Network Configuration Control Via Connectivity Grah Processes Abubakr Muhammad Deartment of Electrical and Systems Engineering University of Pennsylvania Philadelhia, PA 90 abubakr@seas.uenn.edu Magnus Egerstedt School of Electrical and Comuter Engineering Georgia Institute of Technology Atlanta, GA 0 magnus@ece.gatech.edu Abstract In this aer, we discuss how to generate trajectories, modeled as connectivity grah rocesses, on the sace of grahs induced by the network toology. We discuss the role of feasibility, reachability, and otimality in this context. In articular, we study in detail the role of reachability and the comutation of reachable sets by using the cylindrical algebraic decomosition (CAD) algorithm. I. INTRODUCTION The roblem of coordinating multile autonomous agents has attracted significant attention in recent years. Due to advances in many enabling technologies such as advanced communication systems, novel sensing latforms, and chea comutation devices, the realization of large scale networks of cooerating mobile agents has become ossible. An imortant theme in the study of such systems is the use of grah-theoretic models for describing the local interactions in the network. Notable results have been resented in [], [], [], [], []. The conclusion to be drawn from these research efforts is that a number of uestions can be answered in a natural way by abstracting away the continuous dynamics of the individual agents. In [6], [7], the authors have resented a detailed study of grahs that arise due to the limited sensory ercetion or communication of individual agents in a formation. We have shown that the grahs that can reresent formations do in fact corresond to a roer subset of all grahs, denoted by the set of connectivity grahs. We have resented several examles of grahs that fail to exist as connectivity grahs. The idea of infeasibility in the configuration sace has been exlored further in [8], where we have used techniues from semidefinite rogramming to obtain the reuired infeasibility certificates. Based on these results, we have resented in [9] a comutational framework in the context of formation switching for achieving a global objective. At the heart of this framework lies the concet of a connectivity grah rocess, that is made ossible by understanding issues concerning feasibility, reachability and otimality. Several alications of this framework have been outlined in [9], that include the roduction of low-comlexity formations and collaborative beamforming in sensor networks. In this aer, we focus on one articular asect of this framework, namely the uestion of reachability in connectivity grah rocesses. In articular, This work was sonsored by the US Army Research Office through the grant #9988. we give details on the use of the cylindrical algebraic decomosition (CAD) algorithm from real algebraic geometry for obtaining connectivity grah rocesses. This aer is organized as follows. We first summarize our revious work on connectivity grahs (Section II). We introduce the concet of connectivity grah rocesses and use semi-definite rogramming techniues for determining feasible switchings (Section III). We then discuss the comutation of reachable sets from a given initial grah and rovide details on the use of the CAD algorithm (Section IV). We set u the framework for obtaining otimal trajectories as grah rocesses (Section V). We then rovide some simulation results to show that our method is comutable, followed by our conclusions (Section VI). II. FORMATIONS AND CONNECTIVITY GRAPHS Grahs can model local interactions between agents, when individual agents are constrained by limited knowledge of other agents. In this section we summarize some revious results [6] of a grah theoretic formalism for describing formations in which the rimary limitation of ercetion for each agent is the limited range of its sensor. Suose we have N such agents with identical dynamics evolving on R. Each agent is euied with a range limited sensor by which it can sense the osition of other agents. All agents have identical sensor ranges δ. Let the osition of each agent be x n R, and its dynamics be given by ẋ n = f(x n, u n ), () where u n R m is the control for agent n and f : R R m R is a smooth vector field. The configuration sace C N (R ) of the agent formation is made u of all ordered N-tules in R, with the roerty that no two oints coincide, i.e. C N (R ) = (R R... R ), () where = {(x, x,..., x N ) : x i = x j for some i j}. The evolution of the formation can be reresented as a trajectory F : R + C N (R ), usually written as F(t)(x (t), x (t),... x N (t)) to signify time evolution. The satial relationshi between agents can be reresented as a grah in which the vertices of the grah reresent the agents, and the air of vertices on each edge tells us that the corresonding agents are within sensor range δ of each other.
Let G N denote the sace of all ossible grahs that can be formed on N vertices V = {v, v,..., v N }. Then we can define a function Φ N : C N (R ) G N, with Φ N (F(t)) = G(t), where G(t) = (V, E(t)) G N is the connectivity grah of the formation F(t). v i V reresents agent i at osition x i, and E(t) denotes the edges of the grah. e ij (t) = e ji (t) E(t) if and only if x i (t) x j (t) δ, i j. The grahs are always undirected because the sensor ranges are identical. The motion of agents in a formation may result in the removal or addition of edges in the grah. Therefore G(t) is a dynamic structure. Lastly and most imortantly, every grah in G N is not a connectivity grah. The last observation is not as obvious as the others, and it has been analyzed in detail in [6]. A realization of a grah G G N is a formation F C N (R ), such that Φ N (F) = G. An arbitrary grah G G N can therefore be realized as a connectivity grah in C N (R ) if Φ N (G) is nonemty. We denote by G N,δ G N, the sace of all ossible grahs on N agents with sensor range δ, that can be realized in C N (R ). Formations can roduce a wide variety of grahs for N vertices. This includes grahs that have disconnected subgrahs or totally disconnected grahs with no edges. However the roblem of switching between different formations or of finding interesting structures within a formation of sensor range limited agents can only be tackled if no subformation of agents is totally isolated from the rest of the formation. This means that the connectivity grah G(t) of the formation F(t) should always remain connected (in the sense of connected grahs) for all time t. III. FEASIBLE CONNECTIVITY GRAPH TRANSITIONS Connectivity grah rocesses are generated through the movement of individual nodes. For a connectivity grah G j = (V j, E j ) = Φ N (x(t j )) let the nodes be artitioned as V j = Vj 0 Vm m j, where the movement of the nodes in Vj facilitates the transition from G j to the next grah G j+ and Vj 0 is the set of nodes that are stationary. With the ositions x 0 j = {x m(t i )} m V 0 j being fixed, let Feas(G j, Vj m, x0 j ) G N,δ be the set of all connected connectivity grahs that are feasible by an unconstrained lacement of ositions corresonding to Vj m in R. (We will often denote this set as Feas(G j, Vj m), when the the ositions x0 j are understood from context.) It will be aroriate to exlain the reason for keeing track of mobile and stationary nodes at each transition. In rincile, it is ossible to comute this entire set of feasible transitions by an enumeration rocedure. However, in order to manage the combinatorial growth in the number of ossible grahs, it is desirable to let the transitions be generated by the movements of a small subset of nodes only. In fact, we will investigate the situation where only one node is allowed to move at a time. Hence, we let Vj 0 = {,..., k, k +,..., N} and Vm j = {k}. It should be noted that the movement of node k can only result in the addition or deletion of edges that have node k as one of its vertices. Therefore the enumeration of the ossible resulting grahs should count all ossible combinations of G 0 Set of feasible transitions Feas(G 0, {}). Infeasible transitions. Fig.. Feasible and infeasible grahs by movement of node. such deletions and additions. This number can be easily seen to be N for N nodes. Since we are also reuired to kee the grah connected at all times, this number is actually N, obtained after removing the grah in which node k has no edge with any other node. Now, we can use the S-rocedure to evaluate whether each of the new grahs resulting from this enumeration is feasible. Since all nodes are fixed excet for x k = (x, y), the semialgebraic set we need to check for non-feasibility is defined by N olynomial ineualities over R[x, y]. Each of these ineualities has either of the following two forms, [ ] x y 0 x i 0 y i x i y i δ x i y i x y 0, if e ik E, or [ ] 0 x i x y 0 y i x y > 0, if e ik E. x i y i x i + y i δ where i N, E is the edge set of the new grah and we denote x i (t j ) by (x i, y i ) for i k. This comutation can be reeated for all N nodes so that we have a choice of N( N ) grahs. If we let ρ i = δ x i y i then by denoting A i = 0 x i, B j = 0 y i x i y i ρ i 0 x j 0 y j x j y j ρ j and ignoring the lossy asect of the S-rocedure [0], we need to solve the LMI, A α λ αi A αi λ αj B αj 0. i,e αi k E j,e αj k E An examle of such a calculation is given in Figure, where V 0 = {,,, } and V m = {}. The LMI control toolbox [] for MATLAB has been used to solve the LMI for each of these grahs in order to get the aroriate certificates.,
We now give a detailed study of reachability in the context of connectivity grah rocesses as the main contribution of this aer. IV. REACHABILITY AND CONNECTIVITY GRAPH PROCESSES Note that the set Feas(G 0, V0 m ) does not deend on the actual movement of the individual nodes. In fact, even if G Feas(G 0, V0 m ), it does not necessarily mean that there exists a trajectory by which G 0 G or even that G 0 G G... G. The geometrical configuration of nodes may create an obstruction in obtaining a grah rocess that takes G 0 to G. There are two ways by which this obstruction is created: The reuirement to maintain connectivity and conformity to a fixed set of mobile nodes. We therefore need some notion of reachability on the sace G N,δ. We say that a connectivity grah G f is reachable from an initial grah G 0 if there exists a connectivity grah rocess of finite length G 0 G... G f and a seuence of vertex-sets {Vk m} such that each G k+ Feas(G k, Vk m). If Vm k = Vm at each transition, then every G k Feas(G 0, V m ). (In articular, G f Feas(G 0, V m ).) Consider all such G s that are reachable from G 0 with a fixed V m. We will denote this set by Reach(G 0, V m ). It is easy to see that Reach(G 0, V m ) Feas(G 0, V m ). In the revious sub-section, it was shown how to determine the membershi for the set Feas(G 0, V m ). For all such grahs, determining whether they also belong Reach(G 0, V m ) is not very straightforward, secially under the restriction that the intermediate grahs have to be connected. For the secial case of a single mobile node, the situation is manageable as discussed below. We first describe what is called a cylindrical algebraic decomosition or CAD of a semi-algebraic set [], []. Cylindrical Algebraic Decomosition: We first give some definitions. A Nash manifold M R n is an analytic submanifold which is a semialgebraic set. Let U R n. A Nash function f : U R is a smooth function for which there exists a olynomial (x, t) = (x,... x n, t) such that (x, f(x)) = 0 for all x U. A Nash cell in R n is a Nash manifold which is diffeomorhic to an oen box (, ) d of dimension d. Every semialgebraic set can be decomosed into a disjoint union of Nash cells. More recisely []; Theorem.: Let A,, A be semialgebraic subsets of R n. Then there exists a finite semialgebraic artition of R n into Nash cells such that each A j is a union of some of these cell. The existence of such a decomosition is given by a techniue in real algebraic geometry known as the cylindrical algebraic decomosition (CAD) []. A CAD of R n is a artition into finitely many semialgebraic subsets. The CAD of R n is given by induction on n as follows. ) A CAD of R is a subdivision by finitely many oints a <... < a l. The cells are the singletons {a i } and the oen intervals delimited by these oints. ) For n >, a CAD of R n is given by a CAD of R n and for each cell C of R n, the Nash functions ζ C, < < ζ C,lC : C R. The cells of the CAD of R n are the grahs of ζ C,j and the bands in the cylinders C R determined by the grahs. Observe that the cells generated by CAD are Nash, thus roviding a artition satisfying Theorem.. For our need, we only need a CAD of R. For notational convenience, let us denote the collection of Nash cells corresonding to a semialgebraic set S by CAD(S). With V m = {k} and V 0 = {,, k, k +,, N}, consider the semialgebraic set X(V 0 ) = {(x, y) : (x x j ) + (y y j ) δ } R. j V 0 () The first thing to note about this set is that it is comact. This relieves us of any comlicated comactification rocedures that are needed to get the CAD of non-comact sets. Let C be a cell in CAD(X(V 0 )), then we call r C a configuration oint in C if r x j for all j V 0. We then have the following result. Proosition.: Let C CAD(X(V 0 )). If, are any two configuration oints in C then Φ N ((x,..., x k,, x k+,..., x N )) = Φ N ((x,..., x k,, x k+,..., x N )). Proof: If C is a 0-cell we have the result trivially. We therefore assume that dim(c) > 0. For notational convenience denote (x,..., x k,, x k+,..., x N ) by x and the grah Φ N ((x,..., x k,, x k+,..., x N )) by G for a oint C. C B δ (x m ) δ B δ (x m ) dim(c) = dim(c) = Fig.. Toological obstruction used in the roof of Proosition.. First note that any air of ositions x i, x j, where i, j V 0 are always different. This is because they are the roduct of a valid configuration on the configuration sace C N (R ) described in Euation. If is a configuration oint then the N-tule x = (x,, x k,, x k+,, x N ) is a valid configuration on C N (R ), hence the name configuration oint. The configuration oint would corresond to the k- th node in G. This shows that the maing Φ N (x ) is well defined for all configuration oints in C. We now rove the roosition by contradiction. Assume that there exist oints, C such that G G. The grah G would differ from G in at least one edge incident on node k. Without loss of generality, assume that e km is an edge between a node m V 0 and the k-th node corresonding to in G, and that there is no edge between nodes k and m in G. The existence of this edge in G means that is inside the closed ball B δ (x m ), while / B δ (x m ) as shown in Figure. From the definition of δ C
a Nash cell, C is diffeomorhic to (, ) dim(c) which is simly connected. This means that C, in addition to being an oen set, is also simly connected. Therefore the ball B δ (x m ) induces a artition C = (C \ B δ (x m )) (C B δ (x m )), such that B δ (x m )) C. From Figure, it is clear that each connected comonent of B δ (x m )) C induces dim(c) Nash cells that are not already resent in the CAD of CAD(X(V 0 )). Moreover, this means that there exist lower dimensional Nash cells that are roerly contained in C. Both imlications are contrary to the CAD construction described above. Therefore, the connectedness of C creates a toological obstruction in getting G G. This roves the Proosition. x(t0) x x(t) x(t) x x(t) x x(t) x(t) In articular, the -skeleton CAD () (S) consists of all - dimensional Nash cells in CAD(S). Recall that the set X(V 0 ) described by Euation is a union of closed disks in R. Moreover the boundary of of this set X(V 0 ) CAD (0) (X(V 0 )) CAD () (X(V 0 )). If C CAD () (S) and C denotes its closure in R, then by construction of the CAD, C CAD (0) (X(V 0 )) CAD () (X(V 0 )). Therefore, any connected comonent of X(V 0 ) cannot be made u of cells in CAD () (S) alone. In fact, we get the following useful lemma. Lemma.: The set CAD(X(V 0 )) is connected if and only if CAD (0) (X(V 0 )) CAD () (X(V 0 )) is connected. The above lemma suggests the existence of a ath lanning algorithm for moving the mobile node between any two oints of a connected comonent of X(V 0 ). Given, in the same connected comonent of X(V 0 ), we construct a ath γ : [0, T ] X(V 0 ) such that γ(0) = and γ(t ) =. The lacement of both oints, in the same connected comonent ensures that T <. Also, each of the oints, lie in their uniue cells of CAD(X(V 0 )), which we denote by C, C resectively. We build our algorithm from the following observations. Fig.. nodes. Trajectory of a mobile node in the CAD generated by the fixed From this we get the following useful result. Corollary.: Let x(t) = (x,..., x k, γ(t), x k+,..., x N ) be a trajectory on C N (R ). If γ(t) C for all t [t 0, t f ) then the connectivity grah rocess has no transitions while t (t 0, t f ). For a trajectory x(t) = (x,..., x k, γ(t), x k+,..., x N ) on C N (R ) that is not confined to a single Nash cell, the grah may change as the trajectories goes from one cell to another. Moreover, since the connectivity grah remains unchanged inside one cell, the transition must take lace at the boundary of Nash cells. We can now begin to areciate the connection between the CAD decomosition and the geometric origin of transitions in a connectivity grah rocess. We further observe that the trajectory is artitioned into a finite number of ieces, where each iece is confined to a single Nash cell. In more recise terms, for each grah rocess G 0 G... G N with V m i = {k} for 0 i N and transitions at t < t <... < t N, there exist a finite number of Nash cells C x = {C 0, C,..., C M } C, where M N, such that the trajectory x(t) intersects with a finite sub-collection C x,i C x for t (t i, t i+ ). One such trajectory is deicted in Figure. We will construct an exlicit lanning algorithm to go from one oint in the CAD to another. Let us define the k-th skeleton of the CAD of a set S as CAD (k) (S) = {C CAD(S) dim(c) = k}. Toologically, a simly connected set means ath-wise connectedness and the absence of any holes in the set. Fig.. φ s(t) C φ s(t) φ 0 (,) (,) Path between two oints, in a -cell of the CAD. Case : When C = C = C, i.e. when both oints lie in the same cell, consider the diffeomorhism φ : C (, ) dim(c). Then the oints φ() and φ() in the oen cube can be joined by a straight line s(t) = tφ() + ( t)φ(). Now ma this line back to C by γ(t) = φ (s(t)), t [0, ]. This gives us the desired ath. Such an exlicit construction in terms of the diffeomorhism may only be needed when dim(c) =, as deicted in Figure. For lower dimensions the construction of γ(t) is rather obvious. Case : When C C. We study various situations in this case. (a). When dim(c ), dim(c ) <. In this case one can construct the ath γ exlicitly by building a grah G P = (V P, E P ) in the following way. Each v i V corresonds to a 0-cell Ci 0 in the CAD(0) (X(V 0 )) and an edge e ij E if and only if there exists a -cell C CAD () (X(V 0 )) such that Ci 0, C0 j C. Note that by construction of the CAD, if such a -cell exists, it will be uniue. Moreover, each -cell will be maed to a uniue edge in G P. If dim(c ) = dim(c ) = 0 we are done. If not, we modify as follows. We add one vertex for each of the oints G P
γ(t) v C 0 i C 0 j v i w ij v j v x k(t 0) x x x CAD(X(V 0 )) G P x k(t f) Fig.. Path lanning via lanning grah., that belong to a -cell. Suose dim(c ) =. Then C corresonds to an edge e jk = (v j, v k ) induced by 0-cells Cj 0 and Ck 0. We add a vertex v corresonding to. Now modify the grah G P by removing the edge e jk and inserting two edges e j = (v j, v ) and e k = (v, v k ). A similar rocedure modifies G P if dim(c ) =. We call this modified grah our lanning grah G P. An examle of this construction is shown in Figure. We can now convert this grah into a weighted grah, by assigning each edge e ij a real number w ij. One choice of weights can be a constant weight on all edges. We will show later how to assign a more natural set of weights. Once we have that, we can use a standard discrete lanning algorithm such as the Dijkstra s algorithm [] to get a ath on G P that connects v to v by a seuence of edges and vertices. This grahical ath can now be maed back to R to get an exlicit ath γ(t) that connects to. We now give this maing. Note first that a -cell Cj has two 0-cells say C0 j 0 and Cj 0 on its boundary. If this -cell is also a subset of B δ (x k ) for some k V 0 then by the roerties of CAD construction, Cj has a natural arametrization as a curve in R given by c j : s (s, sgn(c j ) δ (s y k ) +x k ), s (x j0, x j ), where sgn(cj ) can be determined uniuely for each such - cell. If w j0j is the weight on the corresonding edge e j0j E P, then we can re-arameterize by a linear maing T j : R R that sends [0, w j0 j ) [x j0, x j ).We can therefore define a arameterized curve γ j0 j (t) = c j (T j (t)), where t [0, w j0j ). In case the -cell Ck is a vertical line segment, the arametrization is more simle. Simly, let γ k0k (t) = T k (t) where t [0, w k0k ) and T k mas [0, w k0k ) [y k0, y k ). In this way we have a method to ma back an edge in the lanning grah to R. We have omitted here details for edges that have either v or v as one of the vertices. It is easy to see that the maing for these edges is very similar. Now, given a ath in G P as a seuence of vertices and edges, one can build exlicitly a ath in R as a concatenation of the curves described above. More recisely let the ath be given by a seuence v, e k, v k, e k k, v k,..., v km, e km, v, then we define a ath that connects to by γ(t) = γ k γ k k... γ km (t), t [0, W ) Fig. 6. Path between two oints that belong to their resective -cells in the CAD. where W = w k + w k k +... w km and is the ath concatenation oeration defined by γ i γ j (t) = { γi (t), t [0, w i ); γ j (t), t [w i, w i + w j ). (b). General Case: We now discuss the most general case. We resent a strategy for the case when either or both of and belong to a -cell. Suose, are inside C, C CAD () (X(V 0 )) resectively. Now consider the line segment γ 0 (s) = s + ( s), where s [0, ]. By the construction of CAD, this line segment cuts at least cells of dimension less than. Let s be the minimum value of s at which the line cuts a lower dimensional cell. Also let s be the corresonding maximum value. Now let = γ 0 (s ) and = γ 0 (s ). Using the rocedure described in the revious case, a lanning grah can be constructed to find a ath that connects to. Concatenating the resulting curve with γ 0 (t) for t [0, s ) in the beginning and γ 0 (t) for t [s, ] at the end we get the resulting ath. One such construction is shown in Figure 6. The above discussion makes it clear that it ossible to find a ath that transorts the k-th node at time t 0 from any initial osition x(t 0 ) to a desired osition inside the same connected comonent of CAD(X(V 0 )) in a finite time. In order to give a meaning to this transortation from the oint of view of control, it enough to assume that the dynamics of the nodes described by Euation are globally controllable. Combining these observations, we get our main result. Theorem.: Assume that the individual nodes are globally controllable. Let Vi m = {k} be fixed. Given two connectivity grahs Φ N (x(0)) and G f, there exists a finite connectivity grah rocess Φ N (x(0)) G... G f, and a corresonding trajectory x(t) C N (R ), t [t 0, t f ] such that x(t f ) Φ N (G f ) if and only if there exist a finite collection of Nash cells C x = {C 0, C,..., C M } CAD(X(V 0 )) such that x(0) C 0, x(t f ) C M and x(t) C j for some C j C x for all t [t 0, t f ] This gives us a way to realize trajectories on the sace of connectivity grahs and a method to characterize the reachable set with one mobile node. In fact, by construction this node only needs two different tyed of motions, namely along constant vector fields and rotations about fixed oints.
V. GLOBAL OBJECTIVES, DESIRABLE TRANSITIONS AND OPTIMALITY The urose of a coordinated control strategy in a multiagent system is to evolve towards the fulfilment of a global objective. This tyically reuires the minimization (or maximization) of a cost associated with each global configuration. Viewed in this way, a lanning strategy should basically be a search rocess over the configuration sace, evolving towards this otimum. If the global objective is fundamentally a function of the grahical abstraction of the formation, then it is better to erform this search over the sace of grahs instead of the full configuration sace of the system. By introducing various grahical abstractions in the context of connectivity grahs, we have the right machinery to erform this kind of lanning. In other words, we will associate a cost or score with each connectivity grah and then work towards minimizing it. Given Reach(G 0, V m ), a decision need to be taken regarding what G f Reach(G 0, V m ) the system should switch to. For this we define a cost function Ψ : G N,δ R and we choose the transition through G f = arg min Ψ(G) G Reach(G 0,V m ) Here Ψ is analogous to a terminal cost in otimal control. If, in addition, we also take into account the cost associated with every transition in the grah rocess G 0 G... G M = G f that takes us to G f, then we would instead consider the minimization of the cost M J = Ψ(G f ) + β(i)l(g i, G i+ ), i=0 where L : G N,δ G N,δ R is the analogue of a discrete Lagrangian, β(i) are weighting constants, and G i+ Reach(G i, V m ) at each ste i. The choice of a Lagrangian lets us control the transient behavior of the system during the evolution of the grah rocess. In rincile, the framework described in this aer can be used for any alication that reuired otimization over connectivity grahs. We have resented such several such alications in a recent work [9]. Here we reroduce one such simulation result in Figure 7, where a low-comlexity formation called as a δ-chain [7] is achieved by a series of maneuvers in which different nodes take the role of the mobile node at aroriate stes. VI. CONCLUSIONS We have resented a generic framework for connectivity grah rocesses. The concets of feasibility and reachability are useful for obtaining otimal trajectories on the sace of connectivity grahs. These grahical abstractions are comutable using the techniues of semi-definite rogramming and CAD, as verified by simulation results. Fig. 7. Snashots of a connectivity grah rocess that generates a δ-chain by choosing different mobile nodes at aroriate intermediate transitions. REFERENCES [] R.Saber and R. Murray, Agreement Problems in Networks with Directed Grahs and Switching Toology, in Proc. IEEE Conference on Decision and Control, 00. [] A. Jadbabaie, J. Lin, and A. Morse, Coordination of grous of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, Vol. 8, No. 6,. 988-00, 00. [] M. Mesbahi, On State-Deendent Dynamic Grahs and their Controllability Proerties, IEEE Conference on Decision and Control, 00 [] E. Klavins, R. Ghrist, and D. Lisky, A Grammatical Aroach to Self-Organizing Robotic Systems, IEEE Transactions on Automatic Control, 00. (To aear) [] Z. Lin, B. Francis, and M. Maggiore, Necessary and sufficient grahical conditions for formation control of unicycles, IEEE Transactions on Automatic Control, 00. [6] A. Muhammad and M. Egerstedt, Connectivity Grahs as Models of Local Interactions. Journal of Alied Mathematics and Comutation, Vol. 68, No.,. -69, Set. 00. [7] A. Muhammad and M. Egerstedt, On the Structural Comlexity of Multi-Agent Agent Formations, in Proc. American Control Conference, Boston, Massachusetts, USA, 00. [8] A. Muhammad and M. Egerstedt, Positivstellensatz Certificates for Non-Feasibility of Connectivity Grahs in Multi-agent Coordination, 6th IFAC World Congress, Prague, July -8, 00. [9] A. Muhammad and M. Egerstedt, Alications of Connectivity Grah Processes in Networked Sensing and Control, Worksho on Networked Embedded Sensing and Control, University of Notre Dame, 00. [0] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Martix Ineualities in Systems and Control Theory, SIAM Studies in Alied Mathematics, 99. [] The Math- Works Inc., LMI Control Toolbox, Version.0.7, May 00. [] S. Basu, R. Pollack and M. Roy, Algorithms in Real Algebraic Geometry, Algorithms and Comutation in Mathematics Series, Vol. 0, Sringer, 00. [] J. Bochnak, M. Coste, M. Roy, Real Algebraic Geometry, Sringer- Verlag, Berlin, 998. [] E. Dijkstra, A Note on Two Problems in Connexion With Grahs, in Numerische Mathematik, vol.,. 697, 99.