Cooperative control of multi-agent systems to locatesource of an odor

Abhinav Sinha, Rishemjit Kaur, Ritesh Kumar and Amol P. Bhondekar

Abstract—This work targets the problem of odor sourcelocalization by multi-agent systems. A hierarchical cooperativecontrol has been put forward to solve the problem of locatingsource of an odor by driving the agents in consensus when atleast one agent obtains information about location of the source.Synthesis of the proposed controller has been carried out in ahierarchical manner of group decision making, path planningand control. Decision making utilizes information of the agentsusing conventional Particle Swarm Algorithm and informationof the movement of filaments to predict the location of the odorsource. The predicted source location in the decision level isthen utilized to map a trajectory and pass that informationto the control level. The distributed control layer uses slidingmode controllers known for their inherent robustness and theability to reject matched disturbances completely. Two cases ofmovement of agents towards the source, i.e., under consensusand formation have been discussed herein. Finally, numericalsimulations demonstrate the efficacy of the proposed hierarchicaldistributed control.

Index Terms—Odor source localization, multi-agent systems(MAS), sliding mode control (SMC), homogeneous agents, coop-erative control.

I. INTRODUCTION

A. Overview

Inspiration of odor source localization problem stems frombehavior of biological entities such as mate seeking by moths,foraging by lobsters, prey tracking by mosquitoes and bluecrabs, etc., and is aimed at locating the source of a volatilechemical. These behaviors have long been mimicked by au-tonomous robot(s). Chemical source tracking has attractedresearchers around the globe due to its applications in bothcivilian and military domains. A plethora of applications arepossible, some of which include detection of forest fire, oilspills, release of toxic gases in tunnels and mines, gas leaksin industrial setup, search and rescue of victims and clearingleftover mine after an armed conflict. A plume containingfilaments, or odor molecules, is generally referred to thedownwind trail formed as a consequence of mixing of con-taminant molecules in any kind of movement of air. Thedynamical optimization problem of odor source localizationcan be effectively solved using multiple robots working incooperation. The obvious advantages of leveraging multi-agent systems (MAS) are increased probability of success,

A. Sinha is with School of Mechatronics & Robotics, Indian Instituteof Engineering Science and Technology; and Central Scientific InstrumentsOrganization (CSIR- CSIO), India.email: sinha.abhinav.pg2016@mechatronics.iiests.ac.inR. Kaur, R. Kumar & A. P. Bhondekar are with CSIR- CSIO.emails: rishemjit.kaur@csio.res.in,riteshkr@csio.res.in,amolbhondekar@csio.res.in

redundancy and improved overall operational efficiency andspatial diversity in having distributed sensing and actuation.

B. Motivation

Odor source localization is a three stage problem– sensing,maneuvering and control. Some of reported literature on odorsource localization date back to 1980s when Larcombe et al.[1] discussed such applications in nuclear industry by con-sidering a chemical gradient based approach. Other works in1990s [2]–[6] relied heavily on sensing part using techniquessuch as chemotaxis [7], infotaxis [8], anemotaxis [9], [10]and fluxotaxis [11]. The efficiency of such algorithms waslimited by the quality of sensors and the manner in which theywere used. These techniques also failed to consider turbulencedominated flow and resulted in poor tracking performance.

Bio-inspired algorithms have been reported to maneuver theagents, some of which include Braitenberg style [12], E. colialgorithm [13], Zigzag dung beetle approach [14], silkwormmoth style [15]–[17] and their variants. A tremendous growthof research attention towards cooperative control has beenwitnessed in the past decade [18], [19] but very few haveaddressed the problem of locating source of an odor. Hayes etal. [20] proposed a distributed cooperative algorithm based onswarm intelligence for odor source localization and experimen-tal results proved multiple robots perform more efficiently thana single autonomous robot. A Particle Swarm Optimization(PSO) algorithm [21] was proposed by Marques et al. [22],[23] to tackle odor source localization problems. To avoid trap-ping into local maximum concentrations, Jatmiko et al. [23]proposed modified PSO algorithms based on electrical chargetheory, where neutral and charged robots has been used. Lu etal. [24] proposed a distributed coordination control protocolbased on PSO to address the problem. It should be notedthat simplified PSO controllers are a type of proportional-onlycontroller and the operating region gets limited between globaland local best. This needs complicated obstacle avoidancealgorithms and results in high energy expenditure. Lu et al.[25] also proposed a cooperative control scheme to coordinatemultiple robots to locate odor source in which a particle filterhas been used to estimate the location of odor source basedon wind information, a movement trajectory has been planned,and finally a cooperative control scheme has been proposed tocoordinate movement of robots towards the source.

Motivated by these studies, we have implemented a robustand powerful hierarchical cooperative control strategy to tacklethe problem. First layer is the group level in which the

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information about the source via instantaneous sensing andswarm intelligence is obtained. Second layer is designed tomaneuver the agents via a simplified silkworm moth algorithm.Third layer is based on cooperative sliding mode control andthe information obtained in the first layer is passed to the thirdlayer as a reference to the tracking controller.

C. Contributions

Major contributions of this paper are summarized below.1) As opposed to existing works on cooperative control

to locate source of odor, we have considered a moregeneral formulation by taking nonlinear dynamics ofMAS into account. When the uncertain function is zero,the problem reduces to stabilizing integrator dynamics.

2) The control layer is designed on the paradigms of slidingmode, a robust and powerful control with inherentrobustness and disturbance rejection capabilities. Thereaching law, as well as the sliding manifold in thisstudy are nonlinear and novel resulting in smoothercontrol and faster reachability to the manifold. Use ofsliding mode controller also helps in achieving a finitetime convergence as opposed to asymptotic convergenceto the equilibrium point. The proposed control providesstability and ensures robustness even in the presence ofbounded disturbances and matched uncertainties.

3) Odor propagation is non-trivial, i.e., odor arrives inpackets, leading to wide fluctuations in measuredconcentrations. Plumes are also dynamic and turbulent.As odor tends to travel downwind, direction of the windprovides an effective information on relative positionof the source. Hence, we have used wind informationbased on a measurement model describing movementof filaments and concentration information from swarmintelligence to locate the source of odor.

4) Formation keeping of agents to locate source of odorhas also been demonstrated in this work.

D. Paper Organization

After introduction to the study in section I, remainderof this work in organized as follows. Section II providesinsights into preliminaries of spectral graph theory and slidingmode control. Section III presents dynamics of MAS andmathematical problem formulation, followed by hierarchicaldistributed cooperative control scheme in section IV. Resultsand discussions have been carried out in section V, followedby concluding remarks in section VI.

II. PRELIMINARIES

A. Spectral Graph Theory for Multi-Agent Systems

A directed graph, also known as digraph is representedthroughout in this paper by G = (V, E ,A). V is the nonemptyset in which finite number of vertices or nodes are containedsuch that V = {1, 2, ..., N}. E denotes directed edge and is

represented as E = {(i, j) ∀ i, j ∈ V & i 6= j}. A is theweighted adjacency matrix such that A = a(i, j) ∈ RN×N.

The possibility of existence of an edge (i, j) occurs iff thevertex i receives the information supplied by the vertex j, i.e.,(i, j) ∈ E . Hence, i and j are termed neighbours. The set Nicontains labels of vertices that are neighbours of the vertex i.For the adjacency matrix A, a(i, j) ∈ R+

0 . If (i, j) ∈ E ⇒a(i, j) > 0. If (i, j) /∈ E or i = j ⇒ a(i, j) = 0.

The Laplacian matrix L [26] is central to the consensusproblem and is given by L = D −A where degree matrix,D is a diagonal matrix, i.e, D = diag(d1, d2, ..., dn) whoseentries are di =

∑nj=1 a(i, j). A directed path from ver-

tex j to vertex i defines a sequence comprising of edges(i, i1), (i1, i2), ..., (il, j) with distinct vertices ik ∈ V , k =1, 2, 3, ..., l. Incidence matrix B is also a diagonal matrixwith entries 1 or 0. The entry is 1 if there exists an edgebetween leader agent and any other agent, otherwise it is 0.Furthermore, it can be inferred that the path between twodistinct vertices is not uniquely determined. However, if adistinct node in V contains directed path to every other distinctnode in V , then the directed graph G is said to have aspanning tree. Consequently,the matrix L + B has full rank[26]. Physically, each agent has been modelled by a vertex ornode and the line of communication between any two agentshas been modelled as a directed edge.

B. Sliding Mode Control

Sliding Mode Control (SMC) [27] is known for its inher-ent robustness. The switching nature of the control is usedto nullify bounded disturbances and matched uncertainties.Switching happens about a hypergeometric manifold in statespace known as sliding manifold, surface, or hyperplane.The control drives the system monotonically towards thesliding surface, i.e, trajectories emanate and move towardsthe hyperplane (reaching phase). System trajectories, afterreaching the hyperplane, get constrained there for all futuretime (sliding phase), thereby ensuring the system dynamicsremains independent of bounded disturbances and matcheduncertainties.

In order to push state trajectories onto the surface s(x),a proper discontinuous control effort uSM(t, x) needs to besynthesized satisfying the following inequality.

sT (x)s(x) ≤ −η‖s(x)‖, (1)

with η being positive and is referred as the reachabilityconstant.

∵ s(x) =∂s

∂xx =

∂s

∂xf(t, x, uSM) (2)

∴ sT (x)∂s

∂xf(t, x, uSM) ≤ −η‖s(x)‖. (3)

The motion of state trajectories confined on the manifold isknown as sliding. Sliding mode exists if the state velocityvectors are directed towards the manifold in its neighbourhood.Under such consideration, the manifold is called attractive,i.e., trajectories starting on it remain there for all future time

and trajectories starting outside it tend to it in an asymptoticmanner. Hence, in sliding motion,

s(x) =∂s

∂xf(t, x, uSM) = 0. (4)

uSM = ueq is a solution, generally referred as equivalentcontrol is not the actual control applied to the system but canbe thought of as a control that must be applied on an averageto maintain sliding motion and is mainly used for analysis ofsliding motion.

III. DYNAMICS OF MULTI-AGENT SYSTEMS & PROBLEMFORMULATION

Consider first order homogeneous MAS interacting amongthemselves and their environment in a directed topology.Under such interconnection, information about the predictedlocation of source of the odor through instantaneous plumesensing is not available globally. However, local informationis obtained by communication among agents whenever at leastone agent attains some information of interest. The governingdynamics of first order homogeneous MAS consisting of Nagents is described by nonlinear differential equations as

xi(t) = f(xi(t)) + uSMi(t) + ςi; i ∈ [1, N ], (5)

where f(·) : R+×X → Rm is assumed to be locally Lipschitzover some fairly large domain DL with Lipschitz constant L,and denotes the uncertain nonlinear dynamics of each agent.Also X ⊂ Rm is a domain in which origin is contained. xi anduSMi

are the state of ith agent and the associated control re-spectively. ςi represents bounded exogenous disturbances thatenter the system from input channel, i.e., ‖ςi‖ ≤ ςmax <∞.

The problem of odor source localization can be viewed as acooperative control problem in which control laws uSMi needto be designed such that the conditions limt→∞ ‖xi−xj‖ = 0and limt→∞ ‖xi−xs‖ ≤ θ are satisfied. Here xs represents theprobable location of odor source & θ is an accuracy parameter.

IV. HIERARCHICAL DISTRIBUTED COOPERATIVECONTROL SCHEME

In order to drive the agents towards consensus to locate thesource of odor, we propose the following hierarchy.

A. Group Decision Making

This layer utilizes both concentration and wind informationto predict the location of odor source. Then, the final probableposition of the source can be described as

ψ(tk) = c1pi(tk) + (1− c1)qi(tk), (6)

with pi(tk) as the oscillation centre according to a simpleParticle Swarm Optimization (PSO) algorithm and qi(tk)captures the information of the wind. c1 ∈ (0, 1) denotesadditional weighting coefficient.

Remark 1. The arguments in (6) represent data captured att = tk instants (k = 1, 2, ...) as the sensors equipped with theagents can only receive data at discrete instants.

It should be noted that ψ is the tracking reference that isfed to the controller. Now, we present detailed description ofobtaining pi(tk) and qi(tk).

Simple PSO algorithm that is commonly used in practicehas the following form.

vi(tk+1) = ωvi(tk) + uPSO(tk), (7)xi(tk+1) = xi(tk) + vi(tk+1). (8)

Here ω is the inertia factor, vi(tk) and xi(tk) represent therespective velocity and position of ith agent. This commonlyused form of PSO can also be used as a proportional-only typecontroller, however for the disadvantages mentioned earlier,we do not use PSO as our final controller. PSO control lawuPSO can be described as

uPSO = α1(xl(tk)− xi(tk)) + α2(xg(tk)− xi(tk)). (9)

In (9), xl(tk) denotes the previous best position and xg(tk)denotes the global best position of neighbours of ith agentat time t = tk, and α1 & α2 are acceleration coefficients.Since, every agent in MAS can get some information about themagnitude of concentration via local communication, positionof the agent with a global best can be easily known. By theidea of PSO, we can compute the oscillation centre pi(tk) as

pi(tk) =α1xl(tk) + α2xg(tk)

α1 + α2, (10)

where

xl(tk) = arg max0<t<tk−1

{g(xl(tk−1)), g(xi(tk))}, (11)

xg(tk) = arg max0<t<tk−1

{g(xg(tk−1)),maxj∈N

aij g(xj(tk))}.

(12)

Thus, from (9), (10)

uPSO(tk) = (α1 + α2){pi(tk)− xi(tk)}, (13)

which is clearly a proportional-only controller with propor-tional gain α1 + α2, as highlighted earlier.

In order to compute qi(tk), movement process of a sin-gle filament that consists several order molecules has beenmodelled. If xf (t) denotes position of the filament at timet, va(t) represent mean airflow velocity and n(t) be somerandom process, then the model can be described as

xf (t) = va(t) + n(t). (14)

Without loss of generality, we shall regard the start time ofour experiment as t = 0. From (14), we have

xf (t) =

∫ t

0

va(τ)dτ +

∫ t

0

n(τ)dτ + xs(0). (15)

xs(0) denotes the real position of the odor source at t = 0.

Assumption IV.1. We assume the presence of a single, sta-tionary odor source. Thus, xs(t) = xs(0).

Implications from remark 1 require (15) to be implementedat t = tk instants. Hence,

xf (tk) =

t∑m=0

va(τm)∆t+

t∑m=0

n(τm)∆t+ xs(tk), (16)

xf (tk) = xs(tk) + v?a(tk) + w?(tk). (17)

In (17),∑tm=0 va(τm)∆t = v?a(tk) and

∑tm=0 n(τm)∆t =

w?(tk).

Remark 2. In (17), the accumulated average of v?a(tk) andw?(tk) can also be considered ∀ possible filament releasingtime.

From (17),

xf (tk)− v?a(tk) = xs(tk) + w?(tk). (18)

The above relationship, (18) can be viewed as the informationabout xs(tk) with some noise w?(tk). Hence,

qi(tk) = xs(tk) + w?(tk). (19)

Therefore, ψ in (6) can now be constructed from (10) & (19).

B. Path Planning

Since, detection of information of interest is tied to thethreshold value defined for the sensors, the next state isupdated taking this threshold value into account. Thus, theblueprints of path planning can be described in terms of threetypes of behavior.

1) Surging: If the ith agent receives data well above thresh-old, we say that some clues about the location of thesource has been detected. If the predicted position of thesource at t = tk as seen by ith agent be given as xsi(tk),then the next state of the agent is given mathematicallyas

xi(tk+1) = xsi(tk). (20)

2) Casting: If the ith agent fails to detect information at anyparticular instant, then the next state is obtained usingthe following relation.

xi(tk+1) =‖xi(tk)− xsi(tk)‖

2+ xsi(tk). (21)

3) Search and exploration: If all the agents fail to detectodor clues for a time segment [tk, tk+l] > δ0 for somel ∈ N and δ0 ∈ R+ being the time interval for whichno clues are detected or some constraint on wait timeplaced at the start of the experiment, then the next stateis updated as

xi(tk+1) = xsi(tk) + zφσ. (22)

In (22), zφσ is some random parameter with σ as itsstandard deviation and φ as its mean.

C. Distributed Control

In the control layer, we design a robust and powerfulcontroller on the paradigms of sliding mode. It is worthyto mention that based on instantaneous sensing and swarminformation, at different times, each agent can take up the roleof a virtual leader whose opinion needs to be kept by otheragents. ψ from (6) has been provided to the controller as thereference to be tracked. The tracking error is formulated as

ei(t) = xi(t)− ψ(tk) ; t ∈ [tk, tk+1[. (23)

In terms of graph theory, we can reformulate the error variableas

εi(t) = (L+ B)ei(t) = (L+ B)(xi(t)− ψ(tk)). (24)

From this point onward, we shall denote L + B as H. Next,we formulate the sliding manifold

si(t) = λ1 tanh(λ2εi(t)), (25)

which is a nonlinear sliding manifold offering faster reach-ability to the surface. λ1 ∈ R+ represents the speed ofconvergence to the surface, and λ2 ∈ R+ denotes the slope ofthe nonlinear sliding manifold. These are coefficient weightingparameters that affect the system performance. The forcingfunction has been taken as

si(t) = −µ sinh−1(m+ w|si(t)|)sign(si(t)). (26)

In (26), m is a small offset such that the argument of sinh−1

function remains non zero and w is the gain of the controller.The parameter µ facilitates additional gain tuning. In general,m << w. This novel reaching law contains a nonlineargain and provides faster convergence towards the manifold.Moreover, this reaching law is smooth and chattering free,which is highly desirable in mechatronic systems to ensuresafe operation.

Theorem IV.1. Given the dynamics of MAS (5) connected ina directed topology, error candidates (23, 24) and the slidingmanifold (25), the stabilizing control law that ensures accuratereference tracking under consensus can be described as

uSMi(t) = −{

(ΛH)−1µ sinh−1(m+ w|si(t)|)sign(si(t))Γ−1

+ (f(xi(t))− ψ(tk))}

(27)

where Λ = λ1λ2, Γ = 1− tanh2(λ2εi(t)), w > supt≥0{‖ςi‖}& µ > sup{‖ΛHςiΓ‖}.Remark 3. As mentioned earlier, λ1, λ2 ∈ R+. This ensuresΛ 6= 0 and hence its non singularity. The argument of tanh isalways finite and satisfies λ2εi(t) 6= πι(κ + 1/2) for κ ∈ Z,thus Γ is also invertible. Moreover the non singularity of Hcan be established directly if the digraph contains a spanningtree with leader agent as a root.

Proof. From (24) and (25), we can write

si(t) = λ1{λ2εi(t)(1− tanh2(λ2εi(t)))} (28)

= λ1λ2εi(t)− λ1λ2εi(t) tanh2(λ2εi(t)) (29)

= λ1λ2εi(t){1− tanh2(λ2εi(t))} (30)

= ΛH(xi(t)− ψ(tk))Γ (31)

with Λ & Γ as defined in Theorem IV.1. From (5), (31) canbe further simplified as

si(t) = ΛH(f(xi(t)) + uSMi(t) + ςi − ψ(tk))Γ. (32)

Using (26), the control that brings the state trajectories on tothe sliding manifold can now be written as

uSMi(t) = −

{(ΛH)−1µ sinh−1(m+ w|si(t)|)sign(si(t))Γ

−1

+ (f(xi(t))− ψ(tk))}. (33)

This concludes the proof.

Remark 4. The control (27) can be practically implementedas it does not contain the uncertainty term.

It is crucial to analyze the necessary and sufficient condi-tions for the existence of sliding mode when control protocol(27) is used. We regard the system to be in sliding mode iffor any time t1 ∈ [0,∞[, system trajectories are brought uponthe manifold si(t) = 0 and are constrained there for all timethereafter, i.e., for t ≥ t1, sliding motion occurs.

Theorem IV.2. Consider the system described by (5), errorcandidates (23, 24), sliding manifold (25) and the controlprotocol (27). Sliding mode is said to exist in vicinity ofsliding manifold, if the manifold is attractive, i.e., trajectoriesemanating outside it continuously decrease towards it. Statingalternatively, reachability to the surface is ensured for somereachability constant η > 0. Moreover, stability can beguaranteed in the sense of Lyapunov if gain µ is designedas µ > sup{‖ΛHςiΓ‖}.

Proof. Let us take into account, a Lyapunov function candi-date

Vi = 0.5s2i . (34)

Taking derivative of (34) along system trajectories yield

Vi = sisi (35)

= si{

ΛH(f(xi(t)) + uSMi(t) + ςi − ψ(tk))Γ

}. (36)

Substituting the control protocol (27) in (36), we have

Vi = si(− µ sinh−1(m+ w|si|)sign(si) + ΛHςiΓ

)= −µ sinh−1(m+ w|si|)‖si‖+ ΛHςiΓ‖si‖={− µ sinh−1(m+ w|si|) + ΛHςiΓ

}‖si‖

= −η‖si‖, (37)

where η = µ sinh−1(m + w|si|) − ΛHςiΓ > 0 is calledreachability constant. For µ > sup{‖ΛHςiΓ‖}, we have

Vi < 0. (38)

Thus, the derivative of Lyapunov function candidate is negativedefinite confirming stability in the sense of Lyapunov.Since, µ > 0, ‖si‖ > 0 and sinh−1(·) > 0 due to the natureof its arguments. Therefore, (37) and (26) together provideimplications that ∀si(0), sisi < 0 and the surface is globallyattractive. This ends the proof.

V. RESULTS AND DISCUSSIONS

Interaction topology of the agents represented as a digraphhas been shown here in figure 1. The associated graph matriceshave been described below. The computer simulation has beenperformed assuming that agent 1 appears as virtual leader toall other agents, making the topology fixed and directed forthis study. It should be noted that, the theory developed so farcan be extended to the case of switching topologies and shallbe dealt in future.

1 2

3

4

5

Fig. 1: Topology in which agents are connected

A =

0 0 1 00 0 0 00 1 0 00 0 1 0

, B =

1 0 0 00 1 0 00 0 0 00 0 0 0

, D =

1 0 0 00 0 0 00 0 1 00 0 0 1

,(39)

L = D−A =

1 0 −1 00 0 0 00 −1 1 00 0 −1 1

,L+ B =

2 0 −1 00 1 0 00 −1 1 00 0 −1 1

(40)

Agents have the following dynamics.

x1 = 0.1 sin(x1) + cos(2πt) + uSM1(t) + ς1, (41)

x2 = 0.1 sin(x2) + cos(2πt) + uSM2(t) + ς2, (42)

x3 = 0.1 sin(x3) + cos(2πt) + uSM3(t) + ς3, (43)

x4 = 0.1 sin(x4) + cos(2πt) + uSM4(t) + ς4, (44)

x5 = 0.1 sin(x5) + cos(2πt) + uSM5(t) + ς5. (45)

In this study, advection model given in [28] has been usedto simulate the plume with both additive and multiplicativedisturbances. The initial conditions for simulation are takento be large values, i.e., far away from the equilibrium point.Time varying disturbance has been taken as ςi = 0.3 sin(π2t2),accuracy parameter θ = 0.001 and maximum mean airflowvelocity vamax = 1 m/s. Other key design parameters arementioned in table 1.

0 2 4 6 8 10 12

time progression for agents (sec)

-0.5

0

0.5

1

1.5

2

2.5po

sitio

n of

age

nts

(m)

10

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

10.9

11

posi

tion

of th

e so

urce

Agents progressing towards the source

x1

x2

x3

x4

x5

source info

True Odor Source

Direction of movement of filaments releasedfrom the source

Direction of movement of agentstowards the source

Fig. 2: Agents in consensus to locate source of odor

0 2 4 6 8 10 12

time progression for agents (sec)

-2

-1

0

1

2

3

4

posi

tion

of a

gent

s (m

)

10

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

10.9

11

odor

sou

rce

loca

tion

Agents progressing towards the source in parallel formationx1

x2

x3

x4

x5

source info (movementof filaments away fromsource)

agent 1 initial point

agent 2 initial point

agent 3 initial point

agent 4 initial point

agent 5 initial point

agent 1 terminal point

agent 2 terminal point

agent 3 terminal point

agent 4 terminal point

agent 5 terminal point

Trueodorsource

Formationgap

Fig. 3: Agents in formation to locate source of odor

0 1 2 3 4 5 6 7 8 9 10

time (sec)

0

0.5

1

1.5

Nor

m o

f err

or v

aria

bles

ei(t

)

Tracking errors

‖e1‖‖e2‖‖e3‖‖e4‖‖e5‖

Fig. 4: Norm of tracking errors

0 1 2 3 4 5 6 7 8 9 10

time (sec)

-10

-8

-6

-4

-2

0

2

4

6

8

ui(t

)

Control signals

u1

u2

u3

u4

u5

Fig. 5: Control signals during consensus

0 1 2 3 4 5 6 7 8 9 10

time (sec)

-1.5

-1

-0.5

0

0.5

1

1.5su

rfac

e va

riabl

es s

i(t)

Sliding manifolds

s1s2s3s4s5

Fig. 6: Sliding manifolds during consensus

TABLE I: Values of the design parameters used in simulation

c1 ωmax α1 α2 λ1 λ2 µ m w

0.5 2 rad/s 0.25 0.25 1.774 2.85 5 10−3 2

Figure 2 shows agents coming to consensus in finite timeto locate the source of odor and figure 3 shows agents movingin parallel formation to locate the odor source. Norm of thetracking errors has been depicted in figure 4. It is evident thatthe magnitude of error is very small. Plot of control signalsduring consensus has been shown in figure 5 and the plot ofsliding manifolds has been shown in figure 6.

VI. CONCLUDING REMARKS

The problem of odor source localization by MAS has beendealt with in a hierarchical manner in this work. The problemtranslates into a cooperative control problem wherein agentsare driven towards consensus to locate the true odor sourcein finite time. Through computer simulations, it has beenconfirmed that the proposed strategy is faster and providesaccurate tracking even in the presence of time varying distur-bances.

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