06268350

7/30/2019 06268350

1/11

IEEE TRANSACTIONS ON ROBOTICS, VOL. 28, NO. 6, DECEMBER 2012 1335

Hierarchical Formation Control Based on a VectorField Method for Wheeled Mobile Robots

Ji-Wook Kwon and Dongkyoung Chwa

AbstractThis paper proposes a hierarchical formation con-trol using a target tracking control law based on the vector fieldmethod such that a decentralized and flexible formation controlcan be achieved without additional motion planning. Previously,many researchers have dealt with the control laws for the rigid for-mation, where the line of sight toward the leader is controlled forthe leaderfollower formation control. However, a width change ora collision of the formation can occur since a limited motion of therigid formation can occur when the formation control maintainsthe line of sight. Therefore, the formation of multiple mobile robotsis required to be flexible, keeping the width and curvature of theformation. To this end, a formation control law based on a vectorfield method is proposed, and a hierarchical formation structure is

introduced in such a way that it consists of a line formation and acolumn formation based on the leaderfollower formation strategy.First, a subgroup, which consists of several robots, is generated us-ing the line formation, and then, the overall formation structure isconstructed from several subgroups using the column formation.Finally, we show the stability of the whole formation. The stabil-ity analysis and simulation results of the proposed hierarchicalformation control using this vector field method are included todemonstrate the practical applicability of the proposed method.

Index TermsFlexible formation, hierarchical formation,leaderfollower control, mobile robots, vector field method.

I. INTRODUCTION

FORMATION control has been studied by many researchgroups with various methods since they can achieve a task

better than a single-robot system. These works on formation

control show that several advantages, such as a reduced cost, ro-

bustness, improved performance, and efficiency, can be achieved

by using a multiple-robot system [1]. The formation control

method can be efficiently employed in various applications,

such as the manipulation of large objects [2], [3], an intelligent

highway system [4], [5], surveillance systems [6], flight forma-

tion systems [7], [8], and surface vehicle formation systems [9].

Since the formation control algorithm can improve the perfor-

mance and efficiency of these systems, the formation control

Manuscript received June 30, 2011; revised December 9, 2011 and March26, 2012; accepted June 24, 2012. Date of publication August 14, 2012; dateof current version December 3, 2012. This paper was recommended for publi-cation by Associate Editor S. Carpin and Editor D. Fox upon evaluation of thereviewers comments. This work was supported by the Basic Science ResearchProgram through the National Research Foundation of Korea funded by theMinistry of Education, Science and Technology (2012006233).

J.-W. Kwon is with the Korea Institute of Industrial Technology, Ansan 426-171, Korea.

D. Chwa is with the Department of Electrical and Computer Engineering,Ajou University, Suwon 443-749, Korea (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2012.2206869

of multiple robots has been studied via various methods, such

as virtual structure methods [3], behavior-based methods [10],

and leaderfollower methods [1], [11][13]. The virtual struc-

ture method maintains a rigid geometric relationship between

the robots and a reference frame. The behavior-based method

defines the group behavior as a combination of the behavior

of each member. In the leaderfollower method, the followers

are controlled to follow the motion of the leader. The followers

usually maintain a line of sight toward the leader and the other

robots whether they use visual sensors or not [13]. Even when

they are in wireless communication environment, the member

robots keep the formation using the method based on the lineof sight. In addition, the width of the formation can increase

abruptly with a turn, because of the decreasing radius of the

follower trajectory. This change in the width of the formation

can result in collisions unless an additional motion planning

algorithm is employed. Therefore, the method keeping the line

of sight cannot provide a flexible formation structure without

additional motion planning.

The formation can be classified as either a rigid or a flexi-

ble one [14]. In many previous studies [1], [11][13], the rigid

formation was frequently used because the representation and

control of the formation becomes easier. However, a rigid forma-

tion can suffer from collisions or a limited mobility, especiallyat corners or narrow passages when the formation size is bigger

than the corner or the passage [14]. In the case of the flexible

formation control, however, the formation of the multiple robots

can be adjusted, depending on the size and shape of the corner,

road, or corridor in order to avoid collisions [14].

To achieve flexible formation control, additional motion plan-

ning and path generation was used in [14] and [15]. Since all

the robots in the formation should know the reference trajectory

information of the other robots for additional motion planning

and path generation, this formation control algorithm requires

a wide bandwidth communication channel and fast processors,

and therefore, it becomes infeasible if the number of the mem-

bers of the formation increases to too great a value [1].

In this paper, we propose a novel hierarchical formation con-

trol law based on the vector field method [18], [19]. In some of

previous research, each wheeled mobile robot (WMR) is con-

trolled for the whole formation. In addition, the hierarchical

formation control was studied in [16] and [17], which, how-

ever, handles the hierarchical formation structure where each

robot follows the other robot using tree structure without group-

ing and considering the flexibility. In the previous results on

the hierarchical structure in [26], the role of each subgroup

and the robot in the hierarchy is unclear, and furthermore, the

flexibility cannot be obtained. On the other hand, as smaller

1552-3098/$31.00 2012 IEEE

7/30/2019 06268350

2/11

1336 IEEE TRANSACTIONS ON ROBOTICS, VOL. 28, NO. 6, DECEMBER 2012

Fig. 1. Hierarchical formation structure. (a) Line formation. (b) Column formation. (c) Hierarchical formation.

subgroups, which construct the whole formation, can be con-

sidered to be a single WMR in the proposed method, a basic for-

mationcan be easily maintained, even if several WMRs aremiss-

ing. In other words, when the hierarchical formation represented

by subgroups is achieved, we can expect that robust perfor-mances, such as the expanded abilities of sensing and commu-

nication, can be achieved even if the whole formation cannot be

maintained.

Accordingly, we introduce the hierarchical structure which

can describe the role of each subgroup and robot more clearly

and also give the flexibility of the formation using a line and

column formation for formation control. The member robots

construct the subgroups which can be represented as the line

formation, and the subgroups construct a group formation by

maintaining the column formation. This way, the polygonal for-

mation can be described by using a combination of column and

line formations. The hierarchical formation structure introduced

in this paper is depicted in Fig. 1. As shown in Fig. 1, in the

line formation, the robots are placed parallel to each other along

the same column, and in the column formation, the robots are

placed along the same line. The more general forms of polygo-

nal formation can be achieved by the appropriate combination

of the line and column formations, as will be shown later in

Section IV. Due to the hierarchical structure, we can obtain the

flexible formation, as depicted in Fig. 1(c).

To maintain the line and column formations for the whole

polygonal formation, we control the robots based on the vec-

tor field method. That is, we generate the desired position of

each robot with respect to the leader of the subgroup to main-

tain the line formation and determine the subgroup positionfrom the preceding subgroup information. This way, the fol-

lower subgroups can be positioned on the route of the leader

subgroup, and the mobile robots in the subgroups can maintain

the subgroup structure. Similarly, a potential-field-based forma-

tion control algorithm was proposed, which could not achieve

the grouping and flexibility [24], [25].

Accordingly, thecontribution of this paper canbe describedas

follows. First, the flexibility of formation can be achieved with-

out additional algorithms unlike the previous works. In virtue

of the flexibility by the vector field method, the member robots

can maintain a curvature even when the formation is in a turning

motion. As mentioned before, this cannot be seen in the results

using the rigid formation which keeps the line of sight [11][13].

The proposed formation control law and hierarchical structure

can maintain the width and flexibility of formation, as can be

seen in Fig. 1(c), without the additional algorithm. Second, the

subgroups can follow the route of the leader group. It can pro-vide the advantage that the followers can avoid the collision in

some particular cases when the robots move with a formation

where the width of the follower group is not larger than that

of the leader group (e.g., column formation) in stationary envi-

ronment, such that the member robots can avoid obstacles while

following the route of the leader if the leader group moves beside

the obstacles. Thus, the usage of the collision avoidance algo-

rithm can be reduced by employing the proposed control law

based on the vector field method. Third, it can be said that the

proposed formation structure and control method is based on a

decentralized strategy, since the subgroups can be controlled by

using just the information obtained from the neighboring sub-

groups and also each robot in the subgroup can be controlled by

using the information from the neighboring robots. In addition,

the communication load of the member robots can be reduced

because those in the line formation just require the information

of the subgroup leader, and in turn, each subgroup leader of line

formations only needs to acquire the information of its preced-

ing subgroup to keep the column formation, unlike l andunlike l and ll methods, where l is the distance betweenrobots, and is the desired relative angle which use informationof more than two robots [9]. The communication flow will be

described in Section IV.

This paper is organized as follows. Thevector field generation

using the route of the leader is described in Section II, and thetarget tracking control using the vector field method is explained

in Section III. In Section IV, the hierarchical formation control

method, which consists of the column and line formation control

algorithms, is proposed. To demonstrate the usefulness of the

proposed control algorithm, the simulation results are presented

in Section V. Finally, the conclusions of the study are given in

Section VI.

II. VECTOR FIELD GENERATION FOR THE TARGET TRACKING

The WMRs considered in this paper are underactuated sys-

tem with two actuations for three degrees of freedom and have a

7/30/2019 06268350

3/11

KWON AND CHWA: HIERARCHICAL FORMATION CONTROL BASED ON A VECTOR FIELD METHOD FOR WHEELED MOBILE ROBOTS 1337

Fig. 2. Kinematics of the target and the actual robot with respect to COR.

nonholonomic constraint in Cartesian coordinates. The kine-

matic model of the robot is given by xiyi

i

=

cosi

sini

0

0

0

1

vi

i

(1)

where xi and yi are position variables, i is the orientation angleof the mobile robot, and vi and i are the linear and rotationalvelocities, respectively. Subscript i will be t, d, or c in the caseof the target, desired, or actual value, respectively.

Before designing the nonlinear controller used for the target

tracking based on the vector field method for formation control,the vector fields should be generated with respect to the route of

the target, as shown in Fig. 2. Although the vector field was gen-

erated from the given path for the path following of unmanned

aerial vehicle (UAV) in [16], we use the vector field for the tra-

jectory tracking problem of the mobile robot. Fig. 2 shows the

vector field generated by the moving target. In Fig. 2, the circle

represents the route of the target, the center of rotation (COR) is

the center of the circle, and the arrows represent the direction of

the vector fields generated from the route of the target. It should

be noted here that a straight route can be represented as a part

of circle with an infinite radius, the validity of which will be

demonstrated later in Scenario 1 in Section V. As can be seenin Fig. 2, the mobile robot can track the circular route of the

target when the mobile robot moves along the direction of the

vector field represented by the arrows. In Fig. 2, x, y, and weredefined previously, is the angular position with respect to theCOR, r is the radial distance from the COR, and the subscriptst, d, or c replacing i mean the target, desired, or actual value ofthe mobile robot, respectively.

The COR can be calculated using the information of the target

position and the linear and angular velocities as

COR(xC r , yC r ) =

xt + rt cos {t + sgn(vt )sgn(t )/2}

yt + rt sin {t + sgn(vt )sgn(t )/2}

(2)

where xC r and yC r are the position of the COR, vt and t arethe linear velocity and the angular velocity of the target, and

rt := |vt /t | is the radius of the circular route of the target. Forclarity, the derivation of (2) is given in Appendix A.

Remark 1: In the case that t becomes zero, t can beapproximated to be a small value without loss of generality.

The validity of this will be shown in the simulation results for

Scenario 1 where the reference trajectory is chosen to include a

straight line.

Remark 2: The objective of the proposed hierarchical forma-

tion control is to make the formation be maintained using the

followers such that the formation can be generated by the leader

which has arbitrary trajectory because the leader is not influ-

enced by the formation control. The leader motion, however,

should be constrained to avoid the rapid turn. This constraint

is described as the limitation on the curvature of the circular

route of the leader. If the rapid turn occurs, the formation can-

not be maintained unless additional algorithms, such as shape

change and collision avoidance, are used. To focus on the for-

mation control, we assume the constraint of the curvature of theformation route.

In [18], the vector field using the relationship between the

trajectory and the mobile robot was generated as follows.

First, when the linear velocities of the trajectory and mo-

bile robot are nonnegative (i.e., vt 0 and vc 0), the desiredorientation angle d can be described as

d (er ) = c +

2sgn(t ) + sgn(t )tan

1 (kd er ) (3)

where

c = atan2yc yC r

xc xC r (4)

er = rc rt (5)

for a constant kd > 0. Here, atan2() is a four-quadrant inversetangent with thevalues in theintervalsof(, ]. Although onlythe positive linear velocity of the robot is considered in [10],

because the robot is an UAV which cannot move in a backward

direction, both the positive and the negative linear velocities of

the mobile robot should be considered because the WMR can

move in both the forward and backward directions. Therefore,

the extended vector field can be obtained as

d (er ) =

c + sgn(t )

2+ tan1 (kd er ) , vt 0

c + sgn(t )

2+ tan1 (kd er )

, vt < 0.

(6)

Here, d (er ) is set in two opposite directions to make the robotbe able to track the target moving in both forward and backward

direction [22]. The derivation of (6) is shown in Appendix B.

III. TARGET TRACKING CONTROLLER DESIGN

USING THE VECTOR FIELD METHOD

The target tracking controller is designed based on the ex-

tended vector field in the previous section, which will be

used later to achieve the formation control. The kinematic

relationship between the target and the WMR is presented in

7/30/2019 06268350

4/11


Fig. 3. Relationship between the target and the actual mobile robot.

Fig. 3. In Fig. 3, lc is the distance between the target and theactual mobile robot, qt is the angle between the orientation angleof the target with respect to the straight line that passes through

the target and the actual mobile robot, and d is the desired

orientation angle obtained in (6).Taking the time derivative ofrc , c , and d , we have

rc = vc cos(c c ) (7)

c =vcrc

sin(c c ) (8)

d =vcrc

sin(c c ) + sgn(t )kd vc cos(c c )

1 + (kd er )2. (9)

When the tracking errors are chosen as

el = lc , e = c d , er = rc rt (10)

the control objective is to make the convergence of the trackingerrors in (10) to zero. The time derivatives of el and e can beobtained as

el = vt cos(qt ) vc cos(qc ) (11)

e = c vcrc

sin(c c ) sgn(t )kd vc cos(c c )

1 + (kd er )2. (12)

Then, we can design the control law for the target tracking as

vc = (qc )1

cos(qc )(vt cos(qt ) + kl el ) (13)

c

=vc

rcsin(

c

c) + sgn(

t)

kd vc cos(c c )

1 + (kd er )2 k

e

(14)

where kl and k are positive constants, and

(qc ) = 1 exp

|qc |

2

2 2

(15)

for a constant > 0. The radial function (qc ) can make thecontrol input in (13) avoid very large value or infinity when qcgoes to /2 and, in particular, can make vc be defined always,as will be shown later.

The designed control laws in (13) and (14) will be used for

both the line and column formation control. The stability of the

proposed hierarchical formation structure using the designed

control law in (13) and (14) is stated in the following theorem.

Theorem 1: Consider the relationship between the target and

the actual mobile robot in Fig. 3, and the control law given

by (13) and (14). Then, the error variables el , e , and er areuniformly bounded, and the ultimate bounds can be made to be

smaller with the choice of the smaller value of and the largervalues ofkl .

Proof: To show that e and el are ultimately bounded and canbe made to be smaller with smaller and larger kl , we choosethe Lyapunov function candidate as

V =1

2

e2 + e

2l

. (16)

Then, the time derivative of (16) becomes

V = e e + el el

= e c vcrc

sin(c c ) sgn(t )kd vc cos(c c )

1 + (kd er )2 + el (vt cos(qt ) vc cos(qc )). (17)

Substituting (13) and (14) into (17) gives

V = k e2 (qc )kl e

2l + el vt cos(qt )(1 (qc )). (18)

Since 0 (qc ) 1, vt is known, and vt cos(qc )(1 (qc ))is bounded, (18) becomes

V = k e2 (qc )kl e

2l + el vt cos(qt )(1 (qc ))

k e2 (qc )kl e

2l + |vt ||el | (19)

which shows that

V is negative outside the set {|el | |vt |/kl (qc )}, such that solutions starting in V c, where c >(|vt |/kl (qc ))/2, remain therein for all time [23]. In particular,when is chosen to be very small such that (qc ) becomesalmost 1 and kl is chosen to be large, the ultimate bounds of eand el become much smaller.

Next, we will show that er converges to zero when e and elconverge to zero. From Fig. 2, the law of cosine can be used

to give the relationship e2l = r2c + r

2t 2rc rt cos(c t ) =

(rc rt )2 + 2rc rt {1 cos(c t )} = e

2r +2rc rt {1 cos(c

t )}. Since 2rc rt {1 cos(c t )} is positive, |er | |el |holds. In other words, as el becomes sufficiently small, erwill become smaller accordingly. Thus, the error variables er ,

e , and el to track the target are ultimately bounded and theirultimate bounds can be made to be sufficiently small with the

appropriate choice of and kl via the designed control lawbased on the vector field method. In particular, when el = 0,it follows that rc = rt (i.e., er = 0) and c = t . In addition,noting that

t =

c + sgn(t )

2, vt 0

c + sgn(t )

2 , vt < 0

(20)

holds from Fig. 3, we can also see that e = el = 0 results in

c = t . (Q.E.D.)

7/30/2019 06268350

5/11


Fig. 4. Communication flow between subgroups and robots (dotted arrow:data transfer).

When is chosen to be sufficiently small, vc in (13) is almostthe same as

vc =1

cos (qc )(vt cos (qt ) + kl el ) (21)

when qc is away from /2. On the other hand, when qc con-verges to /2, it can be easily analyzed that vc converges tozero. Thus, vc can be defined to be zero for qc = /2. In thiscase, it is required that the mobile robot has only the angular

velocity, while the robot has zero linear velocity, since the ori-

entation angle of the robot and the direction of the target are

orthogonal. Thus, the stability of the tracking control law in

Theorem 1 can be maintained using the control law in (13) and

(14), and the performances as well remains satisfactory, as will

be shown in Section V.

IV. HIERARCHICAL FORMATION STRUCTURE

In this section, we introduce the hierarchical formation struc-

ture to extend the capability of the system and achieve formation

flexibility. The whole formation consists of the line and column

formations, as shown in Fig. 1. Each member robot is part of the

line formation, and the generated subgroups in the form of the

line formation make up the column formation. The polygonal

formation can be represented as the appropriate combination of

the line and column formations. To build the proposed hierar-

chical structure, the subgroups and robots have the data transfer

mechanism, as described in Fig. 4. In Fig. 4, the informationof the preceding subgroup, such as position and orientation, is

transferred to the follower subgroup leader, and the information

of the subgroup leader is transferred to the follower robots in

the same subgroup as well. For example, the communication

strategy in Fig. 4 can be readily employed to the actual robot

using RF communication modules including Bluetooth and

Zigbee. This way, the performance of the formation control

for the multiple robots can be improved effectively. As in Fig. 4,

the robots are positioned on the polygonal formation. These po-

sitions can be described by the horizontal and vertical distances

with respect to leader subgroup and robot in the column and line

formation, respectively.

Fig. 5. Line formation structure.

To generate the column formation for each following sub-

group, the position of the following subgroup should be placed

along the route of the preceding subgroup, while keeping the

relative desired distance between them. In addition, each robot

in the same subgroup shares the information of its leader robot

to generate the line formation. The hierarchical formation con-

trol is achieved by the following procedure. First, a trajectory of

the whole formation is set to be the desired position of the leader

subgroup. Then, the leader of the leader subgroup transfers its

information to that of the following subgroup, based on which

the desired position of each member robot is generated. Finally,

they are controlled to track the desired position using the vector

field method in Section III.

Here, it should be noted that the curvature of the leader robotof theformation shouldbe limited because excessivelyrapid turn

of the leader robot can break the formation shape, as described

in Remark 2. This assumption is reasonable, since we can see

that many systems have the curvature constraints even in the

case of the single robot such as car-like robot, ship, fixed-wing

UAV, etc. The details of each line and column formation control

are described in the following sections.

A. Line Formation Control of the Mobile

Robots in a SubgroupTo maintain the line formation, the desired position of the fol-

lowers is generated with respect to the leader in the subgroup. If

the followers track their desired position, then the line formation

can be maintained as in Fig. 5. In Fig. 5, the desired positions of

the mobile robots are generated using the position of the leader

robot of the subgroup. The subgroup size can be adjusted by the

desired distances ddi1 and ddi+1 . Robot i is the leader robot, and

robot i 1 and robot i + 1 are the follower robots; they need tobe controlled to go to their desired positions.

To this end, the desired positions are generated using the kine-

matics of the two-wheeled mobile robots [19], and the desired

positions and linear and angular velocities are expressed using

7/30/2019 06268350

6/11


Fig. 6. Relationship between the leader subgroup and the follower subgroupfor the column formation.

the geometry between the robots in the subgroup as follows:

Robot i 1:

xd

i1ydi1

di1

= x

iyi

i

+ dd

i1cos(

i+ /2)

ddi1 sin(i + /2)

0

vdi1

di1

=

1 ddi1

0 1

vi

i

(22)

Robot i + 1:

xdi+1

ydi+1

di+1

=

xi

yi

i

+

ddi+1 cos (i /2)

ddi+ 1 sin (i /2)

0

vdi+ 1

di+ 1

= 1 ddi+1

0 1 vi

i (23)where the subscripts i 1 and i + 1 mean left and right sides ofthe leader robot (robot i), respectively. As can be seen in Fig. 5and kinematic equations in (22) and (23), the followers are

positioned on the line which connects the COR and the leader,

and outer and inner followers should move faster and slower

than the leader, respectively, to maintain the line structure.

For robot i 1 and robot i + 1 to be able to track the desiredposition to maintain the line formation, we use the tracking

control law that is designed in Section III. To use the control

law in (13) and (14), we employ the error variables er , e , andel , which are defined in (10). When each robot is controlled

well, the line formation can be maintained.

B. Column Formation Control of the Subgroups

in the Multiple Mobile Robot Systems

The position of the subgroups can be determined to be kept

along the route of the leader subgroup, as shown in Fig. 6.

In Fig. 6, x, y, were defined already in Section II, and thesubscripts l and f mean the leader subgroup and the followersubgroup, respectively, ld is the desired distance between thepreceding and follower subgroups, and d is the desired angulardifference between neighboring subgroups derived from ld as

d = ld /rl . From the relationship in Fig. 6, the position of the

Fig. 7. Route of the delta formation of the four robots.

follower subgroup can be set as

xf = xc + rl cos (f) (24a)

yf = yc + rl sin (f) (24b)

f = l sgn (vl ) sgn (l ) d (24c)

f = l sgn (vl ) sgn (l ) d . (24d)

In addition, the linear and angular velocities of the follower

subgroups are set to be the same as those of the leader subgroup,

since each subgroup keeps moving along the same route while

maintaining the desired distance from its preceding subgroup.

This way, the column formation can be constructed, and also

along with the line formation, the desired position of each robotcan be generated from its subgroup position.

V. SIMULATION RESULTS

In this section, the simulation results on the proposed hierar-

chical formation control include various situations. Four mobile

robots that have kinematic model in (1) and control laws in

(13), (14), and (24) are considered for the hierarchical forma-

tion structure in the two scenarios, each of which has delta

and rectangle shape, respectively, as in Figs. 7 and 8. In the

two scenarios, the initial positions of member robots are given

for first robot: (2 m, 0 m, 0 rad), second robot (2 m, 2 m,

0 rad), third robot (2 m, 2 m, 0 rad), and fourth robot (2 m,4 m, 0 rad). Instead of the signum function sgn () in (14), weused the hyperbolic tangent function tanh() to avoid chatteringphenomena. In the results of the two scenarios, we can see that

the formation can move forward and backward in a flexible way.

In Scenario 1, the member robots make the delta forma-

tion. The first robot is the first subgroup leader, the second

and the third robots are in the second subgroup of which leader

is the second robot, and the third subgroup has the fourth one

as the leader, as can be seen in Fig. 7. The desired distances

for the delta formation shape and the first subgroups linear and

angular velocities are set as in Table I. In addition, a zero mean

randomnoisewith themaximum valueof 2% measurement units

7/30/2019 06268350

7/11


TABLE IINFORMATION OF DELTA FORMATION IN SCENARIO 1

TABLE IIINFORMATION OF DELTA AND RECTANGLE FORMATION IN SCENARIO 2

Fig. 8. Formation shape change from delta to rectangle.

(meters and meters per second for position and velocity, respec-

tively) is included in the position and velocity information ac-

quired from the other robots like the simulation in [27] to show

the robustness of the control law without compensation of per-turbations. The noise is included in the simulations to show that

the proposed control law and hierarchical structure can work

even in the presence of the uncertainties in the measurement

process in an actual environment. The tracking performance of

the proposed formation control method in Scenario 1 is pre-

sented in Figs. 7 and 9. Fig. 7 shows the routes of member

robots in the delta formation with conditions in Table I. The

member robots start to move backward and then move forward,

as presented in Table I and Fig. 7. In Fig. 7, we can see that

the simulation results include the straight line and circular route

with the forward and backward motion. When the velocities of

the leader subgroup are changed instantaneously, the errors con-

verge to zero after little transient motion. The errors of robots

are depicted in Fig. 9. In Fig. 9, it can be seen that the distance

and orientation errors of each robot converge to zero. It can be

noted that the error spike at every 10 s occurs because of the

sudden changes of the leader motion, as mentioned in Table I.

In Scenario 2, the member robots start to maintain the delta

formation as in Scenario 1, and then, the formation is changed

to rectangle formation at t = 6 s. The first and second robots arein the first subgroup, and the third and fourth robots are in the

second subgroup. In addition, the 2% random noise is included

in the position and velocity information from the leaders. The

first and third robots are subgroup leaders, and the robots initial

conditions and formation information are shown in Table II. The

performances of the proposed formation control method in this

case are depicted in Figs. 8 and 10. Fig. 8 shows the routes of all

robots in the formation with shape changed delta to rectangle

with the conditions in Table II. Especially, we can ensure that

the formation can show the flexible motion and can be main-

tained in the sharp turn even if the formation shape is changed.

In addition, the change of the sign of the angular velocity of the

leader is included in the form of g 1 (t) = 0.15tanh(t 10).Thus, we can make sure that the formation can be constructed

and maintained by the proposed hierarchical structure and con-trol laws even when there is an immediate change of the sign of

the leader subgroups angular velocity. The errors of robots are

depicted in Fig. 10. In Fig. 10, it can be seen that the distance

and orientation errors of each robot converge to zero. When the

sign of the velocities of the leader subgroup are changed in-

stantaneously, the errors converged to zero after short transient

motion. In addition, the errors can converge to zero, even when

the formation shape is changed immediately.

It is noted that almost the same results for each scenario could

be obtained for many simulations, including the random noise,

and therefore, only one of these cases was presented in Figs. 9

and 10. From these results of the simulation, we can see that

7/30/2019 06268350

8/11


Fig. 9. Tracking performances of the robots in delta formation in Scenario 1. (a) Distance error of the first robot. (b) Orientation error of the first robot.(c) Distance error of the second robot. (d) Orientation error of the second robot. (e) Distance error of the third robot. (f) Orientation error of the third robot.(g) Distance error of the fourth robot. (h) Orientation error of the fourth robot.

the shape of the hierarchical formation structure can be main-

tained by the proposed control law. To show the performance

of the flexibility and the maintenance of the formation width,

these results can be compared with the results presented in [20]and [21]. In these results, compared with the rigid formation, the

width of the formation is changed in order to maintain the line

of sight, as mentioned in Section I. As shown in the previous

works, the followers move inside the route of the leader and pre-

ceding robots in the case of the formation control based on the

line-of-sight information; thus, the additional collision avoid-

ance algorithm can be required even though the formation has

been constructed well because of the movement of the followers

inside the route of the leader. On the other hand, the proposed

formation approach can achieve the flexibility of the formation,

compared with the previous approach. The flexibility achieved

by the proposed structure and control law can provide the advan-

tage that the followers can avoid the collision in some particular

cases when the robots move with a formation where the width

of the follower group is not larger than that of the leader group

(e.g., column formation) in stationary environment, such that themember robots can avoid obstacles while following the route of

the leader if the leader group moves beside the obstacles.

VI. CONCLUSION

To control the formation with flexibility, we have introduced

the decentralized hierarchical formation structure and proposed

the formation control law based on the extended vector field

control method. This way, we could solve the problem of the

previous formation control law without flexibility, where the

width of the formation can be changed due to the line of sight

toward the leader, and thus, the rigid formation cannot avoid

7/30/2019 06268350

9/11


Fig. 10. Tracking performances of the robots in rectangular formation in Scenario 2. (a) Distance error of the first robot. (b) Orientation error of the first robot.(c) Distance error of the second robot. (d) Orientation error of the second robot. (e) Distance error of the third robot. (f) Orientation error of the third robot.(g) Distance error of the fourth robot. (h) Orientation error of the fourth robot.

collisions, especially when the formation is turning a corner.

We constructed the hierarchical formation structure using the

line and column formation and designed the nonlinear control

law based on the vector field. The proposed control law is one

of a decentralized strategy since each subgroup and the mem-

ber robot get the information from their preceding subgroup

and subgroup leader, respectively. To this end, the vector fieldmethod, which was previously valid only for a positive linear

velocity, is extended to the case where both the positive and

negative linear velocities of the robot should be taken into ac-

count. The simulation results showed that the proposed scheme

can achieve the desired flexibility of the formation and main-

tain the width of the formation. In addition, we can see that the

proposed formation control law and hierarchical structure can

work well even when the formation shape is changed. In future

research, the issue for the robustness of the formation in various

environments with communication failure, as well as guaran-

teeing the string/mesh stability under the perturbation of the

leader and/or leader subgroup (e.g., position), will be pursued

with respect to the proposed hierarchical structure and control

law. For real robots, robust control mechanism compensating

for the uncertainties and disturbances of the formation structure

will be pursued, and these will be implemented to actual robot

system. In addition, the collision avoidance between WMRs in

the formation can be studied to consider the safety of the robot

in the formation.

APPENDIX A

DERIVATION OF (2)

In order to derive (2), we need to consider the sign of the

linear and angular velocities, which defines the position of the

COR with respect to the target. The relationship between the

target and COR can be described as the following four cases in

Fig. 11. In Fig. 11, the COR is positioned with respect to the sign

of the linear and angular velocities and the radius of the circular

route of the target. Because the target is on a circular route,

the COR is orthogonal to the direction of the target. Therefore,

7/30/2019 06268350

10/11


Fig. 11. Relationship between the target and COR. (a) vt > 0, t > 0.(b) vt > 0, t < 0. (c) vt < 0, t > 0. (d) vt < 0, t < 0.

if the signs of linear and angular velocities are the same as in

Fig. 11(a) and (d), then the COR can be represented as

COR =

xt + rt cos (t + /2)

yt + rt sin (t + /2)

. (A1)

On the other hand, if the signs of linear and angular velocities

are different as in Fig. 11(b) and (c), the COR can be represented

as

COR =

xt + rt cos (t /2)

yt + rt sin (t /2)

. (A2)

From (A1) and (A2), we can derive the position of COR using

sgn(vt )sgn(t ) as

COR =

xt + rt cos (t + sgn (vt ) sgn (t ) /2)yt + rt sin (t + sgn (vt ) sgn (t ) /2)

.

APPENDIX B

DERIVATION OF (7)

To derive (7), the target and the actual mobile robot in Fig. 5

are considered. There are four cases depending on the sign of

the linear and angular velocities of the target, since the direction

and position of the circular route with respect to the target are

defined.

1) First, we consider the case ofvt > 0 and t > 0. If rc is

significantly larger than rt , then the desired orientation an-gle becomes d (er ) = c + . When rc equals rt , the de-sired orientation angle is c (er ) = c + /2. Therefore,when vt > 0 and t > 0, the desired orientation angle canbe derived as

d (er ) = c +

2+ tan1 (kd er ) (B1)

where kd is a positive constant that determines the rate oftransition from c + to c + /2.

2) When vt > 0 and t < 0, the desired orientation angleis d (rc ) = c if rc is significantly larger than rt .When rc = rt , the desired orientation angle is d (er ) =

c /2. Thus, when vt > 0 and t < 0, the desired

orientation angle is

d (er ) = c

2 tan1 (kd er ) . (B2)

3) When vt < 0 and t > 0, the desired orientation an-gle is d (er ) = c + if rc is significantly larger thanrt . If rc = rt , then the desired orientation angle is

c (er ) = c + /2. Thus, when vt < 0 and t > 0, thedesired orientation angle is

d (er ) = c +

2+ tan1 (kd er ) . (B3)

4) When vt < 0 and t < 0, the desired orientation angle isd (er ) = c ifrc is significantly larger than rt . When rc =rt , the desired orientation angle is d (er ) = c /2.Thus, when vt < 0 and t < 0, the desired orientationangle is

d (rc ) = c

2

tan1 (kd er ) . (B4)

From the above four cases, we can derive the desired orien-

tation angle as

d (er ) =

c + sgn (t )

2+ tan1 (kd er )

, vt 0

c + sgn (t )

2+ tan1 (kd er )

, vt < 0.

(B5)

REFERENCES

[1] H. G. Tanner, G. J. Pappas, and V. Kumar, Leader-to-formation stability,IEEE Trans. Robot. Autom., vol. 20, no. 30, pp. 443455, Jun. 2004.

[2] L. Chaimowicz, V. Kumar, and M. F. M. Campos, A paradigm for dy-namic coordination of multiple robots, Auton. Robots, vol. 17, no. 1,pp. 721, Jul. 2004.

[3] M. A. Lewis and K.-T. Tan, High precision formation control of mobilerobots using virtual structures, Auton. Robots, vol. 4, no. 4, pp. 387403,Oct. 1997.

[4] P. Seiler, A. Pant, and K. Hedrick, Disturbance propagation in vehiclestrings, IEEE Trans. Autom. Control, vol. 49, no. 10, pp. 18351841,Oct. 2004.

[5] A. Renfrew, A. G. Villasenor, and P. Brunn, Contact bridging dampersfor back-up on automated highways, IET Intell. Transp. Syst., vol. 1,no. 2, pp. 167175, Jun. 2007.

[6] D. W. Casbeer, D. B. Kingston, R. W. Beard, and T. W. Mclain, Coopera-tive forest fire surveillance using a team of small unmanned air vehicles,

Int. J. Syst. Sci., vol. 37, no. 6, pp. 351360, May 2006.[7] Y. Gu,B. Seanor, G. Campa,M. R.Napolitano, L. Rowe,S. Gururajan, and

S. Wan, Design and flight testing evaluation of formation control Laws,IEEE Trans. Control Syst. Technol., vol. 14, no. 6, pp. 11051112, Nov.2006.

[8] C.-K. Ryoo, Y.-H. Kim, and M.-J. Tahk, Optimal UAV formation guid-ance law with timing constraint, Int. J. Syst. Sci., vol. 37, no. 6, pp. 415427, May 2006.

[9] F. Fahimi, Sliding-mode formation control for underactuated surfacevessels, IEEE Trans. Robot., vol. 23, no. 3, pp. 617622, Jun. 2007.

[10] T. Balch and R. C. Arkin, Behavior-based formation control for multi-robot teams, IEEE Trans. Robot. Autom., vol. 14, no. 6, pp. 926939,Dec. 1998.

[11] A. K. Das, R. Fierro, V. Kumar, J. P. Ostrowski, J. Spletzer, and C.J. Taylor, A vision-based formation control framework, IEEE Trans.

Robot. Autom., vol. 18, no. 5, p. 813825, Oct. 2002.[12] O. A. A. Orqueda, X. T. Zhang, and R. Fierro, An output feedback

nonlinear decentralized controller for unmanned vehicle co-ordination,Int. J. Robust Nonlinear Control, vol. 17, no. 12, pp. 11061128, Jan.

2007.

7/30/2019 06268350

11/11


[13] G. A. S. Pereira, A. K. Das, V. Kumar, and M. F. M. Campos, Formationcontrol withconfigurationspace constraints, in Proc. IEEE/RSJ Int. Conf.

Intell. Robot. Syst., Las Vegas, NV, Oct. 2003, pp. 27552760.[14] T. D. Barfoot and C. M. Clark, Motion planning for formations of mobile

robots, Robot. Auton. Syst., vol. 42, no. 2, pp. 6578, Feb. 2004.[15] R. Ghabcheloo, A. Pascoal, C. Silvestre, and I. Kaminer, Coordinated

pathfollowing control of multiplewheeled robots using linearization tech-niques, Int. J. Syst. Sci., vol. 37, no. 6, pp. 399414, May 2005.

[16] L. Consolini, F. Morbidi, and D. Prattichizzo, Stabilization of a hierarchi-cal formation of unicycle robots with velocity and curvature constraints,

IEEE Trans. Robot., vol. 25, no. 5, pp. 11761184, Oct. 2009.[17] T.-S. Chio and T.-J. Tarn, Rules and control strategies of multi-robot

team moving in hierarchical formation, in Proc. IEEE Int. Conf. Robot.Autom., Sep. 2003, pp. 27012706.

[18] D. R. Nelson, D. B. Barber, T. W. McLain, and R. W. Beard, Vector fieldpath following for miniature air vehicles, IEEE Trans. Robot., vol. 23,no. 3, pp. 519529, Jun. 2007.

[19] R. Siegwart and I. R. Nourbakhsh, Introduction to Autonomous MobileRobots. Cambridge, MA: MIT Press, 2004.

[20] D. Cruz, J. McClintock, B. Perteet, O. A. A. Orqueda, Y. Cao, andR. Fierro, Decentralized cooperative control, IEEE Control Syst. Mag.,vol. 27, no. 3, pp. 5878, May 2007.

[21] R. Vidal, O. Shakernia, and S. Sastry, Following the flock [formationcontrol], IEEE Robot. Autom. Mag., vol. 11, no. 4, pp. 1420, Dec.2004.

[22] D. Chwa, Tracking control of differential-drive wheeled mobile robotsusing backstepping-like feedback linearization, IEEE Trans. Syst., Man,Cybern. A, Syst., Humans, vol. 40, no. 6, pp. 12851295, Nov. 2010.

[23] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992.[24] L. Barnes, M. A. Fields, and K. Valavanis, Unmanned ground vehi-

cle swarm formation control using potential fields, in Proc. 15th IEEEMediterr. Con f. Control Autom., Athens, Greece, Jun. 2007, pp. 18.

[25] J. Wang, X. Wu, and Z. Xu, Potential-based obstacle avoidance in for-mation control, J. Control Theory Appl., vol. 6, no. 3, pp. 311316, Aug.2008.

[26] J. Yuan and G.-Y. Tang, Formation control for mobile multiple robotsbased on hierarchical virtual structures, in Proc. 8th IEEE Int. Conf.Control Autom., Xiamen, China, Jun. 2010, pp. 393398.

[27] A. Moutinho and J. R. Azinheira, Stability and robustness analysis of theAURORA airship control system using dynamic inversion,in Proc. IEEE

Int. Conf. Robot. Autom., Barcelona, Spain, Apr. 2005, pp. 22652270.

Ji-Wook Kwon received the B.S. and M.S. degreesin electrical and computer engineering from AjouUniversity, Suwon, Korea, in 2005, 2007, and 2012,respectively.

He was a Visiting Researcher with the Instituteof Science and Technology of Yvelines, Universityof Versailles St-Quentin-en-Yvelines, Mante la Jolie,France, in 2009. Since 2012, he has been with theKorea Institute of Industrial Technology, Incheon,Korea. His research interests include mobile robotcontrol and its applications, multiple robot coopera-

tion, and formation control.

Dongkyoung Chwa received the B.S. and M.S. de-

grees in control and instrumentation engineering andthe Ph.D. degree in electrical and computer engineer-ing from Seoul National University, Seoul, Korea, in1995, 1997, and 2001, respectively.

From 2001 to 2003, he was a Postdoctoral Re-searcher with Seoul National University, where hewas also a BK21 Assistant Professor in 2004. Since2005, he has been with the Department of Electricaland Computer Engineering, Ajou University, Suwon,Korea, where he is currently an Associate Professor.

He was a Visiting Scholar with the University of New South Wales at the Aus-tralian Defence Force Academy and the University of Melbourne, Melbourne,Vic., Australia, in 2003 and the University of Florida, Gainesville, in 2011. Hisresearch interests include nonlinear, robust, and adaptive control theories andtheir applications to robotics; underactuated systems, including wheeled mobilerobots; underactuated ships; cranes; and guidance and control of flight systems.

06268350

Documents