A = [aij] which is the weighted adjacency matrix where for i jIcirc I = 12n i sup1 jaij 0 for iIcirc I aij = 0 Let r be the distance between any














zσ= 1ε

[ 1 + z2- 1] （6）


σis



ρh(z) =

igrave

iacute

icirc

iumliuml

iumliuml

1 z Icirc[0h)

12

[1 + cos(π(z - h)(1 - h)

] z Icirc[h1]

0 The others

（7）








2[(a + b)σ1(z + c) + (a - b)] （10）



ψα(z) = dα

z



uαi = cα1 aring

jIcirc N αi


jIcirc N αi

aij(q)( p j - pi)


q j - q

1 + ε q j - qi

2is a vector con




48










m - 1nRG



model





i = 1

M



U reps(ρi(ro)) =

igraveiacuteicirc

iuml

iuml

12η( 1ρi(ro)

- 1ρ(rg)






F reps = aringi = 1

N


modeling



Gravitation

y

xo

AUV

Target


Repulsion

y

xo

AUV

Obstacle





i = 1

N

U reps(ρi(ro))





m（23）


n（24）




Vx0 =［1650980651303230496798038615366915808711991］

Vy0 =［9138121764935859085990980809895651315614539］





-5 0 5 10 15 20 25 30 35xm

35302520151050

-5

ym

35302520151050

-5

ym

-5 0 5 10 15 20 25 30 35xm

50






















0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52













zσ= 1ε

[ 1 + z2- 1] （6）


σis



ρh(z) =

igrave

iacute

icirc

iumliuml

iumliuml

1 z Icirc[0h)

12

[1 + cos(π(z - h)(1 - h)

] z Icirc[h1]

0 The others

（7）








2[(a + b)σ1(z + c) + (a - b)] （10）



ψα(z) = dα

z



uαi = cα1 aring

jIcirc N αi


jIcirc N αi

aij(q)( p j - pi)


q j - q

1 + ε q j - qi

2is a vector con




48










m - 1nRG



model





i = 1

M



U reps(ρi(ro)) =

igraveiacuteicirc

iuml

iuml

12η( 1ρi(ro)

- 1ρ(rg)






F reps = aringi = 1

N


modeling



Gravitation

y

xo

AUV

Target


Repulsion

y

xo

AUV

Obstacle





i = 1

N

U reps(ρi(ro))





m（23）


n（24）




Vx0 =［1650980651303230496798038615366915808711991］

Vy0 =［9138121764935859085990980809895651315614539］





-5 0 5 10 15 20 25 30 35xm

35302520151050

-5

ym

35302520151050

-5

ym

-5 0 5 10 15 20 25 30 35xm

50






















0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52










m - 1nRG



model





i = 1

M



U reps(ρi(ro)) =

igraveiacuteicirc

iuml

iuml

12η( 1ρi(ro)

- 1ρ(rg)






F reps = aringi = 1

N


modeling



Gravitation

y

xo

AUV

Target


Repulsion

y

xo

AUV

Obstacle





i = 1

N

U reps(ρi(ro))





m（23）


n（24）




Vx0 =［1650980651303230496798038615366915808711991］

Vy0 =［9138121764935859085990980809895651315614539］





-5 0 5 10 15 20 25 30 35xm

35302520151050

-5

ym

35302520151050

-5

ym

-5 0 5 10 15 20 25 30 35xm

50






















0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52




i = 1

N

U reps(ρi(ro))





m（23）


n（24）




Vx0 =［1650980651303230496798038615366915808711991］

Vy0 =［9138121764935859085990980809895651315614539］





-5 0 5 10 15 20 25 30 35xm

35302520151050

-5

ym

35302520151050

-5

ym

-5 0 5 10 15 20 25 30 35xm

50






















0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52






















0 10 20 30 40 50 60 70 80ts

403020100x

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

403020100y

directio

ntrack

m

0 10 20 30 40 50 60 70 80ts

0 10 20 30 40 50 60 70 80ts

403020100

-10xdire

ctiontr

ackm

403020100

-10ydire

ctiontr

ackm


























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52

























10509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905105090510509051050905

52


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