LEADER FOLLOWING FORMATION CONTROL FOR

OMNIDIRECTIONAL MOBILE ROBOTS

The Target Chasing Problem

Tiago Pereira do Nascimento, Fernando A. Fontes, Ant´onio Paulo Moreira

INESC Porto, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal

Andr´e Gustavo Scolari Conceic¸˜ao

Department of Electrical Engineering, Federal University from Bahia, Salvador, BA, Brazil

Keywords:

Formation control, Nonlinear model predictive controller, Mobile robots, Omnidirectional mobile robots.

Abstract:

This paper describes a novel approach in formation control for mobile robots. Here, a Nonlinear Model

Predictive Controller (NMPC) is used to maintain the formation of three omnidirectional mobile robots. The

details of the controller structure are presented as well as its functionality in a soccer robot team. Three Middle

Size League Robots are used for evaluation. A case of study based in a soccer robot situation is presented,

developed, and implemented to evaluate the performance of the controller. Simulations results are presented

and discussed.

1 INTRODUCTION

An adaptive framework based in predictive control for

creation and maintaining of a mobile robot team for-

mation was conceived as main objective of this work.

A formation is usually deﬁned as the special arrange-

ment of a set of agents of the same type, where the

relative positions of its elements are steady even if

the formation is moving. The used formation differs

from the usual rigid formations where the relative po-

sition of a team element must be precisely maintained.

Here, the ideal formations are the ones that maximize

the team perception of the environment or of an ele-

ment that can be a leader robot or of a moving target.

A formation in ’V’ is usually normal in sets of

birds. This allows them to maximize the traveled dis-

tance and minimize the friction with the air ﬂying in

the air tunnel left by the front bird. Nevertheless, in

the human sphere, a team formation can be used in

a variety of ways that goes from military operations

to team sports. In soccer for example, a formation

describes the number of players used in each tactic

formation in the ﬁeld (defense, offense or middle). In

this sport, the formation is less rigid and highly dy-

namic.

The ﬁeld of formation coordination and control of

a mobile robot team has been the area of many stud-

ies in recent years (Bicho et al., 2006), (Monteiro and

Bicho, 2008) and (Monteiro and Bicho, 2010). The

advantages of using a team of multiple robots include

robustness, ﬂexibility, and adaptability to unknown

dynamic environments (Lim et al., 2009). These are

clearly important when considering applications such

as saving and rescue missions, deep ocean mapping,

forest ﬁre detection, mine removal detection, or even

soccer robot competition.

In (Kanjanawanushkul and Zell, 2008) a strate-

gic division of formation control in three big groups

is made: leader following, virtual structure, and

behavior-based. There are also approximations purely

based in predictive control in which each robot has

an identical role in the formation. The virtual struc-

ture approach handles the problem as a rigid body

where all robots maintain a steady position, subjected

to physical constrains. In this strategy, any pertur-

bation made to any robot is propagated to the other

robots from the structure. This is due to a relatively

simple controller, as the ones described in (Tan and

Lewis, 1996), (Ghommam et al., 2010) and (Ren and

Beard, 2003). Nevertheless, the virtual structure ap-

proach has a high computational cost as the number of

robots increase. Another problem of this approach is

the velocity of the whole formation which is relatively

low, becoming a real problem in cases of dynamic ob-

135

Pereira do Nascimento T., A. Fontes F., Paulo Moreira A. and Gustavo Scolari Conceição A..

LEADER FOLLOWING FORMATION CONTROL FOR OMNIDIRECTIONAL MOBILE ROBOTS - The Target Chasing Problem.

DOI: 10.5220/0003453701350144

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 135-144

ISBN: 978-989-8425-75-1

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

stacle avoidance.

When using strategies based in a behavior-based

approach, reactive behaviors are used in each robot

to control formation integrated with other behaviors

used for trajectory tracking and obstacle avoidance

and detection. Good examples of this application can

be found in (Balch and Arkin, 1998), (Antonelli et al.,

2009) and (Liang and Xiao, 2010). The problem with

this approach is the description of the group dynamics

and the stability of the team.

Finally, the leader followingapproach is one of the

most studied. It is based on the existence of a leader

(real or virtual) robot that follows the precise desired

trajectory when the other robots of the formation just

follow it, maintaining a preset distance and relative

position. Most of these formation control strategies

employ predictive controllers.

An interesting application of the leader follow-

ing approach using Predictive Control is presented by

Zell and Kanjanawanishkull in (Kanjanawanushkul

and Zell, 2008). In this paper, the problem is solved

by dividing in two sub-problems essentially decou-

pled: the leader robot problem and the follower robots

problem. The leader robot follows a ﬁxed trajectory

using a similar approach to (Kanjanawanushkul and

Zell, 2009), transmitting its velocity, state, and tra-

jectory to the other robots in formation. Therefore,

the followers use this information with the desired

formation to estimate its own reference trajectory.

Then, the follower robot follows this newly created

reference trajectory using the same controller used

by the leader. The formation problem proposed here

is decentralized, dividing the whole system in multi-

ple sub-systems which are independently controlled.

This allows the predictions horizons of control to be

bigger. Each robot solves its own optimization prob-

lem to follow its estimated trajectory.

The same authors presented in (Kanjanawanishkul

and Zell, 2008) another strategy to formation control

based in MPC, where each robot has to follow its own

reference trajectory while maintaining the desired for-

mation with the other robots of the group. It is as-

sumed that each robot has a pre-established reference

trajectory and that what the controller ﬁnds are the

velocities that each robot should have. In this case

the interaction between the formation robots is ac-

complished by coupling the cost function from the

controllers of each agent. Each robot solves locally

its control problem and passes the information to its

closest neighbors. Then, these robots solve their own

problem of minimization taking into account these

data in their cost function.

In (Fontes et al., 2009) a predictivecontroller with

two layers to control the formation of non-holonomic

mobile vehicles is proposed. It is considered that

there are two sub-problems to be solved to fulﬁll the

main objective: the trajectory control problem and the

formation control problem. With non-holonomic ve-

hicles, the problems are strictly different. When only

a single vehicle is moving to follow a ﬁxed trajec-

tory it cannot move in any direction, therefore need-

ing a non-linear controller that would allow discon-

tinued feedback and control laws. However, when the

non-holonomic vehicle is moving as part of a forma-

tion, the relative position between the vehicles can be

modiﬁed in any direction as if they were holonomic.

To deal with this problem, only a linear controller was

needed.

It can be noticed that usually the state of the art

studies look for maintaining a rigid formation with

a robot team in which the relative positions between

the robots are ﬁxed. This paper presents an approach

where the desired formation is not rigid. Therefore,

the proposed controller shall control the formations

that optimize the perception of the environment by the

team. By that, the robots relative positions have to

vary during the formation movement.

Figure 1: Distributed Architecture of a Distributed NMPC

Controller.

2 FORMATION CONTROL

The controller used in this work to formation con-

trol was a Non-linear Model Predictive Controller

(NMPC). The general structure of this controller can

be classiﬁed in three types: distributed, centralized,

or hybrid. These categories are based on the way the

control signals of each robot are calculated.

Here, the distributed architecture was chosen as

can be seen in Fig. 1. In this case, each one of the

robots calculates the total control inputs U

solving

its own optimization problem. This takes away the de-

pendencyfrom a central processing unit, guaranteeing

the functioning of the formation even in cases of com-

munication failure. Therefore, each robot must have

information about the state X

(position and speed) of

each mate of its team. Also, in case of the communi-

cation failure or supervisor failure, the robot uses its

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

136

predicted open-loop strategy to determine these infor-

mations, having, therefore, a tolerance degree to fail-

ure. Nevertheless, it has the disadvantage of putting

a cost in computing the simulation of the entire for-

mation progression, which is done by each one of

the robots. However, this was not a problem, for the

robots only calculate their own control inputs. As

each robot solves its optimization problem in a decen-

tralized architecture, the formation becomes difﬁcult

to stabilize.

Figure 2: Structure of the Formation Controller Projected.

The capacity of the NMPC controller to create and

maintain a formation comes from the fact that cost

functions used by the controllers of each robot in the

team formation are coupled. This coupling is done

while the information about the position and speed of

the other robots are used in the cost function of each

robot to penalize the geometry or desired objective

deviation. This turns the entire group formation stable

where the actions of each robot affect the other mates.

Fig. 2 exposes the structure of the used controller.

This controller can be divided in three parts:

• State of the Formation - The controller contains

structures to keep the formations state (position

and speed of each other robot in the formation or

of any target that should be followed), updating

them in each control loop. These informations

can be received by a supervisor or by other robots

from the team, or even by the robot itself using its

own resources;

• Optimizer - This part uses a numeric minimiza-

tion method to optimize the cost function and ob-

tain the signals of optimal control. Here it is used

a method called Resilient Propagation (RPROP),

which guaranties quick convergence;

• Simulator - This part does the simulation not

only of the robot state evolution but also the state

evolution of the other elements in the formation

(other robots or targets). This element uses a dy-

namic simpliﬁed model to emulate the robot evo-

lution. The speeds of the other robots or targets

are assumed during the entire horizon of predic-

tion as being constant and equals to the actual

speed.

The controller receives as parameters the desired

formation, the position of the robot in formation and

the actual state of the other elements in the formation.

For each formation there are a differentcost functions.

Then each mDec uses its controller optimizer and it

starts to give to the simulator the control input U for

the robot the mDec is controlling. The simulator uses

this information to simulate the complete formation

evolution for the prediction horizons T

. The simu-

lator gives back to the optimizer a cost value to the

control inputs, and the iterative process of minimiza-

tion is repeated cyclic.

The Resilient Propagation algorithm (RPROP) ap-

peared in the learning algorithms category used in

neural networks (Riedmiller and Braun, 1993), be-

ing adapted to this application. This is an adaptive

method where the step value is not proportional to

the gradient function value to be minimized in a de-

sired point (as it happens in the Steepest Descent al-

gorithm), but it keeps adapting with the function be-

havior. Therefore, it becomes immune to the uncer-

tainties of the derivative function value, depending

only on the temporal behavior of its signal. This algo-

rithm was tested initially with the values suggested by

(Riedmiller and Braun, 1993) (η

= 1.5, η

−

= 0.5,

∆

= 0.1) and it revels to be capable to converge

where the Steepest Descent failed.

Figure 3: The 5DPO Robot.

3 PROBLEM FORMULATION

The developed framework was applied to a forma-

tion with three omnidirectional mobile robots from

LEADER FOLLOWING FORMATION CONTROL FOR OMNIDIRECTIONAL MOBILE ROBOTS - The Target

Chasing Problem

137

the FEUP’s 5DPO team (Fig. 3) that can fulﬁll two

main objectives: the optimization of the target relative

velocity perception and relative state perception (ball

relative velocity perception and relative state percep-

tion) using two robots (observers) while a third robot

places itself in an ideal position to receive the ball (re-

ceiver). The robots should maintain this formation

following the ball movement and avoiding the colli-

sions between them or with the target.

Figure 4: The Desired Formation.

The mathematical deﬁnition of the system can be

understood as having three robots and a ball (target).

Taking as base for this formation deﬁnition the ele-

ments presented in Fig. 4. The ball position and speed

vectors in global coordinates are respectively:

ball

(k) =



ball

(k) y

ball

(k)



, (1)

ball

(k) =



ball

(k) vy

ball

(k)



. (2)

It is considered also that the unit vector of the

ball’s velocity ˆv

ball

(k)=[ ˆvx

ball

(k), ˆvy

ball

(k)]

, is such

that:

ˆv

ball

(k) =

ball

(k)

ball

(k) + vy

ball

(k)

. (3)

For each robot n, its state is represented by:

(k) =



(k) y

(k) θ

(k)



, (4)

(k) =



(k) vy

(k) w

(k)



. (5)

The position of the ball with respect to robot n is

given by P

Rn−B

(k)=[x

Rn−B

(k), y

Rn−B

(k)], where:

Rn−B

(k) =



ball

(k) − x

(k)) (y

ball

(k) − y

(k))



(6)

Then, it shall be deﬁned the unit vector

Rn−B

(k)=[ ˆx

Rn−B

(k), ˆy

Rn−B

(k)], which indicates the

direction of the ball with respect to the robot, and its

angle θ

Rb−B

Rn−B

(k) =

Rn−B

(k)

Rn−B

(k) + y

Rn−B

(k)

, (7)

Rb−B

(k) = atan2(y

Rn−B

(k), x

Rn−B

(k)). (8)

Finally, there is also the deﬁnition of the positions of

each robot n with respect to its mates in the formation

y, given by P

Rn−Ry

(k)=[x

Rn−Ry

(k), y

Rn−Ry

(k)], where:

Rn−Ry

(k) =



(k) − x

(k)) (y

(k) − y

(k))



(9)

3.1 The Observer Robot

The estimation of the quality of the ball state is a func-

tion of its moving direction with respect to the robot

and the distance in between. This estimation is done

by using a omnidirectional vision system, the descrip-

tion of which can be found in (Gouveia, 2008). There-

fore, it’s clear that in this case, the robot’s direction is

irrelevant.

Nevertheless, big distances between the robot and

the ball results in failure of the ball’s detection. Con-

sequently, this leads to the failure to estimate its ve-

locity. When the distance is too small, it can occur

that the robot cannot see the entire ball and, therefore,

become incapable to detect correctly its position in-

creasing the risk of undesired collisions. Take as an

example the case in which the distance between the

ball and the robot decreases with time in a straight

line. In this case the robot only sees the ball increas-

ing in size, making it difﬁcult to estimate its velocity.

In the ideal case the ball should move perpendicular

to its position with respect to the robot. Therefore, the

desired formation for the observer robots to be around

the ball in a way to better estimate the ball velocity

possesses the following characteristics:

• Each one of the robots puts itself in opposite sides

of the ball, maintaining a parallel velocity with

respect to the ball, v

ball

, with the same modulus;

• The robot position vector with respect to the ball,

, must be perpendicular to the ball’s velocity

vector, v

ball

;

• Each one of the robots must maintain a distance

| from the ball;

• The robots must not collide between them.

Therefore, taking into account all the elements

previously described, the weights given to each one

of them, and a penalization term to the variation of

control effort, the cost function that represents all this,

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

138

embedded in each of the two observer robots is as fol-

lows:

J(N

, N

) =

∑

i=N

setpoint

− |P

Rn−B

(i)|)

∑

i=N

(

Rn−B

(i) · ˆv

ball

(i))

∑

i=N

((

−d

min

+ |P

Rn−Rm

(i)|

)

(

−d

min

+ |P

Rn−Rm

(i)|

)

∑

i=1

(∆U(i))

(10)

Where,

• N

, N

- prediction horizon limits, in discrete time,

so that N

> 0 e N

≤ N

, where N

is the desired

prediction horizon.

• N

- control horizon.

• λ

, λ

- weights for each component of the

cost function

• ∆U(k) = [v

(k) − v

(k − 1)] + [vn

(k) − vn

(k −

1)] + [w

(k) − w

(k− 1)] - variation of the control

signals, with U(i) being the reference velocities

vector with respect of the center of mass of the

robot.

3.2 The Receiver Robot

The ideal position of the receiver robot with respect to

the ball to have a good reception of it corresponds to

the one in which the robot velocity vector is collinear

with the ball velocity vector, with the same modu-

lus. Also, the robot orientation should be such that the

front of the robot is turn towards the ball. Therefore,

the robot can then slowly decelerated and the distance

between it and the ball can be decreased in a way to

receive the ball in ideal conditions.

Summarizing it, theformation here should possess

the following characteristics:

• The robot’s velocity has to be equal in modulus

and direction to the ball’s velocity v

ball

;

• The robot’s position vector with respect to the

ball, P

Rn−B

, must be collinear to the ball’s velocity

vector, v

ball

;

• The robot’s orientation θ

must be at all times

equal to the vector P

Rn−B

’s angle, deﬁned by

Rn−B

, in a way that the kicker of the robot is al-

ways turn towards the ball;

• The robot must be at a distance |P

R−B

| from the

ball.

Finally, joining all the elements previously de-

scribed, the weights given to each one of them, and

a penalization term to the variation of control effort,

the cost function that represents all this, embedded in

the receiver robot is as follows:

J(N

, N

) =

∑

i=N

setpoint

− |P

Rn−B

(i)|)

∑

i=N

(−1)

∑

i=N

((

−d

min

+ |P

Rn−Rm

(i)|

)

(

−d

min

+ |P

Rn−Rm

(i)|

)

∑

i=N

(dif fAngle(θ

, θ

Rn−B

))

∑

i=1

(∆U(i))

(11)

Where,

• N

, N

- predictionhorizon limits, in discreet time,

so that N

> 0 e N

≤ N

, where N

is the desired

prediction horizon.

• N

- control horizon.

• λ

, λ

- weights for each component of

the cost function

• ∆U(k) = [v

(k) − v

(k − 1)] + [vn

(k) − vn

(k −

1)] + [w

(k) − w

(k− 1)] - variation of the control

signals, with U(i) being the reference velocities

vector with respect of the center of mass of the

robot.

4 RESULTS

Once the formation algorithm was implemented,

some tests in simulation were made to validate the

proposed controller and to test its performance un-

der different conditions. Here, a simulation software

called SimTwo was used to simulate the formation

(Costa, 2010) done by three omnidirectional robots

and the target (ball).

In this simulation the SimTwo has the job of an-

other software called HAL (Hardware Abstraction

Layer), which is an application that receives the sen-

sor signals and communicates with the actuators, and

LEADER FOLLOWING FORMATION CONTROL FOR OMNIDIRECTIONAL MOBILE ROBOTS - The Target

Chasing Problem

139

then with the mDec (software of control of the real

robots) by IP protocol. In the real robots, the HAL

sends to the robot’s mDec the state of the other robots

and the state of the ball. Then, each mDec sends to

the SimTwo the control references of its robot. Each

mDec also communicates with another central com-

puter (the supervisor) that contains the Coach soft-

ware, sending its own state and the state of the ball

while observing it. Finally, the Coach sends to each

mDec individually the state of the other robots in for-

mation, in a way that each robot has the information

of position and velocity of its mates. It can be noticed

that this arrangement is similar the one used in real

experiments, where the only difference is the replace-

ment of the SimTwo for the HAL in each robot.

Figure 5: Communications between applications diagram.

There are many variables that inﬂuences the qual-

ity of the result. Among them are the weights (the

) of each cost function and the optimizer parame-

ters. The cost function values for both observer and

receiver robots can be seen in table 1. In the min-

imization of the cost function values, only the rela-

tionship between the weights given to each element

that meters to the ﬁnal result. Therefore, to the penal-

ization of the relative position with respect to the ball

was given a value ten times bigger than the penal-

ization of the distance between the robot and the ball

, due to the fact that the ﬁrst penalization is harder

to maintain. The weight given to the penalization of

the proximity between robots, λ

, was set very high to

avoid the maximum number of collisions. The penal-

ization for the receiver robot’s orientation, λ

in table

1, was set as half of λ

, for this was not very impor-

tant to the formation. Finally, the value of the control

effort (λ

in the ﬁrst table and λ

in the second one)

was chosen to be the smallest value that could forbid

the robots to move themselves around the target when

this was stationary. The ﬁnal values were a result of

an iterative process. This process did not need to be

very precise, due to the fact that there were a very

large range of weights that could give similar results.

Nevertheless, the NMPC controller parameters were

= 10, N

= 2 and the used reference trajectory to

ﬁnd them was an gate signal extracted in a previous

work done by (Ferreira and Moreira, 2010).

Table 1: Weights for the Observers and Receiver.

Weight Observer Value Receiver Value

λ 1 10 10

λ 2 100 100

λ 3 100 100

λ 4 5 50

λ 5 - 5

The initial parameters used on the RPROP opti-

mization algorithm were the ones suggested by (Ried-

miller and Braun, 1993) where the algorithm descrip-

tion can also be found. The ﬁst tests were done with

these parameters (η

= 1.5 and η

−

= 0.5) and re-

sulted in a very satisfactory performance by the con-

troller. Some changes made on these values were

tested (decrement of η

and increment of η

−

) and

produced visible improvements, having the ﬁnal val-

ues become as the ones shown in table 2.

Table 2: RPROP optimizer parameters.

Variable Value

IT max 20

ε 0.05

1.2

−

0.8

Therefore, the following simulation results made

with the formation control framework evaluate the

proposed controller. First, the simulations for forma-

tion convergence are shown to evaluate the formation

controller. Secondly, simulations for the evaluation of

the formation maintenance were made.

4.1 Formation Convergence Results

The following results show the trajectories followed

for each one of the robots when, starting from dif-

ferent positions, converge to a preset formation. The

target in these cases is stationary during the simula-

tion making the internal product of any vector with

the ball velocity vector equals to zero. It is impor-

tant to notice that the robots number 1 and 2 are the

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

140

observer robots while the robot number three is the re-

ceiver robot. The desired distance between each robot

to the ball was deﬁned to be 1 m.

4.1.1 Simulation 1

In this simulation the robots start at positions per-

fectly opposites and far from the ball. For having less

risk of collision or probability of the robots to inter-

fere with each other, this became the simplest case.

The results can be seen in Fig. 6. The robots con-

vergeperfectly to their positions in formation, making

straight trajectories towards the target. As can be seen

in the plot of the distance with respect to the time, it

can be estimate that the robots have converged to the

desired formation in approximately four seconds.

Figure 6: Convergence into formation, simulation 1.

4.1.2 Simulation 2

Here all robots start from the same side of the ball,

thought separated by a distance of 3 m. The results

can be seen in Fig. 7. From this simulation on it can

be noticed some interaction between the robots. The

robot 3 went directly to the ball, in a straight line.

The robots 1 and 2 went also to the ball avoiding the

robot 3 when starting to get close to this robot. This

formation converged in about 5 seconds.

4.1.3 Simulation 3

The third simulation shows a more complex situation,

where the robots start from the same alignment with

respect to the ball. The results can be seen in Fig.

8. The robot 3, closest to the ball, went to occupy

Figure 7: Convergence into formation, simulation 2.

Figure 8: Convergence into formation, simulation 3.

instantaneously the position in its front. When the

robot number 2 approaches the ball, the robot number

3 moves slightly up, therefore, going around the ball,

and occupying the space behind it. Finally, the robot

number 1 puts itself in the space left by robots number

2 and 3. This process takes about 4.5 seconds.

LEADER FOLLOWING FORMATION CONTROL FOR OMNIDIRECTIONAL MOBILE ROBOTS - The Target

Chasing Problem

141

4.2 Formation Maintenance Results

Now, the following simulations test situations in

which, once the formation is made, the robots must

maintain it moving in formation while tracking the

target (the ball). These simulations were made to

evaluate the capacity of the controller to maintain the

desired formation. Besides the XY plot and the dis-

tance with respect to time graph, the graph of the in-

ternal product between the ball velocity vector and the

each robot position vector with respect to the ball was

also shown.

4.2.1 Simulation 1

In these testes the target moved in a straight line

trajectory with a constant velocity that varied from

0.5m/s to 1.5m/s. These simulations can be seen in

Figs. 9, 10, and 11. This is the simplest situation to

maintain the formation in a target tracking problem,

given the fact that the target velocity has direction and

velocity constant what allows the controller to predict

exactly its progression.

Figure 9: Maintaining formation in a straight line, simula-

tion 1a.

To the target velocity equals to 0.5m/s and 1.0m/s

the performance of the controller is similar and

presents an acceptable accuracy. The distances from

Figure 10: Maintaining formation in a straight line, simula-

tion 1b.

the ball are maintained close to the desired vale of

1 m while the values of the internal product are also

similar to the desired ones (0 to the robots 1 and 2

and -1 to the robot 3). It can be noticed in the graphs

of internal product, that the internal product of two

unit vectors vary much more quickly around 0 then

around -1. Therefore, the bigger variations of the in-

ternal product in robots 1 and 2 do not mean a bad

control (as it can be seen in the plots XY).

With a velocity of 1.5m/s the controller shows

a little difﬁculty in maintaining the formation with

accuracy. The robots 1 and 2, although a little bit

late with respect to the ball, do succeed in main-

taining the desired formation. However, the robot 3

presents some difﬁculty in following the target trajec-

tory, putting itself closer than it should what made it

oscillate around the desired one.

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142

Figure 11: Maintaining formation in a straight line, simula-

tion 1c.

4.2.2 Simulation 2

This last simulation had the objective to evaluate the

behavior of the controller when the target changes

its direction abruptly. It has also the objective to

evaluate if the robots are capable of reset their for-

mation positions correctly avoiding any collision be-

tween them. For that porpoise, two types of corners

were tested: a 135

corner (a less abrupt angle) and a

corner (or a right angle). The results can be seen

in Fig. 12 using a speed of 0.5m/s.

By each change of direction the entire formation

turns around the target and the robots reset their posi-

tion in formation in a way to occupy the correct posi-

tions. Observing the plots of distance with respect to

the time and internal product with respect to the time,

the formation takes about three seconds to reset.

5 CONCLUSIONS

In this paper a novel approach of a Non-linear Model

Figure 12: Maintaining formation in a corner trajectory,

simulation 2a.

Predictive Controller was presented to formation con-

trol of omnidirectional mobile robots. The devel-

oped framework showed to be very ﬂexible and eas-

ily adaptable. The projected controller is capable of

making the team of robots to converge to the de-

sired geometry around a target, even if the robots

are very far apart. The merit of this accomplishment

can be directed to the minimization algorithm used,

the RPROP. Also, due to the high penalization on

the extreme proximity between robots, the collisions

between them are effectively limited or completely

avoided.

In terms of maintaining the formation, the ob-

tained results are highly dependent of the target veloc-

ity characteristics. In low velocities the results were

obviously better due to the fact that the errors from the

incorrect predictions do not result in big deviations in

the desired geometry as it does when the velocities

are high. In general, the controller reacted well to

abrupt changes in the target speed direction, with the

formation circulating the target instantaneously after

the change in direction. Consequently, it made the ge-

ometry to be reset to accommodate the new direction.

LEADER FOLLOWING FORMATION CONTROL FOR OMNIDIRECTIONAL MOBILE ROBOTS - The Target

Chasing Problem

143

ACKNOWLEDGMENTS

The authors thank the INESC Porto for contributing

to this work and to the FCT (Fundac¸˜ao para Ciˆencia e

Tecnologia) from Portugal for supporting the project

PTDC/EEA-CRO/100692/2008 - ”Perception-Driven

Coordinated Multi-Robot Motion Control”.

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