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[6] E. Phipps, R. Casey, and J. Guckenheimer, “Periodic orbits of hybridsystems and parameter estimation via AD,” in Automatic Differentiation:Applications, Theory, and Implementations (Lecture Notes in Compu-tational Science and Engineering, Vol. 50). Berlin, Germany: Springer,2006, pp. 211–223.

[7] T. Debus, P. Dupont, and R. Howe, “Contact state estimation using multiplemodel estimation and hidden Markov models,” Int. J. Robot. Res., vol. 23,no. 4–5, pp. 399–413, 2004.

[8] L. M. Miller and T. D. Murphey, “Simultaneous optimal parameter andmode transition time estimation,” in Proc. IEEE Int. Conf. Intell. Robot.Syst., Oct. 2012, pp. 719–724.

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[10] L. Ojeda, D. Cruz, G. Reina, and J. Borenstein, “Current-based slippagedetection and odometry correction for mobile robots and planetary rovers,”IEEE Trans. Robot., vol. 22, no. 2, pp. 366–378, Apr. 2006.

[11] S. Zibin, Y. Zweiri, L. Seneviratne, and K. Althoefer, “Non-linear observerfor slip estimation of skid-steering vehicles,” in Proc. IEEE Int. Conf.Robot. Autom., May 2006, pp. 1499–1504.

[12] L. Caracciolo, A. de Luca, and S. Iannitti, “Trajectory tracking control ofa four-wheel differentially driven mobile robot,” in Proc. IEEE Int. Conf.Robot. Autom., 1999, vol. 4, pp. 2632–2638.

[13] J. Yi, D. Song, J. Zhang, and Z. Goodwin, “Adaptive trajectory trackingcontrol of skid-steered mobile robots,” in Proc. IEEE Int. Conf. Robot.Autom., Apr. 2007, pp. 2605–2610.

[14] K. Iagnemma, K. Shinwoo, H. Shibly, and S. Dubowsky, “Online ter-rain parameter estimation for wheeled mobile robots with application toplanetary rovers,” IEEE Trans. Robot., vol. 20, no. 5, pp. 921–927, Oct.2004.

[15] L. Ray, “Estimation of terrain forces and parameters for rigid-wheeledvehicles,” IEEE Trans. Robot., vol. 25, no. 3, pp. 717–726, Jun. 2009.

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[18] E. Bakker, L. Nyborg, and H. Pacejka, “Tyre modelling for use in vehicledynamics studies,” SAE Tech. Paper 870421, 1987.

[19] R. Balakrishna and A. Ghosal, “Modeling of slip for wheeled mobilerobots,” IEEE Trans. Robot. Autom., vol. 11, no. 1, pp. 126–132, Feb.1995.

[20] J. Nocedal and S. Wright, Numerical Optimization. New York, NY,USA: Springer-Verlag, 2006.

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[22] J. Stephens, Kempe’s Engineers Year Book. ser. Kempe’s Engineer’sYear Book Series. London, U.K.: CMP Information, 2001.

Consensus in Networks of Nonidentical Euler–LagrangeSystems Using P+d Controllers

Emmanuel Nuno, Ioannis Sarras, and Luis Basanez

Abstract—This paper presents a proportional plus damping controllerthat can asymptotically drive a network composed of N nonidentical Euler–Lagrange (EL) systems toward a consensus point. The agents can be fullyactuated or can belong to a class of underactuated EL-systems. The networkis modeled as a weighted and undirected static interconnection graph thatcan exhibit asymmetric variable time delays. Simulations, using a networkwith ten EL-systems, are reported to support the theoretical contributionsof this study.

Index Terms—Consensus, proportional plus damping (P+d) controllers,time-delays.

I. INTRODUCTION

For networks of multiple agents, the consensus control objective isto reach an agreement between certain coordinates of interest usinga distributed controller. There are mainly two consensus problems:the leader–follower, where a network of follower agents has to besynchronized with a given leader, and the leaderless, where all agentsagree on a certain coordinate values. The solutions to these problemshave recently attracted the attention of the research community indifferent fields, such as biology, physics, control theory, and robotics(see [1]–[3] for solutions with linear agents and [4]–[6] for solutionswith some classes of nonlinear agents).

This paper deals with the leaderless consensus problem for a classof physical systems described by the Euler–Lagrange (EL) equationsof motion d

dt∂ L∂ q − ∂ L

∂ q = Gτ , where L : Rm ×Rm → R is the La-

grangian energy function defined as L(q, q) = 12 q�M(q)q − U (q),

with q ∈ Rm the generalized configuration coordinates. M : Rm →Rm ×m is the generalized inertia matrix, U : Rm → R is the potentialenergy function, G ∈ Rm ×n is a transformation matrix, and τ ∈ Rn

is the vector of external forces that acts on the system with m ≥ n.The reason to study EL-systems reside in the fact that they represent awide area of physical, mechanical, and electrical systems [7].

The practical applications of the solutions to the consensus problemare diverse and range from formation control of multiple unmannedaerial vehicles to the synchronization of swarms of robots [8], [9]. Aparticular example is robot teleoperation, where two manipulators arecoupled by a communication channel that, in general, induces timedelays [10]. The control objective in these systems is that when theoperator moves the local manipulator, the remote manipulator has totrack its position, and the force interaction of the latter with the en-vironment has to be reflected back to the operator [11]. The resultsin this paper are a generalization of the stability condition that is re-ported in [12] to the case of networks of fully actuated and a class of

Manuscript received April 26, 2013; accepted August 16, 2013. Date of pub-lication September 9, 2013; date of current version December 2, 2013. Thispaper was recommended for publication by Associate Editor C. Secchi andEditor G. Oriolo upon evaluation of the reviewers’ comments. This work hasbeen partially supported by the Mexican CONACyT project CB-129079 and theSpanish CICYT projects DPI2010-15446 and DPI2011-22471.

E. Nuno is with the Department of Computer Science, University of Guadala-jara, 44100 Guadalajara, Mexico (e-mail: emmanuel.nuno@cucei.udg.mx).

I. Sarras is with the Department of Control, Signal & Systems, IFP EnergiesNouvelles, 92852 Rueil-Malmaison, Paris, France (e-mail: sarras@ieee.org).

L. Basanez is with the Institute of Industrial and Control Engineering, Tech-nical University of Catalonia, 08028 Barcelona, Spain (e-mail: luis.basanez@upc.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2013.2279572

1552-3098 © 2013 IEEE

1504 IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 6, DECEMBER 2013

underactuated EL-systems (in this case, the number of inputs is strictlyless than the degrees-of-freedom (DOF) and designing a controller ismore complicated). A direct application of the controllers reported hereis the teleoperation of multiple-remote devices and the collaborationof multiple users via a multiple-local multiple-remote system, amongothers [13], [14].

Comparison with existing works: The leaderless consensus problemin networks of EL-systems, without time-delays, has been consideredin [15] and [16] using simple proportional controllers, together withfiltered velocities. Hatanaka et al. [17] solved the consensus prob-lem, in the Cartesian space, with only constant time delays appearingin the communications. The work of Nuno et al. [18] and of Liu andChopra [19] are closely related to this study. In contrast with the work ofNuno et al. [18], where an adaptive controller is used to solve the lead-erless consensus problem, this study deals with variable time delays andit does not assume that the regressor matrix model of the EL-systemsis known. Further, the work of Liu and Chopra [19] does not deal withthe leaderless consensus problem but with the leader–follower one,in the Cartesian space, for a balanced and strongly connected graphwith the assumption that the delays first derivatives are less than one.Furthermore, [18] and [19] only deal with fully actuated EL-systems.Recently, in [20], it has been proved that networks composed by non-identical EL-systems with variable time delays can reach a consensus,using proportional plus damping (P+d) controllers, provided enoughdamping is injected. It should be underscored that, all these previousresults deal with fully actuated EL-systems (fully actuated robots).However, in diverse applications, including space and surgical robots,the use of thin, lightweight, and cable-driven manipulators is increas-ing. These systems exhibit joint or link flexibility, and hence, they areunderactuated mechanical systems. To the authors’ knowledge, theliterature on the control of networks of underactuated EL-systems isalmost nonexistent, few remarkable exceptions being [21] and, more re-cently, [22]. In [21], the controlled-Lagrangian technique is employedto solve the consensus in networks without delays, and in [22], theconsensus problem is solved under the assumption that all of the statevector is measurable, all the physical parameters are known, and alldelays are constant.

Original contributions: The contributions of this paper are 1) a suf-ficient condition for the solution of the leaderless consensus problemin networks, modeled as undirected weighted static graphs, of fullyactuated and a class of underactuated EL-systems controlled by simpleP+d injection schemes; 2) the interconnection graph can exhibit asym-metric variable time delays, with the only assumption that such delaysare bounded and that the bounds are known; and 3) the controller onlydepends on measurements of the actuated part of the state.

Notation: R := (−∞,∞), R> 0 := (0,∞), R≥0 := [0,∞). |x| isthe Euclidean norm of vector x. 1k is a column vector, of size k,with all elements equal to one. Ik is the Identity matrix of size k.For any function f : R≥0 → Rn , the L∞ norm is ‖f‖∞ = supt≥0 |f (t)|and the square of the L2 norm is ‖f‖2

2 =∫ ∞

0 |f (t)|2dt. The L∞ andL2 spaces are the sets {f : R≥0 → Rn : ‖f‖∞ < ∞} and {f : R≥0 →Rn : ‖f‖2 < ∞}, respectively. The subscript i ∈ N := {1, . . . , N},where N is the number of agents.

II. SYSTEM DYNAMICS AND PROBLEM DEFINITION

A. System Dynamics

The network is composed of N nonidentical m-DOF EL-systems,which is compactly written as

Mi (qi )qi + Ci (qi , qi )qi +∂

∂qi

Ui (qi ) = Gτ i (1)

where Ci (qi , qi ) := Mi (qi )qi − 12

∂∂ q i

q�i Mi (qi )qi , defined via the

Christoffel symbols of the first kind, and i ∈ N . The dynamics of eachagent enjoys the following properties [12], [23], [24], which will beinstrumental in the proofs.

P1. The symmetric inertia matrix is positive definite, and there existki , Ki ∈ R> 0 such that ki ≤ ‖Mi (qi )‖ ≤ Ki .

P2. The matrix Mi (qi ) − 2Ci (qi , qi ) is skew symmetric.P3. The Coriolis matrix can be bounded as follows: |Ci (qi , qi )qi | ≤

ci |qi |2 for ci ∈ R> 0 .P4. If qi , qi ∈ L∞, then d

dtCi (qi , qi ) is a bounded operator.

The EL-systems exchange information over a network describedby an undirected weighted graph. The interconnection graph is mod-eled using the standard Laplacian matrix L := [�ij ] ∈ RN ×N , whoseelements are defined as

�ii =∑j∈Ni

wij , �ij = −wij (2)

where wij > 0 if j ∈ Ni and wij = 0, otherwise. Ni is the set ofagents that transmit information to the ith agent. Regarding the inter-connection, this paper makes the following assumptions:A1. The graph is static and connected, i.e., there exists a path between

every pair of vertices on the time-invariant graph.A2. The information exchange, from the jth agent to the ith agent, is

subject to a variable time delay Tj i (t) with a known upper bound∗Tj i . Hence, 0 ≤ Tj i (t) ≤ ∗Tj i < ∞.

Remark 1: By construction, L has zero row sum, i.e., L1N = 0.Moreover, Assumption A1 ensures that rank(L) = N − 1, that L hasa single zero-eigenvalue, and that the rest of the spectrum has positivereal parts [25]. Further, since the graph is undirected, then L = L�,and 1�

N L = 0.

B. Consensus Problem

Consider a network of N nonidentical m-DOF EL-systems of theform (1) whose interconnection graph fulfills Assumptions A1 andA2. Additionally assume that only the actuated part of the state isavailable for measurements. In this scenario, find the controllers whichensure that all EL-agents positions asymptotically reach a consensus,i.e., limt→∞ qi (t) = qc , for some qc ∈ Rm , while the generalizedvelocities asymptotically converge to zero, i.e., limt→∞ qi (t) = 0.

III. SOLUTION TO THE CONSENSUS PROBLEM

A. Fully Actuated Case

Let the P+d controllers be given by

τ i = −pi

∑j∈Ni

wij (qi − qj (t − Tj i (t))) − di qi +∂

∂qi

Ui (3)

where pi , di ∈ R> 0 . In this case, m = n, and G := In .Proposition 1: Controller (3) solves the consensus problem de-

scribed in Section II-B, for fully actuated EL-systems, provided thatthe damping injection satisfies

2di > pi �iiαi + pi

∑j∈Ni

∗T 2ij wj i

αj

∀i ∈ N , j ∈ Ni (4)

for any arbitrary αi , αj > 0. �Proof: Evaluating the derivative of the kinetic energy function Vi =

12 q�

i Mi qi , using Property P2, along the closed-loop system (1) and(3), yields

Vi = −di |qi |2 − pi q�i

∑j∈Ni

wij (qi − qj (t − Tj i (t))) .

IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 6, DECEMBER 2013 1505

Consider the following total scaled energy function

V =12q�(L ⊗ In )q +

N∑i=1

1pi

Vi ,

where q := col(q�1 , . . . ,q�

N ) and⊗ is the standard Kronecker product.Property P1 and Assumption A1 ensure that V is positive definite andradially unbounded with respect to |qi − qj | and qi for all i ∈ N andj ∈ Ni . Using Remark 1, Vi and the fact that

∫ t

t−T j i (t) qj (θ)dθ =

qj − qj (t − Tj i (t)) ensure that V satisfies

V = −N∑

i=1

⎡⎣di

pi

|qi |2 +∑j∈Ni

wij q�i

∫ t

t−T j i (t)qj (θ)dθ

⎤⎦ .

Since V does not qualify as a Lyapunov Function, because V isnot strictly negative, Lyapunov or LaSalle classical theorems cannotbe invoked. Instead, similar to [12], the rest of the proof aims to pro-vide the conditions under which the Barbalat’s Lemma can ensure theasymptotical stability claims. For, let us start by integrating V from 0to t, which yields

V (t) − V (0) = −N∑

i=1

[di

pi

∫ t

0|qi (σ)|2dσ+

+∑j∈Ni

wij

∫ t

0q�

i (σ)∫ σ

σ−T j i (σ )qj (θ)dθdσ

⎤⎦ .

Invoking [12, Lemma 1] on the last term, with αi ∈ R> 0 , usingthe fact that V (t) ≥ 0 and that �ii :=

∑j∈Ni

wij yields V (0) ≥∑N

i=1

∑j∈Ni

wij [( d ip i �i i

− α i2 )‖qi‖2

2 −∗T 2

j i

2α i‖qj ‖2

2 ]. Defining Q :=col(‖q1‖2

2 , . . . , ‖qN ‖22 ) ∈ RN and

Ψ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d1

p 1 −�11α1

2

−∗T 2

21w12

2α1. . . −

∗T 2N 1w1N

2α1

−∗T 2

12w21

2α2

d2

p2− �22α2

2. . . −

∗T 2N 2w2N

2α2

......

. . ....

−∗T 2

1N wN 1

2αN

−∗T 2

2N wN 2

2αN

. . .dN

pN

− �N N αN

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

it holds that V (0) ≥ 1�N ΨQ. This last expression implies that if the

sum of the elements of every column of Ψ is strictly positive, thenthere exist λi > 0 such that V (0) ≥ λ1‖q1‖2

2 + · · · + λN ‖qN ‖22 , and

thus, qi ∈ L2 for all i ∈ N . Simple calculations show that a sufficientcondition for the existence of λi > 0 is to choose di to fulfill (4). Thus,setting di as in (4) ensures that qi ∈ L2 and V ∈ L∞, which in turnimplies that |qi − qj |, qi ∈ L∞ for all i ∈ N and j ∈ Ni .

From the closed-loop system (1) and (3), the fact that qi ∈ L∞ ∩ L2

and Properties P1 and P3 imply that qi ∈ L∞. Barbalat’s Lemma provesthat limt→∞ qi (t) = 0. All these bounded signals together with Prop-erty P4 imply that d

dtqi ∈ L∞, meaning that qi is uniformly continu-

ous. This last property and convergence of velocities to zero show thatlimt→∞ qi (t) = 0, supported again by Barbalat’s Lemma.

Finally, since qi − qj (t − Tj i (t)) = qi − qj +∫ t

t−T j i (t) qj (θ)dθ,all equilibrium points of (1) and (3) satisfy the equation∑

j∈Niwij (qi − qj ) = 0, which is equivalent to (L ⊗ In )q = 0.

Thus, from Remark 1, the only possible solution is limt→∞ q(t) =(1N ⊗ qc ) for any qc ∈ Rn . �

B. Underactuated Case

Here, it is assumed that m = 2n, that G� := [0n In ], andthat the class of underactuated EL-systems in (1) satisfy thefollowing: 1) Mi (qi ) := diag(Mi (qi1 ),Ji ); 2) Ci (qi , qi ) := diag(Ci (qi1 , qi1 ), 0n ); and 3) ∂

∂ q iUi (qi ) := Kiqi , where Ki :=

(Si −Si

−Si Si). The constant matrices Ji ,Si ∈ Rn×n are symmetric

and positive definite. Thus, the nonlinear dynamics (1) can be writtenas

Mi (qi1 )qi1 + Ci (qi1 , qi1 )qi1 = Si (qi2 − qi1 ) (5a)

Ji qi2 + Si (qi2 − qi1 ) = τ i (5b)

where qi has been separated into its unactuated and actuated compo-nents qi1 ∈ Rn and qi2 ∈ Rn , respectively. Some examples of phys-ical systems that fit in (5) are the flexible joint and, for a linearizedelasticity model, flexible-link manipulators. It is worth mentioning thatthese systems appear in diverse practical scenarios that range fromindustrial to space applications [21], [26], [27].

Proposition 2: The P+d controller

τ i = −pi

∑j∈Ni

wij (qi2 − qj 2 (t − Tj i (t))) − di qi2 (6)

with pi , di ∈ R> 0 solves the consensus problem of Section II-B, forthe underactuated EL-system (5), provided that pi and di satisfy (4). �

Proof: This proof follows the same procedure as the proof of Propo-sition 1. Hence, only the main steps are given.

Consider the kinetic and potential energy function

Wi =12(q�

i Mi (qi )qi + q�i Kiqi ).

Its time derivative along (5) and (6), using Property P2, is Wi =−di |qi2 |2 − pi

∑j∈Ni

wij q�i2 (qi2 − qj 2 (t − Tj i (t))).

Let us now propose W = 12 q�

2 (L ⊗ In )q2 +∑N

i=11p i

Wi , where

q2 := col(q�12 , . . . ,q

�N 2 ), which is positive definite and radially un-

bounded w.r.t. the agents position errors |qi2 − qj 2 |, the velocitiesqi , and the actuated and nonactuated position error |qi1 − qi2 | for alli ∈ N and j ∈ Ni . W satisfies

W = −N∑

i=1

di

pi

|qi2 |2 −N∑

i=1

∑j∈Ni

wij q�i2

∫ t

t−T j i (t)qj 2 (θ)dθ.

Integrating W , from 0 to t, and following verbatim the procedureof the proof of Proposition 1, it is shown that if damping injectionsatisfies (4), then qi2 ∈ L2 and qi , |qi1 − qi2 |, |qi2 − qj 2 | ∈ L∞ forall i ∈ N and j ∈ Ni .

The rest of the proof is established with a systematic applicationof the Barbalat’s Lemma, which ensures that the consensus prob-lem of Section II-B holds. Hence, limt→∞

∑j∈Ni

wij (qi2 − qj 2 ) =0 ∀i ∈ N , j ∈ Ni . �

C. Additional Remarks

Remark 2: Using the scaled total energy function, it can be provedthat if the EL-systems network is composed of two different sets offully actuated and underactuated EL-systems, simple P+d controllerssolve the consensus objective if condition (4) and Assumptions A1 andA2 hold.

1506 IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 6, DECEMBER 2013

1

3

4

2

5 76

10

9

8

0.3

0.7

0.4

0.3

0.6

0.8

0.5 0.6

0.3 q1

q2

m1

l1

m2

l2

Fig. 1. Weighted network with ten 2-DOF revolute joint manipulators.

0 1 2 3 4 5

0.3

0.5

0.7

Time (s)

Del

ay(s

)

Fig. 2. Emulated UDP/IP Internet delay for the simulations.

Remark 3: A network composed of two agents, and with the in-

formation exchange modeled by the Laplacian L =[

1 −1−1 1

], cor-

responds to the teleoperation system in [12]. In such a case, con-dition (4) is satisfied if the controller gains are set as 4d1d2 >(∗T12 + ∗T21 )2p1p2 .

Remark 4: Similar to [12], if one or more human-operators injectforces on a single or on multiple EL-systems it can be easily provedthat, if such human interaction is passive, then position errors andvelocities are bounded. This claim also applies when the EL-systemsinteract with passive environments.

IV. SIMULATIONS

Using the network with ten 2-DOF revolute joint manipulators ofFig. 1, this section presents simulations that illustrate the solution tothe control problem reported in this paper.

For simplicity, the interconnection variable time delay for all agentsis the same, and it emulates an ordinary UDP/IP Internet delay with anormal Gaussian distribution with mean, variance, and seed equal to0.45, 0.005, and 0.35, respectively [28]. Such delays are shown in Fig. 2,and for practical purposes, ∗Tij = 0.7 s. It should be underscored thatcompared with the real Internet delays in [12], these delays are larger.

A. Fully Actuated Case

In this case, m = n = 2, Gi = I2 , and each manipulator dy-namics follows the EL equation (1), whose inertia and Coriolis

matrices are given by Mi =[

αi + 2βici2 δi + βici2

δi + βici2 δi

], and Ci =

βi

[−2si2 qi2 −si2 qi2

si2 qi1 0

], respectively. In these expressions, cik

, sik

are a short notation of cos(qik) and sin(qik

); qikis the articular position

of link k of manipulator i, with k ∈ {1, 2}; αi = l2i2mi2 + l2i1

(mi1 +

2

4

6

8

10

0 10 20

0

2

4

6

8

−10

−8

−6

−4

−2

0 10 20−8

−6

−4

−2

0

−10

−8

−6

−4

−2

0 10 20−8

−6

−4

−2

0

q = 2.3213

q = −2.3213 q = −3.2682

q = −4.0828 q = −6.2541

q = 4.0828

(a) (b) (c)

Fig. 3. Results for the fully actuated case. In Column A, q(0) is given by (7).In Column B, q(0) in (7) has been multiplied by −1. In Column C, the initialconditions are the same as in Column B without time delays.

mi2 ), βi = li1 li2 mi2 , and δi = l2i2mi2 , where lik

and mikare the

respective lengths and masses of each link.The network is composed of three different groups of manipulators,

with equal members in each group. The physical parameters are: m1 =4 kg, m2 = 2 kg, and l1 = l2 = 0.4 m, for Agents 1, 2, and 3; m1 =3 kg, m2 = 2.5 kg, l1 = 0.6 m, and l2 = 0.5 m for Agents 4, 5, and 6;m1 = 3.5 kg, m2 = 2.5 kg, l1 = 0.3 m, and l2 = 0.35 m for Agents7, 8, 9, and 10.

The proportional gains pi for the controllers (3) are all 10 Nm.Setting αi = 1, using ∗Tij = 0.7, and pi = 10 Nm, condition (4)transforms to di > 8.5�ii , where �ii corresponds to the ith-diagonalelement of the Laplacian of the graph in Fig. 1. Setting the damp-ing gains as d1 = 12, d2 = 7.7, d3 = 2.6, d4 = 9.5, d5 = 4.3, d6 =5.2, d7 = d9 = 7, d8 = 9, and d10 = 14 ensures that condition (4)holds.

Fig. 3 presents the simulations results for three different scenarios.Column A shows that, under the initial conditions

q(0) = [11, 8, 10, 7, 9, 6, 8, 5, 7, 4, 6, 3, 5, 2, 4, 1, 3, 0, 2,−1]� (7)

the EL-systems asymptotically converge to the consensus point qc =[4.0828, 2.3213]�. Column B depicts the convergence to the pointqc = [−4.0828,−2.3213]�, and in this case, the initial condition (7)have been multiplied by −1. Note that a change of sign in the ini-tial conditions resulted in a change of sign in the consensus point.Finally, in Column C, it can be seen that the consensus point isqc = [−6.2541,−3.2682]� for the case when the initial conditionsare the same as in Column B but without time delays. Other sim-ulations, not included here for sake of space, with larger delays(∗Tij = 10 s) show that the consensus point is qc = [0, 0]�. Inter-estingly, the average of the initial conditions in Column C, of Fig. 3, isave(q(0)) = [−6.5,−3.5]�. Hence, based only on these simulations,we can conjecture that the time delays change the consensus point fromsomewhere near the average of the initial conditions to zero. However,this far from rigorous claim remains to be proven theoretically.

In Fig. 4, the stability condition (4) has not been met, and it can beobserved that the agents do not agree at any point.

B. Underactuated Case

In this case, m = 2n = 4, and Gi = diag(02 , I2 ). The nonlineardynamics follow (5) with Mi , Ci defined as Mi ,Ci in Section IV-A.The interconnection network, the time delays, the physical parameters,and the controller gains are the same of Section IV-A. The additionalparameters are for i ∈ {1, 2, 3}, Si = 4I2 , and Ji = diag(1.5, 1.2);for i ∈ {4, 5, 6}, Si = 6.5I2 , and Ji = diag(1.2, 1); and, for i ∈{7, 8, 9, 10}, Si = 4.5I2 , and Ji = diag(0.8, 0.7).

IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 6, DECEMBER 2013 1507

−10

0

10

q1(r

ad)

0 10 20 30 40 50−5

0

5

10

Time (s)

q2(r

ad)

Fig. 4. Simulation results for the fully actuated case when the damping gainsdo not satisfy the stability condition.

Fig. 5. Results for the underactuated case. In Column A, q2 (0) is given by(7). In Column B, q2 (0) in (7) has been multiplied by −1. In Column C, theinitial conditions are the same as in Column B without time delays.

Fig. 5 shows the behavior of the nonactuated and the actuated po-sitions, in its upper and lower parts, respectively, for the same threescenarios of the fully actuated simulations. For simplicity, the initialconditions for the nonactuated and the actuated positions are the same,i.e., q1 (0) = q2 (0). Similar to the simulations in Section IV-A, theinitial conditions in Column A are given by (7), the initial conditionsin Column B are given by (7), multiplied by −1, and the case withoutdelays is depicted in Column C. Additionally, to the same conclusionsas in Section IV-A, it is interesting to note that compared with thefully actuated case, underactuation does not significantly change theconsensus point.

V. CONCLUSION

This paper presents a simple P+d controller that is capable of asymp-totically driving a network composed of fully actuated, or a class ofunderactuated, Euler–Lagrange systems toward a consensus point ifsufficiently large damping is injected. The network is modeled as anundirected and weighted interconnection graph, and it is assumedthat the interconnection exhibits asymmetric variable time delays.

Simulations with ten 2-DOF EL-systems illustrate the convergenceperformance of the closed-loop system for both cases: the fully actu-ated and the underactuated.

Despite the fact that simulations confirm that initial conditions andtime delays play a major role in the location of the consensus point,an open question is the analytical determination of such a point. Us-ing a linearizing controller and without delays, the authors’ previouswork [29] provides an analytical solution to the quasi-average consen-sus problem via the explicit solution to the resulting linear equation.However, since the closed-loop system in this paper is highly nonlin-ear, the previous approach cannot be invoked. Current research aimsto solve this problem using the simple P+d schemes reported here. Afuture research avenue is the solution to the leader–follower and lead-erless consensus problems for a more general class of underactuatedEL-systems.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and the Re-viewers for improving the quality of this paper with their commentsand suggestions.

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