Time Scales and Stability in Networked Multi-Robot Systems

Transcription

1 !"##$%&&&$%'()*'+(,-'+$/-')*)'1)$-'$2-3-(,14$+'5$67(-8+(,-' 9:+';:+,$%'()*'+(,-'+$/-')*)'1)$/)'()* Time Scales an Stability in Networke Multi-Robot Systems Mac Schwager, Nathan Michael, Vijay Kumar, an Daniela Rus Abstract This paper examines the ynamic interplay between ecentralize controllers an mesh networking protocols for controlling groups of robots A proportional controller is use to maintain robots in a formation base on estimates of the robots states observe through the network The state information is propagate through the network using a flooing algorithm, which introuces topology-epenent time elays The couple interaction of information flow over the network with the ynamics of the robots is moele as a linear ynamical system With this moel it is shown that systems mae up of robots with stable first orer ynamics are stable for all network upate times, positive feeback gains, an connecte communication graphs With higher orer robot ynamics it is foun that stability is a complex an counter intuitive function of feeback gain an network upate time A performance metric is propose for analyzing the convergence rate of the multi-robot system Experiments with flying quarotor robots verify the preictions of the moel an the performance metric I INTRODUCTION In this work, we investigate the problem of controlling a group of robots over a wireless mesh network, specifically focusing on the interplay between the ecentralize controller an the networking protocol Controlling a group of robots over a wireless network involves three interacting components: ecentralize control, ecentralize estimation, an a networking protocol Each of these are traitionally stuie in isparate contexts using mathematical tools that are not mutually compatible One of the chief challenges in multi-robot systems is to unerstan the interaction among these components uner a consistent mathematical moel Most previous stuies have focuse only on ecentralize control, or on the couple interaction of ecentralize control an estimation, usually assuming a trivial network protocol (eg instantaneous communication among 1-hop neighbors) However, in experiments with groups of robots it has been observe that the ynamics of the network protocol play a critical role in the performance an stability of the esire control task Aggressive control tasks require well-connecte networks with fast upate rates to maintain aequate performance an preserve stability, an if the upate rate is too slow or the network too isperse, the system can become unstable This phenomenon has been well ocumente, for This research was one in the General Robotics, Automation, Sensing, an Perception (GRASP) Lab at the University of Pennsylvania, an was fune in part by ONR Grants N , N , an N We are grateful for this financial support Mac Schwager, Nathan Michael, an Vijay Kumar are with the GRASP Lab, University of Pennsylvania, Philaelphia, PA 1916, {schwager, nmichael, kumar}@seasupenneu Daniela Rus is with the Computer Science an Artificial Intelligence Laboratory (CSAIL), MIT, Cambrige, MA 2139, USA, rus@csailmiteu example in [1], [2] In this work we specifically focus on moeling a realistic networking protocol known as flooing [3], [4], couple with a proportional feeback controller for maintaining the robots in a formation Using the flooing algorithm, every robot maintains an estimate of the state of every other robot in the network, which it can use in its controller To moel the separate graphical structures of the controller an the communication network we introuce a graph, istinct from the communication graph, calle the control graph Using this convenient notion we show that the multi-robot system compose of the robots, controller, an networking algorithm takes the form of a linear, iscrete time ynamical system With this moel we prove two theorems concerning the stability of systems with specific robot moels We prove, surprisingly, that systems with first orer robots are stable for all positive control gains, network upate rates, an communication graphs Further, we prove that networks with secon orer robots are stable for all positive control gains an connecte communication graphs, as long as the network upate time is a whole multiple of the open-loop perio of the robots We then aress the question of performance of the multi-robot system by formulating a metric to quantify the convergence rate of the system This metric is a function of both the eigenvalues of the system moel an the network upate time We valiate the preictive power of the moel with extensive experimental results with flying quarotor robots We carrie out thirty two experiments with a network of three quarotors to valiate the performance metric at multiple network upate rates an control gains Details of earlier experimental results were presente in [5] A Relate Work Decentralize control, ecentralize estimation, an mesh networking protocols have all been stuie extensively in their own right, though few works have consiere the effects of coupling two or more of these components together Some recent works have stuie control together with estimation For example, in [6] the authors consier formation control with a consensus-base state estimation algorithm The authors erive conitions for the stability of the couple formation controller an state estimator A relate problem is consiere in [7] where a group of robots estimate the state of a noise riven process using a consensus-base estimation algorithm In [8], a Nyquist stability criterion is erive for formation control, an an estimation algorithm is propose for fining the formation center in a istribute way Other works have presente important results for ecentralize control using a simplifie abstraction of the network protocol, usually capture as a fixe time elay, or as a

2 stochastic time elay Decentralize control is consiere in an H framework in [9], where ecentralize control with time elaye feeback is use as an example In [1] an [11] results are erive for a continuous time system with iscrete network upates moele as time elays between 1-hop neighbors In controlling platoons of vehicles in a line, [12] consiers the effects of time elays between neighbors In the stuy of consensus, there has been consierable work on quantifying the effects of time elays on the stability of the system, such as in [13] an [14] where maximum allowable time elays are erive Formation control where the network protocol is abstracte as ranom elays is consiere in [15] A more etaile moel of network inuce time elays is use in [16], which looks at the coorination of groun an aerial vehicles Also, notions of stability in leaer follower formations have been stuie without explicitly accounting for the network algorithm in [17] There is also a well evelope literature on controlling ecentralize systems over a fixe hub-style network (as oppose to the mesh networks consiere here) for which a thorough synopsis can be foun in [18] Our moel iffers from those above in that we consier a particular iscrete time networking protocol an its couple interaction with a continuous time controller an robot ynamics We intentionally use a simple estimation algorithm (each robot uses the most recent state measurement available for each other robot) so as not to obscur the interacting effects of the network an the controller II MODEL AND PROBLEM FORMULATION Let there be n robots, with states x i (t) R N that evolve in continuous time t R accoring to the linear time invariant (LTI) ynamics ẋ i (t) =A i x i (t)+b i u i (t) (1) where u i (t) R M is the control input Suppose also that each robot can irectly measure an output vector y i (t) =C i x i (t), where y i (t) R P Furthermore, the n robots communicate with one another over an a hoc wireless mesh network Let the communication network be moele in the typical way by an unirecte graph G comm = (V, E comm ), where each robot is a vertex in the vertex set V, an E comm is a set of pairs (i, j) an (j, i) for all robots i an j that are in communication with one another We assume throughout the paper that G comm is connecte with iameter n 1 Each robot has a set of communication neighbors, which consists of the inices of the robots that share an ege with robot i in E comm The network is use to propagate output information for each robot to each other robot The robots then apply an estimator to these outputs to obtain a state estimate for the other robots in the network This architecture can be use with any number of network protocols an state estimators We choose to focus on a simple but realistic protocol known as flooing, along with a very simple estimator Ni comm A Network Protocol A common way to propagate information over a wireless mesh network is to use a flooing algorithm Flooing algorithms have many variations an can be tune to suit specific applications, as in [4] Here we will escribe what is perhaps the simplest flooing algorithm because it can be easily moele in a ynamical systems framework The main principle of the flooing algorithm is that each robot maintains an outate copy of the outputs of all the other robots in the network The algorithm procees in upate steps, each of which takes time T Each upate step represents a cycle of a Time Division Multiple Access (TDMA) networking protocol in which each robot has a fixe time slot uring which to broacast information When it is robot i s turn to broacast, that robot sens a packet containing its outate outputs for all of the robots in the network, an a vector of time elays enoting when each output was current The rest of the time, robot i listens for broacasts from other robots Let ŷ ij (t) be robot i s outate copy of robot j s output y j (t ) at some previous time t t Also, let τ ij enote the number of network upates that have passe since the output ŷ ij (t) was current Denote the vector of all of robot i s output estimates an time elays as ŷ i (t) =[ŷ T i1(t) ŷ T in(t)] T, an τ i (t) =[τ i1 (t) τ in (t)] T, respectively Now when robot i receives ŷ k (t) an τ k (t) from robot k, it replaces its ŷ ij values with those of robot k that are more current than its own This can be interprete as a very simple estimator One might also choose, for example, to use a Kalman filter on the outputs to obtain a state estimate for every robot in the network The flooing algorithm is shown in Algorithm 1 We emphasize that our goal is not to improve upon networking protocols, but to provie a realistic moel an salient analysis of a couple protocol an ecentralize controller The estimate ŷ i (t) is constant over the interval between successive network upates pt t <(p + 1)T for the pth network upate Also, we assume that the robots know their own output without any elay, so that ŷ ii (t) =y i (pt ) for pt t<(p + 1)T, an τ ii (t) = Therefore ŷ ii (t) can be thought of as a zero orer hol 1 on y i (t) with sampling interval T Moreover, after the algorithm has repeate a number of times equal to the iameter of the communication graph, the time elay τ ij associate with robot i s estimate of robot j s output is simply the number of hops that j is from i in the communication graph We assume for the remainer of the paper that the flooing algorithm has run long enough so that this is the case (ie we let the flooing algorithm start at time t = T, an the robots motion starts at time t =) Then we have ŷ ii (t) =C i x i (t) an ŷ ij (t) =C j x j (t τ ij T ) (2) where x i (t) enotes the zero orer hol on x i (t) with sampling interval T 1 In control an signal processing, a zero orer hol is a transform which takes a continuous signal an returns a steppe version of that signal with values that are constant over intervals of a given

3 Algorithm 1 Flooing Algorithm (implemente by robot i) Require: Robot i has a clock t synchronize with the rest of the robots in the network 1: initialize ŷ ii = y i (t) an τ ii = 2: initialize ŷ ij =an τ ij = for j = i 3: initialize ŷij min =an τij min = for j = i 4: loop 5: if it is robot i s turn to broacast then 6: broacast ŷ i an τ i 7: else if broacast receive from robot k with ŷ k an τ k then 8: for j =1to n o 9: if τ kj <τij min 1: upate ŷ min ij then =ŷ kj an τij min 11: en if 12: en for 13: en if 14: if TDMA roun has finishe then 15: upate ŷ ii = y i (t) an τij min 16: upate ŷ ij =ŷ min 17: en if 18: en loop ij = τ min ij an τ ij = τ min ij = τ kj +1 for j = i for j = i The flooing algorithm provies a means by which any robot can use any other robot s output information (albeit outate) in its controller This allows for the istinction between a communication graph G comm an a control graph G cont as escribe in the following section B Communication Graphs an Control Graphs Control laws for our multi-robot system can be seen as having a graphical structure separate from the communication graph We efine the control graph G cont =(V, E cont ) to be a irecte graph in which each robot is ientifie with a noe, an if the elaye output of a robot j is use in the computation of the control law of robot i, then there is a irecte ege (j, i) E cont When consiering the control graph it becomes apparent that a particular control task might be implemente over a number of ifferent control graphs, for the same communication graph For example, if we want robots to maintain the formation of a single file line, we might have a controller that controls each robot to remain a istance δ from the neighbor immeiately in front of it, or we might control the ith robot to remain a istance δ(i 1) from the first robot in the line The first controller inuces a chain shape control graph, as in the top of Fig 1, an the secon inuces a star shape control graph, as in the bottom of Fig 1, however both have the same ultimate objective This provokes the question: what is the best control graph to accomplish a given control task over a given communication graph? We point towars a solution to this question by proposing a performance metric in Sec IV C Couple Network an Control System In moeling the close loop controller an network algorithm, we must account for the fact that when robot i Fig 1 Elementary example of two ifferent control graphs that can be use to carry out the same control objective computes its controller, it is using an output value for j that has been route over the communication network, an that this route will have taken a certain number of upate times T It has been observe in many practical multi-robot control scenarios that this time elay is a significant factor, an can lea to instability uner some communication an control graph topologies an network upate times Consier a formation controller of the form u i = K ij (ŷ ij (t) y i (t) δ ij ), (3) j N cont i where δ ij efines the esire relative outputs y j y i for the formation We consier only proportional output feeback in this paper, though similar results can be obtaine for ynamic controllers as in [8] Further ivie the inex set into subsets Niτ cont of all control neighbors j that are τ hops away from i in the communication graph, so that Ni cont = Niτ cont an Niτ cont Niτ cont = for all τ = τ Some N cont iτ may be empty if there are no control neighbors τ hops away from i This construction is illustrate graphically in Fig 2 Fig 2 This figure illustrates how the control neighborhoo of a robot N cont i can be ivie into sub-neighborhoos N cont iτ of all robots that are τ hops away from i in the communication graph The control input for each robot can be ivie into a continuous component K ii y i (t), where K ii = j N cont K ij i an a component that is constant over the time step of j N cont iτ the network K ij (ŷ ij (t) δ ij ) Recall from Section II-A that the elay of ŷ ij (t) is equal to T times the number of hops that j is away from i in the communication graph We can rewrite the system using (2) as ẋ i (t) =(A i B i K ii C i )x i (t)+ K ij (C j x j (t τt) δ ij ), (4) B i j N cont

4 Now the system looks like a continuous time system with a constant input over intervals of length T We can write the continuous time ynamics as a iscrete time equation by integrating over the intervals of length T This gives the value of the continuous state x i (t) at T secon intervals as where x i (t + T )=Φ i x i (t)+ Γ i B i K ij (C j x j (t τt) δ ij ) (5) j N cont iτ an Γ i = Φ i = e (Ai BiKiiCi)T T e (Ai BiKiiCi)s s, (6) an the exponentials are matrix exponentials [19] We emphasize that this is not an approximation; it is the analytically etermine value of the state at iscrete time instants Now efine a state vector x(t) =[x T 1 (t) x T n (t)] T, which is the concatenation of the states x i (t), an X(t) = [x T (t) x T (t T )] T, which is the concatenation of all the time elaye states up to the maximum number of hops it can take for a packet to cross the network, namely The vector X is the state of the couple network protocol an robots Note that it has N n ( + 1) elements, which is ( + 1) times the number of elements there woul be were we to ignore the network protocol Because the robot ynamics, controller, an network algorithm are linear, the entire couple system is linear an can be written where X(t + T )=AX(t)+, (7) Φ Ψ 1 Ψ I Nn A =, (8) I Nn with Φ=iag([Φ 1 Φ n ]), Ψ τ =[κ τ ij ], κ τ Γi B ij = i K ij C j if j Niτ cont otherwise, an = (Γ 1 B 1 j N cont 1 (Γ n B n j N cont n K 1j δ 1j ) T K nj δ nj ) T T, where I Nn is the Nn Nn ientity matrix, an is a matrix of zeros of the appropriate imensions Our goal in this work is to etermine useful conitions on the network upate time T, the feeback control gains K ij, an the control graph G cont to guarantee the stability an convergence of the system in (7) for a given robot ynamics an a given communication graph III STABILITY CONDITIONS In this section we present the main analytical results of the paper Using the moel from Section II we erive a stability conition for the general case, an then fin more targete results for the case of first orer an secon orer robot ynamics A General Stability Conition The conition for the stability (in the sense of Lyapunov) of the close loop system is that all of the eigenvalues of A lie within the close unit circle, that is λ i (A) 1 i {1,,Nn( + 1)} For asymptotic stability, the conition is λ i (A) < 1 for all i The eigenvalues of A are functions of T an K ij, an the structure of A is ictate by the control an communication graphs (ie the Nτ comm neighborhoos etermine which entries in Ψ τ are non-zero) We can simplify the problem of etermining the eigenvalues of A consierably with the following theorem Theorem 1 (System Eigenvalues): The Nn( + 1) eigenvalues of the multi-robot system transition matrix A in (8) are given by the solutions to the equation et λ +1 I Nn λ Φ+ λ τ Ψ τ = (9) Proof: The eigenvalues of A are given by the solutions of the characteristic equation et(λi Nn(+1) A) = We will simplify this by exploiting the block structure of A, an using the ientity et(m) =et(m 22 )et(m 11 M 12 M22 1 M 21), where M = M11 M 12 M 21 M 22 Define M = λi Nn(+1) A an notice that it has the esire block structure with M 11 = λi Nn Φ, M 12 = [Ψ 1 Ψ ], M 21 = [I Nn ] T, an M 22 = λi Nn I Nn( 1) The block M 22 is lower triangular, so its eterminant is the prouct of the iagonal entries, namely et(m 22 )=λ Nn The term M 12 M22 1 M 21 can be compute to be M 12 M22 1 M 21 = λ τ Ψ τ, an applying the ientity gives the consierably simpler characteristic equation λ Nn et λi Nn Φ λ τ Ψ τ =, an bringing the λ Nn insie the eterminant gives the esire result For stability, the solutions of (9) must lie within the close unit circle One can check this conition straightforwarly by computational means for particular systems an controllers Unfortunately, the conition is too general to be of use as a rule of thumb, or to give an insight into the trae-offs between control gain an network upate rate For this, we seek special cases that are of particular

5 B Relation to the Control an Communication Graphs It is typical in control problems with a graphical structure to have algebraic properties of the graph, such as the graph Laplacian or the ajacency matrix, appear in the system equations One may rightly woner where these graph properties are in our moel, especially because our system involves two graphs, the control graph an the communication graph The ajacency matrix Λ(G) of a irecte graph G is the matrix which has a 1 for the entry (i, j) if there is a irecte ege from j to i in the graph (that is, j s output is use in i s controller), an a zero otherwise 2 Let the ajacency matrix of the control graph be given by Λ cont = Λ(G cont ) Now efine a sequence of ajacency matrices Λ comm τ which are the ajacency matrices of the graph of τ-hop neighbors in the communication graph That is, there is a 1 in element (i, j) of Λ comm τ if the shortest path between j an i in the communication graph is τ hops long We can re-write the equation (9) in terms of these ajacency matrices Let enote the element-wise prouct of two matrices, an the Kronecker prouct Then λ τ Λ comm τ Λ cont is a matrix with λ τ in any entry (i, j) which is in the control ajacency matrix an is τ hops away in the communication graph Furthermore, if we assume that all robots have the same ynamics an the same number of control neighbors, so Γ i =Γ, B i = B, an C i = C for all i, an that all robots use the same feeback gain, so that K ij = K for all (i, j) in the control graph, then we have λ τ Ψ τ = λ τ Λ comm τ Λ cont [κ τ ] which leas to λ τ Ψ τ = λ τ Λ comm τ Λ cont ΓBKC Although this notation shows more clearly the relation of the characteristic equation in (9) to the structure of the communication an control graphs, we will continue to use the previous notation because it is more general an more compact C First Orer Robot Moel Consier the case of n one imensional robots moele as stable first orer systems ẋ i = a i x i + b i u i, y i = cx i, (1) where a i, b i c>, with controllers u i = k ij (ŷ ij (t) y i (t) δ ij ), (11) j N cont j for scalar feeback gains k ij > We require that the scalar c be the same for all robots This class of robot moels, especially the simple sub-case of integrator robots (in which the robot s velocity is controlle irectly), is common in the multi-robot control literature For this moel, the close loop 2 In fact, this is the transpose of how the ajacency matrix is typically efine, but it is better suite to our single robot iscrete time ynamics from (5) take the form x i (t + T )=φ i x i (t)+ (1 φ i ) (a i + b i k ii c) j N cont iτ b i k ij c(x j (t τt) δ ij ), (12) where φ i = e (ai+bikiic)t Notice that < φ i < 1 for a i, b i k ii c>, an T> Now the state transition matrix A for the close loop robot network can be written from (8) as Φ Ψ 1 Ψ I n A =, I n where Φ=iag([φ 1 φ n ]), an Ψ τ =[κ τ ij ] with κ τ ij = (1 φi)b ik ijc a i+b ik iic if j Niτ cont otherwise To give the reaer an iea of the size of this system, epening on the iameter of the communication graph, the state is at least 2n imensions an at most n 2 imensions Surprisingly, the network of first orer robots is stable for any positive network upate time T, any positive control gains k ij, an any control graph, as formalize in the following theorem Theorem 2 (First Orer Robot Networks are Stable): For first orer robots (1) with controllers (11) using the flooing algorithm (Algorithm 1) to upate ŷ ij over a static, connecte communication network, the multi-robot system is Lyapunov stable for all k ij >, T >, an for all control graphs Proof: The proof follows from Ger sgorin s circle theorem, which states that all the eigenvalues of an m m matrix lie within the union of the m close circles on the complex plane centere at the iagonal elements with raii equal to the sum of the absolute value of the off iagonal elements in the row Specifically, given an m m matrix A =[a ij ], for all eigenvalues λ i (A) there exists j {1,,m} such that λ i (A) a jj n k=1,k=j a jk, where is the magnitue of a complex number In our case, the first n iagonal elements of A are φ i, an the absolute sum of the off iagonal elements of the first n rows are (1 φ i)b i k ii c (a i + b i k ii c) (1 φ i), using the fact that a i an b i k ii c > This correspons to n circles centere at φ i extening in the positive real irection to a point less than or equal to 1, since φ i + (1 φi)bikiic (a i+b ik iic) φ i + (1 φ i )=1 They exten in the negative real irection to a point greater than or equal to -1, since φ i (1 φi)bikiic (a i+b ik iic) φ i (1 φ i ) > 1 We conclue that these n circles are containe in the close unit circle The last n(n 1) iagonal elements are, an the off iagonal absolute sums are 1, therefore these circles are also containe in the unit circle (they are equal to it) We therefore

6 conclue that all n 2 eigenvalues lie on or within the unit circle regarless of control gain, network upate time, or control graph topology Integrator robots are a particularly common moel in the multi-robot literature, therefore we state the following corollary Corollary 1 (Integrator Robot Networks are Stable): For integrator robots ẋ i (t) = u i (t), with controllers u i = j N cont k ij (ˆx ij (t) x i (t) δ ij ) an using Algorithm i 1 to upate ˆx ij (t) over a static, connecte communication network, the multi-robot system is Lyapunov stable for all k ij >, T>, an for all control graphs Proof: Apply Theorem 2 to the case with a i =, b i =1, an c =1 Remark 1 (Extension to N-Dimensional Robots): The extension of Theorem 2 to N imensional robots with A i, B i, an K ij iagonal is immeiate When A i, B i, an K ij are not iagonal the situation is consierably more complex Remark 2 (Significance an Interpretation): Theorem 2 inicates that integrator an first orer robot moels, which are common in the multi-robot control literature, are not rich enough to exhibit instability ue to network time elays The main factor lying behin this result is the composition of continuous time robot ynamics with iscrete time network ynamics The continuous to iscrete transform in (12) results in coefficients φ i on the state an (1 φ i ) on the input, which balance each other in such a way that instability is impossible D Secon Orer Robot Moel To investigate the stability properties of networks with more realistic robot ynamics, we turn to the case of a secon orer robot moel Consier a one egree of freeom secon orer robot with position p i (t) an state x i (t) = [p i (t) ṗ i (t)] T Let the ynamics of the robot be given by 1 A i = c b,b i = c,c i = 1, (13) an K ij = k, with b, c, an k positive scalars This is a fair moel of the translational ynamics of a flying quarotor robot that is stabilize about its angular axes with an onboar controller, for example [5], [2] For this system, we can fin the matrices Φ i an Ψ i analytically by solving the secon orer response over the T time winow Suppose for now that k>(b 2 c)/(4 Ni cont c) This will ensure that the system response is oscillatory The response of the position as a function of the upate time T can be foun by solving the secon orer ODE to get p i (t + T )=e σt p i (t) cos( T )+ (σp i (t)/ +ṗ i (t)/ ) sin( T ), where σ = b/2 an = c( N cont i k + 1) (b/2) 2 Differentiating with respect to T yiels ṗ i (t + T )=e σt ṗ i (t) cos( T )+ (σ 2 p i (t)/ + p i (t) σṗ i (t)/ ) sin( T ), from which the transition matrix can be foun to be Φ i = e σt cos(ωi T )+ σ 1 sin( T ) sin( T ) ( σ2 + ) sin( T ) cos( T ) σ sin( T ) (14) The matrix Γ i can be foun from the relation I +A i Γ i =Φ i (which follows from the efinitions of Φ i an Γ i in (6)) In our case, the matrix A is invertible, so we can fin Γ i from Γ i = A 1 (Φ i I), an for j Niτ cont we can compute κ τ ij =Γ ib i kc j to be κ τ ij = e σt N cont i k k+1 e σt cos( T ) σ sin( T ) kc sin( T ) (15) With these matrices in han, we can investigate the eigenvalues of A in a similar way to the integrator case an etermine values of k an T for which the network of robots will be stable, as escribe in the next theorem Theorem 3 (Secon Orer Stability): For robots with secon orer ynamics given by (13), using the controller in (3), an using Algorithm 1 to upate ŷ ij (t) over a static, connecte communication network, the following conitions are sufficient for the stability of the multi-robot system: 1) all noes in the control graph have in-egree N1 cont, 2) k> b2 c 4 N cont, 1 c 4πm 3) T = 4c( N cont for some positive integer m 1 k+1) b2 Proof: We again prove the result using Ger sgorin s circles The first conition ensures that Ni cont = N1 cont for all i, which implies that the transition matrix Φ i is the same for all robots, an that κ τ ij is the same for all i an for all j Niτ cont The secon conition ensures that the system response for all robots is oscillatory, with transition matrix given by (14) an with κ τ ij, given by (15) The thir conition says that T =2πm/ Substituting this into (14) yiels Φ i = I 2 e σt, an in (15) it gives κ τ ij = k N cont 1 k+1 (1 e σt ) if j Niτ cont an κ τ ij = otherwise Therefore, the first 2n rows of A have 1 on the iagonal an have off iagonal absolute row sums of N cont 1 k N cont 1 k +1 (1 e σt ), for the o rows an for the even rows Thus the circles corresponing to the first n o rows are centere at e σt an they exten to e σt N cont 1 k + N cont (1 1 k+1 e σt ) <e σt + (1 e σt )=1in the positive real irection an e σt N cont 1 k N cont (1 1 k+1 e σt ) >e σt (1 e σt ) > 1 in the negative real irection, so they are containe in the close unit circle The circles corresponing to the first n even rows have raius zero an are centere at <e σt < 1,

7 the remaining 2(n 1) circles are unit circles Therefore all eigenvalues of A are containe in the close unit circle Remark 3 (Significance an Interpretation): The theorem exploits the fact that, when sample at their natural perio, the robots look like first orer systems, so we can apply the same kin of analysis as in the first orer case The assumptions that all robots have ientical ynamics an that the control graph is regular ensure that the system has only one natural perio Also, the proof oes not hol if the network is upate at multiples of half the perio, since then the first n o Ger sgorin s circles exten beyon 1 on the real axis Also, Ger sgorin s circles are known to be conservative; there are many cases that o not meet the conitions of the proof that will still be stable IV PERFORMANCE METRIC Until this point we have been focuse primarily on moeling an stability of the group of robots Now we turn to the question of performance If we wish to rive a group of robots in an aggressive manner, this will require fast ynamics for the couple robots an communication system Typically, for a stable iscrete time system of the form x(t + 1) = Ax(t), the spee of response of the system is capture by the moulus of the largest eigenvalue λ max (A) The smaller this value is, the faster the transient response of the system In particular, we can use the boun x(t + 1) λ max (A) x(t), leaing to the conclusion that x(t) λ max (A) t x(), for the initial conition x() That is, the size of the state is boune by a geometric series with convergence rate λ max (A) In our case, however, the sequence is governe by the law X(t + T )= AX(t)+ D As a technicality, we must change this to a homogeneous equation of the form X(t + T )=A X(t) by employing the change of variables X = X X ss, where X ss is the steay state value of X(t), which satisfies the equation (I Nn(+1) A)X ss = D Now the trajectory of the system can be boune as X(t + T ) λ max (A) X(t), which leas to X(t) λ max (A) t/t X(), where t =,T,2T, Therefore, in our case the convergence rate of the system, which is the primary inicator of spee of response, is λ max (A) 1/T For this reason we propose the following performance metric for comparing or optimizing the spee of networke multi-robot systems Definition 1 (Convergence Rate): The convergence rate of the multi-robot system X(t + T )=AX(t)+ is given by α = λ max (A) 1/T Notice that a necessary an sufficient conition for Lyapunov stability of the system is that α 1, an for asymptotic stability the conition is α<1 We can use this metric as a esign criterion, by requiring that α α given some require convergence rate α Example plots of the convergence rate as a function of the network upate time T are shown in Fig 3 for three quarotor robots For sufficiently low control gain (K ij =3 in this case), an for T small (compare to the natural frequency of the robots), the convergence rate agrees with the Fig 3 The convergence rate α = λ max(a) 1/T as a function of network upate time T is shown for two gain values K ij for a network of three quarotor robots The soli curves show the convergences rates preicte by our moel, while the ots inicate experimentally etermine rates, an the re circles correspon to the experiments shown in Fig 4 The inset graph iagram shows the control (otte) an communication (soli) topologies intuition that faster network upates give better performance For higher gains (K ij =4in this case) this intuition breaks own, as a slower network upate time can actually improve performance For example, in Fig 3 for K ij =4, increasing the upate time from T =4s to T =6s significantly improves the system performance (inee, it makes the system stable) These performance preictions were verifie experimentally, as escribe in the next section V QUADROTOR EXPERIMENTS We carrie out 32 experimental trials with three quarotor robots in a Vicon motion capture environment A fourth orer moel was use to represent the quarotors with the A i an B i matrices etermine from a system ientification proceure prior to the experiments The controller use only the relative positions of neighbors, not their full state The convergence rates for all 32 trials are shown in Fig 3, along with an inset iagram showing the control an communication graphs The observe convergence rates track the K ij = 3 curve with a root mean square error (RMSE) of 4, an the K ij =4curve with an RMSE of 12 The K ij =4curve fits the ata less closely because the experiments were unstable or nearly unstable in this regime, making it ifficult to etermine the α value Figure 4 shows the results of three specific trials in which, as preicte by our moel, the convergence rate improve with increasing network upate time, contrary to intuition More extensive iscussion will be presente in a journal paper soon to follow The paper [5] gives an in epth iscussion of earlier experiments using the same experimental apparatus VI CONCLUSIONS In this paper we erive a new moel for networke multi-robot systems, placing special attention on the ynamic interaction between the networking protocol an the ecentralize controller The resulting moel is a iscrete

8 (a) Initial Configuration (b) Divergence (c) Convergence x (m) x (m) x (m) 1 2 robot 1 robot 2 robot t (s) () T =4s t (s) (e) T =5s t (s) (f) T =6s Fig 4 The robots start in the initial configuration in 4(a); for an unstable system they iverge as in 4(b); for a stable system they converge as in 4(c) The x positions of the robots are shown in the bottom plots for three example experiments The gain is K ij =4an the network upate time increases from left to right Notice that the performance improves with increase elay in this case, contrary to conventional intuition We performe 32 experiments, the convergence rates from which are shown in Fig 3, where the re circles give the rates for the three experiments above A vieo of the experiments is available at linear ynamical system whose eigenvalues are functions of the network upate time, the control gain, an the control an communication graphs of the system We erive sufficient conitions for stability for the case of first orer an secon orer robot ynamics We also propose a performance metric to quantify the convergence rate of the system The preictions of the moel were verifie in 32 experiments with three quarotor robots In the future we woul like to erive stronger conitions for the stability of the multi-robot system, an to carry out trajectory following experiments REFERENCES [1] N Michael, M M Zavlanos, V Kumar, an G J Pappas, Maintaining connectivity in mobile robot networks, in Experimental Robotics: The Eleventh International Symposium, ser Springer Tracts in Avance Robotics, O Khatib, V Kumar, an G J Pappas, Es Springer Berlin, 29, vol 54, pp [2] J McLurkin, Analysis an implementation of istribute algorithms for Multi-Robot systems, PhD thesis, Massachusetts Institute of Technology, 28 [3] Y Cai, K Hua, an A Phillips, Leveraging 1-hop neighborhoo knowlege for efficient flooing in wireless a hoc networks, in Proc of the International Performance Computing an Communications Conference (IPCCC), Phoenix, AZ, April 25, 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