A Synchronization Approach to Trajectory Tracking of Multiple Mobile Robots While Maintaining Time-V-JL0

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1074 IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 5, OCTOBER 2009

A Synchronization Approach to Trajectory Trackingof Multiple Mobile Robots While Maintaining

Time-Varying FormationsDong Sun, Senior Member, IEEE, Can Wang, Wen Shang, and Gang Feng, Fellow, IEEE

AbstractIn this paper, we present a synchronization approachto trajectory tracking of multiple mobile robots while maintainingtime-varying formations. The main idea is to control each robotto track its desired trajectory while synchronizing its motion withthose of other robots to keeprelative kinematics relationships, as re-quired by the formation. First, we pose the formation-control prob-lem as a synchronization control problem and identify the synchro-nization control goal according to the formation requirement. Theformation error is measured by the position synchronization error,which is defined based on the established robot network. Second,

we develop a synchronous controller for each robots translation toguarantee that both position and synchronization errors approachzero asymptotically. The rotary controller is alsodesignedto ensurethat the robot is always oriented toward its desired position. Bothtranslational and rotary controls are supported by a centralizedhigh-level planer for task monitoring and robot global localization.Finally, we perform simulations and experiments to demonstratethe effectiveness of the proposed synchronization control approachin the formation control tasks.

Index TermsFormation, multiple mobile robots, synchroni-zation.

I. INTRODUCTION

STUDY on the formation control of swarms of mobile robotshas received increasing attention in recent years. Examples

of formation-control tasks include assignment of feasible forma-

tions, moving into a formation, maintenance of formation shape,

and switching between formations. In this paper, we will focus

on how to maintain the networked robots in a time-varying for-

mation while performing a group task as a whole. The relevant

applications include exploration [1], cooperative robot recon-

naissance [2] and manipulation [3], formation flight control [4],

satellite clustering [5], and control of groups of unmanned ve-

hicles [6], [7].

Numerous approaches to the formation control of multiple

mobile robots have been reported in the literature, such as

Manuscript received January 21, 2009; revised May 5, 2009 andJuly 2, 2009.First published August 4, 2009; current version published October 9, 2009. Thispaper was recommended for publication by Associate Editor G. Antonelli andEditor L. Parker upon evaluation of the reviewers comments. This work wassupported in part by the Research Grants Council of the Hong Kong SpecialAdministrative Region, China, under Grant CityU 119907 and in part by theCity University of Hong Kong under Grant 7002461.

The authors are with the Department of Manufacturing Engineering andEngineering Management,City University of HongKong, Kowloon, HongKong(e-mail: [email protected]; [email protected]; [email protected]; [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.2009.2027384

behavior-based control, virtual structure approach, and leader-

following strategy, to name a few. In the behavior-based con-

trol [2], [8][11], several desired behaviors are prescribed for

each agent, and the final control is derived from a weighting of

the relative importance of each behavior. With this method, it

might be difficult to describe thedynamicsof thegroup andguar-

antee the stability of the whole system [12]. In the virtual struc-

ture approach [13][16], the entire formation is treated as a sin-

gle entity. The desired motion is assigned to the virtual structurethat traces out the trajectory for each member of the formation

to follow. However, the controller is not in decentralized archi-

tecture and may encounter difficulty in some applications [12].

With the leader-following strategy [3], [12], [17][20], some

robots are designed as leaders, while others are designed as fol-

lowers. This strategy is easily implemented by using two con-

trollers only and is suitable to describe the formation of robots,

but it is hard to take into account the functioning capabilities

of different robots, i.e., ability gap of a robot [12]. There also

exist some other approaches, such as artificial-potential-based

methods [21], [22] and graph-theory-based methods [23][29].

A recent approach to convex optimization strategies for coordi-

nating large-scale robot formations was reported in [30], whereshape transitions were discussed for mobile robot teams, albeit

from a high-level planner perspective.

In this paper, we propose to use a synchronization control

strategy to address the multirobot formation control problem,

thus utilizing the concept of cross-coupling approach [31]. Our

basic idea, which was first reported in [32], is that a team of

mobile robots track each individuals desired trajectory while

synchronizing motions among the robots to keep relative kine-

matics relationship for maintaining the desired, and perhaps

time-varying, formation. A relevant work was reported in [33],

where a group of mobile agents stays connected while achiev-

ing some performance objective. Traditionally, the control loopof each robot receives only local feedback from the controlled

robot and aims to achieve the desired tracking task without

responding to any other robots. With the proposed synchroniza-

tion control, the control loop of each robot receives feedback

from itself, as well as the others in trajectory tracking, which

is termed as the first task; meanwhile, it takes care of the other

robots to meet the formation requirement, which is termed as the

second task. In other words, under the synchronization control,

not only the convergence of the position errors to zero but how

these position errors converge to zero will also be considered at

the same time. A pure position control without synchronization

may succeed in the first task of the trajectory tracking but, in

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general, cannot guarantee accomplishment of the second task

of formation maintenance during the motion, due to the lack

of effort in synchronizing the robots motions. A pure position

control can only achieve the desired formation when the posi-

tion errors converge to zero but not during the whole motion.

In many applications, the robots are required to maintain the

desired formation during the whole motion period rather than

at the final time only. The proposed synchronization control

strategy is ideally suited to the formation tasks of underlying

trajectory controls of multiple mobile robots while following

desired time-varying formations.

The cross-coupling control technology provides advantages

and opportunities to design such a synchronized controller. Over

the past decades, the cross-coupling concept has been widely

used in multiaxis motion applications, such as reducing con-

touring errors of computer numerically controlled (CNC) ma-

chines [34][39]. The concept was also incorporated into adap-

tive control architecture to solve position synchronization of

multiple axes [40], [41]. The cross-coupling technology has

been used in robotics, such as controls of mobile robots [42]and robot manipulators [43]. Very recently, a combined synchro-

nization and tracking control law was reported in [44] and [45]

for multiple Lagrangian systems in a general setup that includes

complex configurations and different coupling links, such as di-

rectional couplings. To avoid the usage of the system dynamic

models, model-free cross-coupling controllers were introduced

in [46] and [47]. The effort to examine the stability and robust-

ness of the cross-coupled control system was reported in [48].

This paper has made the following contributions.

First, we successfully extend the synchronization approach to

formation control applications and pose the formation control

problem as a motion synchronization problem. As in [43], asynchronization control goal is determined first, based on the

formation requirement in the established robot network. Then,

we divide the motion control of each robot into two parts: One

is to drive the robot along the desired trajectory to achieve the

tracking control goal, which is defined as the first task, and

the other is to synchronize each robots motion with that of

two nearby robots to achieve the synchronization goal, which

is defined as the second task. To measure the synchronicity of

the networked robots, the concept of the synchronization error is

introduced as in [41] and[43], which is definedas the differential

position error between every pair of two neighboring robots.

Second, we develop a decentralized cross-coupled controller

for each robot to achieve its position tracking while synchro-nizing its motion with those of other robots for the desired

formation. The control algorithm utilizes feedback of both posi-

tion and synchronization errors, requires the information of the

two neighboring robots only, and responds to all linked robots

via the group network. It is proven that the proposed controller

can guarantee asymptotic convergence to zero of both position

and synchronization errors.

Third, simulations and experiments are performed on swarms

of mobile robots to demonstrate the effectiveness of the pro-

posed approach. A generalized superellipse with varying pa-

rameters is utilized to present a range of curves and, therefore,

establish a useful mathematic model to define time-varying for-

Fig. 1. Group of two-wheel mobile robots.

mations. The result for a skewed superellipse formation was

also reported in [49].

The advantages of applying our proposed synchronization

approach to the formation control are threefold. First, the syn-

chronization control goal is determined based on the desired

formation and is then divided into a number of subgoals foreach individual robot, without discrimination in job assignment.

The potential of robots capability is not overlooked. Also, the

formation strategy can be well prescribed. Second, a synchro-

nization controller can be constructed to guarantee asymptotic

convergence to zero of both position tracking and formation

errors in a decentralized architecture for time-varying forma-

tions. The controller consists of two parallel actions for both

trajectory tracking and formation, and the weighting of the two

actions is adjustable via tuning the control parameter. Third, the

synchronization controller can be simplified by synchronizing

the motion of each robot with that of two neighboring robots. In

other words, the control of each robot requires the informationof two nearby robots only.

II. MULTIROBOTFORMATION VIASYNCHRONIZATION

Fig. 1 illustrates a group of two-wheel mobile robots, where

qi = [ xi yi]T

denotes the position coordinate of theith robotinxyplane, andi denotes the heading angle. To simplify theanalysis, we assume that the center of the mass of each robot

locates at the geometrical center of the robot. In this way, the

centripetal and Coriolis effects are not considered in the robot

dynamics, and the robot is further simplified as a point mass

robot with the following dynamics:

Miqi = q i , Ii i =i (1)

whereMiandIidenote the inertia of the robot with fixed terms,and qi andi are torque control inputs with respect to qi andi , respectively.

Consider the control problem of guiding and positioning a

group ofn mobile robots, as shown in Fig. 1, along the boundary(curve) of a 2-D compact set. In a similar manner to [50], we

introduce a time-varying desired shape for each robot, which

is denoted by S(, t), where denotes a 2-D position vectorandt the time. The boundary ofS(, t) is parameterized by a2-D planar curve, which is denoted byS(, t) = 0. Assign the

target position qdi to the ith robot, and q

di must be located on



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the curve such thatS(qdi, t) = 0. Our objective is to determinethe appropriate control inputs for dynamics (1) such that the

ith robot converges to its target position qdi while maintainingits position in the desired shape S(, t). Note that althoughonly the position qi is explicitly controlled in the formation,

the orientationi affects the formation implicitly. In this study,the desired heading

d

i of the

ith robot is defined such that the

robot is always oriented toward the robots desired position qdi.

di keeps updated during the motion. When the robot is exactlylocated at its desired position, the desired heading di is the sameas the actual headingi , and no update is needed.

Define the position and heading errors of the ith robot asei =q

di qi andi =

di i , respectively. Besides the tra-

ditional robot control goal of ei 0 and i 0 as timet , the robots here are also required to achieve a formationcontrol goal of maintaining on the desired curve, which can be

formulated asqi , S(qi , t) = 0, wherei = 1, . . . , n.In this paper, we propose to utilize the synchronization control

concept to solve the formation control problem. The basic idea

of the synchronization control is to regulate the motions of therobots while they track the desired positions qdi to ensure the

robots to maintain in the required boundary (curve) such that

S(qi , t) = 0. Note that a pure position control for qi qdicannot guarantee the formation control goal to be achieved,

because the curveS(qi , t) = 0 is dependent on not only theposition qi but on the relative relationship ofqi as well, with

respect to the others at timet.To measure the synchronicity of the robots, we introduce the

concept of the position synchronization error, which arises from

the synchronization constraint, as required by the formation

goal.The following example will show howthis synchronization

constraint is determined based on the formation control goalS(qi , t) = 0. Another example was reported in [51].ExampleConsider thatnrobots are required to maintain in

an ellipse curve during the motions. The coordinate qi of the

ith robot is required to meet the following constraint:

S(qi , t) = 0 :qi (t) =

xi (t)

yi (t)

=

cos i (t)

sin i (t)

a(t)

b(t)

=Ai (t)

a(t)

b(t)

(2)

where a and b denote the longest and the shortest radii ofthe ellipse, respectively, and i = tanh (b sin i /a cos i ),withi = tanh [yi /xi ], denotes the angle of the robot lying on theellipse with respect to the center of the ellipse. Assume that the

robots are not located in the longest or the shortest axis of the

ellipse such that the inverse ofAiexists. Then, the synchroniza-tion constraint toqi can be derived as follows:

A11 q1 =A12 q2 = = A

1n qn =

a

b

. (3)

From the aforementioned example, it is seen that the synchro-

nization constraint can be generally represented in the form

c1q1 =c2q2 = = cnqn (4)

whereci denotes the coupling parameter of theith robot, and itsinverse exists based on (3). Further, (4) holds at all the desired

coordinatesqdi, namely

c1qd1 =c2q

d2 = = cnq

dn . (5)

Subtracting (4) from (5) yields the following synchronization

goal:

c1e1 =c2e2 = = cnen . (6)

Implicitly, (6) represents the formation control goal, which

can be further divided intonsubgoals ofciei =ci+1 ei+ 1 . Notethat wheni = n, we denoten+1 as 1.

Then, the position synchronization errors can be defined as

a subset of all possible pairs of two neighboring robots in the

following way [52]:

1 =c1e1 c2e2

2 =c2e2 c3e3

...

n =cnen c1e1 (7)

whereidenotes the synchronization error of theith robot. Ob-viously, if the synchronization error i = 0for alli= 1, . . . , n,the synchronization goal (6) is achieved automatically.

It is worth noting that thesynchronization error is used to mea-

sure the formation effect and is not equivalent to the position-

tracking error. For systems having closed-loop chain structures,

employment of the synchronization error provides each robot

with motion information both from itself and from the other

robots, and hence, the motions of all robots in the group arecoordinated.

Remark 1:The topology of the networked robots is designed

in terms of the robots physical positions in the group when

performing the group task. A principle that may be followed is

that the two physically close robots are coded as two neighbors

for easy sensorial connection. As a result, all robots in one group

are linked, either directly or indirectly, as a whole. Thenecessary

requirement for the applied shapeS(, t)is that the shape canbe represented mathematically such that the synchronization

constraint (4) can be given.

Remark 2: In some formations, the two neighborly coded

robots may be spatially far and, hence, not sensorially con-

nected. This usually happens when the formation shape is notgeometrically closed, e.g., in a line formation, the first robot

(robot 1) and the last robot (robot n) are far apart but codedas two neighbors. In case there exists a sensorial problem be-

tween these two robots, we can utilize a high-level planner (as

seen in Fig. 2) to globally localize the two robots such that

the two robots can know the information between each other.

Alternatively, we can also simply remove the direct synchro-

nization request between these two robots, since the two robots

synchronize with the other robots, respectively, and the whole

group of the robots are still linked as a whole. Removal of the

synchronization between the two robots can be mathematically

achieved by setting the synchronization error between these two



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Fig. 2. Block diagram of the overall control architecture.

robots to zero. In the relevant work [33], sensorial connection

was analytically studied.

Now, the control problem becomes to drive both the position

errorei and the synchronization error i to zero. Theith robotcan be designed to approach its desired positionqdi while syn-

chronizing its motion with those of its two connected robots

i 1and i+ 1. In this way, the control of each robot does not

require the information of all robots except for its neighbors,and hence, the implementation is simplified.

III. CONTROLDESIGN

Fig. 2 illustrates a block diagram of a two-level control ar-

chitecture used for the formation control. In level 1, there is

a centralized high-level planer for task command generation,

task monitoring (such as detecting malfunctioned robots and

responding), and global localization of the robots. In level 2,

there are decentralized servo controls of the robots and senso-

rial connection among them. Between the two levels is a data

server. Obviously, these two levels have different functions in

accomplishing complex formation control tasks.This section will focus on the decentralized servo controls of

the robots in level 2. A synchronous tracking controller for the

robots translation is developed. The robots orientation is con-

trolled to be always oriented toward the robots desired position.

The issue of how the robot group responds to the malfunctioned

robot will also be discussed briefly at the end of this section.

Without loss of generality, we assume that each robot can obtain

the position information of its two nearby robots, via global or

local localization in the two-level control frame, as shown in

Fig. 2. The relevant works on localization (such as ours [53])

have been reported in the literatures.

A. Synchronous Formation Controller

To make the position and synchronization errors ei and iboth converge to zero, we define a coupled position error Eithat links these two errors into one equation, i.e.,

Ei =ciei +

t0

(i i1 ) d (8)

whereis a diagonal positive gain matrix. As is seen from (7)and (8), this coupled position error for robot i feeds back theinformation of the two neighboring robotsi 1andi+ 1. Note

that wheni = 1, we denotei 1asn.

DifferentiatingEi with respect to time yields

Ei = ciei + ciei+ (i i1 ) . (9)

To achieve Ei 0 and Ei 0, we introduce a commandvectorui that leads to a combined position and velocity error,

which is expressed as follows:

ui =ciqdi + ciei+ (i i1 ) + Ei (10)

where is a diagonal positive gain matrix. Definition ofui in(10) leads to the following position/velocity vectors:

ri =ui ciqi =ciei+ ciei + (i i1 ) + Ei

= Ei + Ei . (11)

We then design a controller to drive ri to zero, such that

the coupled errors Ei and Ei tend to zero as well. For easyimplementation, such a controller is ideally designed in a de-

centralized architecture, thus considering the synchronization

between each robot and its two neighbors only.

We finally design a torque input for controlling the robots

translation as follows:

qi= Mi c1i (ui ciqi) + Kr i c

1i ri+ c

Ti K (i i1 ) (12)

whereKr i andK are positive feedback control gains. The lastterm in (12) is used to compensate for the effect due to addition

of the cross-coupling control to the overall system dynamics.

The necessity of introducing this term will be shown in the

stability analysis.

Substituting (12) into the robot translational dynamics in (1)

yields the following closed-loop dynamics:

Mi c1i ri+ Kr i c

1i ri + c

Ti K (i i1 ) = 0. (13)

Theorem 1:The proposed synchronous controller (12) leadsto asymptotic convergence of both the position tracking and syn-

chronization errors to zero, namely, ei 0 and i 0as timet , under the conditions that the control gain Kri is largeenough to satisfy mi n(Kr i ) ma x

Mi (d/dt)

c1i

ci

,

where mi n()and ma x()denote the minimum and maximumeigenvalues of the matrices, respectively.

Proof:Define a Lyapunov function candidate as

V =n

i=1

1

2

c1i ri

TMi c

1i ri+

1

2

Ti Ki

+

ni=1

1

2 t

0(i i1 ) d

TK

t0

(i i1 ) d.

(14)

DifferentiatingVwith respect to time yields

V =n

i=1

c1i ri

TMi c

1i ri+

c1i ri

TMi

d

dt

c1i

ri

+Ti Ki

+n

i= 1

(i i1 )TK

t

0

(i i1 ) d. (15)



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Multiplying both sides of (13) by

c1i riT

yields

c1i ri

TMi c

1i ri+

c1i ri

TKr i c

1i ri+ r

Ti K(i i1 )

= 0. (16)

Substituting (16) into (15) yields

V =n

i=1

c1i ri

T Kr i Mi

d

dt

c1i

ci

c1i ri

n

i=1

rTi K(i i1 )

+

ni=1

Ti Ki

+n

i=1

(i i1 )TK

t0

(i i1 ) d. (17)

We now analyze the termn

i= 1rTi K(i i1 )in (17). It

directly follows that

ni= 1

rTi K(i i1 )

=rT1K1 rT1Kn + r

T2K2 r

T2K1

+ rTnKn rTnKn 1

=rT1K1 rT2K1 + r

T2K2 r

T3K2

+ rTnKn rT1Kn

=

ni=1

(ri ri+ 1 )T Ki . (18)

Utilizing (7)(9) and (11), we have

ri ri+1

= Ei + Ei Ei+1 Ei+1

= ciei+ ciei ci+ 1ei+1 ci+1 ei+ 1+ (2i i1 i+ 1 )

+ (ciei ci+1 ei+ 1 ) +

t0

(2i i1 i+1 ) d

= i + (2i i1 i+ 1 ) + i

+

t0

(2i i1 i+ 1 ) d. (19)

Then, substituting (19) into (18) yields

ni= 1

rTi K(i i1 )

=

ni= 1

Ti Ki +

ni= 1

Ti Ki

+

ni=1

(i i+1 )TK(i i+1 )

+n

i=1

(i i+1 )TK

t

0

(i i1 ) d. (20)

In (20), we utilize the result that

ni=1

(2i i1 i+ 1 )Ti =

ni= 1

(i i+ 1 )T (i i+1 ).

Finally, substituting (20) into (17) and utilizing the condition

of Theorem 1, we have

V n

i=1

c1i ri

Tm in (Kr )

ma x

Mi

d

dt

c1i

ci

c1i ri

n

i= 1

Ti Ki

ni= 1

i i+1

TK (i i+1 )

0. (21)

It is thus concluded that the defined Lyapunov function (14)

has negative semidefinite time derivative. From (21), we know

that ri and i are bounded. From (13), we know that ri isbounded. From (19), we know that i is bounded. Therefore, riand i are uniformly continuous since ri and i are bounded.Then, from Barbalats lemma, ri 0 and i 0 as timet . The synchronization goal (6) is achieved.

We now prove ei = 0 when ri = 0 and i = 0. From (11),ri = 0implies Ei = 0whent . Combining all equationsin (8) from 1 ton, we have

c1e1 + c2e2 + +cnen = 0. (22)

Substituting (6) into (22) yields c1e1 =c2e2 = =cnen = 0. Since the inverse of ci exists, ei = 0. Therefore,

Theorem 1 is proved. It appears that the control parametersin (8) andK in (12)

dominate the control of the synchronization error. As seen in

(8), the value ofdetermines the weight of the synchroniza-tion error i in the coupled position error Ei and, thus, affects

the synchronization effort in the whole control action. As increases, the synchronization will be enhanced. In practical

applications,should be chosen by taking a balance betweenposition and synchronization controls. The parameter K , ac-companied by i i1 in (12), ensures the stability of thesystem when adding the cross-coupling control with to theoverall system dynamics. It will be further discussed how to

get the desired boundedness of the synchronization error by

adjustingK in Section III-B.To control the robots heading, we propose to utilize a general

computed torque approach to design of the control input i , i.e.,

i =Ii

di +kv i i + kpi i

(23)

wherekv i andkpi are computed torque control gains. The de-sired headingdi, as said before, is defined such that the robot isalways oriented toward its desired position. In other words, the

orientation control serves to achieve the robots translation.

Substituting (23) into the robot rotational dynamics in (1)

yields the closed-loop dynamics as follows:

i+ kv i i+ kpi i = 0. (24)



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This directly yields i = 0 and i = 0 as time t and guarantees the stability properties of the rotation.

Remark 3: If robot i malfunctions, e.g., lags much behindon its trajectory, the other robots will respond under the syn-

chronization control. The centralized high-level planar in level

1 will monitor the task and make the decision whether robot iis

still suitable to remain in the team. The decision making will bebased on the facts whether the robot loses capabilities to execute

the task (i.e., drive the position error to zero) and communicate

with the other robots. If robot i has to be abandoned by theteam, the other robots will treat robot i as a virtual robot andassume its position error to be zero in the follow-on motion.

In this way, the remaining robots can still follow the previous

network topology.

B. Discussions

Two issues are further discussed in this section. One is the

boundedness on the formation error. Another is the robustness

achieved by adaptive synchronization control.1) Boundedness: We now analyze the boundedness through-

out the transient response. Given a upper bound on the formation

error, in the following, we will discuss the lower bound on the

control parameter that achieves the desired upper bound on the

error.

Since the system is asymptotically stable, as concluded from

Theorem 1, given an upper bound i on i , there exists a finite

timeT >0 such that after t > T, one hasi (t) i . Inaddition, att = T

V(T) =n

i= 1 1

2 c1i (T)ri (T)

TMi c

1i (T)ri (T)

+

ni= 1

1

2

Ti (T)Ki (T)

+n

i= 1

1

2

T0

(i i1 ) d

T

K

T0

(i i1 ) d (25)

which is bounded. If we define the first term on the right-hand

side of (25) as A, it then follows from (25) that

V(T) A

=

ni= 1

1

2

Ti (T)Ki (T)

+n

i= 1

1

2

T0

(i i1 ) d

TK

T0

(i i1 ) d

n

i= 1

1

2K i

2

+

ni= 1

1

2K (2 i T)

2

=K 1

2+ 2 T2

n

i= 1

i 2

. (26)

Finally, the lower bound forK can be determined as

K V(T) A

12

+ 2 T2n

i=1 (i 2 )

. (27)

2) Adaptive Control for Robustness: In the controller (12),

the robot inertia Mi is assumed to be known. In practice, the

inertia Mi may be uncertain. Define Mi (t) as the estimate ofthe robot inertiaMi . An adaptive control can be designed as

qi = Mi (t)c1i (ui ciqi ) +Kr i c

1i ri+ c

Ti K (i i1 ).

(28)

The estimated inertia Mi (t)is subject to the adaptation law

Mi (t) = i c

1i (ui ciqi )c

1i ri (29)

wherei is a diagonal positive-definite control gain. Define theestimation error as

Mi (t) =Mi Mi (t). (30)

Then, the adaptive control law can be rewritten as

Mi (t) =i c1i (ui ciqi )c

1i ri . (31)

Substituting the controller (28) into the robot dynamics model

leads to the following closed-loop dynamics:

Mi c1i ri + Kr i c

1i ri+ c

Ti K (i i1 )

= Mi (t)

c1i (ui ciqi )

. (32)

In a similar manner to Theorem 1 and [43], we can prove that

the aforementioned adaptive control law leads to asymptotic

convergence to zero of both the position tracking and synchro-

nization errors.

The aforementioned analysis indicates that an adaptive con-

trol can be well incorporated into the developed synchronization

scheme. Although only inertia uncertainty is discussed here, the

method can be extended to the cases with more complex mod-

eling uncertainty and/or some external disturbances.

IV. SIMULATIONS

Simulations were performed first to verify the effectiveness

of the proposed synchronization control approach. It is assumed

that all the required formations are composed of regular closed,

smooth, and simple planar curves. A generalized superellipsewith varying parameters, as seen later, is used to represent dif-

ferent kinds of formation curvesxia

2/m + yib

2/m = 1 (33)which can also be represented as

xi =a cosm i

yi =b sinm i

(34)

wheremdenotes the exponent index that may be time-varying,and a, b, and i have been defined in (2). In the simulations,

i is fixed, and its value can be known at the beginning. Based



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Fig. 3. Switch from an ellipse to a rounded rectangle.

oni , each robot can be indexed. Besides the exponent m, thevariation of radiia and b also affects the curve significantly. Astudy on switch between two different ellipses with exchanged

radiiaandb was reported in [32].With different exponent m, (33) and (34) represent a range

of shapes including rectangles, ovals, ellipses, and diamonds, in

categories of hyperellipses (m < 1) and hypoellipses (m >1).The following simulation study will consider a switch from an

ellipse (m0 = 1) to a rounded rectangle(mf = 1/8), as seen inFig. 3, wherea andb are fixed.

In the simulation, there are in total 20 mobile robots, which

are denoted by little squares in Fig. 3, all located on an ellipse

curve at the beginning time. During the switch, all the robots

are required to maintain in a desired time-varying hyperellipse

curve, with the exponent index changed in the following way:

m(t) =m0 + (mf m0 )

1 et

(35)

wherem0 = 1andmf = 1/8denote the initial and final valuesof the exponent index. All the 20 robots have different inertia

Mi , which is expressed by Mi = diag{0.5 + 0.075i, 0.5 +0.075i}, wherei = 1, . . . , 20.

The desired trajectory of the ith robot is designed accordingto the required formation task (34) and is given as

qdi(t) =

xdi(t)

ydi(t)

=

cosm (t) i

sinm (t) i

a

b

=Ai (t)

a

b

. (36)

The coupling parameter matrix was then defined as

ci (t) =A1i (t) =

cosm (t) i

sinm (t) i

1. (37)

The sampling period was set to 0.005 s in the simulation.

Two control algorithms were applied in the simulation for

comparison purpose: One was the proposed synchronous con-

Fig. 4. Formation switchfroman ellipse toa rounded rectangleandtrajectoriesof robots, with synchronous control.

trol (12), and the other was the nonsynchronous control by re-

moving all kinds of coupling among the robots in the controller,

which was equivalent to a standard feedforward plus a feedback

control. The control parameters of the synchronous controller

(12) were chosen as = diag{15, 15}, = diag{35, 35},K = diag{10, 10}, andKr i = diag{120, 120}. To implementthe nonsynchronous control, the control parameters were chosen

as = diag{0, 0}, = diag{35, 35}, K = diag{0, 0}, andKri = diag{120, 120}. Note that the same coupling parametermatrix ciwas used to calculate the synchronization error in bothcases for a fair comparison.

Fig. 4 illustrates the actual formation shapes and the trajec-tories of the robots in the switch from the initial ellipse to the

final rounded rectangle under the synchronous control. Fig. 5(a)

and (b) illustrates the position and synchronization errors inx-and y-direction of the 20 robots under the proposed synchronouscontrol. Both the position and synchronization errors grow from

zero up to some nonzero values and, subsequently, decrease and

converge to zero upon reaching thefinal desired formation. Fig. 6

illustrates the results of the nonsynchronous control. It is seen

that the robots have large transient synchronization errors under

nonsynchronous control, which degrades the performance of

the formation. With the proposed synchronous control, the syn-

chronization errors are greatly reduced, and therefore, a better

formation can be achieved.We further study how the robot group responds to a particular

robot that is observed to malfunction. Assume that robot 18 was

found to lag much behind its desired trajectory soon after the

task started. Due to the use of synchronization control, all the

other robots had to slow down to wait for this robot to catch the

team. After recognizing that robot 18 could not resume working

properly, the group decided to abandon this robot at 0.75 s. The

remaining 19 robots then continued the movement and treated

robot 18 as a virtual robot with an assumption that its position

error was zero. Fig. 7(a) and (b) illustrates the position and

synchronization errors of the case. It is seen that due to failure

of robot 18, the position and synchronization errors of all the



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Fig.5. Results of synchronouscontrol.(a) Positionerrors. (b) Synchronizationerrors.

robots in the group increased significantly until robot 18 was

abandoned after 0.75 s.

V. EXPERIMENTS

We further carried out experiments on a group of three mobile

robots to verify the proposed synchronous control method. The

three robots are P3DX mobile robots, as shown in Fig. 8. The

control inputs qi and i were obtained by the inputs actingon the left and the right driving wheels, which are denoted by

li and r i , as given in the Appendix. The experiments were

performed in two cases.

A. Case 1: Triangle Formation

In this case, the three robots were controlled to maintain a se-

ries of equilateral triangles. The three robots were located at the

three vertices of the equilateral triangle. Define a circumcircle

that connects the three vertices of the triangle. The center of the

circumcircle locates at the geometrical central point of the tri-

angle. The switch among the equilateral triangles was modeled

as the switch among the circumcircles with time-varying radius

and orientation, as shown in Fig. 9. Taking reference of (2), the

coordinateqi of theith robot, wherei = 1, . . . , 3, was subject

Fig. 6. Results of nonsynchronous control. (a) Position errors. (b) Synchro-nization errors.

to the following constraint:

S(qi , t) = 0 :qi (t) =

xi (t)

yi (t)

=

cos i (t)

sin i (t)

R(t)

R(t)

=Ai (t)

R(t)

R(t)

(38)

whereR(t) is the time-varying radius of the circumcircle, andi (t)was defined in (2). Since a = b = R(t),i (t) =i .

In the experiment, both the size and the orientation of the

triangle were changed by varying R(t)and i (t), respectively.R(t)was scheduled to change in the following way:

R(t) =R0 + (Rf R0 ) t

t+e1t (39)

where R0 and Rf denote the initial and the final radii of thecircumcircle and were chosen asR0 = 2.4 mandRf = 4.8 m,respectively.i (t)was scheduled to change as

i (t) =i0 + (if i0 ) t

t+e1t (40)

where i0 and if are the initial and the final values for theith robot and were chosen as10 = 0

,1f= 120,20 = 0

,

2f =120, and 30 = 0

, 3f = 0. After system calibra-

tion, the inertia of each robot was estimated as 1.8 kgm2

.



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Fig. 7. Results when dealing with the failed robot. (a) Position errors.(b) Synchronization errors.

Fig. 8. Three mobile robots in the experiment.

Define the coupling parameter matrix as

ci (t) =A1i (t) =

cos i (t)sin i (t)

1

.

Fig. 9. Switch among triangle formations.

The sampling time period was chosen as 200 ms. The control

parameters used in the synchronous controller are

= diag{1, 1}, = diag{2, 2}, K = diag{5, 5}

Kr1 = diag{5, 5}, Kr 2 = diag{5, 5}, Kr 3 = diag{5, 5}.

As in the simulations, two control algorithms of the pro-

posed synchronous control (12) and the nonsynchronous con-

trol were implemented, respectively. To implement the non-

synchronous controller, the coupling among the robots was re-

moved, and the parameters were chosen as = diag{0, 0}andK = diag{0, 0}, and and Kr i were the same as in the syn-chronous case. Note that the same coupling parameter ci wasused to calculate the synchronization error for both the syn-

chronous and nonsynchronous cases. The robots were localized

based on their initial positions and the moving distances mea-

sured by the motor encoders.

The right-hand side of Fig. 9 illustrates the legend of the tri-

angle formation change in the experiment. Fig. 10 illustrates

the position and synchronization errors of the three robots inx- and y-direction, respectively, under the proposed synchro-nization control (12). Fig. 11 illustrates the position and syn-

chronization errors of the robots under the nonsynchronous

control. It is seen that both the control methods could ensure

good convergence of the robot position errors. The synchro-

nization errors under the synchronous controller were much

smaller than those under the nonsynchronous controller. The

synchronization errors actually represent the formation errors.

Fig. 12 illustrates the heading angle errors of the three robots

under the computed torque controller (23), where the control

gains wereKp1 = diag{10, 10}, Kp2 = diag{10, 10}, Kp 3 =

diag{10, 10}, Kv 1 = diag{10, 10}, Kv 2 = diag{10, 10}, andKv 3 = diag{10, 10}. Fig. 13 illustrates the control inputs liandr i on the two wheels of each robot, respectively, under thesynchronous control.

B. Case 2: Ellipse Formation

To further verify the robustness of the proposed adaptive

synchronization control (28) and (29) to the model uncertainty,

the three robots were controlled to form a part of ellipse (see

Fig. 14) and then switch among a series of ellipse formations.

At the beginning, the robots were in a straight line with the

same heading angle, as shown on the left-hand side of Fig. 15.

Then, the robots spread out to explore a series of formations



10/13


Fig. 10. Synchronous control results. (a) Position errors. (b) Synchronizationerrors.

that are all part of ellipses. The coordinate qi of theith robot issubject to (2). The longest and the shortest radii of the ellipse

were changed as

a(t) =a0 + (af a0 ) t

t+e1t (41)

b(t) =b0 + (bf b0 ) tt+e1t

(42)

wherea0 = 3 mand af = 5 mare the initial and the final de-sired longest radii of the ellipse, and b0 = 1 m andbf = 3 mare the initial and the final desired shortest radii of the ellipse.

In the experiment, leti [as defined in (2)] change to

i (t) =i0 + (if i0 ) t

t+e1t (43)

where we chose 10 = 30, 1f = 75

, 20 = 0, 2f =25

,

and30 =30,3f =75

. All inertias of the robots were

estimated to be zero at the beginning. Define the same cou-

pling coefficient as in case 1. The control gains were chosen as

Fig. 11. Nonsynchronous control results. (a) Position errors. (b) Synchroniza-tion errors.

Fig. 12. Heading angle errors of the three robots.



11/13


Fig. 13. Control torques.

Fig. 14. Three robots in an ellipse.

Fig. 15. Switch among formations.

Fig. 16. Adaptive synchronous control results. (a) Position errors. (b) Syn-chronization errors.

follows:

= diag{1, 1}, = diag{1.3, 1.3}, K = diag{1, 1}

Kr 1 = diag{10, 10}, Kr2 = diag{10, 10}

Kr 3 = diag{10, 10}, i = diag{0.1, 0.1}.

The sampling time period was 200 ms.

The right-hand side of Fig. 15 illustrates the legend of the

ellipse formation change in the experiment. Fig. 16 illustrates

good convergence of the position and synchronization errors

of the three robots under the proposed adaptive synchronous

control, although all inertias of the robots were assumed to be

unknown. These results demonstrate the validity of incorpora-

tion of adaptive control into synchronization approach to solve

the model uncertainty problem.



12/13


VI. CONCLUSION

This paper presents a synchronization control approach to

controlling swarms of mobile robots to track the desired trajec-

tories while synchronizing the motions among them to maintain

relative kinematics relationships, as required by the formation.

The formation control problem is successfully posed as a syn-

chronization control problem. The concept of the position syn-chronization error, which is defined as differential position error

between every pair of two neighboring robots, is introduced to

measure the performance of the formation. The proposed syn-

chronous controller guarantees asymptotic convergence to zero

of both position and synchronization errors of each robot in

translation. A rotary controller drives the robot to always be

oriented toward its desired position. Both simulation and ex-

perimental studies are finally performed to demonstrate the ef-

fectiveness of the proposed approach. The future work includes

incorporation of the robot nonholonomic dynamics into the syn-

chronization control, topology design of the networked robots,

and a study on robustness to the environmental uncertainties.

APPENDIX

Define the torque inputs acting on the left and the right driving

wheels of theith robot asli and r i . The transformation matrixfrom (li ,r i ) to (qi ,i ) is expressed as

Bi (i ) = 1

d

cos i cos i

sin i sin i

R R

(44)

wheredandRhave been shown in Fig. 1. Based on Bi (i ),qiandi can be derived as

qi =

qi (x)

qi (y)

=

cos

r (li + ri )

sin

r (li + r i )

(45)

i =R

r(li r i ). (46)

The inverse relationship from (qi ,i ) to (li ,r i ) can beobtained by

li = r

2qi (x)

cos

+i

R

r i = r

2

qi (x)

cos

iR

(47)

or

li = r

2

qi (y)

sin +

iR

r i = r

2

qi (y)

sin

iR

. (48)

To avoid singularity, (47) should be used when i is aroundzero or 180, and (48) should be used wheni is around 90

or

270

.

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Dong Sun (S95M00SM08) received the Bach-elors and Masters degrees in mechatronics and

biomedical engineering from Tsinghua University,Beijing, China, and the Ph.D. degree in robotics andautomation from the Chinese University of HongKong, Hong Kong.

He was a Postdoctoral Researcher with the Uni-versity of Toronto, Toronto, ON, Canada, where heis currently an Adjunct Professor. He was a Researchand Development Engineer in Ontario industry. Since2000, he has been with the Department of Manufac-

turing Engineering and Engineering Management, City University of HongKong, Kowloon, Hong Kong, where he is currently a Full Professor. His currentresearch interests include robotics manipulation, multirobot systems, motioncontrols, and biological processing automation.

Prof. Sun is a Professional Engineer in the province of Ontario. From 2004to 2008, he was an Associate Editor of the IEEE TRANSACTIONS ONROBOTICS.

Can Wang received the B.S. degree from the De-partment of Electrical Engineering, Zhejiang Uni-versity, Hangzhou, China, in 2001, the M.S. de-gree in system-on-chip from Lund University, Lund,Sweden, in 2005, and the Ph.D. degree with the De-partment of Manufacturing Engineering and Engi-neering Management, City University of HongKong,Kowloon, Hong Kong, in 2009.

His current research interests include coordinationof multiple robot systems, synchronization control,and neural networks.

Wen Shangreceived the B.Sc. degree in automationfrom Shandong University, Jinan, China, in 1999 andthe Ph.D. degree in automatic control from SoutheastUniversity, Nanjing, China, in 2005.

She is currently a Senior Research Associate withSuzhou Research Center, City University of HongKong, Kowloon, Hong Kong. Her current researchinterests include mobile robots, multirobot learningcontrol, and multirobot coordination.

Gang Feng(S90M92SM95F09) received theB.Eng. and M.Eng. degrees in automatic controlfrom Nanjing Aeronautical Institute, Nanjing, China,in 1982 and 1984, respectively, and the Ph.D. de-

gree in electrical engineering from the University ofMelbourne, Melbourne, Vic., Australia, in 1992.

Since 2000, he has been with the City Universityof Hong Kong, Kowloon, Hong Kong, where he iscurrently a Chair Professor and an Associate Provost.During 19921999, he was a Lecturer/Senior Lec-turer with the School of Electrical Engineering, Uni-

versity of New South Wales, Sydney, N.S.W., Australia. He was an AssociateEditor of theJournal of Control Theory and Applications. He is an AssociateEditor ofMechatronics. His current research interests include piecewise linearsystems, robot networks, and intelligent systems and control.

Prof. Feng received an Alexander von Humboldt Fellowship during 19971998 and the IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding PaperAward in 2007. He was an Associate Editor of the IEEE TRANSACTIONS ONSYSTEMS, MAN, AND CYBERNETICS, PARTC and was a member of the Confer-ence Editorial Board of the IEEE Control Systems Society. He is an AssociateEditor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the IEEETRANSACTIONS ONFUZZYSYSTEMS.

A Synchronization Approach to Trajectory Tracking of Multiple Mobile Robots While Maintaining Time-V-JL0

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