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Distributed dynamic coverage and avoidance controlunder anisotropic sensing

Dimitra Panagou, Dusan M. Stipanovic and Petros G. Voulgaris

Abstract—This paper addresses dynamic coverage in multi-agent systems along with certain safety and convergence guaran-tees. We consider anisotropic sensing for each agent, realizedas conical sensing footprints and coverage functionals. Thismodeling results in asymmetric (directed) interactions amongagents, in the sense that connected agents may either all be inthe same mode (avoidance) or in different modes (avoidance andcoverage). We build local and global coverage strategies whichforce the agents to collaboratively search a domain of interest,and avoidance strategies which waive the assumption on onlypairwise interactions among agents. The proposed approach issuitable for surveillance applications where agents explore andgather sufficient information about an environment. The efficacyof the approach is demonstrated through simulation results.

I. INTRODUCTION

Mobile sensor and robotic networks are of major researchinterest, in part due to their usefulness in applications rangingfrom surveillance and monitoring, to situational awareness, toindustrial and domestic robots. A mobile network is typicallyrealized as a group of agents which are characterized by localsensing capabilities and which collaborate in order to achievea global objective, often addressed as coverage.

Coverage formulations can be broadly classified as staticcoverage control and dynamic coverage control. Static cov-erage mainly addresses the optimal placement of sensors tocover a region and reduces to finding control laws whichdeploy mobile sensing agents to the centroids of Voronoi cellsin a Voronoi partitioning of a given domain [1]–[5]. On thecontrary, the term dynamic coverage has traditionally beenused to describe problems where a group of mobile sensingagents is deployed to search an area sufficiently well overtime [6]–[13]. The main concept which realizes sufficient (oreffective) coverage is a parameter C?, which is associatedwith the amount of time that a sensing agent should spendon every point of the area. Similar in spirit is the awarenesscoverage problem, where the desired coverage level is definedas a non-decreasing differentiable function of the unknowndensity distribution [14], [15].

One common thread in the earlier works on dynamiccoverage control is the consideration of isotropic sensing

This work has been supported by Qatar National Research Fund underNPRP Grant 4-536-2-199 and AFOSR grant FA95501210193.

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

Dusan M. Stipanovic is with the Coordinated Science Laboratory andthe Industrial and Enterprise Systems Engineering Department, University ofIllinois at Urbana-Champaign; dusan@illinois.edu.

Petros G. Voulgaris is with the Coordinated Science Laboratory andthe Aerospace Engineering Department, University of Illinois at Urbana-Champaign; voulgari@illinois.edu.

capability for each agent. More specifically, each agent isassumed to have a bell-shaped sensing function over a circularsensing footprint, whose maximum value is attained at thecenter of each agent. However, although this model is suitablefor agents carrying sensors such as laser range finders, itis not realistic for agents equipped with onboard cameras.Anisotropic sensing for coverage control problems has beenconsidered in [16]–[19]; however, the formulation in thesecontributions is related to static coverage control, i.e., refersto agents covering the area in a deployment sense.

This paper is in part motivated by surveillance and ex-ploration applications where vision is the main means of in-formation gathering, and addresses dynamic coverage controlin a multi-agent setting along with certain safety guarantees.We build upon our earlier work [20], [21] and considercoverage functionals which encode field-of-view and rangeconstraints in the forward-looking direction, or in other words,a conical forward sensing footprint for each agent, and thedegradation of sensing close to the boundary of the conicalsensing footprint. The main difference compared to [20], [21]is the derivation of novel avoidance control strategies for eachagent, which waive the earlier assumption on pairwise onlyinteractions among agents. More specifically, we explicitlyconsider the cases when anisotropic sensing results in directedinteractions among agents, and as thus, in agents that operatein different modes. For example, agent i senses agent j andthus gets in avoidance mode, yet agent j does not sense agenti and therefore might be either in coverage mode, or avoidancemode with respect to (w.r.t.) some other neighboring agent k.We provide control strategies along with sufficient conditionson collision avoidance for these cases.

The paper is organized as follows: Section II describesthe system modeling along with the considered sensing func-tionals. The technical analysis on the derivation of the localcoverage and avoidance control strategies is given in SectionsIII and IV, respectively. The supervisory global coveragestrategy is given in Section V and simulation results arereported in Section VI. Section VII summarizes our resultsand thoughts on future research, while Section VIII providessome pertinent acknowledgments. Finally, the Appendix IXincludes the proof of Theorem 2.

II. MODELING AND PROBLEM FORMULATION

Consider a group of N mobile agents, whose motion isgoverned by unicycle kinematics:xiyi

θi

=

cos θi 0sin θi 0

0 1

[uiωi

], (1)

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where i ∈ 1, . . . , N, pi = [xi yi]T is the position vector

and θi is the orientation of an agent i w.r.t. a global Cartesiancoordinate frame G, and ui, ωi are the linear and angularvelocities of agent i, respectively, expressed in the body frameBi. Denote qi =

[piT θi

]Tthe state vector of agent i.

Each agent i is assumed to have a fixed onboard camera oflimited angle-of-view and to be able to detect objects which liewithin a limited range in the forward-looking direction. Thesespecifications define a forward sensing region for each agenti, which is realized as a circular sector SFi of radius Ri > 0and angle 2αi > 0 (Fig. 1).

Furthermore, each agent is assumed to have a rear proximitysensor, whose sensing footprint SBi is for simplicity modeledas the symmetric circular sector of SFi w.r.t. the yBi body-fixed axis of agent i. Thus, each agent i is also able to detectobjects lying within a limited range in the backward direction.The role of the rear proximity sensor is to ensure that eachagent i will be able to detect and avoid objects when forcedto move backwards. The effective sensing area for agent i isthen defined as Si := SFi

⋃SBi.

Fig. 1. The forward sensing footprint SFi for agent i.

Coverage Functionals: Consider the functions cki : R3 →R defined as:

c1i = Ri2 − (x− xi)2 − (y − yi)2, (2a)

c2i = αi − φi, (2b)c3i = αi + φi, (2c)

where k ∈ 1, 2, 3, p = [x y]T is the position vector of a

point on R2, pi = [xi yi]T is the position vector and θi is

the orientation of agent i, and φi = atan2(y−yi, x−xi

)−θi

is the angle of a point p ∈ R2 w.r.t. the body frame Bi ofagent i. The region of the state space where all functions (2)take nonnegative values encodes the circular sector of radiusRi and angle 2αi centered at pi. This circular sector modelsthe forward sensing footprint SFi for agent i (Fig. 1). Theanalytical representation of the backward sensing footprint SBiis derived by considering the mirror of the sensing footprintSFi w.r.t. the body-fixed yBi axis of agent i. This analyticalderivation is straightforward and omitted here in the interest ofspace. In the sequel we assume that all agents have the samesensing capabilities, i.e., Ri = R, αi = α.

Let us now note that the barrier function:

Bi =1

max0, c1i+

1

max0, c2i+

1

max0, c3i

tends to +∞ as the k-th constraint cki → 0+, i.e., as a pointp on the interior of the set SFi approaches the boundary ∂SFiof SFi. To keep notation compact, denote max0, cki = Cki,k ∈ 1, 2, 3, and consider the function:

Si(qi, p) =1

Bi=

C1i C2i C3i

C2i C3i + C1i C3i + C1i C2i, (3)

which is zero on the exterior of the sensing footprint SFi, zeroon the boundary ∂SFi except for the points where any twoof the constraints cki are concurrently zero, and positive onthe interior of SFi. Furthermore, (3) tends to zero as the k-thconstraint cki → 0+ (Fig. 2).

Fig. 2. The sensing model Si(qi, p) (3) for agent i.

Remark 1: The function Si(qi, p) can be used as a realisticmodeling of the vision capability for each agent i, in the sensethat the quality of vision-based sensing typically degrades asan object lies far from the camera or close to the boundariesof the camera angle-of-view. In other words, Si(qi, p) encodesthat “seeing” becomes less effective as objects lie closer to theboundaries of the sensing footprint SFi.

Following our earlier work [6], the coverage functional foragent i is defined as:

Qi(t, p) =

∫ t

0

Si (qi(τ), p) dτ. (4)

This functional depends on the amount of time spent on eachpoint p of the sensing footprint SFi throughout the motionof agent i, realized via the position trajectories pi(τ) and theorientation trajectories θi(τ), τ ∈ [0, t], as well as and on thevalue of the sensing function (3) at each point p ∈ SFi, i.e., onthe quality of sensing at each point p in the sensing footprintSFi. Furthermore, the functional encodes that sensing overtime is not uniform along the sensing footprint SFi. Morespecifically: (4) is zero on the boundary ∂SFi of the sensingfootprint SFi, since the sensing function Si(qi(τ), p) is bydefinition zero there, and depends on the amount of time tand the value of the sensing function Si(qi, p) on the points p

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in the interior of the sensing footprint SFi. In other words, thecoverage level is lower at the points over which the sensingquality is lower. The sensing function (3) considered here takesits maximum value on the points p of the sensing footprintSFi where φi = 0. Finally, let us note that since the initialconditions for the functional (4) are points: (t0, p(t0)), i.e., notfunctions, we are still working in a finite-dimensional space.

Local and Global Coverage Errors: Given a compactdomain D, we are interested in deriving control strategieswhich guarantee that the agents cover (or search) the entiredomain up to a satisfactory level. The term “satisfactory level”means that each point p of D should be sensed by at leastone agent for a sufficient amount of time. The notion ofsatisfactory coverage is mathematically captured via the useof a positive constant C?. The value of this constant is chosendepending on how well we would like to search the area;larger values of C? are meant to force each agent to spendmore time sensing each point in the domain D. To proceedwith the definitions of local and global coverage, let us definethe function h(w) = (max0, w)3, so that its first derivativereads: h′ = dh

dw = 3(max0, w)2, and its second derivativereads: h′′ = d2h

dw2 = 6(max0, w), and denote by Di thecompact set of points p ∈ D contained in the sensing footprintSFi.

Definition 1: The local coverage error for the i-th agent isdefined as:

ei(t) =

∫Di(t)

h(C? −Qi(t, p)

)dp, (5)

where Di(t) is the set the points p ∈ D contained in thesensing footprint SFi at time t and C? is the positive constantrealizing the satisfactory coverage level.

Remark 2: Zeroing the error ei(t) implies that the sensingfunctional Qi(t, p) has reached the satisfactory level C? atsome time t.

Definition 2: The set of points p ∈ D whose coverage levelbecomes at least equal to C? via the motion of agent i at timeinstant t is denoted with Ci(t) and is called the set of coveredpoints, or the covered region by agent i, at time t.

Remark 3: The shape of the covered region Ci(t) dependson the sensing footprint SFi, the value of the sensing func-tional Si(qi, p) at each point p ∈ SFi , the satisfactory coveragelevel C?, and on the position and orientation trajectories pi(t)and θi(t) of agent i, i.e., on the control laws ui(t), ωi(t) whichgovern the motion of agent i.

Definition 3: The global coverage error is defined as:

E(t) =

∫D

h(C? −

N∑i=1

Qi(t, p))dp. (6)

Assumption 1: We assume a compact domain D, populatedwith obstacles whose geometry is known to the agents andwhich are located sufficiently sparse so that at least one agentcan safely move among them and cover the area.

Avoidance Control using Conical Avoidance Functions:Safety for the multi-robot network requires that each agenti should avoid collision with any other agent j 6= i, wherei, j ∈ 1, . . . , N, as well as with any physical static obstacles

in the domain D. In [6] the avoidance control between agentsi and j has been encoded via functions of the form:

vij =

(min

0,‖pi − pj‖2 −R2

‖pi − pj‖2 − r2

)2

, (7)

where pi, pj the position vectors of agents i and j, respectively,R > r > 0 are detection and the safety radii around eachagent, and ‖ · ‖ stands for the standard Euclidean norm. Forreasons that will become clear in the sequel, we consider thescaled function:

Vij = 1− e−vij , (8)

so that Vij takes its values in the interval [0, 1).Here we consider that each agent i is capable of detecting

objects (static obstacles and other agents j 6= i) that lie in itssensing area Si, which by definition is the union SFi

⋃SBi

of the sensing footprints described earlier in Section II. Sincethe sensing area Si is no longer a circular region, collisionavoidance can not be encoded using (7) only. Thus, a newform of avoidance functions is developed.

( ),j jx y

( )2 0:i i i jc tα φ+ =

( )3 0:i i i jc tα φ− =

( )i j tφ

( )i tθ

( ) ( )( ),i ix t y t

α

β γij i jp p p= −

i jp ⊻Fig. 3. An agent j is effectively sensed by agent i only when it enters inthe sensing area Si of agent i.

We initially consider collision avoidance w.r.t. objects lyingin the forward sensing footprint SFi. Consider Fig. 3 and notethat the active constraint function c3i : α− φij(t) = 0, whereφij(t) := atan2

(yj(t)− yi(t), xj(t)− xi(t)

)− θi(t) encodes

that at some time instant t, agent j lies on the boundary of SFiand becomes visible to agent i. A meaningful maneuvering foragent i could be then to move so that agent j gets out of SFi.To this end, we consider the angle 0 < β < α shown in Fig. 3and penalize the position trajectories pj(t) = [xj(t) yj(t)]

T

of agent j from approaching the interior of the set SFi where|φij(t)| < β. This requirement can be encoded in a similarspirit to the definition of the avoidance function (7) by definingthe function:

wij =

(min

0,α− φijβ − φij

+α+ φijβ + φij

)2

. (9)

The function wij is: (i) zero for |φij | ≥ α, i.e., at the exteriorof the open set SFi, (ii) positive when the second argument

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−2

0

2

4

6

8

10

−6

−4

−2

0

2

4

6

0

0.2

0.4

0.6

0.8

1

x axisy axis

Aij=

σ ijPij+

(1−σ

ij)Vij

Fig. 4. The conical avoidance function Aij for agent i w.r.t. an agent jlying in the sensing footprint SFi, given by (13).

in the minimum function gets negative values, which occursfor β < |φij | < α, and (iii) tends to +∞ as |φij | tends to β.

In this sense, it can be used as a Lyapunov-like functionencoding that the value of the angle trajectories |φij(t)| shouldalways remain greater than β. We take the scaled function:

Wij = 1− e−wij , (10)

so that Wij takes its values in the interval [0, 1).In order to combine equations (8) and (10) into a single

function, we consider their product:

Pij = Vij Wij . (11)

The function (11) vanishes when at least one of (8), (10) iszero, and varies in (0, 1) anywhere else. Note that this functiondoes penalize system trajectories with β < |φij(t)| < α fromtending to φij(t) = β, yet does not penalize system trajectorieswith |φij(t)| < β from tending to ‖pij‖ = r. In simpler words,this function does not encode that agent i maintains a safedistance ‖pij‖ > r w.r.t. agent j when |φij(t)| < β. For thisreason, we consider a bump function of the form:

σij =

0, if |φij | < β

|φij |−βα−β , if β ≤ |φij | ≤ α

1, if |φij | > α

(12)

and encode the transition between Pij and Vij as:

Aij = σij Pij + (1− σij) Vij . (13)

This function encodes that for |φij | < β one has σij = 0and as thus Aij = Vij . This further reads that the term (8) isactive in this region of the state space, rendering the desirableavoidance objective. The function (13) is depicted in Fig. 4for agent i positioned at pi = [0 0]

T with orientation θi = 0rad. In summary, the conical avoidance function Aij for agenti w.r.t. an agent j lying in the sensing footprint SFi in termsof the bump function (12) reads:

Aij =

0, if σij = 1σij Pij + (1− σij) Vij , if 0 ≤ σij ≤ 1

Vij , if σij = 0.(14)

The construction of a conical avoidance function for agent iw.r.t. an agent j lying in the backward sensing footprint SBifollows exactly the same logic, with the difference that theangle φij takes values in the interval [π+ θi−α, π+ θi +α],instead of the interval [θi − α, θi + α], and is omitted here inthe interest of space.

III. LOCAL COVERAGE CONTROL STRATEGY

We first consider the local coverage control problem andseek control strategies which force the local coverage error(5) of each agent i converge to zero.

We consider an error function ei(t) that is equal to the areaintegral part of the time-derivative of the local coverage errorfunction (5); this is shown to be sufficient for the design andanalysis of the local coverage control laws. More specifically,we consider the error function:

ei(t) = −∫Di

h′ (C? −Qi(t, p))Si(qi(t), p

)dp. (15)

The time derivative of (15) reads as in (16).1 We can furtherwrite:

˙ei(t) = ai0(t)− ui(t)ai1(t)− ωi(t)ai2(t), (17)

where ai0(t), ai1(t), ai2(t) are given as:

ai0(t) =

∫Di

h′′ (C? −Qi(t, p))(Si(qi(t), p

))2dp, (18a)

ai1(t) =

∫Di

h′ (C? −Qi(t, p))(∂Si∂xi

cos θi(t)+∂Si∂yi

sin θi(t))dp,

(18b)

ai2(t) =

∫Di

h′ (C? −Qi(t, p))∂Si∂θi

dp. (18c)

We are now ready to design our local coverage control strategyand state the following theorem:

Theorem 1: If agent i moves under the control law:

uicov = −kcovu ai1(t), (19a)

ωicov = −kcovω ai2(t), (19b)

where kcovu , kcovω > 0, and ai1(t), ai2(t) are given by (18),then the local coverage error ei(t) defined by (5) converges tozero.

Proof: To draw conclusions on the convergence of thecoverage error (5) to zero, we study the evolution of the errorfunction (15) and its time derivative (16).

1In general, the derivative of the area integral J(t) =∫∫

D(t)

f(t, x, y)dxdy

of a function f(t, x, y) over a compact region D(t) which is boundedby a closed curve C(t), which is continuous and consists of smooth arcs,is given as: dJ(t)

dt=

∫∫D(t)

∂f(t,x,y)∂t

dxdy + I(t), where I(t) =∮C(t)

f(t, x, y)(

∂xdt

dy − ∂ydt

dx), with the integration performed along the

counterclockwise direction. The considered sensing function Si(t, x, y) isexactly zero almost everywhere on the boundary C(t), hence the term I(t) iszero. This justifies why the term involving the line integral has been droppedin the time derivative (16) of the error function (15).

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˙ei(t) = −∫Di

(−h′′ (C? −Qi(t, p))Si

(qi(t), p

)(Si(qi(t), p

))+ h′ (C? −Qi(t, p))

d

dt

(Si(qi(t), p

)))dp

=

∫Di

h′′ (C? −Qi(t, p))(Si(qi(t), p

))2dp −

∫Di

h′ (C? −Qi(t, p))d

dt

(Si(qi(t), p

))dp, (16)

where:d

dt

(Si(qi(t), p

))=∂Si∂xi

xi(t) +∂Si∂yi

yi(t) +∂Si∂θi

θi(t)(1)=

(∂Si∂xi

cos θi(t) +∂Si∂yi

sin θi(t)

)ui(t) +

∂Si∂θi

ωi(t).

First, note that Di = int(SFi)⋃∂SFi, and that by defi-

nition: ∂Si∂xi

= ∂Si∂yi

= ∂Si∂θi

= 0, Si = 0 on ∂SFi. Thus theintegrals (18) over the domain Di are equal to the integrals ofthe same functions over the interior of the sensing footprintSFi. Now, since the error function (15) is non-positive, itfollows that for having this signal converging to zero, itsuffices that its time derivative (16) is positive [6]. Under thecontrol law (19) the time derivative (16) reads:

˙ei(t) = ai0(t) + kcovu (ai1(t))2

+ kcovω (ai2(t))2,

which is clearly nonnegative, since the term ai0(t) out of (18a)is nonnegative. Thus, the error function (15) is non-decreasing.Now, its time derivative (16) vanishes when the integrals(18a), (18b), (18c) are concurrently zero. This condition istrue only on the set: P = p ∈ int(SFi) | h′′(C? −Qi(t, p)) = 0

⋂h′(C? − Qi(t, p)) = 0, which reduces to

the set: P = p ∈ int(SFi) | Qi(t, p) ≥ C?. Consequently,the error function (15) is a negative increasing function thatvanishes only on the set P , where the value of (15) is zero,that is, h′(C? − Qi(t, p)) = 0. This further implies thath(C? −Qi(t, p)) = 0, i.e., that the local coverage error (5) isdriven to zero. This completes the proof.

Remark 4: The physical interpretation of the control law(19) is that agent i does not stop moving unless at sometime t it holds that all the points p ∈ int(SFi(t)) which arecontained in the interior of the sensing footprint SFi have beeneffectively covered.

Remark 5: The control law (19) depends only on thecurrent state of each agent; thus, it can be implemented ina decentralized fashion.

Remark 6: An agent moving under (19) is said to be incoverage mode.

IV. AVOIDANCE CONTROL STRATEGY

The function Aij given by (14) offers an encoding ofcollision avoidance for agent i w.r.t. an agent j lying in theforward sensing footprint SFi. Avoidance functions of theform (14) can be defined w.r.t. an agent j lying in the backwardsensing footprint SBi as well, with the difference that theangle φij in this case is an angle taking values in the interval[π + θi − α, π + θi + α]. In the sequel, we denote with Aijthe avoidance function for agent i w.r.t. an agent i lying in thesensing area Si = SFi

⋃SBi. Then, the function:

Ai =∑j∈Ni

Aij (20)

encodes collision avoidance for agent i w.r.t. all its neighboragents j, where Ni denotes the set of agents j 6= i lying inSi = SFi

⋃SBi, while the function:

A =

N∑i=1

Ai =

N∑i=1

∑j∈Ni

Aij

(21)

encodes collision avoidance for the multi-robot system.We consider the case when interaction between a pair of

agents is directed, i.e., agent j lies in the sensing region ofagent i 6= j, but not vice versa. In this case, agent i getsin avoidance mode and needs to ensure that collisions areavoided, despite that agent j might not be in avoidance mode.

Theorem 2: Assume that interaction among pairwise con-nected agents is directed, i.e., at least one agent j lies in thesensing region of at least one agent i 6= j, but not vice versa.Then the motion of all N agents is collision-free under thecontrol law:

(ui, ωi) =

(ucovi , ωcovi ), if

∏j 6=i

σij = 1

(uavi , ωavi ), if 0 <

∏j 6=i

σij < 1

(uavi , ωcovi ), if

∏j 6=i

σij = 0

, where:

uavi = kavi sgn

∑j∈Ni

(∂Vij∂pi

[cos θisin θi

]) , (22)

kavi >3 kcovu α(C?R)2

cosα, kavi >

maxj∈Nnoni,A

kj

cosα

ωavi is given by (32), ucovi and ωcovi are given by (19), andNnoni,A denotes the set of neighbor agents j of agent i that

are in avoidance mode w.r.t. their neighbor agents k 6= i. Theproof is given in Appendix IX.

V. GLOBAL COVERAGE CONTROL STRATEGY

The global coverage controller addresses the case when thelocal coverage errors become less than a predefined threshold,ei(t) ≤ ε, ∀i ∈ 1, 2, . . . , N, where ε > 0 is a sufficientlysmall number. This condition expresses that either the agentshave almost stopped moving while in coverage mode due tothe very small control inputs out of (19), or that the agentshave ended up in deadlocks due to the switching between localcoverage and avoidance controllers. The question is how toensure that the entire domain D will be covered up to thesatisfactory coverage level C?.

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To address this problem, we build upon the logic developedin our earlier works [10], [12]. The idea there is that, atsome time instant tm when the absolute rate of change|E(tm)−E(tm−1)

tm−tm−1| of the global coverage error E(tm) becomes

sufficiently small, each agent i selects a new waypoint p?,i tomove to, based on a heuristic which picks an uncovered pointclose to each agent. This heuristic selection is based on theformula [10], [12]:

wi = µ1‖pi − p‖+ µ21

(C? − C|p)2, (23a)

p?,i = arg minp

(wi), (23b)

where C|p represents the coverage level at a point p ∈ D,and µ1, µ2 > 0 denote weights leveraging between distanceand the current coverage level C|p at points p. As soon as theagents reach their waypoints in the uncovered regions, theyget back into their local coverage mode, and so on.

The procedure we use here is similar to the spirit describedabove. More specifically we implement the following globalcoverage strategy:

(i) If at time instant tk ≥ 0 the global coverage errorE(tk) 6= 0 and its absolute rate of change |E(tk)−E(tk−1)

tk−tk−1|

is less than a predefined level, then the agents have eitherdriven their local coverage errors ei(t) very close to zero,or have ended in a deadlock situation.

(ii) The agents decide on new waypoints within the uncov-ered regions based on the (heuristic) formula (23). Notealso the agents’ orientations when reaching the selectedwaypoints should be such that the sensing footprints lie inthe uncovered regions, where the local coverage errors arenon-zero. Hence the desired orientation for each agent atits waypoint is selected in a randomized manner, whilewaypoints lying closer than the safe distance to otheragents or obstacles are excluded. The motion of the agentstowards their waypoints is guaranteed to be safe underalgorithms such as those in [22], [23].

(iii) Once an agent has reached its assigned waypoint, itswitches to its coverage control law (19) and startsmoving towards driving its coverage error (5) to zero.

(iv) If the rate of change of the global coverage error becomesagain smaller than the predefined threshold, then the sameprocess is repeated, and so on.

The proposed global coverage control strategy forces E(t)to decrease, provided that appropriate waypoints exist, i.e.,provided that the uncovered points returned by formula (23)do not lie within the safe region around the static obstaclesand other agents. Guaranteeing the existence of such points apriori depends on the initial conditions of the agents, as wellas on the sensing/coverage parameters and the control gainsof the agents, which dictate the accomplished local coveragelevel per agent, as well as the resulting agent trajectories.As thus, obtaining explicit conditions on global coverageaccomplishment is a rather intractable task. The explorationof special cases for which explicit conditions can be obtainedis left open as a topic for future research.

VI. SIMULATION RESULTS

The proposed coverage control strategy is evaluated throughnumerical simulations. We consider a scenario of 4 agentswith coverage capabilities realized via the coverage functionals(4), and with avoidance capabilities realized via the avoidancefunctions (14). The agents need to sufficiently explore an areaof dimension da = 15. The satisfactory coverage level isset equal to C? = 1, and this selection is guided based onthe maximum value of the considered sensing function Si(·),which in this scenario is approximately 0.2. The sensing andcoverage parameters are R = 6, r = 3, α = π

6 . The agents canbroadcast information to other agents regarding the areas theyhave individually covered, so that at least one agent (calledthe supervisor) has access to the globally covered area.

In Fig. 5, blue stands for totally uncovered area (i.e., zerotime has been spent on these points of D by the agentsfootprints), yellow stands for totally covered area (i.e., agentsfootprints have stayed on these points of D for the satisfactoryamount of time encoded via C?), while color variationsbetween blue and yellow stand for partially covered area,i.e., for points of D which have been sensed by the agentsfootprints but not for the satisfactory amount of time.

The agents mostly move under their coverage control laws(19), with the control gains set equal to kcovu = 0.535,kcovω = 0.05. The evolution of achieving the satisfactorycoverage level is shown throughout Fig. 5(a)-5(o), while theevolution of the (normalized) global coverage error E(t) isdepicted in Fig. 6(b). About 55% of the total area of thedomain D has been covered at simulation time t = 50 sec.Up to this time, all agents have been driven by their localcoverage control laws. Naturally, the area of the satisfactorilycovered regions by a given number of agents under thecontrol law (19) depends on the coverage (sensing) function(3), the parameter C?, the control law gains in (19) andthe initial conditions.2 The global coverage controller takesaction for the first time at simulation time t = 50 sec andremains active for one simulation cycle, driving the agents towaypoints in the uncovered areas of the domain D. The localcoverage errors of the agents at these new initial conditionsare non-zero and the local coverage controllers get active,forcing the global coverage error to decrease. The motionof the agents towards their waypoints is not simulated here.The global coverage controller activates again at simulationtimes t = 100, 130, 150, 180, 200, 220, 240, . . . sec, remainingactive for one simulation cycle each time, and eventuallydriving the normalized global coverage error to 0.003921 atsimulation time t = 600 sec. In other words, 99.61% of thetotal area of the domain D has been effectively covered atsimulation time t = 600 sec. Finally, the evolution of the inter-agent distances is illustrated in Fig. 6(a), which demonstratesthat all pairwise distances dij remain greater than the criticaldistance r (depicted in red color), implying that the motion ofthe agents is collision-free.

2Note also that the shape of the uncovered areas can not be predicted apriori, as this would involve the computation of the forward reachable sets ofall agents for any possible initial condition, something which is impossibledue to the curse of dimensionality (or state explosion) even when only fewagents are concerned.

IEEE TRANSACTIONS ON 7

(a) t = 1 sec (b) t = 6 sec (c) t = 11 sec

(d) t = 16 sec (e) t = 21 sec (f) t = 26 sec

(g) t = 31 sec (h) t = 51 sec (i) t = 71 sec

(j) t = 91 sec (k) t = 101 sec (l) t = 151 sec

(m) t = 201 sec (n) t = 401 sec (o) t = 601 sec

Fig. 5. The evolution of the coverage level.

IEEE TRANSACTIONS ON 8

Time (seconds)0 100 200 300 400 500 600

Dis

tanc

e

0

5

10

15

20

25

30Inter-Agent Distances Avoidance

Detection

(a) Inter-agent Distances

Time (seconds)0 100 200 300 400 500 600

Err

or

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Normalized Coverage Error

X: 600Y: 0.003921

(b) Normalized Global Coverage Error

Fig. 6. The evolution of the inter-agent distances and the normalized global coverage error over time.

VII. CONCLUSIONS

This paper presented a dynamic coverage and avoidancecontrol design for agents with anisotropic sensing. Anisotropicsensing, realized as conic coverage functionals for each agent,models the degradation of effective vision-based sensing usingforward-looking onboard cameras and imposes directed inter-actions among agents which may be either in coverage modeor in avoidance mode. A switching control strategy comprisinglocal coverage, avoidance and global coverage controllers wasdesigned and its efficacy was demonstrated through numericalsimulations for unicycle robots operating in an environmentwith known circular obstacles. The key difference compared toour earlier work is the design of collision avoidance algorithmswhich waive the pairwise interaction assumption. Current workfocuses on the consideration of supervisory logics to orches-trate efficient global coverage, as well as on the considerationof agents with more complicated dynamics and constraints,such as Dubins-like vehicles.

VIII. ACKNOWLEDGMENTS

We would like to acknowledge the contribution of Dr.Gokhan Atinc and Dr. Christopher Valicka through the fruitfuldiscussions and their help on the simulation implementations.

IX. APPENDIX: PROOF OF THEOREM 2

Proof: We first provide some useful calculations regard-ing the considered gradient vectors. The gradients ∂vij

∂qi=[

∂vij∂xi

∂vij∂yi

∂vij∂θi

]and ∂vij

∂qj=[∂vij∂xj

∂vij∂yj

∂vij∂θj

]of (7),

and the gradients ∂wij∂qi

=[∂wij∂xi

∂wij∂yi

∂wij∂θi

]and ∂wij

∂qj=[

∂wij∂xj

∂wij∂yj

∂wij∂θj

]of (9) w.r.t. the state vectors qi =[

piT θi

]Tand qj =

[pjT θj

]T, respectively, are given on

page 9. With these at hand, note that:

∂vij∂qi

= −∂vij∂qj

, (24a)

∂wij∂pi

= −∂wij∂pj

. (24b)

After some algebraic manipulations we also have:

∂σij∂pi

= −∂σij∂pj

.

Then, it is straightforward to verify that the following sym-metry relation between the gradient vectors of Aij w.r.t. theposition vectors pi and pj holds everywhere:

∂Aij∂pi

= −∂Aij∂pj

. (25)

The time derivative of (20) reads as in (26). Of course, theanalytical expressions of ∂Aij

∂qi, ∂Aij∂qj

are different in each ofthe regions of the state space of agent i. Let us consider thesecases separately.

Avoidance strategies for σij = 1, ∀j 6= i: Out of (14) onehas: Aij = 0, thus (20) yields: Ai = 0. This equivalently readsthat agent i does not take into account the collision avoidanceobjective w.r.t. agents j 6= i which lie out of the sensing areaSi. The linear and angular velocity for agent i may thus bedictated by the coverage control law (19).

Avoidance strategies for σij = 0, ∀j 6= i: Out of (14) onehas: Aij = Vij , thus (20) yields: Ai =

∑j∈Ni

Vij . The time

derivative (26) of Ai under the avoidance control law (22)reads as in (27). The evolution over time of the avoidancefunction Ai for agent i depends on the motion of the connectedagents j 6= i. The first term in (27) is non-positive, implyingthat agent i is attempting to avoid collisions. The second termin (27) depends on the state trajectories qj(t) of the neighboragents j 6= i of agent i and on their control laws uj , whichfurther depend on the state trajectories qk(t) and the control

IEEE TRANSACTIONS ON 9

∂vij∂qi

=

[0 0 0] , if ‖pij‖ ≥ R;

[Ωij xij Ωij yij 0] , if r < ‖pij‖ < R;

not defined, if ‖pij‖ = r;

[0 0 0] , elsewhere

,∂vij∂qj

=

[0 0 0] , if ‖pij‖ ≥ R;

[−Ωij xij − Ωij yij 0] , if r < ‖pij‖ < R;

not defined, if ‖pij‖ = r;

[0 0 0] , elsewhere

where: Ωij =4 (R2 − r2) (‖pij‖2 −R2)(

‖pij‖2 − r2)3 , xij = xi − xj , yij = yi − yj ,

∂wij∂qi

=

[0 0 0

], if |φij | ≥ α;[

Ψij yij‖pij‖2

−Ψij xij‖pij‖2

Ψij

], if β < |φij | < α;

not defined, if |φij | = β;[0 0 0

], elsewhere

,∂wij∂qj

=

[0 0 0

], if |φij | ≥ α;[

−Ψij yij‖pij‖2

Ψij xij‖pij‖2

0], if β < |φij | < α;

not defined, if |φij | = β;[0 0 0

], elsewhere

,

where: Ψij = −16 φij (αβ − β2) (αβ − φij2)

(β − φij)3 (β + φij)3.

dAidt

=∑j∈Ni

dAijdt

=∑j∈Ni

(∂Aij∂qi

dqidt

+∂Aij∂qj

dqjdt

)=∑j∈Ni

(∂Aij∂pi

dpidt

+∂Aij∂θi

dθidt

+∂Aij∂pj

dpjdt

+∂Aij∂θj

dθjdt

)(25),(1)

=∑j∈Ni

∂Aij∂pi

(dpidt− dpj

dt

)+∑j∈Ni

∂Aij∂θi

ωi +∑j∈Ni

∂Aij∂θj

ωj . (26)

dAidt

(26),(8)=

∑j∈Ni

e−vij∂vij∂pi

(dpidt− dpj

dt

)(1)=∑j∈Ni

e−vij Ωij pijT

[cos θisin θi

]ui −

∑j∈Ni

e−vij Ωij pijT

[cos θjsin θj

]uj . (27)

laws uk(t) of the neighbor agents k 6= j of agent j, and soon. We will get back to this case later on.

Avoidance strategies for 0 < σij < 1: Out of (14) one has:Aij = σij Pij +(1−σij) Vij . The expression of the gradientsinvolved in (26) in this case are given by (29), where:

∂σij∂pi

= − 1

(α− β)Ψij

∂wij∂pi

,

∂σij∂θi

= − 1

(α− β)Ψij

∂wij∂θi

= − 1

α− β, (30)

Plugging (29) into (26) yields (31). Furthermore, setting theangular velocity ωi

av of agent i equal to (32) has the effectof canceling out the last terms in (31), yielding (27). Thusthe analysis on the time derivative of Ai reduces to the casebefore.

Let us now proceed with studying the evolution of thetime derivative of Ai given by (27), and consider the caseof pairwise directed interactions between agents, i.e., the casewhen an agent i senses at least one agent j 6= i, but not viceversa. Denote M ≤ N−1 the number of neighbor agents j toagent i, MC the number of neighbor agents in coverage modeand MA the number of neighbor agents in avoidance mode,such that M = MC + MA. Now, the neighbors in avoidancemode can be of the following two types: they can either beagents j which sense and avoid agent i (in this case, pairwise

undirected interactions are formed between agent i and anyof its neighbors j), or they can be agents which do not senseagent i, yet they are in avoidance mode because they senseand avoid at least one agent k 6= i (in this case the interactionbetween agent i and agent j is directed). Denote Mund

A thenumber of neighbor agents j in avoidance mode which senseagent i, forming therefore undirected interactions, and Mdir

A

the number of neighbor agents j in avoidance mode which donot sense agent i, forming therefore directed interactions withagent i, so that MA = Mund

A +MdirA .

Let us initially assume that all neighbor agents j of an agenti are in coverage mode, i.e., M = MC . We are interested inidentifying sufficient conditions so that agent i avoids agentsj. For the coverage control inputs one has that:

|ujcov| ≤ kcovu 3(C?)2 (αR2). (33)

Then the time derivative (27) of the avoidance functionAi of agent i w.r.t. its neighbor agents j under the linearvelocity controller (22), and the angular velocity controllergiven either by (19b), or by (32) (depending on the valueof the functions σij), reads as in (34). The worst-case sce-nario refers to the system configuration which renders thefirst term in (34) as large as possible. It is straightforwardto verify out of the geometry of the sensing system that:

IEEE TRANSACTIONS ON 10

∂Aij∂pi

= σij∂ (VijWij)

∂pi+∂σij∂pi

(VijWij) + (1− σij)∂Vij∂pi− ∂σij

∂piVij =

= σij

(Wij

∂Vij∂pi

+ Vij∂Wij

∂pi

)+∂σij∂pi

(VijWij) + (1− σij)∂Vij∂pi− ∂σij

∂piVij =

= σij

(Wij e

−vij ∂vij∂pi

+ Vij e−wij ∂wij

∂pi

)+∂σij∂pi

Vij (Wij − 1) + (1− σij) e−vij∂vij∂pi

=

(30)= σij

(Wij e

−vij ∂vij∂pi

+ Vij e−wij ∂wij

∂pi

)− 1

(α− β)Ψij

∂wij∂pi

Vij (Wij − 1) + (1− σij) e−vij∂vij∂pi

=

=∂vij∂pi

e−vij(1 + σij (Wij − 1)

)+∂wij∂pi

Vij

(σij e

−wij − 1

(α− β)Ψij(Wij − 1)

)=

=∂vij∂pi

e−vij(1− σij e−wij

)+∂wij∂pi

Vij

(σij e

−wij +1

(α− β)Ψije−wij

)=

=∂vij∂pi

e−vij(1− σij e−wij

)+∂wij∂pi

(1− e−vij ) e−wij(σij +

1

(α− β)Ψij

), (29a)

∂Aij∂θi

= σij

(Wij

∂Vij∂θi

+ Vij∂Wij

∂θi

)+∂σij∂θi

(VijWij) + (1− σij)∂Vij∂θi− ∂σij

∂θiVij

∂Vij∂θi

=0

= σij Vij e−wij ∂wij

∂θi+∂σij∂θi

Vij (Wij − 1) = σij Vij e−wij ∂wij

∂θi− ∂σij

∂θiVij e

−wij =

= Vij e−wij

(σij

∂wij∂θi

− ∂σij∂θi

)(30)= Ψij (1− e−vij ) e−wij

(σij +

1

(α− β)Ψij

), (29b)

∂Aij∂θj

= 0. (29c)

dAidt

=∑j∈Ni

(e−vij

(1− σije−wij

) ∂vij∂pi

+ e−wij(1− e−vij

)(σij +

1

(α− β)Ψij

)∂wij∂pi

)(dpidt− dpj

dt

)+∑j∈Ni

Ψij e−wij (1− e−vij )

(σij +

1

(α− β)Ψij

)ωi

= ui∑j∈Ni

e−vij∂vij∂pi

[cos θisin θi

]+ ui

∑j∈Ni

(−σije−wij

∂vij∂pi

+ e−wij(1− e−vij

)(σij +

1

(α− β)Ψij

)∂wij∂pi

)[cos θisin θi

]− uj

∑j∈Ni

e−vij∂vij∂pi

[cos θjsin θj

]− uj

∑j∈Ni

(−σije−wij

∂vij∂pi

+ e−wij(1− e−vij

)(σij +

1

(α− β)Ψij

)∂wij∂pi

)[cos θjsin θj

]+ ωi

∑j∈Ni

Ψij e−wij (1− e−vij )

(σij +

1

(α− β)Ψij

)(31)

ωiav =

−ui∑j∈Ni

(−σije−wij ∂vij

∂pi+ e−wij (1− e−vij )

(σij + 1

(α−β)Ψij

)∂wij∂pi

) [cos θisin θi

]+uj

∑j∈Ni

(−σije−wij ∂vij

∂pi+ e−wij (1− e−vij )

(σij + 1

(α−β)Ψij

)∂wij∂pi

) [cos θjsin θj

]∑j∈Ni

Ψij e−wij (1− e−vij )(σij + 1

(α−β)Ψij

) (32)

minpj∈Si

(∣∣∣∣pijT [cos θisin θi

]∣∣∣∣) = ‖pij‖ cosα. To see why, consider

that this inner product expresses the projection of the vectorpij onto the xBi body-fixed axis of agent i, and the value of

this projection becomes minimum for an agent j lying on theboundary of the sensing region of agent i, i.e. when φij = α.Therefore (34) further reads as in (35), which in turn yieldsthe following sufficient condition on avoiding collisions for the

IEEE TRANSACTIONS ON 11

dAidt

=∑j∈Ni

e−vij∂vij∂pi

(dpidt −

dpjdt

)= −

∑j∈Ni

e−vij |Ωij |(pij

T[

cos θisin θi

]ui−pijT

[cos θjsin θj

]uj

)

= −ui∑j∈Ni

e−vij |Ωij | pijT[

cos θisin θi

]+∑j∈Ni

e−vij |Ωij | pijT[

cos θjsin θj

]uj = −kavi

∣∣∣∣∣ ∑j∈Ni e−vij |Ωij | pijT[cθisθi

]∣∣∣∣∣+ ∑j∈Ni

e−vij |Ωij | pijT[cθjsθj

]uj

≤ −kavi∣∣∣∣∣ ∑j∈Ni e−vij |Ωij | pijT

[cθisθi

]∣∣∣∣∣+ ∑j∈Ni

e−vij |Ωij | ‖pij‖ |ucovj |

= −kavi

∣∣∣∣∣ ∑j∈Ni e−vij |Ωij | pijT[cθisθi

]∣∣∣∣∣+ kcovu 3(C?√aR)2

∑j∈Ni

e−vij |Ωij | ‖pij‖ . (34)

dAidt ≤−k

avi cosα

∑j∈Ni

e−vij |Ωij | ‖pij‖+ kcovu 3(C?√aR)2

∑j∈Ni

e−vij |Ωij | ‖pij=−(kavi cosα−kcovu 3(C?√αR)2)

∑j∈Ni

e−vij |Ωij | ‖pij‖ (35)

case of an agent i being connected to only covering agents j:

kavi >3 kcovu α(C?R)2

cosα.

Let us now consider the case when an agent i has MC

neighbors in coverage mode and MA neighbors in avoidancemode. Denote NiC = 1, 2, . . . ,MC the subset of coveringneighbors, N und

iA = MC + 1,MC + 2, . . . ,MC +MundA the

subset of undirected avoiding agents, and N diriA = MC +

MundA + 1,MC + Mund

A + 2, . . . ,M the subset of directedavoiding neighbors, NiA = N und

iA

⋃N diriA = MC + 1,MC +

2, . . . ,M the subset of both undirected and directed avoidingagents, where Ni = NiC

⋃NiA = NiC

⋃N undiA

⋃N diriA .

The time derivative (27) of the avoidance function Ai ofagent i w.r.t. its neighbor agents j reads as in (36). Again,the worst-case scenario refers to the system configurationwhich renders the first term in (36) as large as possible, i.e.,for agents j lying on the boundary of the sensing regionof agent i. Then (36) further reads as in (37). Now notethat the time evolution of (37) depends highly on its lastterms, i.e., on the interactions of agent i w.r.t. its neighboragents in avoidance mode. It is very important to stress thatneighbor agents j in directed interaction with agent i are notnecessarily “cooperative” to agent i, or in other words, donot necessarily render their corresponding term in the lastterm of (37) non-positive. Everything depends on the signumof the avoidance control law uavj given by (22), which in turndepends on the interactions of agent j w.r.t. its own neighbors,etc. These interactions are captured within the sgn(·) term of(22), yielding either negative or positive bounded velocities forany agent, i.e., uavj = −kj or uavj = kj . With slight abuse ofnotation, we classify the neighbor in avoidance mode agents jas “cooperative” and “non-cooperative” to agent i as follows:

• If sgn(pij

T[

cos θjsin θj

]uavj

)= −1 then agent j is “cooper-

ative” to agent i since it renders the corresponding termin (37) non-positive.

• If sgn(pij

T[

cos θjsin θj

]uavj

)= 1 then agent j is “non-

cooperative” to agent i since it renders the correspondingterm in (37) non-negative.

Let us denote by N coAi, Nnon

Ai the sets of cooperative and non-cooperative avoiding agents, respectively. Then (37) furtherreads as in (38). Thus we conclude that: If all avoiding agents

are cooperative, then collisions are avoided. If at least oneof the avoiding agents is non-cooperative, then a sufficientcondition on the choice of the gain kavi of agent i is

(kavi cosα− maxj∈NnoniA

kj) > 0.

This completes the proof.

REFERENCES

[1] J. Cortes, S. Martınez, T. Karatas, and F. Bullo, “Coverage control formobile sensing networks,” IEEE Trans. on Robotics and Automation,vol. 20, no. 2, pp. 243–255, 2004.

[2] T. M. Cheng and A. V. Savkin, “Decentralized control of mobile sensornetworks for triangular blanket coverage,” in Proc. of the 2010 AmericanControl Conference, Baltimore, MD, USA, Jun. 2010, pp. 2903–2908.

[3] S. Ferrari, G. Zhang, and T. A. Wettergren, “Probabilistic track coveragein cooperative sensor networks,” IEEE Trans. on Systems, Man, andCybernetics, vol. 40, no. 6, pp. 1492–1504, Dec. 2010.

[4] M. Zhong and C. G. Cassandras, “Distributed coverage control and datacollection with mobile sensor networks,” IEEE Trans. on AutomaticControl, vol. 56, no. 10, pp. 2445–2455, Oct. 2011.

[5] M. Schwager, J. Slotine, and D. Rus, “Unifying geometric, probabilisticand potential field approaches to multi-robot coverage control,” Interna-tional Journal on Robotics Research, vol. 30, no. 3, pp. 371–383, Mar.2011.

[6] D. M. Stipanovic, C. Valicka, C. J. Tomlin, and T. R. Bewley, “Safe andreliable coverage control,” Numerical Algebra, Control and Optimiza-tion, vol. 3, pp. 31–48, 2013.

[7] I. I. Hussein and D. M. Stipanovic, “Effective coverage control formobile sensor networks with guaranteed collision avoidance,” IEEETrans. on Control Systems Technology, vol. 15, no. 4, pp. 642–657,Jul. 2007.

[8] P. Hokayem, D. Stipanovıc, and M. Spong, “On persistent coveragecontrol,” in Proc. of the 2007 IEEE Conference on Decision and Control,New Orleans, LA, USA, Dec. 2007, pp. 6130–6135.

[9] C. Franco, G. Lopez-Nicolas, D. M. Stipanovic, and C. Sagues,“Anisotropic vision-based coverage control for mobile robots,” in IROSWorkshop on Visual Control of Mobile Robots, Vilamoura, Algarve,Portugal, Oct. 2012, pp. 31–36.

[10] G. Atinc, D. M. Stipanovıc, P. G. Voulgaris, and M. Karkoub, “Su-pervised coverage control with guaranteed collision avoidance andproximity maintenance,” in Proc. of the 2013 IEEE Conference onDecision and Control, Florence, Italy, Dec. 2013, pp. 3463–3468.

[11] ——, “Swarm-based dynamic coverage control,” in Proc. of the 2014IEEE Conference on Decision and Control, Los Angeles, CA, Dec. 2014,pp. 6963–6968.

[12] G. Atinc, D. M. Stipanovıc, and P. G. Voulgaris, “Supervised coveragecontrol of multi-agent systems,” Automatica, vol. 50, no. 11, pp. 2936–2942, 2014.

[13] C. Franco, D. M. Stipanovic, G. Lopez-Nicolas, C. Sagues, andS. Llorente, “Persistent coverage control for a team of agents withcollision avoidance,” in provisionally accepted in the European Journalof Control, 2014.

IEEE TRANSACTIONS ON 12

dAidt

=∑j∈Ni

e−vij∂vij∂pi

(dpidt− dpj

dt

)= −

∑j∈Ni

e−vij |Ωij |(pij

T

[cos θisin θi

]ui − pijT

[cos θjsin θj

]uj

)= −ui

∑j∈Ni

e−vij |Ωij | pijT[cos θisin θi

]+∑j∈Ni

e−vij |Ωij | pijT[cos θjsin θj

]uj

= −kavi

∣∣∣∣∣∣M∑j=1

e−vij |Ωij | pijT[cos θisin θi

]∣∣∣∣∣∣+

MC∑j=1

e−vij |Ωij | pijT[cos θjsin θj

]ucovj +

M∑j=MC+1

e−vij |Ωij | pijT[cos θjsin θj

]uavj

≤ −kavi

∣∣∣∣∣∣M∑j=1

e−vij |Ωij | pijT[cos θisin θi

]∣∣∣∣∣∣+

MC∑j=1

e−vij |Ωij | ‖pij‖ |ucovj |+M∑

j=MC+1

e−vij |Ωij | pijT[cos θjsin θj

]uavj

≤ −kavi

∣∣∣∣∣∣M∑j=1

e−vij |Ωij | pijT[cos θisin θi

]∣∣∣∣∣∣+ kcovu 3(C?√aR)2

MC∑j=1

e−vij |Ωij | ‖pij‖+

M∑j=MC+1

e−vij |Ωij | pijT[

cos θjsin θj

]uavj .

(36)

dAidt≤ −kavi cosα

M∑j=1

e−vij |Ωij | ‖pij‖+ kcovu 3(C?√aR)2

MC∑j=1

e−vij |Ωij | ‖pij‖+

M∑j=MC+1

e−vij |Ωij | pijT[cos θjsin θj

]uavj

= −(kavi cosα− kcovu 3(C?

√aR)2

)MC∑j=1

e−vij |Ωij | ‖pij‖ − kavi cosα

M∑j=MC+1

e−vij |Ωij | ‖pij‖

+

M∑j=MC+1

e−vij |Ωij | pijT[

cos θjsin θj

]uavj . (37)

dAidt≤ −

(kavi cosα− kcovu 3(C?)2 (αR2)

)MC∑j=1

e−vij |Ωij | ‖pij‖

−(kavi cosα+ min

j∈N coiAkj

) ∑j∈N coiA

e−vij |Ωij | ‖pij‖ −(kavi cosα− max

j∈NnoniA

kj) ∑j∈NnoniA

e−vij |Ωij | ‖pij‖. (38)

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