robot formations

8/16/2019 Robot Formations

1/10

Robot formation modeling and control based on the relative

kinematics equations

F. Belkhouchea, K. Bendjilalib and B. Belkhouche c

aTexas A&M Intl University

[email protected] Lehigh University,cUAE University

December 17, 2007

Abstract

This paper deals with the problem of modeling, ini-tialization, and control of mobile robots formation.We suggest to use a new family of methods that con-sists of a combination between classical guidance lawsand kinematics rules. These methods allow modelingand controlling a dynamic robotic formation usingsets of differential equations that give the relative mo-tion between the robots. These differential equationsgive the range rate and the visibility angle between

leaders and followers. Graph theory is used to storethe relationship leader-follower and plan the forma-tion by using three different matrices. The configu-ration of the formation is based on these matrices.Initialization of formation is also considered, wheredifferent approaches are suggested. Because of thenature of the kinematics laws, it is easy to model,initialize, and keep any formation shape. Simulationis provided to illustrate the method.

1 Introduction

Swarm intelligence is an important part of distributedartificial intelligence, where teams of autonomousagents are used to accomplish various difficult andcomplex tasks in a cooperative fashion. Over the pastdecade, different models of team formation and coop-eration have been suggested and studied in the liter-ature. The pattern formation problem can be defined

as a cooperative behavior between various robotsthat allows the robots to move in a stable configu-ration. Controlling the formation of air and groundvehicles is becoming an important research topic inthe robotics and control communities. The ability of mobile robots to navigate autonomously in a stableconfiguration and avoid obstacles is central to vari-ous applications in both the military and civilian do-mains. These applications range over transportation,exploration, surveillance, and rescue. Various multi–robot configurations are used for distributed sensing

and coverage. Pattern formation is observed in birds(flocks), fish (schools) and herds. This topic has at-tracted researchers from different areas such as con-trol theory, computer science, biology and physics,where, in general the agents (particles) are requestedto perform different types of operations and motions.Various topics are discussed in the literature:

1. Modeling different types of formations of multi-robot systems.

2. Design control laws for formation keeping, sta-

bilization, and initialization. Formation in com-plex and unstructured environments.

3. Formation under different types of constraintsand limitations, such as the nonholonomic con-straint, limited sensing and communications ca-pabilities, and large scale formation.

1


2/10

Formation modeling goal’s is to model the forma-

tion using mathematical and/or graphical tools. Thistask is a necessary step for the control algorithm.The main goal of the formation model is to estab-lish the relationship between different elements in theformation. Even though the formation is seen as awhole, modeling allows to perform coordination be-tween elements. Coordination is a real time processthat can be accomplished based on sensing or com-munication. Formation modeling has the followingtask objectives:

1. Deduce the relationship between different agentsin the formation, coordination, and communica-

tion channels. Capture the architecture of theformation

2. Decide the formation modes by establishing aswitching mechanism between modes in realtime.

Formation Control aims to develop control archi-tectures and algorithms that allow to plan and con-trol the formation. Path following is an important as-pect of formation control, where it is necessary to en-sure that the trajectory of each vehicle is tracked with

a desired precision by the actuators system. Collisionavoidance is another important aspect that must betaken into account. The control block requires infor-mation regarding the formation architecture and theenvironment. This information is obtained throughsensing or communications between agents. Forma-tion control satisfies the following task objectives:

1. Robustness: Formation should be robust evenunder faulty conditions. This includes robust-ness to external noise and unmodeled dynamics.

2. Autonomous operations and mode switching,since the vehicles move in different modes, itis necessary to switch between modes in an au-tonomous fashion.

3. Cooperative behavior between agents and thestability of the formation. Dynamic and kine-matics constraints must be taken into account.

Here we provide an analytical study with computa-

tional tools of methods that allow the representationand construction of formations of many vehicles per-forming various types of motions. The modeling isbased on the relative kinematics equations, and graphrepresentation. This approach allows to model differ-ent architectures for formation dynamics. The con-trol law is distributed and allows to perform both loseand tight formations. Initialization of the formationis also discussed based on linear navigation functions.Similar control laws are used to control mobile forma-tion. The collision cone approach is used for collisionavoidance. Various methods are discussed in the lit-erature to solve various problems in formation and

cooperative control. The most important approachesare based on different strategies: Model-based meth-ods and behavioral- based methods. A nice com-promise would be to use model-based methods andsensor-based methods combined together ([1]) so thatthe planning and modeling are based on the sensor’sinformation.

Multi-Robot systems are used for cooperativetransportation. This is a difficult task that requirescoordination between agents. This problem is consid-ered by various authors ([6, 7, 8, 16, 17]) . In ([2]),humanoid robots are used to learn and perform a

cooperative transportation task. Transportation of large object is considered in ([3]). Stabilization is dis-cussed in ([6, 5, 4]). Stability of the motion and taskallocation are very important in cooperative trans-portation.

A potential field approach that is quite similar tothe classical potential field is suggested in ([19]) tocontrol a distributed autonomous multi-robot sys-tem. Here artificial force laws are defined betweenpairs of robots or robot groups. The authors calledthe method social potential field, since it deals withmultiple agents. Social potential fields methods arealso used in ([20]), where the authors particularly

consider more challenging cases such as imperfectsensory input. Note that not only wheeled mobilerobots are considered for formation, but other typesof robots as well. For example, formation of walkingrobots is considered in ([11]), air vehicles are consid-ered in ([12]). Navigation of multiple robots is alsowidely considered ([9]). Leader–follower approaches

2


3/10

are widely used for modeling and controlling different

types of formations ([13, 14]). Graph–based methodsare widely used to model formations ([15]).Our goal in this paper is to contribute to the prob-

lem of robotic formation modeling, initialization andcontrol. We consider both static and dynamic ini-tialization and control. Our strategy is based on thecombination of the geometric rules with the kinemat-ics equations.

2 Kinematics and relative ge-

ometry

Our method is based on the relative kinematics andgeometry between the robots in the formation. Inthis section we define the geometric quantities usedin modeling and controlling the robot formation. Wealso derive kinematics models for the robots in polarcoordinates. This polar representation allows to facil-itate the modeling and controlling of the formation.The working space is attached to a global referenceframe of coordinates whose origin is point O. Sup-pose we have N robots in the formation. Robot Ri(i = 1,...,N ) is moving according to the followingkinematics equations

ẋi = vi cos θiẏi = vi sin θiθ̇i = ωi

(1)

where xi, yi represent the coordinates of the refer-ence point of robot Ri in the global reference frameof coordinates, vi represents the linear velocity of Ri,and θi represents its orientation angle in the refer-ence frame, ωi is the angular velocity of the robot.The kinematics model given by (1) is a simple modelbut it captures the nonholonomic constraint. Eachrobot is characterized by two control inputs: the lin-ear velocity and the orientation angle. The workingspace is cluttered with obstacles denoted by Bi. Thefollowing geometric quantities will be used for themodeling and control.

1. We define the visibility line between robots Riand Rj as the imaginary straight line that starts

Figure 1: Relative geometry between two robots

from Ri and is directed towards Rj . This line is

denoted by S ij .

2. Based on the visibility line, we define the visibil-ity angle between Ri and Rj . This is the anglebetween the reference line (positive x-axis) andthe visibility line. The visibility angle is denotedby γ ij .

3. The relative distance between the referencepoints of the robots in the Cartesian coordinates

is given by

xij = xi − xj , yij = yi − yj (2)

The relative range is given by sij . These geometricquantities are illustrated in figure 1.

The kinematics equations in (1) represent the mo-tion of the robots in the Cartesian frame of refer-ence. Our goal is to derive an equivalent model inpolar coordinates. Using a simple transformation of coordinates, it is possible to write

si = x2i + y2i , γ i = a tan2(yi/xi) (3)where si is the distance from the origin to the refer-ence point of the robot, and γ i is the visibility angleO − Ri. By taking the derivatives with respect totime, we obtain

ṡi = ẋixi + ẏiyis2i γ̇ i = −ẋiyi + xi ẏi

(4)

3


4/10

Knowing that

xi = si cos γ i, yi = si sin γ i (5)

we obtain

ṡi = vi cos (θi − γ i)γ̇ i = vi sin (θi − γ i) s

−1

i

(6)

The kinematics model given in (6) is the polar equiv-alent of the Cartesian model given in (1). It is alsoimportant to note that, the visibility angle, the visi-bility line, and the range are defined for the obstaclestoo. The range between robot Ri and obstacle B isdenoted by sib, and the visibility angle is given by γ ib.The pair (sib, γ ib) is used for the collision avoidance.

3 Modeling the formation

Modeling the formation is an important step in theprocess. The relationship between two robots in theformation is established based on two variables: therange and the visibility angle. These two variablesgive the relative position between the robots with re-spect to each other. Consider the relative velocitybetween Ri and Rj given by

vij = vi − vj (7)

In polar coordinates, the relative velocity is expressedas follows

ṡij = vi [cos (θi − γ ij) − k cos(θj − γ ij)]γ̇ ij = vi [sin (θi − γ ij) − k sin(θj − γ ij)] s

−1

ij

(8)

where k is the speed ratio, it is given by k = vjvi

.System (8) is a system of nonlinear differential equa-tions. The closed form solution for this system isdifficult but possible in certain particular situations.The first equation in system (8) gives the solution forthe range. The second equation gives the rate of turnof the robots with respect to each other. If γ̇ ij = 0,then the motion of Ri seen by Rj is a straight line.This means that the visibility lines between Ri andRj are parallel to the initial visibility line. In mostcases, a stable configuration of the formation is char-acterized by constant range and constant line of sightangle (γ̇ ij = 0) . Two different approaches for control-ling the formation are possible: (1) local leader ap-proach, and (2) global leader approach.

3.1 Local leader approach

This approach is decentralized or distributed ([10]).The model in this case establishes a relationship be-tween each leader and its follower. Clearly, a followercannot have more than one leader. Each agent usesa decentralized control law to determine its motionand perform a given relative motion with respect toits local leader. This approach is more flexible andmore robust to external noise. It is likely that flocks,herds, and schools perform their sophisticated motion([18]) by using distributed local control paradigms.

3.2 Global leader approach

The formation has a single global leader, and a cen-tral law (or laws) for the entire system. The centralunit oversees the whole system. In our approach, theformation is modeled by 2(N − 1) equations. Thisapproach suffers from various drawbacks such as thelack of flexibility and high complexity.

3.3 Formation planning

Planning the formation deals with the shape andthe interconnections between robots. The formation

shape is characterized by the ranges and the anglesbetween different elements and agents. Our kinemat-ics model given by (8) can be easily adapted to cap-ture any formation shape, since it is based on angleand ranges. The formation is modeled as an orientedgraph G (V, E ), where V = {v1, v2,...,vn} is a set of vertices, E = {e1, e2,...,en} is a set of edges. Eachrobot can be viewed as a node in the graph. Thisallows to formally define formation of various vehi-cles while considering the uniqueness of the forma-tion. The architecture of the formation is translatedto an adjacent matrix and the formation is repre-sented by three square matrices. The first matrix

A gives the relationships or interconnections leader–follower. The size of the matrix is n ×n. The entriesare either zero or one as follows:aij = 1, if there is a leader follower relationship be-tween Ri and Rj.aij = 0, if there is no leader follower relationship be-tween Ri and Rj.

4


5/10

The second matrix is the distance matrix. This ma-

trix gives the range between each leader—follower.This matrix is denoted by D. The third matrix isthe visibility angle matrix. This matrix gives the vis-ibility angle between each leader–follower. This ma-trix is denoted by Γ. The dynamics of the formationis captured by a sequence of graph transformations,which affects matrix A. The shape of the formationis changed by changing matrices D and Γ. The graphformulation allows to split and rejoin the formationeasily. It allows to add or delete robots from the for-mation. Also, this approach presents good propertiesin terms of the flexibility, where it allows to catch andsupport diverse formation shapes and geometries. To

achieve the desired formation, we need to control therelative position, and the line of sight angle of thefollower to the desired values

sdesij , γ

desij

. Sugihara

and Suzuki [21] suggested an algorithm for patternformation where each robot has the global positionsof all the others. In our approach, each robot is re-quired to know the local position of its local leaderonly.

4 Initialization of formation

Initialization of the formation is a difficult task, es-pecially in the presence of kinematics or dynamicsconstraints. Most researchers discuss the problem of formation keeping, but omit the initialization of theformation. In our approach, various geometric con-figurations of the formation are obtained based onthe linear configuration. This approach allows to ob-tain complex systems from the simplest configuration(line formation). Note that even a simple line forma-tion becomes a difficult task under the nonholonomicconstraint. Figure 2, shows two different line configu-rations with different orientation angles of the robots.The configuration in figure 2–b is more convenient for

a robotic convoy formation. This type of formationcan be easily handled by using our strategy [23].

Our strategy for the initialization of formationsof nonholonomic robots is based on a new approachcalled linear navigation functions [22]. For more de-tails, the reader is referred to [22]. Let θi0 be theinitial orientation angle of robot Ri, and θif be the

Figure 2: Example of two final line configurations

final desired orientation angle. Consider figure 3, letP i be the desired position of robot Ri on the forma-tion line. The coordinates of P i in the reference frameof coordinates are (x pi, y pi). The distance between Riand P i is given by s pi, and the visibility angle is givenby γ pi. It is easy to prove that s pi and γ pi vary asfollows

ṡ pi = −vi cos (θi − γ pi)γ̇ pi = −vi sin (θi − γ pi) s

−1

pi

(9)

The control laws for the linear velocity and orienta-

tion angle for robot Ri are given by

vi = ms pi (10)

and

θi = 2γ pi + c0e−at + ci (11)

respectively, where m > 0 is chosen such that vmax >msmax, where vmax and smax are the maximum val-ues of vi and s pi, respectively. a is a real positiveconstant. c0 and c1 are constant numbers that sat-isfy

θi0 = 2γ pi0 + c0 + ci

θif = −c1(12)

The first term on the right hand side in equation (11)represents a proportionality term, the second termrepresents a heading regulation term, and the thirdterm gives the value of the robot desired final orienta-tion angle. The control laws given by (10) and (11) al-low to drive Ri from its initial configuration given by

5


6/10

Figure 3: Illustration of the final desired configura-

tion in a line formation

(xi0, yi0, θi0) to its final configuration (xif , yif , θif ).Note that under equations (10) and (11), we have

s pi → 0, γ pi → −ci, θi → −ci (13)

This state characterizes the desired final configura-tion. We have the following result.

Proposition:Under the control laws given by (10) and (11) the

robot reaches its final position P i on the formation

line. Furthermore, the robot’s final orientation angleis θif = −ci.Proof The proof that Ri reaches its final position on the

formation line can be achieved easily by proving that(0,−ci) is an asymptotic equilibrium point for system(9). The closed loop system is given by

ṡ pi = −ms pi cos (γ pi + c0e−at + ci)

γ̇ pi = −m sin(γ pi + c0e−at + ci)

(14)

The equilibrium position of the system is given by(0,−ci − c0e

−at). Clearly, the heading regulation

terms go to zero with time, resulting in an equilib-rium position given by (0,−ci) . Our goal is to provethat this equilibrium position is asymptotically sta-ble. The classical linearization near this equilibriumposition gives the following matrix

DF =

−m 00 −m

(15)

Figure 4: Construction of a circle formation from a

line formation

Since the eigenvalues of DF are negative, the equi-librium position is asymptotically stable, therefore

s pi → 0, γ pi → −ci

and the final position of the robot’s orientation angleis

θi → 2γ pi + ci = −ci (16)

4.0.1 Line Formation

Line formation is accomplished by using the controllaw described above by moving the robots one af-ter one. An algorithm for collision avoidance is inte-grated with the control law.

4.0.2 Circle formation

We construct circle formation from line formation bymoving the robots one after one as shown in figure4. Half circle and other arc formations are similar tocircle formation.

4.0.3 Polygon formation

Our approach is adequate for polygon formation.This is due to the fact that our approach uses anglesand ranges, which can easily characterize any type of polygons. Figure 5 shows a diamond formation. Theformation is characterized by angles γ 1, γ 2, γ 3, andranges s1, s2, s3. The goal here is to drive the robots

6


7/10

Figure 5: Illustration of a diamond configuration

R1, R2, R3 and R4 to their final position in the forma-tion, i.e., P 1, P 2, P 3, and P 4, respectively. The controllaws suggested for line formation work in this case aswell.

5 Initialization of mobile for-

mation

Initialization of mobile formation is more difficultthan static formation. A leader follower approach is

used for the initialization of mobile formations. Eachfollower tracks its desired position with respect to itsleader. Once the follower reaches its position in theformation, it follows another control law to keep theformation.

5.0.4 Initialization based on the direct ap-

proach

The follower knows its position in the formation,therefore it navigates towards this position, whichis represented by a virtual moving point denoted byP i (t) = (xi (t) , yi (t)). The relative kinematics equa-

tions between the follower and the virtual movingpoint is given

ṡ pi = v p cos (θ p − γ pi) − vi cos (θi − γ pi)s piγ̇ pi = v p sin (θ p − γ pi) − vi sin (θi − γ pi)

(17)

where s pi, γ pi are the distance and visibility angle be-tween the robot and the virtual moving point P i (t).

The control law for the robot follower is

θi = 2γ pi + c0e−at + c1 (18)

The robot can reach its relative position only if it isfaster than the virtual point P i.

5.0.5 Initialization based on the relative ap-

proach

The follower aims to reach a given position withrespect to its leader. Thus, the relative equationsleader-follower (8) are used. The control law in thiscase is given by

vi = m

sij − sdesi

(19)

andθi = 2γ ij + c0e

−at + c1 (20)

where sdesi is the desired relative range leader followerand −c1 is the desired visibility angle leader–follower.

6 Formation control

The aim here is to control the formation to keepa given configuration. For each robot follower, we

derive two control laws for the orientation and thespeed.

6.0.6 Deviated pursuit approach

Let Ri be a leader robot and Rj is the follower, thenthe position of Rj in the formation with respect to Riis described by sij and γ ij . In the deviated pursuitapproach, there is a constant angle between the lineof sight and the velocity vector of the follower. Thisgives the follower’s orientation angle as follows

θj = γ ij + c0 (21)

where c0 is a constant deviation angle. The secondstep is to design a control law for the speed of Rj tokeep constant distance with Ri. A constant distancebetween Ri and Rj corresponds to ṡij = 0, whichgives

vi cos (θi − γ ij) − vj cos (c0) (22)

7


8/10

from which we get the control input for the velocity

of Rj as follows

vj = vi cos (θi − γ ij)

cos c0(23)

6.0.7 Proportional angle approach

This approach is quite similar to the deviated pur-suit; however, the follower’s orientation angle is pro-portional to the line of sight angle that joins Rj andits leader. That is

θj = K γ ij (24)

with 1 ≤ K ≤ 2. In order to keep the visibilityline between the leader and the follower constant, thefollower controls its linear velocity as follows

vj = vi cos (θi − γ ij)

cos Kγ ij(25)

7 Simulation

Here, we show simulation examples to illustrate themethod. Initialization and control of various types of formations is considered.

Example 1: This example shows an illustrationof the initialization of a line formation, where therobots move parallel to each other. Four robots start-ing from different initial configurations aim to reachtheir final configurations with a final orientation an-gle equal to -45 degrees. Linear navigation functionsare used to accomplish the task. This scenario isshown in figure 6.

Example 2: Linear navigation functions are also

used here to initialize a static diamond formationwith a final orientation angle equal to zero. Thisscenario is shown in figure 7.

Example 3: This example shows the initializationof a mobile line formation using the deviated pursuit.The same technique is used to keep the formation.This scenario is shown in figure 8.

Figure 6: Initialization of a static line formation usinglinear navigation functions

Figure 7: Initialization of a static diamond formationusing linear navigation functions

8


9/10

Figure 8: Illustration of initialization and control of

a line formation using the deviated pursuit

8 Conclusion

We presented a method for formation initialization,control, and keeping. Our method is based on therelative kinematics equations. The modeling is ac-complished by using graph theory. The configurationof the formation is described by using three differentmatrices: One for the relationship leader–follower.The second one is for the range leader–follower, andthe third one is for the visibility line leader–follower.

Initialization of different types of formation is ac-complished using linear navigation functions. Themethod is illustrated using extensive simulation.

References

[1] H. Bunke, T. Hanade, and H. Noltemeier, “Mod-eling and planning for sensor based intelligentrobots systems”, Series in Machine Perceptionand Artificial Intelligence , vol. 21, World Scien-tific, 2002.

[2] Y. Inoue, T. Tohge, and H. Iba, “Learning forcooperative transportation by autonomous hu-manoid robots”, in Studies in Fuzzyness and SoftComputing, N. Nedjah and L. Mourette, editors,Springer, 2005.

[3] A. Yamashita, M. Fukuch, J. Ota, T. Arai andH. Asama, “ Motion planning for cooperative

transportation of a large object by multiple mo-

bile robots in a 3D environment”, in the Pro-ceedings of IEEE International Conference onRobotics and Automation, April 2000, San Fran-cisco, pp. 3144 - 3151.

[4] A. Zhu and X. Yang, “Path planning of multi-robot systems with cooperation”, in the Pro-ceedings of IEEE International Symposium onComputational Intelligence in Robotics and Au-tomation, July 2003, pp. 1028 - 1033.

[5] L. Yangmin and Xin Chen, “ Stability onmulti-robot formation with dynamic interaction

topologies”, in the Proceedings of IEEE/RSJ In-ternational Conference on Intelligent Robots andSystems, August 2005, pp. 394 - 399.

[6] K. Tsuji and T. Murakami , “Load distribu-tion control for cooperative transportation bymultiple mobile robots”, in the Proceedings of IEEE International Workshop on Advanced Mo-tion Control, March 2004, pp. 249-252

[7] N. Miyata, J. Ota, Y. Aiyama and T. Arai,“Real-time task assignment for cooperativetransportation by multiple mobile robots”, in

the Proceedings of IEEE/RSJ InternationalConference on Intelligent Robots and Systems,October 1999, pp. 1167 - 1174.

[8] A. Yamashita, T. Arai, Jun Ota and H. Asama,“Motion planning of multiple mobile robots forCooperative manipulation and transportation”,IEEE Transactions on Robotics and Automa-tion, vol. 19, n. 2, pp. 223 - 237.

[9] D.T. Pham and Dayal R. Parhi , “Navigation of multiple mobile robots using a neural networkand a Petri Net model”, Robotica, Volume 21,Issue 01, Jan 2003, pp 79-93 .

[10] K. Munawar, M. Esashi and M. Uchiyama, “Anapproach towards decentralized control of co-operating non-autonomous multiple robots ”,Robotica, Volume 18, Issue 05, Sep 2000, pp 495-504.

9


10/10

[11] Huosheng Hu and Dongbing Gu, “A Multi-

Agent System for Cooperative Quadruped Walk-ing Robots”, in the Proceedings of the IASTEDInternational Conference Robotics and Applica-tions, August 14-16, 2000 , Honolulu, Hawaii,pp. 318-064 -1.

[12] K.M. Krishna, H. Hexmoor, S. Pasupuleti andJ. Llinas, “Parametric control of multiple un-manned air vehicles over an unknown hostile ter-ritory”, in the Proceedings of the InternationalConference on Integration of Knowledge Inten-sive Multi-Agent Systems, April 18-21, 2005, pp.117 - 121.

[13] L. Consolini,F. Morbidi, D. Prattichizzo andM, Tosques, “A Geometric Characterization of Leader-Follower Formation Control”, in the Pro-ceedings of IEEE International Conference onRobotics and Automation, April 2007, pp. 2397-2402.

[14] S. Marianne and G. Dongbing, “A Fuzzy Leader-Follower Approach to Formation Control of Mul-tiple Mobile Robots ”, in the Proceedfings of IEEE/RSJ International Conference on Intelli-gent Robots and Systems, October 2006, pp.2515-2520.

[15] Jaydev P. Desai, “A graph theoretic approachfor modeling mobile robot team formations”,Journal of Robotic Systems, Volume 19, Issue11, 2002, pp 511-525 .

[16] X. Yang, K. Watanabe, K. Kiguchi and K.Izumi, “Coordinated transportation of a singleobject by two nonholonomic mobile robots”, Ar-tificial Life and Robotics, Volume 7, Numbers1-2, March, 2003, pp. 48-54.

[17] Majid Nili Ahmadabadi, M. R. Barouni

Ebrahimi and Eiji Nakano, “Compliance: en-coded information and behavior in a team of cooperative object-handling robots”, AdvancedRobotics, Volume 17, Number 5 / August, 2003,pp. 427-446.

[18] C. W. Reynolds, “Flocks, Herds, and Schools:A Distributed Behavioral Model”, Computer

Graphics, volume,21, issue 4, July 1987, pp. 25-

34.[19] J. Reif and H. Wamg. “Social potential fields:. A.

distributed behavioral control for autonomousrobots”, Robotics and Autonomous Systems,volume 27, pp. 171-194.

[20] Donald D. Dudenhoeffer and Michael P. Jones,“A formation behavior for large-scale micro-robot force deployment”, in the Proceedingsof the 32nd Winter Conference on Simulation,Florida, 2000, pp. 972 - 982.

[21] K. Sugihara and I. Suzuki, “Distributed algo-

rithms for formation of geometric patterns withmany mobile robots”, Journal of Robotic Sys-tems, volume, 13, issue 3, 1996, pp. 127–139.

[22] Fethi Belkhouche and Boumediene Belkhouche,“Wheeled mobile robot navigation using the pro-portional navigation”, Advanced Robotics, Vol-ume 21, Number 3-4,, 2007, pp. 395-520.

[23] Fethi Belkhouche and B. Belkhouche, ”Modelingand controlling a robotic convoy using guidancelaws strategies”, IEEE Transactions on Systems,Man and Cybernetics, Part B, Volume 35, Num-

ber 4, pp. 813-825, 2005.

10

robot formations

Documents