Journal of Robotics, Networking and Artificial Life, Vol. 2, No. 1 (June 2015), 30-33

Distributed Terminal Backstepping control for Multi-Agent Euler-Lagrange Systems
Hu ChenDep. of Electronic Eng., Pusan National University, Jangjeon-dong, Geumjeong-gu
Busan, 609-735,Korea Republic
Seong-Ik Han
Dep. of Electronic Eng., Pusan National University, Jangjeon-dong, Geumjeong-gu
Busan, 609-735,Korea Republic
Jang-Myung Lee
Dep. of Electronic Eng., Pusan National University, Jangjeon-dong, Geumjeong-gu
Busan, 609-735,Korea Republic
E-mail: chenhu@pusan.ac.kr,skhan@pusan.ac.kr,jmlee@pusan.ac.kr
www.pusan.ac.kr

Abstract
This paper presents a distributed terminal (finite-time) backstepping consensus control for multi-agent EulerLagrange systems. Terminal virtual error surfaces and virtual controls are proposed to guarantee the finite-timeerror consensus and formation convergence of a group of one-leader and multi-follower cooperative tracking EulerLagrange system.Finite-time stability including infinite-time stability was proved by the finite-time Lyapunov
candidate function. Simulation example shows the effectiveness of the proposed finite-time backstepping
coordinated tracking controller.
Keywords: Euler-Lagrange multi-agent system, backstepping control, Terminal virtual error surface.

1. Introduction
In recent years, there has been a great interest for
researches of multi-agent systems, whose applications
include spacecraft, mobile robots, sensor networks, etc.
Interesting research directions are containment control,
consensus, formation, and flocking control [1]. These
problems focus on two cases, namely, the case that
there does not exist a leader and the case where thereexists a leader. The coordinate tracking problems totrack a single leader have been investigate for followers
with single-integrator, double-integrator, high-order
dynamics, nonlinear or Euler-Lagrange dynamics [2-5].
Linear control theory and variable structure control
methods in most researches are used. On the other hand,
there are few examples that use the backstepping
control technique [6] for nonlinear or Euler-Lagrange
multi-agent system. In this method,the problem of
unmatched uncertainty and neglecting the efficientnonlinearities is overcomevia adopting step-bysteprecursiveprocess.
However,although a controller designed using thistheorem guarantees the infinite-time stability of aclosed-loop system, it has drawbacks such as a slow
convergence rate and reduced robustness to uncertainty.
On the other hand, systems with finite-time settlingtime design possess attractive features such as
improved robustness and disturbance rejection
properties [7],In this paper, terminal backstepping
control based multi-agent consensus control for EulerLagrange system with one-leader and multi-followers isdeveloped.
2. Background and Preliminaries
2.1. Concept of Graph Theory
In this paper, multi-agent robot Euler-Lagrange systemsconsisting of one leader and n followers are considered.

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Graph theory is introduced to solve the coordination
K iG xi ,1 − M i−1Gi ( xi ,1 ) ≤ δ gi , and δ ci + δ gi + δ di ≤ δ hi ,problem and model information exchange between
where K iC and K iG are positive definite diagonalagents. The communication topology is a directedmatrices and vectors, respectively, and δ hi > 0 aregraph, G ={V ,E } , where V ={0,1, 2, ...,n} is the set ofupper bounds.
nodes, node i represents the ith agent, E is the set ofedges, and an edge in G is denoted by an ordered pair
3. Distributed
Terminalbackstepping(i, j ) . (i, j ) ∈ E if and only if the ith agent can send
Controller Design and Stability Analysis
information to jth agent directly, but not necessarily
3.1.Controller Design
vice versa.A directed tree is a directed graph, where
The tracking errorsand virtual error surfaces are defined
every node has exactly one parent except for the root,
as follows:
and the root has directed paths to every other node. An(3)zi ,1 ∑ j =1 aij ( yi − y j ) + bi ( yi − x0 ) ,=directed graph , G ={V ,E } , has a directed spanning treeγ
if and only if {V ,E } has at least one node with azi ,2 =xi ,2 + ci ,1 sig ( zi ,1 ) i ,1 − α i ,1 , i = 1,..., n ,(4)=A [ai , j ] ∈ R ( n +1)×( n +1) isdirected path to all other nodes.
where x0 is the position of the leader, α i ,1 are the virtual
called the weighted adjacency matrix of G ,γ i ,1where aii = 0 and aij ≥ 0 with aij > 0 if there is an edge
controls, sig ( zi ,1 ) = zi ,1 sgn( zi ,1 ) , ci ,1 > 0 are
between the ith agent and jth . The Laplacian of theconstants, and γ i ,1 = ξi ,1 / ζ i ,1 , ξi ,1 and ζ i ,1 are positive
weighted
graphcanbedefinedas( n +1)×( n +1)
odd numbers, ξi ,1 < ζ i ,1 < 2ξi ,1 , sgn( zi ,1 ) is a sign,whereL = D − A∈ R( n +1)× ( n +1)function,. (3) can be changed for the formation control
is the degree matrix
D = diag (d 0 , d1 ,..., d n ) ∈ Rncase
as follows:
and di = ∑ j = 0 aij for i = 0,1,..., n . For simplicity, it isnassumed that aij = 1 if (i, j ) ∈ E and 0 otherwise. The =zi ,1 ∑ j =1 aij ( yi + ∆ i − y j − ∆ j ) + bi ( yi + ∆ i − x0 − ∆ 0 )connection weight between agent i and the leader is(5)denoted by bi such that bi = 1 if agent i connected to theThe time derivative of the first error surfaces zi ,1 alongleader and 0 otherwise.
(2) isγzi ,1 =(di + bi )( zi ,2 − ci ,1 sig ( zi ,1 ) i ,1 + α i ,1 )n
2.2.Multi-Agent Euler-Lagrange Systems−∑ j =1 aij xi ,2 − bi x0 .
(6)
The nonlinear dynamics of a group of n + 1 fully
The Lyapunov function candidate Vi ,1 = ziT,1 zi ,1 / 2 toactuated Euler-Lagrange systems are described as
design the distributed virtual controller. Differentiating
follows:
Vi ,1 yields
M i (qi )qi + Ci (qi , qi )qi + Gi (qi ) + τ di =
τ i , i 1,..., n + 1 ,=(1)where M i (qi ) is a symmetric and positive definite
matrix; Ci (qi , qi ) is

inertiaavelocity-dependent

centripetal and Coriolis forcesmatrix; Gi (qi ) is a
gravitational vector; τ di is a bounded unknown
disturbance including unmodelled dynamics andexogenous disturbance; and τ i is an input torque. The
simple dynamic equation can be expressed as thefollowing state space model:xi ,1 = xi ,2 ,=xi ,2 f i ( x2 ) + gi ( x2 )ui ,yi = x=
1,..., n + 1 ,i ,1 , i

(2)

where
=xi ,1 q=qi , x2 = [ xi ,1 , xi ,2 ] , f i ( x2 ) =i , xi ,2TγVi ,1 = ziT,1[(di + bi )( zi ,2 − ci ,1 sig ( zi ,1 ) i ,1 + α i ,1 )−∑ j =1 aij xi ,2 − bi x0 ] .
nChoosing the distributed virtual control as
1nα i ,1=−ki ,1 zi ,1 + ∑ i =1 aij xi ,2 + bi x0 ,di + bi(7) becomes
where ki ,1 > 0 are constants. Differentiating
Lyapunov function
Vi ,2 =Vi ,1 + ziT,2 zi ,2 / 2 + δhi2 / 2ηi ,along (2) and (4),−δhiδˆhi / ηi .

Assumption 1. M τ≤ δ di , K − M C ( xi ,2 ) ≤ δ ci ,theγ i ,1 +1Vi ,2 =−ki ,1 ziT,1 zi ,1 − (di + bi )ci ,1 zi ,1+ (di + bi ) ziT,1 zi ,2ui = τ i .
−1ii(8)γ i ,1 +1Vi ,1 =−ki ,1 ziT,1 zi ,1 − (di + bi )ci ,1 zi ,1+ (di + bi ) ziT,1 zi ,2 ,(9)+ ziT,2 [ fi ( x2 ) + gi ( x2 )ui + ci ,1γ i ,1 zi ,1Ci)(− M i−1Ci ( xi ,2 ) xi ,2 − M i−1Gi ( xi ,1 ) − M i−1τ di , gi = M i−1 ,and−1idi(7)γ i ,1 −1zi ,1 − αi ,1 ]Choosing the control inputs and adaptive laws as

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ui = gi−1[−ki ,2 zi ,2 − (di + bi ) ziT,1 + KiC xi ,2 + KiG xi ,1−ci ,1γ i ,1 zi ,1+δˆhi zi ,2γ i ,1 −1γzi ,1 − ci ,2 sig ( zi ,2 ) i ,2+ αi ,1 ] ,zi ,2 + κ i ,2indexed by 1, 2, 3, and 4, respectively as shown in Fig.
1. The strict feedback state equations of each agent are
expressed asxi ,1 = xi ,2 ,(11)()

where2=δˆhi ηi zi ,2 / ( zi ,2 + κ i ,2 ) − ηi′δˆhi ,(12)where ki ,2 > 0 , ηi > 0 , ηi′ > 0 , ci ,2 > 0, and 0.5 < γ i ,2 < 1δ hi − δˆhi , δˆhi are estimates of δ hi ,are constants, δ=hi
we obtain the following expression:
2γ i ,1 +1
γ i ,2 +1Vi ,2 ≤ −∑ ki ,1 ziT, k zi , k − (di + bi )ci ,1 zi ,1− ci ,2 zi ,2k =1

+δhi ( ziT,2 − δˆhi / ηi )2γ +1
η ′δ 2 2≤ −∑ ki ,1 ziT, k zi , k − i i − ∑ β i , k zi , k+ηi′δ i2 / 22k 1=k 1
γ +122 2′ηδη ′δ 2 i ,k
≤ −  ∑ ki , k ziT, k zi , k + i i  −(∑ β i, k ziT, k zi , k + i i ) 22  k =12
 k =1+ µii ,kγ i ,k +1

γ i ,k +1≤ −aVi ,2 −b 2 Vi ,2 2

=xi ,2 f i ,2 ( xi ,1 ) + gi ,2 ui ,+ µi ,(16)f i ,2 =−[Gi (qi ) + τ di ] / J i , gi = 1/ J i , ui = τ i

J i = mi L2i / 3 , Gi (qi ) = mi Li cosqi ,the mass of the link

mi = 1kg , and the length of link Li = 0.25m .Let theinitial condition of four followers be=x1,1 1,=x1,2 0 ,, x3,1 2,==x2,1 1.2,
=x2,2 0=x3,2 0 , x4,1 = −1.2, x4,2 = 0 . The
Laplacian can be written as00L=0 −1 −1030

0−110−1−10
0−10

100
−1,=b3000 1,=
b4 1 .

Simulation results are obtained with thetime-varying control input to the leader being designed− sin( x0,1 ) / (1 + e − t ) , x0,1 = π / 2 , and x0,2 = 0 .
as u0 =(13)whereβi , k min[(di + bi )ci ,1 , ci ,2 ], a = min[2ki ,1 , 2ki ,2 ,ηi′] ,=22
 η ′δ 2  η ′δ 2b = min[2 β i ,1 , 2 β i ,2 ,ηi′] , µi = ηi′ zi , k δi +  i i  + i i2 2 

3.2. Finite-Time Stability Analysis((13) can be rewritten the following two forms:
 γ +1
 γ +1
 i ,k2µi  i ,k2
Vi ,2 ≤ −aVi ,2 −  b− γ i ,k +1  Vi ,2
Vi ,2 2 
= −a ′Vi ,2 −b′Vi γ,2′ ,(14)γ i ,k +1

γ i ,k +1

where a ′= a − µi / Vi ,2 =, b′ b 2 − µi / Vi ,2 2γ′=

γ i,k + 12Fig. 1. Directed graph of the manipulator group

The error functions for the illustration of the formation)
controlarechangedinto(5),where∆1 = −1 , ∆2 = −2 , ∆3 = −3 , and ∆4 = −4 . Simulation
results are presented in Fig. 2 (consensus control) andFig. 3 (formation control), where the settling time of theproposed TBSC system is 31% faster than that of theBSC system. In addition, the steady state errors of theTBSC system are smaller compared to the BSC system.

, and.From (14), if a and b is selected such2γ

that a > µi / Vi ,2 and b > µ i ,k+1/ Vi ,2 , respectively. Then,from the definition of finite-time stability [7], theequilibrium point x = 0 is globally finite-time stable andthe settling time ts can be given by′ts ≤aV 1−γ ( x0 ) + b′
1ln.
a (1 − γ ′)b′(15)4. Simulation Example
To validate the proposed control scheme, the following
group of one leader indexed by 0 and four followers
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Acknowledgements
This research was supported by the MOTIE (Ministryof Trade, Industry & Energy), Korea, under theIndustry Convergence Liaison Robotics CreativeGraduates Education Program supervised by the KIAT
(N0001126).
This work was supported by the National Research
Foundation of Korea(NRF) Grant funded by the Korean
Government(MSIP) ( NRF-2013R1A1A2021174 )This research was supported by the MOTIE (Ministryof Trade, Industry & Energy), Korea, under theIndustry Convergence Liaison Robotics Creative
Graduates Education Program supervised by the KIAT
(N0001126)
Fig. 2. Consensus control simulation results. (a) Tracking
outputs of BSC system. (b) Tracking outputs of TBSC system.
(c) z1,1 of BSC. (d) z1,1 of TBSC.

Table 1. Settling time (sec) of BSC and TBSC systems
Consensus
Formation
BSC
TBSC
BSC
TBSCz1,1 ≤ 0.01
2.03 s0.77 s2.23 s0.83 sz2,1 ≤ 0.05

2.20 s

0.67 s

2.63 s

0.91 sz3,1 ≤ 0.01

2.07 s

0.76 s2.83 s0.86 s2.15 s100%0.42 s32%

1.46 s100%0.29 s31%

z4,1 ≤ 0.05
Mean (%)

References
1. W. Ren, Distributed consensus in multi-vehicle
cooperativecontrol (Springer-Verlag, London,2008).
2. H. Arcak, and M. Wen, Adaptive motion coordination:using relative velocity feedback to track a reference
velocity, Automatica, 45(2) (2009) 1020-1025.
3. H. Su, G. Chen, and Z. Lin, Adaptive second-orderconsensus of networked mobile agents with nonlinear
dynamics, Automatica, 47(2) (2011) 368-375
4. H. Zhang, and F. L. Lewis, Adaptive cooperative
tracking control of high-order nonlinear systems withunknown dynamics, Automatica, 48 (2012) 1432-1439.
5. S. Chung, and J. J. Slotine, Cooperative robot control
and concurrent synchronization of Lagrangian systems,IEEE Tran. Robotics, 25(3) (2012) 686-700.
6. K. Kristic, I.Kanellakopoulos, and P. V. Kokotovic,Nonlinear and adaptive control design (Wiley, NewYork, 1995).
7. S. Yu, X. Yu, B. Shirinzadeh, and Z. Man, Continuous
finite-time control for robotic manipulators with terminal
sliding mode, Automatica, 41(2005)1975–1964.

5. Conclusion
A terminal backstepping control scheme to guarantee
the fast error convergence and small tracking error
performance for a multi-agent Euler-Lagrange system
is developed in this paper. A virtual finite-time error
surface is defined to design a virtual control. The finitetime convergence is proved by the finite-time stability
analysis of Lyapunov function. Simulation for one-link
manipulator agents confirms the theoretical proposal.

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