On rendezvous for visually-guided agents in a nonconvex polygon

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1 CDC 2005, To aear On rendezvous for visually-guided agents in a nonconvex olygon Anurag Ganguli Jorge Cortés Francesco Bullo Abstract This aer resents coordination algorithms for mobile autonomous agents euied with line-of-sight sensors in a nonconvex olygon. The objective of the roosed algorithms is to achieve rendezvous, that is, agreement over the location of the agents in the network, using only information from the line-of-sight sensors. Two key novel comonents of the algorithms are the notions of locally-cliueless visibility grah and of convex continuous constraint set. Initial osition of the agents Evolution of the network Final osition of the agents I. INTRODUCTION Consider a grou of robotic agents moving in a nonconvex environment. For simlicity, we model the environment as a simle olygon and the agents as oint masses. Assume that each member of the grou is euied with omnidirectional line-of-sight sensors. By a line-of-sight sensor, we mean any device or combination of devices that can be used to determine, in its line-of-sight, (i) the osition or state of another agent, and (ii) the distance to the boundary of environment. By omnidirectional, we mean that the fieldof-vision for the sensor is 2π radians. We assume that the algorithm regulating the agents motion is memoryless, i.e., we consider static feedback laws. Given this model, the goal is to design a rovably correct discrete-time algorithm which ensures that the agents converge to a common location within the environment. See Fig. 1 for a grahical descrition of our objective. Ideally, the algorithm would work asynchronously but here we confine ourselves to the synchronous case. This work is motivated by the recent surge of interest in the study of grous of mobile autonomous robots. The multiagent rendezvous roblem and the first circumcenter algorithm have been introduced in [1]. The algorithm roosed in [1] has been extended to various asynchronous strategies in [2], [3]. A related algorithm, in which connectivity constraints are not imosed, is roosed in [4]. One imortant difference between these works and the resent one is that we consider visually-guided robots. In fact, technical advancement in sensor technology and mobile robotics have facilitated the imlementation of these algorithms on real systems. Examles of anoramic deth sensors relevant to our work are (1) omnidirectional cameras, e.g., [5], and (2) laser scanners with accurate distance measurements at high angular density. We conclude our literature review by mentioning that the roblem of rendezvousing at a secified location for visually-guided agents was first Anurag Ganguli is with the Coordinated Science Laboratory, University of Illinois at Urbana-Chamaign, and with the Deartment of Mechanical and Environmental Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA, aganguli@uiuc.edu Jorge Cortés is with the Deartment of Alied Mathematics and Statistics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA, jcortes@ucsc.edu Francesco Bullo is with the Deartment of Mechanical and Environmental Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA, bullo@engineering.ucsb.edu Fig. 1. Execution of the Circumcenter Algorithm described in Section IV-C on a network of agents distributed in a olygon shaed like a tyical floor lan. The algorithm is run over the visibility grah G vis,q (see Section III). introduced in [6]. However, the roosed solution was not distributed, in the sense that each agent reuired the knowledge of the locations of all other network agents. The contribution of this aer is threefold. First, we develo a geometric framework which makes it ossible to aly recently develoed results on convergence analysis of nonlinear systems, e.g., the LaSalle Invariance Princile for set-valued mas, on a network of visually-guided agents in a nonconvex environment. More exlicitly, we constrain the motion of agents to sets that (i) ensure that the visibility between two agents is reserved, and (ii) changes continuously as a function of the osition of the agents. We call such sets convex continuous constraint sets and characterize their roerties. Second, based on a discussion on visibility grahs, we define a new roximity grah, called the locallycliueless visibility grah, which contains fewer edges than the visibility grah, and has the same connected comonents. This construction can be, in general, useful for any roblem where the connectivity of the visibility grah is imortant and fewer constraints on the agents, in terms of number of neighbors, is beneficial. Examles of such roblems might include line-of-sight wireless routing and consensus roblems over line-of-sight wireless communication networks. Third, we roose a coordination algorithm to solve the rendezvous roblem and rovide a convergence roof. II. CONVEX CONTINUOUS CONSTRAINT SETS Here we design motion constraint sets for air of agents mutually visible to one another. By constraining the motion of agents, we aim to reserve the connectivity of the network. Additionally, we reuire that motion constraint sets change continuously as a function of the osition of the agents. This turns out to be a critical roerty for the convergence analysis of algorithms based on these sets. We emhasize that the construction roosed here may be alied to any distributed algorithm for a network of visually-guided agents. We begin by reviewing some notation for standard geometric objects. For R 2, let B(,r) denote the closed ball centered at of radius r R +. We let R + and R + denote the

2 ositive and the nonnegative real numbers, resectively. For a bounded set X R 2, we let co(x) denote the convex hull of X. For, R 2, we let ],[= {λ+(1 λ) 0 < λ < 1} and [,] = co({,}) denote the oen and closed segment with extreme oints and, resectively. For a closed convex set X R 2 and R 2, let roj X () denote the orthogonal rojection of onto X. For a bounded set X R 2, we let CC(X) denote the circumcenter of X, i.e., the center of the smallest-radius circle enclosing X. The comutation of the circumcenter is a strictly convex roblem. Let X denote the cardinality of a finite set X in R 2. Next, we define continuous set-valued mas; see [7]. Definition II.1 Let X and Y be toological vector saces (real and Hausdorff). A set-valued ma f : X 2 Y with non-emty and comact values is continuous at a oint x 0 X if given any ɛ > 0, there exists a δ > 0 such that for all x B(x 0,δ), we have f(x) B(y,ɛ) and f(x 0 ) B(y, ɛ) y f(x 0) y f(x) Now let us turn our attention to the environment. A olygon is simle if its vertices are the only oints in the lane common to two olygon edges and every vertex belongs to at most two olygon edges. Such a olygon has a well defined interior and exterior. Note that a simle olygon can contain holes. Let Q denote the set of all simle olygons. Let Q Q and let Ve(Q) = (v 1,...,v n ) be the list of vertices of Q ordered counterclockwise. The interior angle of a vertex v of Q is the angle formed inside Q by the two edges of the boundary of Q incident at v. The oint v Ve(Q) is a reflex vertex if its interior angle is strictly greater than π radians. Let Ve r (Q) denote the list of reflex vertices of Q ordered counterclockwise. If X is a finite set of oints in Q n, let MPP(X,Q) be the minimal erimeter olygon containing X which is a subset of Q (see Fig. 2 for an examle). Note that Q does not necessarily have to be Definition II.2 Let v be a reflex vertex of Q, and let w Ve(Q) be visible from v. The (v, w)-generalized inflection segment I(v,w) is the set I(v,w) = { S(v) = λv + (1 λ)w,λ 1}. If w Ve r (Q), then we call I(v,w) a bitangent of Q. Let {I α } α A be the set of bitangents of Q. A reflex vertex v of Q is an anchor of Q if it is visible from and { S(v) = λv + (1 λ),λ > 1}. In other words, a reflex vertex is an anchor of if it occludes a ortion of the environment from. Next we define and characterize certain useful convex sets deicted in Fig. 3. Definition II.3 Given Q Q, let, Q such that [,] Q. Let v Ve r (Q). Let e v and e v be the edges of Q determining v. Then we define H v (,) R 2 as follows: (i) if v / [,], then H v (,) is the half-lane with the following roerties: (a) the boundary of H v (,) contains v and is erendicular to the line assing through v and roj [,] v, and (b) and belong to the interior of H v (,); (ii) if v = with, then H v (,) is the halflane with the following roerties: (a) the boundary of H v (,) contains v and is erendicular to the line assing through and, and (b) belongs to the interior of H v (,) (Note: a similar definition holds when we interchange and ); (iii) if v ],[ with, then H v (,) is the halflane with the following roerties: (a) the boundary of H v (,) contains the line assing through and, and (b) the interior of H v (,) intersected with e v or with e v is emty; (iv) if v = =, then H v (,) is the set H v H v. H v is a half-lane with the following roerties: (a) the boundary of H v contains the edge e v, and (b) the interior of H v intersected with e v is emty. We define H v similarly with e v interchanged with e v. roj [,] v v = v Fig. 2. Minimal erimeter olygon of a set of oints inside nonconvex olygonal environments. The environments are reresented by dashed lines, while the olygons reresented by the solid lines are the minimal erimeter olygons of the oints reresented by the solid circles. On the left, the environment is a simle olygon whereas on the right the environment is olygonal, not simle, and still has a well-defined interior and exterior. simle for the minimal erimeter olygon to be defined; it only needs to have a well defined interior and exterior. A oint Q is visible from Q if [,] Q. The visibility olygon S() Q from a oint Q is the set of oints in Q visible from. We can also think of S() as a ma from Q to the set of olygons contained in Q. v e v v = = e v Fig. 3. Definition of the sets H v(, )

3 Remark II.4 With the above definition, wherever defined, H v (,) is a closed and convex set containing and. Also, if V Q is convex and comact, then H v (,) is welldefined everywhere in (V) 2 and (,) H v (,) is a setvalued ma over the domain (V) 2 with range 2 (R2). Lemma II.5 Given any v Ve r (Q) and a convex and comact subset V of Q, the set-valued ma (,) H v (,) Q restricted to (V \ Ve r (Q)) 2 is continuous. v V Lemma II.6 Let V Ve r (Q) and V be a convex and comact subset of Q. The following statements are true: (i) the set-valued ma (,) S() H v (,) restricted to (V \ (Ve r (Q) ( α A I α ))) 2 is continuous; (ii) the set-valued ma S() H v (,) re- v V stricted to V \ (Ve r (Q) ( α A I α )) is continuous. Definition II.7 (Convex Continuous Constraint Sets) Let, Q have the roerty that [, ] Q and let I Q (,) = Ve r (Q) S() S(). The convex continuous constraint set between and is C Q (,) = S() H v (,). v I Q(,) Fig. 4 illustrates the constraint set. v k1 v k2 v k3 Fig. 4. The figure on the left is an examle of the constraint set C Q (, ) where I Q (, ) = {v k1, v k2, v k3 }. The figure on the right is an examle of C Q (, ) where I Q (, ) = Ve r(q). In addition, we show that this grah can be comuted based on the information obtained only from the visibility grah. We begin by introducing some concets regarding roximity grahs for oint sets in R 2. We assume the reader is familiar with the standard notions of grah theory. We recall that a cliue of a grah is a comlete subgrah of it. A maximal cliue of an edge is a cliue of the grah that (i) contains the edge and (ii) is not a strict subgrah of any other cliue of the grah that also contains the edge. Given a vector sace V, let F(V) be the collection of finite subsets of V. Accordingly, F(R 2 ) is the collection of finite oint sets in R 2 ; we shall denote an element of F(R 2 ) by P = { 1,..., n } R 2, where 1,..., n are distinct oints in R 2. Let G(R 2 ) be the set of undirected grahs whose vertex set is an element of F(R 2 ). A roximity grah function G : F(R 2 ) G(R 2 ) associates to a oint set P an undirected grah with vertex set P and edge set E G (P), with E G : F(R 2 ) F(R 2 R 2 ) such that E G (P) P P \ diag(p P) for any P. Here, diag(p P) = {(,) P P P}. In other words, the edge set of a roximity grah deends on the location of its vertices. General roerties of roximity grahs are defined in [8], [9]. Here, we define: (i) a Euclidean Minimum Sanning Tree of a roximity grah G, denoted G EMST,G, assigns to each P a minimum-length sanning tree of G(P) whose edge ( i, j ) is assigned a length i j. If G(P) is not connected, then G EMST,G (P) is simly the union of Euclidean Minimum Sanning Trees of its connected comonents. For simlicity, when G is the comlete grah (P, P P \ diag(p P)), we denote the Euclidean Minimum Sanning Tree by G EMST ; (ii) the visibility grah G vis,q, for Q Q, with ( i, j ) E Gvis,Q (P) if the line segment [ i, j ] Q; (iii) the locally-cliueless visibility grah G lc-vis,q, for Q Q, with ( i, j ) E Glc-vis,Q (P) if ( i, j ) E Gvis,Q (P) and ( i, j ) belongs to a set E GEMST (P ) for any maximal cliue P of the edge ( i, j ) in G vis,q. Fig. 5 contains some examles of roximity grahs in a nonconvex olygon Q shaed like a tyical floor lan. Theorem II.8 Let V Q be convex and comact. For any two oints, V, the following statements are true: (i) C Q (,) is convex, C Q (,) = C Q (,), and (ii) the set-valued ma (,) C Q (,) C Q (,) restricted to (V \ Ve r (Q)) 2 is continuous. III. THE LOCALLY-CLIQUELESS VISIBILITY GRAPH In Section II we roosed the construction of motion constraint sets to reserve the connectivity of the network. The number of such constraints for an agent is the number of the agents visible to it. It is intuitively clear that the lesser the number of such constraints, the faster will be the convergence of the algorithm. Here we introduce the notion of locally-cliueless visibility grah, which is a subgrah of the visibility grah. In general, it contains fewer edges than the visibility grah but has the same connected comonents. Fig. 5. From left to right, visibility grah, Euclidean Minimum Sanning Tree for the five agents in the center, and locally-cliueless visibility grah. To each roximity grah function G, we associate the set of neighbors ma N G : R 2 F(R 2 ) F(R 2 ), defined by N G (, P) = { P (,) E G (P {})}. Also, for R 2, define N G, : F(R 2 ) F(R 2 ) by N G, (P) = N G (, P). Let G 1 and G 2 be two roximity grah functions. G 1 is satially distributed over G 2 if, for all P, N G1,(P) = N G1,( NG2,(P) ).

4 It is straightforward to deduce that if G 1 is satially distributed over G 2, then G 1 is a subgrah of G 2, that is, G 1 (P) G 2 (P) for all P F(R 2 ). Two roximity grah functions G 1 and G 2 have the same connected comonents if, for any P F(R 2 ), G 1 (P) and G 2 (P) have the same number of connected comonents consisting of the same vertices. Theorem III.1 For Q Q, the following statements hold: (i) G EMST,Gvis,Q G lc-vis,q G vis,q ; (ii) G lc-vis,q is satially distributed over G vis,q, for the case when Q does not contain any hole; (iii) G lc-vis,q, G vis,q have the same connected comonents. In general, the inclusions in Theorem III.1(i) are strict. Fig. 6 shows an examle where G EMST,Gvis,Q G lc-vis,q G vis,q. Fig. 6. From left to right, visibility grah, locally-cliueless visibility grah and Euclidean Minimum Sanning Tree of the visibility grah. IV. RENDEZVOUS VIA PROXIMITY GRAPHS Here we state the model, the control objective, the coordination algorithm, and the closed-loo system roerties. A. A synchronous network of visually-guided agents By a visually-guided agent, we refer to any agent, occuying a location in Q Q, and caable of measuring the relative osition of every other agent visible to it, i.e., within line-of-sight. In addition to this, it can also sense the boundary of Q. Each agent has a rocessor with the ability of allocating continuous and discrete states and erforming oerations on them. A collection of finite number, say n, of such agents form a network. Note that as a conseuence of the above, whenever Q contains no hole, the rocessor on any agent has the caability to answer the uery as to whether two agents visible to it are mutually visible to one another. The ith agent in such a network is caable of moving at any time m N, for any unit eriod of time, according to the synchronized discrete-time control system i (m + 1) = i (m) + u i. (1) We also assume that there is a maximum ste size s max R + for all agents, that is, u i s max, for i {1,...,n}. B. The rendezvous motion coordination roblem We now state the control design roblem for the network of visually-guided agents. The rendezvous objective is to steer each agent to a common location. This objective is to be achieved with the limited information flow described in the model above. Tyically, it will be imossible to solve the rendezvous roblem if the agents are laced in such a way that they do not form a connected grah. Arguably, a good roerty of any algorithm to rendezvous is that of maintaining some form of connectivity between agents. C. The Circumcenter Algorithm Here is an informal descrition of what we shall refer to as the Circumcenter Algorithm over a roximity grah G: Each agent erforms the following tasks: (i) it detects its neighbors according to G; (ii) it comutes the circumcenter of the oint set comrised of its neighbors and of itself, and (iii) it moves toward this circumcenter while maintaining connectivity with its neighbors. This algorithm is insired by the one introduced in [1]. Let us clarify which roximity grahs are allowable and how connectivity is maintained. Firstly, we are allowed to design over any roximity grah G that is satially distributed over G vis,q. This is a direct conseuence of our modeling assumtion that each agent can acuire the location of every other agent visible to it. Secondly, we maintain connectivity by restricting the allowable motion of each agent. In articular, if agents i and j are neighbors in the roximity grah G, then their subseuent ositions are reuired to belong to C Q ( i, j ) as defined in Theorem II.8. If an agent i has its neighbors at locations { 1,..., l }, then define M i = { 1,..., l } { i }. We define the constraint set C i,q(m i ) by C i,q(m i ) = C Q ( i,). M i Remark IV.1 C i,q(m i ) is a convex subset of Q containing i. This follows from the definition of C i,q(m i ) and Theorem II.8 (i). If M i Ve r (Q) is emty and the set of neighbors of i is fixed, then C i,q(m i ) changes continuously as a function of i and of the ositions of its neighbors. This follows from the fact that for each j M i, i is constrained to remain in C Q ( i, j ) which is a convex and comact subset of Q. The statement is then a conseuence of Theorem II.8 (iii) and the fact that C i,q(m i ) is an intersection of continuous mas. With this, we are ready to formally describe the algorithm. Name: Circumcenter Algorithm over G Assumes: (i) s max R + is maximum ste size (ii) Q Q (iii) G is a satially distributed roximity grah over G vis,q For i {1,...,n}, agent i executes the following at each time instant in N: 1: acuire { 1,..., k } := N Gvis,Q, i (P) 2: comute M i := N G,i ({ 1,..., k }) { i } 3: comute X i := C i,q(m i ) co(m i ) 4: comute i := roj Xi (CC(M i )) 5: u i := min(smax, i i ) i i ( i i)

5 See Fig. 7 for examles of the constraint sets C i,q(m i ). Fig. 7. Constraint sets C i,q(m i ) generated by the algorithm encoded as described in Section V In what follows we shall refer to the Circumcenter Algorithm over the roximity grah G as the ma T G : Q n Q n. D. Asymtotic correctness of the Circumcenter Algorithm Henceforth, P shall refer to tules of elements in Q of the form ( 1,..., n ). With a slight abuse of notation, we shall use P interchangeably with a oint set P of the form { 1,..., n }. Before roceeding to analyze the convergence roerties of the Circumcenter Algorithm, let us first define a candidate Lyaunov function V erim,q : Q n R +, by V erim,q (P) = erimeter(mpp(p,q)). Lemma IV.2 For any olygon Q with a well-defined interior and exterior, we have the following: (i) for P Q n, MPP(P,Q) contains all the visibility edges of G vis,q (P); (ii) for P 1 Q n and P 2 Q m, we have that MPP(P 1,Q) MPP(P 1 P 2,Q); (iii) for any olygon X Q with a well-defined interior and exterior and P X n, we have that MPP(P,X) MPP(P,Q); (iv) C i,q(m i ) MPP(M i,s( i )) = C i,q(m i ) co(m i ), where i and M i are as in the descrition of the Circumcenter Algorithm in Section IV-C; (v) C i,q(m i ) MPP(M i,s( i )) is convex; (vi) if MPP(P,Q) is a strict subset of MPP(P,Q), then V erim,q (P ) < V erim,q (P ). Finally, we state an imortant lemma that is crucial in characterizing the set to which the seuence of the ositions of the agents converges. Lemma IV.3 Let P (Q \ Ve r (Q)) n. Let G(P) be any grah satially distributed over G vis,q (P). There exists at least one agent i with i Ve(MPP(P,Q)) \ Ve r (MPP(P,Q)) such that the following are true: (i) there exists X i such that i and [ i,] X i ; (ii) i roj Xi CC(M i ) > 0. We shall also reuire, at some times, to make the following assumtion on a seuence {P m } m N {0} Q n : (A) There exists a comact set X (Q \ Ve r (Q)) such that {P m } m N {0} X n. We are now ready to state the following convergence result. Theorem IV.4 Let 1,..., n be a network of visuallyguided agents in Q Q, with maximum ste size s max R +. Assume that Q does not contain any holes, and that the roximity grah G is satially distributed over G vis,q and has the same connected comonents as G vis,q. Then, any trajectory {P m } m N {0} of T G has the following roerties: (i) if the locations of two agents belong to the same connected comonent of G vis,q (P k ) for some k N {0}, then they remain in the same connected comonent of G vis,q (P m ) for all m k, (ii) V erim,q (P m+1 ) V erim,q (P m ), for all m N {0}, (iii) if {P m } m N {0} satisfies (A), then {P m } m N {0} converges to a oint P X n such that either i = j or [ i, j ] Q for all i,j {1,...,n}. The roof for Theorem IV.4 is based on the following useful results. The technical aroach in what follows is similar to the one in [9]. To a roximity grah function G that is satially distributed over G vis,q, and a configuration P Q n, one may associate a grah G G(P) = ({1,...,n},E) by defining (i,j) E if ( i, j ) is an edge of G(P). Clearly, for each P Q n, N GG(P) (i) is eual to the set of neighbors of i with resect to the grah G(P). Given an undirected grah G = ({1,...,n},E), define the Circumcenter Algorithm at Fixed Toology T G : Q n Q n whose ith comonent is (T G ) i ( 1,..., n ) = (T G ) i ( 1,..., n ). Lemma IV.5 For G = ({1,...,n},E), the ma T G : Q n Q n has the following roerties: (i) The ma P T G (P) restricted to (Q \ Ve r (Q)) n is continuous, and (ii) MPP(T G (P),Q) MPP(P,Q), for P Q n. Given Q Q, define the Circumcenter Algorithm at All Connected Toologies T : Q n 2 (Qn) by T(P) = {T G (P) Q n G = ({1,...,n},E) is connected}. Proosition IV.6 For Q Q, the ma T : Q n 2 (Qn) has the following roerties: (i) the ma P T(P) restricted to X, a comact subset of (Q \ Ve r (Q)) n, is uer semicontinuous, and (ii) MPP(T(P),Q) MPP(P,Q), for P Q n if there exists i, j P such that. i j. Now that we have analyzed the smoothness of T, let us study the roerties of the function V erim,: Q n R +. Lemma IV.7 The function V erim,: Q n R + has the following roerties: (i) V erim,q is continuous, and is invariant under ermutations of its arguments; (ii) V erim,q (P) = 0 if and only if i = j for all i P,i {1,...,n}; (iii) V erim,q is strictly decreasing along T as long as V erim,q (P) > 0.

6 We now resent the asymtotic convergence roerties of the algorithm T. The roof of this relies on a discrete-time LaSalle Invariance Princile for set-valued mas; see [9]. Initial osition of the agents Evolution of the network Final osition of the agents Lemma IV.8 Let Q Q. Assume that Q does not contain any holes, and that the roximity grah G is satially distributed over G vis,q and has the same connected comonents as G vis,q. Then, any seuence {P m } m N {0}, defined by P m+1 T(P m ) and satisfying Assumtion (A), converges to a oint P X n such that i = j for all i,j {1,...,n}. E. A variant of the Circumcenter Algorithm In Section IV-D, we conjecture that the Circumcenter Algorithm solves the rendezvous roblem for visually-guided agents if the network evolves in a comact subset of Q \ Ve r (Q). In what follows we describe an algorithm that we conjecture guarantees convergence without this assumtion. Name: Modified Circumcenter Algorithm over G Assumes: (i) s max R + is maximum ste size (ii) Q Q (iii) G is a satially distributed roximity grah over G vis,q with the roerty that two agents at the same location have identical sets of neighbors. For i {1,...,n}, agent i executes the following at each time instant in N: 1: acuire { 1,..., k } := N Gvis,Q, i (P) 2: comute W i := { j j = i, j {1,...,n}} 3: comute B i := (N G,i ({ 1,..., k }) \ W i ) 4: comute M i := B i { i } 5: if B i = {v}, for v Ve r (Q), and i / Ve r (Q) then 6: comute i := v 7: else 8: comute X i := C i,q(m i ) MPP(M i ) 9: comute i := roj Xi (CC(M i )) 10: end if 11: u i := min(smax, i i ) i i ( i i) Remark IV.9 The grah G lc-vis,q fulfills assumtion (iii) in the statement of the Modified Circumcenter Algorithm. V. SIMULATION RESULTS To conduct exeriments, a two-layer simulation environment has been develoed in Matlab R. Figs. 1, 8 and 9 illustrate the erformance of the Circumcenter Algorithm in Section IV-C. VI. CONCLUSIONS This aer focuses on the distributed control of synchronous networks of visually-guided robotic agents. We have defined some useful geometric uantities, such as continuous constraint sets and generalized visibility grahs, and studied circumcenter algorithms for rendezvous. We have rovided a convergence roof as well as successful numerical simulations. Possible future work involves coordination algorithms for deloyment and search for visuall-guided agents. Fig. 8. Simulation results of the Circumcenter Algorithm on a network of agents distributed in a siral olygon. The locations of the agents, at all times, do not belong to reflex vertices. However, at some instants, reflex vertices are aroached very closely. The algorithm is run over G vis,q. Initial osition of the agents Evolution of the network Final osition of the agents Fig. 9. Simulation results of the Circumcenter Algorithm on a network of agents distributed in a olygon shaed like a tyical floor lan. The algorithm is run over G lc-vis,q. Acknowledgments: This material is based uon work suorted in art by AFOSR through Award F , by ONR through YIP Award N , and by faculty research funds granted by the University of California, Santa Cruz. The authors thank Prof. Jana Kosěcká for an early insiring discussion on the subject of this aer. REFERENCES [1] H. Ando, Y. Oasa, I. Suzuki, and M. Yamashita, Distributed memoryless oint convergence algorithm for mobile robots with limited visibility, IEEE Transactions on Robotics and Automation, vol. 15, no. 5, , [2] J. Lin, A. S. Morse, and B. D. O. Anderson, The multi-agent rendezvous roblem: An extended summary, in Proceedings of the 2003 Block Island Worksho on Cooerative Control, ser. Lecture Notes in Control and Information Sciences, V. Kumar, N. E. Leonard, and A. S. Morse, Eds. New York: Sringer Verlag, 2004, vol. 309, [3] P. Flocchini, G. Prencie, N. Santoro, and P. 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