Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 1

ISS N 2229-5518

Churning Multiple Communication Sources in

Cooperative Control of a Mobile Robot

Chimpalthradi R. Ashokkumar

.

—————————— ——————————

Nomenclature:

a : Op en loop sy stem matrix.

A : Closed loop system matrix.

b : Control influence matrix.

C : Communication matrix elements.

c : Interconnections

f : Feed forward control.

k : Feedback controller.

N : Number of robots engaged in the formation.

p : Position.

p : Velocity .

r : Radius of a circle for coop eration.

t : Time instants.

T : Formation matrix.

u : Control inp ut.

x : State vector.

y : Communication vector.

: Coop erative control index indicatin g velo city directions.

Subscripts:

a : Lead er.

i, j, 1, 2 : Followers.

i : Follower who gives information to its associate j.

1 : Follower who gives information to its associate 2.

j : Follower receivin g information from a and i.

2 : Follower receivin g information from *a *and 1.

r : Lead er or follower.

Cooperative control of multi -vehicle s ys tems has been an active res earch area in recent years [1,2]. Higher level control is gene r- ally mixed with decis ion ma king polic ies [3,4]. Various lower level control methods for multi -vehicle s ys tems are dis cuss ed in the firs t paper [5] of the s pecial is s ue [1]. So me of the control methods are demons trated experimentally [6]. Such an exerc is e for an air vehic le or a mobile robot cooperating with mu ltiple s ys tems with multip le commun ication s ources require further understanding of churning [7]. Churn ing is a co mputational o p- tion to invite data hierarchically fro m mult iple s ources and process them at the centralized or decentralized p latform o n- board a system cooperating with thos e s ystems providing the data. In this paper, a decentralized a rchitecture for leader- follower fra mework is cons idered. All the fo llo wers are as s umed to be cooperating with the leader which is ass igned to s eek a target. The leader is pro viding its informat ion to all the fo llo w- ers . In addition, one of the follower s ystems further receives

———— ——— ——— ——— ———

informat ion fro m its ass ociate. While s eeking the target, this paper develops a cooperative control algorithm and inves t igates the cooperative behavior o f the follower s ys tem that receives informat ion fro m the mu ltiple s ources . It is s hown that churning s ignificantly affects performance. Further, the initia l conditions and the direction of the velocity vectors are critica l to develop a formation with cooperation among multiple co mmun ication s ources .

In leader-follower fra mework, the informat ion flow a mong the s ys tems is shown to be process ed at the feed forward path of the repres entative systems [8,9]. Thus , a feed forward gain propo r- tional to a communication pattern is required to cooperate with the s ys tems providing the information. In doing s o, s ince the informat ion s haring among the s ystems is fro m mu ltip le s ources , it is required to cons ider a s pecific co mmun ication pattern and determine gains of the feed forward controller. Thus , various options aris e to represent the communication pattern and churn the data for a cooperative performance. In this paper, the info r- mat ion for cooperation is as sumed to be available a lternatively. Accordingly, a control alg orith m fo r the follower that receives s uch informat ion is developed to s eek a target along with the leader. Un like a s warm which s eeks a moving target [10], in order to unders tand the effects of churning on coo peration, the

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 2

ISS N 2229-5518

target is as sumed to be fixed. A lthough cooperative performance deteriorates at the expens e of data churning, the sys tem with mu ltip le communication s ources is capable of performing a ce r- tain tas k ass igned to either one of the s ys tems s haring the info r- mat ion.

The paper is organized us ing an exa mp le that enhances the t u- torial va lue. Ma inly, an interco nnection in terms of the velocity vector direction for cooperation a mong the dis tributed s ystems is dis cus sed. Then, various iss ues to process multiple co mmunic a- tions at the local control s tru cture of the s ys tem are pres ented. In decentralized platform, for a followe r with mult iple co mmunic a-

move ments in Eq. (2). The circle c rite rion is further chos en to s tudy the influence of the changing velocity directions of the follower s ys tem with res pect to the leader. Cons ider a linear time-invariant feedback controller *k *j and define a two-degree of freedom control law,

u j (t) k j x j (t) f j (t) . (3) The clos ed loop dynamics becomes ,

x j (t 1) Aj x j (t) b j f j

Aj (a j b j k j ) ;

*p *j (0)

tion s ources , an algorith m for churning the data is developed and

e xa mples are illus trated. Co nclus ions and future directions in

x j (0) p

(0)

this res earch area are s umma rized.

In leader-follower fra mework, the pos ition ( pr ) and velocity

2

Each followe r s ys tem fro m its initia l condition *x*r(0) is driven to origin in a certain duration of time a long with their leader. The feed forward control input *f*j(*t*) is introduced to enhance coopera- tion between the leader and its followers . Given the leader in- formation, the feed forward control proportional to the leader informat ion is applied at the follower s ys tems . However, at the

follower s ys tem *x*j(*t*), the feed forward control is proportional to

( p r ) of a finite s et of mobile robots in *R*

are shown in Fig.1.

the leader as well as its as sociate inputs . The objective of this

The dynamics of each robot is ass umed as an integrator. The

dis crete-time model with s amp ling one time unit is ,

paper is determine s uch a feed forward control for the follower s ys tem *x*j(*t*) s uch that the cooperative constraint in Eq. (2) is

*p*r (*t * 1) 1

1 *p*r (*t*)

0.5

(1a)

met.

*u*r (*t*)

*p* r (*t * 1)

or,

0 1 *p* r (*t*)

1

2

The cooperation given by Eq. (2) is augmented in the s ys tems as

follows . An interconnection is developed by differentiat ing Eq. (2) with res pect to time.

xr (t 1) ar xr (t) br ur (t),

xr (t) R

(1b )

pa p a

p j p j

0 pa

p j

*p* j

p a

(4)

*p*i (0)

Eq.(4) leads to *x*a(*t*) – *x*j(*t*) interconnection as follo ws ,

*p*3 (0) R2

*p* i (0)

1

c (t)

0 *p*a (*t*)

a

0 *p *j (*t*)

(5)

*p* 3 (0)

j

*p*a (0)

0 a *p* a (*t*) 0

1 *p* j (*t*)

*p* a (0)

Ta Tb

*p *j (0)

Us ing the aggregate model dis cuss ed in Ref.11 and its interco n-

nections in Eq. (5), feed forward control at the followe r s ys tem

*x*i(*t*) is applied for a=1 and a= 1, res pectively. Further, Eq.(4)

*p* j (0)

Fig. 1. Fo llo wer *x*j(*t*) cooperating wit *x*a(*t*) and *x*i(*t*).

Here *t *refe rs the time units for the dis crete time mode l taking the values 0,1, . In Figure 1, the leader is denoted by the s ub

s cript *a. *Cons istent with the Fig. 1, the followe rs are denoted by the subs cripts *i, j *and 3, res pectively. The leader or fo llower s ys tem is denoted by *r*[*a*,1, ,*j*,*N*]. Sys tems *x*a(*t*), *x*i(*t*) and *x*j(*t*) are as sumed to be in a formation to s eek a target at the ori- gin. Leader *x*a(*t*) commun icates with its followers *x*i(*t*) and *x*j(*t*)

res pectively. In addition, *x*i(*t*) also communicates with its ass o-

ciate *x*j(*t*) for further cooperation. A cooperation is cons idered by an algebraic cons traint given by,

implies that when a=1, the follower s eeks the origin by coop e- rating with its leader in s uch a way the velocity co mponents are the same in both direction and magnitude. Fig.2a and Fig.2c for various initia l conditions in firs t and third quadrants depict this s ituation, where the follower s ystem attempts to align its elf with the leader us ing a feed forward control. Thus the initial cond i- tions (15,1) and (-15,-1) a re concluded as cooperative. In the abs ence of feed forward co ntrol, it is obs erved that the follower s ys tem becomes uncooperative to both the initia l conditions . The interconnections a= 1 in Fig.2b and Fig.2d draw the s ame conclus ions . However, the formations are diffe rent with the v e-

locity vectors of the leader and follower s ystems actin g in oppo- s ite directions .

If the des ign cons traints such as the s tate and control input s at u- ration limits are impos ed, majority of the inte rconnections for

comple x maneuvers become infeas ible. Cons ider the des ign

p 2 (t) p 2 r 2

(2)

a j a

cons traints ,

That is , the follower move ments are such that at each time in- s tant they s atis fy a circ le criterion with res pect to the leader

{25 *p*r (*t*) 25, 5 *p* r (*t*) 5} .

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 3

ISS N 2229-5518

Clearly, the initia l condition (15, 1) with a=1 is cooperative with feed forward control. But it becomes invalid becaus e the maneuver is comple x. In addition, if the s ys tem begins to coop e- rate with other ass ociate following the leader, cooperative pe r- formance with init ial conditions such as (15,1) may further det e- riorate. This as pect of the s tudy is formulated ne xt.

Problem Definition **: **A more comp le x p roble m is pos ed as fo l-

lows . In addition to the leader to followe r co mmunicat ion, let the followe r *x*j(*t*) receives information fro m its ass ociate *x*i(*t*) s uch that,

a) *x*j(*t*) cooperates with *x*i(*t*) and

b) Both *x*i(*t*) and *x*j(*t*) cooperate with *x*a(*t*).

In this cas e, the cooperative cons traints in terms of the two alg e-

braic cons traints are,

p 2 (t) p 2 r 2 and

a j a

p 2 (t) p 2 r 2 .

fro m both is pres ented. In the following s ections , a decentralized computational platform is ass umed to s tudy the problem.

In the previous section, modeling of s patially dis tributed s ys tems with co mmunication fro m mu ltiple s ources is pres ented for a decentralized architecture. Depending upon the c ommunication s ources , a procedure to handle mult iple dyna mic mode ls pre- s ented in Eq. (9) becomes neces sary. This in turn pos es s ome computational is s ues in the feed forward path. Cons ider the fo r- mat ion of a leader *x*a(*t*) with two associates , namely *x*1(*t*) and *x*2(*t*). That is , the three systems , *i *=1 and *j *= 2 are cons idered. As a result, follower 2 receives informat ion fro m its leader *x*a(*t*) and its ass ociate *x*1(*t*). In Fig.3, a typical co mputational platform with feed back controlle r *k *2 and reconfigurable feed fo rwa rd controllers {ga,g1} for the s ystem *x*2(*t*) is pres ented. The co m-

municat ions are accordingly received and process ed at the feed

i j i

Now, fo llo wer *j *s atis fies the follo wing t wo independent inte r- connections , one to cooperate with the leader and another to cooperate with its ass ociate,

forwa rd path.

1

0 *p*a (*t*)

a

0 *p *j (*t*)

c j (t)

*p*

(*t*)

(6)

0 a *p* a (*t*) 0

1 j

Ta

Tb

c~ t

1 0 *p*i (*t*)

i

0 *p *j (*t*)

j ( )

*p*

(*t*)

(7)

0 i *p* i (*t*) 0

1 j

~

Ta

~

Tb

Upon trans formation, the aggregate dynamics for the follower xj(t) that need to cooperate with both xa(t) and xi(t) is as follo ws : x j (t 1) Aj x j (t) xa (t) b j f j (t) , (8a)

1 1

Fig. 3. A Co mputational plat form for a fo llo wer s ystem receiv-

where,

*T*b

Ta Aa AjTb

1

Ta .

ing information fro m mu ltip le co mmunication s ources .

The trans formation

x j Tb

c j is with res pect to the fo rmation

matrix *T*b . The leader co mmunication is denoted us ing a s tru c- tured matrix *T*a . Similarly, the dynamics of *x*j(*t*) with res pect to

~

Suppos e a centralized architecture is required. The integrated

dynamics of the dis tributed s ys tems for the centralized archite c- ture is as follo ws :

its as sociate’s communication

Ti , is derived. The follower -

follower interconnection with respect to the trans fo rmation

*x*a (*t * 1)

*a*a 0

0 *x*a (*t *)

*b*a 0

0 *u*a (*t*)

~ 1

~*x *j *T*b*c*~j is given by,

*x*1 (*t * 1) 0 *a*1

0 *x*1 (*t*) 0 *b*1

0 *u*1 (*t*)

~*x *(*t *1) *A *~*x *(*t*) ~*x *(*t*) *b*

f (t) , (8b)

*x*2 (*t * 1)

0

0 *a*2 *x*2 (*t*)

0

0 *b*2 *u*2 (*t*)

j j j i j j

~ ~ 1 ~

~ 1 ~

(10a)

where, *T*b

Ti Ai AjTb

Ti .

It can accommodate unlimited info rmation in a vector s imilar to

The feed forward controller at the followe r *x*j(*t*) with mu ltiple communicat ions becomes ,

the meas urement vector in conventional control, which is re-

ferred as a co mmunicat ion vector,

f j (t) ga (t)xa (t) (or)

*C*aa 0

0 *x*a

(*t*)

f j (t) gi (t)xi (t)

(9)

y(t) C1a

*C*

C11

C

0 *x*1 (*t*) . (10b)

C x (t)

Eq.(9) blends the information it rece ives fro m both *x*a(*t*) and

2a

21 22 2

xi(t). Accordingly, the feed forward gains

ga , gb

*R*12

are re-

The diagonal entries are the standard meas ured s ignals applic a-

ble for a conventional control sy stem. The off -diagonal entries

configured. Various poss ibilities do exis t to churn the data from

s ys tems *x*a(*t*) and *x*i(*t*). In the pres ence of trans mis s ion delay, the problem becomes comp le x. In order to avoid conges tion, a s im- ple procedure that accepts data from either *x*a(*t*) or *x*i(*t*) but not

indicate the communication s ources among the dis tributed s ys- tems . The triangular s tructure in Eq.(10b) has a s pecial reference to the decentralized architecture. The control law for this s pecif- ic co mmunicat ion vector will be of the fo rm,

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 4

ISS N 2229-5518

*u*a (*t*)

*k*aa 0

0 *x*a (*t*)

Case (a): x1(t) and x2(t) receive informat ion fro m xa(t) and coo-

perate with *x*a(*t*). This is the s tandard leader-fo llo wer control

*u*1 (*t*) *k*1a

k11

0 *x*1 (*t*) . (10c)

problem where each fo llo wer receives informat ion fro m s ingle

*u*2 (*t*)

*k*2a

k21

k22 x2 (t)

s ource, that is , the leader *x*a.

The output vector is given by,

*c*1 (*t*)

Ta Tb

0 *x*a (*t*)

Case (b): As in Cas e a. In addition, x2(t) begins to communicate

*c *(*t*)

0 *T * *x*

t . (10d)

with *x*1(*t*) to develop cooperation (may be to perform a d ifferent

2 *T*a

~ ~

1 ( )

tas k when *x*1(*t*) takes a leaders hip).

*c*~ (*t*)

0 *T*

T x2 (t)

Fig.5 s hows Cas e a, wherein, the followers *x *(*t*) and *x *(*t*) coope-

2 *i b *2 1

Although the centralized platform is s uitable to accommodate communicat ion sources in the s tate-s pace model, it is difficu lt to synthes ize a s tructured controller as in Eq.(10c) . Fu rther, the output vector is not dimens ionally s table and its s tructure is e x- pected to vary. Thus the s tudy confines to the decentralized computing platform.

When a communicat ion delay is not pres ent, a communication pattern adopted to develop the cooperative control algorith m is s hown in Fig. 4. He re the s ystem rece iving informat ion fro m

xa (t )

x1 (t )

rate with *x*a(*t*). The initial condition for each sys tem is cons i-

dered in the firs t quadrant as (10,2). The e ffect of in itia l cond i-

tions when followers are in third quadrant is illus trated in Fig. 2. The trajectories of the fo llowe rs cooperating with *x*a(*t*) s uggest that the cooperative co ntrol cons traint is s atis fied after a fin ite number of time s teps . The feed forward control input his tories (*f*1(*t*) at *x*1(*t*) and *f*2(*t*) at *x*2(*t*)) are also compared. In each of thes e s imulations , the interco nnection is defined us ing

a 1 1 . That is , the speed of each s ys tem does not exceed the s peed of the other system. In Fig.6, Cas e b is illus trated. Be- caus e *x*2(*t*) receives commun ication fro m both *x*1(*t*) and *x*a(*t*), it is trying to cooperate with both of them. Further, s a me in itia l co n- dition for each fo llo wer is as sumed. Thus , for the as sumed communicat ion pattern in Fig. 4, the type of cooperation exis ts between the s ystems *x*2(*t*) and *x*1(*t*), and between the s ys tems *x*2(*t*) and *x*a(*t*) does not change. When the initial conditions are

modified, the cooperation is s ignificantly mod ified as shown in

Sampling @ Processor

x2 (t )

t

Fig.7. As before, the interconnections are chos en us ing

a 1 1 . When multiple co mmunicat ion s ources are intro- duced, the initial cond itions s ignificantly affect the formations . The cooperative cons traints of s ys tem *x*2(*t*) with its leader *x*a(*t*)

Initial time Final time

Fig. 4. A co mmunicat ion pattern for cooperative control alg o-

and its associate *x*1(*t*) can be inferred. With the exis ting commu-

nication pattern, it is obs erved that *x*2(*t*) in Fig.7 is uncooperative to the initial condition (10,2). Although *x*2(*t*) is cooperative to

rith m

(10,2), the s peed cons traint with

a 1 1 becomes effect ive

more than one s ys tem is identified as *x*2(*t*). When informat ion fro m *x*a(*t*) is received, it is as sumed that no data fro m *x*1(*t*) is received. Thus no data conges tion occurs at s ys tem *x*2(*t*). The trans mis s ion time for this type of data trans fer is further a s- s umed twice the s ampling time of the process or at each homo- geneous s ys tem engaged in the format ion.

Als o, system *x*2(*t*) receives both pos ition and velocity informa- tion fro m *x*a(*t*) and *x*1(*t*), res pectively. The control s tructure at the feed forwa rd path is fixed as a proportional plus derivative co n- trolle r. An algorithm is developed to minimize the error b etween the leader trajectory and the followe r tra jectory. The feed fo r- ward gains for the co mmunication pattern in Fig. 4 a re acco r- dingly determined. If *x*2(*t*) makes an attempt to cooperate with

and churning occurs as it approaches the ta rget.

Control of s patially dis tributed s ys tems with mu ltip le commun i- cation s ources pres ents major challenges . This paper focus es on mu ltip le models res ulting fro m mult iple co mmun ication s ources . Unlike d is tributed s ys tems with s ingle communication s ource as in a s tandard leader-fo llo wer fra me work, when more than one communicat ion s ource is introduced, churning inevitably occurs and cooperative performance deteriorates . Us ing an a commun i- cation pattern, multip le models among the dis tributed s ys tems are us ed to develop a cooperative control algorithm. Various e xa mples with in itia l condition uncertainty are illu s trated. It is

concluded that format ion is s ens itive to initial conditions and is

xa(t), then interconnection c2

becomes effective with gain *g*a(*t*).

difficult to achieve es pecially when mult iple co mmun ication

Similarly, if *x*2(*t*) ma kes an attempt to cooperate with *x*1(*t*), then

~

s ources are present. However, such sys tems can cooperate with

the interconnection c2

becomes effective with gain *g*1(*t*). The

either one of the s ystem provid ing the informat ion a nd perform

cooperative control algorith m is compared in t he fo llo wing t wo cas es :

diffe rent tas ks . Currently, churning in s warms is under inves t i-

gation.

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 5

ISS N 2229-5518

Fig. 2. Fo llo wer * x*i(

Fig. 5. Cooperative s ys tems with s ingle co mmun ication s ource.

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 6

ISS N 2229-5518

Fig. 6. Cooperative Sys tems With Multip le Co mmun ication Sources .

Fig. 7. In itia l Condition Effects on Cooperative Control.

IJSER © 2012

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 7

ISS N 2229-5518

[1] Special Is s ue on Analys is and Control of Multi-Agent Dynamic Sys tems , *Journal of Dynamic Systems, Measurement and Control*, Vol 5, pp: 129, 2007.

[2] Special Is s ue on Cooperative Control Approaches for Multiple Autonomous Vehic les , *International Journal of Systems Science*, Vol. 37, No. 6, 2006.

[3] J. How, E. King, and Y. Ku wata, ―Flight De mons trations of Cooperative Control for UA V Tea ms ,‖ AIAA 3rd "*Unmanned Unlimited*" *Technical Conference, Work shop and Exhibit 20 *- 23 September 2004, Ch icago, Illinois , AIAA 2004-6490.

[4] G. Vachts evanos , L. Tang, G. Dro zes ki, and L. Gut ierrez,

―Intelligent Control of Unmanned Aerial Veh icles for Improved Autonomy and Relaib ility,‖ *5th IFAC/EUR ON Symposium on Intelligent Autonomous Vehicles*, Instituto Superior Técn ico, Lis boa, Port ugal, July 5-7, 2004.

[5] R.M. Murray, ― Recent Res earch in Cooperative Control of

Multi-Vehic le Sys tems ,‖ *Journal of Dynamic Systems, Measurement, and Contro*l, Vo l. 129, No. 5, pp. 571-584,

2007

[6] E. King, Y. Kuwata, and J.P. How, ― Expe rimental

De mons tration of Coordinated Control of Multi-Vehic le

Teams ,‖ *International Journal of Systems Science *, Vol.

37, No. 6, pp. 385-398, 2006.

[7] W.J. Curt is , ―Churning: Repeated Optimization and Cooperative Ins tability,‖ In *Recent Developments in Cooperative Control and Optimization *. Sergiy Butenko, Robert Murphey and Panos Pardalos , Eds ., Kluver Academic Publis hers , Norwe ll, Mas s achus etts , pp: 105-

114, 2004.

[8] A.J. Fa x, and R.M. Murray, ― Graph Lap lacians and Stabilization of Vehicle Fo rmations ,‖ The 15th IFAC World Congress, June 2002.

[9] A.J. Fa x, and R.M. Murray, ―Information Flow and Cooperative Control of Veh icle Format ions ,‖ *IEEE Transactions on Automatic Control*, Vol. 49, No. 9, pp:

1465-1476, 2004.

[10] J. Yao, R. Ordóñez, and V. Ga zi, ― Swa rm T racking Us ing Artificia l Potentials and Sliding Mode Control,‖ *Journal of Dynamic Systems, Measurement, and Control *Vol. 129, No. 5, pp. 749-755, 2007.

[11] C.R. As hokku mar, and D.E. Je ffcoat, ―Cooperative

Systems Under Communication Delay,‖ *AIAA Guidance, Control, and Navigation Conference*, Aus tin, TX 11-13 Aug

2003.

IJSER © 2012