Cooperative detection of areas of rapid change in spatial fields

Transcription

1 Cooperative detetion of areas of rapid hange in spatial fields Jorge Cortés Mehanial and Aerospae Engineering, University of California, San Diego, CA 9093, USA Abstrat This paper proposes a distributed oordination algorithm for roboti sensor networks to detet boundaries that separate areas of rapid hange of planar spatial phenomena. We onsider an aggregate objetive funtion, termed wombliness, to measure the hange of the spatial field along the losed polygonal urve defined by the loation of the sensors. We enode the network task as the optimization of the wombliness and haraterize the smoothness properties of the objetive funtion. In general, the omplexity of the spatial phenomena may make the gradient flow ause self-intersetions in the polygonal urve desribed by the network. We design the hybrid wombling algorithm that allows for network splitting and merging and guarantees loal onvergene to the ritial onfigurations of the wombliness, while monotonially optimizing it. The tehnial approah ombines ideas from statistial estimation, dynamial systems, and hybrid modeling and design. Key words: Roboti sensor networks; hybrid systems; distributed detetion; spatial fields; wombling 1 Introdution Consider a network of mobile sensors moving in a planar environment where a spatial proess takes plae. Our aim is to design a distributed algorithm that allows the group of sensors to determine boundaries separating areas with large differenes in the spatial proess. Suh boundary phenomena are relevant in multiple appliations, inluding upwelling boundaries in oeanographi studies, see e.g., Sousa et al. [008] and loud boundaries in weather foreasting, see e.g., [Nowak et al., 008]. Literature review: Our work has onnetions with several domains. In statistial estimation [Fagan et al., 003, Banerjee and Gelfand, 006], wombling boundaries are urves that delimit areas of rapid hange of some sientifi phenomena of interest. Algorithms for deteting these boundaries are used in biology [Barbujani et al., 1989], omputational eology [Fagan et al., 003], and mediine [Jaquez and Greiling, 003]. Banerjee and Gelfand [006] use Monte Carlo Markov Chain methods to detet wombling boundaries in spatial proess models. In omputer vision [Osher and Paragios, 003, Kimmel, 003, Paragios et al., 005], image segmentation and edge detetion problems are enoded as optimization problems for a variety of objetive funtionals. These problems are solved using PDE-based approahes that build on variational information of the funtion- Researh supported by NSF award ECS A onferene version of this work appeared as [Cortés, 009]. address: ortes@usd.edu (Jorge Cortés. als. Work in this area assumes the availability of full, omplete information, and is not diretly appliable to motion oordination of distributed sensors with partial information. Finally, we use modeling tools from hybrid systems [van der Shaft and Shumaher, 000, Liberzon, 003, Sanfelie et al., 008] in the algorithm design. Statement of ontributions: The spatial phenomena is desribed by a time-independent twie ontinuously differentiable funtion on a planar ompat set. We onsider a network of agents with first-order dynamis apable of measuring the gradient and the Laplaian of the field along finite segments. The wombliness of a not selfinterseting, losed urve is a measure of the alignment of its normal diretion with the gradient of the field. We use the wombliness assoiated to a losed polygonal urve to formulate the network objetive as a distributed optimization problem. Our first ontribution is the study of the smoothness properties of the wombliness, an expliit expression for its gradient and a haraterization of its ritial points. If the network were to follow a gradient asent, situations may arise where the polygonal urve desribed by the sensors beomes self-interseting and the ensuing flow ill-posed. To prevent this, our seond ontribution is the synthesis of the hybrid wombling algorithm for distributed wombliness optimization. This strategy allows for splitting and merging of urves and may require the inlusion of additional agents. Our third ontribution is the haraterization of its onvergene properties. Preprint submitted to Automatia 7 September 011

2 Preliminaries Let us start with some notation. For S R d, we denote by int(s, S, and S its interior, losure, and boundary, respetively. Let S be the ardinality of a finite set S. Let unit : R R be the map defined by unit(x = x/ x for x 0 and unit(0 = 0. A totally ordered S is a set paired with a total order, i.e., a transitive, skewsymmetri and total binary relation. Given a S, we let a + (resp. a denote the smallest (with respet to b S suh that a b (resp. largest b S suh that b a. Given a, b S, let a, b = S : a b} if a b and a, b = S : a or b} if b a..1 Planar geometri notions Given a vetor v = (v 1, v R, let v = (v, v 1 R denote the vetor perpendiular to v to the right, i.e., the 90 degree lokwise rotation of v. Given p q R, let ]p, q[ and [p, q] be, respetively, the open and losed segments with end points p and q. We let [p, q[= [p, q] \ q}. Let u [p,q] = unit(q p be the unit vetor in the diretion from p to q and n [p,q] = u [p,q] the unit normal vetor to the right. In oordinates, 1 u [p,q] = q p (q 1 p 1, q p, 1 n [p,q] = q p (q p, p 1 q 1, where p = (p 1, p and q = (q 1, q. We denote by H[p,q] out = z R : (z p T n [p,q] 0} the halfplane of points in the positive diretion of the normal vetor with respet to the losed segment [p, q]. Likewise, we denote H[p,q] in = z R : (z p T n [p,q] 0}. Given p R and v R, let ray(p, v = z R : z = p + tv, t R 0 }. The wedge wedge(p, (v 1, n 1, (v, n is the open one with vertex p and axes ray(p, v 1 and ray(p, v, i.e., the set of points towards whih n 1 points along ray(p, v 1 and n points along ray(p, v, see Fig. 1 for an illustration. For the wedge to be well-defined, the normal vetors n 1 and n need to speify it uniquely. Given a simply onneted set D R with nonempty n 1 v 1 v p (a n v p n 1 v 1 (b Fig. 1. Wedges determined by p, (v 1, n 1 and (v, n. interior and v R with origin q in D, let pr D (v be the orthogonal projetion of v onto the tangent spae T q D of D at q if q D and v points outside D, and v otherwise. n. Curve parameterizations A urve C in R is the image of a map γ : [a, b] R. The map γ is alled a parametrization of C. We often identify a urve with its parametrization. We only deal with pieewise smooth and ontinuous urves. A urve C is self-interseting if γ is not injetive on (a, b. A urve C is losed if γ(a = γ(b. For a urve C, γ denotes the tangent diretion to C and n C = unit( γ the unit normal vetor to C. A losed, not self-interseting urve C partitions R into two disjoint open and onneted sets, Inside C and Outside C, suh that n C along C points outside Inside C and inside Outside C, respetively. The orientation of C affets the definition of n C and Inside C, Outside C. Given a urve C parametrized by a pieewise smooth map γ : [a, b] C, the line integral of a funtion h : C R R over C is C h = C h(qdq = b a h(γ(t γ(t dt, (1 and is independent of the seleted parametrization..3 Modeling of hybrid systems We briefly review the framework for modeling hybrid systems in [Goebel et al., 004].A hybrid system is defined by a tuple H = (C, F, D, G, where C R d is the flow set, F : R d R d with C Dom(F, is the flow (set-valued map, D R d is the jump set, and G : R d R d with D Dom(G is the jump (set-valued map. The dynamis is given by ẋ F (x, x C, x + G(x, x D. ( The notation ẋ indiates the time derivative of the state, whereas x + indiates the value of the state after an instantaneous hange. In this paper, we use singletonvalued flow maps F (x = f(x}, with f : R d R d. Equation ( reflets the fat that solutions an evolve aording to the differential inlusion ẋ F (x when they belong to C and aording to the differene inlusion x + G(x when they belong to D. The onept of solution an be formalized using the notions of hybrid time domain and hybrid ar [Goebel et al., 004]. The set E R 0 N is a ompat hybrid time domain if E = J 1 j=0 ([t j, t j+1 ], j, with 0 = t 0 t 1 t J. The set E R 0 N is a hybrid time domain if for all (t, j E, E ([0, t] 0, 1,..., j} is a ompat hybrid time domain. The length of E is length (E = sup t E + su E, i.e., the sum of the supremum of the t oordinates and the supremum of the j oordinates. A funtion φ : E R d is a hybrid ar if E is a hybrid time domain and, for eah j N, the funtion t φ(t, j is loally absolutely ontinuous on I j = t R 0 : (t, j E}. Finally, a hybrid ar φ : E R d is a solution of ( if φ(0, 0 C D and the following holds:

3 (i for j N with int(i j, φ(t, j C for t int(i j and φ(t, j F (φ(t, j for almost all t I j ; (ii for (t, j E suh that (t, j + 1 E, φ(t, j D and φ(t, j + 1 G(φ(t, j. Of partiular interest are omplete (length (E = and Zeno (su E = and sup t E < trajetories. 3 Problem statement Let D R be a ompat, simply onneted set with nonempty interior, and let Y : D R be a twie ontinuously differentiable funtion modeling a planar timeindependent field. Consider a network Σ of agents with positions p 1,..., p n moving in D. Our objetive is to find regions in D where loally maximal hanges our in Y by determining their boundaries. We begin by defining a measure of how fast the field hanges aross a given urve. The wombliness or alignment of a urve C is W(C = Y, n C, (3 C see e.g., [Kimmel, 003, Banerjee and Gelfand, 006]. The interpretation of W is as follows. At eah point of the urve, we look at how muh Y is hanging along the normal diretion to C (i.e., how muh Y is flowing through C. The integral sums this hange throughout the urve. We are interested in using the roboti network to find urves whose value of W is large. For a losed not self-interseting urve, W an be written, W(C = Y, n C = div Y = Y, (4 C using the Gauss Divergene Theorem [Courant and John, 1999], where D is the set in R whose boundary is C, and Y = Y x + Y y is the Laplaian of Y. Note that W is a bounded funtion on the set of losed not self-interseting urves in D. Observe that, in general, the level urves of Y are not optimizers of W. An example is given by Y (x 1, x = e x 1 x, whose level sets are the ellipses x 1 + x =, 0. One has Y (x 1, x = Y (x 1, x (x 1 + 8x 3, and hene, using (4, we dedue that the ellipse x 1 + 8x 3 = 0 (whih is not a level urve of Y maximizes W. In general, the optimization of (3 is an infinitedimensional problem. Our approah is to order ounterlokwise the agents aording to their unique identifier, and onsider the losed polygonal urve that results from joining the positions of onseutive robots, i.e., for (p 1,..., p n D n, let γ p be the losed polygonal urve that results from the onatenation of [, 1 ], i 1,..., n 1} and [p n, p 1 ]. In general, suh urves may be self-interseting. Therefore, we restrit our attention to the following open subset of D n, S = (p 1,..., p n D n : γ p is not self-interseting}. D D Define W : S R by W (p 1,..., p n = W(γ p, i.e., W (p 1,..., p n = n i=1 [,1] Y, n [pi,1]. (5 The optimization of (5 is now a finite-dimensional problem. Our objetive is then to have the network Σ loally optimize W. Note that W an be expressed in terms of the polygon determined by the onatenated straight segments. If P(p 1,..., p n denotes this polygon, then W (p 1,..., p n = P(p 1,...,p n Y. (6 For reasons that will beome lear later, we assume that, at eah network onfiguration, agent i 1,..., n} an measure Y and Y along [1, ] and [, 1 ]. 4 Smoothness of the wombliness measure In this setion we analyze the smoothness properties of the wombliness measure, provide expliit expressions for the gradient, and haraterize the ritial points. Proposition 4.1 (Gradient of W The funtion W : S R is ontinuously differentiable. For eah i 1,..., n}, at (p 1,..., p n S, we have W ( = ( + [,1] [1,] 1 q 1 Y (q n [pi,1]dq (7 q 1 1 Y (q n [pi 1,]dq. PROOF. To ompute the gradient of W, we use (6. For a parameterization γ of the boundary of P(p 1,..., p n, aording to the generalized onservation-of-mass lemma, f. [Bullo et al., 009, Proposition.3], W = Y γ n P dγ. (8 P Next, we develop this expression. Let γ ii+1 : [0, 1] R be a parametrization of [, 1 ], γ ii+1 (t = + t(1. (9 The olletion of these parameterizations for i 1,..., n} defines a parameterization γ of the boundary of P(p 1,..., p n. Substituting into (8, we get W = + [,1] [1,] Y γ ii+1 n [pi,p p i+1] dγ ii+1 i Y γ i 1i n [pi 1,] dγ i 1i. 3

4 Aording to (9, γ ii+1 / = (1 ti and γ i 1i / = ti. Therefore, W = (1 t Y n [pi,1] 1 dt 0 t Y n [pi 1,] 1 dt. (10 Finally, from (9 and using t [0, 1], we have (1 t 1 = 1 γ ii+1 (t, t 1 = γ i 1i (t 1. The result follows from using these formula in (10. Aording to Proposition 4.1, agent i needs to know the loation of its neighbors in the ring graph (agents i 1 and i + 1 and the Laplaian of the field along the orresponding segments to ompute W. The following haraterization of the ritial onfigurations of W follows from observing that if three agents 1,, and 1 are not aligned, then n [pi,1] and n [pi 1,] are linearly independent. In the following result, with a slight abuse of notation, we let W : S R denote the extension by ontinuity, see e.g., [Pedrik, 1994], of W to S. Corollary 4. (Critial points of W Let (p 1,..., p n S be a ritial ( onfiguration of W. Then, for i W 1,..., n}, pr D = 0. Moreover, if (p 1,..., p n int(d n and no three onseutive agents are aligned, this an be alternatively desribed by, for i 1,..., n}, [,1] [,1] 1 q Y = 0, q Y = 0. 5 The hybrid wombling algorithm (11a (11b Following Setion 3, we seek to optimize the funtion W. The approah we propose is to implement a distributed gradient flow using the result of Proposition 4.1. However, the evolution under this flow of the losed polygonal urve γ p defined by the network agents may beome self-interseting. To address this problem, we design here the hybrid wombling algorithm on Σ. This strategy presribes state transitions that resolve the urve intersetions. We will see that, in some ases, the state transition requires the inlusion of additional agents. In what follows, we desribe the resulting hybrid system in detail within the framework of Setion.3. The flow set. Given N N, let l = l 1,..., l l } be a partition of 1,..., N} into disjoint totally ordered subsets with l 1 1 and l 3 for,..., l }. Let F(1,..., N} denote the (finite lass of all suh olletions. For P R N and l F(1,..., N}, let γ p be the losed polygonal urve defined by p k } k l,,..., l }. The agents in l 1 will be used later in the definition of the jump map. We define the flow set by C = (P, l R N F(1,..., N} for µ =,..., l, γ µ p S w/ disjoint inside or disjoint outside sets}. Note the slight abuse of notation in this expression when using S, sine eah urve γ µ p might onsist of a different number of points. Note also that the set C is losed. The flow map. For eah urve γp, let P be the vetor whose omponents are p k } k l. With a slight abuse of notation, let W (P = W(γp. The flow map f : C R N+1 is defined as follows. For k l 1 N + 1}, f k (P, l = 0. For,..., l } and k l, its kth omponent is ( W pr D p f k (P, l = k if W (P 0, ( W (1 pr D if W (P < 0. p k Note that f k is ontinuous exept possibly at onfigurations for whih p k D for some k l (due to the projetion operation and at onfigurations where W (P = 0 (due to the sign hange. The jump set and the jump map. As mentioned above, the evolution under the gradient flow presribed by f on C might lead to a urve self-intersetion or, if there is more than one urve evolving in D, to the intersetion among several urves. A urve self-intersetion orresponds to D 1 = (P, l C : there exists,..., l } and k, z 1, z l with p k [p z1, p z ]}. An intersetion of urves orresponds to D = (P, l C : there exist β,..., l } with k l and p k γp}. β Both sets are losed. Among these onfigurations, we need to identify the ones where f points outside C, and hene a state transition is neessary to have a welldefined network evolution. This is what we study next. Setion 5.1 deals with urve self-intersetion and Setion 5. deals with intersetion of urves. In both ases, we identify the onfigurations in the jump set D C and define the ation of the jump map G. 5.1 Curve self-intersetion Let (P, l D 1. Here we onsider the ase of a single urve self-intersetion. This disussion forms the basis for the treatment of the multiple self-intersetion ase in Setion Let γp denote the self-interseting urve. When a self-intersetion ours, either Inside γp or Outside γp beome disonneted. We refer to these two ases as inside and outside self-intersetions, respetively, see Fig.. Setions and 5.1. desribe the onfigurations in the jump set when the self-intersetion ours at an open segment or at a point s loation, respetively. Setion desribes the jump map. 4

5 5.1.1 Self-intersetion at an open segment For i j l suh that ], [, define λ [0, 1 by = (1 λ + λ and onsider v i = (1 λu j + λu j+, u k = sgn(w (P pr D p k, where k i, j, j + }. The onditions haraterizing when (P, l belongs to the jump set D depend on the type of self-intersetion, and are detailed next. Inside self-intersetion. If the self-intersetion is of inside type, the segment [, ] belongs to H[n j, ] and there exists the possibility of rossing from H[n j, ] to H[p out j, ], see Fig. (a. Then, (P, l D iff (i (P, l D 1 with inside self-intersetion at an open segment and (u i v i T n [pj, ] 0. Inluding (u i v i T n [pj, ] = 0 (orresponding to evolving along the segment [, ] in the definition of D simplifies the onvergene analysis of Setion 6. pj+ pi pj γ (a inside pi+ pi γ (b outside Fig.. γ p defined by p 1,..., p n is self-interseting at an open segment. (a inside self-intersetion and (b outside self-intersetion. γ p is the union of two not self-interseting urves and γ. Outside self-intersetion. If the self-intersetion is of outside type, the segment [, ] belongs to H[p out j, ] and there exists the possibility of rossing from H out [, ] to Hin [, ], see Fig. (b. Then, (P, l D iff (ii (P, l D 1 with outside self-intersetion at an open segment and (u i v i T n [pj, ] Self-intersetion at a point For i j l suh that =, onsider u i = sgn(w (P pr D, u j = sgn(w (P pr D. γ (a inside γ (b outside Fig. 3. γ p defined by p 1,..., p n is self-interseting at a point s loation. (a inside self-intersetion and (b outside self-intersetion. γ p is the union of two not self-interseting urves and γ. Inside self-intersetion. If the self-intersetion is of inside type, see Fig. 3(a, define the vetors u[pi v 1 =,pi] if [, ] H in u [pj, ] if [, ] H in u[pi+,pi] if [, ] H in v = u [pj, ] Then, (P, l D iff [,], [,], [, ], if [, ] H in [, ]. (iii (P, l D 1 with inside self-intersetion at a point and u i u j wedge(, (v 1, v 1, (v, v. Outside self-intersetion. If the self-intersetion is of outside type, see Fig. 3(b, define the vetors u[pj,p j v 1 = ] if [, ] H in u [pi,] if [, ] H in u[pj,p j+ ] if [, ] H in v = u [pi+,] Then, (P, l D iff [,], [,], [, ], if [, ] H in [, ]. (iv (P, l D 1 with outside self-intersetion at a point and u i u j wedge(, (v 1, v 1, (v, v Jump map Here, we speify the ation of the jump map G on (P, l D D 1 with a single self-intersetion. In this ase, γ p an be deomposed into two polygonal urves and γ, see Figs. and 3. The urve is defined by the onatenation of the segments [p k, p k+ ] : k i, j } [, ], if ], [, and [p k, p k+ ] : k i +, j } [, ], if =. The urve γ is defined by the onatenation of the segments [p k, p k+ ] : k j +, i } [, ], if ], [, and [p k, p k+ ] : k j +, i } [, ], if 5

6 =. Let k 1 and k denote the indies of the agents with m l 1. Here, k is the result of substituting i by m in k. This transition is illustrated in Fig. 5. G may not pj pj pi pi+ pi pi pi+ γ γ pj+ pj+ γ γ pi pi Fig. 4. Agent re-positioning. Agents in get re-positioned onto γ. defining and γ, respetively. In the ase of a selfintersetion at an open segment, note that i k 1 k. The wombliness of γ p is split between and γ as W(γ p = W( + W(γ. Depending on the sign of W( and W(γ, we define: Agent re-positioning. If W( and W(γ have different signs, we only keep the urve whose wombliness has the same sign as γp and reposition the agents of the other urve along it arbitrarily. Formally, if sgn(w( sgn(w(γ 0, then G(P, l is the set of onfigurations (P, l suh that P satisfies p k = p k for k k µ or k k µ, p k γ µ otherwise, where sgn(w(γ µ = sgn(w(γ p and µ 1, }\µ}. Note that if both urves have nonzero wombliness, the absolute value of the wombliness of the resulting not selfinterseting urve is stritly larger than the value of the wombliness of the original self-interseting urve γ p. Fig. 4 illustrates this transition. Curve splitting. If W( and W(γ have the same sign as W(γ p (note that this implies n 5, then the urves split. If the self-intersetion ours on an open segment, one additional agent must be added to the network at the intersetion loation. Formally, if sgn(w( sgn(w(γ > 0 and the self-intersetion is at a point, G(P, l = (P, l } with l = l + 1 and l β = l β, for β 1,..., l } \ }, l = k 1, l l +1 = k. If sgn(w( sgn(w(γ > 0 and the self-intersetion is on an open segment, G(P, l is the set of onfigurations (P, l that satisfy l = l + 1 and p k = p k, for k m, p m =, l β = l β, for β,..., l } \ }, l 1 = l 1 \ m}, l = k 1, l l +1 = k, Fig. 5. Curve splitting. γ p is split into and γ, and these urves evolve independently afterward. be outer semiontinuous relative to D at onfigurations where the wombliness of one of the urves is zero Multiple self-intersetions Let (P, l D 1. Here we disuss the ase when γp has multiple self-intersetions. One proeeds by haraterizing the self-intersetions that need to be resolved beause the flow map f points outside C, in an analogous way to ases (i-(ii in Setion and (iii-(iv in Setion If e is the number of self-intersetions that need to be resolved, then γp an be deomposed into e + 1 not self-interseting urves,..., γ e+1. The wombliness of γp is split as W(γp = e+1 µ=1 W(γµ. Similarly to Setion 5.1.3, the ation of G is: For µ 1,..., e+1} with sgn(w(γ µ = sgn(w(γ p, γ µ splits and remains after the jump as an independent urve. At least a urve always falls in this ase; For µ 1,..., e+1} with sgn(w(γ µ sgn(w(γ p, all agents in γ µ are repositioned along urves in the previous ase. The repositioning is arbitrary. For spae reasons, we leave to the reader the formal desription of the ation of the jump map. Finally, if (P, l is a onfiguration where multiple urves self-interset, the jump map ats on eah one as desribed above. 5. Intersetion between urves Let (P, l D. Here we onsider the ase of a single intersetion of two urves. This disussion is the basis for the treatment of the multiple intersetion ase in Setion Let γ p and γ β p denote the two interseting urves. When an intersetion ours, either Inside γ p Inside γ β p (inside intersetion or Outside γ p Outside γ β p (outside intersetion is nonempty, see Fig. 6. Setions 5..1 and 5.. desribe the onfigurations in the jump set when the intersetion ours at an open segment or at a point s loation, respetively. Setion 5..3 desribes the jump map. 6

7 5..1 Intersetion at an open segment For i l and j l β suh that ], [, define λ [0, 1 by = (1 λ + λ. Consider Inside intersetion. If the intersetion is of inside type, see Fig. 7(a, define v i = (1 λu j + λu j+, u i = sgn(w (P pr D, u k = sgn(w β (P β pr D β p k, γp γp β γ β p γ p where k j, j + }. The onditions haraterizing when (P, l belongs to the jump set D depend on the type of self-intersetion, and are detailed next. γ β p (a inside (b outside Fig. 7. γ p and γ β p interset at a point s loation. (a inside intersetion and (b outside intersetion. In both ases, γ p and γ β p an be merged into a new self-interseting urve γ. γ β p γ p (a inside γ p (b outside Fig. 6. γ p and γ β p interset at an open segment. (a inside intersetion and (b outside intersetion. In both ases, γ p and γ β p an be merged into a new self-interseting urve γ. Inside intersetion. If the intersetion is of inside type, the segment [, ] of γp belongs to H[n j, ] and there exists the possibility of rossing from H[n j, ] to H[p out j, ], see Fig. 6(a. Then, (P, l D iff (i (P, l D with inside intersetion at an open segment and (u i v i T n [pj, ] 0. Outside intersetion. If the intersetion is of outside type, the segment [, ] of γp belongs to H[p out j, ] and there exists the possibility of rossing from H[p out j, ] to H[n j, ], see Fig. 6(b. Then, (P, l D iff (ii (P, l D with outside intersetion at an open segment and (u i v i T n [pj, ] Intersetion at a point For i l and j l β suh that =, onsider u i = sgn(w (P pr D, u j = sgn(w (P β pr D β. u[pi v 1 =,pi] if [, ] H in u [pj, ] if [, ] H in u[pi+,pi] if [, ] H in v = u [pj, ] Then, (P, l D iff [,], [,], [, ], if [, ] H in [, ]. (iii (P, l D with inside intersetion at a point and u i u j wedge(, (v 1, v 1, (v, v. Outside intersetion. If the intersetion is of outside type, see Fig. 7(b, define u[pj,p j v 1 = ] if [, ] H in u [pi,] if [, ] H in u[pj,p j+ ] if [, ] H in v = u [pi+,] Then, (P, l D iff [,], [,], [, ], if [, ] H in [, ]. (iv (P, l D 1 with outside intersetion at a point and u i u j wedge(, (v 1, v 1, (v, v Jump map Here we speify the ation of G on (P, l D D with a single intersetion of urves. In this ase, the two urves an be merged into a single one, γ, see Figs. 6 and 7, defined by the onatenation of the segments [p k, p k+ ] : k l } [, ] [p k, p k+ ] : k l β \ j}} [, ], if ], [, and [p k, p k+ ] : k l } [p k, p k+ ] : k l β }, if =. If the urve intersetion is at an open segment, appears twie in γ. The wombliness of γ p and γ β p is summed up as W(γ = W(γ p + W(γ β p. Depending on the sign of W(γ p and W(γ β p, we have: 7

8 Agent re-positioning. If W(γ p and W(γ β p have different signs, we only keep the urve whose wombliness has the same sign as γ and reposition the agents of the other urve along it arbitrarily. Formally, if sgn(w(γ p sgn(w(γ β p 0, then G(P, l is the set of onfigurations (P, l with l = l 1 and l η = l η, for η 1,..., l } \, β}, l µ = l l β, p k = p k, for k l µ, p k γ µ p, for k l µ, where sgn(w(γ µ = sgn(w(γ p and µ, β} \ µ}. If both urves have nonzero wombliness, the absolute value of the wombliness of the resulting not selfinterseting urve is stritly larger than the value of the wombliness of γ. Curve merging. If W(γ p and W(γ β p have the same sign, then we merge the respetive urves into γ. If the intersetion ours on an open segment, one additional agent must be added to the network at the intersetion loation. Formally, if sgn(w(γ p sgn(w(γ β p > 0 and the self-intersetion is at a point, G(P, l = (P, l } with l = l 1 and l η = l η, for η 1,..., l } \, β}, l = l l β. If sgn(w( sgn(w(γ > 0 and the self-intersetion is on an open segment, G(P, l is the set of onfigurations (P, l with l = l 1, m l 1, and p k = p k, for k m, p m =, l η = l η, for η,..., l } \, β}, l = l l β m}, l 1 = l 1 \ m}. G may not be outer semiontinuous relative to D at onfigurations where the wombliness of one of the urves is zero. Suh onfigurations are never attained by the ontinuous flow of hybrid wombling algorithm Multiple intersetions Let (P, l D. Having desribed above the ase of a single intersetion of urves, we disuss next the ase when two or more urves interset at multiple points. One proeeds by haraterizing the intersetions that need to be resolved beause the flow map f points outside C, in an analogous way to ases (i-(ii in Setion 5..1 and (iii- (iv in Setion 5... The ation of G an be desribed along the same lines of Setion The sum of the wombliness of the urves involved in the intersetions plays a key role. Agents in urves whose wombliness has the opposite sign as the total sum get re-positioned (this an be done in an arbitrary way. Curves whose wombliness has the same sign as the total sum and are still interseting after the agent re-positioning are merged together. If the urve resulting from this operation is selfinterseting, then it is handled aording to Setion 5.1. For reasons of spae, we leave to the reader the formal desription of the ation of the jump map. Remark 5.1 (Algorithm implementation and distributed harater The implementation of the hybrid wombling algorithm by the network requires agents to be able to detet intersetions that need resolving (an event of loal nature and to be apable of aggregating the wombliness of the urves to whih they belong or are involved in an intersetion. This aggregation an be done in a distributed way with a number of algorithms that rely on network onnetivity and interation among neighbors, see e.g., [Lynh, 1997, Bullo et al., 009]. This apability allows individual agents to be able to determine the type of transition. Exeuting a given transition is another event of loal nature, sine the agents involved simply have to hange their neighbors to the left and to the right. Finally, the network waits for agents whih are repositioning to travel to their new positions before ontinuing with its evolution. 6 Convergene analysis Here, we haraterize the onvergene properties of a roboti network evolving under the hybrid wombling algorithm. In our analysis, we only onsider trajetories whih are not Zeno. In partiular, this exludes the possibility of bloking transitions, i.e., a urve splitting jump resulting in a new onfiguration that satisfies the onditions for a urve merging; or a urve merging jump resulting in a new onfiguration that satisfies the onditions for a urve splitting. The wombliness W Σ : C R assoiated to a network Σ with state (P, l is given by W Σ (P, l = W(γ p + + W(γ l p. Sine the wombliness is bounded on the set of losed not self-interseting urves in D, W Σ is bounded. Theorem 6.1 A not Zeno network trajetory that evolves under the hybrid wombling algorithm and undergoes a finite number of transitions resulting in agent additions onverges to the set of ritial onfigurations of W Σ. Moreover, W Σ is monotonially inreasing along the network trajetory. PROOF. Sine the network trajetory undergoes a finite number of transitions that result in agent additions, there exists N N with N n suh that the network evolution is ontained in C. Consider the hybrid system (C, f, D, G. Note that C and D are losed by onstrution. However, as noted above, the map f is not ontinuous and the set-valued map G is not outer semiontinuous. Hene, we use their Krasovskii regularizations f and Ĝ, defined respetively by, see [Sanfelie et al., 008], f(x = δ>0 onf((x + δb(0, 1 C, x C, Ĝ(x = δ>0 G((x + δb(0, 1 D, x D. (13a (13b 8

9 Here, B(0, 1 is the losed ball entered 0 with radius 1, and on is the losed onvex hull. Sine f and G are loally bounded, so are their regularizations. Moreover, f is outer semiontinuous relative to C and Ĝ is outer semiontinuous relative to D, f. [Sanfelie et al., 008]. Sine f(x f(x and G(x Ĝ(x, solutions of (C, f, D, G are solutions of (C, f, D, Ĝ too. The effet of the regularization (13 is to extend the set of allowed network evolutions. Speifially, in (C, f, D, Ĝ, a urve with zero wombliness an flow in either the positive or negative diretion of the gradient; if a urve self-intersets and one of the urves has zero wombliness and non-empty interior, then both agent re-positioning and urve splitting are allowed; and similarly, if a urve of zero wombliness intersets another urve, then both agent re-positioning and urve merging are allowed. The bounded funtion V = W Σ is a Lyapunov funtion for (C, f, D, Ĝ, i.e., for eah fixed l F(1,..., N}, V is monotonially non-inreasing along the ontinuous evolution of the network, and V is monotonially non-inreasing when network transitions take plae. Moreover, the network trajetory has the property that all but (at most a finite of the elapsed times between jumps are bounded below by a positive onstant. Formally, this means that there exists τ > 0 suh that, for all but a finite number of j N, sup t t : (t, j, (t, j E} τ. This fat an be shown by ontradition. Suppose the above statement does not hold, i.e, for τ m = 1/m, there exists j m N suh that sup t t : (t, j m, (t, j m E} < 1/m. This implies that lim m sup t t : (t, j m, (t, j m E} = 0. Note that a jump orresponds to either (i an agent re-positioning, (ii a urve splitting, or (iii a urve merging. Note that, sine the trajetory is not Zeno, the new network onfiguration after (ii takes plae does not satisfy the requirements to exeute (iii, and vie versa. In ase (i, one the jump takes plae and using the fat that f is upper bounded, one an dedue that there exists M > 0 suh that the length of the ensuing ontinuous time interval is lower bounded by M min γ p \ (], [ ], [. In ase (ii, one all agent additions have taken plae (whih, by hypothesis, ours after a finite number of jumps, the only ensuing jumps that an our orrespond to network onfigurations with (self-intersetion at a point. In ase (ii, this implies that, after splitting, if the urves slide along eah other (hene orresponding to a onfiguration where two urves interset, this annot result in a jump (sine it annot orrespond to either an agent repositioning beause of the signs of the wombliness of eah urve or a urve merging beause the intersetion is at an open segment, and hene a similar bound to ase (i an be found. This bound is also valid if the urves instead move away from eah other, given the definition of f. An analogous argument an be made for ase (iii. Therefore, in order for lim m sup t t : (t, j m, (t, j m E} = 0 to hold, the inter-agent distanes in at least one of the urves should derease all the way to zero. This ontradits the fat that suh onfigurations an never be reahed under the hybrid wombling algorithm. The onvergene result then follows from the Hybrid Invariane Priniple [Sanfelie et al., 007, Theorem 4.7 and Corollary 5.9], whih implies that there exists r R suh that the trajetory onverges to the largest invariant set ontained in V 1 (r (D n F(1,..., N} S 1, with S 1 = (P, l C : pr D ( W Σ (P, l = 0}. The statement of the result does not prelude the possibility of the trajetory experiening an infinite number of transitions. Also, onvergene to global optima is not guaranteed. Figs. 8 and 9 present illustrations of the exeution of the hybrid wombling algorithm. Remark 6. (Resolving situations that lead to multiple instantaneous jumps Theorem 6.1 an be extended to trajetories with a finite number of bloking transitions. This is beause suh transitions an be resolved at the ost of adding more agents to the network. We desribe why next. First, note that it is suffiient to disuss a bloking transition that takes plae at a point, say q = =. Now, if Y (q > 0, then two new neighbors must be added to i in the ring graph at q 1 ], [ and q ], [ suh that ( [,q ] ( = q q q Y (q n [pi,q ]dq [q 1,] q q 1 q 1 Y (q n [q1,]dq > 0, and two new neighbors must be added to j in the ring graph at q 3 ], [ and q 4 ], [ suh that ( [,q 4] ( = q 4 q q 4 Y (q n [pj,q 4]dq [q 3,] q q 3 q 3 Y (q n [q3,]dq > 0. Beause of the geometry of the interseting urves, it is not diffiult to show that the resulting network only satisfies the requirements for a urve merging and not for a urve splitting, and hene the transition beomes nonbloking. A similar proedure an be done if Y (q < 0 to yield a network that only satisfies the onditions for a urve splitting and not for a urve merging. 7 Conlusions We have proposed the hybrid wombling algorithm for roboti sensor networks that seek to detet areas of rapid hange of a spatial phenomena. Our algorithm design ombines notions from statistial estimation and omputer vision with tools from hybrid systems. The strategy allows for network splitting and re-grouping, and is guaranteed to monotonially inrease the wombliness of the overall ensemble. In future work we plan to study 9

10 (a (b ( Fig. 8. Exeution of the hybrid wombling algorithm with 10 agents in D = [ 3, 3] [ 3, 3]. (a initial onfiguration, (b robot trajetories, and ( final onfiguration. The field is Y (x1, x = 1.5e (x (x e (x1.75 (x +.. Along the evolution, 1 outside self-intersetion and then inside self-intersetions are triggered, with all transitions resulting in agent re-positionings. (a (b ( Fig. 9. Exeution of the hybrid wombling algorithm with 10 agents in D = [ 4, 4] [ 4, 4]. (a initial onfiguration, (b robot trajetories, and ( final onfiguration. The field is Y (x1, x = e (x1 + x + 1.5e (x1 x. Along the evolution, an inside self-intersetion is triggered that results in a urve splitting. After this, eah new urve undergoes an inside self-intersetion resulting in agent re-positionings. extensions to three dimensions and onsider stohasti senarios, where agents take point measurements of the field and build estimates of its gradient and Laplaian. Referenes S. Banerjee and A. E. Gelfand. Bayesian wombling: Curvilinear gradient assessment under spatial proess models. Journal of the Amerian Statistial Assoiation, 101(476: , 006. G. Barbujani, N. L. Oden, and R. R. Sokal. Deteting areas of abrupt hange in maps of biologial variables. Systemati Zoology, 38: , F. Bullo, J. Corte s, and S. Martı nez. Distributed Control of Roboti Networks. Applied Mathematis Series. Prineton University Press, 009. ISBN Eletronially available at J. Corte s. Distributed wombling by roboti sensor networks. In R. Majumdar and P. Tabuada, editors, HSCC 09, volume 5469 of Leture Notes in Computer Siene, pages , New York, 009. Springer. R. Courant and F. John. Introdution to Calulus and Analysis II/. Classis in Mathematis. Springer, New York, W. F. Fagan, M. J. Fortin, and C. Soykan. Integrating edge detetion and dynami modeling in quantitative analyses of eologial boundaries. BioSiene, 53: , 003. R. Goebel, J. P. Hespanha, A. R. Teel, C. Cai, and R. G. Sanfelie. Hybrid systems: generalized solutions and robust stability. In IFAC Symposium on Nonlinear Control Systems, pages 1 1, Stuttgart, Germany, G. M. Jaquez and D. A. Greiling. Geographi boundaries in breast, lung, and oloretal aners in relation to exposure to air toxins in long island, new york. International Journal of Health Geographis, :1, 003. R. Kimmel. Fast edge integration. In S. Osher and N. Paragios, editors, Geometri Level Set Methods in Imaging, Vision, and Graphis, pages Springer, 003. D. Liberzon. Swithing in Systems and Control. Systems & Control: Foundations & Appliations. Birkha user, 003. ISBN N. A. Lynh. Distributed Algorithms. Morgan Kaufmann, ISBN D. Nowak, D. Ruffieux, J. L. Agnew, and L. Vuilleumier. Detetion of fog and low loud boundaries with ground-based remote sensing systems. Journal of Atmospheri and Oeani Tehnology, 5(8: , 008. S. Osher and N. Paragios, editors. Geometri Level Set Methods in Imaging, Vision, and Graphis. Springer, New York, 003. N. Paragios, Y. Chen, and O. Faugeras, editors. Handbook of Mathematial Models in Computer Vision. Springer, New York, 005. G. Pedrik. A first ourse in analysis. Undergraduate Texts in Mathematis. Springer, New York, R. G. Sanfelie, R. Goebel, and A. R. Teel. Invariane priniples for hybrid systems with onnetions to detetability and asymptoti stability. IEEE Transations on Automati Control, 5(1:8 97, 007. R. G. Sanfelie, R. Goebel, and A. R. Teel. Generalized solutions to hybrid dynamial systems. ESAIM: Control, Optimisation & Calulus of Variations, 14(4:699 74, 008. F. M. Sousa, S. Nasimento, H. Casimiro, and D. Boutov. Identifiation of upwelling areas on sea surfae temperature images using fuzzy lustering. Remote Sensing of Environment, 11 (6:817 83, 008. A. J. van der Shaft and H. Shumaher. An Introdution to Hybrid Dynamial Systems, volume 51 of Leture Notes in Control and Information Sienes. Springer, 000. ISBN