Robot Ecology: Constraint-Based Control Design for Long Duration Autonomy

Magnus Egerstedta,∗, Jonathan N. Paulib, Gennaro Notomistac, Seth Hutchinsond

aSchool of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USAbDepartment of Forest and Wildlife Ecology, University of Wisconsin, Madison, WI, USAcSchool of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USAdSchool of Interactive Computing, Georgia Institute of Technology, Atlanta, GA, USA

Abstract

Long duration autonomy considers robots that are deployed over long time scales. As a result, the robots’ ability tooperate across changing environmental conditions and perform tasks that require more time to complete than what canbe supported on a single battery charge, have implications for how the control design task should be approached. Bydrawing inspiration from ecology, which concerns itself with the strong coupling between organism and the environmentit inhabits, we encode these “survival constraints” through Boolean compositions of multiple control barrier functions,and show how this construction supports long duration autonomy in the context of persistent environmental monitoringon a team of mobile robots.

1. Introduction

Robots are increasingly leaving the confines of theirhighly structured and curated environments within cageson manufacturing floors, academic laboratories, and pur-posefully arranged warehouses. This robot migration is5

taking place over large temporal and large spatial domains.For example, in precision agriculture, it is envisioned thatrobots will be embedded in fields, tending to individualplants by monitoring, and eventually meeting, their fer-tilizer, pesticide, or water needs, e.g., Ball et al. (2017);10

Sistler (1987). These agricultural robots will be presentthroughout the entire growing cycle, i.e., over an entireseason, Arkin and Egerstedt (2015). Similarly, a numberof environmental monitoring scenarios have been pursuedand discussed, whereby robotic sensor nodes are monitor-15

ing aspects of a natural environment, Dunbabin and Mar-ques (2012); Steinberg et al. (2016). Examples include:searching for the possibly extinct Ivory-billed Woodpeckerin the forests of Louisiana, Song et al. (2008), employ-ing underwater robots for tracking marine pollution or the20

spread of invasive species, Hollinger et al. (2012); Tokekaret al. (2013), or for monitoring the effects of climate changeon polar ice caps, Spears et al. (2014).

Deploying robots over truly long time scales poses prob-lems that are fundamentally different from those faced by25

robots deployed in factories or other controlled environ-ments, in which operating conditions exhibit only limitedvariability, power is readily available, and regularly sched-uled maintenance routines ensure that, for all practical

∗Corresponding authorEmail addresses: magnus@gatech.edu (Magnus Egerstedt),

jnpauli@wisc.edu (Jonathan N. Pauli), g.notomista@gatech.edu(Gennaro Notomista), seth@gatech.edu (Seth Hutchinson)

purposes, robots can function indefinitely. To achieve long30

term autonomy, robots must be able not only to adapttask performance to changing environmental conditions,but must also confront issues related to their very surviv-ability. Much research has focused on the former (e.g.,Campbell et al. (2010); Steinberg et al. (2016)), but rel-35

atively little has focused on the latter. The basic issueof survivability over long time periods remains an openproblem for autonomous robotic systems in uncontrolledenvironments.

To date, survivability issues have been addressed either40

by imposing scheduled behaviors (e.g., periodic visits tothe recharging station) during task execution, or by explic-itly encoding survivability goals into performance criteria.Neither approach is ideal. For example, when issues suchas limited battery life are dealt with by imposing power45

management schemes (e.g., Kaplan et al. (2017); Setterand Egerstedt (2017); Stramigioli (2015)), systems mustadhere to fixed policies during execution, without regardto implications on achieving task goals. If survivability isencoded as a performance objective, it must be somehow50

combined with other performance goals (e.g., via scalar-ization), or multi-objective optimization approaches mustbe used. In the former case, the primacy of survivabilityis not ensured, raising the possibility of catastrophic fail-ure in the opportunistic pursuit of short-term gain. In the55

latter case, none of the available equilibium or optimalityconcepts (e.g., Nash or Stackelberg equilibria, or Paretooptimality) ensure survivability, instead balancing, in oneway or another, the degree of survivability against othertask performance criteria. In short, previous approaches60

fail to adequately recognize (or exploit) the fact that sur-viving is prerequisite to thriving, a view which suggestsconstrained optimization as the appropriate semantics for

Preprint submitted to Annual Reviews in Control September 26, 2018

robot programming in long duration autonomy applica-tions.65

In this paper, we explore such a constraint-based ap-proach to control design, an approach that ensures sur-vival while optimizing task performance. Our approachis inspired by ecological studies of biological systems, andimplemented using the engineering tools of constrained op-70

timization and control barrier functions. Specifically, wedraw inspiration from a field of study that explicitly inves-tigates the tight couplings between “autonomous agents”and the environments in which they are deployed, namelyecology, where richness of behavior is a direct function75

of environmental constraints, Odum and Barrett (2004);Ricklefs (2008); Smith et al. (2012), including the abun-dance and distribution of resources, favorable microcli-mates, suitable mates, and predators. Indeed, when ecol-ogists study the distribution of species, the composition of80

populations and communities or individual habitat selec-tion, space use and movement, the environmental realityand the associated ecological constraints are as important,if not more so, than any goal driven behaviors, Ricklefs(2008); Smith et al. (2012).85

Based on this observation that constraints are funda-mentally important to animal behavior, we will investigateif this vantage-point translates to effective control designprinciples for engineered systems as well. As such, we willapproach the design problem as one where the robots’ be-90

haviors are mostly constraint-driven, as opposed to goal-driven, following that many behaviors emerge among or-ganisms to manage and balance survival constraints. Themost extreme instantiation of this view would be robotsthat are tasked with doing nothing at all, subject to long-95

term survivability constraints, such as avoiding collisionswith obstacles or other robots, or never completely deplet-ing the battery. Below, these types of constraints will bederived (albeit subject to a slight robotic re-interpretation)from basic ecological principles. In fact, as ecology is ulti-100

mately aimed at understanding the interaction of specieswith their environment and other organisms, this seemslike a particularly fruitful metaphor also for robots leavingthe highly curated laboratory or factory settings, and en-tering dynamic, unstructured, natural environments over105

long temporal and spatial scales.The outline of this paper is as follows. In Section 2,

we introduce constraint-based control design that encodesa number of safety and longevity constraints as controlbarrier functions. We, moreover, show how these can be110

composed to provide a complete Boolean logic using toolsfrom non-smooth analysis. In Section 3, we introduceour notion of “robot ecology.” We instantiate the ecolog-ical metaphor, first as mathematical specification of con-straints on robot behavior (Section 3.1), and then as a con-115

strained optimization formulation (Section 3.2) to effectgoal-directed behaviors (e.g., analogous to reproduction inbiological systems). We illustrate our approach using apersistent environmental monitoring scenario, whereby anumber of real robots are deployed in such a way that they120

effectively cover an area while staying safe and never beingstranded away from a charging station with completely de-pleted batteries. The paper concludes in Section 4, wherethe robot ecology idea is used to formulate an “Auton-omy On-Demand” framework, where users recruit robots,125

deployed over long periods of time, and specify their goal-driven behaviors, while the survival constraints are usedto ensure longevity.

2. Constraint-Based Control Design

One fundamental fact that makes robots deployed in an130

environment over truly long time scales (weeks, months,and even years) different from targeted, short term de-ployments, is that any assumptions about environmentalstationarity, or even predictability, are doomed to be incor-rect. As a result, carefully tuned, optimal control strate-135

gies, which are only optimal given precise model assump-tions, are unlikely to transfer well across environmentalconditions. In fact, as observed in Carlson and Doyle(2002); Csete and Doyle (2002), optimality is closely re-lated to fragility, i.e., when the environmental changes are140

sufficiently disruptive and the modeling assumptions nolonger hold, the controllers are not only not-optimal, butthey may be infeasible. In contrast to this, a constraint-based design would allow for a controller to be used as longas low-level constraints, related to safety and/or longevity145

of the system, are satisfied. In this section, we discusswhat such a constraint-based design methodology entailsat a technical level, while the connections between longduration autonomy, ecology, and constraint-based controldesign is left to subsequent sections.150

2.1. Control Barrier Certificates

Consider a robot (or collection of robots), whose dy-namics are given by

x = f(x) + g(x)u, (1)

where x ∈ Rn is the state of the robot (or possibly thecomposite state of multiple robots), f encodes the uncon-trolled drift, g corresponds to the robot’s actuation modal-ities (both f and g are assumed to be locally Lipschitz155

everywhere), and u ∈ Rm is the control input. This con-trol affine structure is quite standard and large classes ofrobots – may they be on the ground, under water, or inthe air – can be modeled on this form, e.g., Choset et al.(2005).160

One way of approaching constraint-based control de-sign is to select a control input u that ensures that therobot stays in some set, S ⊂ Rn, for all times. Assumingthat we can encode the set membership through a contin-uously differentiable level set function h : Rn → R, in thesense that

h(x) ≥ 0⇔ x ∈ S, (2)

2

the constraint that must be satisfied, for all times, is apositivity constraints on h, i.e., the constraint is satisfiedif h(x(t)) ≥ 0, ∀t ≥ 0.

If, somehow, the control input u was selected such that

d

dth(x) = −α(h(x)),

for some locally Lipschitz, extended class-K function α :Rn → Rn (strictly increasing and α(0) = 0), the resultwould be that h approaches 0 from above (as long as itstarts positive), e.g., Ames et al. (2014). Consequently, ifu had instead been selected to ensure that

d

dth(x) ≥ −α(h(x)),

the Comparison Lemma (e.g., Rudin (1976)) immediatelytells us that in this case h(x) ≥ 0 for all times as long165

as it starts positive, as observed, for example, in Ameset al. (2014); Wang et al. (2017). As such, this is exactlythe type of constraint we would like to enforce throughoutthe life time of the robot as a way of rendering the set Sforward invariant; if the system starts in S, it stays in S.170

But, the above expression is contingent on the selectionof the control input u, even though the input does not seemto explicitly play a role in the constraint. The way theinput shows up is by incorporating the robot dynamics,i.e.,

d

dth(x) =

∂h(x)

∂xx = Lfh(x) + Lgh(x)u,

where the Lie derivatives of h(x) in the f and g directionsare

Lfh(x) =∂h(x)

∂xf(x), Lgh(x) =

∂h(x)

∂xg(x).

What we have shown is that the set S is rendered for-ward invariant if, at each point in time, u is chosen tosatisfy the following linear constraint (linear in the deci-sion variable u)

Lgh(x)u ≥ −Lfh(x)− α(h(x)). (3)

Under this formulation, h is called a Control Barrier Func-tion (CBF), which is akin to a Control Lyapunov Functionin that a point-wise (in time) constraint on the controlinput u is sufficient to say something about the systemperformance for all times. In fact, the resulting inequalityconstraint in Equation (3) for ensuring forward invarianceof S is a Control Barrier Certificate (CBC), as discussedin Ames et al. (2014); Prajna et al. (2007); Romdlony andJayawardhana (2016). As it will be more convenient todiscuss the constraints directly in terms of the CBFs, h,rather than the differential constraints obtained throughthe CBCs, the notation that will be used throughout thispaper is

C(h[x;u]) = Lgh(x)u+ Lfh(x) + α(h(x)) ≥ 0

as shorthand for the inequality constraint in Equation (3).Such CBFs have been used in a variety of applicationsto encode a number of constraints for single and multi-ple robots, such as collision avoidance, Wang et al. (2017),adaptive cruise controllers Ames et al. (2014), bipedal walk-175

ing, Nguyen et al. (2016), and quadrotor control, Wu andSreenath (2016).

Common to these aforementioned applications is thenotion of a nominal controller, e.g., a controller obtainedthrough an optimal control process, with a CBC imposed180

on the nominal controller as a way of enforcing safety con-straints. In this paper, we will explore the view that theconstraints are not only safe-guards added to the nominalcontroller, but are in fact the key to getting to robots thatare deployed over long periods of time. To this end, we185

will consider a few examples of such constraints for teamsof robots that are of relevance to long duration autonomy.

2.2. Constraints for Longevity

To make matters concrete, assume that the state xi ofeach robot is given in part by a directly controlled pla-nar position, pi ∈ R2. Additionally, as the availability ofenergy plays a crucial role to longevity, we add the energy-level, Ei, to the state of each robot, i.e.,

xi =

[piEi

]∈ R3,

with the simplified dynamics (pure energy-decay)

xi =

[ui−κEi

].

We moreover follow Ames et al. (2014); Wang et al. (2017)in choosing the class-K function, α, as the cube α(h) =190

γh3, for some γ > 0.For this choice of system dynamics, we can now intro-

duce a number of constraints that are fundamental for longduration autonomy:

(i) Safety (based on the developments in Wang et al. (2017)):195

Given a safety distance ∆s > 0, two robots, i, j, withpositions pi, pj , are safe as long as

hsafe(xi, xj) = ‖pi − pj‖2 −∆2s ≥ 0.

Under the model assumptions above, the correspondingCBC, C(hsafe[x;u]), becomes

C(hsafe[x;u]) = (pi−pj)T (ui−uj)+γ

2

(‖pi−pj‖2−∆2

s

)3 ≥ 0.

(ii) Connectivity (as discussed in Wang et al. (2016)):Two robots i, j are connected if they are close enough, i.e.,if

hconnect(xi, xj) = ∆2c − ‖pi − pj‖2 ≥ 0,

given a connectivity distance ∆c > 0. Or, formulated as aCBC, C(hconnect[x;u]) ≥ 0, where

C(hconnect[x;u]) = −(pi−pj)T (ui−uj)+γ

2

(∆2

c−‖pi−pj‖2)3.

3

(ii) Coverage (based on the locational cost in Cortes et al.(2002)):200

A robot that is tasked with monitoring a domain Di ⊂ R2,where the importance of point q ∈ Di is given by thedensity function φ(q) ∈ R+, and where the quality of thesensor coverage associated with this point, as achieved bythe robot located at pi ∈ Di, can be encoded by a qualityfunction Q(‖pi − q‖). (Typically, the further away thepoint is, the worse the coverage is.) Additionally, let c bea lower limit for acceptable coverage performance, whichgives the coverage constraint

hcoverage(xi) =

∫Di

Q(‖pi − q‖)φ(q)dq − c ≥ 0.

Note that despite this looking like a rather complicatedexpression, the corresponding CBC is still linear in thedecision variable ui, i.e., it is actually not a particularlyhard constraint to handle.

(iv) Energy (following the task persistification ideas in No-205

tomista et al. (2018)):If Ei is the energy-level associated with robot i and d(pi)is the distance to the closest “charging station” (as in-terpreted broadly), one can let E(d(pi)) be the energy re-quired to travel that distance under suitable assumptionson the motion model, Setter and Egerstedt (2017). Therequirement that the robot never gets stranded away froma charging station thus becomes

henergy(xi) = E(d(pi))− Ei − Emin ≥ 0,

given some smallest acceptable energy-level Emin.

(v) Charging (based on Notomista et al. (2018)):To prevent over-charging, one would be tempted to intro-duce the constraint

hcharge(xi) = Emax − Ei ≥ 0,

where Emax is the maximum energy-level supported. How-ever, viewing hcharge as an output of the system, the rela-tive degree associated with this output is not well-definedin that the input ui never shows up in any of the timederivatives of hcharge, which makes it a poor choice fora constraint. The problem is that charging only happenswhen the robot is close to the charging station, which isnot reflected in the constraint. As such, if we, as before,let d(pi) be the distance to the charging station, one candefine an additional, smooth approximation of a step func-tion s∆(d(pi)) that is 1 when d(pi) = ∆ and zero whend(pi) = 0. The charging constraint thus becomes

hcharge(xi) = Emaxs∆(d(pi))− Ei ≥ 0.

As is the case when handling constraints, there mightnot be any feasible solutions. This could for example be210

the case if a robot “wakes up” with almost completely de-pleted batteries too far away from a charging station, inwhich case the CBC associated with henergy ≥ 0 would

not have a solution. Or, if there simply is not enoughdensity in the domain Di for any point xi ∈ Di to satisfy215

hcoverage ≥ 0. These are ultimately issues pertaining tothe choice of constraints, and they may need to be tweakedin order to fit the particulars of the scenarios under con-sideration. Regardless, these types of constraints will beused in this paper in order to ensure that robots do not220

get stranded away from charging stations without suffi-cient energy, while, at the same time, they remain safe,connected, or exhibit other types of relevant behaviors,such as covering a sufficiently large domain. But, to thisend, we must be able to combine multiple constraints into225

a single CBF.

2.3. Composition of Constraints

Assume that we would like the robot to satisfy bothh1(x) ≥ 0 and h2(x) ≥ 0 for all times. This correspondsto a Boolean conjunction, and in Glotfelter et al. (2017) itwas observed that if the smallest of h1 and h2 is positivethen both of them are positive. In other words, the logicalAND, h1∧2, is given by

h1(x) ≥ 0∧h2(x) ≥ 0 ⇔ h1∧2(x) = min{h1(x), h2(x)} ≥ 0.

Similarly, Boolean disjunction is obtained by notingthat h1 or h2 are positive if the largest of the two is posi-tive. In other words, the logical OR, h1∨2, is given by

h1(x) ≥ 0∨h2(x) ≥ 0 ⇔ h1∨2(x) = max{h1(x), h2(x)} ≥ 0.

The Boolean composition of constraints is made completeby including the negation as well, as was done in Glotfelteret al. (2017), through

¬h(x) ≥ 0 ⇔ − h(x) ≥ 0,

where one has to accept the ambiguity associated with theboundary h(x) = 0.

Although this way of composing constraints together230

captures what we need in terms of Boolean logic, thereare complications that arise due to the fact that max andmin are non-smooth operators. As such, turning h1∧2 andh1∨2 into a CBC requires some additional machinery.

For continuously differentiable CBFs, we have that

d

dth(x) =

∂h(x)

∂xx = ∇h(x)T x.

But this is no longer well-defined without a clarification of235

what the gradient of h actually entails in the non-smoothcase. To this end, we use the Clarke generalized gradient,∇Ch(x), e.g., Cortes (2008), which is a set-valued operatorthat maps from x to all possible “gradients” achievablethrough a limiting sequence converging to x, as long as240

h is lower semi-continuous. (Luckily, the min and maxoperators satisfy this property.)

As shown in Glotfelter et al. (2017), the non-smoothversion of CBCs states that the inequality has to hold for

4

all points in the generalized gradient, i.e., the CBC be-comes

min{∇Ch(x)T f(x) +∇Ch(x)T g(x)u} ≥ −α(h(x)). (4)

And, despite the potential complications from having theu-term placed inside the minimization operation over theset of gradient directions, this still translates to a linear245

constraint in the decision variable u, when h is composedtogether of min and max operations over continuously dif-ferentiable functions, Glotfelter et al. (2017).

We can thus combine constraints through arbitraryBoolean compositions. For example, given the sets S1,S2,S3

(with associated CBFs h1, h2, h3, and where we use S asshorthand for the proposition x ∈ S), the CBF associatedwith the formula S1 ∧ (S2 ∨ ¬S3) thus becomes

h1∧(2∨¬3)(x) = min{h1(x),max{h2(x),−h3(x)}

}.

Similarly, secondary operations, like the material implica-tion S1 → S2, which is logically equivalent to ¬S1∨S2, andthe exclusive or S1⊕S2 (equivalent to (S1∨S2)∧¬(S1∧S2))become

h1→2(x) = max{−h1(x), h2(x)

},

h1⊕2(x) = min{

max{h1(x), h2(x)},−min{h1(x), h2(x)}}.

In particular, the material implication provides a wayto manage situations where constraints may be violated250

due to a lack of feasibility. For example, the constraint¬S1 → S2 encodes the proposition that the state of thesystem should satisfy the constraint x ∈ S1. But, if itdoes not, then x ∈ S2 should be satisfied. This ability toexpress arbitrary Boolean compositions of constraints will255

be key for formulating a constraint-based design strategyfor long duration autonomy. A step towards this end willbe taken in the next section, where ecological considera-tions are coupled to the types of constraints discussed inthis section.260

3. Robot Ecology

Ecological constraints give rise to organismal traits thatare uniquely adapted to habitat, Alcock and Rubenstein(2009); Danchin et al. (2008). Ultimately, this boils downto two components of an individual’s fitness on the land-265

scape: its ability to survive and reproduce successfully.Organisms must first meet the demands of the environ-ment to survive, and then successfully reproduce to passon their genotypic and phenotypic traits. Survival, then,involves navigating energetic landscapes, meeting nutri-270

tional needs, avoiding predation; reproduction involves thosestrategies to successfully find a mate and produce viableoffspring. All of these components are typically consid-ered when trying to characterize and understand an or-ganism’s ecology and behavior, Alcock and Rubenstein275

(2009). As the purpose with this paper is to try to under-stand the tight coupling between environment and robot,we will initially discuss this in the context of ecological con-straints, and then map these onto corresponding types ofconstraints relevant to long duration autonomy. We then280

augment our constraint-based approach to include goal-directed behaviors encoded via optimality criteria.

3.1. Ecological Constraints

Foraging corresponds to one of the most obvious behav-iors that result from ecological constraints, i.e., an animalhas to acquire the necessary energy and nutrition in or-der to survive. For a robot (or group of robots) deployedin an environment, foraging would correspond to recharg-ing/refueling, i.e., the CBF hcharge from the previous sec-tion (or some similar type of constraint) could capture thisneed. But, this in turn has important implications for anorganism’s space use and selection of particularly habitattypes. In other words, an organism must forage across asufficient area to support the energetic needs of the ani-mal, i.e., hcoverage may be of relevance. Finally, if criticalresources (e.g., particular diet items or water sources) areonly available at select locations, the organism is“spatiallyanchored”, Sih (2005), and is forced to return to those lo-cations, i.e., henergy may be needed as well. As a result,if one were to produce a CBF for encoding the foragingconstraint, it could look something like

hforage(x) = henergy(x) ∧ (hcoverage(x) ∨ hcharge(x))

i.e., the animal/robot should always have enough energyto return to the spatially anchored feeding locations, it285

should either be covering a sufficiently large territory orbe engaged in the process of charging/eating. One impor-tant aspect of this formulation is that the environmentalcoupling is explicitly taken into account through henergy,which relates the position of the robot to the position of290

available feeding sources, e.g., sunshine (if using solar pow-ered robots), charging stations for batteries, or refuelingstations. Additionally, the coverage term also relates tohabitat quality – prevalence of objects of interest. In otherwords, the density function φ : Di → R+ encodes how295

“good” the habitat is for an animal to make a living there.Competitive interactions and territoriality within species

produce other fundamental ecological constraints. Thatis, within many species, individuals avoid one another tolimit competition and to limit “interference” with one an-other. We also wish to encode this interspecific interac-tion through a constraint. For robots deployed over longtime scales, there might be reasons why the robots shouldinteract, e.g., for the purpose of exchanging information,which would be encoded through the hconnect CBF. But,at the same time, territoriality could correspond to robotsnot getting too close, encoded through hsafe, which wouldproduce the following CBF for the ecological interactionconstraint

hinteract(x) = hsafe(x) ∧ hconnect(x).

5

To ground these two constraints derived from ecolog-ical constraints in a robotic application, consider a situ-ation where a team of robots, indexed by i = 1, . . . , N ,are supposed to cover an area over a long period of timewithout running out of energy. This means that they haveto spread out and cover the area while, at the same time,avoid collisions and, more importantly, be able to return toa charging station to recharge the batteries. These spec-ifications can be encoded through a combination of theforaging and interaction constraints

hmonitor(x) = hforage(x) ∧ hinteract(x).

It must be noted that the constraints themselves arenot sufficient to get the robots to act as there is no auto-matic way of deciding which u to use from the set of con-trollers that satisfy the CBC C(hmonitor[x;u]) ≥ 0 withoutan additional selection criteria. As such, one could askto keep the energetic expenditures to a minimum, whichwould result in the following problem, to be solved at eachinstant of time,

minu ‖u‖2

subject to C(hmonitor[x;u]) ≥ 0.(5)

The robots would then, subsequently, be executing theminimizing control input at each point in time.

This formulation may indeed produce the intended be-havior, but it is not guaranteed; what is guaranteed is that300

the robots never run out of battery, but not that any mean-ingful environmental monitoring actually occurs. The rea-son for this is due to the interplay between the size of thedomains Di, the amount of mass in each domain, as en-coded through the density function φ, the cutoff for accept-305

able coverage c, and the disjunction between hcoverage andhcharge in hforage. In fact, if these parameters are not se-lected carefully, it is entirely possible that the robots wouldsimply linger forever near the charging stations since thatway, hcharge ≥ 0 while hcoverage < 0 for all times, thereby310

rendering the disjunction in hforage true for all times.One can resolve this by a careful tweaking of the rel-

evant parameters. But, that does not seem like the rightway to go if one wants to produce robot systems that canbe deployed robustly over long periods of time across envi-315

ronmental conditions. In fact, any strategy that hinges oncarefully selected parameters is almost certainly doomedto fail in the context of long duration autonomy. A moreappropriate path forward can instead, once again, be foundin ecology.320

3.2. Goal-Driven Behaviors

Central to individual fitness is reproductive success andthe associated mating ecology, whereby certain behaviorsoccupy a special place in the behavioral taxonomies in thatan individual’s reproductive success should be viewed as325

an objective, or goal, rather than as a constraint, Alcockand Rubenstein (2009). For these reasons, the introduced,

constraint-based design should be augmented to includegoal-driven behaviors as well.

Since it is not entirely clear what “mating” would ac-tually entail for robots, we can simply take the idea of amating behavior as a proxy for any type of goal-driven be-haviors, such as coverage, search, perimeter protection, orsignal tracking behaviors. As such, the ecological take onlong duration autonomy would be

minu ‖u− ugoal‖2

subject to C(hsurvive[x;u]) ≥ 0,(6)

where hsurvive encodes the set of ecological constraints,330

and ugoal is the nominal control input that corresponds tothe goal-driven behavior of the animal/robot.

Returning to the persistent environmental monitoringtask, the problematic aspect of specifying this as a collec-tion of robots asked to literally do nothing subject to themonitoring constraint was that the coverage behavior wasnot sufficiently well executed due to the disjunctive bal-ance between the coverage and charging constraints. Tothis end, we can simply remove hcoverage and instead letthe constraint be

hsurvive(x) = henergy(x) ∧ hcharge(x) ∧ hinteract(x). (7)

3.3. Persistent Environmental Monitoring

As coverage is the behavior that will have to be encodedas a goal-driven behavior in the context of persistent en-vironmental monitoring, we start by observing that thisproblem is quite well-studied. As before, given a numberof robots with positions p1, . . . , pN , we let the region ofdominance associated with robot i be the set of all pointsclosest to robot i, i.e., the Voronoi cell Di(x) = {q ∈D | ‖pi − q‖ ≤ ‖pj − q‖, ∀j 6= i}, where D is the totaldomain over which the robots are deployed. Moreover, letthe quality of the measurement deteriorate quadraticallywith the distance to the point that is being measured. Un-der this formulation, the performance of the entire teamcan be expressed using the so-called Locational Cost, e.g.,Cortes et al. (2002),

J(x) =

N∑i=1

∫Di(x)

‖pi − q‖2φ(q)dq.

As all agents are trying to minimize this cost in a co-ordinated manner, gradient descent is a natural choice:

pi = ui = −∂J(x)

∂pi

T

= −2

∫Di(x)

(pi − q)φ(q)dq.

But, as observed in Cortes and Egerstedt (2017), we canscale this using a positive, state-dependent gain to arrive ata continuous-time version of Lloyd’s celebrated algorithmfor producing Centroidal Voronoi Tessellations:

pi = ui = k(ρi(x)− pi), (8)

6

where k > 0 and ρi is the weighted center of Voronoi cellDi(x), i.e.,

ρi(x) =

∫Di(x)

qφ(q)dq∫Di(x)

φ(q)dq.

This construction allows us to formulate the persistent en-vironmental monitoring problem as

minu ‖u− ucoverage‖2

subject to C(hsurvive[x;u]) ≥ 0,(9)

where ucoverage is given in Equation (8) and hsurvive isgiven in Equation (7).335

An example of this in action is shown in Figure 1,where six differential-drive mobile robots are deployed onthe Robotarium, Pickem et al. (2017) – a remotely acces-sible swarm-robotics lab housed at the Georgia Instituteof Technology. In this example scenario, the robots are in-340

strumented with two extending prongs at different heightsthat, when charging, connect to two aluminum strips em-bedded in the arena walls. One of the metal strips suppliesa 5V input voltage to the robots, while the other serves asground, which allows the robots to drive up to the charging345

stations and recharge without human intervention. Thedistance to the closest charging location, d(pi), is thus thedistance to the point on the charging strip that is closestto the robot.

To show that the constraint-based formulation holds350

out promise to make the robots achieve persistent envi-ronmental monitoring, the measured battery voltage dataare shown in Figure 2. Note that even though the batterymodel used is highly idealized (and arguably even wrong),the fact that it is encoded as a constraint and not as an355

optimal controller, makes the system still satisfy the con-straint for all times thanks to the robustifying effects ofthe CBCs, Ames et al. (2014).

4. Conclusions: Persistent Autonomy On-Demand

This paper presents an ecologically inspired control360

design methodology for long duration autonomy, wherethe idea is that “survival conditions”, i.e., conditions forthe robots to be effective over long spatial and tempo-ral scales, are encoded as constraints. These constraintsare interpreted in terms of ecological constraints that al-365

low for the robots to do whatever they are supposed tobe doing – goal-driven behaviors – subject to persistencyconstraints. The efficacy of the proposed method is illus-trated in the context of persistent environmental modeling,where the goal-driven behavior is given in terms of a cover-370

age controller, and the constraints ensure that the robotsnever collide and that they never get stranded away froma charging location with completely depleted batteries.

This way of formulating the long duration autonomyproblem lends itself to an autonomy on-demand interpreta-375

tion, where the survival constraint is always in force whileusers can recruit robots to have them perform particular,

Figure 1: A sequence of salient frames from the persistent monitor-ing scenario from Equation (9) is shown. Six miniature robots aredeployed in the Robotarium and projected onto tthe Robotariumarena are the Voronoi cells associated with each robot. In the pro-gression, the top left robot is moving towards the charging strip atthe left edge of the arena in order to recharge.

goal-driven tasks. In other words, the on-demand aspect isgiven in terms of the performance-cost ‖u−uuser‖2, whilethe long duration autonomy piece is encoded through the380

ecological pressures in hsurvive(x). This way of formulat-ing the long duration autonomy tasks are of relevance toa number of application domains, where robots are de-ployed over long time scales, such as environmental moni-toring, precision agriculture, robotic warehousing, search-385

and-rescue robotics, and traffic monitoring, just to namea few.

Acknowledgements

This work was supported by the U.S. Office of NavalResearch under Grant #N00014-15-1-2115 on “Robot Ecolo-390

gies: Biologically Inspired Heterogeneous Teams”.

7

0 400 800 1200 1600 2000 2400 2800 3200 3600t [s]

3500

3650

3800

3950

4100

V [m

V]

Figure 2: Measured battery voltage data of the 6 robots executingthe monitoring scenario in Fig. 1. The top and bottom red hor-izontal lines depict the values of Emin and Emax in henergy andhcharge, respectively. As can be seen, the voltage always stays inthe prescribed region.

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