Constraint-Driven Coordinated Control of Multi-Robot Systems

Gennaro Notomista1 and Magnus Egerstedt2

Abstract— In this paper we present a reformulation–framedas a constrained optimization problem–of multi-robot taskswhich are encoded through a cost function that is to beminimized. The advantages of this approach are multiple.The constraint-based formulation provides a natural way ofenabling long-term robot autonomy applications, where re-silience and adaptability to changing environmental conditionsare essential. Moreover, under certain assumptions on the costfunction, the resulting controller is guaranteed to be decen-tralized. Furthermore, finite-time convergence can be achieved,while using local information only, and therefore preserving thedecentralized nature of the algorithm. The developed controlframework has been tested on a team of ground mobile robotsimplementing long-term environmental monitoring.

I. INTRODUCTION

Robotic swarms are gradually leaving academic laborato-ries, e. g. [1], [2], in favor of industrial settings, as in thecase of [3], to reach even less structured and more dynamicenvironments, like agricultural lands and construction sites[4]. This presents new challenges that robots have to face,which come either from unexpected and/or unmodeled phe-nomena or from changing environmental conditions. Theseissues become even more pronounced when the robots aredeployed on the field for long-term applications, like persis-tent environment surveiling [5] or plant growth monitoring[6]. Therefore, a way of encoding survivability [7], i. e., theability to remain alive (in a robotic sense), is needed nowmore than ever. In this paper, we introduce a method whichcan be used to ensure survivability, and which makes use ofoptimization tools that allow to only minimally influence thetask which the robots are asked to perform.

Several solutions have been proposed in order to makerobots robust to unknown or changing environmental con-ditions and to ensure their applicability to unstructured oreven hazardous environments [8]. Moreover, in order to letthe robots survive for as much time as possible, some ofthe proposed methods entail scheduled periodic maintenance[9], or path optimization with the aim of maximizing thetime spent in the field/minimizing the consumed energy[10]. Some other methods employ a power-dependent multi-objective optimization to ensure that the robots execute theirtask, while maintaining a desired energy reserve, as done in[11]. In both cases, a careful parameters tuning is required

*This work was sponsored by the U.S. Office of Naval Research throughGrant No. N00014-15-2115.

1G. Notomista is with the School of Mechanical Engineering, Institute forRobotics and Intelligent Machines, Georgia Institute of Technology, Atlanta,GA, USA g.notomista@gatech.edu

2M. Egerstedt is with the School of Electrical and Computer Engineer-ing, Institute for Robotics and Intelligent Machines, Georgia Institute ofTechnology, Atlanta, GA, USA magnus@gatech.edu

in order to prevent situations in which the robots tradesurvivability for shorter-term rewards.

As a matter of fact, goal-oriented control strategies maynot be ideal for long-term applications, where robustnessto changing environmental conditions is required. Indeed,control policies obtained using optimal control strategiesare characterized by a fragility related to the precise modelassumptions [12]. These are likely to be violated duringthe long time horizons over which the robots are deployedin the field. For this reason, in this paper, we consider aconstraint-oriented approach for multi-robot systems, wherethe survivability of the swarm is enforced as a constraint onthe robots’ task, encoded by the nominal input to the robots,unom. The control input u is, then, synthesized, at each pointin time, by solving the optimization problem

minu‖u− unom‖2 (1)

s.t. csurv(x, u) ≥ 0,

where csurv(·, ·) is the survivability constraint, which is alsoa function of the robots’ state x [7].

It is informative to note that ecological studies have shownthat the constraints imposed by an environment stronglydetermine the behaviors developed by animals living init [13]. Inspired by this concept, we ask whether roboticswarms can be purely constraint-based controlled. What thisentails is that robots are programmed to do nothing, subjectto task and survivability constraints. This is formalized bythe following optimization problem

minu‖u‖2 (2)

s.t. csurv(x, u) ≥ 0

ctask(x, u) ≥ 0,

where ctask(·, ·) encodes the task constraint, which is equiva-lent to executing the nominal input unom in (1). Thus, whatwe could call a robot-ecological formulation [7] naturallylends itself to be implemented using optimization-basedcontrol techniques.

In this paper, we first give sufficient conditions for turningcertain classes of multi-robot tasks into constraints within anoptimization problem. Then, we present a systematic wayof doing this. And, finally, we propose an effective taskprioritization technique obtained by combining hard and softconstraints, such as survivability and task execution in (2).

The remainder of the paper is organized as follows. InSection II, we briefly recall the control techniques thatwill be used to synthesize the constraint-driven coordinatedcontrol policies for multi-robot systems. Then, we introduce

an optimization program for executing minimum-energygradient flow, which will be used, in Section III, to for-malize the constraint-based control of multi-robot systems.Moreover, we show how to achieve decentralized finite-timeminimization algorithms using the presented approach. Twoapplications are presented in Section IV, namely formationcontrol and coverage control. Section V reports the results ofexperiments executed with a team of ground mobile robotsimplementing the proposed controller to execute a long-termenvironmental monitoring task.

II. CONSTRAINT-BASED CONTROL DESIGN

In this section, we review the concepts related to controlLyapunov functions and control barrier functions which willthen be used to formulate optimization problems whosesolution corresponds to the execution of decentralized co-ordinated controllers for multi-robot systems.

A. Control Lyapunov and Control Barrier Functions

In order to design controllers that allow the executionof multi-robot tasks, we make use of control Lyapunovfunctions and, in particular, we resort to methods from finite-time stability theory of dynamical systems.

Consider the dynamical system in control affine form

x = f(x) + g(x)u, (3)

with x ∈ Rn, u ∈ U ⊆ Rm, and f and g locally Lipschitzcontinuous vector fields. One of the results we will use isgiven by the following theorem.

Theorem 1 (Based on Theorem 4.3 in [14]). Given adynamical system (3) and a continuous, positive definitefunction V : Rn → R, a continuous controller u such that

infu∈U{LfV (x) + LgV (x)u+ c(V (x))γ} ≤ 0 ∀x ∈ Rn,

where LfV (x) and LgV (x) denote the Lie derivatives ofV in the directions of f and g, respectively, c > 0 andγ ∈ (0, 1), renders the origin x = 0 finite-time stable.

Moreover, an upper bound for the settling time T is givenby

T ≤ 1

c(1− γ)(V (x0))

1−γ,

where x0 is the value of x(t) at time t = 0.

Proof. Similar to Theorem 4.3 in [14].

To enforce constraints, such as survivability or task ex-ecution for multi-robot systems, we employ control bar-rier functions. These, as will be shown, are suitable forsynthesizing constraints that can be encoded in terms ofset-membership. Conceptually similar to control Lyapunovfunctions, control barrier functions have been introduced in[15] with the objective of ensuring safety in a provablycorrect way. In this context, ensuring safety means ensuringthe forward invariance of a set S ⊂ Rn, in which we wantto confine the state x(t), ∀t ≥ 0. Without loss of generality,here we consider forward complete systems, for which x(t),solution to (3), exists for all t ≥ 0.

Suppose we can find a continuously differentiable functionh : Rn → R, such that the safe set S can be defined as thezero-superlevel set of h, i. e.,

S = {x ∈ Rn | h(x) ≥ 0}, (4)

and

∂S = {x ∈ Rn | h(x) = 0}, S◦ = {x ∈ Rn | h(x) > 0},

where ∂S and S◦ denote the boundary and the interior of S,respectively. If the following condition is satisfied

supu∈U{Lfh(x) + Lgh(x)u+ α(h(x))} ≥ 0 ∀x ∈ Rn, (5)

with α an extended class K function [16], then h is calleda (zeroing) control barrier function. Conditions to ensureforward invariance of the set S are given in the followingtheorem.

Theorem 2 (Safe set forward invariance [17]). Given adynamical system (3) and a set S ⊂ Rn defined by acontinuously differentiable function h as in (4), any Lipschitzcontinuous controller u such that (5) holds renders the setS forward invariant.

Proof. See [17].

The existence of the control barrier function h also ensuresthe asymptotic stability of the set S, as shown in thefollowing theorem.

Theorem 3 (Safe set asymptotic stability [17]). Under thesame hypotheses as in Theorem 2, any Lipschitz continuouscontroller u such that (5) holds renders the set S asymp-totically stable, that means x(t)|t=0 /∈ S ⇒ x(t) →∈ S ast→∞.

Proof. See [17].

B. Minimum-Energy Gradient Flow

In this section we consider the problem of minimizinga cost function J . We present a method to reformulate theclassic gradient flow algorithm using the tools introducedin Section II-A. This allows us to synthesize a constrainedoptimization program that is equivalent to the minimizationof the cost J .

Consider the single integrator dynamical system x = u,where x, u ∈ Rn are the state and the control input, re-spectively. Assume that the objective consists in minimizinga cost J(x), where J : Rn → R+ is a continuouslydifferentiable function. Applying gradient flow algorithms,the problem

minu

J(x) (6)

can be directly minimized by choosing

u = −∂J∂x

T

(x). (7)

In fact, with this choice of input, applying chain rule leadsto:

J(x) =dJ

dt=∂J

∂xx =

∂J

∂xu = −

∥∥∥∥∂J∂x∥∥∥∥2 ≤ 0. (8)

We now show that the minimization problem (6) can beformulated as a minimum-energy problem that achieves thesame objective of minimizing the cost J .

To this end, let us define the barrier function

h(x) = −J(x) (9)

and its zero-superlevel set, i. e. the safe set,

S = {x | h(x) ≥ 0} = {x | J(x) ≤ 0} = {x | J(x) = 0} .

By Theorems 2 and 3, the differential constraint

h(x) =∂h

∂xu ≥ −α(h(x)),

where α is an extended class K function, ensures that the setS is forward invariant when h(x(0)) ≥ 0 and asymptoticallystable when h(x(0)) ≤ 0, x(0) being the value of the statex at time t = 0.

Observation 4. Theorem 3 shows the existence of theLyapunov function

V (x) =

{−h(x) if x /∈ S0 if x ∈ S,

i. e. V (x) ≡ J(x). Indeed, from (8), since J(x) ≥ 0, ∀x ∈Rn, one can see that J(x) is a Lyapunov function. In fact, if xbelongs to X ⊂ Rn compact, LaSalle’s Invariance Principleensures that the state will converge to a stationary point ofJ(x), namely, x→ x∗, with ∂J

∂x (x∗) = 0.

We can now introduce the following optimization problem:

minu,δ‖u‖2 + |δ|2

s.t.∂h

∂xu ≥ −α(h(x))− δ,

(10)

δ ∈ R, which solves the problem in (6), as shown in thefollowing proposition.

Proposition 5. The solution of the optimization problem(10), where h(x) is given by (9) and α is an extended classK function, solves (6), driving the state x of the dynamicalsystem x = u to a stationary point of the cost J .

Proof. The KKT conditions for the problem in (10) are

−∂h∂xu∗ − α(h(x))− δ∗ ≤ 0

λ∗ ≥ 0

λ∗(−∂h∂xu∗ − α(h(x))− δ∗

)= 0[

2u∗

2δ∗

]+ λ∗

−∂h∂xT−1

= 0,

(11)

where u∗, δ∗ and λ∗ are primal and dual optimal points [18].First of all, we note that, if λ∗ = 0, then u∗ = 0 by thefourth equation in (11). Therefore, from the first equation in(11), −α(h(x)) ≤ 0. This is equivalent to −α(−J(x)) ≤ 0and, since J(x) ≥ 0, this implies that J(x) = 0. Incase λ∗ > 0, from the third and fourth equation in (11),

one has λ∗ = −2α(h(x))(1 + ‖∂h∂x‖

2)−1

, and therefore,u∗ = −α(h(x))∂h∂x

T (1 + ‖∂h∂x‖

2)−1

. Since J ≥ 0, J(0) = 0and J is continuously differentiable, one can show that ∂J

∂xvanishes at x = 0. Thus, we can unify the two cases, λ∗ = 0and λ∗ > 0, and write the expression of the optimal u asfollows:

u∗ =α(−J(x))∂J∂x

T

1 + ‖∂J∂x‖2. (12)

With this expression of the input u, the evolution in time ofthe cost J is given by

J =∂J

∂xx =

∂J

∂xu∗ =

α(−J(x))‖∂J∂x‖2

1 + ‖∂J∂x‖2.

So, ∂J/∂x 6= 0⇒ J < 0 and ∂J/∂x = 0⇒ J = 0. Hence,as t→∞, x(t)→ x∗, such that ∂J

∂x (x∗) = 0.

In summary, we saw that the expression for u given in(12) solves the initial optimization problem (6), which canbe equivalently solved following the gradient flow of the costJ using (7).

We now illustrate that, besides the advantages related tolong-term autonomy applications discussed in Section I, theformulation in (10) can be used to design decentralized costminimization algorithms that are faster than gradient descent.In the optimization literature, there are plenty of methodsthat can be employed to improve the convergence speedof gradient flow algorithms (see, e. g., [19]). Nevertheless,these second order methods, such as Newton’s methodor conjugate gradient, suffer from their centralized nature.Only in some cases, this issue can be partially mitigatedby resorting to distributed optimization techniques, such as[20], [21]. The above-mentioned methods are all suitable forminimizing a cost function. However, we insist on having aconstrained optimization formulation–where we encode costminimization as a constraint–because of the flexibility androbustness properties discussed in Section I, useful for long-term robot autonomy applications.

The following proposition shows that, using the formula-tion in (10), it is possible to minimize the cost J and to benot just faster than gradient descent, but actually to reach astationary point in finite time.

Proposition 6. Given the dynamical system x = u and theobjective of minimizing the cost function J , the solution ofthe optimization problem

minu,δ‖u‖2 + |δ|2

s.t.∂h

∂xu ≥ −c(h(x))γ − δ,

where h(x) is given by (9), c > 0 and γ ∈ (0, 1), will drivethe state x to a stationary point of the cost J in finite time.

Proof. Similarly to what has been done in Propositon 5, itcan be shown that

J =−c(J(x))γ‖∂J∂x‖

2

1 + ‖∂J∂x‖2.

Thus, by Theorem 1, we conclude that ∂J/∂x 6= 0⇒ J → 0in finite time, and ∂J/∂x = 0⇒ J = 0. Hence, x → x∗,with ∂J/∂x(x∗) = 0, in finite time. Indeed, as shown in [22],h(x) such that h ≥ −c(h(x))γ is a finite-time convergencecontrol barrier function for the system characterized by singleintegrator dynamics, x = u.

Using the results derived in this section, the next sectionpresents a procedure to synthesize decentralized optimizationproblems whose solutions result in coordinated control ofmulti-robot systems.

III. CONSTRAINT-BASED CONTROL OFMULTI-ROBOT SYSTEMS

Local, scalable, safe and emergent are four essentialfeatures that decentralized multi-robot coordinated controlalgorithms should possess [23]. Many algorithms that satisfythese properties have been developed for applications rangingfrom social behavior mimicking [24], formation assembling[25] and area patrolling [26]. In [23], the authors analyzethe common features among these algorithms and discusstheir decentralized implementations in robotic applications.In this section, we apply the results derived in Section IIwith the aim of obtaining constrained optimization problemsequivalent to the decentralized execution of multi-robot tasks.

Consider a collection of N robots, whose position isdenoted by xi ∈ Rd, i ∈ {1, . . . , N}, where d = 2 forplanar robots and d = 3 in the case of aerial robots. Assumeeach robot is equipped with an omni-directional range sensorthat allows it to measure the relative position of neighboringrobots, namely robot i is able to measure xj−xi, when robotj is within its sensing range. These interactions among therobots are described by a graph G = (V, E), where V ={1, . . . , N} is the set of vertices of the graph, representingthe robots, and E ⊆ V × V is the set of edges between therobots, encoding adjacency relationships. If (i, j) ∈ E , thenrobot i can measure robot j’s position. For the purposes ofthis paper, we assume that the graph is undirected, namely(i, j) ∈ E ⇔ (j, i) ∈ E . In order to obtain decentralizedalgorithms, we want each robot to act only based on localinformation, by which we mean the relative positions of itsneighbors. By construction, this leads to inherently scalablecoordinated control algorithms.

Denoting the ensemble state of the robotic swarm by x =[xT1 , . . . , x

TN ]T ∈ RNd, a general expression for the cost that

leads to decentralized control laws is given by

J(x) =

N∑i=1

∑j∈Ni

Jij(‖xi − xj‖), (13)

where Ni is the neighborhood set of robot i, and Jij : R→R, Jij(‖xi−xj‖) = Jji(‖xj−xi‖) is a symmetric, pairwisecost between robots i and j. We assume that Jij(x) ≥0, ∀(i, j) ∈ E , ∀x ∈ Rn, so that J(x) ≥ 0, ∀x ∈ Rn.Assuming we can directly control the velocity of robot i,xi, we can employ a gradient descent flow policy like (7) to

minimize J , obtaining

ui = −∑j∈Ni

∂Jij∂‖xi − xj‖

xi − xj‖xi − xj‖

=∑j∈Ni

wij(xj − xi).

This is nothing but a weighted consensus protocol, and itis decentralized insofar as the input ui only depends onrobot i’s neighbors. The construction shown in Section IIcan be then applied to minimize the cost given in (13) byformulating the following minimum-energy problem:

minu,δ‖u‖2 + |δ|2

s.t. − ∂J

∂xu ≥ −α(−J(x))− δ,

(14)

where u = [uT1 , . . . , uTN ]T ∈ RNd is the vector of robots’

inputs, and a single integrator dynamics, xi = ui, is assumedfor each robot. Solving (14) leads to the accomplishmentof the task, by which we mean that a stationary point ofthe cost J has been reached. As explicitly shown by (12)in Proposition 5, a minimum-energy formulation, initiallyintroduced in [15], allows the robots to move towards lowervalues of the cost J until the task is accomplished (u ≡ 0).

The following proposition gives the expression of the op-timization problems whose solutions lead to a decentralizedminimization of the cost J in (13).

Proposition 7 (Constraint-driven decentralized task execu-tion). Given the pairwise cost function J defined in (13),a collection of N robots, characterized by single integratordynamics, minimizes J in a decentralized fashion, if eachrobot executes the control input, solution of the followingoptimization problem:

minui,δi

‖ui‖2 + |δi|2

s.t. − ∂Ji∂xi

ui ≥ −α(−Ji(x))− δi,(15)

where Ji(x) =∑j∈Ni

Jij(‖xi − xj‖) and α is an extendedclass K function, α : x ∈ R 7→ α(x) ∈ R, superadditivefor x < 0, i. e. α(x1 + x2) ≥ α(x1) + α(x2), ∀x1, x2 < 0.If α(x) = cxγ , c > 0, γ ∈ (0, 1), a stationary point of thecost J is reached in finite time, with the upper bounds onthe settling time given in Theorem 1.

Proof. Proposition 5 ensures that, by imposing the globalconstraint −∂J∂xu ≥ −α(−J(x)), constructed using thewhole state vector x, the cost J is decreasing towardsa stationary point. We want to show that, by imposingonly local constraints (i. e., such that robot i only needsinformation about its neighbors), the multi-robot system isable to enforce the global constraint and, hence, to minimizethe cost J in a decentralized fashion.

We proceed by starting to sum up the constraints for eachrobot, obtaining:

N∑i=1

(−∂Ji∂xi

ui

)≥

N∑i=1

(−α(−Ji(x))− δi)

≥− α

(−

N∑i=1

Ji(x)

)− δ ≥ −α (−J(x))− δ,

where we used the superadditivity property of α, and we setδ =

∑Ni=1 δi. Moreover, since the graph G, which encodes

the neighboring relations between the robots, is undirected,we have that ∂Ji

∂xi= 1

2∂J∂xi

. Thus,

∂J

∂xu ≥ −2α (−J(x))− 2δ = −α′ (−J(x))− δ′,

where ∂J∂x =

[∂J∂x1

, . . . , ∂J∂xN

], u =

[uT1 , . . . , u

TN

]T, and α′

an extended class K function. Hence, by Proposition 5, xwill converge to a stationary point of J .

Finally, we note that a class K function α(x) = cxγ ,defined for x < 0 is convex, and hence superadditive, forx < 0. Applying Proposition 6, the last statement holds.

The structure of the cost function J(x), even though quitespecific, allows us to encode a rich set of multi-robot tasks,by carefully choosing the weights wij as a function of thestate x. The following section shows two variations on thecost function which allow a multi-robot system to performformation control, i. e., assembling particular shapes, andcoverage control, consisting in spreading out the roboticswarm in the environment in an optimal way.

IV. APPLICATIONS

In this section we recall the expression of the cost Jfor two specific multi-robot tasks: formation control andcoverage control.

A. Formation Control

In formation control applications, the robots are asked toassemble a predefined shape, specified in terms of inter-agent distances. In order to frame this problem as a costminimization problem, let J be the formation error

J(x) =

n∑i=1

∑j∈Ni

1

2(‖xi − xj‖ − dij)2 =

n∑i=1

Ji(‖xi − xj‖),

where dij is the desired distance between robots i and j. Jmeasures how far the robots are from assembling the desiredformation characterized by the relative distances dij . J = 0corresponds to the robots forming the desired shape. Notethat, as Ji(x) is a sum of squares, Ji(x) ≤ J(x), ∀i, requiredas a hypothesis in order for Proposition 7 to hold.

The gradient the Ji(x) evaluates to

∂Ji∂xi

=∑j∈Ni

‖xi − xj‖ − dij‖xi − xj‖

(xi − xj)T . (16)

This can be interpreted as follows: if the distance betweenrobots i and j is smaller than dij , then the weight wij =‖xi−xj‖−dij‖xi−xj‖ is negative, and the robots experience a repelling

effect. Conversely, if the two robots are further than dij apart,the positive weight wij will attract one towards the other.The special case in which dij = 0, ∀i, j corresponds to thewell-known consensus problem.

The expression of the gradient in (16) is decentralized asrobot i has to compute relative distances only with respectto its neighboring robots.

B. Coverage Control

In coverage control, the task given to the robots is thatof covering a domain D. Given a coverage performancemeasure, the robots should spread over the domain in an op-timal way. As shown in [27], each robot should be in chargeonly of a subset of the domain D that, more specifically,is its Voronoi cell, defined as Vi = {p ∈ D | ‖p − xi‖ ≤‖p− xj‖ ∀i 6= j}.

Let us introduce the measure of how bad a domain is beingcovered:

J(x) =

N∑i=1

1

2‖xi −Gi(x)‖2 =

n∑i=1

Ji(x), (17)

where Gi denotes the centroid of the Voronoi cell Vi. Thisform is just a reformulation of the locational cost originallyintroduced in [26]. Taking the derivative of Ji with respectto xi, required in the optimization problem (15), one obtains:

∂Ji∂xi

= (xi −Gi(x))T(I − ∂Gi(x)

∂xi

),

where I is the identity matrix. Note that, even if Gi(x)virtually depends on the entire ensemble state, x, of therobotic swarm robot i, in order to compute it, only requiresinformation from the robots with which it shares part of theboundary of its Voronoi cells.

We deployed the optimization-based control algorithmswith the expressions of the costs J derived in Sections IV-A and IV-B on a real multi-robot system and, in the nextsection, we show the experimental results. Moreover, theconstraint-driven formulation of Section III is used to achievelong-term environmental monitoring, where the robots aretasked with covering a domain over a time-horizon whichis much longer than their battery life, and during whichthe robots will also have to avoid collisions with obstaclesmoving around in the domain.

V. EXPERIMENTS

The coordinated control approach presented in this paperhas been tested on the Robotarium [1], a remotely acces-sible swarm robotics testbed. The Robotarium is populatedby small-scale differential-drive robots which can be pro-grammed by uploading code scripts via a web interface.

Throughout the paper, we assumed we can directly controlthe velocity of the robots, by modeling them using singleintegrator dynamics. In [28], it is shown that it is possibleto derive a near-identity diffeomorphism which can be usedto partially feedback linearize the unicycle dynamics use tomodel differential-drive robots. This way, the unicycle canbe abstracted as a single integrator.

Regarding the implementation of the optimization program(15) needed for the execution of the tasks presented in theprevious section, the function α has been chosen to beα(x) = 3

√x, which is an extended class K function, convex

for x < 0. This implies it is also superadditive for x < 0, asrequired by the hypotheses in Proposition 7.

A. Combining and Prioritizing Tasks

In this section, we present the application of the proposedconstraint-driven coordinated control to long-term environ-mental monitoring. The setup of the experiment is as follows.Six robots are asked to monitor an area by performingcoverage control. While executing this task, the robots mustnot run out of energy and must not collide with two dynamicobstacles, embodied by two additional robots moving in theenvironment. In order to do so, we define constraints thatallow the robots to always keep enough residual energy intheir batteries and to be always a minimum distance apartfrom the obstacles.

To accomplish the first goal, we use a method similarto the one developed in [29]. Assuming that the domainis endowed with charging stations, i. e. locations where therobots can recharge their batteries, let us define the followingbarrier function:

he,i(xi, Ei) = Ei − Emin − k(‖xc,i − xi‖ − dchg)2, (18)

where xi is the position of robot i, Ei is the energy in itsbattery, Emin is the minimum residual energy we want therobots to keep, xc,i is the location of the charging stationdedicated to robot i, dchg is the minimum distance fromthe charging station at which the robots can recharge theirbatteries (typical behavior of wireless charging technologies),and k is a constant such that k(‖xc,i − xi‖ − dchg)2 upper-bounds the energy required to reach a charging station. Werefer to [29] for a rigorous analysis.

As far as obstacle avoidance is concerned, we define, foreach obstacle, the following barrier function, which ensurescollision-free operations in multi-robot systems [30]:

ho,i(xi) = ‖xi − xo‖2 − d2o,

where xo is the position of the obstacle and do is theminimum distance we want the robots to maintain from theobstacle.

Combining the energy constraint he,i ≥ he,i(xi, Ei) andthe obstacle constraint ho,i ≥ ho,i(xi) together with the cov-erage task constraints, the following optimization problemcan be formulated:

minui,δi

‖ui‖2 + δ2i

s.t. − ∂Ji∂xi

ui ≥ −α(−Ji(x))− δi

he,i ≥ he,i(xi, Ei)ho,i ≥ ho,i(xi).

(19)

The variable δi in the coverage constraint acts as a relaxationparameter, which allows the constraints related to energyand collisions to be fulfilled. This translates to trading taskexecution for survivability. Consequently, this formulationallows tasks prioritization obtained by combining hard andsoft constraints.

The results of the long-term environmental monitoringexperiment are shown in Fig. 1. The Voronoi partitiongenerated by the robots is projected down onto the testbed.

Arranged vertically along the left edge of the domain, thereare six charging stations depicted as blue circles that turnyellow when the robots are charging. The two robots circle inred are moving in the environment acting solely as obstacles.The sequence of snapshots shows the robots starting toperform coverage (Fig. 1a), two robots avoiding a obstacles(top right and bottom left in Fig. 1b), and two robotsrecharging their batteries (Fig. 1c). In Fig. 1d the robots havereached a configuration corresponding to a local minimumof the locational cost (17). A video of the experiments canbe found online [31].

Due to the limited amount of time that each experimentsubmitted to the Robotarium is allowed to last, we simulatethe battery dynamics in such a way that the robots experiencemultiple charging cycles during the course of a single exper-iment. Fig. 2 shows the energy levels of the robots employedto perform coverage. The minimum desired energy level,Emin, and the value corresponding to fully charged battery,Echg , are depicted as black thick lines. Enforcing the energyconstraints using (18) allows the robots to keep their energylevel always above Emin.

With this experiment, we have shown how the constraint-driven control formulation can be used to build a minimum-energy optimization problem, whose constraints encode boththe task that the robots are asked to perform and the surviv-ability specifications, thus enabling the robust deployment ofrobots for long-term applications.

VI. CONCLUSIONS

In this paper we presented a reformulation of optimization-based multi-robot tasks in terms of constrained optimization.Identifying a multi-robot task with a cost function that needsto be minimized, we leverage control barrier functions tosynthesize decentralized optimization-based controllers thatachieve the desired goal. The advantages of this approachinclude its flexibility of encoding several multi-robot tasksand the ease of combining them with different types ofconstraints. We showed how this flexibility can be usedto enforce robot survivability and achieve long-term robotautonomy, where robustness and resilience are indispensableproperties that robots have to possess. A systematic way offormulating the optimization problems for each agent of arobotic swarm is derived. Its effectiveness is demonstratedthrough experiments with a team of ground mobile robotsshowcasing a long-term environmental monitoring applica-tion.

ACKNOWLEDGMENT

The authors would like to thank Professor Jonathan N.Pauli for helpful discussions about ecology.

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(a) (b) (c) (d)

Fig. 1. A team of six robots is tasked with monitoring a rectangular domain on the Robotarium, by performing coverage control as in Fig. ??. This time,however, the robots are asked to perform this task over a time horizon which is much longer than their (simulated) battery life. Additionally, two morerobots (circled in red) act as obstacles which have to be avoided by the remaining six robots. These execute (19) to avoid the obstacles, go and rechargetheir batteries at the dedicated charging stations (blue circles on the left of the figures that turn yellow when the robots are charging), while always coveringthe given domain. A video of the experiments is available online [31].

0 100 200 300

Emin

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E

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