Decentralized Goal Assignment and Safe TrajectoryGeneration in Multi-Robot Networks via Multiple

Lyapunov FunctionsDimitra Panagou, Matthew Turpin and Vijay Kumar

Abstract—This paper considers the problem of decentralizedgoal assignment and trajectory generation for multi-robot net-works when only local communication is available, and proposesan approach based on methods related to switched systems andset invariance. A family of Lyapunov-like functions is employedto encode the (local) decision making among candidate goalassignments, under which a group of connected agents choosesthe assignment that results in the shortest total distance to thegoals. An additional family of Lyapunov-like barrier functionsis activated in the case when the optimal assignment may leadto colliding trajectories, maintaining thus system safety whilepreserving the convergence guarantees. The proposed switchingstrategies give rise to feedback control policies that are com-putationally efficient and scalable with the number of agents,and therefore suitable for applications including first-responsedeployment of robotic networks under limited information shar-ing. Simulations demonstrate the efficacy of the proposed method.

I. INTRODUCTION

Task (target) assignment problems in multi-agent systemshave received great interest within the robotics and controlscommunities in the past couple of years, in part becausethey encode the accomplishment of various objectives inapplications such as surveillance, exploration and coverage[1]. A common thread towards solving assignment prob-lems is the development of algorithms that assign targets toagents by optimizing a predefined criterion [2]–[6]. Standardgoal assignment strategies are the greedy assignment [7],the iterative assignment [8], the multiple Traveling Salesmanassignment [9], the MinPoS assignment [10] and market-basedmechanisms [11]–[13].

The main focus in this line of research is mostly relatedto the optimization aspects of the problem and the associatedcomputational complexity, so that suboptimal solutions withcertain performance guarantees are derived. A hybrid controlsolution to multi-robot goal assignment is given in [14], whilea game-theoretic approach is presented in [15]. More recently,

Dimitra Panagou is with the Department of Aerospace Engineering, Uni-versity of Michigan, Ann Arbor, MI, USA. dpanagou@umich.edu

Matthew Turpin was with the GRASP Lab, School of Engineering and Ap-plied Science, University of Pennsylvania, Philadelphia, PA, USA when thiswork was done. He is now with Qualcomm. mturpin@seas.upenn.edu

Vijay Kumar is with the GRASP Lab, School of Engineering andApplied Science, University of Pennsylvania, Philadelphia, PA, USA.kumar@seas.upenn.edu

We gratefully acknowledge the support of ARL grant W911NF-08-2-0004, ONR grant N00014-09-1-1051, and TerraSwarm, one of six centersof STARnet, a Semiconductor Research Corporation program sponsored byMARCO and DARPA.

the problem of assigning a group of mobile robots to an equalnumber of static targets under the assumption that the robotshave limited range-based communication and target-sensingcapabilities was studied in [16], and in [17], respectively.

Nevertheless, such optimization-based techniques are oftenderived under relaxed assumptions, such as point or staticagents, which are in principle not acceptable in realisticsettings. For instance, first-response or search-and-rescue mis-sions using robotic agents (e.g., aerial robots) typically involvemultiple tasks that, on one hand, may need to be performedas quickly as possible, yet on the other hand they can beaccomplished only when information sharing is available, andunder tight safety guarantees. In response to this challenge, inour earlier work [18], [19] we proposed a centralized approachon the Concurrent Assignment and Planning of Trajectories(CAPT) for multi-robot systems, that produces minimumdistance traveled trajectories for non-point robots, that are ad-ditionally collision-free under mild assumptions on the initiallocations of the robots and the locations of the goals. Hencewe simultaneously addressed two challenges: (i) the combina-torially complex problem of finding a suitable assignment ofrobots to goal locations, and (ii) the generation of collision-free trajectories for every robot under given assumptions. Morespecifically, we considered unlabeled (interchangeable) robotsand goal locations, and proposed algorithmic solutions forthe centralized CAPT problem, as well as a decentralizedimplementation of the centralized algorithm that producessuboptimal solutions in terms of total distance traveled. Itshould be noted that although the combinations of N robotsto N goals result in intractable enumeration for all but thesmallest problems, fortunately, the O(N3) Hungarian Algo-rithm from the Operations Research community [20] optimallysolves this problem in a centralized manner, while extensionsto distributed systems have been also developed [21]. Anextensive treatment of assignment problems and algorithmscan be found in [22]. Our simulated teams of N robots underthe CAPT algorithm on a single computer take approximately10−7N3 seconds to compute the optimal assignment, i.e.,a team of 100 robots takes about 0.1 seconds to plan itstrajectories. This is efficient for the considered multi-robotapplications. However, the decentralized implementation of theCAPT algorithm does not in general ensure safety for themulti-robot system, i.e., that the trajectories of the robots arecollision-free.

Let us note that some technical ingredients of the problemconsidered here might be applicable or useful to other types

of decentralized/distributed multi-robot planning and controlproblems as well, such as, for instance, robotic team reconfig-uration [23], control of modular robots, or formation/flockingcontrol. However, the main novel characteristics that differ-entiate our work from similar problems and approaches are:(i) the casting of both the assignment and safety problemsfor a dynamic multi-agent/multi-robot system into a switchedsystems stability problem, and (ii) the development of decen-tralized feedback control algorithms that generate convergentand collision-free trajectories with provable guarantees. Thecontributions of this work are described in more detail in thefollowing section.

A. Overview and Contributions

In this paper we build upon our earlier work [18], [19] thatconsiders a computationally efficient way of safely assigninggoals in a multi-robot system in a centralized manner, andwe focus on the problems of concurrently assigning goalsand generating collision-free trajectories for the decentralizedcase, i.e., for when the robots have limited sensing andcommunication capabilities and hence exchange informationonly locally, i.e., when they become connected. We developalgorithms that concurrently address the problems of:(P1) Assigning goal locations to agents by minimizing a cost

function expressing distance to the goals.(P2) Designing feedback control policies guaranteeing:

(i) the convergence of the agents to their assigned goals,(ii) that the resulting trajectories are collision-free.

The key specifications in the proposed formulation are:(S1) Agents and goal locations are interchangeable, which

means that the mission is considered accomplished wheneach agent has converged to some goal location.

(S2) Information exchange between a pair of agents is feasibleand reliable only when they lie within a certain commu-nication range, which means that the decision makingon the optimal goal assignment (in terms of distanceto the goals) can be performed only locally, i.e., in adecentralized fashion.

(S3) Agents are modeled as non-point robots.The adopted decentralized formulation, and the underlying

goal assignment and control design techniques in the currentpaper offer several improvements compared to the earlier workof the authors’ in the conference paper [24], namely: Theconservative conditions on the initial positions of the agentsand of the goal locations in [24] are now relaxed via the designof vector-field controllers instead of gradient-based controllers;more specifically, the proposed vector fields are always well-defined everywhere in each agent’s state space, opposed to thebarrier functions utilized in [24] that exhibit some pathologicalsituations when the distances between neighbor agents andgoal locations become smaller than a certain distance. Thisway, the overall switched system can be implemented by forc-ing the agents switch among two control policies (describedbelow), instead of three in [24], where the third controllerwas taking care of the pathological situations of the barrierfunctions. In addition, the initial positions of the robots canbe selected to be R apart, compared to the conference version

that required them to be 2R√

2 apart. More specifically, thenovelties of the proposed approach are as follows:

Contribution 1: We consider the goal assignment and trajec-tory generation problem posed in [19] that seeks to minimizethe distance traveled by all robots, and formulate it into aswitched systems framework [25]; this provides a novel andnatural way of encoding multiple tasks via Lyapunov-likefunctions, and obtaining guarantees on the accomplishment ofthe encoded control objectives. We build our approach basedon multiple Lyapunov-like functions [26] and set invariance[27] techniques. We first encode the decision making on theoptimal goal assignment (which in the sequel we call theOptimal Goal Assignment (OGA) policy) as a state-dependentswitching logic among a family of candidate Lyapunov-likefunctions. Each Lyapunov-like function encodes a cost undera candidate goal assignment, that is, the sum of squareddistances to the goals. The switching logic dictates that, whena set of agents becomes connected at some time instant t, theydecide to switch to the Lyapunov-like function of minimumvalue at time t. We show that this decision making gives riseto a Globally Asymptotically Stable (GAS) switched systemwhich furthermore does not suffer from Zeno trajectories.Then, based on our recent work in [28], [29], we build anadditional state-dependent switching logic which employs afamily of Lyapunov-like barrier functions and vector fieldsencoding both inter-agent collision avoidance and convergenceto the goal locations determined by the OGA policy. Thiscontrol policy (in the sequel called the Collision Avoidance(CA) policy) provides sufficient conditions on determiningwhether the OGA policy is safe, and furthermore serves as asupervisor that takes action only when safety under the OGApolicy is at stake. We show that the switching between the OGApolicy and the CA policy results in asymptotically stable andsafe trajectories for the multi-robot system, at the expense ofpossibly resorting to suboptimal paths in terms of total distancetraveled; this situation appears only in the cases when the CApolicy forces the agents to deviate from their optimal (straightline) paths to the goals, in order to maintain system safety. Tothe best of our knowledge, the proposed approach of encodingassignment and motion tasks via Lyapunov-like functions is anovel approach in multi-robot task and motion planning, andintroduces a framework of correct-by-construction switched(hybrid) systems that satisfy the encoded tasks, along with thefeedback control policies that accomplish the encoded tasks.

Contribution 2: Part of this work appeared in [24]. Com-pared to the conference paper, here we develop a differentconflict resolution and collision avoidance strategy that isbased on directly defining feedback vector fields for eachone of the agents, instead of barrier functions. The methodgenerates safe reference directions and feedback controllersfor the agents, and alleviates (i) the issues often raised in puregradient-based solutions, where the behavior of the gradient-based controller is hard to tune, as well as (ii) the use ofthe third family of Lyapunov-like functions in the switchingstrategy in [24]. Furthermore, (iii) the new design providessafety and convergence guarantees for the decentralized multi-robot system under less restrictive assumptions on the spatialdistributions of the agents and the goal locations, compared to

the assumptions adopted in [19], [24]. Finally, (iv) we providea robustness result on the safety of the resulting multi-robottrajectories in the case of communication failures.

The paper is organized as follows: Section II presents themathematical formulation of the problem, while the adoptednotions and tools from switched systems theory are reviewedin Section III. The proposed goal assignment and collisionavoidance policies are given in Sections IV and V, respec-tively, along with the mathematical proofs that verify theircorrectness. Simulation results to evaluate their efficacy areincluded in Section VI, along with a discussion on the com-munication requirements and planning times in the connectedgroups of agents, and a comparison with the results in the con-ference version [24]. Section VII summarizes our conclusionsand thoughts on future research.

II. PROBLEM FORMULATION

We consider N agents i and equal number of goal locationsGi, i ∈ N = {1, 2, . . . ,N}. The objective of each agent i isto safely converge to some goal Gi, while collaborating withits locally connected agents (see definition below) so that theirtotal traveled distance is minimized. The motion of each agenti is governed by single integrator dynamics:

ri = ui, (1)

where ri =[xi yi

]Tis the position vector of agent i with

respect to (w.r.t.) a global Cartesian coordinate frame G, and uiis its control vector comprising the velocities uxi, uyi w.r.t. theframe G. We denote the speed of agent i, i.e., the magnitudeof the velocity vector ui, as ui = ‖ui‖ =

√u2xi + u2yi.

We assume that each agent i: (i) has access to its positionri and velocities ui via onboard sensors, (ii) can reliablyexchange information with any agent j 6= i which lies withinits communication region Ci : {ri ∈ R2, rj ∈ R2 | ‖ri −rj‖ ≤ Rc}, where Rc is the communication range. In otherwords, a pair of agents (i, j) is connected as long as thedistance dij = ‖ri − rj‖ ≤ Rc.

Remark 1: Note that maintaining connectivity for the entirenetwork is not fundamental for the proposed assignment andcollision avoidance strategies; in other words, maintaining aconnected graph is neither necessary, nor a control objectivein the proposed problem and approach.

Remark 2: The consideration of equal number of robots andgoal locations is mainly due to the fact that we do not considerthe problem of globally assigning goals to robots at the initialtime of the deployment, but rather we consider the problemof locally re-assigning goals to robots whenever/if the robotsget connected along with safety and convergence guarantees.The initial goal assignment is random and hence, suboptimalin terms of total traveled distance in general. In the case of Mgoal locations and N robots with M > N, our approach wouldbe initialized by randomly assigning a goal location per robot,and from there on the selected N goals would remain the sameover the evolution of the system (i.e., the remaining M − Nwould never be selected by any robot). Similarly, in the caseof M < N, only M robots would be randomly assigned a goal

location each, while the remaining N−M robots would not beassigned any goal.

The task is considered completed as long as each agentconverges to some goal location, with goal locations beinginterchangeable – each agent can converge to any goal loca-tion. This specification defines N! possible goal assignmentsk ∈ {1, 2, . . . ,N!}. Each goal assignment can be encoded viaan assignment matrix:

Ak = {αim} =

α11 . . . α1 N...

......

αN 1 . . . αN N

,with the rows representing agents and the columns repre-senting goal locations. For agent i ∈ {1, . . . ,N} assignedto goal Gm, m ∈ {1, . . . ,N}, one has αim = 1 and theremaining elements of the i-th row equal to zero. For instance,for 2 agents and 2 goal locations, the possible N! = 2 goalassignments are encoded via the assignment matrices:

A1 =

[1 00 1

], A2 =

[0 11 0

].

In this case, the matrix A1 encodes that agent 1 goes to G1

and agent 2 goes to G2, while the matrix A2 encodes thatagent 1 goes to G2 and agent 2 goes to G1.

Denote rGm =[xGm yGm

]Tthe position vectors of the

goal locations Gm w.r.t. the global frame. Then, the assignedgoal position for agent i under the Ak assignment matrix (orunder the k-th goal assignment) can be expressed as:

r(k)gi =

N∑m=1

α(k)im rGm ,

where α(k)im is the i-th row of Ak.

The next lemma establishes that under any assignment, therobots will converge to the respective goal locations. This isessential for the switching control strategy developed later.

Lemma 1: The position trajectories ri(t) of agent i areGlobally Exponentially Stable (GES) w.r.t. the k-th goalassignment under the control law:

u(k)i = −λi

(ri − r(k)gi

), (2)

where i ∈ {1, . . . ,N}, λi > 0.Proof: Consider the candidate Lyapunov function V

(k)i

for each agent i under the k-th assignment as:

V(k)i =

∥∥∥ri − r(k)gi

∥∥∥2 ,which encodes the distance of agent i from its goal r(k)gi

under the goal assignment k. The function V(k)i is continu-

ously differentiable, positive definite, and radially unbounded.Furthermore, the closed-loop system vector field (1) under thecontrol law (2) is globally Lipschitz continuous. Given thatrGi = 0, one has that the time derivative of V (k)

i along thesystem trajectories under the control law (2) reads:

V(k)i = −2λi

∥∥∥ri − r(k)gi

∥∥∥2 ≤ −2λiV(k)i ,

yielding that the position trajectories ri(t) of each agent i areGES w.r.t. the assigned goal r(k)gi . Furthermore, by consideringthe Lyapunov function:

V (k) =N∑i=1

V(k)i , (3)

encoding the sum of distances of the agents to their assignedgoals, one has that (2) render its time derivative:

V (k) =N∑i=1

−2λiV(k)i ≤ −2 min

i{λi}

N∑i=1

V(k)i =

= −2 mini{λi}V (k),

which implies that the sum of squared distances of agents ito their assigned goals r(k)gi is GES to zero.

Problem Description: Our goal is to develop decentralizedalgorithms that force Nc ≤ N connected agents to decidewhether they keep their current goal assignment or swapgoals. The criterion on the decision making is to choosethe assignment k ∈ {1, . . . ,Nc!} which minimizes the sumof total distance (i.e., results in the shortest paths) to thegoal locations. In the sequel, we refer to this assignment asthe optimal assignment. The challenge in the case of non-point robots is to ensure that the optimal assignment resultsnot only in stable, but also in collision-free trajectories w.r.t.the assigned goal locations. We develop a switching policybetween optimal (shortest) and safe (collision-free) trajectorieswith provable guarantees. This policy is realized via switchingbetween the OGA and CA policies defined in the sequel, whilethe switched systems and control framework is employed tofacilitate the stability and safety analysis for the resultingtrajectories of the multi-robot system. We first review somebasic results on switched systems theory.

III. NOTIONS FROM SWITCHED SYSTEMS THEORY

Following [26], let us consider the prototypical example ofa switched system:

x(t) = fk(x(t)), k ∈ K = {1, 2, . . . ,K}, (4)

where x(t) ∈ Rn, each fk is globally Lipschitz continuous andthe k’s are picked in such a way that there are finite switchesin finite time. Consider a strictly increasing sequence of times:

T = {t0, t1, . . . , tn, . . . , }, n ∈ N,

the set: I(T ) =⋃n∈N[tn, tn+1] denoting the interval comple-

tion of the sequence T , and the switching sequence:

Σ = {x0; (k0, t0), (k1, t1), . . . , (kn, tn), . . .},

where t0 is the initial time, x0 is the initial state, N isthe set of nonnegative integers, and the elements in the set{k0, k1, . . . , kn, . . .} take values in K. The switching sequenceΣ along with (4) completely describes the trajectory of theswitched system according to the following rule: (kn, tn)means that the system evolves according to:

x(t) = fkn(x(t), t)

Fig. 1. Branicky’s “decreasing sequence” condition. Image from [31].

for tn ≤ t < tn+1. Equivalently, for t ∈ [tn, tn+1) one has thatthe kn-th subsystem is active. We assume that the switchingsequence is minimal in the sense that kn 6= kn+1.

Now, for any subsystem k ∈ K, denote:

Σ | k = {t0|k, t1|k, . . . , tn|k, tn+1|k, . . . , }

the sequence of switching times when the k-th subsystem isswitched on or switched off. Hence,

I(Σ | k) =⋃n∈N

[tn|k, tn+1|k]

is the set of times that the k-th subsystem is active. DenoteE(T ) = {t0, t2, t4, . . . , } the even sequence of T ; then,

E(Σ | k) = {t0|k, t2|k, . . . , tn|k, . . . | n ∈ N}

denotes the sequence of the switched on times of the k-thsubsystem. In the sequel, the notation tn|k will be equivalent tok(tn), denoting the time instance tn when the k-th subsystembecomes active.

Definition 1: [26] A C1 function V : Rn → R+ withV (0) = 0 is called a Lyapunov-like function for a vector fieldf(·) and the associated trajectory x(·) over a strictly increasingsequence of times T if:

(i) V (x(t)) ≤ 0, ∀t ∈ I(T ),(ii) V is monotonically non-increasing on E(T ).

Theorem 1: [26] If for each k ∈ K, Vk is a Lyapunov-like function for the k-th subsystem vector field fk(·) and theassociated trajectory over Σ | k, then the origin of the systemis stable in the sense of Lyapunov.If in addition, for any tp, tq ∈ E(Σ | k), where p < q,

Vk(tq)(x(tq))− Vk(tp)(x(tp)) ≤ −W (x(tp)) , (5)

holds for some positive definite continuous function W , thenthe origin of the system is GAS [30].

The theorem essentially states that: Given a family offunctions {Vk | k ∈ K} such that the value of Vk decreaseson the time interval when the k-th subsystem is active, if forevery k the value of the function Vk at the beginning of suchinterval exceeds the value at the beginning of the next intervalwhen the k-th system becomes active (see Fig. 1), then theswitched system is GAS.

A fundamental assumption of the theorem is the decreasingcondition of V on E(Σ | k). This is quite conservative and,in general, hard to check [32]. The concept of generalizedLyapunov-like functions relaxes this condition.

Definition 2: [32] A C0 function V : Rn → R+

with V (0) = 0 is called a generalized Lyapunov-like func-tion for a vector field f(·) and the associated trajectoryx(·) over a strictly increasing sequence of times T ={t0, t1, . . . , tn, . . . , }, n ∈ N, if there exists a functionh : R+ → R+ satisfying h(0) = 0, such that: V (x(t)) ≤h(V (x(tn))), for all t ∈ [tn, tn+1) and all n ∈ N.

Theorem 2: [32] Consider the prototypical switched sys-tem (4) and assume that for each k ∈ K there exists a pos-itive definite generalized Lyapunov-like function Vk(x) w.r.t.fk(x(t)) and the associated trajectory x(t), t ∈ [tn, tn+1).Then:

(i) The origin of the switched system is stable if and only ifthere exist class GK functions αk(·) satisfying

Vk(x(t2q|k))− Vk(x(t0|k)) ≤ αk(‖x0‖), (6)q ≥ 1, k = 1, 2, ...,K.

(ii) The origin of the switched system is asymptotically stableif and only if (6) holds and there exists k such thatVk(x(t2q|k))→ 0 as q →∞.

The theorem essentially states that stability of the switchedsystem is ensured as long as Vk(x(t2q|k))−Vk(x(t0|k)), whichis the change of Vk between any “switched on” time t2q|k andthe first active time t0|k, is bounded by a class GK function,regardless of where Vk(x(t0|k)) is.

Remark 3: The condition (6) is equivalently rewritten as:∞∑q=1

(Vk(x(t2q|k))− Vk(x(t2(q−1)|k))

)≤ αk(‖x0‖),

for any k ∈ K. Note that here Vk(x(t2q|k))−Vk(x(t2(q−1)|k)),q ≥ 1, stands for the change of Vk(x) at two adjacent“switched on” times. Thus, the condition means that Vk isallowed to grow on E(Σ | k) but the total such change on anyfinite time interval should be bounded from above by a classGK function.

Remark 4: Theorem 2 relaxes the decreasing requirementof Vk(x(t)) on the corresponding active intervals. Instead, weonly need that Vk(x(t)) on an active interval does not exceedthe value of some function of Vk at the “switched on” time.

IV. THE OPTIMAL GOAL ASSIGNMENT POLICY

A. The N = 2 Robot Case

For simplicity, we first consider N = 2 agents i, j that needto safely move towards goal locations G1, G2, and build aswitching logic that locally decides on the OGA.

Assume that the agents initiate at t0 = 0 such that dij(t0) >Rc under a random assignment k ∈ {1, 2} (i.e., not necessarilythe optimal one) and move under the control laws (2). If theinter-agent distance happens to remain always greater than thecommunication range Rc, i.e., if dij(t) > Rc, ∀t ∈ [0,∞),we say that the agents are never involved in a meeting event,and as thus, no decision on the goal assignment needs to be

made. Out of Lemma 1, the position trajectories ri(t), rj(t)are GES w.r.t. the assigned goal locations. Clearly, in suchcases, no inter-agent collisions occur.

Assume now that the agents initiate at t0 = 0 such thatdij(t0) ≤ Rc, or that at some time instant td > 0 they liewithin distance dij(td) ≤ Rc under some goal assignment k ∈{1, 2} (again, not necessarily the optimal one). We say that theagents are involved in a meeting event and a decision regardingon the goal assignment has to be made. The proposed decisionmaking strategy is defined as follows: The agents:

1) Exchange information on their current positions ri(td),rj(td) and goal locations rgi(td), rgj (td).

2) Compare the values of the Lyapunov functions V (k)(td)given out of (3) for each possible assignment k ∈ {1, 2}.

3) Move under: V (td) = min{V (k)(td) : k ∈ {1, 2}

}.

In other words: Without loss of generality, let k = 1 bethe goal assignment before the decision making. The agentsimplement the following switching logic:• If V (1)(td) ≤ V (2)(td), then: V (td) = V (1)(td), i.e., they

keep the same goal assignment,• If V (1)(td) > V (2)(td), then: V (td) = V (2)(td), i.e., they

switch goal assignment.In the sequel, we refer to the decision making based on thelogic described above as to the OGA policy. Note that ingeneral for N > 2 agents, this policy results in locally optimalpaths since it re-assigns goals among the connected agentsonly (which may be Nc < N in number), while it does notnecessarily apply at the initial time t0, hence to the initialpositions of the robots, but rather is triggered at any timetd ≥ t0 that Nc ≤ N robots may happen to become connected.

Problem 1: Establish that the OGA policy renders asymp-totically stable trajectories for the multi-robot system, i.e., thateach agent does converge to some goal location.

B. Stability Analysis on the Switched Multi-robot System

We resort to control analysis tools for switched systems.The closed-loop dynamics of agent i read:

ri(t) = u(k)i = −λi

(ri(t)− r(k)gi

), (7)

while similarly one may write the closed-loop dynamics forthe trajectories rj(t) of agent j. The OGA policy gives riseto switched dynamics for each agent, in the sense that fork = 1 each agent moves under assignment matrix A1, whilefor k = 2 each agent moves under assignment matrix A2.

Denote r =[riT rj

T]T

the state vector of the multi-robotsystem, which is governed by the switched dynamics:

r(t) = fk(r(t)), (8)

where fk =

[u(k)i

u(k)j

], k ∈ K = {1, 2}.

Assumption 1: We assume for now that there are only afinite number of switches per unit time.

Denote n ∈ N and consider the sequence of switchingtimes T = {t0, t1, t2, t3, . . . , tn, . . .} and the switching se-quence: Σ = {r0; (k0, t0), (k1, t1), . . . , (kn, tn), . . .}, kn ∈

K,∀n ∈ N. Assuming that the switching sequence is min-imal, i.e., that the switching signal does not imply a se-quential switching between the same subsystems1, it followsthat kn 6= kn+1. Without loss of generality, assume thatk0 = 1. Then the (minimal) switching sequence reads:Σ = {r0; (1, t0), (2, t1), (1, t3), (2, t4), . . .}.

Theorem 3: The trajectories r(t) of the switched multi-robot system (8) are GAS w.r.t. the goal assignment k underthe OGA policy. This implies that agents i, j converge to theirgoal locations r(k)gi , r(k)gj .

Proof: We consider the candidate Lyapunov-like func-tions V (k), k ∈ {1, 2}, given out of (3), which encode themotion of the agents under the goal assignment k. Out ofLemma 1 one has that each individual k-th subsystem, i.e. themotion of the multi-robot system under the k-th assignment,is GES i.e., each V (k) is decreasing during the time intervalswhen the k-th subsystem is active. Furthermore, the OGApolicy dictates that at the switching times {t0, t1, t2, t3, . . . , }one has:

V (1)(t0) > V (2)(t1) > V (1)(t2) > V (2)(t3) > . . . ,

i.e., the value of each Lyapunov-like function V (k) at thebeginning of the time intervals when the k-th subsystembecomes active satisfies the decreasing condition. This provesthat the switched system is stable.

To draw conclusions on asymptotic stability, consider thetime intervals [t2n, t2n+1), [t2n+1, t2n+2), n ∈ N, when thek = 1 and k = 2 subsystems are active, respectively. Then thefollowing inequalities hold:

V (1)(t0) > V (1)(t1),since the subsystem 1 is

active on [t0, t1) and GES

V (1)(t1) ≥ V (2)(t1), out of the OGA Policy

V (2)(t1) > V (2)(t2),since the subsystem 2 is

active on [t1, t2) and GES

V (2)(t2) ≥ V (1)(t2), out of the OGA Policy

V (1)(t2) > V (1)(t3),since the subsystem 1 is

active on [t2, t3) and GES

V (1)(t3) ≥ V (2)(t3), out of the OGA Policy...

which imply that: V (1)(t2n+2) < V (1)(t2n), ∀n, i.e.:

V (1)(t2n+2) = ρ1V(1)(t2n), 0 < ρ1 < 1. Then:

V (1)(t2n+2)− V (1)(t2n) = −(1− ρ1) V (1)(t2n)︸ ︷︷ ︸W (r,r

(1)g )

,

where 1 − ρ1 > 0, and W (r, r(1)g ) = g

(‖r − r(1)g ‖2

)is

a continuous, positive definite function w.r.t. the goals r(1)g

1Note that this condition is met in our case since in the stability analysis weconsider only those switching times when the goal assignment does change.

imposed by the k = 1 assignment. Similarly we obtain:V (2)(t2n+1) < V (2)(t2n+3), ∀n, which leads to

V (2)(t2n+3)− V (2)(t2n+1) = −(1− ρ2) V (2)(t2n+1)︸ ︷︷ ︸W (r,r

(2)g )

,

where 0 < ρ2 < 1. Therefore, the switched system is GAS.

C. Avoiding Zeno Behavior

In the sequel we establish that the switching policy doesnot result in Zeno trajectories [33], i.e., in trajectories thatconverge to some point or switching surface other than thedesired equilibrium; such trajectories in practice would resultin robots not converging to their respective goal locations.

Given the sequence T = {t0, t1, . . . , tn, . . . }, n ∈ N, ofswitching times, a switched system is Zeno if there exists somefinite time tZ such that:

limn→∞

tn =∞∑n=0

(tn+1 − tn) = tZ .

Zeno behavior means that the switching times have a finiteaccumulation point, i.e., that infinite amount of switchingsoccurs in a finite time interval. In general, the task of detectingpossible Zeno trajectories and extending them beyond theiraccumulation points is far from trivial [25] and depends onthe problem at hand.

The definition above results in two qualitatively differenttypes of Zeno behavior. A Zeno switched system is [33]:

(i) Chattering Zeno if there exists a finite C ∈ N such thattn+1 − tn = 0 for all n > C.

(ii) Genuinely Zeno if tn+1 − tn > 0 for all n ∈ N.The difference between these is prevalent especially in theirdetection and elimination. Chattering Zeno behavior resultsfrom the existence of a switching surface on which the vectorfields “oppose” each other. This behavior can be eliminatedby defining either (i) Filippov solutions which force the flowto “slide” along the switching surface; this results in a slidingmode behavior, or (ii) hysteresis switching, in order to, notonly approximate a sliding mode, but also to maintain theproperty that two consecutive switching events are alwaysseparated by a time interval of positive length.

The proposed OGA policy dictates that a switch amongvector fields fk may occur when ‖ri(t) − rj(t)‖ ≤ Rc, i.e.,when the multi-robot system trajectories r(t) hit the surfaceSc(t) :=

{r ∈ R2 N | ‖ri(t)− rj(t)‖ = Rc

}. Denote td the

time instant when the system trajectories lie on the surfaceSc(td) and the agents are involved in the decision making,t−d , t+d the time instants before and after the decision making,respectively, and k

(t−d), k(t+d)

the assignment before andafter the decision making, respectively. The decision makingon Sc(td) results in two different cases:

1) The agents decide to keep their goal assignment, i.e.k(t−d)

= k(t+d), in which case no switching occurs.

2) The agents decide to swap goals, i.e. k(t−d)6= k

(t+d)

and a switching occurs.

Theorem 4: The switched multi-robot system (8) under theOGA policy does not suffer from Zeno points.

Proof: We employ the results in [34], Theorem 2. Let usassume that the switched multi-robot system (8) has a Zenopoint r. Then it holds that: r ∈ Sc(td), r is an accumulationpoint of the set S = {r ∈ Sc(td) : fk

(t−d)

= fk(t+d)}, and

satisfies: ∇Sc(r)fk(t−d)

= ∇Scfk(t+d)

= 0, where td is adecision making and switching time instant.

Denote rG1=[xG1

yG1

]T, rG2

=[xG2

yG2

]Twhere

rG16= rG2

the position vectors of the goal locations G1, G2,assigned to the agents i, j respectively, at time t−d . The agentsimplement the OGA policy at time td and decide to swap goals.Then, the corresponding vector fields read:

fk(t−d)

=

[−λi(ri − rG1)−λj(rj − rG2)

], fk

(t+d)

=

[−λi(ri − rG2)−λj(rj − rG1)

],

while ∇Sc(td) =[rij

T −rijT], where rij = (ri−rj). It is

easy to verify that S = ∅, since fk(t−d)

= fk(t+d)⇒ rG1

=rG2

, a contradiction. Thus, the set of accumulation points ofS is empty, which implies that no Zeno points r are containedthere, a contradiction. Thus, the OGA policy does not sufferfrom Zeno points.

This result establishes that the decision making under theOGA renders trajectories that do not accumulate on, i.e., alwaysescape, the switching surface Sc(td); hence the robots do notget stuck away from their assigned goals. Finally, to maintainthe property that there always will be a positive time intervalτ > 0 between consecutive switching times, we implement ahysteresis-like switching logic; the idea is that after the (i, j)decision making at time td and while connected, the agentsi, j do not get involved in a new (i, j) decision making,unless the agents have become disconnected first. Note thatin addition, this hysteresis logic validates our Assumption 1on finite amount of switches per unit time interval.

D. The N > 2 Robot Case

The results on the stability and non-Zeno behavior of themulti-robot system extend to the case of N > 2 agents. Con-sider N agents that need to converge to N goal locations. Thisgives rise to N! possible goal assignments k ∈ {1, 2, . . . ,N!},or in other words, N! candidate switched subsystems fk forthe switched multi-robot system:

r(t) = fk(r(t)), (9)

where: r =

r1...rN

, fk =

u(k)1...

u(k)N

, k ∈ K = {1, . . . ,N!}.

For N > 2 agents the decision making on the optimal goalassignment involves the Nc ≤ N connected agents at time td.The agents exchange information on their positions and goallocations at time td, compare the values of the Nc! Lyapunovfunctions V (k)(td), where k ∈ Kc(td) ⊆ K, and pick the goalassignment k which corresponds to min{V (k)(td)}.

1) Stability Analysis: Consider the sequence of switchingtimes T = {t0, t1, . . . , tn, . . .} and the switching sequenceΣ = {r0; (k0, t0), (k1, t1), . . . , (kn, tn), . . .}, kn ∈ K. Then,the reasoning and analysis followed in Theorem 3 apply to theN > 2 case as well.

Theorem 5: The trajectories r(t) of the switched multi-robot system (9) are GAS w.r.t. the goal assignment k.

Proof: We consider the candidate Lyapunov-like func-tions V (k), k ∈ {1, . . . ,N!}, encoding the motion of the agentsunder goal assignment k. Out of Lemma 1, each individual k-th subsystem, i.e. the motion of the multi-robot system underthe k-th assignment, is GES. This implies that each V (k)

is decreasing on the time intervals that the k-th subsystemis active. Furthermore, the OGA policy dictates that at theswitching times {t0, t1, t2, t3, . . . , } one has:

V (k(t0))(t0) > V (k(t1))(t1) > V (k(t2))(t2) > . . . ,

where k(tn) denotes the k-th subsystem that becomes active attime tn, n ∈ N; in other words, the value of each Lyapunov-like function V (k) at the beginning of the time intervals whenthe k-th subsystem becomes active satisfies the decreasing con-dition, i.e., the switched system is Lyapunov stable. To drawconclusions on the asymptotic stability, consider any pair ofswitching times tn1 < tn2 when the k-th subsystem becomesactive, the corresponding time intervals [tn1 , tm1), [tn2 , tm2)while the k-th subsystem remains active, with tm1

< tn2, and

follow the reasoning as in Theorem 3, i.e.:

V (k)(tn1) > V (k)(tm1),since the subsystem k is

active on [tn1 , tm1) and GES

V (k)(tm1) ≥ V (l)(tm1

), out of the OGA Policy, l 6= k

V (l)(tm1) ≥ V (k)(tn2

), out of the OGA Policy

V (k)(tn2) > V (k)(tm2

),since the subsystem k is

active on [tn2, tm2

) and GES

which imply that: V (k)(tm2) < V (k)(tm1), i.e., that:

V (k)(tm2) = ρkV(k)(tm2), 0 < ρk < 1. Then:

V (k)(tm2)− V (k)(tm1) = −(1− ρk) V (k)(tm1),

where 1−ρk > 0. Following the same reasoning as in Theorem3, we have that the switched system is GAS.

2) Non-Zeno Behavior: Finally, following the same patternas for the N = 2 case, one may verify that:

Theorem 6: The switched multi-robot system (9) under theOGA policy does not suffer from Zeno behavior.

Proof: Assume that the switched system (9) has a Zenopoint r. Then, r ∈ Sc(td) and r is an accumulation point ofthe set S = {r ∈ Sc(td) : fk

(t−d)

= fk(t+d)}, where Sc(td)

is the switching surface at the decision and switching time td.The OGA policy dictates that at least one pairwise goal swapoccurs at time td, since if the agents decided to keep the goalsthey had at time t−d , then td would not have been a switchingtime, a contradiction. Since at least one pair of agents (i, j)switches goal locations, denoted as rGi , rGj , the conditionfk(t−d)

= fk(t+d)

on the switched vector fields holds trueonly when rGi = rGj , see the analysis in Theorem 4, i.e.,

j ir

2j =

1i =

cR

A

B

jδ

iδ

jδ⊻ iδ

⊻iφ

jφ

Fig. 2. Collision may occur under the OGA policy.

a contradiction by construction. Thus, the set S = ∅, whichfurthermore implies that the set of its accumulation points isempty, implying that no Zeno points can be contained there.Thus, the switched multi-robot system (9) does not exhibitZeno behavior.

V. COLLISION AVOIDANCE POLICY

The OGA policy does not ensure that inter-agent collisions,realized as keeping dij(t) ≥ ds, where ds is a safety distance,are avoided ∀t ∈ [0,∞). To verify this, consider the scenarioin Fig. 2 and assume that the agents move under k = 1, withagent i assigned to goal A and agent j assigned to goal B. Atsome time instant td, one has dij(td) = ‖ri(td) − rj(td)‖ =Rc, and the agents run the OGA policy. The total distance-to-go under k = 1 is δ(td) = δi(td) + δj(td), while underk = 2 the total distance-to-go is δ?(td) = δ?i (td) + δ?j (td).Since δ(td) > δ?(td) the agents switch to k = 2, with agent jmoving to goal A and agent i moving to goal B. If, however,agent j moves relatively fast to its goal location compared toagent i, collision may occur.

Problem 2: Establish (sufficient) conditions under whichthe OGA policy is collision-free.

A. Detecting Conflicts

We are referring to time t > td, i.e. after Nc ≤ N connectedagents have decided on a goal assignment k based on theOGA policy, and move towards their goal locations r(k)gi . Inthe sequel we drop the notation ·(k), in the sense that the goalassignment k is kept fixed.

We would like to identify a metric (a “supervisor”) deter-mining whether the OGA policy results in collisions. Let usconsider the collision avoidance constraint:

cij(ri, rj) = (xi − xj)2 + (yi − yj)2 − ds2 > 0, (10)

encoding that the inter-agent distance dij = ‖ri − rj‖ shouldalways remain greater than a safety distance ds.

To facilitate the analysis using Lyapunov-like approaches,we encode the constraint (10) as a Lyapunov-like function.

Recall that a barrier function [35] is a continuous functionwhose value on a point increases to infinity as the pointapproaches the boundary of the feasible region; therefore, abarrier function is used as a penalizing term for violationsof constraints. The concept of Recentered Barrier Function(RBF) in particular was introduced in [36] in order to not onlyregulate the solution to lie in the interior of the constrainedset, but also to ensure that, if the system converges, then itconverges to a desired point.

In this respect, we define the logarithmic barrier functionbij(·) : R2×R2 → R for the constraint (10) as:

bij(ri, rj) = − ln (cij(ri, rj)) , (11)

which tends to +∞ as cij(·)→ 0, i.e. as dij → ds. The RBFof (11) is defined as [36]:

rij = bij(ri, rj)− bij(rgi , rj)−∇bijT∣∣rgi

(ri − rgi), (12)

where rgi =[xgi ygi

]Tis the goal location for agent i,

bij(rgi , rj) is the value of (11) evaluated at rgi , ∇bij =[∂bij∂xi

∂bij∂yi

]Tis the gradient vector of the function bij(·), and

∇bijT∣∣rgi

is the transpose of the gradient vector, evaluated atrgi . To ensure that we have a nonnegative function to be used

as a Lyapunov-like function, we define [28]:

wij(·) = (rij(ri, rj , rgi))2, (13)

which now is a positive definite function. To furthermoreencode that the position trajectories of agent i remain boundedin a prescribed region (for reasons that will be explained inthe technical analysis later on), for each agent i we define theworkspace constraint:

ci0(ri, r0) = R20 − (xi − x0)2 − (yi − y0)2 > 0, (14)

which encodes that the position ri =[xi yi

]Tof agent

i should always lie in the interior of a circle of centerr0 =

[x0 y0

]Tand radius R0 > Rc. This can be thought

as the workspace where the agents operate. Then, in the samereasoning followed before, we define the barrier function:

bi0(ri, r0) = − ln (ci0(ri, r0)) , (15)

and its corresponding RBF:

ri0 = bi0(ri, r0)− bi0(rgi , r0)−∇bi0T∣∣rgi

(ri − rgi), (16)

which vanishes only at the goal location rgi and tends toinfinity as the position ri tends to the workspace boundary.To get a positive definite function we consider:

wi0(·) = (ri0(ri, r0, rgi))2. (17)

Therefore, an encoding that agent i stays ds away fromall its neighbor agents j, while staying within the boundedworkspace, can be now given by an approximation of themaximum function of the form:

wi =

(wi0)δ +∑j∈Ni

(wij)δ

1δ

, (18)

where δ ∈ [1,∞), and Ni ⊆ N is the set of connected(neighbor) agents j 6= i of agent i. The function wi vanishesat the goal location rgi and tends to +∞ as at least one ofthe terms wi0, wij tends to +∞, i.e., as at least one of theinter-agent distances dij tends to ds, or as agent i tends tothe workspace boundary. Finally, to ensure that we have aLyapunov-like function for agent i which uniformly attains itsmaximum value on constraints’ boundaries we take:

Wi =wi

1 + wi, (19)

which is zero for wi = 0, i.e., at the goal position rgi ofagent i, and equal to 1 as wi → ∞. For more details on theanalytical construction the reader is referred to [28], [37].

The level sets of the Lyapunov-like function (19) can nowbe used as a sufficient criterion on determining whether thecontrol inputs out of the OGA policy jeopardize safety. Let usconsider the time derivative of (19):

Wi =[∂Wi

∂xi∂Wi

∂yi

] [xiyi

]+∑j∈Ni

([∂Wi

∂xj∂Wi

∂yj

] [xjyj

])=

= ζiTui +

∑j∈Ni

(ζij

Tuj

), (20)

where ζi ,[∂Wi

∂xi∂Wi

∂yi

]T, ζij ,

[∂Wi

∂xj∂Wi

∂yj

]T. This time

derivative is defined almost everywhere, except for a set ofmeasure zero corresponding to the case when the gradient term∇bij |rgi is undefined (namely, this corresponds to rj = rgi ).Not surprisingly, the evolution of the time derivative (20)along the trajectories of agent i depends not only on its ownmotion (through ui), but also on the motion of its neighboragents j 6= i through their velocities uj . This time derivativeprovides sufficient conditions on establishing safety under theOGA policy. To see how, let us consider the following Lemma.

Lemma 2: If the control inputs ui, uj , where j ∈ Ni, outof the OGA policy satisfy the following condition for all i ∈{1, . . . ,N}:

ζiTui +

∑j∈Ni

(ζij

Tuj

)< 0, (21)

then the OGA policy is asymptotically stable w.r.t. the currentgoal assignment k and furthermore collision-free.

Proof: To verify the argument, consider the properties ofthe Lyapunov-like function (19) and denote the constrainedset for each agent i as Ki = {r ∈ R2 N | cij(·) ≥ 0}, wherej ∈ Ni ∪ {0}. The set Ki is by construction compact (i.e.,closed and bounded), with the level sets of Wi being closedcurves contained in the set Ki. Then, the condition (21) impliesthat the system trajectories ri(t) under the OGA control inputui evolve along the negative direction of the gradient vector∇Wi =

[∂Wi

∂xi∂Wi

∂yi

]of Wi, i.e. always remain in the interior

of the constrained set Ki; this further reads that the inter-agentdistances dij(t) never become smaller than ds.

Problem 3: Seek a policy that becomes active when theOGA policy is about to result in colliding trajectories, andensures that the inter-agent distances dij remain greater thanthe critical distance ds.

We develop the CA policy described in the following section.

B. Resolving Conflicts

Let us consider the switching surface:

Qi = ζiTui +

∑j∈Ni

(ζij

Tuj

). (22)

Then for Qi < 0 one has out of Lemma 2 that the OGApolicy is both GAS and collision-free. For Qi > 0 one hasWi > 0, which implies that the position trajectories ri(t)evolve along the positive direction of the gradient vector ∇Wi.This condition may jeopardize safety and dictates the definitionof an additional policy to ensure that the trajectories ri(t)remain in the constrained set Ki.

At this point, one may be tempted to use a CA controllaw for agent i along the gradient-descent direction of Wi, torender the trajectories of the multi-robot system collision-freein the set Qi > 0. However, such control laws typically growunbounded for system trajectories approaching local minimumpoints, i.e., as ‖∇Wi‖ → 0. This may lead to implementationissues when it comes to realistic robotic agents and thereforeis undesirable from a practical point-of-view.

We thus emulate the gradient descent performance alongthe level sets of (19) with control inputs that are always well-defined. We define a vector field Fi : R2 → R2 to be used as afeedback motion plan for each agent i, in the spirit developedin our earlier work [29]. The analytical expression is given as:

Fi =∏j∈N〉

(1− σij)Fig +∑j∈N〉

σijFioj , (23)

where:

σij =

1, for ds ≤ dij ≤ Rs;a dij

3 + b dij2 + c dij + d, for Rs < dij < Rc;

0, for dij ≥ Rc;(24)

the attractive to the goal position rgi =[xgi ygi

]Tvector

field Fig is defined as:

Fixg = − x− xgi√(x− xgi)2 + (y − ygi)2

, (25a)

Fiyg = − y − ygi√(x− xgi)2 + (y − ygi)2

, (25b)

the repulsive vector field Fioj around each agent j 6= i isdefined as:

Fixoj =xi − xj√

(xi − xj)2 + (yi − yj)2, (26a)

Fiyoj =yi − yj√

(xi − xj)2 + (yi − yj)2, (26b)

Rs the radius of the circle where the repulsive vector field isactive, the coefficients a, b, c and d have been computed as:

a = − 2

(Rs −Rc)3, b =

3(Rs +Rc)

(Rs −Rc)3,

c = − 6 RsRc(Rs −Rc)3

, d =Rc

2(3Rc −Rs)(Rs −Rc)3

,

so that (24) is a C1 function. The vector field (23) for eachagent i has integral curves that are:

1) convergent to the goal location rgi as long as dij > Rcfor all j 6= i, i.e., when no agent j lies in the communi-cation region Ci of agent i, and

2) repulsive w.r.t. agents j which lie within distance dij <Rs w.r.t. agent i, i.e., within the communication regionCi of agent i.

Most importantly, the integral curves of (23) are always well-defined within the constrained set Ki of agent i, in contrastto the gradient vector field out of the Lyapunov-like barrierfunction (19) that is undefined at the points where the function(19) is not defined. The vector field (23) vanishes only atthe goal location rgi , which is its unique singular point. Inthis respect, the vector field (23) can be used as a referencefeedback motion plan dictating a safe direction of motionφi = atan2(Fiy,F

ix) for agent i, where Fix, Fiy are the

components of the vector field (23). Now, in the ideal case of astatic environment, i.e., when agents j 6= i are not moving, thecontrol policy on the linear speed ui of agent i can naturallybe independent of the linear speeds uj of other agents j 6= i, inthe sense that these are identically zero. In the practical caseof dynamic environments however, i.e., when agents j 6= iare moving, the control policy on the magnitude ui of thelinear velocity vector ui of each agent i should take intoaccount the motion of neighbor agents as well, so that theinter-agent distances dij(t) are guaranteed to remain greaterthan the lower bound ds, ∀t ≥ 0.

We use the following velocity coordination protocol togovern the speed ui for each agent i:

Lemma 3: Let Nc ≤ N agents become connected at sometime instant td ≥ 0 while moving under some goal assignmentk ∈ {1, . . . ,Nc!}. The motion of each agent i in the connectedgroup is collision-free under the control law:

ui =

(min

j∈Ni|Ji<0{ui|j}

)ηi, (27)

where: ηi =[cosφi sinφi

]Tis the unit vector along the

direction of the vector field (23), rij = ri− rj , Ji = rijTui,

uis|j =rij

TujrijTηi

, 0 < εi < 1, and

ui|j = uidij − dsRc − ds

+ εirij

TujrijTηi︸ ︷︷ ︸uis|j

Rc − dijRc − ds

. (28)

Remark 5: In effect, the control law (27) forces agent ito adjust its speed w.r.t. the worst-case neighbor j ∈ Ni, i.e.,w.r.t. the neighbor j towards whom the rate of change of inter-agent distance dij is maximized via the motion of agent i.

Proof: Our goal is to show that the constrained set Kiencoded via the outermost level set of the Lyapunov-likebarrier (19) is rendered a positively invariant set under thecontrol law (27). We employ the fundamental characterizationof positively invariant sets [27] through the necessary and suf-ficient conditions of Nagumo’s theorem on the boundary of theconstrained set Ki of each agent i, namely: d

dt cij(ri, rj) ≥0, ∀rj ∈ ∂Ki, ∀(i, j).

Consider that at some time td ≥ 0, the distance dij(td)between a pair of connected agents (i, j) is dij(td) ≤ Rc.

Fig. 3. If Jj , rijTηi < 0, i.e., if agent i moves towards agent j, then

agent i adjusts its linear speed according to the velocity profile shown here,given analytically by (27).

Consider the time derivative of the inter-agent distance, whichafter some calculations reads:

d

dtdij

(1)=rij

T (ui − uj)dij

. (29)

Under the control law (27), agent i adjusts its linear speedui according to the velocity profile shown in Fig. 3, givenanalytically out of (27), so that the distance dij w.r.t. theworst case neighbor j remains greater than ds. The worst caseneighbor is the agent j ∈ {Ni | Ji < 0} towards whom therate of change of relative distance dij given by (29), due tothe motion of agent i, is maximum. More specifically: Theterm j ∈ {Ni | Ji < 0} describes the set of neighbor agentsj of agent i towards whom agent i is moving [28]. Agenti computes safe velocities ui|j given out of (28) w.r.t. eachneighbor in this set, and picks the minimum min{ui|j} of thesafe velocities ui|j so that the first term in (29) is as small aspossible. Now, the value of the safe velocity ui|j (28) whendij = ds is equal to εiuis|j = εi

rijTuj

rijTηi. Plugging this value

into (29) reads:

dij∣∣dij=ds

=(εi − 1)uj rij

Tηjdij

≥ 0.

To see why this condition is true, recall that εi−1 < 0, uj > 0,and rijTηj ≤ 0: this is because agent j is either following avector field Fj that points away from agent i, or happens tomove away from agent i in the first place. This implies thatthe inter-agent distance dij can not become less than ds, i.e.,that collisions are avoided.

C. Ensuring both Stability and Safety

We now define the switching policy that realizes thecollision-free motion for each agent i as follows:

Let Nc ≤ N agents become connected at some timeinstant td ≥ 0 while moving under some goal assignmentk ∈ {1, . . . ,Nc!}. Define

Ri(td) = cos(θi(td)) cos(φi(td))

the decision-making surface for agent i, where θi(td) is thedirection of the velocity vector ui at time td given out of theOGA controller (2), and φi(td) is the direction of the vectorfield vector Fi at time td, given out of (23). Then:• If Ri(td) > 0, the trajectories under the OGA policy are

safe, and agent i keeps moving under (2).• If Ri(td) ≤ 0, the trajectories out of the OGA policy

might violate safety, and agent i moves under (27).The switching between the OGA policy and the CA policy

across the switching surface Ri = 0 gives rise to closed-loopdynamics with discontinuous right-hand side for each agent i:

ri(t) = v(p)i , (30)

where p ∈ P = {1, 2}, and v(p)i is the closed-loop dynamicsof agent i under the OGA policy (2) and the CA policy (27),respectively. More specifically, for p = 1, agent i moves underthe OGA policy, while for p = 2, agent i moves under the CApolicy. We formalize the switching logic between the resultingclosed-loop subsystems via a hysteresis technique, similar tothe one used in Section IV-C; i.e., after making the decision be-tween either subsystem p = 1 or subsystem p = 2, the agentsdo not get involved in a new decision making unless theybecome disconnected first. The hysteresis logic gives rise toa switching sequence of times T = {τ1, τ2, τ3, . . . , τn, . . . , },where the time interval τn− τn−1 is finite ∀n ∈ N, and to theswitching sequence:

Σ? = {ri(τ0); (1, τ0), (2, τ1), . . . , (1, τ2q), (2, τ2q+1), . . . , },

where τ0 > td, q ∈ N, τ2q are the time instants when thesubsystem p = 1 (OGA policy) is “switched on”, τ2q+1 arethe time instants when the subsystem p = 2 (CA policy) is“switched on”.

Problem 4: Establish that the proposed switching strategyΣ? between the OGA policy and the CA policy renders themulti-robot trajectories asymptotically stable w.r.t. to the as-signed goals, and also collision-free.

For the stability analysis of this switching strategy, it isinefficient to employ Branicky’s decreasing condition on the“switched-on” time intervals of each individual subsystem; themain reason is that in principle the CA strategy might resultin trajectories that either are not monotonically decreasing, orare even increasing with a bounded growth for a finite amountof switches. Therefore we resort to results on the stability ofswitched systems that remove this condition in order to drawconclusions on asymptotic stability.

Theorem 7: The trajectories of the switched multi-robotsystem under the switching logic Σ? are (i) collision-free, and(ii) asymptotically stable w.r.t. the assigned goals.

Proof: The first argument is proved in Lemmas 2 and3. The second argument can be verified by directly applyingTheorem 2. To this end, let us first note that Lemma 1 impliesthat the function Vi for each agent i serves as generalizedLyapunov-like function for the subsystem p = 1. Furthermore,Wi is a valid generalized Lyapunov-like function almosteverywhere for each agent i for the subsystem p = 2. Tosee why, consider that the vector field (23) is by constructiontransverse to the boundary of the constrained set Ki, and

convergent to the goal location rgi , except for a set of initialconditions of measure zero [29]. Hence, during the timeintervals when the subsystem p = 2 is active, the growth,if any, of the function Wi along the system trajectory isbounded. Furthermore, the stability condition (6) for eachsubsystem p is satisfied out of the boundedness of the solutionsri(t) within the constrained set Ki, which is by constructionclosed and bounded. Therefore, the switched system is stable.To establish asymptotic stability, it should hold that for atleast one of the individual subsystems p, the value of thecorresponding generalized Lyapunov-like function decreasesalong the sequence of switching times. Let us assume thatthis is not the case; then this would imply that the closed-loopswitched vector fields v(p)i , p ∈ {1, 2}, “oppose” each otherand cancel out on the switching surface Ri, forcing the systemtrajectories to get stuck there. In other words, this wouldimply that the switching policy Σ? exhibits Zeno trajectories.A Zeno point r ∈ Sc(td) would be an accumulation pointof the set S = {r ∈ Sc(td) : v(p)

(t−d)

= v(p)(t+d)}, and

satisfies: ∇Sc(r)v(p)(t−d)

= ∇Scv(p)(t+d)

= 0, where tdis a decision making and switching time instant. It is easyto verify that the second Zeno condition after some algebrareads: (ui+uj)‖rij‖ = 0, which for positive speeds holds trueonly when both agents have reached at their goal locations.Hence the switching strategy between the OGA policy and theCA policy does not exhibit Zeno trajectories. Therefore, theswitched multi-robot system under the switching strategy Σ?

is asymptotically stable.

D. Robustness against Communication Failures

We note that establishing robustness against uncertainty(e.g., against sensor noise, communication dropouts and/ordelays) is beyond the scope of the current paper. As a specialcase, we only consider how the case of failed communicationcan be handled within the centralized version of the proposedapproach. Note that finding the set of initial conditions thatresult in collision-free trajectories for a given distribution ofgoals and for every possible decentralized assignment is essen-tially a backwards reachability problem, which is intractablefor even low-dimensional systems, i.e., for teams of no morethan 2-3 robots. However, the centralized case as treated inthe earlier work of the authors [18] offers a (conservative,yet guaranteed) robustness measure for the case of failedcommunication in a decentralized setting; more specifically,we have the following corollary:

Corollary 1: Let the multi-robot system under switchedagent dynamics given by (30), and the switching sequence Σ?.If for all ri(0), rj(0), rGi , rGj , i, j ∈ {1, . . . , N}, it holds that‖ri(0)− rj(0)‖ ≥ 2R

√2, and ‖rGi − rGj‖ ≥ 2R

√2, where

R is the radius of the robots, then under failed communicationthe multi-robot trajectories are collision-free.

Proof: The corollary states that if the initial robot po-sitions and goal locations are 2R

√2 apart, then the result-

ing multi-robot trajectories are collision-free, even when theagents’ communication fail and no goal swap or collisionavoidance controllers are implemented. Note that under failedcommunication, the affected agents will not be involved in any

decision on re-assigning their goals (i.e., will keep their currentgoals), and will not update their control laws. Hence, whetherthe trajectories will be collision-free depends on the initialconditions (positions) of the robots and the spatial distributionof the goal locations. The analysis of the centralized versionof the OGA policy in [18] proves that under the stated assump-tions, the OGA policy results in collision-free trajectories. Thiscompletes the proof.

VI. SIMULATION RESULTS

A. Simulations for N = 15 and N = 40 agentsSimulation results are provided to evaluate the performance

of the switched multi-robot system under the proposed switch-ing logic and control policies. In the interest of space, andgiven that the representation of large groups of robots results inrather congested figures, here we consider only two scenariosinvolving N = 15 and N = 40 agents, respectively.

The initial positions and the assigned goal locations areselected in very close proximity so that the OGA policy on itsown is not sufficient to ensure collision-free trajectories, seeFig. 4 and 5, respectively. The agents start from the initial con-ditions shown in Fig. 4(a), 5(a) towards goal locations markedwith “x”; as discussed earlier, the initial goal assignment isnot necessarily the optimal one. The communication rangesare denoted by the rings around each agent, and set equal toRc = 4R, where R is the radius of the agents. The straightline paths from the starting locations to the initially assignedgoals are depicted as the dotted lines, whereas the actual pathsfollowed by the switching strategy are shown in Fig. 4(b) and5(b), respectively. The final locations of the robots are shownin Fig. 4(c) and 5(c), respectively. As indicated by the colorsof the initially assigned goals, the followed paths, and the finalgoal locations of the robots, the robots performed goal swapsduring the duration of the simulation under the OGA policy,while they also moved under the CA policy during the pathsegments that are not straight. Figures 6 and 7, respectively,show the minimum distance between any two pairs of agentsat each time instance of the simulations. As this distance isalways greater than the safety distance ds, there is never acollision between any robots.

Additional videos with teams of 6, 30, 80, 100, and 150agents are available through the following links:https://www.dropbox.com/s/a5hykza2wla7x3n/6.mp4?dl=0https://www.dropbox.com/s/1gazjlt7nell0mn/30.avi?dl=0https://www.dropbox.com/s/48bnwsnlqibkgaj/80.avi?dl=0https://www.dropbox.com/s/e6wljow9ft2ziap/100.avi?dl=0https://www.dropbox.com/s/uif0yahjlhv0bsf/150.avi?dl=0

It is noteworthy that the simulation videos demonstrate theexpected performance of the overall algorithm in terms of theconvergence rate; more specifically, the slower convergenceof the agents to their goal locations towards the end of thesimulation, compared to their faster motion in the beginning ofthe simulation, is consistent with the exponential convergencerate of the OGA policy. 2

2Note however that the videos run at different frames-per-second or havedeliberate pauses to explain the algorithm, and as thus the time lengths ofthe videos shall not be used for a direct comparison of the convergence timeamong the cases of different N.

-15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

(a) Initial positions of the agents and the straight paths to the goalsdetermined by the random initial goal assignment.

-10 -5 0 5 10 15

-15

-10

-5

0

5

(b) The actual paths to the goals under the proposed control strategy.

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

(c) The final locations of the agents.

Fig. 4. A simulation with N = 15 robots. The paths followed are straightwhen in the OGA segments, but are potentially curved when using the CAcontrol law.

-20 -10 0 10 20 30

-20

-15

-10

-5

0

5

10

15

20

(a) Initial positions of the agents and the straight paths to the goalsdetermined by the random initial goal assignment.

-20 -15 -10 -5 0 5 10 15 20 25 30

-20

-15

-10

-5

0

5

10

15

(b) The actual paths to the goals under the proposed control strategy.

-20 -10 0 10 20 30-25

-20

-15

-10

-5

0

5

10

15

20

(c) The final locations of the agents.

Fig. 5. A simulation with N = 40 robots. The paths followed are straightwhen in the OGA segments, but are potentially curved when using the CAcontrol law.

Time [sec]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Min

imum

Pai

rwis

e In

ter-

agen

t Dis

tanc

e d

ij

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

Minimun Pairwise Interagent DistanceSafety Distance

Fig. 6. The minimum clearance between any two robots for a simulated trialwith N = 15, where clearance is defined as the free space between robots.Note that as this is always positive, there were never any collisions betweenrobots.

Time [sec]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Min

imum

Pai

rwis

e In

ter-

Age

nt D

ista

nce

dij

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

Minimum Pairwise Interagent DistanceSafety Distance

Fig. 7. The minimum clearance between any two robots for a simulated trialwith N = 40, where clearance is defined as the free space between robots.Note that as this is always positive, there were never any collisions betweenrobots.

B. Notes on communication requirements and planning time

The implementation of the proposed algorithm in a realisticsetting would require the capability of exchanging messages(e.g., via wireless communication links) among the subsetof agents that are considered connected at some time in-stance, i.e., among robots that lie within Rc distance andtheir neighbor agents. Once a connected group is formed,the OGA policy would run in a synchronous manner. Recallthat in our approach the initial assignment is random, i.e.,the focus of this work is not on determining the globallyoptimal assignment of the robots to the goals, but insteadwe focus on re-assigning goals among connected robots.Thus, since the decision-making mechanism (i.e., the OGApolicy) on re-determining the goal assignment is triggeredwhenever (a subset of) agents become connected, it followsthat the number of re-plan operations, and hence the number

of messages that are exchanged among the connected agentson deciding whether they swap goals, depends greatly on thecommunication range, the number of robots, and their initialdistribution, i.e., the quality (in terms of sub-optimality) ofthe initial assignment. Note that for sufficiently large radiusRc, which results in all agents being connected at the sametime, the centralized case is recovered, meaning that the agentsdecide among N! assignments.

The time for a re-plan operation also directly depends onthe number of agents in the connected component. Planningtimes on simulated teams of Nc robots using a single computertake approximately 10−7Nc3 seconds, which reads that acomponent of 100 connected robots takes about 0.1 seconds tore-plan. The required computational time is typically sufficientfor the proposed multi-robot applications.

C. Comparison with earlier work in [24]

As discussed earlier, the current approach dominates ourearlier work [24] in the following aspects: (i) The initialpositions of the robots and the goal locations can be close asds = 2R, in contrast to [24] that requires the initial locationsand goals to be 2R

√2 apart for safety. (ii) The collision-free

trajectories of the current approach are rendered via switchingamong two control policies, opposed to [24] that involvesswitching among three control policies. The approach herenevertheless does not consider the initially optimal assignment,and as thus, depending on the connected subcomponentsat initial time and throughout the evolution of the system,might result in suboptimal paths in terms of total distance.For completeness, we include simulation results between thecurrent approach and [24]; in the scenario illustrated in Fig. 8,the resulting paths are suboptimal; the total computational timefor the entire simulation in the former case is was 2.625 sec,while in the latter case was 1.375 sec. This demonstrates thateven with suboptimal paths, the algorithm is computationallyefficient for the considered applications.

VII. CONCLUDING REMARKS

We presented a switched systems approach on the decen-tralized concurrent goal assignment and trajectory generationfor multi-robot networks which guarantees safety and globalstability to interchangeable goal locations. The proposedswitching logic relies on multiple Lyapunov-like functions thatencode goal swap among locally connected agents based onthe total traveled distance, avoidance of inter-agent collisions,and convergence to the assigned goal locations. As such,the proposed methodology renders feedback control policieswith local coordination only, and therefore is suitable forapplications such as in first-response deployment of roboticnetworks under limited information sharing. Our future workwill focus on agents with more complicated dynamics, as wellas on the robustness of our algorithms under communicationfailures and uncertainty.

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Dimitra Panagou received the Diploma and PhDdegrees in Mechanical Engineering from the Na-tional Technical University of Athens, Greece, in2006 and 2012, respectively. Since September 2014she has been an Assistant Professor with the De-partment of Aerospace Engineering, University ofMichigan. Prior to joining the University of Michi-gan, Dr. Panagou was a postdoctoral research as-sociate with the Coordinated Science Laboratory,University of Illinois, Urbana-Champaign (2012-2014), a visiting research scholar with the GRASP

Lab, University of Pennsylvania (June 2013, fall 2010) and a visiting researchscholar with the University of Delaware, Mechanical Engineering Department(spring 2009). Her research interests include the fields of planning, coor-dination, and distributed control and estimation for complex systems, withapplications in unmanned aerial systems, robotic networks and autonomousmulti-vehicle systems (ground, marine, aerial, space). She was a recipient ofthe NASA 2016 Early Career Faculty Award and of the Air Force Office ofScience Research 2017 Young Investigator Award.

Matthew Turpin received his BS degree in Me-chanical Engineering from Northwestern University,and his PhD degree in Mechanical Engineeringfrom University of Pennsylvania. During his doctoralstudies at the University of Pennsylvania he focusedon formation control as well as on the couplingof task assignment and motion planning in multi-robot systems. He is currently a research scientist atQualcomm Technologies Inc. where he has workedon research and development of software for multi-rotor MAVs, and currently works on planning and

control for autonomous cars.

Vijay Kumar is the Nemirovsky Family Dean ofPenn Engineering with appointments in the De-partments of Mechanical Engineering and AppliedMechanics, Computer and Information Science, andElectrical and Systems Engineering at the Univer-sity of Pennsylvania. He served as the assistantdirector of robotics and cyber physical systems atthe White House Office of Science and TechnologyPolicy (2012-2014) and is the founder of ExynTechnologies, a company that develops solutions forautonomous flight.

Dr. Kumar is a Fellow of the American Society of Mechanical Engineers(2003), a Fellow of the Institute of Electrical and Electronic Engineers (2005)and a member of the National Academy of Engineering (2013). Among hisrecent awards are the 2012 World Technology Network award, a 2013 PopularScience Breakthrough Award, a 2014 Engelberger Robotics Award, the 2017IEEE Robotics and Automation Society George Saridis Leadership Award inRobotics and Automation and the 2017 ASME Robert E. Abbott Award. Hehas won best paper awards at DARS 2002, ICRA 2004, ICRA 2011, RSS2011, RSS 2013, and BICT 2015 and has advised doctoral students who havewon Best Student Paper Awards at ICRA 2008, RSS 2009, and DARS 2010.