Flocking Control of Mobile Robots with Obstacle Avoidance Based...

Research ArticleFlocking Control of Mobile Robots with Obstacle AvoidanceBased on Simulated Annealing Algorithm

Jin Cheng and Bin Wang

University of Jinan Jinan China

Correspondence should be addressed to Jin Cheng cse_chengjujneducn

Received 20 March 2020 Revised 5 May 2020 Accepted 28 May 2020 Published 27 June 2020

Guest Editor Carlos Llopis-Albert

Copyright copy 2020 Jin Cheng and Bin Wang )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

Flocking control problem of mobile robots under environment with unknown obstacles is addressed in this paper Based onthe simulated annealing algorithm a flocking behaviour for mobile robots is achieved which converges to alignment whileavoiding obstacles Potential functions are designed to evaluate the positional relationship between robots and obstaclesUnlike the existing analytical method simulated annealing algorithm is utilized to search the quasi-optimal position ofrobots in order to reduce the potential functions Motion control law is designed to drive the robot move to the desiredposition at each sampling period Experiments are implemented and the results illustrate the effectiveness of the proposedflocking control method

1 Introduction

Flocking problems are studied and applied in many re-search fields such as self-organized mobile sensor net-works in [1] cooperative unmanned aerial vehicles(UAVs) in [2ndash5] and military reconnaissance in [6]Research of flocking behaviour stems from the coordi-nation phenomena in nature from swarming of bacteriabiochemical cellular networks up to flocking of birdsschooling of fish and herds of land animals [7] Reynoldsproposed a computer model of coordinated animal mo-tion named ldquoboidsrdquo in [8] and defined three concepts todescribe the behaviours of boids

(1) Collision avoidance avoid collisions with nearbyflockmates

(2) Velocity matching attempt to match velocity withnearby flockmates

(3) Flock centering attempt to stay close to nearbyflockmates

)ese behaviours are seen in astonishing amount ofcoordinated systems which exhibit unbelievable efficient

and robust coordination [9 10] )ese inspired a lot ofwork in statistical physics and control theories aboutstability Flocking behaviour has been addressed in thecontext of nonequilibrium phenomena in many degree-of-freedom dynamical systems and self-organization insystems of self-propelled particles [7 11 12]

Booming-related work has been done worldwide in thelast decades Some works focus on the dynamic model ofmobile robots [13] In [14] a method based on the in-trinsic properties of the Euclidean plane is proposed forthe calculation of the moving mechanism on a trajectoryIn [15] a direct analytical method for solving the lineartransformation of the characteristic equation for indus-trial robots is proposed A model predictive flockingcontrol scheme for second-order multiagent systems withinput constraints is designed in [16] In [17] a dynamicpinning control algorithm (DPCA) is developed to gen-erate stable flocking motion for all the agents without theassumption of connectivity or initial connectivity of thenetwork In [18] a distributed control protocol with in-dividual local information is proposed for the second-order multiagent system with interference )e problem offlocking and velocity alignment in a group of kinematic

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7357464 9 pageshttpsdoiorg10115520207357464

nonholonomic agents in 2 and 3 dimensions is addressedin [19] A geodesic control law that minimizes a mis-alignment potential and results in velocity alignment andflocking is proposed In [20] the flocking of multipleagents which have significant inertias and evolve on abalanced information graph is studied Passive decom-position method is used to incorporate this inertial effectwhich can cause unstable group behaviour )e proposedflocking control law is proven to be stable and can ensurethat the internal group shape is exponentially stabilized toa desired state

In recent years the flocking problem of mobile robotshas become an attractive subject in the research field of themultiagent system and cooperative robots In [21] thecooperative motion coordination of leader-follower for-mations of nonholonomic mobile robots under visibilityand communication constraints in known polygonal ob-stacle environments is studied Each robot can take care ofconverging to a desired configuration while maintainingvisibility with its target Flocking with obstacle avoidance isan emerging research interest in recent years [22ndash26] Anew distributed coordination algorithm with Voronoipartitions for multivehicle systems is presented in [22]which results not only in obstacle avoidance and motion tothe goal but also in a desirable geographical distribution ofthe vehicles A flocking algorithm in the presence of ar-bitrary shape obstacle avoidance is presented in [23] bothin the 2D space and 3D space In [27] learning-basedmodel predictive control approach is applied to theflocking problem of UAVs in the presence of uncertaintiesand obstacles A distributed framework that ensures reli-ably collision-free behaviours in large-scale multirobotsystems with switching interaction topologies is presentedin [28]

In practical applications such as formulation of themobile robot or surface vessels the agent model is under-actuated and the kinematic motion is constricted withnonholonomic constraints In our research project Ami-goBot is adopted as an α minus agent in the flocking controlproblem )e kinematic model of AmigoBot is a kind ofnonholonomic system in which the number of control in-puts is less than the degree of freedom and cannot betransformed into a second-order linear system It is nec-essary to redesign the control algorithms for the flocking ofAmigoBots

In this paper a control method for the flocking problemof mobile robots is proposed Simulated annealing (SA)algorithm is used to design a simple behaviour for eachrobot while flocking can be self-established SA algorithm isa probabilistic technique for approximating the globaloptimum of a given function Specifically it is a meta-heuristic to approximate global optimization in a largesearch space for an optimization problem It is often usedwhen the search space is discrete (eg the travelingsalesman problem) For problems where finding an ap-proximate global optimum is more important than findinga precise local optimum in a fixed amount of time sim-ulated annealing may be preferable to alternatives such asgradient descent

)e paper is organized as follows Section 2 gives thekinematic model of AmigoBot and the problem formulationDesign process of the flocking control law is presented inSection 3 Simulation is implemented to illustrate the ef-fectiveness of the proposed control law and results areshown in Section 4 Finally the conclusions are made inSection 5

2 Problem Formulation

21Model ofFlockingAgents )efirst model which is widelyconsidered as a flocking agent is constructed in [12] whichwas proposed by Vicsek et al to interpret the flockingphenomenon such as clustering and convection

xi(t + 1) xi(t) + vi(t)Δt

θ(t + 1) langθ(t)rangr + Δθ(1)

where xi is the position of particle i on the plane vi is thevelocity with an absolute value v and an angle given byθ(t + 1) i 1 2 middot middot middot n langθ(t)rangr is the average direction of theparticles in the neighbourhood of particle i and Δθ is therandom noise

From the perspective of theoretical research the dy-namics of agents in flocking is also often modelled as asecond-order linear system [29]

_xi vi _vi ui i 1 2 n (2)

where xi vi and ui are the position velocity and controlforce respectively

)e other second-order linear system of the flockingagent is proposed in [30 31])e agents moving on the planeare modelled with the following dynamics

_ri vi _vi ui i 1 2 n (3)

where ri (xi yi)T is the position of agent i vi ( _xi _yi)

T isits velocity and ui (ux uy)T is its control inputs )eheading angle of agent i is defined with θi arctan( _yi _xi)

22 KinematicModel of theMobile Robot Different from theagent models proposed above which are treated with thelinear system the kinematic model of the mobile robot ismuchmore complex As shown in Figure 1 the mobile robotis controlled only by the velocities of two drive wheelsAssuming that there exists no slipping motion between thewheels and the ground the dynamics of AmigoBot can bemodelled as

_xi vi cos θi

_yi vi sin θi

θ

i ωi

(4)

where vi and ωi are the linear and angular velocity re-spectively (xi yi) is the position of AmigoBot in Cartesiancoordinates and θi is the orientation angle of theAmigoBot

2 Mathematical Problems in Engineering

Unlike the models of (1)ndash(3) mobile robot like AmigoBotis usually considered as a nonholonomic system due to the factthat the system has less dimension of control inputs than itsdegrees of freedom For AmigoBot it has 3 degrees of freedomie the lateral and longitudinal motion on the horizontal planeand rotatory motion along the axis vertical to the horizontalplane but with only two control inputs

For the differential wheel drive mobile robot it has vi

(vRi + vLi)2 and ωi (vRi minus vLi)L vRi and vLi are the ve-locities of the right and left wheel respectively and L is thedistance between the two wheels as described in Figure 1Constrained by limitations on the control inputs theexisting flocking laws or algorithms are not suited for theflocking control problem of mobile robots

3 Design of the Flocking Control Law

31 Potential Function ofCoordination Each mobile robot isconsidered as an α minus agent Define the position vector of theith AmigoBot with ri [xi yi]

T )e relative positions be-tween the ith and jth AmigoBots are defined withrij ri minus rj )e distance between these two AmigoBots isthen given with rij A potential function Vij is defined toevaluate the relative positions between any pair of neigh-bouring AmigoBots as follows

Vij rij

1113874 1113875 d2

r

rij

2 + log rij

2

1113874 1113875 + erij

minus dr1113872 1113873

2

(5)

where dr gt 0 is a design parameter )e curve of function Vij

is shown in Figure 2 dr determines the desired distancebetween the agents When rij dr it hasminVij(rij) 2 + log(d2

r) Unlike the potential functionproposed in [31] also shown in Figure 2 as traditional Vijthe third term in (5) is helpful to get a flexible behaviour suchas split and squeezing maneuver in an environment withobstacles

)e potential function of AmigoBot i is then definedwith the sum of all Vij

Viα 1113944

jisinAVij rij

1113874 1113875 (6)

whereA is the set that includes all other AmigoBots that theith AmigoBot can detect with sensors

32 Potential Function of Obstacle Avoidance Obstacles canbe detected by AmigoBot with sonar sensors in an unknownenvironment Define the position of the obstacle withOk [xk yk]T To keep a safe distance from the obstacle Okit is assumed that the obstacle Ok has a circular boundarywith radius ds β minus agent is then defined with the projectionof the α minus agent on the boundary of obstacle Ok [32]

Define the projection of the position ri of robot i onOk as

qik arg minqkisinΩk

ri minus qk

(7)

where Ωk is the boundary of OkDefine the distance between robot i and its projection qik

on the obstacle Ok with rik ri minus qik A potentialfunction Vik is designed to evaluate the relative positionbetween robot i and obstacle Ok as follows

Vik rik

1113872 1113873

1rik

minus1ds

1113888 1113889

2

rik

ltds

0 rik

geds

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Define

(a)

L2

L2(x y)

Drive wheel

Caster

(b)

Figure 1 Mobile robot with sonar sensors (a) AmigoBot (b) Model of AmigoBot

Modified VijOriginal Vij

0

10

20

30

40

50

60

Vij

05 1 15 2 25 30r

Figure 2 Potential function Vij

Mathematical Problems in Engineering 3

Viβ 1113944

kisinOi

Vik rik

1113872 1113873

(9)

where Oi is the set of obstacles detected by robot i

33 Flocking Behaviour via Random Search AlgorithmWith the aforementioned designed potential functions Vi

αand Vi

β the flocking control problem of the mobile robot isconsidered as an optimization problem in this paper In [33]Cheng et al designed a flocking control law that can drivethe agents to evolve into the flock centering state Motioncontrol laws are proposed to synchronize the velocities andorientation angles via the Newton method in an environ-ment without obstacles

For the flocking problem of the mobile robot in acomplex environment with unknown obstacles define thetotal potential function for robot i as

Vi Viα + V

iβ (10)

Obviously if all the robots move towards the direction ofthe gradient descent of Vi at each sample period the robotswill keep a desired distance among each other finally

θ lowasti arg minθiisinΘi

Vi foralltk (11)

where Θi is the orientation space of robot i tk is the sampleperiod θlowasti is the optimal orientation angle which has themaximal extent of the gradient descent of Vi at tk Due tothat the space Θi is continuous methods based on thegradient descent are generally adopted [30 33] Methodsbased on the random search algorithm provide anotherdependable solution In this paper SA algorithm is appliedto searching of the minimal solution of Vi which is designedto regulate the flocking state

Compared to other approximation methods such asthe Newton method gradient descent method and

LevenbergndashMarquardt method SA can avoid being trappedin the local minimal in early iterations and is able to exploreglobally for better solutions in finite iterations [34] SA is amethod for solving unconstrained and bound-constrainedoptimization problems )e method models the physicalprocess of heating a material and then slowly lowering thetemperature to decrease defects thus minimizing the systemenergy [35 36] )ere is an essential similarity between theflocking control problem and the simulated annealingcourse Both of them aim to explore the optimal solution thathas minimal energy or desired homogeneity

)e flocking control problem of mobile robots isconsidered as an approximation of the optimal solution ofeach Vi With full information of all the other robots andthe obstacles detected by sonar sensors each robot tries tokeep a quasi-optimal distance from each other in themotion while avoiding the obstacles Firstly the optimalposition rlowasti with minimal Vi is approximated via the SAalgorithm )en the optimal orientation angle θlowasti is cal-culated with rlowasti Finally a motion control law is designed todrive the robot move to the optimal position rlowasti at eachsampling period tk

)e pseudo-code of the random search method of rlowasti viathe SA algorithm is shown in Algorithm 1 whererlowasti [xlowasti ylowasti ]T is the desired position vector for the ith

AmigoBot Define the change of energy level with

ΔE Vi rnew( 1113857 minus Vi 1113957ri( 1113857 (12)

)e initial position 1113957ri in Neighbor(1113957ri) is not generatedrandomly but given with ri Also the function Neighbor(1113957ri)

is defined with

rnew 1113957ri + Steplowast [Rand( )Rand( )]T (13)

where Step is a parameter decided by the maximal velocity ofthe robot and Rand is a random function which can generatea value randomly in the range between 0 and 1

Input objective functionVi and ri i 1 2 n

Output quasi minus optimal solution⟶ rlowasti(1) for robot i do(2) Initialize SA (T Tmin α and N)(3) rbest⟵ ri

(4) 1113957ri⟵ ri

(5) While TltTmin do(6) for i 1 N do(7) rnew⟵Neighbor(1113957ri)

(8) if ΔElt 0 then(9) 1113957ri⟵ rnew(10) rbest⟵ 1113957ri

else(11) if exp(minusΔET)gtRand then(12) 1113957ri⟵ rnew(13) T⟵ αT

(14) rlowasti ⟵ rbest(15) return rlowasti

ALGORITHM 1 Flocking behaviour search


Obviously if rlowasti is well tracked Vi would converge to theminimal value and the distance between the agents wouldalso converge to a constant value which in fact is determinedby dr )at is the flocking centering state can be guaranteed

Once the quasi-optimal position rlowasti is achieved theoptimal orientation angle can be calculated withθlowasti arctan((ylowasti minus yi)(xlowasti minus xi))

Define θlowasti arctan((ylowasti minus yi)(xlowasti minus xi)) and it haszθ lowastizxi

minusyi minus ylowasti

xi minus xlowasti( 11138572

+ yi minus ylowasti( 11138572

zθ lowastizyi

xi minus xlowasti



(14)

Define θe θi minus θlowasti and design the control law for ωi

with

ωi minuskw θi minus θ lowasti( 1113857 + vi

zθ lowastizxi

cos θlowasti + vi

zθ lowastizyi

sin θlowasti (15)

where kw gt 0 is a design parameterWith the control inputωi

in (15) it has

θe

minuskw θi minus θlowasti( 1113857 minuskwθe (16)

Define a candidate function Vθe (12)θ2e then it has

_Vθe minuskwθ

2e le 0 With Lyapunov stability and LaSallersquos in-

variant principle it can be proven that the equilibrium θe

0 is asymptotically stable )at is with the control law for ωi

in (15) the ith AmigoBot can be steered towards the desiredposition rlowasti

Define δri [xi minus xlowasti yi minus ylowasti ]T Let

vi 1

N minus 11113944jisinA

vj + kv δri

Vi (17)

Table 1 Parameters in the flocking control method

Variables Value

Potential function dr 1000mmds 1000mm

Motion control kw 00001kv 0003

SA algorithm

T 100Tmin 0001α 05N 20

Step 100

(a) (b) (c) (d) (e)

(f ) (g) (h) (i)

Figure 3 Experiment inMobileSim (a) tk 0 (b) tk 50 (c) tk 80 (d) tk 110 (e) tk 120 (f ) tk 150 (g) tk 170 (h) tk 190 (i) tk 270


where vr is the linear speed of the c minus agent or the speedrequired for the flocking motion and kv gt 0 is a designparameter

4 Simulation Results

)e effectiveness of the proposed flocking control method isillustrated by experiments in MobileSim MobileSim issoftware for simulating mobile robots and their environ-ments for debugging and experimentation with ARIA orother software that support mobile robot platforms In thisexperiment five AmigoBots are controlled to form a flockingcentering state while avoiding the obstacles )e robots areconnected by software with TCP ports from 8101 to 8105Software is coded in VS2008 )e data in the control processare stored in a text file at each sampling time and the resultsare depicted in MATLAB figures

)e initial positions of the five AmigoBots are set with(0 2000mm) (0 1000mm) (0 0) (0 minus1000mm) and(0 minus2000mm) All the headings of the AmigoBots are setwith 0deg Velocities of robots are set with 20mmsec at thebeginning

)e parameters in the method proposed above aredesigned as given in Table 1

)e initial and some intermediate processes of robots areshown in Figure 3 Obstacles are plotted with squareshexagons and triangles With sonar sensors the robot candetect the obstacles at each sampling period (500ms) )epositions of the other α-agents are shared at a period of 1 secFrom Figures 3(c)ndash3(f) it can be seen that the robots smartlyavoid the obstacles As shown in Figures 3(g)ndash3(i) all therobots attempt to assemble with a desired distance afterpassing through the obstacles which brings out a flockingcenter state

)e detailed traces of the five AmigoBots during theflocking process are depicted in Figure 4 It proves that the

designed potential function of Viα and Vi

β can regulate therelative position between robots and obstacles effectively)e robots keep a safe distance from obstacles while acrossthe gap and a desired distance from each other in the ob-stacle-free space Connection graph of the robots markedwith the red circle is shown in Figure 5 which illustrates theevolution of the flocking center state during the wholeprocess It can be seen that the five AmigoBots are driven to astable pattern of geometry Each AmigoBot has a stableneighbouring relationship and keeps a stabilized motion

For the 10 connection graphs in Figure 5 define Vαk asthe average of the 5 potential function Vi

α for each graph Ithas Vαk [16052 574 889 62 1480 50 39 39 36 62])e potential function Vi

α reduces greatly from the begin-ning and increases highly while the robots go across theobstacles and then reduces to a low value when robots are inthe obstacle-free space

times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)

Figure 4 Trajectories of robots

times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)

Figure 5 Connection graph

Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times

Figure 6 Potential function Viα


)e responses of potential functions Viα and Vi

β aredepicted in Figures 6 and 7 which illustrate how Vi

α and Viβ

affect the behaviour of the robot It can be seen that when thedistances between robots are farther than the desired dis-tance dr or the distances to the obstacles are nearer than thesafe distance ds Vi

α and Viβ have large values which impel

the robot to move to a desired position to reduce them withthe SA algorithm It has t0 Vi

α(t0) [79325

6580 1390 6640 79317](i 1 2 3 4 5) at the startingtime t0 and Vi

α(tk) [1657 1238 1682 2080 2063] atthe ending time tk 350 All the robots have approximatedthe minimal values of Vi

α

Figures 8 and 9 show the flocking behaviour of align-ment )e orientation angles and velocities of five robotsalmost converge to the same value

5 Conclusion

In this paper the flocking behaviour of the mobile robot inan unknown environment with obstacles is designed via theheuristic search algorithm With the potential functionsdefined for the α minus agent and β minus agent the mobile robot canacquire the quasi-optimal direction with the SA algorithm infinite search times Control laws for vi and ωi are alsodesigned )e system of mobile robots acts out separation

0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times

Figure 7 Potential function Viβ

θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times

Figure 8 Orientation angles of robots

Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times

Figure 9 Velocities of robots


and alignment behaviours obviously Moreover the robotscan avoid obstacles detected with sonar sensors by a safedistance Future work will address the issue of flockingcontrol of mobile robots in the field environment with thisproposed method

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 work was supported by the National Nature ScienceFoundation of China under Grant nos 61203335 and61603150

References

[1] P Antoniou A Pitsillides A Engelbrecht et al ldquoMimickingthe bird flocking behavior for controlling congestion in sensornetworksrdquo in Proceedings of the 2010 3rd InternationalSymposium on Applied Sciences in Biomedical and Commu-nication Technologies (ISABEL 2010) pp 1ndash7 Rome ItalyNovember 2010

[2] A Chapman and M Mesbahi ldquoUAV flocking with windgusts adaptive topology and model reductionrdquo in Proceedingsof the 2011 American Control Conference pp 1045ndash1050 June2011 San Francisco CA USA

[3] S A P Quintero G E Collins and J P Hespanha ldquoFlockingwith fixed-wing uavs for distributed sensing a stochasticoptimal control approachrdquo in Proceedings of the 2013American Control Conference pp 2025ndash2031 WashingtonDC USA June 2013

[4] B K Sahu and B Subudhi ldquoFlocking control of multiple auvsbased on fuzzy potential functionsrdquo IEEE Transactions onFuzzy Systems vol 26 no 5 pp 2539ndash2551 2018

[5] C Speck and D J Bucci ldquoDistributed UAV swarm formationcontrol via object-focused multi-objective sarsardquo in Pro-ceedings of the 2018 Annual American Control Conference(ACC) pp 6596ndash6601 Milwaukee WI USA June 2018

[6] M R KhanMM Billah M Ahmed et al ldquoA novel algorithmfor collaborative behavior of multi-robot in autonomousreconnaissancerdquo in Proceedings of the 2011 4th InternationalConference on Mechatronics (ICOM) pp 1ndash6 Kuala LumpurMalaysia May 2011

[7] J Toner and Y Tu ldquoFlocks herds and schools a quantitativetheory of flockingrdquo Physical Review E vol 58 no 4pp 4828ndash4858 1998

[8] C W Reynolds ldquoFlocks herds and schools A distributedbehavioral modelrdquo in Proceedings of the 14th annual con-ference on Computer graphics and interactive techni-quesmdashSIGGRAPH lsquo87 vol 21 ACM Anaheim CA USA July1987

[9] A Okubo ldquoDynamical aspects of animal grouping swarmsschools flocks and herdsrdquo Advances in Biophysics vol 22pp 1ndash94 1986

[10] C C Ioannou ldquoSwarm intelligence in fish )e difficulty indemonstrating distributed and self-organised collective

intelligence in (some) animal groupsrdquo Behavioural Processesvol 141 pp 141ndash151 2017

[11] H G Tanner A Jadbabaie and G J Pappas Flocking inTeams of Nonholonomic Agents Springer Berlin HeidelbergBerlin Heidelberg 2005

[12] T Vicsek A Czirok E Ben-Jacob I Cohen and O ShochetldquoNovel type of phase transition in a system of self-drivenparticlesrdquo Physical Review Letters vol 75 no 6 pp 1226ndash1229 1995

[13] A Kilin P Bozek Y Karavaev et al ldquoExperimental inves-tigations of a highly maneuverable mobile omniwheel robotrdquoInternational Journal of Advanced Robotic Systems vol 14no 6 2017

[14] P Bozek P Pokorny J Svetlik et al ldquo)e calculations ofJordan curves trajectory of the robot movementrdquo Interna-tional Journal of Advanced Robotic Systems vol 13 no 52016

[15] P Bozek Z Ivandic A Lozhkin et al ldquoSolutions to thecharacteristic equation for industrial robotrsquos elliptic trajec-toriesrdquo Tehnicki Vjesnik-Technical Gazettevol 23 no 4pp 1017ndash1023 2016

[16] H-T Zhang Z Cheng G Chen and C Li ldquoModel predictiveflocking control for second-order multi-agent systems withinput constraintsrdquo IEEE Transactions on Circuits and SystemsI Regular Papers vol 62 no 6 pp 1599ndash1606 2015

[17] J Gao X Xu N Ding and E Li ldquoFlocking motion of multi-agent system by dynamic pinning controlrdquo IET ControlBeory amp Applications vol 11 no 5 pp 714ndash722 2017

[18] F F Chu H Y Yang F J Han et al ldquoAnti-disturbancecontrol of flocking moving of networked systemsrdquo in Pro-ceedings of the 33rd Chinese Control Conference pp 1425ndash1429 Nanjing China July 2014

[19] N Moshtagh and A Jadbabaie ldquoDistributed geodesic controllaws for flocking of nonholonomic agentsrdquo IEEE Transactionson Automatic Control vol 52 no 4 pp 681ndash686 2007

[20] D Lee and M W Spong ldquoStable flocking of multiple inertialagents on balanced graphsrdquo IEEE Transactions on AutomaticControl vol 52 no 8 pp 1469ndash1475 2007

[21] D Panagou and V Kumar ldquoCooperative visibility mainte-nance for leader-follower formations in obstacle environ-mentsrdquo IEEE Transactions on Robotics vol 30 no 4pp 831ndash844 2014

[22] M Lindhe P Ogren and K H Johansson ldquoFlocking withobstacle avoidance a new distributed coordination algorithmbased on voronoi partitionsrdquo in Proceedings of the 2005 IEEEInternational Conference on Robotics and Automationpp 1785ndash1790 Barcelona Spain April 2005

[23] B Dai and W Li ldquoFlocking of multi-agents with arbitraryshape obstaclerdquo in Proceedings of the 33rd Chinese ControlConference pp 1311ndash1316 Nanjing China July 2014

[24] B R Trilaksono ldquoDistributed consensus control of robotswarm with obstacle and collision avoidancerdquo in Proceedingsof the 2015 2nd International Conference on InformationTechnology Computer and Electrical Engineering (ICITA-CEE) p 2 October 2015

[25] A M Burohman A Widyotriatmo and E JoeliantoldquoFlocking for nonholonomic robots with obstacle avoidancerdquoin Proceedings of the 2016 International Electronics Sympo-sium (IES) pp 345ndash350 Denpasar Indonesia September2016

[26] D Sakai H Fukushima and F Matsuno ldquoFlocking formultirobots without distinguishing robots and obstaclesrdquoIEEE Transactions on Control Systems Technology vol 25no 3 pp 1019ndash1027 2017


[27] A Hafez and S Givigi ldquoFormation reconfiguration of co-operative uavs via learning based model predictive control inan obstacle-loaded environmentrdquo in Proceedings of the 2016Annual IEEE Systems Conference (SysCon) pp 1ndash8 OrlandoFL USA April 2016

[28] G He and H Li ldquoDistributed control for multirobot systemswith collision-free motion coordinationrdquo in Proceedings of the2017 10th International Symposium on Computational Intel-ligence and Design (ISCID) vol 2 pp 72ndash76 HangzhouChina December 2017

[29] H-T Zhang C Zhai and Z Chen ldquoA general alignmentrepulsion algorithm for flocking of multi-agent systemsrdquoIEEE Transactions on Automatic Control vol 56 no 2pp 430ndash435 2011

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part i fixed topologyrdquo in Proceedings of the42nd IEEE International Conference on Decision and Control(IEEE Cat No 03CH37475) vol 2 pp 2010ndash2015 IEEEMaui HI USA December 2003

[31] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part ii dynamic topologyrdquo in Proceedings ofthe 42nd IEEE International Conference on Decision andControl (IEEE Cat No 03CH37475) vol 2 pp 2016ndash2021IEEE Maui HI USA December 2003

[32] R O Saber and R M Murray ldquoFlocking with obstacleavoidance cooperation with limited communication in mo-bile networksrdquoin Proceedings of the 42nd IEEE InternationalConference on Decision and Control vol 2 pp 2022ndash2028Maui HI USA December 2003

[33] J Cheng B Wang and Y Xu ldquoFlocking control of amigobotswith Newtonrsquos methodrdquo in Proceedings of the 2017 IEEEInternational Conference on Robotics and Biomimetics(ROBIO) pp 2372ndash2376 Macau China December 2017

[34] D H Wolpert and W G Macready ldquoNo free lunch theoremsfor optimizationrdquo IEEE Transactions on Evolutionary Com-putation vol 1 no 1 pp 67ndash82 1997

[35] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimizationby simulated annealingrdquo Science vol 220 no 4598pp 671ndash680 1983

[36] X-S Yang ldquoOptimization and metaheuristic algorithms inengineeringrdquo Metaheuristics in Water Geotechnical andTransport Engineering pp 1ndash23 2013


nonholonomic agents in 2 and 3 dimensions is addressedin [19] A geodesic control law that minimizes a mis-alignment potential and results in velocity alignment andflocking is proposed In [20] the flocking of multipleagents which have significant inertias and evolve on abalanced information graph is studied Passive decom-position method is used to incorporate this inertial effectwhich can cause unstable group behaviour )e proposedflocking control law is proven to be stable and can ensurethat the internal group shape is exponentially stabilized toa desired state

In recent years the flocking problem of mobile robotshas become an attractive subject in the research field of themultiagent system and cooperative robots In [21] thecooperative motion coordination of leader-follower for-mations of nonholonomic mobile robots under visibilityand communication constraints in known polygonal ob-stacle environments is studied Each robot can take care ofconverging to a desired configuration while maintainingvisibility with its target Flocking with obstacle avoidance isan emerging research interest in recent years [22ndash26] Anew distributed coordination algorithm with Voronoipartitions for multivehicle systems is presented in [22]which results not only in obstacle avoidance and motion tothe goal but also in a desirable geographical distribution ofthe vehicles A flocking algorithm in the presence of ar-bitrary shape obstacle avoidance is presented in [23] bothin the 2D space and 3D space In [27] learning-basedmodel predictive control approach is applied to theflocking problem of UAVs in the presence of uncertaintiesand obstacles A distributed framework that ensures reli-ably collision-free behaviours in large-scale multirobotsystems with switching interaction topologies is presentedin [28]

In practical applications such as formulation of themobile robot or surface vessels the agent model is under-actuated and the kinematic motion is constricted withnonholonomic constraints In our research project Ami-goBot is adopted as an α minus agent in the flocking controlproblem )e kinematic model of AmigoBot is a kind ofnonholonomic system in which the number of control in-puts is less than the degree of freedom and cannot betransformed into a second-order linear system It is nec-essary to redesign the control algorithms for the flocking ofAmigoBots

In this paper a control method for the flocking problemof mobile robots is proposed Simulated annealing (SA)algorithm is used to design a simple behaviour for eachrobot while flocking can be self-established SA algorithm isa probabilistic technique for approximating the globaloptimum of a given function Specifically it is a meta-heuristic to approximate global optimization in a largesearch space for an optimization problem It is often usedwhen the search space is discrete (eg the travelingsalesman problem) For problems where finding an ap-proximate global optimum is more important than findinga precise local optimum in a fixed amount of time sim-ulated annealing may be preferable to alternatives such asgradient descent

)e paper is organized as follows Section 2 gives thekinematic model of AmigoBot and the problem formulationDesign process of the flocking control law is presented inSection 3 Simulation is implemented to illustrate the ef-fectiveness of the proposed control law and results areshown in Section 4 Finally the conclusions are made inSection 5

2 Problem Formulation

21Model ofFlockingAgents )efirst model which is widelyconsidered as a flocking agent is constructed in [12] whichwas proposed by Vicsek et al to interpret the flockingphenomenon such as clustering and convection

xi(t + 1) xi(t) + vi(t)Δt

θ(t + 1) langθ(t)rangr + Δθ(1)

where xi is the position of particle i on the plane vi is thevelocity with an absolute value v and an angle given byθ(t + 1) i 1 2 middot middot middot n langθ(t)rangr is the average direction of theparticles in the neighbourhood of particle i and Δθ is therandom noise

From the perspective of theoretical research the dy-namics of agents in flocking is also often modelled as asecond-order linear system [29]

_xi vi _vi ui i 1 2 n (2)

where xi vi and ui are the position velocity and controlforce respectively

)e other second-order linear system of the flockingagent is proposed in [30 31])e agents moving on the planeare modelled with the following dynamics

_ri vi _vi ui i 1 2 n (3)

where ri (xi yi)T is the position of agent i vi ( _xi _yi)

T isits velocity and ui (ux uy)T is its control inputs )eheading angle of agent i is defined with θi arctan( _yi _xi)

22 KinematicModel of theMobile Robot Different from theagent models proposed above which are treated with thelinear system the kinematic model of the mobile robot ismuchmore complex As shown in Figure 1 the mobile robotis controlled only by the velocities of two drive wheelsAssuming that there exists no slipping motion between thewheels and the ground the dynamics of AmigoBot can bemodelled as

_xi vi cos θi

_yi vi sin θi

θ

i ωi

(4)

where vi and ωi are the linear and angular velocity re-spectively (xi yi) is the position of AmigoBot in Cartesiancoordinates and θi is the orientation angle of theAmigoBot








Vij rij

1113874 1113875 d2

r

rij

2 + log rij

2

1113874 1113875 + erij

minus dr1113872 1113873

2

(5)





Viα 1113944

jisinAVij rij

1113874 1113875 (6)





ri minus qk

(7)



Vik rik

1113872 1113873

1rik

minus1ds

1113888 1113889

2

rik

ltds

0 rik

geds

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Define

(a)

L2

L2(x y)

Drive wheel

Caster

(b)



0

10

20

30

40

50

60

Vij

05 1 15 2 25 30r



Viβ 1113944

kisinOi

Vik rik

1113872 1113873

(9)



αand Vi



Vi Viα + V

iβ (10)



Vi foralltk (11)









is defined with





(4) 1113957ri⟵ ri













zθ lowastizyi

xi minus xlowasti



(14)


with


zθ lowastizxi

cos θlowasti + vi

zθ lowastizyi

sin θlowasti (15)


in (15) it has

θe



_Vθe minuskwθ






vi 1


vj + kv δri

Vi (17)


Variables Value



SA algorithm

T 100Tmin 0001α 05N 20

Step 100

(a) (b) (c) (d) (e)

(f ) (g) (h) (i)















times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)


times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)


Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times





α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References














































Vij rij

1113874 1113875 d2

r

rij

2 + log rij

2

1113874 1113875 + erij

minus dr1113872 1113873

2

(5)





Viα 1113944

jisinAVij rij

1113874 1113875 (6)





ri minus qk

(7)



Vik rik

1113872 1113873

1rik

minus1ds

1113888 1113889

2

rik

ltds

0 rik

geds

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Define

(a)

L2

L2(x y)

Drive wheel

Caster

(b)



0

10

20

30

40

50

60

Vij

05 1 15 2 25 30r



Viβ 1113944

kisinOi

Vik rik

1113872 1113873

(9)



αand Vi



Vi Viα + V

iβ (10)



Vi foralltk (11)









is defined with





(4) 1113957ri⟵ ri













zθ lowastizyi

xi minus xlowasti



(14)


with


zθ lowastizxi

cos θlowasti + vi

zθ lowastizyi

sin θlowasti (15)


in (15) it has

θe



_Vθe minuskwθ






vi 1


vj + kv δri

Vi (17)


Variables Value



SA algorithm

T 100Tmin 0001α 05N 20

Step 100

(a) (b) (c) (d) (e)

(f ) (g) (h) (i)















times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)


times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)


Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times





α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References








































Viβ 1113944

kisinOi

Vik rik

1113872 1113873

(9)



αand Vi



Vi Viα + V

iβ (10)



Vi foralltk (11)









is defined with





(4) 1113957ri⟵ ri













zθ lowastizyi

xi minus xlowasti



(14)


with


zθ lowastizxi

cos θlowasti + vi

zθ lowastizyi

sin θlowasti (15)


in (15) it has

θe



_Vθe minuskwθ






vi 1


vj + kv δri

Vi (17)


Variables Value



SA algorithm

T 100Tmin 0001α 05N 20

Step 100

(a) (b) (c) (d) (e)

(f ) (g) (h) (i)















times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)


times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)


Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times





α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References














































zθ lowastizyi

xi minus xlowasti



(14)


with


zθ lowastizxi

cos θlowasti + vi

zθ lowastizyi

sin θlowasti (15)


in (15) it has

θe



_Vθe minuskwθ






vi 1


vj + kv δri

Vi (17)


Variables Value



SA algorithm

T 100Tmin 0001α 05N 20

Step 100

(a) (b) (c) (d) (e)

(f ) (g) (h) (i)















times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)


times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)


Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times





α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References




















































times104

Robot1Robot2Robot3

Robot4Robot5

ndash5000

0

5000

10000

15000

y (m

m)

05 1 15 20x (mm)


times104

ndash2000

0

2000

4000

6000

8000

10000

12000

14000

y (m

m)

05 1 15 20x (mm)


Viα

Robot1Robot2Robot3

Robot4Robot5

0

200

400

600

800

1000

50 100 150 200 250 3503000Times





α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References










































α and Viβ




α(t0) [79325



α


5 Conclusion


0

1

2

3

4

5

6

7

~Vb

times105

Robot1Robot2Robot3

Robot4Robot5

50 100 150 200 250 300 3500Times


θ1θ2θ3

θ4θ5

ndash150

ndash100

ndash50

0

50

100

150

Orie

ntat

ion

angl

es (d

eg)

50 100 150 200 250 300 3500Times


Robot1Robot2Robot3

Robot4Robot5

45

50

55

Vel

ociti

es (m

ms

) Vi

50 100 150 200 250 300 3500Times




Data Availability




Acknowledgments


References









































Data Availability




Acknowledgments


References








































Flocking Control of Mobile Robots with Obstacle Avoidance Based...

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