Hybrid Control Structure for Multi-robot Formation

K. Diamantaras, W. Duch, L.S. Iliadis (Eds.): ICANN 2010, Part II, LNCS 6353, pp. 307 – 316, 2010.

© Springer-Verlag Berlin Heidelberg 2010

Hybrid Control Structure for Multi-robot Formation

Daniela Cernega and Razvan Solea

Department of Control Systems and Industrial Informatics,

Computer Science Faculty, “Dunarea de Jos” University of Galati,

Domneasca 47, 800008, Galati, Romania

{Daniela.Cernega, Razvan.Solea}@ugal.ro

Abstract. In this paper a hybrid control structure to control a multi-robot forma-

tion is proposed. The hybrid control structure consists of two control levels: the

discrete control level and the continuous control level. The discrete control level

ensures the supervisory control and the continuous control level ensures the tra-

jectory tracking control. The trajectory tracking problem is solved using the slid-

ing mode control. The syntheses of the supervisor and of the sliding mode con-

trollers for each discrete state are presented. Simulation example is used to

evaluate the sliding-mode algorithm and to show the application of the algorithm

in practice. The controller is simply structured and easy to implement.

Keywords: Hybrid control, Sliding-mode control, Multi-robot formation control.

1 Introduction

The multi robots systems is an important robotics research field. Such systems are of

interest for many reasons; tasks could be too complex for a simple robot to accom-

plish; using several simple robots can be easier, cheaper and more flexible than a

single powerful robot [1], [2], [3], [4].

Formation control has been one of the important research topics in multiple robot

systems as it is applicable to many areas such as geographical exploration, rescue

operations, surveillance, mine sweeping, and transportation. Different approaches

have been developed recently, for example, behavior-based control, LQ control, vis-

ual servoing control, Lyapunov-based control, input and output feedback linearization

control, graph theory, and nonlinear control.

In leader-follower formation control, the most widely used control technique is

feedback linearization based on the kinematics model of the system.

In this study the hybrid leader-follower robot formation control is considered. The

referenced robot is called leader, and the robot following it, is called follower. Thus,

there are many pairs of leaders and followers and complex formations can be

achieved by controlling relative positions of these pairs of robots respectively. This

approach is characterized by simplicity, reliability and no need for global knowledge

and computation.

The hybrid control structure consists of two levels: the discrete control level im-

plementing the supervisory control, and the numerical control level using a sliding-

mode controller to solve the trajectory tracking problem.

308 D. Cernega and R. Solea

To control a multi-robot formation as a discrete event system means to follow a de-

sired behavior described through the imposed constraints. The desired behavior is

modeled as discrete event system for the entire formation and a supervisor is designed

to achieve this behavior. The approach used to design the supervisor is proposed in

[5], and in [6].

The discrete control level is coupled with the numerical control level and it detects

the functioning situations. Each functioning situation is a discrete state. Each discrete

state is characterized with a continuous model as shown in [7]. For each of the situa-

tions detected at the discrete event level the appropriate continuous model is selected

together with the corresponding continuous controller. For each discrete state the

references of the corresponding continuous controllers are also established.

The model for a hybrid automaton (HA), is defined with:

( )1 2 3 0, , , , , , , fHA X Q Q Qµ µ µ= Σ

(1)

where: X is the vector space of the system state x, the continuous state vector of the

system, denoted by [ ]1...

T

nx x x X= ∈ , supposed to be continuous observable vector;

Q the set of discrete states corresponding to all the possible phases, Q = {qi, i=1…m};

the hybrid state of the system is defined within the pair (x, l)∈X x Q; µ1 is the set of the

m vector fields associated to each discrete phase; µ2 is the set of the constraints associ-

ated to each discrete phase; Σ is the set of the events; µ3 is the set of functions associ-

ated with the events; Q0 is the set of the initial states and Qf is the set of final sates.

In the next section the hybrid control system is presented. The discrete dynamic

event system model for the multi-robot formation is obtained. The supervisor to en-

sure the desired behavior of the system is designed. The continuous models for each

discrete state are also presented.

The wheeled mobile robot is a nonlinear system. The continuous control level is

dedicated to the trajectory tracking control [8] - [12]. The trajectory tracking control

problem is solved using the sliding mode control. In the Section 4 this problem is

solved and the control laws together with the references for the discrete states are

obtained. Section 5 is dedicated to the results obtained after implementing this hybrid

structure.

2 The Hybrid Control Structure

The hybrid control structure proposed in this paper is shown in Fig.1.

To control the multi-robot formation as a discrete event system leads to a supervisor

design. The discrete event model used is the automaton called G, defined as follows:

G = (Q, , !, q0, Qm) (2)

where Q is the set of the discrete states physically possible of the system, is the set

of all the events, ! is the transition function of the automaton, q0 is the initial state and

Qm is the set of the marked states of the system. The events in the follow the leader-

formation are from two distinctive categories: u is the set of the uncontrollable

events and c is the controllable events set. The controllable events are subject of the

Hybrid Control Structure for Multi-robot Formation 309

Fig. 1. Hybrid Control Structure Fig. 2. The sensitivity sphere of robot

control action and these events can be enabled and disabled at any time i.e. from any

state. The uncontrollable events cannot be enabled or disabled by the control action.

Supervisory control for this discrete event system has the objective to ensure the

desired behavior of the follower robot according to some constraints imposed.

The supervisor design for this problem is based on sonar data. The sensitivity

sphere is a concept defined in order to establish the smallest distance equal to the

length of the follower robot in order to avoid collision with the leader or with other

obstacles. The sensitivity sphere is represented in Fig. 2.

The analysis of the robot motion according to the defined sensitivity sphere, for

this problem generates seven cases:

-Case 1: the sonar 3 and 4 detect an object inside the sensitivity sphere and the ro-

bot will receive references to move ahead;

-Case2: sonar 2 and 3 detect an object inside the sensitivity sphere and the supervi-

sor generates references for the robot motion the constant distance d and the angle ;

- case3: sonar 4 and 5 detect an object inside the sensitivity sphere and the supervi-

sor generates references for the robot motion the constant distance d and the angle - ;

-Case4: sonar 1 and 2 detect an object inside the sensitivity sphere and the supervi-

sor generates references for the robot motion the constant distance d and the angle 2 ;

-Cas5: sonar 5 and 6 detect an object inside the sensitivity sphere and the supervisor

generates references for the robot motion the constant distance d and the angle -2 ;

-Case6: any sonar detects an object inside the sensitivity sphere closer then the

minimum allowed distance and the supervisor generates references for the robot to

stop;

-Case7: no sonar pair detects an object and the supervisor generates references for

the robot circular motion in order to search the leader.

These cases are generating the discrete states set, Q, of the automaton G, model of

the process defined in (2).

The objective of the supervisory control for the follower is to track the leader when

in the environment there are some unknown obstacles identified within the sensitivity


sphere, or another leader appears inside the sensitivity zone. The discrete events gen-

erating discrete state transitions in this system are:

- c = { ec0, ec1, ec2, ec3, ec4}, where ec0 - the start command, ec1 - the distance es-

tablished between the two robots is respected, ec2 – start the distance evaluation, ec3 –

command the robot movement with speed references inside the established limits, ec4

- the sonar data are valid;

- u = {e1, eu1, eu2, eu3, eu4}, where e1 - end initialization, eu1 - an obstacle appeared

in the interior of the sensitivity sphere, eu2 the leader is lost, eu3 reading errors from

sonar detected, eu4 another leader appeared inside the sensitivity sphere.

The discrete state set, Q contains the states defined as follows: q1 robot initializa-

tion; q2 sonar reading; q3 nearest limit verification for all the sonar; q4 data analysis

from sonar 3 and 4; q5 trajectory tracking algorithm for case 1, q6 movement accord-

ing to case1 references, q7 data analysis from sonar 2 and 3, q8 trajectory tracking

algorithm for case 2, q9 movement according to case2 references; q10 data analysis

from sonar 4 and 5; q11 trajectory tracking algorithm for case 3; q12 movement accord-

ing to case3 references; q13 data analysis from sonar 1 and 2, q14 trajectory tracking

algorithm for case 4, q15 movement according to case4 references, q16 data analysis

from sonar 5 and 6, q17 trajectory tracking algorithm for case 5, q18 movement accord-

ing to case5 references, q19 180 degrees rotation, q20 STOP, q21 corresponds to the

situation when an obstacle appears inside the sensitivity sphere during the movement

corresponding to one the states q6, q9, q12, q15, q18; this state provides the supervisor

the ability to avoid collisions, q22 is the state to be avoided with the supervisor con-

trol: the collision state.

The transition function, δ, of the automaton is represented in figure 3.

q1 q2

q3

q4 q5 q6

q7 q8 q9

q10 q11 q12

q13 q14 q15

q16 q17 q18

q19

q20

q21 q22

ec1

ec0

ec2 ec2

eu1

eu1

ec4

ec2 ec3

ec2 ec3

ec3

ec3

ec3

ec2

ec2

ec2

ec1

ec1

ec1

ec1

ec1eu3

eu3

eu3

eu3

eu1

eu1

eu1

eu1

eu1

ec3

eu1

eu1

Fig. 3. The automaton G, model of the discrete event system

3 Leader-Following Formation Models

Figure 4 is a leader-following control model where the formation pattern is specified

by the separate distance d and the relative bearing ψ for two robots r1 and r2. The

desired formation pattern can be defined as the desired separate distance dd and the


relative bearing ψd. The follower r2 regulates the formation state errors of the separate

distance and the relative bearing through its speed control signals ur2 = [vxr2 ωr2]T.

!

"#$

%−

!

"#$

%=

!

"

##$

%

ψψψ

dddd

d

~

~

(3)

The relative distance between the leader and the follower robot is denoted as d, the

separation bearing angle is ψ, and they are given by:

( ) ( )2

21

2

21 crcr yyxxd −+−= ; ( )[ ]21211 ,2arctan crcrr xxyy −−−−= θπψ

(4)

where: )sin(),cos( 222222 rrcrrc lyylxx θθ ⋅+=⋅+= (5)

The formation control can be investigated by modeling the formation state error as

follows [13]:

2112 ,~

~

rrrr uFuGd ωωφψ

−=⋅+⋅= !

"

##$

%

(6)

and

( ) ( )( ) ( )

!

"

##

$

%

−=

!

"

##

$

%+⋅

−+

+⋅−+−=

1)sin(

0)cos(

,cossinsincos

d

F

d

l

d

l

G ψψ

ψφψφψφψφ

where: [ ]Tririri vu ω= , 21 rr θθφ −= and l is the distance between the robot position

(xr2, yr2) and the robot hand position (xc2, yc2) as shown in Fig. 4.

Fig. 4. Leader-following formation models


4 Sliding-Mode Controller Design

In a leader-follower configuration, with the leader’s position given and once the

follower’s relative distance and angle with respect to the leader are known, the fol-

lower’s position can be determined. To use the leader-following approach, it is as-

sumed that the angular and linear velocities of the leader are known. In order to

achieve and maintain the desired formation between the leader and follower, it is only

need to control the follower’s angular and linear velocities to achieve the relative

distance and angle between them as specified. Therefore, the leader-following based

mobile robot formation control can be considered as an extension of the tracking

control problem of the nonholonomic mobile robot.

A practical form of reaching the control law (proposed by Gao and Hung [14]) is

defined as:

.2,1,0,),sgn( =>⋅−⋅−= iqpsqsps iiiiiii

(7)

By adding the proportional rate term – pi⋅si, the state is forced to approach the switch-

ing manifolds faster when si is large. It can be shown that the reaching time for x to

move from an initial state x0 to the switching manifold si is finite, and is given by

i

iii

i

iq

qsp

pt

+⋅= ln

1. A new design of sliding surface is proposed, such that dis-

tance between the leader and the follower robot, d, and the separation bearing angle,

ψ, are internally coupled with each other in a sliding surface leading to convergence

of both variables. For that purpose the following sliding surfaces is proposed:

dkds d

~~1 ⋅+=

; φψψψ ψ ⋅⋅+⋅+= )~sgn(~~

02 kks

(8)

where k0, kd and kψ are positive constant parameters and φψ ,~,~d are defined by (3).

If s1 converge to zero, trivially d~

converge to zero. If s2 converge to zero, in

steady-state it becomes ( ) φψψψ ψ ⋅⋅−⋅−= ~sgn~~0kk . Since φ is always bounded, the

following relationship between ψ~ and ψ ~ holds: IF 0~0~ >&< ψψ and IF

0~0~ <&> ψψ .

From the time derivative of (8) and using the reaching law defined in (7) yields:

)sgn(~~

11111 sqspdkds d ⋅−⋅−=⋅+=

(9)

)sgn()sgn()~sgn(~~222202 sqspkks ⋅−⋅−=⋅⋅⋅+⋅+= φφψψψ ψ

(10)

After some mathematical manipulation, one can achieve:

)cos(

~)sgn( 11111

2ψφ +

−⋅+⋅+⋅=

Cdksqspv d

c

; ( )

)cos(

~)sgn( 22222

2ψφ

ψω ψ

+⋅

−⋅⋅+⋅+⋅=

l

Cdksqspc

(11)


where

)sin()cos(

)()()sin()~sgn(

)sin()cos()()()sin(

11

1202

11121

ψφψ

ωψψφψφφφψ

ψφψωψψφψφω

⋅⋅−⋅−

−+⋅+⋅++⋅−⋅⋅⋅⋅=

⋅⋅−⋅−+⋅+⋅−+⋅⋅=

rr

rr

rrrr

vv

dvdkC

vvdlC

The sgn functions in the sliding surface were replaced by saturation functions, to

reduce the chattering phenomenon [15].

5 Simulation Results

In this section, some simulation results are presented to validate the proposed control

law. To show the effectiveness of the proposed sliding mode control law numerically,

experiments were carried out on the multi-robot formation control problem.

High-level control algorithms (including desired motion generation) are written in

C++ and run with a sampling time of Ts = 100 ms on a embedded PC, which also

provides a user interface with real-time visualization and a simulation environment.

All the simulations were made using the MobileSim. MobileSim is a software for

simulating MobileRobots’ platforms and their environments, for debugging and ex-

perimentation with ARIA. The ARIA software can be used to control the mobile

robots like Pionner, PatrolBot, PeopleBot, Seekur etc. ARIA (Advanced Robot Inter-

face for Applications) it is an object-oriented Applications Programming Interface

(API), written in C++ and intended for the creation of intelligent high-level client-side

software.

Figure 5 shows a block diagram of the proposed sliding-mode controller.

Fig. 5. Block diagram

Wheel velocity commands, R

Lv

R

Lv cc

L

cc

R

2222 ;ω

ωω

ω⋅−

=⋅+

=

(12)


are sent to the power modules of the follower mobile robot, and encoder measures NR

and NL are received in the robots pose estimator for odometric computations.

Two simulation experiments were carried out to evaluate the performance of the

sliding mode controller presented in Section 4. The first simulation refers to the case

of circular trajectory (vr1 = 0.4 [m/s] and wr1 = 0.1 [rad/s]). The initial conditions of

the leader and the follower are, xr1(0) = 0 , yr1(0) = 0, θr1(0) = 0, xr2(0) = -1, yr2(0) = -

1, θr1(0) = 0, dd = 1 [m], ψd

= 135[deg] (see Fig. 6).

In the second simulation the leader robot execute a linear trajectory but with a non-

zero initial orientation (θr1 = 45 [deg]). The initial conditions of the leader and the

follower in this second case are, xr1(0) = 0, yr1(0) = 0, θr1(0) = pi/4, xr2(0) = -1, yr2(0) =

-1, θr2(0) = 0, dd = 1 [m], ψd

= 120 [deg] (see Fig. 6).

The good performance for controlling the formation with the developed control

law can be observed from Figs. 6 - 9. The outputs of the formation system ( d~

and ψ~ )

asymptotically converge to zero, as shown in Figs. 8 and 9.

Fig. 6. Simulation results using Aria and MobileSim software (case I and II)

Fig. 7. Sliding surfaces s1 and s2 for case I and II


Fig. 8. Separate distance ( d~

) for case I and II

Fig. 9. Relative bearing (ψ~ ) for case I and II

6 Conclusions

In this study a hybrid control structure to control a multi-robot formation is proposed.

The hybrid control structure consists of two control levels: the discrete control level

and the continuous control level. The discrete control level ensures the supervisory

control and the continuous control level ensures the trajectory tracking control.

The desired formation, defined by two parameters (a distance and an orientation

function) is allowed to vary in time. The effectiveness of the proposed designs have

validated via simulation experiments.

Simulation example is used to evaluate the sliding-mode algorithm and to show the

application of the algorithm in practice. The controller is simply structured and easy

to implement. From the simulation results, it is concluded that the proposed strategy

achieves the effectiveness of desired performance.

Future research lines include the experimental validation of our control scheme

and the extension of our results to skid-steering mobile robots. For the sake of

simplicity in the present paper a single-leader, single follower formation has been

considered. Future investigations will cover the more general case of multi leader,

multi-follower formations.

Acknowledgments. This work was supported by CNCSIS-UEFISCSU, project PNII-

IDEI 506/2008.


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Hybrid Control Structure for Multi-robot Formation