Decentralized cooperative control of multiple nonholonomic dynamic systems with uncertainty

Automatica 45 (2009) 706–710

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Decentralized cooperative control of multiple nonholonomic dynamic systems with uncertaintyI Wenjie Dong ∗ , Jay A. Farrell Department of Electrical Engineering, University of California, Riverside, CA 92521, USA

article

info

Article history: Received 20 July 2007 Received in revised form 2 May 2008 Accepted 11 September 2008 Available online 17 December 2008

a b s t r a c t This paper considers feedback control of a group of nonholonomic dynamic systems with uncertainty. Decentralized cooperative controllers are proposed with the aid of Lyapunov techniques, results of graph theory, and backstepping techniques. Robustness of the control laws with respect to communication delays is analyzed. An application of the proposed results is discussed. Simulation results show the effectiveness of the proposed controllers. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Cooperative control Consensus Decentralized control Nonholonomic system Formation control Mobile robots

1. Introduction Cooperative control of multiple agents has received considerable attention in recent years due to its challenging features and many applications, e.g., rescue mission, movement of large objects, troop hunting, formation control, and satellite clustering. Various methods have been proposed, which include behavior based (Lawton, Beard, & Young, 2003), virtual structure based (Beard, Lawton, & Hadaegh, 2001), leader–follower (Desai, Ostrowski, & Kumar, 2001), artificial potential based (Leonard & Fiorelli, 2001), graph theory based (Fax & Murray, 2004; Jadbabaie, Lin, & Morse, 2003; Olfati-Saber and Murray, 2004) methods, and so on. Cooperative control of wheeled mobile robots has been studied in Desai et al. (2001), Justh and Krishnaprasad (2004), Lawton et al. (2003) and Lin, Francis, and Maggiore (2005). Several cooperative control laws have been proposed. For cooperative control of multiple general nonholonomic kinematic systems, consensus based control laws were proposed in Dong and Farrell (2008). However, most of the existing results on the cooperative control of wheeled mobile robots are based on the kinematics of the systems. The dynamics of wheeled mobile robots and

I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Huaguang Zhang under the direction of Editor Toshiharu Sugie. This work was supported by the National Science Foundation under Grant No. ECCS-0701791. ∗ Corresponding author. Tel.: +1 951 827 0987; fax: +1 951 827 2425. E-mail address: [email protected] (W. Dong).

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.09.015

possible uncertainty in the dynamics were not considered. In practice, most of nonholonomic mechanical systems are dynamic systems. The dynamics of systems usually cannot be neglected in the control when high performance of the closed system is required. In this paper we consider the cooperative control of multiple nonholonomic systems with dynamics and uncertainty and propose cooperative control laws such that the state of each system converges to a target point which moves along a desired trajectory. To solve the cooperative control problem, we propose cooperative controllers in two steps. In the first step, decentralized cooperative control laws are proposed for multiple kinematic systems with the aid of backstepping techniques and results from graph theory. In the second step, cooperative control laws for multiple dynamic systems are proposed with the aid of backstepping techniques and passivity properties of the dynamic systems. It is shown that the proposed control laws make the state of the group of systems converge to a desired trajectory if the communication graph is bidirectional, fixed, and strongly connected. Since communication delays are inevitable between neighboring systems, we analyze the effects of time delays on the proposed cooperative control laws. It is shown that our proposed cooperative control laws are robust to finite communication delays. As an application, we show that the proposed results can be successfully applied to formation control of multiple mobile robots with a desired trajectory. The contributions of this paper are as follows: (1) Cooperative control of multiple general nonholonomic dynamic systems with uncertainty is first considered and solved; (2) Decentralized cooperative control laws are systematically

W. Dong, J.A. Farrell / Automatica 45 (2009) 706–710

707

proposed with the aid of backstepping techniques and results from graph theory; (3) The proposed controllers are analyzed to show that they are robust to constant communication delay during the control.

To solve the cooperative control problem, we convert (1)–(2) into a suitable form. Let the vector fields g1 (q∗j ), . . . , gs+1 (q∗j ) form a basis for the null space of J (q∗j ). Then, by (2), there exists an (s + 1)vector u∗j = [u1j , . . . , us+1,j ]> such that

2. Problem statement

q˙ ∗j = g (q∗j )u∗j = g1 (q∗j )u1j + · · · + gs+1 (q∗j )us+1,j

Consider a group of m mechanical systems subjected to nonholonomic constraints. For system j, its motion is defined in the following form

where g (q∗j ) = [g1 (q∗j ), . . . , gs+1 (q∗j )] ∈ Rn×(s+1) . Differentiating both sides of (5) and substituting it into (1) and multiplying on the left by g > (q∗j ), we have

Mj (q∗j )¨q∗j + Cj (q∗j , q˙ ∗j )˙q∗j + Gj (q∗j )

= Bj (q∗j )τj + J > (q∗j )λj , J (q∗j )˙q∗j = 0

¯ j (q∗j )˙u∗j + C¯ j (q∗j , q˙ ∗j )u∗j + G¯ j (q∗j ) = B¯ j (q∗j )τj M (1) (2)

where q∗j = [q1j , . . . , qnj ]> is the state of system j, Mj (q∗j ) is an n × n bounded positive-definite symmetric matrix, Cj (q∗j , q˙ ∗j )˙q∗j presents centripetal and Coriolis torque, Gj (q∗j ) is gravitational torque, Bj (q∗j ) is an n × r input transformation matrix, τj is an rvector of control input. J (q∗j ) is (n − s − 1) × n full row rank matrix with s = n − 1 − Rank(J (q∗j )), 2 ≤ s + 1 < n, r ≥ s + 1. λj is an (n − s − 1)-vector Lagrange multiplier which expresses the constraint force on system j, and the superscript > denotes the transpose. In system (1)–(2), the constraint (2) is assumed to be completely nonholonomic for each system (Murray & Sastry, 1993). Dynamics (1) has the following two properties (Dong & Huo, ˙ j − 2Cj are skew-symmetric for proper 1999): (1) Matrices M definitions of Cj ; (2) For any differentiable vector ξ ∈ Rn , Mj (q∗j )ξ˙ + Cj (q∗j , q˙ ∗j )ξ + Gj (q∗j ) = Yj (q∗j , q˙ ∗j , ξ , ξ˙ )aj where aj are unknown inertia parameter vectors and the regressor matrices Yj (q∗j , q˙ ∗j , ξ , ξ˙ ) are known functions. During the control each system knows its own state and the states of some of the other systems by communication. For simplicity, we assume that the communications between systems are bidirectional. If each system is considered as a node, the communication between the systems can be described by a graph G = {V , E }, where V = {1, 2, . . . , m} is a node set, E is an edge set with element (i, j) which describes the communication from node i to node j. If the state of system i is available to system j, there will be an edge (i, j) in E . We call system i a neighbor of system j if the state of system i is available to system j. Since the communication is bidirectional, (i, j) ∈ E implies (j, i) ∈ E . For system j, the indexes of its neighbors form a set which is denoted by Nj . Therefore, the available states to system j for control are the state of system j and the state of system i for all i ∈ Nj . We call a graph G strongly connected if for any two different nodes i and j in V there exist a series of nodes l1 (= i), l2 , . . . , lk (= j) such that (ls , ls+1 ) ∈ E for 1 ≤ s ≤ k − 1. In this paper, we make the following assumption on the communication graph.

(5)

(6)

¯ j (q∗j ) = where we have used the fact that g > (q∗j )J > (q∗j ) = 0, M > > ¯ ˙ g (q∗j )Mj (q∗j )g (q∗j ), Cj (q∗j , q∗j ) = g (q∗j )Mj (q∗j )˙g (q∗j ) + ¯ j (q∗j ) = g > (q∗j )Gj (q∗j ), and B¯ j (q∗j ) = g > (q∗j )Cj (q∗j , q˙ ∗j )g (q∗j ), G g > (q∗j )Bj (q∗j ). Based on the properties of dynamics (1), the following two ˙¯ − 2C¯ is skewfacts can be easily proved: (1) Matrix M j j symmetric; (2) For any differentiable vector ξ ∈ R(s+1)×(s+1) , ¯ j (q∗j )ξ˙ + C¯ j (q∗j , q˙ ∗j )ξ + G¯ j (q∗j ) = Y¯j (q∗j , q˙ ∗j , ξ , ξ˙ )aj , where M
d Y¯j (q∗j , q˙ ∗j , ξ , ξ˙ ) = g > (q∗j )Yj q∗j , q˙ ∗j , g (q∗j )ξ , dt (g (q∗j )ξ ) . The

regressor matrix Y¯j is known while the parameter vector aj is not. The reduced system (5)–(6) describes the motion of the system (1)–(2). Therefore, the cooperative control problem can be considered based on the reduced system (5)–(6) instead of system (1)–(2). In order to completely actuate each nonholonomic system, B¯ j (q) is assumed to be a full rank matrix. To simplify the cooperative control problem, we assume that Eq. (5) has two inputs and is in the following chained form after suitable state and input transformations q˙ 1j = u1j ,

q˙ 2j = u2j ,

q˙ ij = u1j qi−1,j ,

3 ≤ i ≤ n.

(7)

Since the desired trajectory qd satisfies the constraint J (qd )˙qd = 0, noting the assumptions made above, trajectory qd satisfies q˙ d1 = w1 ,

q˙ d2 = w2 ,

q˙ di = w1 qdi−1 ,

3≤i≤n

(8)

where w1 and w2 are known time-varying functions. By the analysis shown above, the defined cooperative control problem with uncertainty is equivalent to finding a control law τj for system (6)–(7) with unknown inertia parameter vector aj using qd , q∗j , and the relative state information between its neighbors such that (3)–(4) are satisfied. 3. Cooperative controller design

Assumption 1. The communication graph G is bidirectional, fixed, and strongly connected.

To facilitate the controller design, the following assumptions are made on the desired trajectory.

Given a differentiable desired trajectory qd (t ) = [qd1 (t ), . . . , (t )]> which satisfies J (qd )˙qd = 0. The cooperative control problem discussed in this article is defined as follows.

Assumption 2. The signals qdi (2 ≤ i ≤ n − 1) are bounded. the

qdn

Cooperative control problem: Design a control law τj for system j with unknown inertia parameter vector aj using its own state q∗j , the relative state information between system j and system i for i ∈ Nj , and qd such that lim (q∗j − q∗i ) = 0,

t →∞

lim

t →∞

d

q −

m 1 X

m l =1

1 ≤ i 6= j ≤ m

(3)

= 0.

dl w1 dt l

(0 ≤ l ≤ n − 3;

following condition: and for all t ≥ 0.

(4)

R t +T t

d0 w1 dt 0 2n−4 1

w

= w1 ) are bounded and satisfy the (s)ds ≥  for some T > 0,  > 0,

Since the target point is required to move along the desired trajectory, we introduce the following tracking error variables: zij = qij − qdi − φij ,

! q∗l

signals

where

1 ≤ i ≤ n, 1 ≤ j ≤ m

(9)

708

W. Dong, J.A. Farrell / Automatica 45 (2009) 706–710

 φ1j = 0, φnj = 0, φn−1,j = −kn znj w12n−5       ˙1 ∂φn−1,j w   φn−2,j = −kn−1 zn−1,j w12n−5 − znj +   ∂w1 w1   
∂φn−1,j  2n−5  + − k w z + z  n 1 nj n−1,j   ∂ znj  φij = −ki+1 zi+1,j w12n−5 − zi+2,j  nX −i−2  ∂φi+1,j w1[l+1] ∂φi+1,j    + + (zn−1,j  [l]  w1 ∂ znj ∂w1  l =0   n −1  X  ∂φi+1,j  2n−5  (zl−1,j − k w z ) +  n nj 1   ∂ zlj  l=i+2  − kl w12n−5 zlj − zl+1,j ), i = n − 3, . . . , 2;

systems (6) and (11) such that the requirements in Lemma 2 are satisfied. Noting the structure of systems (6) and (11), we apply backstepping techniques in this paper. We first design the cooperative controllers for the kinematics (7) with the aid of Eq. (11). Then, we propose the cooperative controllers for system (6)–(7). (10)

η1j = −

[i]

[i]

di w1 ). dt i

These definitions yield

z˙1j = u1j − w1      ∂φ2j   (w1 zn−1,j − kn w12n−4 znj ) z˙2j = u2j − w2 −   ∂ znj    n −1  X  ∂φ2j   − (w1 zl−1,j − kl w12n−4 zlj − w1 zl+1,j )    ∂ z lj  l =3   n −3 n  X X  ∂φ2j [l+1] ∂φ2j   − w − ( u − w ) elj 1j 1  [l] 1 ∂ zlj l=0 ∂w1 l =3 (11)      z˙3j = −k3 z3j w12n−4 − z4j w1 + z2j w1     +(u1j − w1 )e3j    ..    .     ˙ z = −kn−1 zn−1,j w12n−4 − znj w1 + zn−2,j w1  n−1,j    + (u1j − w1 )en−1,j     z˙nj = −kn w12n−4 znj + w1 zn−1,j + (u1j − w1 )enj Pn ∂φ where enj = qn−1,j , eij = qi−1,j − l=i+1 ∂ z ij elj for i = n − 1, . . . , 2 lj and 1 ≤ j ≤ m. Eq. (11) can be obtained by the backstepping procedure, see e.g., Dong and Farrell (2008). That reference also proves the following lemma. Lemma 1. Under Assumption 2, if limt →∞ (z∗j − z∗i ) = 0 for 1 ≤ i 6= j ≤ m, then (3) holds. Furthermore, if limt →∞ z∗j = 0 for 1 ≤ j ≤ m, then (3)–(4) hold. Lemma 2. For system (11), under Assumption 2, if lim (u1j − w1 )eij = 0 and

t →∞

lim z2j = 0,

(12)

t →∞

for 3 ≤ i ≤ n and 1 ≤ j ≤ m, then limt →∞ zij = 0. 2 Proof. Define the positive Lyapunov function V1 = 21 i=3 zij . Differentiating V along the solutions of Eq. (11), we have 1 Pn Pn V˙ 1 = − i=3 ki w12n−4 zij2 + (u1j − w1 ) i=3 zij eij + z2j z3j w1 ≤

Pn

√ −2kw12n−4 V1 + 2f1 V1 , where k = min{ki , 3 ≤ i ≤ n}, f1 = Pn √1 √1 i=3 |(u1j −w1 )eij |+ 2 |z2j w1 | which is a nonnegative function 2 and converges to zero to zero √ because each term +in f1 converges by (12). Let V2 = V1 , thus we have D V2 ≤ −kw12n−4 V2 + Rt

R t − kw2n−4 (s)ds t 2n−4 f1 . So, V2 (t ) ≤ e− 0 kw1 (s)ds V2 (0) + 0 e % 1 f1 (%)d%. Noting Assumption 2, we can prove that V2 converges to zero. Furthermore, limt →∞ zij = 0 (3 ≤ i ≤ n, 1 ≤ j ≤ m).  R

X

bji z1j − z1i + ∆j − ∆i − µj (z1j + ∆j ) + w1




(13)

i∈Nj

the constants ki > 0 for 3 ≤ i ≤ n, and the notation w1 indicates the ith derivative of w1 (i.e. w1 = the following dynamic equations,

Lemma 3. Consider system (7), under Assumptions 1 and 2, the decentralized controllers u1j = η1j and u2j = η2j make (3)–(4) hold, where

Based on the results in Lemmas 1 and 2, the cooperative control problem can be solved by designing cooperative controllers for

η2j = −

X

bji z2j − z2i − µj z2j +




i∈Nj

n−3 X ∂φ2j l=0

∂w1[l]

w1[l+1]


∂φ2j −kn w12n−4 znj + w1 zn−1,j + w2 ∂ znj n−1 X ∂φ2j (−kl w12n−4 zlj − w1 zl+1,j + w1 zl−1,j ), (14) + ∂ zlj l =3 Pm with constants bji = bij > 0, µj ≥ 0 and l=1 µl > 0, and Pn Pn ∂φ2j ∆j = −z2j l=3 ∂ z elj + l=3 zlj elj . lj Pm Proof. Let the positive definite Lyapunov function V = 21 j=1 Pn 2 i=1 zij . Differentiating V along solutions of system (11) with the Pm Pn Pm 2 control laws, we have V˙ = − j=1 i=3 ki w12n−4 zij2 − j=1 µj [z2j + > 2 > (z1j + ∆j ) ] − z2∗ Lz2∗ − (z1∗ + ∆) L(z1∗ + ∆) ≤ 0, where L is the +

Laplacian matrix of the communication graph G with the weight matrix B = [bji ]. Therefore, V is bounded. Furthermore, zij are bounded. By Barbalat’s Lemma, limt →∞ V˙ = 0. So, lim ki w12n−4 zij2 = 0,

t →∞

lim z2>∗ Lz2∗ = 0,

t →∞

lim

t →∞

m X j =1

µj z2j2 = 0,

(3 ≤ i ≤ n, 1 ≤ j ≤ m)

(15)

lim (z1∗ + ∆)> L(z1∗ + ∆) = 0.

(16)

t →∞

lim

t →∞

m X

µj (z1j + ∆j )2 = 0.

(17)

j =1

Since the graph G is strongly connected, limt →∞ (z2∗ (t )− c2 (t )1) = 0 and limt →∞ (z1∗ (t ) + ∆(t ) − c1 (t )1) = 0 where c1 and c2 are bounded variables. Since at least one of µj is greater than zero, say µp > 0, then by (17) we have limt →∞ z2p = 0 and limt →∞ (z1p + ∆p ) = 0. Noting z2j converges to c2 , we see c2 = 0. Since (z1j + ∆j ) converges to c1 , we see c1 = 0. So, limt →∞ (u1j − w1 )eij = 0 for 3 ≤ i ≤ n and 1 ≤ j ≤ m. By Lemma 2, limt →∞ zij = 0 (3 ≤ i ≤ n, 1 ≤ j ≤ m). Noting the definitions of φij and ∆j , we can prove that limt →∞ z1j = 0 (1 ≤ j ≤ m) with the aid of Assumption 2. By Lemma 1, Eqs. (3) and (4) hold.  Remark 1. In (13)–(14), the first term is a weighted sum of the relative state information between system j and its neighbors. The terms µj (z1j + ∆j ) and µj z2j in (13)–(14) are called the damping terms which are used to make µj (z1j + ∆j ) and µj z2j converge to zero. The motion of the system is driven by the relative information between neighbors, which distinguishes the cooperative control laws from the tracking control laws for a single nonholonomic system in the literature. Control laws (13)–(14) include a tracking control law for a single system as a special case. In (13)-(14), the control parameters are bjl , ki , and µj . Generally, increasing ki and µj will increase the convergence rate of zij for 3 ≤ i ≤ n. The value of bji and the topology of the communication graph determine λ2 (L). Large λ2 (L) means that (z1j − z1i ) and (z2j − z2i ) converge to zero fast. The value of λ2 (L) and ki also affect the convergence rate of q∗j to qd .

W. Dong, J.A. Farrell / Automatica 45 (2009) 706–710

709

The Eqs. (13) and (14) provide the controllers for kinematics (7). Below we propose decentralized cooperative controllers for dynamics (6) using backstepping techniques. Theorem 1. For system (6)–(7), under Assumptions 1 and 2, the control laws 1 τj = B¯ − −Kj (u∗j − η∗j ) + Y¯j (q∗j , q˙ ∗j , η∗j , η˙ ∗j )ˆaj − Λj j




(18)

and update laws a˙ˆ j = −Γj Y¯j> (q∗j , q˙ ∗j , η∗j , η˙ ∗j )(u∗j − η∗j )

(19)

for 1 ≤ j ≤ m make (3)–(4) hold and aˆ j bounded, where constant matrices Kj and Γj are symmetric and positive definite, η∗j = [η1j , η2j ]> , η1j and η2j are defined in (13)–(14), u∗j = [u1j , u2j ]> , Λj = [z1j + ∆j , z2j ]> , and the other control parameters are defined in Lemma 3. Proof. Define u˜ ∗j = u∗j − η∗j and a˜ j = aj − aˆ j . Let the nonnegPm Pn 2 −1 > ¯ ˜> ative function V = 12 a˜ j ], difj Γj j=1 [ i=1 zij + u∗j Mj u∗j + a ferentiating V along solutions of the closed-loop system, we have
Pm Pn 2 2n−4 ˜ ∗j − µj z2j2 − µj (z1j + ∆j )2 V˙ = − u˜ > ∗j K j u j =1 − i=3 ki zij w1

− (z1∗ + ∆)> L(z1∗ + ∆) − z2>∗ Lz2∗ ≤ 0, following the proof of Lemma 3, we can prove that limt →∞ u˜ ∗j = 0 and limt →∞ zij (t ) = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ m. By Lemma 1, Eqs. (3) and (4) hold. 

Remark 2. The control law τj for system j consists of its own state, the relative state information with its neighbors, and the desired trajectory. Therefore, it is decentralized. Similarly, the update law (19) is decentralized too. The motion of the closed-loop system is driven by the relative information between neighbors. The control parameters are bji , Kj , Γj , ki (3 ≤ i ≤ n). The bji can be interpreted as the weights of the links of the communication digraph. Their values are user defined. Increasing Kj and ki will increase the convergence rate of the closed-loop system. However, large control parameters will result in large control inputs. The estimated values aˆ j are bounded. They are not guaranteed to converge to their real values. Considering the communication delay during the control, we have the following results. Theorem 2. For system (6)–(7), under Assumptions 1 and 2, control laws (18)–(19) make (3)- (4) hold and aˆ j bounded, where η1j and η2j are defined by

η1j (t ) = −µj (z1j (t ) + ∆j (t )) −

X

bji (z1j (t )

i∈Nj

− z1i (t − δi ) + ∆j (t ) − ∆i (t − δi )) + w1 (t ) X η2j (t ) = −µj z2j (t ) − bji (z2j (t ) − z2i (t − δi ))

(20)

i∈Nj

+ w2 (t ) +

n−3 X ∂φ2j (t ) l =0

∂w1[l]

(t )w1[l+1] (t )

∂φ2j (t ) (w1 (t )zn−1,j (t ) − kn w12n−4 (t )znj (t )) ∂ znj (t ) n −1 X ∂φ2j (t ) + (w1 (t )zl−1,j (t ) − kl w12n−4 (t )zlj (t ) ∂ z ( t ) lj l =3 − w1 (t )zl+1,j (t )) (21) Pm with constants bji = bij > 0, µj ≥ 0 and l=1 µl > 0, communication delays δi (≥ 0) are constants, and the other variables +

and control parameters are defined in Theorem 1.

Fig. 1. Desired geometric pattern.

Pm Pn 2 1 [ z (t ) + 2 Rj=t 1 i=1 ij P −1 > ¯ ˜ ˜ b (( z u> ( t ) M ( t ) u ( t ) + a ( t ) Γ a ( t ) + 1i (s) + j ∗j j ∗j j j i∈Nj t −δi ji 2 2 ∆i (s)) + z2i (s))ds], the theorem can be proved by following the Proof. Let the nonnegative function V (t ) =

proof of Theorem 1.



Remark 3. Eqs. (3) and (4) hold for large constant delays. However, the cooperative performance is bad if communication delays are large. 4. Formation control with a desired trajectory and simulations Consider a group of m wheeled mobile robots which move on a plane. Let Ξ∗j = [xj , yj , θj ]> be the kinematic portion of the state of robot j, where (xj , yj ) is the coordinate of the middle point of the front wheels of robot j in the fixed coordinate frame OXY , θj is the orientation of robot j with respect to the X -axis of the coordinate frame O-XY , the dynamics of each robot can be described as (1)–(2) with the state Ξ∗j used instead of q∗j , where Cj = 0, Gj = 0, J (Ξ∗j ) = [sin θj , − cos θj , 0], Mj = diag[mj , mj , Ij ],   Bj = R1 [cos θj , sin θj , Lj ]> , [cos θj , sin θj , Lj ]> , mj is the mass of j

robot j, Ij is its inertia moment around the vertical axis at the middle point between the two driving wheels, Rj is the radius of the two driving wheels, and 2Lj is the length of the axis of the driving wheels (see Dong and Huo (1999)). Given a desired formation P described by constant centroid offset vectors (pjx , pjy ) (1 ≤ j ≤ m) and a desired trajectory Ξ d = (xd , yd , θ d ) which satisfies x˙ d sin θ d − y˙ d cos θ d = 0. Assume the communication between m robots is described by a graph G, we consider the following problem. Formation control with a desired trajectory: Design a control law for robot j using Ξ d , Ξ∗j , and the relative information between Ξ∗j and Ξ∗i for i ∈ Nj such that



xl − xj lim t →∞ yl − yj





lim (θl − θj ) = 0,

t →∞

m X xi

m i =1

(22)

1 ≤ l 6= j ≤ m

t →∞

lim



p − pjx = lx , ply − pjy

! − xd

= 0,

lim

t →∞

(23) m X yi

m i =1

! − yd

= 0.

(24)

To solve this problem, following the procedure in Section 2, we have x˙ j = v1j cos θj , y˙ j = v1j sin θj , and θ˙j = v2j , where v1j and v2j are the speed and angular rate of robot j, respectively.

710

W. Dong, J.A. Farrell / Automatica 45 (2009) 706–710

desired trajectory. If there are constant communication delays in control, the control laws achieve the same objectives according to Theorem 2. Due to space limit, we omit them here. 5. Conclusion This paper discusses the cooperative control of multiple nonholonomic dynamic systems with uncertainty. Cooperative control laws are proposed. Simulation results show the effectiveness of the proposed control laws. References

Fig. 2. Communication graph G.

Fig. 3. Paths of the centroid of the five robots and the geometric patterns at several times.

This system can be converted into system (7) with n = 3 by the state transformations (58) and (62) in Dong and Farrell (2008). The reduced dynamics of each robot is (6). Define new variables as in (63) in Dong and Farrell (2008), we have Eq. (8) with n = 3. Simple calculation derives the following result. Lemma 4. By the transformations in Eqs. (58) and (62)–(63) in Dong and Farrell (2008), if (3) holds, then Eq. (22) is satisfied. Furthermore, if (3)–(4) hold, then Eqs. (22)–(24) are satisfied. By Lemma 4, the formation control problem with a desired trajectory can be solved by the results obtained in Section 3. To verify the effectiveness of the proposed results, we choose m = 5. Assume the desired formation P is shown in Fig. 1 and the communication graph G is shown in Fig. 2. The desired trajectory (xd , yd , θ d ) = (40 sin 2t , −40 cos 2t , 2t ). The cooperative controllers can be obtained by Theorem 1. Fig. 3 shows the path of the centroid of the five robots and the geometric patterns of the five robots at several times. It is shown that the five robots come into the desired formation and the centroid of the group of robots converges to the

Beard, R. W., Lawton, J., & Hadaegh, F. Y. (2001). A coordination architecture for spacecraft formation control. IEEE Transactions on Control Systems Technology, 9(6), 777–790. Desai, J. P., Ostrowski, J. P., & Kumar, V. (2001). Modeling and control of formations of nonholonomic mobile robots. IEEE Transactions on Robotics and Automation, 17, 905–908. Dong, W., & Farrell, J. A. (2008). Cooperative control of multiple nonholonomic mobile agents. IEEE Transactions on Automatic Control, 53(7). Dong, W., & Huo, W. (1999). Adaptive stabilization of uncertain dynamic nonholonomic systems. International Journal of Control, 72(18), 1689–1700. Fax, J. A., & Murray, R. M. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49, 1465–1476. Jadbabaie, A., Lin, J., & Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48, 988–1001. Justh, E. W., & Krishnaprasad, P. S. (2004). Equilibrium and steering laws for planar formations. System and Control Letters, 52, 25–38. Lawton, J. R. T., Beard, R. W., & Young, B. (2003). A decentralized approach to formation maneuvers. IEEE Transactions on Robotics and Automation, 19(6), 933–941. Leonard, N.E., & Fiorelli, E. (2001). Virtual leaders, artificial potentials and coordinated control of groups. In Proc. of the IEEE conf. on decision and control (pp. 2968–2973). Lin, Z., Francis, B., & Maggiore, M. (2005). Necessary and sufficient graphical conditions for formation control of unicycles. IEEE Transactions on Automatic Control, 50(1), 121–127. Murray, R. M., & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using sinusoids. IEEE Transactions on Automatic Control, 38(5), 700–716. Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49, 101–115.

Wenjie Dong is interested in robotics and automation. His research areas include robot control, internet based teleoperation, nano-manipulation and assembly, mobile sensor networks, swarms, insect flight control system analysis and applications, information dynamics in networked feedback systems, nonlinear system control, model predictive control, etc. He is particularly interested in insect control systems and other biological control systems and their applications in engineering.

Jay A. Farrell received B.S. degrees (1986) in physics and electrical engineering from Iowa State University, and M.S. (1988) and Ph.D. (1989) degrees in electrical engineering from the University of Notre Dame. At Charles Stark Draper Lab (1989–1994), he was the principal investigator on projects involving intelligent and learning control systems for autonomous vehicles. Dr. Farrell received the Engineering Vice President’s Best Technical Publication Award in 1990, and Recognition Awards for Outstanding Performance and Achievement in 1991 and 1993. He is a Professor and former Chair of the Department of Electrical Engineering at the University of California, Riverside. He is the author of over 140 technical publications. He is also the co-author of the books ‘‘The Global Positioning System and Inertial Navigation’’ (McGraw-Hill, 1998) and ‘‘Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches’’ (John Wiley 2006).