MAINTAINING LIMITED-RANGE CONNECTIVITY AMONG SECOND-ORDER AGENTS

MAINTAINING LIMITED-RANGE CONNECTIVITY AMONG SECOND-ORDER AGENTS KETAN SAVLA, GIUSEPPE NOTARSTEFANO AND FRANCESCO BULLO Abstract. In this paper we consier a-hoc networks of robotic agents with ouble integrator ynamics. For such networks, the connectivity maintenance problems are: (i) o there exist control inputs for each agent to maintain network connectivity, an (ii) given esire controls for each agent, can one compute the closest connectivity-maintaining controls in a istribute fashion? The propose solution is base on three contributions. First, we efine an characterize amissible sets for ouble integrators to remain insie isks. Secon, we establish an existence theorem for the connectivity maintenance problem by introucing a novel state-epenent graph, calle the ouble-integrator isk graph. Specifically, we show that one can always maintain connectivity by maintaining a spanning tree of this new graph, but one will not always maintain connectivity of a particular agent pair that happens to be connecte at one instant of time. Finally, we esign a istribute flow-control algorithm for istribute computation of connectivity-maintaining controls. 1. Introuction. This work is a contribution to the emerging iscipline of motion coorination for a-hoc networks of mobile autonomous agents. This loose terminology refers to groups of robotic agents with limite mobility an communication capabilities. It is envisione that such networks will perform a variety of useful tasks incluing surveillance, exploration an environmental monitoring. The interest in this topics arises from the potential avantages of employing arrays of agents rather than single agents in certain applications. For example, from a control viewpoint, a group of agents inherently provies robustness to failures of single agents or of communication links. The motion coorination problem for groups of autonomous agents is a control problem in the presence of communication constraints. Typically, each agent makes ecisions base only on partial information about the state of the entire network that is obtaine via communication with its immeiate neighbors. One important ifficulty is that the topology of the communication network epens on the agents locations an, therefore, changes with the evolution of the network. In orer to ensure a esire emergent behavior for a group of agents, it is necessary that the group oes not isintegrate into subgroups that are unable to communicate with each other. In other wors, some restrictions must be applie on the movement of the agents to ensure connectivity among the members of the group. In terms of esign, it is require to constrain the control input such that the resulting topology maintains connectivity throughout its course of evolution. In [], a connectivity constraint was evelope for a group of agents moele as first-orer iscrete time ynamic systems. In [] an in the relate references [3, 4], this constraint is use to solve renezvous problems. Connectivity constraints for line-of-sight communication are propose in 1 Submitte to the SIAM Journal on Control an Optimization, Special Issue on Control an Optimization in Cooperative Networks, on November 15, 006. This material is base upon work supporte in part by the ARO MURI Awar W911NF-05-1-019, by the NSF Awar CMS-044041, an by the RECSYS Project IST-001-37170. Giuseppe Notarstefano an Francesco Bullo thank Ruggero Frezza for his kin support. A preliminary version of this manuscript appeare at the 006 American Control Conference in Minneapolis, Minnesota, as reference [1]. Laboratory for Information an Decision Systems, Massachusetts Institute of Technology, Cambrige, MA 0139, Unite States, ksavla@mit.eu, http://web.mit.eu/ksavla/www 3 Dipartimento i Ingegneria ell Innovazione, Università el Salento, Via per Monteroni 73100 Lecce, Italy, giuseppe.notarstefano@unile.it, http://www.ei.unip.it/ notarste/ 4 Center for Control, Dynamical Systems an Computation, University of California, Santa Barbara, CA 93106, Unite States, bullo@engineering.ucsb.eu, http://motion.mee.ucsb.eu 1

[5]. Another approach to connectivity maintenance for first-orer systems is propose in [6]. In [7], a centralize proceure to fin the set of control inputs that maintain k-hop connectivity for a network of agents is given. However, there is no guarantee that the resulting set of feasible control inputs is non-empty. In this paper we fully characterize the set of amissible control inputs for a group of agents moele as secon orer iscrete time ynamic systems, which ensures connectivity of the group in the same spirit as escribe earlier. The contributions of the paper are threefol. First, we consier a control system consisting of a ouble integrator with boune control inputs. For such a system, we efine an characterize the amissible set that allows the ouble integrator to remain insie isks. Secon, we efine a novel state-epenent graph the ouble-integrator isk graph an give an existence theorem for the connectivity maintenance problem for networks of secon orer agents with respect to an appropriate version of this new graph. Specifically, we show that one can always maintain connectivity by maintaining a spanning tree through a subset of all eges without necessarily maintaining connectivity of any particular agent pair that happens to be connecte at an instant of time. Remarkably, this conclusion is ifferent from the one for the single integrator kinematic agent moel. Finally, we formulate an solve an optimization problem for the istribute computation of connectivity-maintaining controls. Specifically, given a set of esire control inputs for all the agents, we aim to compute the set of connectivity-mantaining inputs that are closest to the esire ones. We set up this esign problem as a stanar quaratic programming problem an provie a istribute flow-control algorithm to solve it. As an example application, we solve this optimization problem for a particular set of esire controls to achieve a behavior reminiscent of the well-stuie flocking behavior (e.g., see [8]) among secon orer agents with boune controls while maintaining connectivity, something that has not been reporte in the literature so far. The paper is organize as follows. In Section, we efine an characterize the amissible sets for a ouble integrator to remain insie a isk an base on this we efine a new graph the ouble-integrator isk graph. In Section 3, we provie an existence theorem for the set of control inputs for the whole network of secon orer agents that maintains connectivity with respect to an appropriately scale version of this new graph. We also characterize an give an inner polytopic representation of the constraint set for these connectivity-maintaining control inputs. In Section 4, we propose an optimization problem to compute connectivity-maintaining controls. We also provie some illustrative simulations which suggest an alternative way of achieving a weak form of flocking of the agents. Finally we conclue with a few remarks about future work in Section 5.. Preliminary evelopments. We begin with some notations. We let N, N 0, an R + enote the natural numbers, the non-negative integer numbers, an the positive real numbers, respectively. For N, we let 0 an 1 enote the vectors in R whose entries are all 0 an 1, respectively. We let p enote the Eucliean norm of p R. For r R + an p R, we let B(p,r) enote the close ball centere at p with raius r, i.e., B(p,r) = {q R p q r}. For x,y R, we let x y enote component-wise inequality, i.e., x k y k for k {1,...,}. We let f : A B enote a set-value map; in other wors, for each a A, f(a) is a subset of B. We ientify R R with R.

.1. Maintaining a ouble integrator insie a isk. For t N 0, consier the iscrete-time control system in R p[t + 1] = p[t] + v[t], v[t + 1] = v[t] + u[t], (.1) where the norm of the control is upper-boune by r ctr R +, i.e., u[t] B(0,r ctr ) for t N 0. We refer to this control system as the iscrete-time ouble integrator in R or, more loosely, as a secon-orer system. Given (p,v) R an {u τ } τ N0 B(0,r ctr ), let φ(t,(p,v), {u τ }) enote the solution of (.1) at time t N 0 from initial conition (p,v) with inputs u 1,...,u t 1. In what follows we consier the following problem: assume that the initial position of (.1) is insie a isk centere at 0, fin inputs that keep it insie that isk. This task is impossible for general values of the initial velocity. In what follows we ientify assumptions on the initial velocity that rener the task possible. For r pos R +, we efine the amissible set at time zero by A 0(r pos ) = B(0,r pos ) R. For r pos,r ctr R +, we efine the amissible set for m time steps by A m(r pos,r ctr ) = { (p,v) R {u τ } τ [0,m 1] B(0,r ctr ) an the amissible set by A (r pos,r ctr ) = { (p,v) R {u τ } τ N0 B(0,r ctr ) s.t. φ(t,(p,v), {u τ }) A 0(r pos ) t [0,m] }, s.t. φ(t,(p,v), {u τ }) A 0(r pos ), t N 0 }. With slight abuse of notation we shall sometimes suppress the arguments in the efinitions of amissible sets. The following theorem establishes some important properties of the amissible sets. Theorem.1 (Properties of the amissible sets). For all N an r pos,r ctr R +, the following statements hol: (i) for all m N, A m(r pos,r ctr ) A m 1(r pos,r ctr ) an A (r pos,r ctr ) = lim m + A m(r pos,r ctr ) = lim m + m k=1 A k(r pos,r ctr ); (ii) A (r pos,r ctr ) is a convex, compact set an is the largest controlle-invariant 1 subset of A 0(r pos ); (iii) A (r pos,r ctr ) is invariant uner orthogonal transformations in the sense that, if (p,v) A (r pos,r ctr ), then also (Rp,Rv) A (r pos,r ctr ) for all orthogonal matrices R in R ; (iv) if 0 < r 1 < r, then A (r pos,r 1 ) A (r pos,r ) an A (r 1,r ctr ) A (r,r ctr ). Proof. The two facts in statement (i) are irect consequences of the efinitions of A m an A. Regaring statement (ii), each A m, m N, is close, the intersection 1 A set is controlle invariant for a control system if there exists a feeback law such that the set is positively invariant for the close-loop system. A matrix R R is orthogonal if RR T = R T R = I. 3

of close sets is close, an, therefore, A = lim m + m k=1 A k is close. To show that A is boune it suffices to show that A 1 is boune. Note that (p,v) A 1 implies that there exists u B(0,r ctr ) such that (p,v) A 0 an (p + v,v + u) A 0. This, in turn, implies that p B(0,r pos ) an p + v B(0,r pos ). Therefore, A 1 is boune. Next, we prove that A m is convex. Given (p 1,v 1 ) an (p,v ) in A m, let u 1 an u be controls that ensure that φ(t,(p i,v i ), {u i }) A 0, t [0,m], i {1,}. For λ [0,1], consier the initial conition (p λ,v λ ) = (λp 1 + (1 λ)p,λv 1 + (1 λ)v ) an the input u λ = λu 1 + (1 λ)u, an note that, by linearity, φ(t,(p λ,v λ ),u λ ) = λφ(t,(p 1,v 1 ), {u 1 }) + (1 λ)φ(t,(p,v ), {u }), t [0,m]. Because φ(t,(p 1,v 1 ), {u 1 }) an φ(t,(p,v ), {u }) belong to the convex set A 0, then also their convex combination oes. Therefore, (p λ,v λ ) belongs to A m, an A m is convex. Finally, A is convex because the intersection of convex sets is convex. Next, we show that A is controlle invariant. Given (p,v) A (with corresponing control sequence {u τ } τ N0 ), we nee to show that there exists a control input x B(0,r ctr ) such that φ(1,(p,v),x) A. Such input can be chosen as x = u 0. Inee, the control sequence {u τ+1 } τ N0 keeps the trajectory starting from φ(1,(p,v),x) insie A 0 an, therefore, φ(1,(p,v),x) A. Aitionally, it is easy to see that A A 0. Finally, we nee to prove that A is the largest controlle-invariant subset of A 0. Assume that there exists A with the same properties an larger than A. This means that there exists (p,v) A \ A. This is equivalent to saying that τ N 0 such that, for every choice of the input u, φ(τ,(p,v),u) / A 0. But, since A A 0, this leas to a contraiction. Regaring statement (iii), observe that, if (p,v) A 0, then (Rp,Rv) A 0 an, if u B(0,r ctr ), then Ru B(0,r ctr ). Therefore, using again the linearity of the maps φ, the proof follows. Regaring statement (iv), the two results follow from the efinition of A (r pos,r ctr ) an the facts that, for all 0 < r 1 < r, B(0,r 1 ) B(0,r ) an A 0(r 1 ) A 0(r ). Next, we stuy the set-value map that associates to each state in A (r pos,r ctr ) the set of control inputs that keep the state insie A (r pos,r ctr ) in one step. We efine the amissible control set U (r pos,r ctr ) : A (r pos,r ctr ) B(0,r ctr ) by U (r pos,r ctr ) (p,v) = {u B(0,r ctr ) (p + v,v + u) A (r pos,r ctr )}, or, equivalently, U (r pos,r ctr ) (p,v) = B(0,r ctr ) {w v (p + v,w) A (r pos,r ctr )}. (.) Lemma. (Properties of the amissible control set). For all (p,v) A (r pos,r ctr ), the set U (r pos,r ctr ) (p,v) is non-empty, convex an compact. For generic (p,v) A (r pos,r ctr ), the set U (r pos,r ctr ) (p,v) oes not contain 0. Proof. The non-emptiness of the set U (r pos,r ctr ) (p,v) erives irectly from the efinition of A (r pos,r ctr ). Clearly, from equation (.), U (r pos,r ctr ) (p,v) is close (it is the intersection of two close sets). It is also boune (U (r pos,r ctr ) (p,v) B(0,r ctr )), hence it is compact. To prove that it is convex, we nee to show the following: given (p,v) A (r pos,r ctr ), if there exist u 1 an u in U (r pos,r ctr ) (p,v) such that φ(1,(p,v),u 1 ) an φ(1,(p,v),u ) belong to A (r pos,r ctr ), then u 1 = λu 1 + (1 λ)u, λ [0,1], belongs to U (r pos,r ctr ) (p,v), that is, φ(1,(p,v),u 1 ) A (r pos,r ctr ). But this fact follows irectly from the linearity of φ an the convexity of A (r pos,r ctr ). This proves that U (r pos,r ctr ) (p,v) is convex. The fact that it oes 4

not necessarily contain the origin can be proven by contraiction as follows. Consier a (p,v) A (r pos,r ctr ) such that v 0 an U (r pos,r ctr ) (p,v) contains 0. This means that (p+v,v) also belongs to A (r pos,r ctr ). Now, either U (r pos,r ctr ) (p+v,v) oes not contain 0, in which case we have prove the statement, or A (r pos,r ctr ) also contains (p + v,v). Continuing along these lines, if it were true that U (r pos,r ctr ) (p,v) contains the origin for all (p,v) A (r pos,r ctr ), then one coul show that (p+tv,v) belongs to A (r pos,r ctr ) for all t N. However, A (r pos,r ctr ) is boune by Theorem.1. Hence, one can always fin a t N such that (p+t v,v) A (r pos,r ctr ) but (p + (t + 1)v,v) / A (r pos,r ctr ), thereby proving that U (r pos,r ctr ) (p + t v,v) oes not contain 0... Computing amissible sets. We characterize A for = 1 in the following result an we illustrate the outcome in Figure.1. Lemma.3 (Amissible set in 1 imension). For r pos,r ctr R +, the following hols: (i) A 1 (r pos,r ctr ) is the polytope containing the points (p,v) R satisfying r pos m m 1 r ctr v + p m r pos m + m 1 r ctr, (.3) for all m N, an p [ r pos,r pos ]; (ii) If m(r pos,r ctr ) N is efine by m(r pos,r ctr ) = then A 1 = A 1 m = A 1 bm(r pos,r ctr), for m m(r pos,r ctr ). 1 1 + 4 + 4r pos, (.4) r ctr Proof. Regaring statement (i), it suffices to show that, for m N, A 1 m(r pos,r ctr ) is the set of points in A 1 m 1(r pos,r ctr ) that satisfy equation (.3). If we show that this property hols for all m, then we can use statement (i) of Theorem.1 to complete the proof. Consier the set of equations (.1) for m consecutive time inices after t. The solution of the linear system can be written in terms of the state at instant t as [ ] p[t + m] = v[t + m] [ 1 m 0 1 ][ p[t] v[t] ] + m 1 τ=0 [ 1 (m 1 τ) 0 1 ] [ 0 1 ] u[t + τ]. (.5) It is clear that the points on the bounary of A 1 m have the property that the maximum control effort is neee to enforce the constraint. In other wors we look for the points (p[t],v[t]) A 1 0 with v[t] 0 (the case v[t] 0 can be solve in a similar way) such that the points p[t + m] r cmm are reache by using the maximum control effort u[t + τ] = r ctr, τ {0,...,m 1}. Substituting the expression of the control in (.5) we obtain m 1 p[t + m] = p[t] + mv[t] r ctr (m 1 τ), v[t + m] = v[t] mr ctr, τ=0 an using the equality m 1 m(m 1) τ=0 (m 1 τ) =, we have m(m 1) p[t + m] = p[t] + mv[t] r ctr, v[t + m] = v[t] mr ctr, (.6) 5

In orer to belong to A 1 m, the point (p[t],v[t]) must satisfy the constraint p[t + τ] r cmm, τ {1,...,m}, or equivalently v[t] p[t] τ + r cmm τ (τ 1) + r ctr, τ {1,...,m}. Using the same proceure for the points in the half plane v[t] 0 (in this case the control is u[t + τ] = r ctr, τ {0,...,m 1}), it turns out that A 1 m is equal to the set of all pairs (p,v) A 1 0 satisfying p τ r cmm τ τ 1 r ctr v p τ + r cmm τ + τ 1 r ctr, τ {1,...,m}. By using statement (i) of Theorem.1 the proof is complete. Regaring statement (ii), let us consier the case v[t] 0 an evaluate the points on the bounary such that (p[t + m],v[t + m]) = (r cmm,0), m N. These points are obtaine by substituting the above value of (p[t + m],v[t + m]) in (.6). The points obtaine are (p,v) such that p = r cmm m (m + 1) r ctr, m N 0. It is easy to see that m(r pos,r ctr ), as efine in equation (.4), is the lowest m such that p r cmm. v m = 3 m = m = 1 r pos r pos p m = 1 m = m = 3 Fig..1: The amissible set A 1 for generic values of r pos an r ctr Remarks.4. (i) If r ctr r pos, then A 1 = A 1 1, that is, for sufficiently large r ctr /r pos, the amissible set is equal to the amissible set in 1 time step. (ii) The methoology for constructing A 1 (r pos,r ctr ) is relate to the proceure for constructing the so-calle isochronic regions for iscrete time optimal control of ouble integrators, as outline in [9]. Next, we introuce some efinitions useful to provie an inner approximation of A when. Given p R an v R \ {0 }, efine p R an p R by v p = p v + p, 6

where p v = 0. For r pos,r ctr R +, efine A (r pos,r ctr ) = { (p,v) B(0,r pos ) R v = 0 or (p, v ) A 1( r pos p,r ctr ) }. (.7) Lemma.5. For r pos,r ctr R +, A (r pos,r ctr ) is a compact subset of A (r pos,r ctr ). Proof. We begin by showing that efinition (.7) is equivalent to A (r pos,r ctr ) = { (p,v) A 0 v = 0 or {u τ } τ N0 [ r ctr,r ctr ] ( s.t. φ t,(p,v), {u τ } v ) A } v 0(r pos ), t N 0. (.8) To establish this equivalence, we use the efinition of the set A 1. For v 0, we rewrite the solution of the system as φ(t,(p,v), {u τ }) = φ (t,(p,v), {u τ }) v v + φ (t,(p,v), {u τ }), where φ (t,(p,v), {u τ }) v = 0 for all t N 0. It is easy to see that, if {u τ } τ N0 = {u τ } τ N0 v v, then φ (t,(p,v), {u τ }) = (p,0 ) for all t N 0. Therefore, φ(t,(p,v), {u τ }) = φ (t,(p,v), {u τ }) v v + (p,0 ). v Note that, if p = p v + p, then p r pos if an only if p rpos p. ( ) Therefore, the property φ t,(p,v), {u τ } v v A 0(r pos ) is equivalent to ( φ t,(p,v), {u τ } v ) ) A 1 0 ( r pos v p, an, in turn, efinitions (.7) an (.8) are equivalent. In orer to prove that A (r pos,r ctr ) is compact, we simply observe that it is a close subset of the compact set A (r pos,r ctr ). Remark.6. In what follows we use our representation of A to compute an inner approximation for the convex set A, for. For example, for fixe p B(0,r pos ), we compute velocity vectors v such that (p,v) A by consiering a sample of unit-length vectors w R an computing the maximum allowable velocity v parallel to w accoring to equation (.7). Furthermore, we perform computations by aopting inner polytopic representations for the various compact convex sets..3. The ouble-integrator isk graph. Let us introuce some concepts about state epenent graphs an some useful examples. For a set X, let F(X) be the collection of finite subsets of X; e.g., P F(R ) is a set of points. For a finite set X, let G(X) be the set of unirecte graphs whose vertices are elements of X, i.e., whose vertex set belongs to F(X). For a set X, a state epenent graph on X is a map G : F(X) G(X) that associates to a finite subset V of X an unirecte graph with vertex set V an ege set E G (V ) where E G : F(X) F(X X) satisfies 7

E G (V ) V V. In other wors, what eges exist in G(V ) epens on the elements of V that constitute the noes. The following three examples of state epenent graphs play an important role. First, given r pos R +, the isk graph G isk (r pos ) is the state epenent graph on R efine as follows: for {p 1,...,p n } R, the pair (p i,p j ) is an ege in G isk (r pos ) ({p 1,...,p n }) if an only if p i p j r pos p i p j B(0,r pos ). Secon, given r pos,r ctr R +, the ouble-integrator isk graph G i-isk (r pos,r ctr ) is the state epenent graph on R efine as follows: for {(p 1,v 1 ),...,(p n,v n )} R, the pair ((p i,v i ),(p j,v j )) is an ege if an only if the relative positions an velocities satisfy (p i p j,v i v j ) A (r pos,r ctr ). Thir, it is convenient to efine the isk graph also as a state epenent graph on R by stating that ((p i,v i ),(p j,v j )) is an ege if an only if (p i,p j ) is an ege of the isk graph on R. We illustrate the first two graphs in Figure.. Fig..: The isk graph an the ouble-integrator isk graph in R for 0 agents with ranom positions an velocities. Remark.7. As is well known, the isk graph is invariant uner rigi transformations an reflections. Loosely speaking, the ouble integrator isk graph is invariant uner the following class of transformations: position an velocities of the agents may be expresse with respect to any rotate an translate frame that is moving at constant linear velocity. These transformations are relate to the classic Galilean transformations in geometric mechanics. 3. Connectivity constraints among secon-orer agents. In this section we state the moel, the notion of connectivity, an a sufficient conition that guarantees connectivity can be preserve. 3.1. Networks of robotic agents with secon-orer ynamics an the connectivity maintenance problem. We begin by introucing the notion of network of robotic agents with secon-orer ynamics in R. Let n be the number of agents. Each agent has the following computation, motion control, an communication capabilities. For i {1,...,n}, the ith agent has a processor with the ability 8

of allocating continuous an iscrete states an performing operations on them. The ith agent occupies a location p i R, moves with velocity v i R, accoring to the iscrete-time ouble integrator ynamics in (.1), i.e., p i [t + 1] = p i [t] + v i [t], v i [t + 1] = v i [t] + u i [t], (3.1) where the norm of all controls u i [t], i {1,...,n}, t N 0, is upper-boune by r ctr R +. The communication moel is the following. The processor of each agent has access to the agent location an velocity. Each agent can transmit information to other agents within a istance r cmm R +. We remark that the control boun r ctr an the communication raius r cmm are the same for all agents. Remarks 3.1. (i) Our network moel assumes synchronous execution, although it woul be important to consier more general asynchronous networks. (ii) We will not aress in this paper the correctness of our algorithms in the presence of measurement errors or communication quantization. We now state the control esign problem of interest. Problem 3. (Connectivity Maintenance). Choose a state epenent graph G target on R an esign (state epenent) control constraints sets with the following property: if each agent s control takes values in the control constraint set, then the agents move in such a way that the number of connecte components of G target (evaluate at the agents states) oes not increase with time. This objective is to be achieve with the limite information available through message exchanges between agents. This problem was originally state an solve for first-orer agents in []. 3.. A known result for agents with first-orer ynamics. In [], a connectivity constraint was evelope for a set of agents moele by first-orer iscretetime ynamics: p i [t + 1] = p i [t] + u i [t]. Here the graph whose connectivity is of interest, is the isk graph G isk (r cmm ) over the vertices {p 1 [t],...,p n [t]}. Network connectivity is maintaine by restricting the allowable motion of each agent. In particular, it suffices to restrict the motion of each agent as follows. If agents i an j are neighbors in the r cmm -isk graph G isk (r cmm ) at time t, then their positions at time t+1 are require to belong to B ( p i[t]+p j[t] ), rcmm. In other wors, connectivity between i an j is maintaine if ( pj [t] p i [t] u i [t] B ( pi [t] p j [t] u j [t] B, r ) cmm,, r cmm The constraint is illustrate in Figure 3.1. Note that this constraint takes into account only the positions of the agents; this fact can be attribute to the first-orer ynamics of the agents. We wish to construct a similar constraint for agents with secon orer ynamics. It is reasonable to expect that this new constraint will epen on positions as well as velocities of the neighboring agents. 9 ).

p j p i Fig. 3.1: Starting from p i an p j, the agents are restricte to move insie the isk centere at p i+p j with raius rcmm. 3.3. An appropriate graph for the connectivity maintenance problem for agents with secon-orer ynamics. We begin working on the state problem with a negative result regaring two caniate target graphs. Lemma 3.3 (Unsuitable graphs). Consier a network of n agents with ouble integrator ynamics (3.1) in R. Let r cmm be the communication range an let r ctr be the control boun. Let G target be either G isk (r cmm ) on R or G i-isk (r cmm,r ctr ). There exist states {(p i,v i )} i {1,...,n} such that (i) the graph G target at {(p i,v i )} i {1,...,n} is connecte, an (ii) for all {u i } i {1,...,n} B(0,r ctr ), the graph G target at {(p i +v i,v i +u i )} i {1,...,n}, is isconnecte. Proof. The proof of the statement for G target = G isk (r cmm ) is straightforwar. Consier two agents whose relative position an velocity belong to A 0 \ A 1. Then, after one time step, the two agents will not be connecte in G isk (r cmm ) no matter what their controls are. Next, consier the case G target = G i-isk (r cmm,r ctr ). For = 1, let v be the maximal velocity in A 1 (r cmm,r ctr ) at p = 0, that is, v = min{r cmm /m + (m 1)r ctr m N}. Take three agents with positions p 1 = p = p 3 = 0 an velocities v 1 = v, v = 0, an v 3 = v. At this configuration, the graph G i-isk (r cmm,r ctr ) contains two eges: between agent 1 an an between agent an 3. Connectivity is preserve after one time step if agent remains connecte to both agents 1 an 3 after one time step. However, to remain connecte with agent 1, its control u must be equal to r ctr an, analogously, to remain connecte with agent 3, its control u must be equal to +r ctr. Clearly this is impossible. Remarks 3.4. (i) The result in Lemma 3.3 on the ouble integrator graph has the following interpretation. Assume that agent i has two neighbors j an k in the graph G i-isk (r cmm,r ctr ). By efinition of the neighboring law for this graph, we know that there exists boune controls for i an j to maintain the ((p i,v i ),(p j,v j )) link an that there exists boune controls for i an k to maintain the ((p i,v i ),(p k,v k )) link. The lemma states that, for some states of the agents i, j, an k, there might not exist controls that maintain both links simultaneously. (ii) In other wors, Lemma 3.3 shows how the isk graph G isk (r cmm ) an the ouble integrator isk graph G i-isk (r cmm,r ctr ) are not appropriate choices for the connectivity maintenance problem. The following theorem suggests that an appropriate scaling of the control boun is helpful in ientifying a suitable state epenent graph for Problem 3.. 10

Theorem 3.5 (A scale ouble-integrator isk graph is suitable). Consier a network of n agents with ouble integrator ynamics (3.1) in R. Let r cmm be the communication range an let r ctr be the control boun. For k {1,...,n 1}, efine ν(k) = k. Assume that k {1,...,n 1} an the state {(p i,v i )} i {1,...,n} together have the property that the graph G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n} contains a spanning tree T with iameter at most k. Then there exists {u i } i {1,...,n} B(0,r ctr ) such that if ((p i,v i ),(p j,v j )) is an ege of T, then ((p i + v i,v i + u i ),(p j + v j,v j + u j )) is an ege of G i-isk (r cmm,ν(k)r ctr ) at {(p i + v i,v i + u i )} i {1,...,n}. These results are base upon Shostak s Theory for systems of inequalities, as expose in [10]. We provie the proof in the Appenix. The following results are immeiate consequences of this theorem. Corollary 3.6 (Simple sufficient conition). With the same notation in Theorem 3.5, efine ν min = (n 1). Assume that the state {(p i,v i )} i {1,...,n} has the property that the graph G i-isk (r cmm,ν min r ctr ) is connecte at {(p i,v i )} i {1,...,n}. Then (i) there exists {u i } i {1,...,n} B(0,r ctr ), such that the graph G i-isk (r cmm,ν min r ctr ) is also connecte at {(p i + v i,v i + u i )} i {1,...,n} ; an (ii) if T is a spanning tree of G i-isk (r cmm,ν min r ctr ) at {(p i,v i )} i {1,...,n}, then there exists {u i } i {1,...,n} B(0,r ctr ), such that, for all eges ((p i,v i ),(p j,v j )) of T, it hols that ((p i +v i,v i +u i ),(p j +v j,v j +u j )) is an ege of G i-isk (r cmm,ν min r ctr ) at {(p i + v i,v i + u i )} i {1,...,n}. Remark 3.7 (Scaling of ν min with n). The conition ν min = (n 1) implies that accoring to the sufficient conitions in Corollary 3.6, as the number of agents grows, the velocities of the agents must be closer an closer in orer for the agents to be able to remain connecte. If G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n} is not connecte for some k, then Theorem 3.5 applies to its connecte components. In what follows we fix k an without loss of generality assume the graph G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n} to be connecte. Remark 3.8 (Distribute computation of connectivity an of spanning trees). Each agent can compute its neighbors in the graph G i-isk (r cmm,ν(k)r ctr ) just by communicating with its neighbors in G isk (r cmm ) an exchanging with them position an velocity information. Alternatively, this computation may also be performe if each agent may sense relative position an velocity with all other agents at a istance no larger than r cmm. It is possible to compute spanning trees in networks via stanar epth-first search istribute algorithms. It is also possible [11] to istributively compute a minimum iameter spanning tree in a network. 3.4. The control constraint set an its polytopic representation. We now concentrate on two agents with inices i an j. For t N 0, we efine the relative position, velocity an control by p ij [t] = p i [t] p j [t], v ij [t] = v i [t] v j [t] an u ij [t] = u i [t] u j [t], respectively. It is easy to see that p ij [t + 1] = p ij [t] + v ij [t], v ij [t + 1] = v ij [t] + u ij [t]. 11

Assume that agents i,j are connecte in G i-isk (r cmm,ν(k)r ctr ) at time t. By efinition, this means that the relative state (p ij [t],v ij [t]) belongs to A (r cmm,ν(k)r ctr ). If this connection is to be maintaine at time t + 1, then the relative control at time t must satisfy u i [t] u j [t] U (r cmm,ν(k)r ctr ) (p ij [t],v ij [t]). (3.) Also, implicit are the following bouns on iniviual control inputs u i [t] an u j [t]: u i [t] B(0,r ctr ), u j [t] B(0,r ctr ). (3.3) This iscussion motivates the following efinition. Definition 3.9. Given r cmm,r ctr,ν(k) R + an given a set E of eges in G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n}, the control constraint set is efine by U E(r cmm,r ctr,ν(k)) ({p i,v i } i {1,...,n} ) = {(u 1,...,u n ) B(0,r ctr ) n ((p i,v i ),(p j,v j )) E, u i u j U (r cmm,ν(k)r ctr ) (p i p j,v i v j )}. In other wors, the control constraint set for an ege set E is the set of controls for each agent with the property that all eges in E will be maintaine in one time step. Remark 3.10. We can now interpret the results in Theorem 3.5 as follows. (i) To maintain connectivity between any pair of connecte agents, we shoul simultaneously hanle constraints corresponing to all eges of G i-isk (r cmm,ν(k)r ctr ). This might rener the control constraint set empty. (ii) However, if we only consier constraints corresponing to eges belonging to a spanning tree T of G i-isk (r cmm,ν(k)r ctr ), then the set UT (r cmm,ν(k)r ctr ) ({p i,v i } i {1,...,n} ) is guarantee to be nonempty. Let us now provie a concrete representation of the control constraint set. Given a pair i,j of connecte agents, the amissible control set U (r cmm,ν(k)r ctr ) (p ij,v ij ) is convex an compact (Lemma.). Hence, we can fit a polytope with N poly sies insie it. This approximating polytope leas to the following tighter version of the constraint in (3.): (C η ij )T (u i u j ) w η ij, η {1,...,N poly}, (3.4) for some appropriate vector C η ij R an scalar w η ij R. Similarly, one can compute an inner polytopic approximation of the close ball B(0,r ctr ) an write the following linear vector inequalities: (C η iθ )T u i w η iθ, η {1,...,N poly}, (3.5) where the symbol θ has the interpretation of a fictional agent. In this way, we have cast the original problem of fining a set of feasible control inputs into a satisfiability problem for a set of linear inequalities. Remark 3.11. Rather than a network-wie control constraint set, one might like to obtain ecouple constraint sets for each iniviual agent. However, (1) it is not clear how to esign a istribute algorithm to perform this computation, () 1

such an algorithm will likely have large communication requirements, an (3) such a calculation might lea to a very conservative estimate for the ecouple control constraint sets. Therefore, rather than explicitly ecoupling the control constraint sets, we next focus on a istribute algorithm to search the control constraint set for feasible controls that are optimal accoring to some criterion. 4. Distribute computation of optimal controls. In the previous section, we erive sufficient conitions for the existence of connectivity-maintaining control inputs. In this section, we utilize these analysis results to tackle a esign problem. We provie an algorithm to compute connectivity-maintaining control inputs an we o so satisfying two requirements: we require the algorithm to be istribute an to be optimal in the following sense. We assume a high level controller is available to compute esire control inputs for the agents to achieve a specific task inepenent of the connectivity-maintaining constraints. We then esign a low level filter which computes, insie the set of connectivity-mantaining inputs, the closest inputs to the esire ones. We set up this filter esign problem in the form of an optimization problem with the performance criteria being the minimization of the (square) Eucliean norm of the eviation away from the esire inputs. The resulting quaratic optimization problem can be solve through a istribute flow control algorithm. As an example application, we illustrate by simulations that solving this optimization problem for a simple choice of the esire inputs achieves a weak form of connectivity-preserving flocking behavior among the agents. 4.1. Problem formulation. We consier the following optimization problem: given an array of esire control inputs U es = (u es,1,...,u es,n ) T (R ) n, fin, via local computation, the array U = (u 1,...,u n ) belonging to the control constraint set, that is closest to the esire array U es. To formulate this problem precisely, we nee some aitional notations. Let E be a set of eges in the unirecte graph G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n}. To eal with the linear inequalities of the form (3.4) an (3.5) associate to each ege of E, we introuce an appropriate multigraph. A multigraph (or multiple ege graph) is, roughly speaking, a graph with multiple eges between the same vertices. More formally, a multigraph is a pair (V mult,e mult ), where V mult is the vertex set an the ege set E mult contains numbere eges of the form (i,j,η), for i,j V an η N, an where E mult has the property that if (i,j,η) E mult an η > 1, then also (i,j,η 1) E mult. In what follows, we let G mult enote the multigraph with vertex set {1,...,n} an with ege set E mult = {(i,j,η) {1,...,n} {1,...,N poly } ((p i,v i ),(p j,v j )) E, i > j}. Note that to each element (i,j,η) E mult is associate the inequality (C η ij )T (u i u j ) w η ij. We are now reay to formally state the optimization problem at han in the form of the following quaratic programming problem: minimize subj. to 1 n u i u es,i, i=1 (C η ij )T (u i u j ) w η ij, for (i,j,η) E mult, (C η iθ )T u i w η iθ, for i {1,...,n},η {1,...,N poly}. (4.1) Here, somehow arbitrarily, we have aopte the -norm to efine the cost function. Remark 4.1 (Feasibility). If E is a spanning tree of G i-isk (r cmm,νr ctr ) at a connecte configuration {(p i,v i )} i {1,...,n}, then the control constraint set UE (r cmm,r ctr,ν(k)) ({p i,v i } i {1,...,n} ) is guarantee to be non-empty by Theorem 3.5. In turn, this implies that the optimization problem (4.1) is feasible. 13

4.. Solution via uality: the projecte Jacobi metho. The problem (4.1) can be written in a compact form as: minimize 1 U U es, subj. to B T multu w, for appropriately efine matrix B mult an vector w. A ual projecte Jacobi metho algorithm for the solution of this stanar quaratic program is escribe in [1]. Accoring to this algorithm, let λ be the value of Lagrange multipliers at optimality. Then the global minimum for U is achieve at U = U es B mult λ. (4.) The projecte Jacobi iteration for each component of λ is given by { τ ( λ α (t + 1) = max 0,λ α (t) (Bmult T B (w BmultU T es ) α mult) αα N poly (e+n) + β=1 )} (BmultB T mult ) αβ λ β (t), (4.3) where α {1,...,N poly (e + n)} an τ is the step size parameter. We can select 1 τ = N poly (e+n) to guarantee convergence. 4.3. A istribute implementation of the ual algorithm. Because of the particular structure of the matrix Bmult T B mult, the iterations (4.3) can be implemente in a istribute way over the original graph G. To highlight the istribute structure of the iteration we enote the components of λ by referring to the noes that they share an the inequality they are relate to. In particular for each ege in G mult, the corresponing Lagrange multiplier will be enote as λ η ij if the ege goes from noe i to noe j, i > j, an the ege is associate to the inequality constraint C η ij (u i u j ) w η ij. This makes up the first N poly e entries of the vector λ. To be consistent with this notation, the next N poly n entries will be enote λ 1 1θ,...,λN poly 1θ,...,λ 1 nθ,...,λn poly nθ. Aitionally, efine N(i) = {j {1,...,n} {(p i,v i ),(p j,v j )} E} {θ}. The symbol θ has the interpretation of a fictional noe. Defining λ η ij := λη ji an Cη ij := Cη ji for i < j, we can write equations (4.) an (4.3) in components as follows. Equation (4.) reas, for i {1,...,n}, u i = u es,i N poly k N(i) η=1 C η ik λη ik. One can easily work an explicit expression for matrix prouct B T mult B mult in (4.3). 14

Then, equation (4.3) reas, for (i,j,η) E mult, λ η ij (t + 1) = max 0,λη ij (t) τ (C η ij )T C η ij N poly k N(i) σ=1 ( ) (C η ij )T Cikλ σ σ ik + k N(j) together with, for i {1,...,n}, η {1,...,N poly }, λ η iθ (t + 1) = max {0,λ η iθ (t) τ ( (C η iθ )T C η iθ N poly k N(i) σ=1 N poly σ=1 ( ) (C η ji )T Cjkλ σ σ jk + w η ij (Cη ij )T (u es,i u es,j ), ((C η iθ )T Cikλ σ σ ik) + w η iθ (Cη iθ )T u es,i )}. We istribute the task of running iterations for these N poly (e + n) Lagrange multipliers among the n agents as follows: an agent i carries out the upates for all quantities λ η iθ an all λη ij for which i > j. By means of this partition an by means of iterate one-hop communication among agents, it is possible to compute the global solution for the optimization problem (4.1) in a istribute fashion over the ouble integrator isk graph. 4.4. Simulations. To illustrate our analysis we focus on the following scenario. For the two imensional setting, i.e., for =, we assume that there are n = 5 agents with (ranomly chosen) initial conition an such that they are connecte accoring to G i-isk. The boun for the control input is r ctr = an the communication raius is r cmm = 10. We assigne to one of the agents a erivative feeback control u x [p,v] = (v x ), u y [p,v] = (v y 5) as esire input. For the other agents the esire input is set to zero. We show the agent trajectories in Figure 4.1a, the velocities of the agents with respect to time in Figure 4.1b, an the istances between agents which are neighbors in the spanning tree in Figure 4.1c. Notice that the agents move with approximately ientical velocity reaching a configuration in which all of them are at the limit istance r cmm = 10. The interesting aspect of this simulation is that the maintenance of connectivity leas to the accomplishment of apparently unrelate coorination tasks as velocity alignment an cohesiveness. This numerical result illustrate how our connectivity maintenance approach might inee be a starting point for novel investigations into the problem of flocking with connectivity. 5. Conclusion. We provie some istribute algorithms to enforce connectivity among networks of agents with ouble-integrator ynamics. Future irections of research inclue (i) evaluating the communication complexity of the propose istribute ual algorithm an possibly esigning faster ones, (ii) stuying the relationship between the connectivity maintenance problem an the platooning an mesh stability problem, an (iii) investigating the flocking phenomenon an esigning flocking algorithms which o not rely on a blanket assumption of connectivity. 15

py i 70 60 50 40 30 0 10 0 v i 6 5 4 3 1 0 r ij 10 9 8 7 6 5 4 3 10 0 50 100 150 px i (a) Positions 1 0 5 10 15 0 5 30 35 t (b) Velocities (v x an v y) 1 0 5 10 15 0 5 30 35 t (c) Inter-agent istances Fig. 4.1: Velocity alignment an cohesiveness for 5 agents in the plane ( = ) REFERENCES [1] G. Notarstefano, K. Savla, F. Bullo, an A. Jababaie, Maintaining limite-range connectivity among secon-orer agents, in American Control Conference, (Minneapolis, MN), pp. 14 19, June 006. [] H. Ano, Y. Oasa, I. Suzuki, an M. Yamashita, Distribute memoryless point convergence algorithm for mobile robots with limite visibility, IEEE Transactions on Robotics an Automation, vol. 15, no. 5, pp. 818 88, 1999. [3] J. Lin, A. S. Morse, an B. D. O. Anerson, The multi-agent renezvous problem, in IEEE Conf. on Decision an Control, (Maui, HI), pp. 1508 1513, Dec. 003. [4] J. Cortés, S. Martínez, an F. Bullo, Robust renezvous for mobile autonomous agents via proximity graphs in arbitrary imensions, IEEE Transactions on Automatic Control, vol. 51, no. 8, pp. 189 198, 006. [5] A. Ganguli, J. Cortés, an F. Bullo, On renezvous for visually-guie agents in a nonconvex polygon, in IEEE Conf. on Decision an Control an European Control Conference, (Seville, Spain), pp. 5686 5691, Dec. 005. [6] D. P. Spanos an R. M. Murray, Motion planning with wireless network constraints, in American Control Conference, (Portlan, OR), pp. 87 9, June 005. [7] M. M. Zavlanos an G. J. Pappas, Controlling connectivity of ynamic graphs, in IEEE Conf. on Decision an Control an European Control Conference, (Seville, Spain), pp. 6388 6393, Dec. 005. [8] H. Tanner, A. Jababaie, an G. J. Pappas, Stable flocking of mobile agents, Part I: Fixe topology, in IEEE Conf. on Decision an Control, (Maui, HI), pp. 010 015, Dec. 003. [9] Z. Gao, On iscrete time optimal control: A close-form solution, in American Control Conference, (Boston, MA), pp. 5 58, 004. [10] B. Aspvall an Y. Shiloach, A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality, SIAM Journal on Computing, vol. 9, no. 4, pp. 87 845, 1980. [11] M. Bui, F. Butelle, an C. Lavault, A istribute algorithm for constructing a minimum iameter spanning tree, Journal of Parallel an Distribute Computing, vol. 64, pp. 571 577, 004. [1] D. P. Bertsekas an J. N. Tsitsiklis, Parallel an Distribute Computation: Numerical Methos. Belmont, MA: Athena Scientific, 1997. Appenix A. Shostak s test. This section provies a proof for Theorem 3.5. The proof amounts to showing that if E is the ege set of a spanning tree T in G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n}, then the control constraint set U E (r cmm,r ctr,ν(k)) ({p i,v i } i {1,...,n} ) is non-empty. We first consier a polytopic approximation of constraints (3.) an (3.3). Among all 16

possible choices, we use the conservative orthotope approximation that allows us to ecouple the constraints into inepenent sets of linear inequalities (one for each imension). Then we use Shostak s theory to obtain sufficient conitions for the feasibility of these linear inequalities. For brevity, we rop the epenence of the quantities on t an we assume that the variables u i are scalars, for all i {1,...,n} an t 0. The resulting sets of linear inequalities for one particular imension are δ l i,j u i u j δ u i,j, an r ctr u i r ctr. (A.1) where ν(k)r ctr δ l i,j δu i,j ν(k)r ctr, for all i,j {1,...,n} an i j. A.1. Shostak Theory. In this section we present Shostak s theory for feasibility of linear inequalities involving at most two variables, similar to the ones in (A.1). These ieas will then be use to prove Theorem 3.5. The notations use in [10] aapte to our case are presente next. Let u 0 be an auxiliary zero variable that always occurs with zero coefficient - the only variable that can o this. Without loss of generality, we can thus assume that all the inequalities in L contain two variables. As a result of this, the inequalities in (A.1) can be succinctly written as u i u j δ i,j, i,j {0,...,n}, (A.) where for all i,j {1,...,n}, i j, ν(k)r ctr δ i,j ν(k)r ctr an for all i {1,...,n},δ i,0 = δ 0,i = rctr. Also implicit in this formulation is the relation that δ i,j + δ j,i 0 for all i,j {0,...,n} an i j. Let L enote the system of inequalities in (A.). We construct the graph G(L) with n + 1 vertices an (n 1) eges as follows: (a) For each variable u i occurring in L, a a vertex i to G(L). (b) For each inequality of the form a i,j u i + b i,j u j δ i,j in L, a an unirecte ege between i an j to G(L), an label the ege with the inequality (see Figure A.1). It is easy to see the following relation between the spanning tree T in G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n} that is use to erive the constraints in the inequalities (A.) an the graph G(L): (a) The vertex set of G(L) is the union of the vertex set of T an the auxiliary vertex 0 (b) For every ege {i,j} in T, there are two eges between the vertices i an j in G(L) (c) Aitionally, G(L) contains two eges between 0 an every other vertex i, for all i {1,...,n}. u j rctr 0 u i rctr ui rctr i u i u j δ i,j u j u i δ j,i j u j rctr Fig. A.1: Snippet of the graph G(L) for the system of inequalities in (A.) To every ege represente by the inequality of the form a i,j u i + b i,j u j δ i,j, we associate a triple a i,j,b i,j,δ i,j. Note that b i,j,a i,j,δ i,j is also a triple associate 17

with the same ege. Without loss of generality, consier a path of G(L) etermine by the vertices {1,,...,l + 1} an the eges e 1,,e,3,...,e l,l+1 between them. A triple sequence, P, associate with the path is efine as a 1,,b 1,,δ 1,, a,3,b,3,δ,3,..., a l,l+1,b l,l+1,δ l,l+1, where, for 1 i l, a i,i+1 u i + b i,i+1 u j δ i,i+1 is the inequality associate with the ege e i,i+1. If a i+1,i+ an b i,i+1 have opposite signs for 1 i < l, then P is calle amissible. Define a P,b P,δ P, the resiue of P, as a P,b P,δ P = a 1,,b 1,,δ 1, a,3,b,3,δ,3... a l,l+1,b l,l+1,δ l,l+1, where is the associativity binary operator efine on triples by a,b,δ a,b,δ = κaa, κbb,κ(δa δ b), where κ = a / a. Intuitively, the operator takes two inequalities an erives a new inequality by eliminating a common variable; e.g., ax+by δ an a y+b z δ imply aa x+bb z (δa δ b) if a < 0 an b > 0. Note that the signs of a P an a 1, agree, as o the signs of b P an b 1,. A path is calle a loop if the initial an final vertices are ientical. (A loop is not uniquely specifie unless its initial vertex is given.) If all the intermeiate vertices of a path are istinct, the path is simple. An amissible triple sequence P associate with a loop with initial vertex x is infeasible if its resiue satisfies a P + b P = 0 an δ P < 0. A loop which contains an infeasible triple sequence is calle an infeasible loop. Thus if G(L) has an infeasible loop, the system of inequalities L is unsatisfiable. However, the converse is not true in general. Next, we show how to exten L to an equivalent system L such that G(L ) has an infeasible simple loop if an only if L is unsatisfiable. For each vertex i of G(L) an for each amissible triple sequence P with a P +b P 0 associate with a simple loop of G(L) an initial vertex i, a a new inequality (a P + b P )u i δ P to L. This new system L is referre to as the Shostak extension of L. We now state the necessary an sufficient conition on the extene system of inequalities L for the satisfiability of the original system L. Theorem A.1 (Shostak s Theorem [10]). Let L be the Shostak extension of L. The system of inequalities L is satisfiable if an only if G(L ) contains no infeasible simple loop. A.. Satisfiability test. In this section we use the Shostak criterion to erive conitions for the satisfiability of the inequalities in (A.). Lemma A.. Let L be the system of inequalities of the form (A.) obtaine by consiering pairwise neighbors in a spanning tree T in G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n}. Then the Shostak extension of L is itself. Proof. Consier a simple loop of G(L) with the initial vertex i {0,1,...,n}. Consier an amissible triple sequence P associate with the loop. Since a i,j,b i,j { 1,+1}, for all i,j {1,...,n},i j, an a 0,i,a i,0,b i,0,b 0,i { 1,0,+1}, for all i {1,...,n}, the resiue of P, a P,b P,δ P, is such that a p + b p = 0. Hence, no new inequality must be ae to obtain the Shostak extension of L. 18

Lemma A.3. Let L be the system of inequalities of the form (A.) obtaine by consiering pairwise neighbors in a spanning tree T of epth at most k in G i-isk (r cmm,ν(k)r ctr ) at {(p i,v i )} i {1,...,n}. If ν(k) = k, then there is no infeasible simple loop in G(L). Proof. Looking at figure A.1 it is clear that there are two types of simple loops with amissible triple sequences in G(L): (i) +1, 1,δ i,j, +1, 1,δ j,i or 1,+1,δ i,j, 1,+1,δ j,i, where i,j {0,...,n 1} an {i,j} is an ege in T. (ii) 0, 1, rctr, +1, 1,δ i1,i,..., +1, 1,δ il 1,i l, +1,0, rctr or 0,+1, rctr, 1,+1,δ i,i 1,..., 1,+1,δ il,i l,l 1, 1,0, rctr, where i l {1,...,ζ} for all l {1,...,ζ} an {i l,i l+1 } is an ege in T. The resiue for the first set of loops is +1, 1,δ i,j + δ j,i or 1,+1,δ i,j + δ j,i. The feasibility conition is trivially satisfie by construction since δ i,j + δ j,i 0. For the secon set of loops, the resiue is: 0, 1, r ctr +1, 1,δ i1,i... +1, 1,δ iζ 1,i ζ + 1,0, r ctr = 0,0, r ζ 1 ctr + δ il,i l+1, or 0,+1, r ctr 1,+1,δ i,i 1... 1,+1,δ iζ,i ζ 1 1,0, r ctr = l=1 0,0, r ζ 1 ctr + δ il,i l+1. In orer to guarantee the feasibility of the secon set of loops, we nee that rctr + ζ 1 l=1 δ i l,i l+1 0. We erive conitions for the worst case which occurs when the loop is written for the longest path in T, i.e., when ζ = k+1 an when δ il,i l+1 = ν(k)r ctr, for all l {1,...,k}. In this case, there is no infeasible simple loop if an only if r ctr kν(k)r ctr 0, l=1 that is, if an only if ν(k) = k. Finally, the proof of Theorem 3.5 follows from Theorem A.1, Lemma A. an Lemma A.3. 19