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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 29 (2012) 2539 – 2544 www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

Attitude Synchronization for Multiple Rigid Bodies with Time Delays Jianting Lva*, Dai Gaob, Siming Menga a a School of Mathematical Science, Heilongjiang University, Harbin 150080, China b Research Center of Satellite Technology, Harbin Institute of Technology, Harbin, 150080, China

Abstract In this paper we consider distributed attitude synchronization problem for multiple rigid bodies with time delays in the intercommunication. Based on graph theory and Lyapunov stability theory, we propose two distributed control laws. The first control law can guarantee attitude synchronization with zero final angular velocities when time delays exist. The second control law can guarantee attitude synchronization with non-zero final angular velocities when time delays exist. Throughout the paper, the communication flow among rigid bodies is assumed to be fixed and

undirected.

© 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Keywords: multiple rigid bodies; attitude synchronization; time delays.

1. Introduction Coordinated attitude control of formation flying spacecraft or rigid bodies have been the interest of many researchers. Various strategies and approaches, such as leader-follower, behavioral and virtual structure, are discussed in many technical notes [1-5]. Recently, compared to [1-5], graph theory which was often used in dealing with single or double integrator dynamics has been applied in analyzing coordinated attitude problem. In [6], distributed attitude synchronization problem for multiple spacecraft through local information exchange was studied. In [7], distributed attitude synchronization and tracking problem in the presence of a time-varying reference state was addressed. In [8], leader-follower control strategy for coordinated

* Corresponding author. Tel.: +86-451-86221980 E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.347

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attitude control of multiple rigid bodies was proposed. In [9], a behavioral approach was used for attitude synchronization without angular velocity measurement. In [10], authors presented a solution to the problem of tracking relative attitude in a leader-follower spacecraft formation without angular velocity measurements. In [11], authors proposed a passivity based control law for distributed attitude synchronization under undirected communication graph. The aforementioned literature, its results apply to the system dynamics without time delays. Due to the ubiquitous existence of communication constraints such as delays and message dropouts in the networked environments, cooperative control algorithms with time delays have been studied by many researchers [12-16]. In [12], authors developed a decentralized variable structure coordinated attitude control law for spacecraft formation in the presence of time delays, and adopted nonlinear relative attitude expression to describe the relative attitudes and assumed the communication flow was undirected. In this paper, we propose two distributed control laws with communication time-delays for multiple rigid bodies. We first show that attitude synchronization with zero final angular velocities can be guarantee under an undirected communication graph. The global asymptotical stability for the resulting closed-loop system is proved theoretically. Furthermore, attitude synchronization with non-zero final angular velocities is also discussed. The rest of the paper is organized as follows. In section 2 we introduce background and preliminaries. In Section 3 we propose distributed attitude control laws for attitude synchronization among multiple rigid bodies. Section 4 contains our conclusion. 2. Background and preliminaries Consider a group of N rigid body with the equations of motion given by J i ω& i + ωi× J i ωi = ui = σ& i G (σi= )ωi

⎞ 1 ⎛ 1 − σ iT σi I 3 + σ i× + σi σiT ⎟ ωi ⎜ 2⎝ 2 ⎠

(1) (2)

where J i ∈ R 3×3 is the constant, positive-definite, symmetric inertia matrix, ωi ∈ R 3 is the angular velocity vector of the body frame with respect to the inertial frame, expressed in the body frame, ui ∈ R 3 is the control torque, σ i ∈ R 3 denotes the MRP that represents the orientation of the body frame with respect to the inertial frame, ωi× denotes a 3 × 3 skew-symmetric matrix. For an N rigid bodies system, there is information interchange among rigid bodies. We assume that information flow among rigid bodies is directed and balanced and is described by the graph G = {V , E } . V = {1, 2,L , N } is the set of nodes and E ∈ V × V is set of edges. An edge (i, j ) in an undirected graph denotes that node i and node j can obtain information from one another. The weighted adjacency matrix

= A [aij ] ∈ � n×n of a graph G is defined as aii = 0 and aij > 0 if (i, j ) ∈ E . A graph is called connected

if for any two nodes there exists a set of edges that connect the two nodes.

3. Control Law Design In this section, we consider distributed attitude synchronization for multiple rigid bodies with communication time-delays. We assume the communication graph G is fixed and undirected. T ji is the time delays from the jth rigid body to the ith and it is assumed that each T ji is constant and uniquely

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defined. We first develop a distributed control law to guarantee attitude synchronization with zero final angular velocities when time delays exists and the result is stated in the theorem 1. Theorem 1: For the system given by (1)~(2), using the following control law N

N

ui = −ki G iT σ& i − G iT ∑ aij (σi − σ j (t − T ji )) − G iT ∑ aij (σ& i − σ& j (t − T ji ))

(3)

if the undirected graph G is connected and control gains satisfy ki >

1 N ∑ aij (μ 2 + 1) for T ji ≤ μ , then 2 j =1

=j 1 =j 1

σi → σ j (t − T ji ) , ωi → ω j (t − T ji ) → 0 , as t → ∞ .

Proof: Define = Fi F= (σi ) Gi−1 (σi ) and substitute (2) into (1), and then it follows that J i*σ&&i + Fi*σ& i = ui*

(4)

in which = Fi F= (σi ) G −1 (σi ) , J i* = FiT J i Fi , Fi* = − FiT ( J i Fi G& (σ i ) + ( J i Fi σ& i )× ) Fi , ui* = FiT ui . Consider the Lyapunov function candidate t 1 N T * 1 N N 1 N N σ& i J i σ& i + ∑∑ aij (σ i − σ j )T (σi − σ j ) + ∑∑ aij ∫ σ& Tj ( τ )σ& j ( τ )dτ ∑ t −T ji 2=i 1 4=i 1 =j 1 2=i 1 =j 1

= V

t 0 1 N N + μ ∑∑ aij ∫ ∫ σ& Tj (α )σ& j (α )dαdτ − +τ μ t 2 =i 1 =j 1

(5)

The time derivative of V is given by = V&

N

1

∑ ( 2 σ&

=i 1

+

T i

( J&i∗ − 2Fi ∗ )σ& i + σ& iT ui∗ ) +

1 N N ∑∑ aij (σ&i − σ& j )T (σi − σ j ) 2=i 1 =j 1

t 1 N N 1 N N aij (σ& j σ& Tj − σ& Tj (t − T ji )σ& j (t − T ji )) + ∑∑ aij ( μ 2 σ& Tj σ& j − μ ∫ σ& Tj (τ )σ& j (τ )dτ ) ∑∑ t −μ 2=i 1 =j 1 2=i 1 =j 1

(6)

Because the communication graph is fixed and undirected, we have that N

N

∑∑ a σ

=i 1 =j 1

ij

T j

N

N

σ j = ∑∑ aij σiT σi

(7)

=i 1 =j 1

1 N N (σ i − σ j ) ∑∑ aij (σ& i − σ& j )T = 2=i 1 =j 1

N

N

∑∑ a σ&

=i 1 =j 1

ij

T i

(σ i − σ j )

(8)

Using (7) and (8), (6) becomes N T i =i 1 =j 1

V&=

N

∑ σ& N

N

+ ∑∑ aij σ& iT (σi − σ j ) + =i 1 =j 1

+

N

(−∑ aij (σi − σ j (t − T ji )) − ∑ aij (σ& i − σ& j (t − T ji )) − ki σ& i ) =j 1

N

N 1 aij (σ& iT σ& i − σ& Tj (t − T ji )σ& j (t − T ji )) ∑∑ 2=i 1 =j 1

t 1 N N aij ( μ 2 σ& iT σ& i − μ ∫ σ& Tj (τ )σ& j (τ )dτ ) ∑∑ t −μ 2=i 1 =j 1

(9)

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Here we have used the fact that ( J&i∗ − 2Fi ∗ ) is skew-symmetric.

Using the following inequality T ji ∫

t

σ& Tj (τ )σ& j (τ )dτ ≤ μ ∫

∫

t

t −T ji

t

t −μ

t −T ji

σ& j (τ )dτ ≤ T ji ∫

t

σ& Tj (τ )dτ ≤ σ& iT σ& i + T ji ∫

t

σ& Tj (τ )dτ ∫

t

t −T ji

−2σ& iT ∫

t

t −T ji

σ& Tj (τ )σ& j (τ )dτ

t −T ji

t −T ji

(10)

σ& Tj (τ )σ& j (τ )dτ σ& Tj (τ )dτ ∫

t

t −T ji

(11)

σ& j (τ )dτ

(12)

Then we know that N

N

N

V& ≤ ∑ σ& iT (−∑ aij (σ i − σ j (t − T ji )) − ∑ aij (σ& i − σ& j (t − T ji ) − ki σ& i )

=i 1 =j 1 N

=j 1

N

1 N N + ∑∑ aij σ& i (σi − σ j ) + ∑∑ aij (σ& iT σ& i − σ& Tj (t − T ji )σ& j (t − T ji )) 2=i 1 =j 1 =i 1 =j 1 N N 1 N N 1 N N + ∑∑ aij μ 2 σ& iT σ& i + ∑∑ aij σ& iT σ& i + ∑ ∑ aij σ& iT (σ j − σ j (t − T ji )) 2=i 1 =j 1 2=i 1 =j 1 i =1 =j 1

(13)

N

1 N 1 N aij ( μ 2 + 1))σ& iT σ& i − ∑ (σ& i − σ& j (t − T ji ))T (σ& i − σ& j (t − T ji )) ∑ 2 j1 2i1 =i 1 = = = − ∑ ( ki −

1 N Note that 0 ≤ V (t ) ≤ V (0) < ∞ , and V& ≤ 0 with ki > ∑ aij ( μ 2 + 1) . In addition, we can verify that 2 j =1 V&& is bounded. Using Barbalat’s lemma [17], this implies that limt →∞ V& = 0 . Therefore, it follows that σ& i → 0 , σ& i → σ& j (t − T ji ) as t → ∞ .

With the direct manipulation, we have det ( 4G (σi ) ) = (1 + σi2 )3 ≠ 0 , which implies that matrix G (σ i ) is nonsingular, therefore we get ωi → 0 and ωi → ω j (t − T ji ) . Furthermore, through Barbalat's lemma we obtain ω& i → 0 . Using the above results, the dynamics (1), with (3) reduces to

N

∑a j =1

ij

(σi − σ j (t − T ji )) → 0 . Therefore,

we can conclude that σi → σ j (t − T ji ) , ωi → ω j (t − T ji ) → 0 asymptotically. Remark 1: The proposed control scheme is model independent. The implementation of the control scheme doesn’t require the accurate values of the inertia matrix. Next we show attitude synchronization but with non-zero final angular velocities can be guaranteed when communication time-delays exist. The result is stated in the following theorem. Theorem 2: For the system given by (1) and (2), using the following control law N

N

u= ωi× J i ωi − ki J i G (σi )ωi − ki ∑ aij (σi − σ j (t − T ji )) − ∑ aij (ωi − ω j (t − T ji )) i

(14)

=j 1 =j 1

if the undirected graph G is connected and control gains satisfy ki > 0 , then σi → σ j (t − T ji ) , ωi → ω j (t − T ji ) , as t → ∞ .

Proof: Consider the Lyapunov function candidate

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et al. / Procedia 29 (2012) 2539 – 2544 JiantingJianting Lv,Dai Lv Gao,Siming Meng /Engineering Procedia Engineering 00 (2011) 000–000 t k N N 1 N T 1 N N si J i si + i ∑∑ aij σijT σij + ∑∑ aij ∫ sTj ( τ ) s j ( τ )dτ V= ∑ t −T ji 2=i 1 2 =i 1 =j 1 2=i 1 =j 1

(15)

ωi + ki σi , σij = σ j (t − T ji ) − σi (t ) . where s= i

The time derivative of V is given by = V&

N

∑s

T i

=i 1 N

N

N

J i s&i + ki ∑∑ aij σ& ijT σij + =i 1 =j 1

1 N N ∑∑ aij ( sTj s j − sTj (t − Tji ) s j (t − Tji )) 2=i 1 =j 1

1 N N = ∑ s (−ω J i ωi + ui + ki J i σ& i ) + ki ∑∑ aij σ& ij σij + ∑∑ aij ( sTj s j − sTj (t − T ji ) s j (t − T ji )) 2=i 1 =j 1 =i 1 =i 1 =j 1 T i

×

N

N

N

N

(16)

T

Using (14), (16) becomes = V&

N

N

∑ s (∑ a T i

=i 1

+

ij

=j 1

N

N

N

N

( sTj (t − T ji ) − si )) + ki ∑∑ aij σ& ijT σij =i 1 =j 1

1 ∑∑ aij ( sTj s j − sTj (t − T ji ) s j (t − T ji )) 2=i 1 =j 1

= ki ∑∑ aij σ& ijT σij − =i 1 =j 1 N

N

= ki ∑∑ aij σ& ijT σij − =i 1 =j 1

1 N N ∑∑ aij (sTj (t − T ji ) − si )T ( sTj (t − T ji ) − si ) 2=i 1 =j 1

(17)

1 N N ∑∑ aij (σ&ij + ki σij )T (σ& ij + ki σij ) 2=i 1 =j 1

1 N N = − ∑∑ aij (σ& ijT σ& ij + ki2 σijT σ ij ) 2=i 1 =j 1

Note that 0 ≤ V (t ) ≤ V (0) < ∞ . In addition, we can verify that V&& is bounded. Using Barbalat’s lemma, this implies that lim V& = 0 . Therefore, it follows that σ → 0 , σ& → 0 as t → ∞ . t →∞

ij

ij

Therefore, we can conclude that σi → σ j (t − T ji ) , ωi → ω j (t − T ji ) asymptotically. 4. Conclusion We address the problem of attitude synchronization with time delays for multiple rigid bodies. Distributed control schemes are presented. The global asymptotic stability of closed-loop system is shown through Lyapunov analysis attitude synchronization can be guaranteed. Further work will consider the extension of the proposed control algorithms in the presence of control input constraints. Acknowledgements This work is supported by the Fund of Heilongjiang Education Committee under Grant No. 12511402, and the Fund of Heilongjiang University for Young Teachers under Grant No.QL201008. References [1] Krogstad TR, Gravdahl JT. Coordinated attitude control of satellites in formation. In Group Coordination and Cooperative

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Jianting Lv et al.Meng / Procedia Engineering 29 (2012) 2539000–000 – 2544 Jianting Lv,Dai Gao,Siming / Procedia Engineering 00 (2011)

Control, Lecture Notes in Control and Information Sciences, vol 336, chapter 9, pages 153-170. Springer-Verlag, 2006. [2] VanDyke MC, Hall CD. Decentralized coordinated attitude control within a formation of spacecraft. Journal of Guidance, Control and Dynamics, 2006, 29(5): 1101-1109. [3] Kang W, Yeh H. Coordinated attitude control of multi-satellite formation. International Journal of Robust and Nonlinear Control, 2002, 12: 185-205. [4] Ren W, Beard RW. Decentralized scheme for spacecraft formation flying via the virtual structure approach. Journal of Guidance, Control, and Dynamics. 2004, 27(1):73~82. [5] Cong B, Liu X, Chen Z. Distributed attitude synchronization of formation flying via consensus-based virtual structure. Acta Astronautica, 2011, 68(11-12): 1973-1986. [6] Ren W. Distributed attitude alignment in spacecraft formation flying. International Journal of Adaptive Control and Signal Processing. 2007, 21: 95-113. [7] Ren W. Formation keeping and attitude alignment for multiple spacecraft through local interactions. Journal of Guidance, Control and Dynamics. 2007, 30(2): 633-638. [8] Dimarogonas DV, Tsiotras P, Kyriakopoulos, KJ. Leader–follower cooperative attitude control of multiple rigid bodies. Systems & Control Letters, 2009, 58(6):429-435. [9] Mehrabian AR, Tafazoli S, Khorasani K. Coordinated attitude control of spacecraft formation without velocity feedback: A decentralized approach. AIAA Guidance, Navigation, and Control Conference, Chicago, 2009: 1-15. [10] Kristiansen R, Loria A, Chaillet A, Nicklasson P. Spacecraft relative rotation tracking without angular velocity measurements. Automatica, 2009, 45(3): 750-756. [11] Ren W. Distributed cooperative attitude synchronization and tracking for multiple rigid bodies. IEEE Transactions on Control Systems Technology. 2010, 18(2): 383-392. [12] Jin E, Jian X, Sun Z. Robust decentralized attitude coordination control of spacecraft formation. Systems & Control Letters, 2008, 57(7): 567-577. [13] Jin E, Sun Z. Robust attitude synchronization controllers design for spacecraft formation. IET Control Theory & Applications, 2009, 3(3): 325-339. [14] Wang N, Zhang T, Xu J. Formation control for networked spacecraft in deep space: with or without communication delays and with switching topology. Information Sciences. 2011, 54(3):469-481. [15] Nuno E, Basanez L, Ortega R. Passivity-based control for bilateral teleportation: A tutorial. Automatica, 2011, 47:485-495. [16] Min H, Sun F, Wang S, Li H. Distributed adaptive consensus algorithm for networked euler-lagrange systems. IET Control Theory and Applications, 2011, 5(1):145-154. [17] Eslotine J, Li W. Applied nonlinear control. New Jersey: Prentice Hall, Englewood Cliffs; 1991.