multi-target localization and circumnavigation by a single agent

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control0000;00:1–12Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc

Multi-Target Localization and Circumnavigation by a SingleAgent Using Bearing Measurements

Mohammad Deghat1,∗, Lu Xia2, Brian D. O. Anderson1 and Yiguang Hong3

1 Research School of Engineering, Australian National University and National ICT Australia, Canberra, ACT 2601,Australia

2 University of Melbourne, Melbourne, VIC, Australia3 Key Lab of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,

Beijing 100190, China

SUMMARY

This paper considers the problem of localization and circumnavigation of a group of targets, which are eitherstationary or moving slowly with unknown speed, by a single agent. An estimator is proposed, initially forthe stationary target case, to localize the targets and the center of mass of them as well as a control law thatforces the agent to move on a circular trajectory around the center of mass of the targets such that both theestimator and the controller are exponentially stable. Then the case where the targets might experience slowbut possibly steady movements is studied. The system inputsinclude the agent’s position and the bearingangles to the targets. The performance of the proposed algorithms is verified through simulations. Copyrightc© 0000 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: target localization, circumnavigation, persistence of excitation.

1. INTRODUCTION

A common surveillance problem is to arrange for one or possibly several agents to navigate arounda single or a group of targets on a circular trajectory of prescribed radius. A simple scenario forsuch a problem is that there is a single agent whose goal is to circle around a stationary target withknown position. The task is to find a control law that causes the agent to move to and then arounda circle in an agreed sense (i.e. clockwise or counterclockwise) with prescribed radius centered onthe target. There are various ways in which the problem can bemade more complex, for examplewhen there is a group of agents/targets or when the target(s)is moving. Prior literature dealing withsuch problems includes, but is not limited to, [2–7].

When the target(s) has an unknown initial position, an estimator of the target position as well as acontrol algorithm which forces the agent to move on the desired circular trajectory are required forthe surveillance task. In such problems ideas of adaptive control or dual control can appear and withcertain controls, no estimation is in fact possible; this sort of phenomenon is associated with dualcontrol, and it is the persistence of excitation concept of adaptive control [8,9] which clarifies whatneeds to happen. Simultaneous control and estimation are required, and the quality of estimationcan heavily depend on the control used.

∗Correspondence to: Mohammad Deghat, Australian National University, Canberra, ACT 2601, Australia.†E-mail: [email protected]‡An abbreviated conference version of this paper has been presented in [1].

Copyright c© 0000 John Wiley & Sons, Ltd.

Prepared usingrncauth.cls [Version: 2010/03/27 v2.00]

2 M. DEGHAT ET AL.

Problems of this type with single-agent or multi-agent collaborative circumnavigation algorithmshave recently been studied. In [10–12], localization and circumnavigation of a moving target havebeen studied when the agent(s) can measure the relative position of the target, i.e. its range andbearing. In some applications, however, it is preferred to employ localization and circumnavigationalgorithms that require less sensed knowledge about the target so that the proposed algorithm canbe used to control a UAV with limited payload capacity (and thus limited sensing capability).There have been some research efforts to study such localization and circumnavigation problemsusing distance-only measurements [13–15], bearing-only measurements [16,17] and received signalstrength (RSS) measurement [18]. In the scenarios where the agent has to maintain radio silence forthe fear that its position will be detected, it is usually preferred not to use distance measurements.This is because of the fact that distance measurement techniques are usually active methods inwhich the agent must transmit signals. In contrast, RSS measurement techniques and usuallybearing measurement techniques are passive methods. RSS based localization techniques measurethe strength of the received signal and use a log-normal radio propagation model to estimate thedistance to the target. The path loss exponent is a key parameter in the log-normal model whichdepends on the environment in which the sensor is deployed. The problem with this method isthat an accurate knowledge of the path loss exponent is required in order to convert signal strengthmeasurements to range, and it can be difficult to obtain [19].

The problem of bearings-only target localization has been studied in the literature using statisticalestimators such as extended Kalman filter (EKF) and unscented Kalman filter [20,21]. It is assumedin such estimators that the agent knows the system model as well as the noise model. Although theseestimators work well when the agent knows the motion characteristics of the target, they cannot beused when the target can move freely while the agent is not aware of the target motion. Many ofthe current results in the literature assumed that the target is either stationary or moving with aknown constant velocity. We however consider the scenario that the target is allowed to move onany directions and the agent does not know the motion characteristics of the target.

In this paper, we investigate the case where there is a singleagent but multiple targets whichcan be either stationary or moving. We assume that the agent has a single integrator modeland propose estimation and control algorithms using bearing measurements for determining theestimated positions of the targets and making the agent circle around them. The algorithms proposedin this paper are inspired by the algorithms in [16] which are for the single target case. However, thealgorithms proposed here are not trivial extensions of the single target case where the target itselfacts as the center of the circle around which the agent moves.In the multiple targets case, the centerof the circle is not naturally or automatically defined; therefore we need to add a center estimationalgorithm. This changes the structure of the control algorithm and adds another layer of conceptualand computational complexity to the problem.

As noted, the algorithms we propose in this paper use bearingbut not range measurementsand, as is generally desirable and usual, avoid using derivatives of measurements, i.e. do notmeasure angular velocity, to avoid high-frequency noise effect. In the stationary target case, theestimation and control algorithms exhibit exponentially fast convergence. In terms of robustnessagainst noise and system uncertainties, although filteringalgorithms like EKF can be applied whenthe measurements are noisy or when the target is moving slowly, the proposed method, withoutusing any filtering algorithm, can tolerate measurement noise and slow movement of the target.This results from the fact that exponentially stable systems are robust against many types of systemuncertainties.

The rest of this paper is structured as follows. In Section2, the problem is formally defined andthe proposed solution is provided in Section3. Section4 contains results from Matlab simulationsdemonstrating the feasibility of the proposed algorithm. Finally, conclusions and proposals forfuture work are presented in Section5.

Copyright c© 0000 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control(0000)Prepared usingrncauth.cls DOI: 10.1002/rnc

MULTI-TARGET LOCALIZATION AND CIRCUMNAVIGATION 3

2. PROBLEM STATEMENT

Suppose there aren targets with unknown positionspTi(t) ∈ R

2, i ∈ 1, 2, 3, · · · , n at timet andthere is also an agent moving on a known trajectorypA(s) ∈ R

2 for s ≤ t. Until further notice, weshall assume allpTi

(t) are constant. Both the targets and the agent are assumed to bemodellable aspoints. Letϕi(t), i = 1, 2, · · · , n be the unit vectors in the direction of the line going frompA(t) topTi

(t), that is

ϕi(t) =pTi

(t)− pA(t)

‖pTi(t)− pA(t)‖

(1)

and letϕi(t) be the unit vector obtained byπ/2 clockwise rotation ofϕi(t). LetpT (t) be the pointaround which the agent is seeking to move. We call this point thevirtual targetand define it as

pT (t) =1

n

∑

i

pTi(t). (2)

Also letρ(t) beρ(t) = ||pA(t)− pT (t)|| (3)

andρd(t) be the desired radius of a circle withpT (t) as the center on which the agent should move.The value ofρd(t) will be estimated by the agent and should be such that the circle encloses alltargets. Of course, just as the agent does not initially knowthe positions of the targets, it does notinitially know the position of the virtual target and the radius of the circle. The cases = t wheren = 5 is depicted in Figure1. Let pTi

(t) be the estimated position of targeti and pT (t) be theestimate ofpT (t) at timet. The discussion on how to calculatepTi

(t) andpT (t) appears in the nextsection. We suppose the agent can measure the bearing anglesto all targets and defineρd(t) as

ρd(t) = maxi

‖pT (t)− pTi(t)‖ + d (4)

whered > 0 is a constant scalar. The constantd makes the desired circle be larger than the maximumdistance of the targets to the center of the circle so that alltargets are inside the circle and not on theperimeter.

There are two tasks to be done by the agent. The first is to estimate the position of the virtualtargetpT (t) and the radiusρd(t) such that the estimation error given in (5) converges exponentiallyfast to zero when the targets are stationary and to a small neighborhood of zero when the targets aremoving slowly:

pT (t) = pT (t)− pT (t). (5)

Note that when the target positionspTi(t) are constant,pT (t) will also be taken as constant.

The second task is to move toward and then on the desired circle around the virtual target suchρ(t)− ρd(t) converges to zero exponentially fast in the stationary target case and to a neighborhoodof zero in the moving target case.

3. PROPOSED SOLUTION

We previously studied the case where there was only one target and one agent (see [16]). It is shownin [16] that both estimator and controller are exponentially stable. Before considering the case wherethere is more than one target, we first recall the estimator and controller for the case where there isonly one target located atpT1

(t):

˙pT1(t) = kest

(

I −ϕ1(t)ϕ⊤1 (t)

)(

pA(t)− pT1(t)

)

(6)

andpA(t) = u(t) =

(

ρ1(t)− ρd(t)

)

ϕ1(t) + αϕ1(t) (7)


4 M. DEGHAT ET AL.

Figure 1. An illustration of the problem and the relationship between variables.

wherepT1(t) is the estimate ofpT1

(t) at timet, I is the2× 2 identity matrix,ρ1(t) = ||pA(t)−

pT1(t)||, u(t) is the control input, andkest and α are positive constants. When the estimator

converges,ρ1(t) → ρ

1(t) = ||pA(t)− pT1

(t)|| and according to (7), the agent moves toward thedesired circle ifρ

1(t) 6= ρd(t). Once it reaches the circle, it moves with the tangential speed ofα

around the target. Note that these equations give exponential convergence whenpT1(t) is constant,

and robust behavior for slow motion of the target.When there is more than one target, the first step is to find the virtual targetpT (t) around which

the agent is going to move and the radius of the circle,ρd(t).

3.1. Stationary targets

We continue to assume the targets are all stationary. We willlater consider the case where the targetsmove slowly in Section3.2. Our first approach is to estimate all target positions at thesame timeand then use the individual estimatespTi

(t) to calculatepT (t) as

pT (t) =1

n

∑

i

pTi(t). (8)

Note that when all estimators converge,pT (t) also converges topT . Similarly to (6), the estimatorfor targeti can be written as

˙pTi(t) = kest

(

I −ϕi(t)ϕ⊤i (t)

)(

pA(t)− pTi(t)

)

(9)

and after some calculations, the estimation error dynamicscan be written as

˙pTi(t) = −kest

(

I −ϕi(t)ϕ⊤i (t)

)

pTi(t)

= −kestϕi(t)ϕ⊤i (t)pTi

(t).(10)

wherepTi(t) = pTi

(t)− pTi(t). Let ρ(t) and the unit vectorϕ(t) be defined as

ρ(t) = ||pA(t)− pT (t)|| (11)

ϕ(t) =pT (t)− pA(t)

‖pT (t)− pA(t)‖. (12)

Then the controller for the multi-target case can be writtenas

pA(t) = u(t) =(

ρ(t)− ρd(t))

ϕ(t) + αϕ(t) (13)

whereϕ(t) is the unit vector obtained byπ/2 clockwise rotation ofϕ(t). Note that the unit vectorϕ(t) is not a measurement of the bearing angle of an actual target,but represents the bearing



angle ofpT (t) which is the estimated position of the virtual targetpT . The reason why we usethe bearing angle ofpT (t) rather than the bearing angle to an actual target is that we donot haveany measurement from the center of the circle around the targets (virtual target) in the multi-targetcase.

There is an important distinction now which should be made. For the single target case, the unitvectorϕ(t) in (7) always points toward the target but in the multi-target case explained above,it points to pT (t) rather thanpT (t). Hence in the multi-target case with the additional level ofcomplexity, it is harder to estimate the circle center. Thiscan be seen by comparing Figure2 andFigure3. In Figure2, we simulated a single target case and used the estimator (6) and the controller(7). In this case, the unit vectorϕ1(t) used in both the estimator and controller points toward thetarget. In Figure3, we used the same initial conditions for the agent and the target estimate, andsimulated the case where the estimator (9) and the controller (13) are used, and there is only a singletarget. The value ofρd(t) in both cases is set to2 as there is only one target and there is no needto estimate the desired radius of the circle. Thus the only difference is that in Figure2 we used thecontroller (7) in which the unit vectorϕ1(t) is used and points toward the target, but in Figure3 weused the controller (13) in which the unit vectorϕ(t) is used and points toward the target estimate.

−5 0 5 10 15 20

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12

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16

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20

X (m)

Y (

m)

PA(t) (solid blue), PT1(dashed-dotted black), and PT1

(red +)

Figure 2.Simulation of a single target case when the controller makesthe agent move around the target.

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X (m)

Y (

m)

PA(t) (solid blue), PT (dashed-dotted black), and PT (red +)

Figure 3.Simulation of a single target case when the controller makesthe agent move around the target estimate.

It can be seen that the motion of the agent changes a lot when the controller changes from (7) to(13). Furthermore, the motion of the virtual target estimate also persists for longer, or is a biggermotion (i.e. the estimate of the center of the target circle wanders more before convergence).

In what follows, we show that by using the estimator (9) and the controller (13), the estimationerror pT (t) converges to zero exponentially fast. To this end, we recallthe following definition andproposition [8].


6 M. DEGHAT ET AL.

Definition 1w(t) : R+ → R

n×r ∀n, r ≥ 1 is persistently exciting if there exist some positiveα1, α2, δ such that

α1I ≤

∫ t0+δ

t0

w(t)w⊤(t)dt ≤ α2I for all t0 ≥ 0 (14)

Proposition 1Consider the differential equation

x = −w(t)w⊤(t)x (15)

wherew(t) : R+ → Rn×r ∀n, r ≥ 1 is a regulated matrix function (i.e., one-sided limits exist for all

t ∈ R+). Then (15) is exponentially asymptotically stable if and only ifw(t) is persistently exciting.

The persistence of excitation condition in (14) requires thatw(t) rotates sufficiently in space suchthat the integral of the matrixw(t)w⊤(t) is uniformly positive definite over any interval of somelengthδ [8,9]. Consider now the estimation error equation of targeti in (10). If the unit vectorϕi(t)(or the unit vectorϕi(t)) rotates sufficiently in space, i.e., if the bearing angle totargeti changessufficiently fast, then according to Proposition1, pTi

converges to zero exponentially fast. The onlysituation whereϕi(t) does not rotate sufficiently is when the agent moves on a straight line towardthe targeti or when it converges to such a straight line. Consider the case shown in Figure4. Ifthe agent moves on the dotted line shown in Figure4, ϕi(t) does not change and is not persistentlyexciting. But the other unit vectorsϕj(t) to other targets rotate and therefore otherpTj

(t) convergeto zero exponentially fast. The following theorem shows that by using the estimator (9) and thecontroller (13), the agent cannot move on such a straight line as shown in Figure4 and all of theunit vectors pointing from the agent to the targets are persistently exciting.

Figure 4.An example of the trajectory of the agent along whichϕi(t) and alsoϕi(t) are not persistently exciting.

Theorem 1Adopt the notation above and assume that all targets are stationary. Then by using the estimator (9)and the controller (13), the estimation error for each target converges to zero exponentially fast andconsequentiallypT (t) has the same property.

ProofIf we show that allpTi

(t) converge to zero exponentially fast, then we can conclude that pT (t) alsoconverges to zero exponentially fast. We prove by contradiction and assume that at least for targeti, ϕi(t) is not persistently exciting and thereforepTi

(t) does not converge to zero exponentiallyfast. Consider Proposition1 and letw(t) in (14) bew(t) = ϕi(t). Note thatϕi(t)ϕi(t)

⊤ is singularfor all t and the persistency of excitation condition in (14) requires thatϕi(t) rotates sufficiently inspace that the integral of the matrixϕi(t)ϕi(t)

⊤ is uniformly positive definite overany intervalofsome lengthδ > 0. So we consider that the motion of the agent is such thatϕi(t) does not changeor changes very slowly that there is noδ > 0 such that the integral of the matrixϕi(t)ϕi(t)

⊤ overδis uniformly positive definite†.

Note that the estimated position of targeti always converges to a constant value (which might notbe the correct position of targeti) exponentially fast. This can be seen from (9) and Figure5, as the

†This means that agenti moves on a straight line toward the target or converges to such a straight line.



estimator always forces the estimated position of targeti to go to the pointX shown in Figure5which is on the line passing through the agent and targeti.

So the estimated position of the virtual target which is the average of the estimated position of alltargets converges to a constant value exponentially fast. Then according to (13), the agent tries tomove around the estimated position of the virtual target andhas a nonzero tangential velocity equaltoα. Thus, the agent cannot keep moving on the straight line shown in Figure4 and thereforeϕi(t)also rotates, which ensures that the estimated position of targeti converges to its actual position.

Figure 5.An illustration on how the estimator in (9) works.

Having established that the estimation process proceeds satisfactorily, it remains to demonstratethat the control law achieves the required objective.

Theorem 2Using the estimator (9) and the controller (13), ρ(t)− ρd(t) converges to zero exponentially fast.

ProofConsidering (11), (12) and (13), one has

˙ρ(t) =

(

pA(t)−˙pT (t)

)⊤(

pA(t)− pT (t))

ρ(t)

= −(

pA(t)−˙pT (t)

)⊤

ϕ(t)

= −ρ(t) + ρd(t) + ˙pT (t)⊤ϕ(t).

(16)

Since the targets are stationary,˙pTi(t) = ˙pTi

(t) and thus the last term on the right hand side

of the above equation is−kest

n

∑

i

(

ϕi(t)ϕ⊤i (t)pTi

(t))⊤

ϕ(t). Since allpTi(t) converge to zero

exponentially fast (see Theorem1) and thereforeρd(t) converges exponentially fast to a constantvalue, then in the light of (16), ρ(t)− ρd(t) converges to zero exponentially fast.

Consider now a triangle with vertices atpA(t), pT (t) andpT (t). Then by the triangle inequalityone has

ρ(t) ≤ ρ(t) + ‖pT (t)‖. (17)

Sinceρ(t) and‖pT (t)‖ converge respectively tolimt→∞ ρd(t) and zero exponentially fast,ρ(t) alsoconverges tolimt→∞ ρd(t) exponentially fast.

Remark 1Although we assumed the bearing angles to all targets are available to the agent at all time,pT (t)andρd(t) still converge to their desired values if the agent is not able to measure the bearing angleto one or some of the targets for some period of time. In this case the convergence rate might beslower. An example of this scenario is presented in Section4.


8 M. DEGHAT ET AL.

3.2. Slowly moving targets

We have discussed in the previous subsection how the proposed algorithm works whenpTi(t), ∀i =

1. · · · , n are assumed constant. Now we would like to show that when the targets are moving slowly,the estimation errorpT (t) converges to a neighborhood of zero. To this end, we recall the followingproposition (Theorem 8.3 of [22]):

Proposition 2If the coefficient matrixA(t) is continuous for allt ∈ [0,∞) and constantsa > 0, b > 0 exist suchthat for every solution of the homogeneous differential equation

x(t) = A(t)x(t)

one has‖x(t)‖ ≤ b‖x(t0)‖e

−a(t−t0), 0 ≤ t0 < t < ∞

then for eachf(t) bounded and continuous on[0,∞), every solution of the nonhomogeneousequation

x(t) = A(t)x(t) + f(t), x(t0) = 0

is also bounded fort ∈ [0,∞).

It is also shown in [22] that if ‖f(t)‖ ≤ Kf < ∞, for some positive constantKf , then the solutionof the perturbed system satisfies

‖x(t)‖ ≤ b‖x(t0)‖e−a(t−t0) +

bKf

a

(

1− e−a(t−t0))

(18)

We make the following assumption on the motion of the targets.

Assumption 1The trajectories of the targets are such that‖pTi

(t)‖ ∀i = 1, · · · , n are bounded and piecewisecontinuous fort ≥ 0 and there exists a sufficiently smallε such that‖pTi

(t)‖ < ε. Furthermore,there exists some positive scalarD > 0 such thatmax

i‖pT (t)− pTi

(t)‖ ≤ D for all t ≥ 0.

The conditionmaxi

‖pT (t)− pTi(t)‖ ≤ D guarantees that the distance between the targets is

bounded which is a necessary condition for the agent to circumnavigate the targets. We show inthe following theorem that the estimation and control errorconverge to a neighborhood of zerowhen the targets are moving such that the above assumption holds.

Theorem 3Adopt the notation above and suppose Assumption1 holds. Then by using estimator (9)and controller (13) the estimation errorpT (t) and the control errorρ(t)− ρd(t) converge toneighborhoods of zero exponentially fast.

ProofThe key assumption in this theorem is that the speed of the targets is sufficiently low and the distancebetween any two targets is less than2D for all t ≥ 0. This assumption should always hold as theagent should always move significantly faster than the targets and the distance between any twotargets should be bounded so that the agent can circumnavigate the targets. Now when the targetsare moving, the estimation error dynamics for targeti changes from (10) to

˙pTi(t) = −kestϕi(t)ϕ

⊤i (t)pTi

(t)− pTi(t). (19)

Based on the results in the stationary target case, we know that (10) is asymptotically exponentiallystable. SincepTi

(t) is bounded and can be regarded as a non-vanishing perturbation applied toan exponentially stable system, we can conclude according to Proposition2 that pTi

(t) convergesexponentially fast to a neighborhood of the zero. ThuspT (t) also converges to a neighborhood ofzero exponentially fast and the size of this neighborhood isproportional to the maximum speed ofthe targets.



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−1

0

1

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3

4

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8

9

X (m)

Y (

m)

Agent trajectory (solid blue) and positions of the targets (+)

Figure 6.Agent trajectory in X-Y plane, for the case where the targetsare stationary.

When the targets are non-stationary,˙ρ(t) in (16) also changes fromρ(t) = −ρ(t) + ρd(t) +˙pT (t)

⊤ϕ(t) to

˙ρ(t) = −ρ(t) + ρd(t) + ˙pT (t)⊤ϕ(t) + pT (t)

⊤ϕ(t). (20)

Becauseρd(t) is bounded, it can be shown thatρ(t)− ρd(t) converges to a neighborhood of zeroexponentially fast. Then according to (17), ρ(t)− ρd(t) also converges to a neighborhood of zeroexponentially fast.

4. SIMULATIONS

In this section we consider two different scenarios corresponding to whether the targets arestationary or moving. First we consider the case where thereare three stationary targets atpT1

= [2, 4]⊤, pT2= [1, 2]⊤ andpT3

= [3, 3]⊤. We assume that the initial target estimates arepT1

(0) = [3, 2]⊤, pT2(0) = [3, 0]⊤ andpT3

(0) = [7, 3]⊤. We also assume thatα = 5, kest = 5 andd in (4) is 0.5. Simulation results for the case where the bearing angles toall targets are available tothe agent for allt > 0 are shown in Figure6 and Figure7. It can be seen that the estimation errorexponentially converges to zero and the circling radius exponentially converges toρd = 1.91m. Wethen suppose that the measurements are taken in segments such that the agent can only measureone bearing angle at a time and will switch between the targets at intervals ofτ = .05 sec. Thusthe estimated position of only one of the targets is updated at any time. The results are shown inFigure8 and Figure9. If we compare the results of segmented and continuous measurement cases,we see that in both cases the estimation and control errors converge exponentially to zero, howeverin the segmented measurement case, the convergence is slower.

We then consider the case where the bearing angles to the targets are perturbed by normallydistributed random noises. We simulate two different scenarios by applying noises with two differentstandard deviations to the bearing measurements. We assumethe noises are zero mean with standarddeviation of0.1 and0.2. Simulation results for these cases are shown in Figure10. It can be seenthat the errors go to neighborhoods of zero and the size of these neighborhoods is larger when thestandard deviation of noise is larger.

Now we consider the case where the targets are moving slowly such that pT1= [2 +

.025t, 4 + sin(.03t) + .025t]⊤, pT2= [1 + .03t, 2 + sin(.025t) + .03t]⊤ andpT3

= [3 + .025t, 3 +sin(.035t) + .025t]⊤. Simulation results for this case are shown in Figure11 and Figure12. Notethat the desired radius of the circle in Figure12 changes by time as the targets move in differentdirections with different speed and thus the radius of the circle changes by time. It can be seen thatthe estimation and control error do not converge to zero, butconverge to neighborhoods of zero.


10 M. DEGHAT ET AL.

0 10 20 30 40 501.6

1.8

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2.2

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time (sec)

ρd(t)

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time (sec)

||PT (t)|| (dashed-dotted black line) and ρ(t)− ρd(t) (solid blue line)

Figure 7. The left hand side figure shows the estimated radius of the circleρd(t), and the right hand side figure shows‖pT (t)‖ andρ(t) − ρd(t) for the case where the targets are stationary.

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Agent trajectory (solid blue) and positions of the targets (+)

Figure 8. Agent trajectory in X-Y plane, for the case where the targetsare stationary and the bearing measurements aretaken in segments.

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4.5

time (sec)

ρd(t)

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time (sec)


Figure 9. The left hand side figure shows the estimated radius of the circle ρd(t), and the right hand side figureshows‖pT (t)‖ andρ(t) − ρd(t) for the case where the targets are stationary and the bearingmeasurements are taken in

segments.

5. CONCLUSION AND FUTURE WORK

In this paper, we considered the localization and circumnavigation problem of multiple targets. Weproposed estimator and control algorithms and showed that with stationary targets, the estimationerror and the control error converge to zero exponentially fast. It is shown that for the moving target



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time (sec)


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Figure 10.Simulation results for the case where the bearing measurements are noisy. The left hand side figure shows thecase where the standard deviation of noise is0.1 while the right hand side figure is for case where the standarddeviation

is 0.2.

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Trajectories of the agent (blue) and the targets (red)

Figure 11.Agent trajectory in X-Y plane for the case where the targets are moving.

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1.5

2

2.5

3

3.5

time (sec)

ρd(t)

0 50 100 150 200 250 300−1

0

1

2

3

4

5

6

time (sec)


Figure 12. The left hand side figure shows the estimated radius of the circle ρd(t), and the right hand side figureshows‖pT (t)‖ andρ(t) − ρd(t) for the case where the targets are stationary and the bearingmeasurements are taken in

segments.

case, the estimator and controller can tolerate slow motionof targets with only modest affects onaccuracy. It also appears that the larger the speed of the targets, the larger the estimation error.

Future directions of research include general collision avoidance; considering more realistic agentmodels; having three or more agents forming a polygon formation circling a target/multiple targets;estimating the speed of targets if they are moving at an unknown constant speed; and using methodslike EKF or IPDA-FR [23] to further reduce the effect of noise and to increase accuracy.


12 M. DEGHAT ET AL.

ACKNOWLEDGEMENT

This work is supported by NICTA, which is funded by the Australian Government as represented by theDepartment of Broadband, Communications and the Digital Economy and the Australian Research Councilthrough the ICT Centre of Excellence program. Lu Xia’s work was done while at ANU. Yiguang Hong issupported by the National Science Foundation of China (grant number: 61333001).

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