Containment Control in Mobile Networks
M.Ji,G.Ferrari-Trecate,M.Egerstedt,and A.Buffa Abstract—In this paper,the problem of driving a collection of mobile robots to a given target destination is studied.In particular,we are inter-ested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader-agents,while the remaining agents,only employ local interaction rules.To this aim we exploit the theory of partial difference equations and propo hybrid con-trol schemes bad on stop-go rules for the leader-agents.Non-Zenoness, liveness and convergence of the resulting system are also analyzed.
Index Terms—Containment problems,decentralized control,graph theory,leader-following,multi-agent systems,partial difference equations.
I.I NTRODUCTION
This paper investigates a particular subarea of multi-agent control, namely the so-called containment problem where a collection of au-tonomous,mobile agents are to be driven to a given target location while guaranteeing that their motion satisfies certain geometric con-straints.The constraints are there to ensure that the agents are con-tained in a particular area during their transportation.Such is
sues ari for example when a collection of autonomous robots are to cure and then remove hazardous materials.This removal must be cure in the n that the robots should not venture into populated areas or in other ways contaminate their surroundings.
We approach this problem from a leader-follower point-of-view [1]–[3].In particular,we will let the agents move autonomously bad on local,connsus-like interaction rules,commonly found in the liter-ature under the banner of algebraic graph theory[4]–[6].However,we will augment this control structure with the addition of leader-agents or anchor nodes[7].The leaders are to define vertices in a convex polytope(the leader-polytope)and they are to move in such a way that the target area is reached while ensuring that the follower-agents stay in the convex polytope spanned by the leaders,up to a given tolerance. As such,the followers movements are calculated in a decentralized manner according to afixed interaction topology,while the leaders are assumed to be able to detect if any of the followers violate the containment property.
For the leaders,we will u a hybrid Stop-Go policy[8],[9],in which the leaders move according to a decentralized formation control strategy until the containment property is about to be violated.At this point,they stop and let the followers ttle back into the leader-polytope before they start moving again.For such a strategy to be successful, a number of results are needed,including a guarantee th
at the Lapla-cian-bad follower-control will in fact drive the followers back into the leader-polytope.Moreover,we must also ensure that such a control Manuscript received August24,2006;revid September3,2007.Current version published September24,2008.The work by G.Ferrari-Trecate was par-tially supported by the European Commission under the Network of Excellence HYCON,contract number FP6-IST-511368.The work by M.Egerstedt and M.Ji was supported by the U.S.Army Rearch Office through Grant#99838. Recommended by Associate Editor J.Hespanha.
M.Ji and M.Egerstedt are with the Georgia Institute of Technology,School of Electrical and Computer Engineering,Atlanta,GA30332USA(e-mail: magnus@ece.gatech.edu;mengji@ece.gatech.edu).
G.Ferrari-Trecate is with the Dipartimento di Informatica e Sistemistica,Uni-versitàdegli Studi di Pavia,27100Pavia,Italy and also with INRIA,Domaine de V oluceau,Rocquencourt—B.P.105,78153,Le Chesnay Cedex,France(e-mail: giancarlo.ferrari@unipv.it).
A.Buffa is with the Istituto di Matematica Applicata e Tecnologie Infor-matiche,C.N.R.,27100Pavia,Italy(e-mail:annalisa@imatir.it).
mac快捷键Digital Object Identifier10.1109/TAC.2008.930098strategy is feasible in the n of non-Zeno,live in
the n of not staying in the Stop mode indefinitely,and convergent in the n that the target area is in fact reached.This approach can also be generalized to hierarchial networks,as was illustrated by our preliminary work in [10].
II.B ACKGROUND AND M ATHEMATICAL P RELIMINARIES
农业致富好项目
In this ction we will prent the basic mathematical framework and some enabling results in multi-agent control.
We start with basic notions of graph theory.For more details we refer the reader to[11].An undirected graph G is defined by a t N G= f1;...N g of nodes and a t E G N G2N G of edges.We will also
u jN G j for denoting the cardinality of N G.Two nodes x and y are neighbors if(x;y)2E G.The neighboring relation is indicated with x y and P(x)=f y2N G:y x g collects all neighbors to the node x.A L is afinite quence of nodes such that x i01 x i,i=1;...;L.A graph G is connected if there is a path connecting every pair of distinct nodes.
Definition1:Let S=(N S;E S)be an undirected host graph and N S N S.The subgraph S0associated with N S is the pair (N S;E S)where E S=f(x;y)2E S:x2N S;y2N S g
Definition1allows basic operations in t theory to be extended to graphs.For instance,if S1and S2are two subgraphs of the graph S, then S1[S2,S1\S2,S1n S2are the graphs associated with N S[N S, N S\N S,and N S nN S,respectively.For our purpos,we will often u graphs with a boundary.
Definition2:Let S be a subgraph of G.The boundary of S is the subgraph@S G associated with N@S
:=f y2N
G n N S:9x2 N S:x y g.The closure of S is S=@S[S.
Note that the definition of the boundary of a graph depends upon the host graph G.This implies that if one considers three graphs S0 S G,the boundaries of S0in S and in G may differ.
In the context of multi-agent systems,the nodes of the host graph G reprent agents and the edges are communication links.In particular, an agent x has access to the states of all its neighbors and can u this piece of information to compute its control law.Although a complete graph is not necessary for a distributed control algorithm,we always assume that the host graph is connected.
In order to model the collective behavior of the agents we will u functions f:N G
!d defined over a graph G[12].The partial derivative of f is defined as@y f(x)
:=f(y)0f(x)and the Laplacian of f is given by
1f(x):=
y
2N;y x
桁打一成语
@2y f(x)=
+
y
2N;y x
@y f(x);(1)
where the last identity follows from the fact that@2y f(x)=0@y f(x). The integral and the average of f are defined,respectively,as
G
fdx:
=
x
2N
f(x);h f i:=1
jN G j G
fdx:(2) Let L2(G
j d)be the Hilbert space compod by all functions f: N G
!d endowed with the norm k f k2L =
G
k f k2.We will u the shorthand notation L2when there is no ambiguity on the underlying domain and range of the functions.
Let S be a subgraph of G and@S be its boundary in G.We as-sume that S[@S=G.As in[12],we also consider the Hilbert space H10(S)
=f2L2(G):f j@S=0(e[12]for the definition of a suitable norm on H10(S)).Note that a function f2H10(S)is defined on S and possibly non null only on S.
0018-9286/$25.00©2008IEEE
The next theorem,proved in [12],characterizes the eigenstructure of
the Laplacian operator de fined on H 1
(S ).Theorem 1:Let G be a connected graph and S a proper subgraph
of G .Then,the operator 1:H 10
(S
j d )!L 2( S
j d )has jN S j d strictly negative eigenvalues.Moreover,the corresponding eigenfunc-tions form a basis for H 1
0(S
j d ).
III.M ULTIPLE S TATIONARY L EADERS
In this ction,we u PdEs for modelling and analyzing a group of agents with multiple leaders.A leader is just an agent that moves toward a prede fined goal,and who control policy is independent of the motion of all the followers.However,followers that are neighbors to the leader can u the leader state in order to compute their control inputs.
Let r (x;t )be the position of the agent x at time t 0,where 1r 2L 2
.The communication network is reprented by the undirected and connected graph G .For distinguishing between leaders and followers,we consider two subgraphs S F and S L of G and assume that S L =@S F and S F [S L =G ,where the subscripts denote ”Leaders ”and ”Followers ”respectively.Note that we assume that all agents are either designated as leaders or followers.
As already mentioned in the introduction,we will assume that the
followers obey the simple dynamics _r
(x;t )=u (x;t ),where u (x;t ):
=1r (x;t )
(3)
is the Laplacian control law.Let ^r
(x;t ),x 2N
S be the trajectory of the leaders.Then,the collective dynamics is reprented by the model
_r (x;t )=1r (x;t )x 2N
S (4a)r (x;t )=^r (x;t )x 2N
S
(4b)
endowed with the initial conditions r (1;0)=~r
2L 2(S F ).Model (4)is an example of a continuous-time Partial difference Equation (PdE)with non-homogeneous Dirichlet boundary conditions.We refer the reader to [12]–[14]for an introduction to PdEs.
The main results on Laplacian control available in the literature and specialized to model (4)are:
•in the leaderless ca (i.e.,S L =;),the Laplacian control solves the rendezvous ,r (x;t )!r 3
2d ;8x 2N G as t !+1.Moreover,the agents converge exponentially to r 3=h ~r i thus achieving average connsus.The results have been established in [15],[16]through the joint u of tools in control theory and algebraic graph theory.A formal analysis of the PdE (4a)has been conducted in [13],[14],[17]showing a complete accordance with results available within the theory of the heat equation [18];
•in the ca of a single leader (i.e.,N
S =f x L g )with fixed posi-tion (i.e.,^r (x L ;t )= r
2d ),Laplacian control solves the ren-dezvous problem with r 3= r [15].This property has also been shown in [13],[14]within the PdE framework,thus highlighting the profound links between model (4)an
d the heat equation with Dirichlet boundary conditions [18].
The first attempt of this paper is to characterize the asymptotic be-havior of the followers in the prence of multiple leaders with fixed positions.To this end,for the remainder of this ction,we will assume that ^r (x;t )= r (x )2L 2(S L ).The equilibria of (4)are then given by the solutions to the PdE
1h (x )=0x 2N
S (5a)h (x )= r (x )x 2N
S
(5b)
1For
sake of conciness,for a function
祖先是什么意思()
:we
will often
write
instead of
(
).and they have been studied in [12].In particular,[12,Theorem 3.5]shows that if G is connected and N S =;then,the PdE (5)has a unique solution 2h (x ).By analogy with the jargon of Partial Differen-tial Equations,h is termed the harmonic extension of the boundary con-ditions r
.Our next aim is to verify that r !h as t !+1.Let us consider the decomposition
r (x;t )=r 0(x;t )+h (x );熊银匠
r 02H 1
0(S F ):
(6)
Since h does not depend upon time and 1h =0,8x 2N
S ,the PdE
(4)is equivalent to the following one
_r 0(x;t )=1r 0(x;t )x 2N
S (7a)r 0(x;t )=0x 2N
S :
(7b)
From (6),it is apparent that the problem of checking if r !h as
日期英语t !+1can be recast into the problem of studying the convergence to zero of the solutions to the PdE (7).The fact that r 0!0as t !+1follows from Theorem 1and it can be shown by proceeding exactly as in the proof of [17,Theorem 5]3.
The next Theorem,proved in [19],highlights a key geometrical fea-ture of h (x ).For a t X of points
in d ,Co (X )will denote its convex hull.Moreover,the t L is the convex hull of leaders ,
L :
=Co (f r
(y );y 2N
S g ).Theorem 2:Let S 1be a nonempty connected subgraph of S F and @S 1be its boundary in G .Then,8x 2N S it holds
h (x )2Co (f h (y );y 2N @S g ):
(8)
Moreover,one has that h (x )2 L ,i.e.,that the position of each fol-lower lies in the convex hull of the leaders positions.Finally,if L is
full-dimensional 4,then h (x )2 L n @ L ,8x 2N
S .
Another geometrical feature which we need is the following:
Theorem 3:Suppo that L is full-dimensional and that r (x;t )is evolving according to (4).Suppo that,at a given time t =t ,there is an agent x 2N S such that r (x;t )2@ L and r (y;t )2 L ,8y 2P (x ).Then,two situations may occur:
1)there exists an (af fine)hyperplane such that
r (x;t )2 \@ L ;and r (y;t )2 \@ L
8y 2P (x ):
Then
9 >0:r (x;t )+ _r (x;t )2 \@ L
(9)
2)otherwi
9 >0:r (x;t )+ _r (x;t )2 L n @ L :
(10)
Note that (9)means that the velocity of x will be along the hyperplane
(in other words,the agent may slide on the boundary @ L ),whereas (10)means that the velocity of x is pointing inside the polytope L .While Theorem 2and the fact that r !h as t !+1guarantee that followers asymptotically enter L ,Theorem 3ensures that once all followers are in L they cannot exit from this t and therefore containment will be never violated.
2[12,
Theorem 3.5]assumes that the
subgraph is induced (e [12]for the
de finition of induced subgraphs).However,a careful examination of the proof,reveals that this assumption is unnecessary.
3Actually,[17,Theorem 5]proves a stronger property,namely that the origin of (7)is “exponentially stable on the
space ()”.The de finition of stability of equilibria on subspaces is provided in [17].4The t
is full-dimensional if the dimension of the af fine hull generated by
is
(e [20]).
Proof:(Theorem 3):Since r (x;t )obeys to (4),by rearranging terms we obtain
_r (x;t )=0jP (x )j r (x;t )
+
y 2P (x )
r (y;t ):
Then,tting =jP (x )j 01,it holds
r (x;t )+ _r (x;t )=jP (x )j 0
1
y 2P (x )
r (y;t );
<,r (x;t )+ _r (x;t )is the barycenter b (Y x )of the polytope Y x :
=
Co (f r (y;t );y 2P (x )g ).Note that,if r (y; t
)2 L ,8y 2P (x )one has Y x 2 L .Moreover,thanks to convexity,the barycenter of Y x lies in the relative interior of Y x .Thus,if all y 2P (x )verify that r (y;t )2 \@ L then Y x \@ L and so does b (Y x ),i.e.,b (Y x )2 \@ L ;otherwi b (Y x )2 L n @ L .
IV .L EADER -F OLLOWER C ONTAINMENT C ONTROL
Containment of all the followers is achieved in the ca of static leaders in the last ction.However,if the leaders are moving,this prop-erty might be violated.In order to prevent the followers from leaving the polytope spanned by the leaders,appropriate control strategies need to be designed for the leaders to guarantee the containment.In what fol-lows,we propo a hybrid strategy for this purpo and analyze liveness and reachability of the resluting clod-loop system.A.Hybrid Control Strategy
For the sake of containment,we de fine two distinctly different con-trol modes for the evolution of the leaders.The first of the two control modes is the STOP mode that corresponds to the leaders halting their movements altogether in order
to prohibit a break in the containment:
ST OP :(4a);(4b)and _^
r (x;t )=0;x 2N S :
(11)
It is clear that in order to execute this mode,no information is needed
for the leaders whatsoever.
The cond control mode under consideration is the GO mode,in which the leaders move toward a g
iven target formation.A number of different control laws can be de fined for this,but,for the sake of conceptual uni fication,we let the
GO mode be given by a Laplacian-bad control strategy as well.
GO :
(4a);(4b)and
_^r (x;t )=1S (^r
(x;t )0r T (x ));x
2N S
(12)
where r T (x );x 2N S denotes the
desired target position of leader x
and 1
S denotes the Laplacian operator
de
fined solely over the sub-graph S L ,i.e.,
1S f (x ):
=0
y x;y 2N
@2y f (x ):
Under the assumption that S L is connected,and by exactly the same
reasoning as for the standard rendezvous problem,under the in flu-ence of the GO mode alone the leaders will converge exponentially
to r L (x )=h ^r
(1;0)0r T (1)i +r T (x ),i.e.,9k >0; >0such that k ^r (1;t )0r L (x )k L ke 0 t k ^r (1;0)0r L (x )k L .In other words,no convergence to a prede fined point is achieved.Rather,this control law ensures that the leaders arrive at a translationally invariant target formation.
Note that the details of the leaders ’motion is not crucial and this par-ticular choice is but one of many possibilities.However,this choice is appealing in that it makes the information flow explicit,and the leaders
only need access to the positions (and target locations)of their neigh-boring leaders in order to compute their motion.As such the decentral-ized character of the algorithm is maintained.
In order to fully specify the hybrid Stop-Go leader policy transition rules are needed as well.As befor
e,
let L denote the leader-polytope and let d
( ; L )denote the signed distance
d ( ; L ):
= (
)min x 2@
k 0x k 2;(13)
where k 1k 2denotes the Euclidean 2-norm,and where ( )=01if 2 L
and +1otherwi.Using this distance measure we let the two ,transition conditions,be given by
GO 2ST OP :9y 2N S j d (r (y;t ); L
) 0?(14a)ST OP 2GO :
d (r (y;t ); L )<0
8y 2N S ?
(14b)
where a transition from GO to STOP triggers when the conditions in GO2STOP are met,and similarly for STOPT2GO ,and where >0is a threshold.
Note that the guard STOP2GO is crosd only if the following as-sumptions are veri fied:
Assumption 1:Let ^h
(1;t )be the solution to (5)for r (1)=^r (1;t ),8t 0and consider the t
L (t )=f y 2 L (t ):d (y;@ L (t ))<0 g .Then
1)
L
(t )is nonempty,8t 0;2)Co (f ^h
(x;t );x 2N S g ) L (t ).Note that,for a given time t 0,the uniqueness of ^h
(1;t )follows from the uniqueness of the solution to (5).In particular,Assumption 1im-plies that L must be full-dimensional at all times and “suf ficiently fat ”along every direction (e condition 1).Conditions relating property 2of Assumption 1to the graph topology are currently under investiga-tion.A few comments must be made about the computation and com-munication requirements that the guards give ri to.If two leaders are located at the end-points of the same face of L ,then they must be able to determine if any of the followers are in fact on this face.This can be achieved through a number of range nsing devices,such as ultra-sonic,infra-red,or lar-bad range-nsors.Moreover,in order for all leaders to transition between modes in unison,they must communicate between them,which means that either S L is a complete graph,or that multi-hop strategies are needed.In either way,a minimal requirement for the mode transitions to be able to occur synchronously,without having to rely on information flow across follower-agents,is that S L
must be connected.
The hysteresis threshold >0in the STOP2GO guard and the next assumption are needed in order to avoid Zeno
behaviors.Let de-note the supremum of the diameter of L during an execution.Assumption 2:9M <1such that M .
It is easy to check that Assumption 2is veri fied when Laplacian con-trol governs the leaders ’motion in the GO mode as in (12).Indeed,the
exponential convergence of ^r
(x;t )to r L (x )=h ^r (1;0)0r T (1)i +r T (x )implies that ^r (x;t )is bounded at all times.However,Laplacian control is but one of many possible control strategies and can be re-placed by other control schemes (e.g.,plan-bad leader control laws)without generating Zeno executions as long as Assumption 2is veri fied.Theorem 4:Under Assumptions 2and 1,the hybrid automaton de-fined by (11),(12)and (14)is non-Zeno.
Proof:Let the system
be in the STOP mode.Under Assumption 2we have
k _r
(x;t )k =k
1r (
x;t )k
y
x
k @y r (x )k
y x
N ;8x 2N S :
(15)
From Assumption 1,in order for the system to leave the STOP mode,at least one follower agent must have travelled at least a distance ,which in turn implies that the system will always stay for a time greater than or equal to =N
in the STOP mode.In order for the system to exhibit Zeno executions,a necessary condition is that the difference between the transition times must approach zero [21].Since this is not the ca here,the non-Zeno property is established.B.Liveness and Reachability
As already mentioned,the propod solution is non-Zeno.However,as it is currently de fined,the Stop-Go policy may be blocking in the n that the system never leaves the STOP mode.One remedy to this problem is to allow the containment to be slightly less tight.In other words,we can lect different ,
GO 2ST OP :9y 2N
S j d (r (t;y ); L )>2 ?(16a)ST OP 2GO :
d (r (t;y ); L )
8y 2N
S ?
(16b)
where >0.What this means is that we do not enter the STOP mode until a follower is 2 outside L .Let us de fine
L; :
=f y
2d :d (y; L ) g :Note that,one has L L; .The next Theorem summarizes the main properties of the resulting hybrid automaton.A remarkable fea-ture of the guards (16)is that Assumption 1is no longer needed in order to guarantee liveness.
Theorem 5:Under Assumption 2,the hybrid automaton by (11),(12)and (16)is non-Zeno,live,in the n of always leaving the
STOP mode eventually,and convergent in the n that ^r
(x;t )!h ^r (1;0)0r T (1)i +r T (x ).
Proof:We first prove liveness.Assume that the system is in the STOP mode.From Theorem 2we have that h 2 L .Since 8x 2S F ,r (x;t )!h ,and L L; ,every follower will eventually get back in L; in finite time (recall that the leaders are stationary in the STOP mode)hence triggering a transition to the GO mode.
Under Assumption 2,it holds k _r
(x;t )k N (
+2 )and we can repeat the non-Zeno argument in the proof of Theorem 4in order to e that the system always stays in the GO mode for a time greater than or equal to =(N (
+2 )).
As a result,in a non-blocking system the leaders will be given in fin-itely many opportunities to move during a finite (bounded away from zero)time horizon,which implies convergence to the target location as long as the leaders would in fact end up at the target location under the in fluence of the GO mode alone.
V .C ONCLUSIONS
In this paper,we prented a hybrid Stop-Go control policy for the leaders in a multi-agent containment scenario.In particular,the control strategy allows us to transport a collection of follower-a
gents to a target area while ensuring that they stay in the convex polytope spanned by the leaders.The enabling results needed in order to achieve this is that,for stationary leaders,the followers in a connected interaction graph will always converge to locations in the leader-polytope.Extensions to the propod control strategy are moreover given in order to ensure certain liveness properties.
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