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On Attitude Synchronization of Multiple Rigid Bodies with Time Delays
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
On Attitude Synchronization of Multiple Rigid Bodies with Time Delays ⋆ Hanlei Wang and Yongchun Xie Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing, 100190, China (Tel: +86-010-68744801; e-mail: wanghanlei01@ yahoo.com.cn). Abstract: This paper investigates the attitude synchronization problem under communication time delays. The encountered difficulty in synchronizing rigid body attitudes is non-convexity of the rotation group. Through exploiting the geometrical structure of the rigid body attitude kinematics, we propose a PD-like synchronization control law. Using Lyapunov stability analysis, it is shown that the attitude synchronization errors converge asymptotically to zero, and that the proposed controller achieves robust stability for communication time delays. Simulation results are presented to show the performance of the proposed synchronization control algorithm. Keywords: Attitude synchronization; time delay; PD controller. 1. INTRODUCTION Synchronization of multi-agent systems has received great attention in recent years and it has important applications in various areas such as spacecraft formation control (Wang, Hadaegh, and Lau, 1999; Ren, 2007; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006), robot telemanipulation (Lee and Spong, 2006; Chopra, Spong, and Lozano, 2008), cooperative control of multiple robot manipulators (Rodrigues-Angeles and Nijmeijer, 2004; Chung and Slotine, 2009), unmanned air vehicles (UAVs) and mobile robots (Cheah, Hou, and Slotine, 2009), and attitude synchronization of multiple rigid bodies (Sarlette, Sepulchre, and Leonard, 2009; Meng, Ren, and You, 2009; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006; Ren, 2010; Chung, Ahsun, and Slotine, 2009). Attitude synchronization of rigid bodies or multiple spacecraft, as a special control problem, has some distinct features incurred by its complex rigid body attitude kinematics. Moreover, it is impossible to obtain a rotation matrix via adding two different rotation matrices, that is, the space formed by rotation matrices is non-convex, which is well-known in Lie group literature. To tackle this problem, several approaches via exploiting the geometric structure of the attitude kinematics have been proposed (Sarlette, Sepulchre, and Leonard, 2009; Smith, Hanssmann, and Leonard, 2001). And other control schemes have also been presented, which are based on various representations of the attitude synchronization error (Meng, Ren, and You, 2009; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006; Ren, 2010; Chung, Ahsun, and Slotine, 2009; VanDyke and Hall, 2006; Abdessameud and Tayebi, 2009). However, most of these control schemes cannot tolerate communication delays, which is thought to be ubiquitous in the attitude synchronization of multiple rigid bodies or master-slave teleoperator end-effectors. ⋆ This work is supported in part by the National Natural Science Foundation of China under grant 61004058.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
One attitude synchronization scheme that provides robust stability for constant time delay is from Igarashi, Hatanaka, Fujita, and Spong (2007; 2009). Assuming the positive definiteness of the rotational matrix, it achieves asymptotic convergence of attitude synchronization errors at the kinematic level. Nevertheless, the assumption of positive definite rotational matrix implies that the controller can only work for the case of attitude deviation that is below 90 degree rotation. In addition, Igarashi, Hatanaka, Fujita, and Spong (2007; 2009) did not take the attitude dynamics into consideration, and thus it is unclear about the total system’s behavior if taking both the attitude kinematics and dynamics into account. Although there is only limited work in delayed attitude synchronization, many synchronization-based control approaches have been proposed to deal with communication time delays in bilateral teleoperation (Lee and Spong, 2006; Chopra, Spong, and Lozano, 2008; Chopra, Spong, Ortega, and Barabanov, 2006; Nuno, Ortega, Basanez, and Barabanov, 2008) and synchronization of multi-robot networks (Chung and Slotine, 2009). Master-slave joint position synchronization (joint space of the manipulator Rn is Euclidean and convex) is achieved via injecting joint dissipation both at the master and the slave sides. However, in attitude synchronization problem, the rotational group SO(3) is non-Euclidean and non-convex, which presents a major obstacle to accommodate the time delays. Inspired by the results for bilateral teleoperation (Lee and Spong, 2006; Chopra, Spong, Ortega, and Barabanov, 2006; Nuno, Ortega, Basanez, and Barabanov, 2008; Wang and Xie, 2011), this paper attempts to investigate a new attitude synchronization strategy under communication delays, and both the attitude dynamics and kinematics are taken into consideration. It is shown that the proposed new approach can ensure robust stability for constant time delays provided the dissipation gains satisfy some inequality. The proposed delay-robust controller bears simple PD structure, and furthermore it is singularity-free
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10.3182/20110828-6-IT-1002.00144
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
and has explicit physical meaning in the sense that the attitude synchronization error is computed in the rotational group SO(3) with non-Euclidean geometric structure, rather than in Euclidean form. These features make our result distinctive from the wave-variable based delayrobust Euclidean-form attitude synchronization strategy (Chung, Ahsun, and Slotine, 2009), and the robot synchronization algorithms in Euclidean space (Chopra, Spong, and Lozano, 2008; Chung and Slotine, 2009). 2. RIGID BODY NETWORKS AND ITS KINEMATICS AND DYNAMICS 2.1 Network topology This paper considers a rigid body network that consists of n rigid bodies. We assume that these n rigid bodies form a connected undirected graph, that is, every two different bodies is connected by an undirected path (Jungnichel, 2008). The vertex of the network i (i = 1, 2, · · · , n) represents the rigid body i and its edge set E defines the corresponding information flow between rigid bodies. The set formed by the neighbours of node i is denoted by Ni = { j ∈ ν|(i, j) ∈ E} , where ν is the vertex set of the network. In an undirected network, the communicating interaction between rigid body agents bears symmetric features, that is, if j ∈ Ni , then i ∈ Nj . 2.2 Rigid body kinematics and dynamics
With the operator ∨¡ : so(3) → R3 , we obtain the ¢ attitude error skew∨ RjT Ri from the skew-symmetric ¢ ¡ matrix skew RjT Ri , where ∨ is the inverse of the cross product operator S(·). The passivity property related to the attitude error repre¡ ¢ sentation skew∨ RjT Ri is given by the following lemma. ¢ ¡ Lemma 1. The mapping from skew∨ RjT Ri to the error T angular velocity ωij = ωi −R respect¢ ¡ iTRj ω¢j is passive with ¡ to the storage function ϕ Rj Ri = (1/2) tr I − RjT Ri (Bullo and Murray, 1999), Z t ¢ ¡ T ωij skew∨ RjT Ri dr 0 ¡ ¢ ¡ ¢ = ϕ RjT (t)Ri (t) − ϕ RjT (0)Ri (0) (5)
where tr(·) denotes the trace of the matrix ·.
3. ATTITUDE SYNCHRONIZATION CONTROL In this section, we attempt to propose control laws for attitude synchronization of multiple rigid bodies. To say attitude synchronization, we mean that skew∨ (RjT Ri ) → 0 for ∀i, j ∈ ν as t → ∞. To achieve the goal of attitude synchronization, we propose the following delay-robust synchronization control law, X ¡ ¢ kv ωi − RiT Rj (t − T )ωj (t − T ) τi = − j∈Ni
The i-th rigid body kinematics can be written as (Egeland and Godhavn, 1994), R˙ i = Ri S(ωi ) (1) where Ri ∈ SO(3) denotes the orientation matrix of the ith rigid body with respect to some inertial frame, ωi is its angular velocity expressed and # " in body-fixed frame, 0 −ωi,z ωi,y ωi,z 0 −ωi,x , where S(ωi ) is defined as S(ωi ) = −ωi,y ωi,x 0 T ωi = [ ωi,x ωi,y ωi,z ] . The equations of motion of the attitude dynamics can be expressed as (Egeland and Godhavn, 1994), Hi ω˙ i − S(Hi ωi )ωi = τi (2) 3×3 where Hi ∈ R is the constant, positive definite, and symmetric inertia matrix, τi ∈ R3 is the applied control torque vector. 2.3 Attitude synchronization error representation and its basic properties Consider two different rigid bodies i, j ∈ ν, and their error attitude matrix can be written as RjT Ri . Differentiating the error attitude matrix yields (Egeland and Godhavn, 1994), ¡ ¢ d £ T ¤ Rj Ri = RjT Ri ωi − RiT Rj ωj (3) dt Using the representation from Bullo and Murray (1999), we obtain the following attitude error representation, ¡ ¢ RjT Ri − RiT Rj skew RjT Ri = (4) 2
−kP
X
j∈Ni
¡ ¢ skew∨ RjT (t − T )Ri − (KD,i + Pi ) ωi (6)
where T is the communication delay, which is assumed to be constant and bounded, and for simplicity we assume that all rigid body agents suffer the same time delay. kv , kP > 0 are positive scalar feedback gains, −Pi ωi is the i-th rigid body’s self-dissipation for achieving the asymptotic stabilization purpose, and the dissipation term −KD,i ωi is included to compensate for the effects of the communication time delay, where Pi , KD,i are both symmetric positive definite matrices. Now we present the main result of this paper. Theorem 1. If the rigid bodies form a tree network, and furthermore P the controller gains are chosen such that KD,i ≥ T kP ·I, then, the control (6) results in attitude j∈Ni
synchronization of the rigid body networks (1) (2), i.e., skew∨ (RjT Ri ) → 0 for ∀i, j ∈ ν as t → ∞. Before proving Theorem 1, we first give a useful lemma. Lemma 2. Let A(t) and B(t) be n × n matrices, for ∀t ≥ 0, ¸ 12 ·Z t ·Z t ¸ 21 Z t ¡ T ¢ ¡ T ¢ ¡ T ¢ tr A A dr tr A B dr ≤ tr B B dr 0
0
0
(7)
tr
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(·Z
t
A (r) dr
t
A (r) dr
0
0
≤ t · tr
¸T ·Z
·Z
0
t T
A (r) A (r) dr
¸
¸) (8)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Proof. The matrices A(t), B(t) can be expressed as the cluster of column vectors, A(t) = [ f1 (t) f2 (t) · · · fn (t) ] (9) B(t) = [ g1 (t) g2 (t) · · · gn (t) ] (10) Then, using Schwarz inequality, we have, Z t Z tX Z t n ¡ ¢ f¯T g¯dr fkT gk dr = tr AT B dr = ≤
t
·Z
f¯T f¯dr
·Z
¸ 12 ·Z
t
g¯T g¯dr
0
0
=
0
0 k=1
0
t
0
T
¡
¢ tr A A dr
¸ 12 ·Z
t
¸ 12
¡
T
¢
tr B B dr
0
¸ 21
(11)
¤T ¤T £ £ , g¯ = g1T g2T · · · gnT . where f¯ = f1T f2T · · · fnT This completes the proof of the inequality (7). Next, we show the inequality (8). In fact, from Schwarz inequality, we obtain, (·Z ¸T ·Z t ¸) t tr A (r) dr A (r) dr 0
= ≤
0
n ·Z t X
i=1 n X
fi (r) dr
·Z
fi (r) dr
t
0
i=1
= t · tr
Z
t
0
0
t·
¸T Z
fiT (r)fi (r)dr
t
AT (r)A(r)dr
0
¸
(12)
Proof of Theorem 1. We consider the following LyapunovKrasovskii function candidate, n n X kP X X ¡ T ¢ 1 T ϕ Rj Ri ω i Hi ω i + V = 2 2 i=1 i=1 j∈Ni Z n kv X X t + ωjT ωj dr (13) 2 i=1 t−T
Using (15), we can rewrite (14) as, n ¤T kv X X £ ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) V˙ = − 2 i=1 j∈Ni £ ¤ T · ωi (r) − Ri (r)Rj (r − T )ωj (r − T ) n X n X X ωiT ωiT (KD,i + Pi ) ωi + kP − i=1 j∈Ni
i=1
£ ¡ ¢ ¡ ¢¤ · skew∨ RjT Ri − skew∨ RjT (t − T )Ri (16) ¢ ¤ ¡ ¢ ¡ £ Let Xij = ωiT skew∨ RjT Ri − skew∨ RjT (t − T )Ri in (16) be rewritten as, o 1 n T Xij = − tr (Rj − Rj (t − T )) Ri S (ωi ) (17) 2
Integrating Xij in time interval [0, t] and using Lemma 2 and some techniques in Chopra, Spong, Ortega, and Barabanov (2006) and Nuno, Ortega, Basanez, and Barabanov (2008), we obtain, Z t Xij (r)dr 0 #T Z t "Z T 1 = − tr R˙ j (r − s) ds Ri S (ωi ) dr 2 0 0 "Z # ) 12 Z Z T T 1 t T tr R˙ j (r − s) ds · R˙ j (r − s) ds dr ≤ 2 0 0 0 (Z t h i ¾ 12 · tr R˙ iT (r)R˙ i (r) dr 0
≤
Z 1
t
T · tr
2 0 (Z
T
0
#
R˙ jT (r − s) R˙ j (r − s)ds dr
) 21
i ¾ 12 T ˙ ˙ tr Ri (r)Ri (r) dr
t
h
·
j∈Ni
"Z
0
) 21 Z TZ t h Differentiating V with respect to time and using the skewi T ˙ ˙ symmetric property of S (Hi ωi ) and Lemma 1, we have, = T· tr Rj (r − s) Rj (r − s) drds 2 n X 0 0 X £ ¤ k T v (Z ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) V˙ = − t i ¾ 12 h 2 i=1 j∈Ni T · tr R˙ i (r)R˙ i (r) dr £ ¤ 0 · ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) ) 21 n n Z TZ t h X X X ¡ T ¢ i T ∨ T T ωi skew Rj (t − T )Ri ≤ 1 T · kP ωi (KD,i + Pi ) ωi − − tr R˙ j (r) R˙ j (r) drds 2 i=1 i=1 j∈Ni 0 0 n (Z X ¡ ¢ kP X t i ¾ 12 h + (ωi − RiT Rj ωj )T skew∨ RjT Ri (14) T ˙ ˙ 2 · tr Ri (r)Ri (r) dr 1
i=1
j∈Ni
0
Since the rigid bodies form an undirected network topology, thus, we can rewrite the last term on the right side of equation (14) as the following well-known form (based on the result of VanDyke and Hall (2006)), n ¡ ¢T ¡ ¢ 1X X kP ωi − RiT Rj ωj skew∨ RjT Ri 2 i=1 j∈Ni
=
n X i=1
kP
X
j∈Ni
ωiT skew
¡ ∨
RjT Ri
¢
(15)
(Z
1 = T 2 (Z ·
0
T 2
Z
t
t
tr
0
t
h
R˙ jT
i ¾ 21 (r) R˙ j (r) dr
i ¾ 12 T ˙ ˙ tr Ri (r)Ri (r) dr h
ωiT (r)ωi (r) + ωjT (r)ωj (r)dr 0 h i where we use the fact that tr R˙ iT R˙ i = 2ωiT ωi .
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≤
(18)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Now integrating V˙ in (16) with respect to time and using the result (18), we have, V (t) − V (0) Z n ¤T kv X X t £ ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) ≤− 2 i=1 j∈Ni 0 £ ¤ · ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) dr n Z t X X − T · I ωi dr ωiT KD,i − kP 0
i=1
−
n Z t X 0
i=1
j∈Ni
ωiT Pi ωi dr
where Di = KD,i − kP
(19) P
T · I is positive semi-definite
j∈Ni
due to the controller gain choice given in Theorem 1. Thus, from (19), we obtain, Z n ¤T kv X X t £ ωi (r) − RiT Rj (r − T )ωj (r − T ) V (t) + 2 i=1 j∈Ni 0 £ ¤ T · ωi (r) − Ri Rj (r − T )ωj (r − T ) dr n Z t n Z t X X + ωiT Pi ωi dr + ωiT Di ωi dr ≤ V (0) (20) i=1
0
i=1
0
Equation (20) means that V (t) ∈ L∞ , implying the boundedness of ωi for i = 1, 2, ..., n. And hence ωi − RiT Rj (t − T )ωj (t − T ) must be bounded. Then the control input τi is bounded. Using the rigid body attitude dynamics (2), we obtain the boundedness of ω˙ i . The boundedness of ω˙ i (t) gives rise to the uniform continuity of ωi − RiT Rj (t − T )ωj (t − T ) and ωi . Moreover, from (20), we have ωi − RiT Rj (t − T )ωj (t − T ) ∈ L2 , ωi ∈ L2 . Therefore, ωi − RiT Rj (t − T )ωj (t − T ) → 0 and ωi → 0 as t → ∞. From the rigid body attitude dynamics, we get, ω˙ i = Hi−1 [τi + S (Hi ωi ) ωi ]. Due to the uniform continuity of τi and S (Hi ωi ) ωi , ω˙ i must also be uniformly continuous. Using Barbalat Lemma (Slotine and Li, 1991), we have ω˙ i → 0 as t → ∞. Based on the closed-loop attitude dynamics, Hi ω˙ i − S (Hi ωi ) ωi = X kv (ωi − RiT Rj (t − T )ωj (t − T )) − j∈Ni
− kP
X
j∈Ni
¡ ¢ skew∨ RjT (t − T )Ri − (KD,i + Pi ) ωi (21)
we arrive at the conclusion that, as t → ∞, X ¡ ¢ skew∨ RjT (t − T )Ri → 0
undirected path and ¡ there¢are no cycles (Jungnichel, 2008), we obtain skew∨ RjT Ri → 0 for ∀i ∈ ν, j ∈ Ni as t → ∞ in a similar way as Abdessameud and Tayebi (2009). In fact, the incidence matrix B ∈ Rn×2(n−1) for a tree network with n vertices has the rank ¡of n −¢ 1, see Jungnichel (2008). In addition, since skew∨ RjT Ri = ¢ ¡ −skew∨ RiT Rj for j ∈ Ni , thus, the original synchronization error vector y whose dimension is 2 (n − 1) can be reduced to a vector x with dimension n − 1. Corre¯ = 0, spondingly, By = 0 is transformed equivalently to Bx ¯ ∈ R(n−1)×(n−1) is of full rank, which means that where B x = 0. x = 0 and the connectedness of the tree network result in attitude synchronization of the networked rigid ¢ ¡ bodies, i.e., skew∨ RjT Ri → 0 for ∀i, j ∈ ν as t → ∞.
For a general connected graph (e.g., it may contain loops or cycles), the case becomes much more complex and we have the following result.
Theorem 2. With the controller gains chosen as KDi ≥ P T kP · I and the rigid body network topology being j∈Ni
a general connected graph, if the error attitude matrix RlT Ri + RiT Rl (∀i, l ∈ ν) is positive definite, attitude synchronization of the rigid body networks (1) ¡(2) will¢ be achieved under the control law (6), i.e., skew∨ RjT Ri → 0 for all i, j ∈ ν as t → ∞.
Proof. Using skew∨T (RlT Ri ) to multiply both sides of (24) and computing the summation with respect to l, k yields, n X n X ¡ ¢ ¡ ¢X skew∨ RjT Ri skew∨T RlT Ri i=1 l=1 n n X X
1 = 2 =
1 4
j∈Ni
X
i=1 l=1 j∈Ni n X n X X i=1 l=1 j∈Ni
tr
"
RiT Rl
− RlT Ri RjT Ri − RiT Rj 2 2
£ ¤ tr RlT Ri − RlT Ri RjT Ri = 0
#
(25)
Next, we follow a similar procedure from Igarashi, Hatanaka, Fujita, and Spong (2009) to proceed the proof of Theorem 2. Since the network is connected and undirected, then, n X n X n X n X X £ ¤ X £ ¤ tr RlT Rj = tr RlT Ri (26) i=1 l=1 j∈Ni
i=1 l=1 j∈Ni
Using (26), we can reformulated (25) as follows, n n ¡ ¢ ¤ 1 XX X £ T tr Rl Ri + RiT Rl − RlT Ri + RiT Rl RjT Ri 4 i=1 l=1 j∈Ni
(22)
j∈Ni
Using finite increment theorem, we have, RjT (t − T )Rj − I = RjT (t − T )R˙ j (ξ)T, ξ ∈ [t − T, t] (23) Since R˙ j (ξ) = Rj (ξ) S (ωj (ξ)) → 0 as t → ∞, thus, RjT (t − T ) Rj (t) → I as t → ∞. Then, equation (22) can be further written as (at the extreme state t = ∞), X ¡ ¢ skew∨ RjT Ri = 0 (24)
=0
(27)
Since the error orientation matrix RlT Ri + RiT Rl (∀i, l) is positive definite, we obtain the following result using similar techniques as Igarashi, Hatanaka, Fujita, and Spong (2009), n n X X ¡ ¢ ¡ ¢X skew∨ RjT Ri skew∨T RlT Ri
j∈Ni
For a tree network topology, i.e., an undirected graph in which every two different bodies is connected by an 8777
i=1 l=1 n n X X
≥
1 4
j∈Ni
X
i=1 l=1 j∈Ni
¡ ¢ £ ¤ λmin RlT Ri + RiT Rl tr I − RjT Ri
(28)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where λmin (·) denotes the¡minimum ¢eigenvalue of ·. From (25) and (28), we have tr I − RjT Ri → 0 for ∀i ∈ ν, j ∈ Ni as t → ¡∞, which further results in the conclusion ¢ that skew∨ RjT Ri → 0 for ∀i, j ∈ ν as t → ∞ due to the connectedness of the rigid body network. Thus, delayrobust attitude synchronization is achieved for general connected graph case.
1
5
6 2
4. SIMULATION RESULTS In this section, we present simulation results to examine the performance of the proposed attitude synchronization strategy under communication time delays. In simulation, we employ six rigid body agents and they are required to synchronize their attitudes asymptotically. The network topology of the six rigid bodies is assumed to be a tree for simplicity, as shown in Fig. 1. Due to the limited space, we did not perform simulation study for the general network topology case. The communication delay between rigid body agents is T = 1s.
Fig. 1. Network topology formed by the six rigid bodies
agent sk
For general connected graphs, we just obtain a sufficient condition for ensuring attitude synchronization, which, however, seems somewhat restrictive. Our future research will focus on how to relax or remove this condition.
x
agent attitude sk
x
agent sk
1.5
4
x
agent sk 5
1
x
agent sk 6
x
0.5 0 −0.5 −1 0
20
40 time(s)
60
80
Fig. 2. The attitude coordinate skx of the six rigid body agents (without additional dissipation) 0.8 0.6 x
0.4
5. CONCLUSION In this paper, we studied the time-delayed attitude synchronization problem of the rigid body networks with undirected graph topology. Via exploitation of the geometrical structure of rigid body attitude kinematics, we proposed a synchronization control law which is composed of angular velocity and attitude synchronization terms and an additional dissipation term. It is shown that if the dissipation gain satisfies some inequality, the closedloop networked system is stable and the attitude synchronization errors converge to zero. And the convergence of the attitude synchronization errors holds with any boundknown constant communication time delay. A six-rigidbody network with tree topology is employed as a simulation example to show the performance of the proposed attitude synchronization controller.
x
agent sk 3
agent attitude sk
1, 2, · · · , 6, and kv , kP are chosen to be same as the first simulation. The time history of the six agents’ attitudes is shown in Fig. 3 to Fig. 5, from which we can see that attitude synchronization with communication time delay is achieved via adding dissipation −KD,i ωi according to Theorem 1.
2
2
In the first simulation, the controller parameters are chosen as kv = 5, KD,i = 0, Pi = 5I, kP = 20. The element skx of the i-th agent attitude representation skew∨ (Ri ) = T [ skx sky skz ] , i = 1, 2, ..., 6 is shown in Fig. 2. It can be seen that without additional dissipation term, i.e., KD,i = 0, attitude synchronization cannot be achieved and instability phenomenon is observed.
j∈Ni
agent1 skx
2.5
For simplicity, we assume that " the six rigid # bodies have 30 8 9 the same inertia matrix Hi = 8 21 6 , i = 1, 2, · · · 6. 9 6 25
In the second Psimulation, we choose the dissipation gains as KDi = T kP · I, the damping gain Pi = 5I, i =
3
4
agent sk
0.2
1
x
agent sk
0
2
x
agent sk 3
−0.2
x
agent sk 4
x
−0.4
agent5 skx
−0.6
agent6 skx
−0.8 0
20
40 time(s)
60
80
Fig. 3. The attitude coordinate skx of the six rigid body agents (with additional dissipation) REFERENCES Abdessameud, A. and Tayebi, A. (2009). Attitude synchronization of a group of spacecraft without velocity measurements. IEEE Transactions on Automatic Control, 54(11), 2642-2648. Bullo, F. and Murray, R. M. (1999). Tracking for fully actuated mechanical systems: a geometrical framework. Automatica, 35, 17-34.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
1
agent attitude sky
0.5 agent sk 1
y
agent2 sky
0
agent3 sky agent4 sky −0.5
agent5 sky agent sk 6
−1 0
20
40 time(s)
y
60
80
Fig. 4. The attitude coordinate sky of the six rigid body agents (with additional dissipation) agent sk 1
1
z
agent2 skz agent3 skz agent sk
agent attitude skz
0.5
4
z
agent5 skz agent6 skz 0
−0.5
−1 0
20
40 time(s)
60
80
Fig. 5. The attitude coordinate skz of the six rigid body agents (with additional dissipation) Cheah, C. C., Hou, S. P., and Slotine, J. J. E. (2009). Region-based shape control for a swarm of robots. Automatica, 45, 2406-2411. Chopra, N., Spong, M. W., Ortega, R., and Barabanov, N. E. (2006). On tracking performance in bilateral teleopera-tion. IEEE Transactions on Robotics, 22(4), 861-866. Chopra, N., Spong, M. W., and Lozano, R. (2008). Synchronization of bilateral teleoperators with time delay. Automatica, 44, 2142-2148. Chung, S.-J., Ahsun, U., and Slotine, J. J. E. (2009). Application of synchronization to formation flying spacecraft: Lagrangian approach. Journal of Guidance, Control and Dynamics, 32(2), 512-526. Chung, S.-J. and Slotine, J. J. E. (2009). Cooperative robot control and concurrent synchronization of Lagrangian systems, IEEE Transactions on Robotics, 25(3), 686-700. Dimarogonas, D. V., Tsiotras, P., and Kyriakopoulos, K. J. (2006). Laplacian cooperative attitude control of multiple rigid bodies, In Proceedings of the 2006 IEEE International Symposium on Intelligent Control, Munich, Germany. pp. 3064 -3069. Egeland, O. and Godhavn, J.-M. (1994). Passivity based adaptive attitude control of a rigid spacecraft. IEEE Transactions on Automatic Control, 39(4), 842-846.
Hughes, P. C. (1986). Spacecraft attitude dynamics. New York, John Wiley & Sons. Igarashi, Y., Hatanaka, T., Fujita, M., and Spong, M. W. (2007). Passivity-based 3D attitude coordination: convergence and connectivity. In Proceedings of IEEE International Conference on Decision and Control, New Orleans, LA, USA. pp. 2558-2565. Igarashi, Y., Hatanaka, T., Fujita, M., and Spong, M. W. (2009). Passivity-based attitude synchronization in SE(3). IEEE Transaction on Control Systems Technology, 17(5), 1119-1134. Jungnichel, D. (2008). Graphs, networks and algorithms. Berlin Heidelberg, Springer-Verlag. Lee, D. and Spong, M. W. (2006). Passive bilateral teleoperation with constant time delays. IEEE Transactions on Robotics, 22(2), 269-281. Meng, Z., Ren, W., and You, Z. (2009). Decentralized cooperative attitude tracking using modified rodriguez parameters, In Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China. pp. 853-858. Nuno, E., Ortega, R., Basanez, L., and Barabanov, N. (2008). A new proportional controller for nonlinear bilateral teleoperators. In Proceedings of 17th IFAC World Congress, Seoul, Korea. pp. 15660-15665. Ren, W. (2006). Distributed attitude consensus among multiple networked spacecraft. In Proceedings of the American Control Conference, Minneapolis, Minnesota, USA.pp. 1760-1765. Ren, W. (2007). Formation keeping and attitude alignment for multiple spacecraft through local interacttions, Journal of Guidance, Control and Dynamics, 30(2), 633-638. Ren, W. (2010). Distributed cooperative attitude synchronization and tracking for multiple rigid bodies, IEEE Transactions on Control Systems Technology, 18(2), 383-392. Rodrigues-Angeles, A. and Nijmeijer, H. (2004). Mutual synchronization of robots via estimated state feedback: a cooperative approach, IEEE Transactions on Control Systems Technology, 12(4), 542-554. Sarlette, A., Sepulchre, R., and Leonard, N. E. (2009). Autonomous rigid body attitude synchronization, Automatica, 45, 572-577. Slotine, J. J. E. and Li, W. (1991). Applied nonlinear control. Englewood Cliffs NJ: Prentice-Hall. Smith, T. R., Hanssmann, H., and Leonard, N. E. (2001). Orientation control of multiple underwater vehicles with symmetry-breaking potentials. In Proceedings of 40th IEEE Conference on decision and control, Orlando, Florida, USA. pp. 4598-4603. VanDyke, M. C. and Hall, C. D. (2006). Decentralized coordinated attitude control within a formation of spacecraft, Journal of Guidance, Control and Dynamics, 29(5), 1101-1109. Wang, P. K. C., Hadaegh, F. Y., and Lau, K. (1999). Synchronized formation rotation and attitude control of multiple free-flying spacecraft, Journal of Guidance, Control and Dynamics, 22(1), 28-35. Wang, H. and Xie, Y. (2011). Passivity based task-space bilateral teleoperation with time delays, In IEEE International Conference on Robotics and Automation, accepted.
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On Attitude Synchronization of Multiple Rigid Bodies with Time Delays ⋆ Hanlei Wang and Yongchun Xie Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing, 100190, China (Tel: +86-010-68744801; e-mail: wanghanlei01@ yahoo.com.cn). Abstract: This paper investigates the attitude synchronization problem under communication time delays. The encountered difficulty in synchronizing rigid body attitudes is non-convexity of the rotation group. Through exploiting the geometrical structure of the rigid body attitude kinematics, we propose a PD-like synchronization control law. Using Lyapunov stability analysis, it is shown that the attitude synchronization errors converge asymptotically to zero, and that the proposed controller achieves robust stability for communication time delays. Simulation results are presented to show the performance of the proposed synchronization control algorithm. Keywords: Attitude synchronization; time delay; PD controller. 1. INTRODUCTION Synchronization of multi-agent systems has received great attention in recent years and it has important applications in various areas such as spacecraft formation control (Wang, Hadaegh, and Lau, 1999; Ren, 2007; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006), robot telemanipulation (Lee and Spong, 2006; Chopra, Spong, and Lozano, 2008), cooperative control of multiple robot manipulators (Rodrigues-Angeles and Nijmeijer, 2004; Chung and Slotine, 2009), unmanned air vehicles (UAVs) and mobile robots (Cheah, Hou, and Slotine, 2009), and attitude synchronization of multiple rigid bodies (Sarlette, Sepulchre, and Leonard, 2009; Meng, Ren, and You, 2009; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006; Ren, 2010; Chung, Ahsun, and Slotine, 2009). Attitude synchronization of rigid bodies or multiple spacecraft, as a special control problem, has some distinct features incurred by its complex rigid body attitude kinematics. Moreover, it is impossible to obtain a rotation matrix via adding two different rotation matrices, that is, the space formed by rotation matrices is non-convex, which is well-known in Lie group literature. To tackle this problem, several approaches via exploiting the geometric structure of the attitude kinematics have been proposed (Sarlette, Sepulchre, and Leonard, 2009; Smith, Hanssmann, and Leonard, 2001). And other control schemes have also been presented, which are based on various representations of the attitude synchronization error (Meng, Ren, and You, 2009; Dimarogonas, Tsiotras, and Kyriakopoulos, 2006; Ren, 2010; Chung, Ahsun, and Slotine, 2009; VanDyke and Hall, 2006; Abdessameud and Tayebi, 2009). However, most of these control schemes cannot tolerate communication delays, which is thought to be ubiquitous in the attitude synchronization of multiple rigid bodies or master-slave teleoperator end-effectors. ⋆ This work is supported in part by the National Natural Science Foundation of China under grant 61004058.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
One attitude synchronization scheme that provides robust stability for constant time delay is from Igarashi, Hatanaka, Fujita, and Spong (2007; 2009). Assuming the positive definiteness of the rotational matrix, it achieves asymptotic convergence of attitude synchronization errors at the kinematic level. Nevertheless, the assumption of positive definite rotational matrix implies that the controller can only work for the case of attitude deviation that is below 90 degree rotation. In addition, Igarashi, Hatanaka, Fujita, and Spong (2007; 2009) did not take the attitude dynamics into consideration, and thus it is unclear about the total system’s behavior if taking both the attitude kinematics and dynamics into account. Although there is only limited work in delayed attitude synchronization, many synchronization-based control approaches have been proposed to deal with communication time delays in bilateral teleoperation (Lee and Spong, 2006; Chopra, Spong, and Lozano, 2008; Chopra, Spong, Ortega, and Barabanov, 2006; Nuno, Ortega, Basanez, and Barabanov, 2008) and synchronization of multi-robot networks (Chung and Slotine, 2009). Master-slave joint position synchronization (joint space of the manipulator Rn is Euclidean and convex) is achieved via injecting joint dissipation both at the master and the slave sides. However, in attitude synchronization problem, the rotational group SO(3) is non-Euclidean and non-convex, which presents a major obstacle to accommodate the time delays. Inspired by the results for bilateral teleoperation (Lee and Spong, 2006; Chopra, Spong, Ortega, and Barabanov, 2006; Nuno, Ortega, Basanez, and Barabanov, 2008; Wang and Xie, 2011), this paper attempts to investigate a new attitude synchronization strategy under communication delays, and both the attitude dynamics and kinematics are taken into consideration. It is shown that the proposed new approach can ensure robust stability for constant time delays provided the dissipation gains satisfy some inequality. The proposed delay-robust controller bears simple PD structure, and furthermore it is singularity-free
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10.3182/20110828-6-IT-1002.00144
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
and has explicit physical meaning in the sense that the attitude synchronization error is computed in the rotational group SO(3) with non-Euclidean geometric structure, rather than in Euclidean form. These features make our result distinctive from the wave-variable based delayrobust Euclidean-form attitude synchronization strategy (Chung, Ahsun, and Slotine, 2009), and the robot synchronization algorithms in Euclidean space (Chopra, Spong, and Lozano, 2008; Chung and Slotine, 2009). 2. RIGID BODY NETWORKS AND ITS KINEMATICS AND DYNAMICS 2.1 Network topology This paper considers a rigid body network that consists of n rigid bodies. We assume that these n rigid bodies form a connected undirected graph, that is, every two different bodies is connected by an undirected path (Jungnichel, 2008). The vertex of the network i (i = 1, 2, · · · , n) represents the rigid body i and its edge set E defines the corresponding information flow between rigid bodies. The set formed by the neighbours of node i is denoted by Ni = { j ∈ ν|(i, j) ∈ E} , where ν is the vertex set of the network. In an undirected network, the communicating interaction between rigid body agents bears symmetric features, that is, if j ∈ Ni , then i ∈ Nj . 2.2 Rigid body kinematics and dynamics
With the operator ∨¡ : so(3) → R3 , we obtain the ¢ attitude error skew∨ RjT Ri from the skew-symmetric ¢ ¡ matrix skew RjT Ri , where ∨ is the inverse of the cross product operator S(·). The passivity property related to the attitude error repre¡ ¢ sentation skew∨ RjT Ri is given by the following lemma. ¢ ¡ Lemma 1. The mapping from skew∨ RjT Ri to the error T angular velocity ωij = ωi −R respect¢ ¡ iTRj ω¢j is passive with ¡ to the storage function ϕ Rj Ri = (1/2) tr I − RjT Ri (Bullo and Murray, 1999), Z t ¢ ¡ T ωij skew∨ RjT Ri dr 0 ¡ ¢ ¡ ¢ = ϕ RjT (t)Ri (t) − ϕ RjT (0)Ri (0) (5)
where tr(·) denotes the trace of the matrix ·.
3. ATTITUDE SYNCHRONIZATION CONTROL In this section, we attempt to propose control laws for attitude synchronization of multiple rigid bodies. To say attitude synchronization, we mean that skew∨ (RjT Ri ) → 0 for ∀i, j ∈ ν as t → ∞. To achieve the goal of attitude synchronization, we propose the following delay-robust synchronization control law, X ¡ ¢ kv ωi − RiT Rj (t − T )ωj (t − T ) τi = − j∈Ni
The i-th rigid body kinematics can be written as (Egeland and Godhavn, 1994), R˙ i = Ri S(ωi ) (1) where Ri ∈ SO(3) denotes the orientation matrix of the ith rigid body with respect to some inertial frame, ωi is its angular velocity expressed and # " in body-fixed frame, 0 −ωi,z ωi,y ωi,z 0 −ωi,x , where S(ωi ) is defined as S(ωi ) = −ωi,y ωi,x 0 T ωi = [ ωi,x ωi,y ωi,z ] . The equations of motion of the attitude dynamics can be expressed as (Egeland and Godhavn, 1994), Hi ω˙ i − S(Hi ωi )ωi = τi (2) 3×3 where Hi ∈ R is the constant, positive definite, and symmetric inertia matrix, τi ∈ R3 is the applied control torque vector. 2.3 Attitude synchronization error representation and its basic properties Consider two different rigid bodies i, j ∈ ν, and their error attitude matrix can be written as RjT Ri . Differentiating the error attitude matrix yields (Egeland and Godhavn, 1994), ¡ ¢ d £ T ¤ Rj Ri = RjT Ri ωi − RiT Rj ωj (3) dt Using the representation from Bullo and Murray (1999), we obtain the following attitude error representation, ¡ ¢ RjT Ri − RiT Rj skew RjT Ri = (4) 2
−kP
X
j∈Ni
¡ ¢ skew∨ RjT (t − T )Ri − (KD,i + Pi ) ωi (6)
where T is the communication delay, which is assumed to be constant and bounded, and for simplicity we assume that all rigid body agents suffer the same time delay. kv , kP > 0 are positive scalar feedback gains, −Pi ωi is the i-th rigid body’s self-dissipation for achieving the asymptotic stabilization purpose, and the dissipation term −KD,i ωi is included to compensate for the effects of the communication time delay, where Pi , KD,i are both symmetric positive definite matrices. Now we present the main result of this paper. Theorem 1. If the rigid bodies form a tree network, and furthermore P the controller gains are chosen such that KD,i ≥ T kP ·I, then, the control (6) results in attitude j∈Ni
synchronization of the rigid body networks (1) (2), i.e., skew∨ (RjT Ri ) → 0 for ∀i, j ∈ ν as t → ∞. Before proving Theorem 1, we first give a useful lemma. Lemma 2. Let A(t) and B(t) be n × n matrices, for ∀t ≥ 0, ¸ 12 ·Z t ·Z t ¸ 21 Z t ¡ T ¢ ¡ T ¢ ¡ T ¢ tr A A dr tr A B dr ≤ tr B B dr 0
0
0
(7)
tr
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(·Z
t
A (r) dr
t
A (r) dr
0
0
≤ t · tr
¸T ·Z
·Z
0
t T
A (r) A (r) dr
¸
¸) (8)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Proof. The matrices A(t), B(t) can be expressed as the cluster of column vectors, A(t) = [ f1 (t) f2 (t) · · · fn (t) ] (9) B(t) = [ g1 (t) g2 (t) · · · gn (t) ] (10) Then, using Schwarz inequality, we have, Z t Z tX Z t n ¡ ¢ f¯T g¯dr fkT gk dr = tr AT B dr = ≤
t
·Z
f¯T f¯dr
·Z
¸ 12 ·Z
t
g¯T g¯dr
0
0
=
0
0 k=1
0
t
0
T
¡
¢ tr A A dr
¸ 12 ·Z
t
¸ 12
¡
T
¢
tr B B dr
0
¸ 21
(11)
¤T ¤T £ £ , g¯ = g1T g2T · · · gnT . where f¯ = f1T f2T · · · fnT This completes the proof of the inequality (7). Next, we show the inequality (8). In fact, from Schwarz inequality, we obtain, (·Z ¸T ·Z t ¸) t tr A (r) dr A (r) dr 0
= ≤
0
n ·Z t X
i=1 n X
fi (r) dr
·Z
fi (r) dr
t
0
i=1
= t · tr
Z
t
0
0
t·
¸T Z
fiT (r)fi (r)dr
t
AT (r)A(r)dr
0
¸
(12)
Proof of Theorem 1. We consider the following LyapunovKrasovskii function candidate, n n X kP X X ¡ T ¢ 1 T ϕ Rj Ri ω i Hi ω i + V = 2 2 i=1 i=1 j∈Ni Z n kv X X t + ωjT ωj dr (13) 2 i=1 t−T
Using (15), we can rewrite (14) as, n ¤T kv X X £ ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) V˙ = − 2 i=1 j∈Ni £ ¤ T · ωi (r) − Ri (r)Rj (r − T )ωj (r − T ) n X n X X ωiT ωiT (KD,i + Pi ) ωi + kP − i=1 j∈Ni
i=1
£ ¡ ¢ ¡ ¢¤ · skew∨ RjT Ri − skew∨ RjT (t − T )Ri (16) ¢ ¤ ¡ ¢ ¡ £ Let Xij = ωiT skew∨ RjT Ri − skew∨ RjT (t − T )Ri in (16) be rewritten as, o 1 n T Xij = − tr (Rj − Rj (t − T )) Ri S (ωi ) (17) 2
Integrating Xij in time interval [0, t] and using Lemma 2 and some techniques in Chopra, Spong, Ortega, and Barabanov (2006) and Nuno, Ortega, Basanez, and Barabanov (2008), we obtain, Z t Xij (r)dr 0 #T Z t "Z T 1 = − tr R˙ j (r − s) ds Ri S (ωi ) dr 2 0 0 "Z # ) 12 Z Z T T 1 t T tr R˙ j (r − s) ds · R˙ j (r − s) ds dr ≤ 2 0 0 0 (Z t h i ¾ 12 · tr R˙ iT (r)R˙ i (r) dr 0
≤
Z 1
t
T · tr
2 0 (Z
T
0
#
R˙ jT (r − s) R˙ j (r − s)ds dr
) 21
i ¾ 12 T ˙ ˙ tr Ri (r)Ri (r) dr
t
h
·
j∈Ni
"Z
0
) 21 Z TZ t h Differentiating V with respect to time and using the skewi T ˙ ˙ symmetric property of S (Hi ωi ) and Lemma 1, we have, = T· tr Rj (r − s) Rj (r − s) drds 2 n X 0 0 X £ ¤ k T v (Z ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) V˙ = − t i ¾ 12 h 2 i=1 j∈Ni T · tr R˙ i (r)R˙ i (r) dr £ ¤ 0 · ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) ) 21 n n Z TZ t h X X X ¡ T ¢ i T ∨ T T ωi skew Rj (t − T )Ri ≤ 1 T · kP ωi (KD,i + Pi ) ωi − − tr R˙ j (r) R˙ j (r) drds 2 i=1 i=1 j∈Ni 0 0 n (Z X ¡ ¢ kP X t i ¾ 12 h + (ωi − RiT Rj ωj )T skew∨ RjT Ri (14) T ˙ ˙ 2 · tr Ri (r)Ri (r) dr 1
i=1
j∈Ni
0
Since the rigid bodies form an undirected network topology, thus, we can rewrite the last term on the right side of equation (14) as the following well-known form (based on the result of VanDyke and Hall (2006)), n ¡ ¢T ¡ ¢ 1X X kP ωi − RiT Rj ωj skew∨ RjT Ri 2 i=1 j∈Ni
=
n X i=1
kP
X
j∈Ni
ωiT skew
¡ ∨
RjT Ri
¢
(15)
(Z
1 = T 2 (Z ·
0
T 2
Z
t
t
tr
0
t
h
R˙ jT
i ¾ 21 (r) R˙ j (r) dr
i ¾ 12 T ˙ ˙ tr Ri (r)Ri (r) dr h
ωiT (r)ωi (r) + ωjT (r)ωj (r)dr 0 h i where we use the fact that tr R˙ iT R˙ i = 2ωiT ωi .
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≤
(18)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Now integrating V˙ in (16) with respect to time and using the result (18), we have, V (t) − V (0) Z n ¤T kv X X t £ ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) ≤− 2 i=1 j∈Ni 0 £ ¤ · ωi (r) − RiT (r)Rj (r − T )ωj (r − T ) dr n Z t X X − T · I ωi dr ωiT KD,i − kP 0
i=1
−
n Z t X 0
i=1
j∈Ni
ωiT Pi ωi dr
where Di = KD,i − kP
(19) P
T · I is positive semi-definite
j∈Ni
due to the controller gain choice given in Theorem 1. Thus, from (19), we obtain, Z n ¤T kv X X t £ ωi (r) − RiT Rj (r − T )ωj (r − T ) V (t) + 2 i=1 j∈Ni 0 £ ¤ T · ωi (r) − Ri Rj (r − T )ωj (r − T ) dr n Z t n Z t X X + ωiT Pi ωi dr + ωiT Di ωi dr ≤ V (0) (20) i=1
0
i=1
0
Equation (20) means that V (t) ∈ L∞ , implying the boundedness of ωi for i = 1, 2, ..., n. And hence ωi − RiT Rj (t − T )ωj (t − T ) must be bounded. Then the control input τi is bounded. Using the rigid body attitude dynamics (2), we obtain the boundedness of ω˙ i . The boundedness of ω˙ i (t) gives rise to the uniform continuity of ωi − RiT Rj (t − T )ωj (t − T ) and ωi . Moreover, from (20), we have ωi − RiT Rj (t − T )ωj (t − T ) ∈ L2 , ωi ∈ L2 . Therefore, ωi − RiT Rj (t − T )ωj (t − T ) → 0 and ωi → 0 as t → ∞. From the rigid body attitude dynamics, we get, ω˙ i = Hi−1 [τi + S (Hi ωi ) ωi ]. Due to the uniform continuity of τi and S (Hi ωi ) ωi , ω˙ i must also be uniformly continuous. Using Barbalat Lemma (Slotine and Li, 1991), we have ω˙ i → 0 as t → ∞. Based on the closed-loop attitude dynamics, Hi ω˙ i − S (Hi ωi ) ωi = X kv (ωi − RiT Rj (t − T )ωj (t − T )) − j∈Ni
− kP
X
j∈Ni
¡ ¢ skew∨ RjT (t − T )Ri − (KD,i + Pi ) ωi (21)
we arrive at the conclusion that, as t → ∞, X ¡ ¢ skew∨ RjT (t − T )Ri → 0
undirected path and ¡ there¢are no cycles (Jungnichel, 2008), we obtain skew∨ RjT Ri → 0 for ∀i ∈ ν, j ∈ Ni as t → ∞ in a similar way as Abdessameud and Tayebi (2009). In fact, the incidence matrix B ∈ Rn×2(n−1) for a tree network with n vertices has the rank ¡of n −¢ 1, see Jungnichel (2008). In addition, since skew∨ RjT Ri = ¢ ¡ −skew∨ RiT Rj for j ∈ Ni , thus, the original synchronization error vector y whose dimension is 2 (n − 1) can be reduced to a vector x with dimension n − 1. Corre¯ = 0, spondingly, By = 0 is transformed equivalently to Bx ¯ ∈ R(n−1)×(n−1) is of full rank, which means that where B x = 0. x = 0 and the connectedness of the tree network result in attitude synchronization of the networked rigid ¢ ¡ bodies, i.e., skew∨ RjT Ri → 0 for ∀i, j ∈ ν as t → ∞.
For a general connected graph (e.g., it may contain loops or cycles), the case becomes much more complex and we have the following result.
Theorem 2. With the controller gains chosen as KDi ≥ P T kP · I and the rigid body network topology being j∈Ni
a general connected graph, if the error attitude matrix RlT Ri + RiT Rl (∀i, l ∈ ν) is positive definite, attitude synchronization of the rigid body networks (1) ¡(2) will¢ be achieved under the control law (6), i.e., skew∨ RjT Ri → 0 for all i, j ∈ ν as t → ∞.
Proof. Using skew∨T (RlT Ri ) to multiply both sides of (24) and computing the summation with respect to l, k yields, n X n X ¡ ¢ ¡ ¢X skew∨ RjT Ri skew∨T RlT Ri i=1 l=1 n n X X
1 = 2 =
1 4
j∈Ni
X
i=1 l=1 j∈Ni n X n X X i=1 l=1 j∈Ni
tr
"
RiT Rl
− RlT Ri RjT Ri − RiT Rj 2 2
£ ¤ tr RlT Ri − RlT Ri RjT Ri = 0
#
(25)
Next, we follow a similar procedure from Igarashi, Hatanaka, Fujita, and Spong (2009) to proceed the proof of Theorem 2. Since the network is connected and undirected, then, n X n X n X n X X £ ¤ X £ ¤ tr RlT Rj = tr RlT Ri (26) i=1 l=1 j∈Ni
i=1 l=1 j∈Ni
Using (26), we can reformulated (25) as follows, n n ¡ ¢ ¤ 1 XX X £ T tr Rl Ri + RiT Rl − RlT Ri + RiT Rl RjT Ri 4 i=1 l=1 j∈Ni
(22)
j∈Ni
Using finite increment theorem, we have, RjT (t − T )Rj − I = RjT (t − T )R˙ j (ξ)T, ξ ∈ [t − T, t] (23) Since R˙ j (ξ) = Rj (ξ) S (ωj (ξ)) → 0 as t → ∞, thus, RjT (t − T ) Rj (t) → I as t → ∞. Then, equation (22) can be further written as (at the extreme state t = ∞), X ¡ ¢ skew∨ RjT Ri = 0 (24)
=0
(27)
Since the error orientation matrix RlT Ri + RiT Rl (∀i, l) is positive definite, we obtain the following result using similar techniques as Igarashi, Hatanaka, Fujita, and Spong (2009), n n X X ¡ ¢ ¡ ¢X skew∨ RjT Ri skew∨T RlT Ri
j∈Ni
For a tree network topology, i.e., an undirected graph in which every two different bodies is connected by an 8777
i=1 l=1 n n X X
≥
1 4
j∈Ni
X
i=1 l=1 j∈Ni
¡ ¢ £ ¤ λmin RlT Ri + RiT Rl tr I − RjT Ri
(28)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where λmin (·) denotes the¡minimum ¢eigenvalue of ·. From (25) and (28), we have tr I − RjT Ri → 0 for ∀i ∈ ν, j ∈ Ni as t → ¡∞, which further results in the conclusion ¢ that skew∨ RjT Ri → 0 for ∀i, j ∈ ν as t → ∞ due to the connectedness of the rigid body network. Thus, delayrobust attitude synchronization is achieved for general connected graph case.
1
5
6 2
4. SIMULATION RESULTS In this section, we present simulation results to examine the performance of the proposed attitude synchronization strategy under communication time delays. In simulation, we employ six rigid body agents and they are required to synchronize their attitudes asymptotically. The network topology of the six rigid bodies is assumed to be a tree for simplicity, as shown in Fig. 1. Due to the limited space, we did not perform simulation study for the general network topology case. The communication delay between rigid body agents is T = 1s.
Fig. 1. Network topology formed by the six rigid bodies
agent sk
For general connected graphs, we just obtain a sufficient condition for ensuring attitude synchronization, which, however, seems somewhat restrictive. Our future research will focus on how to relax or remove this condition.
x
agent attitude sk
x
agent sk
1.5
4
x
agent sk 5
1
x
agent sk 6
x
0.5 0 −0.5 −1 0
20
40 time(s)
60
80
Fig. 2. The attitude coordinate skx of the six rigid body agents (without additional dissipation) 0.8 0.6 x
0.4
5. CONCLUSION In this paper, we studied the time-delayed attitude synchronization problem of the rigid body networks with undirected graph topology. Via exploitation of the geometrical structure of rigid body attitude kinematics, we proposed a synchronization control law which is composed of angular velocity and attitude synchronization terms and an additional dissipation term. It is shown that if the dissipation gain satisfies some inequality, the closedloop networked system is stable and the attitude synchronization errors converge to zero. And the convergence of the attitude synchronization errors holds with any boundknown constant communication time delay. A six-rigidbody network with tree topology is employed as a simulation example to show the performance of the proposed attitude synchronization controller.
x
agent sk 3
agent attitude sk
1, 2, · · · , 6, and kv , kP are chosen to be same as the first simulation. The time history of the six agents’ attitudes is shown in Fig. 3 to Fig. 5, from which we can see that attitude synchronization with communication time delay is achieved via adding dissipation −KD,i ωi according to Theorem 1.
2
2
In the first simulation, the controller parameters are chosen as kv = 5, KD,i = 0, Pi = 5I, kP = 20. The element skx of the i-th agent attitude representation skew∨ (Ri ) = T [ skx sky skz ] , i = 1, 2, ..., 6 is shown in Fig. 2. It can be seen that without additional dissipation term, i.e., KD,i = 0, attitude synchronization cannot be achieved and instability phenomenon is observed.
j∈Ni
agent1 skx
2.5
For simplicity, we assume that " the six rigid # bodies have 30 8 9 the same inertia matrix Hi = 8 21 6 , i = 1, 2, · · · 6. 9 6 25
In the second Psimulation, we choose the dissipation gains as KDi = T kP · I, the damping gain Pi = 5I, i =
3
4
agent sk
0.2
1
x
agent sk
0
2
x
agent sk 3
−0.2
x
agent sk 4
x
−0.4
agent5 skx
−0.6
agent6 skx
−0.8 0
20
40 time(s)
60
80
Fig. 3. The attitude coordinate skx of the six rigid body agents (with additional dissipation) REFERENCES Abdessameud, A. and Tayebi, A. (2009). Attitude synchronization of a group of spacecraft without velocity measurements. IEEE Transactions on Automatic Control, 54(11), 2642-2648. Bullo, F. and Murray, R. M. (1999). Tracking for fully actuated mechanical systems: a geometrical framework. Automatica, 35, 17-34.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
1
agent attitude sky
0.5 agent sk 1
y
agent2 sky
0
agent3 sky agent4 sky −0.5
agent5 sky agent sk 6
−1 0
20
40 time(s)
y
60
80
Fig. 4. The attitude coordinate sky of the six rigid body agents (with additional dissipation) agent sk 1
1
z
agent2 skz agent3 skz agent sk
agent attitude skz
0.5
4
z
agent5 skz agent6 skz 0
−0.5
−1 0
20
40 time(s)
60
80
Fig. 5. The attitude coordinate skz of the six rigid body agents (with additional dissipation) Cheah, C. C., Hou, S. P., and Slotine, J. J. E. (2009). Region-based shape control for a swarm of robots. Automatica, 45, 2406-2411. Chopra, N., Spong, M. W., Ortega, R., and Barabanov, N. E. (2006). On tracking performance in bilateral teleopera-tion. IEEE Transactions on Robotics, 22(4), 861-866. Chopra, N., Spong, M. W., and Lozano, R. (2008). Synchronization of bilateral teleoperators with time delay. Automatica, 44, 2142-2148. Chung, S.-J., Ahsun, U., and Slotine, J. J. E. (2009). Application of synchronization to formation flying spacecraft: Lagrangian approach. Journal of Guidance, Control and Dynamics, 32(2), 512-526. Chung, S.-J. and Slotine, J. J. E. (2009). Cooperative robot control and concurrent synchronization of Lagrangian systems, IEEE Transactions on Robotics, 25(3), 686-700. Dimarogonas, D. V., Tsiotras, P., and Kyriakopoulos, K. J. (2006). Laplacian cooperative attitude control of multiple rigid bodies, In Proceedings of the 2006 IEEE International Symposium on Intelligent Control, Munich, Germany. pp. 3064 -3069. Egeland, O. and Godhavn, J.-M. (1994). Passivity based adaptive attitude control of a rigid spacecraft. IEEE Transactions on Automatic Control, 39(4), 842-846.
Hughes, P. C. (1986). Spacecraft attitude dynamics. New York, John Wiley & Sons. Igarashi, Y., Hatanaka, T., Fujita, M., and Spong, M. W. (2007). Passivity-based 3D attitude coordination: convergence and connectivity. In Proceedings of IEEE International Conference on Decision and Control, New Orleans, LA, USA. pp. 2558-2565. Igarashi, Y., Hatanaka, T., Fujita, M., and Spong, M. W. (2009). Passivity-based attitude synchronization in SE(3). IEEE Transaction on Control Systems Technology, 17(5), 1119-1134. Jungnichel, D. (2008). Graphs, networks and algorithms. Berlin Heidelberg, Springer-Verlag. Lee, D. and Spong, M. W. (2006). Passive bilateral teleoperation with constant time delays. IEEE Transactions on Robotics, 22(2), 269-281. Meng, Z., Ren, W., and You, Z. (2009). Decentralized cooperative attitude tracking using modified rodriguez parameters, In Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China. pp. 853-858. Nuno, E., Ortega, R., Basanez, L., and Barabanov, N. (2008). A new proportional controller for nonlinear bilateral teleoperators. In Proceedings of 17th IFAC World Congress, Seoul, Korea. pp. 15660-15665. Ren, W. (2006). Distributed attitude consensus among multiple networked spacecraft. In Proceedings of the American Control Conference, Minneapolis, Minnesota, USA.pp. 1760-1765. Ren, W. (2007). Formation keeping and attitude alignment for multiple spacecraft through local interacttions, Journal of Guidance, Control and Dynamics, 30(2), 633-638. Ren, W. (2010). Distributed cooperative attitude synchronization and tracking for multiple rigid bodies, IEEE Transactions on Control Systems Technology, 18(2), 383-392. Rodrigues-Angeles, A. and Nijmeijer, H. (2004). Mutual synchronization of robots via estimated state feedback: a cooperative approach, IEEE Transactions on Control Systems Technology, 12(4), 542-554. Sarlette, A., Sepulchre, R., and Leonard, N. E. (2009). Autonomous rigid body attitude synchronization, Automatica, 45, 572-577. Slotine, J. J. E. and Li, W. (1991). Applied nonlinear control. Englewood Cliffs NJ: Prentice-Hall. Smith, T. R., Hanssmann, H., and Leonard, N. E. (2001). Orientation control of multiple underwater vehicles with symmetry-breaking potentials. In Proceedings of 40th IEEE Conference on decision and control, Orlando, Florida, USA. pp. 4598-4603. VanDyke, M. C. and Hall, C. D. (2006). Decentralized coordinated attitude control within a formation of spacecraft, Journal of Guidance, Control and Dynamics, 29(5), 1101-1109. Wang, P. K. C., Hadaegh, F. Y., and Lau, K. (1999). Synchronized formation rotation and attitude control of multiple free-flying spacecraft, Journal of Guidance, Control and Dynamics, 22(1), 28-35. Wang, H. and Xie, Y. (2011). Passivity based task-space bilateral teleoperation with time delays, In IEEE International Conference on Robotics and Automation, accepted.
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