j; i) 62 _vi = ui; i = 1;... ; n - Semantic Scholar
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009
Collective Motion From Consensus With Cartesian Coordinate Coupling Wei Ren, Member, IEEE
Abstract—Collective motions including rendezvous, circular patterns, and logarithmic spiral patterns can be achieved by introducing Cartesian coordinate coupling to existing consensus algorithms. We study the collective motions of a team of vehicles in 3-D by introducing a rotation matrix to an existing consensus algorithm for double-integrator dynamics. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound, and the Euler angle is below, equal, or above a critical value, the vehicles will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane perpendicular to the Euler axis. In particular, when the vehicles eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the vehicles on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric Laplacian matrix. Simulation results are presented to demonstrate the theoretical results. Index Terms—Collective motion, consensus, cooperative control, distributed algorithms, multi-vehicle systems.
I. INTRODUCTION Coordination of robotic networks has received significant attention in recent years due to its potential impact in numerous civilian, homeland security, and military applications. Consensus plays an important role in achieving distributed coordination. The basic idea of consensus is that a team of vehicles reaches an agreement on a common value by negotiating with their neighbors. Consensus algorithms are studied for both single-integrator kinematics [1]–[3] and double-integrator dynamics [4]–[9], to name a few. Related to consensus is the cyclic pursuit strategy, where each vehicle pursues only one other vehicle with the network topology forming a unidirectional ring. Cyclic pursuit is studied for single-integrator kinematics in [10], [11] while for wheeled vehicles subject to nonholonomic constraints in [12]. Ref. [13] generalizes the cyclic pursuit strategy by letting each vehicle pursue one other vehicle along the line of sight rotated by a common offset angle. It is shown that depending on the common offset angle, the vehicles can achieve different symmetric formations, namely, convergence to a single point, a circle, or a logarithmic spiral pattern. Other researchers also study symmetric formations by adopting models based on the Frenet-Serret equations of motion [14] or by exploring the connections between phase models of coupled oscillators and kinematic models of self-propelled particle groups [15]. While the strategy proposed in [13] generates interesting symmetric formation patterns, there are limitations. First, the results in [13] are
Manuscript received March 27, 2008; revised October 04, 2008 and January 14, 2009. First published May 27, 2009; current version published June 10, 2009. This work was supported by a National Science Foundation CAREER Award (ECCS-0748287). Recommended by Associate Editor M. Egerstedt. The author is with the Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322 USA (e-mail: [email protected]. edu). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TAC.2009.2015544
limited to 2-D. However, for applications involving unmanned aerial vehicles, it will be more natural to study motions in 3-D. Second, the results in [13] primarily focus on single-integrator kinematics. While an extension to double-integrator dynamics is proposed to deal with a formation control problem, the extension relies not only on relative positions but also on relative velocities between neighbors. However, the requirement for the knowledge of relative velocities might be too restrictive for some applications. Third, the results in [13] rely on a unidirectional ring topology and the resulting circular and logarithmic spiral patterns are evenly spaced. However, for a team consisting of heterogenous vehicles with different sensing/communciation capabilities, it might not always be desirable that the vehicles are evenly spaced and move along the same orbit with an identical radius. It will also be interesting to study the motions resulting from a general (not necessarily unidirectional ring) network topology. To address these limitations, the strategy proposed in [13] needs to be extended. In this note, we extend the results in [13] threefold, namely, i) extension from 2-D to 3-D, ii) extension from single-integrator kinematics to double-integrator dynamics without the knowledge of relative velocities, and iii) extension from a unidirectional ring topology to a general network topology to generate possibly non-evenly-spaced circular and logarithmic spiral patterns on concentric orbits with possibly nonidentical radii. In particular, we introduce Cartesian coordinate coupling to an existing consensus algorithm for double-integrator dynamics through a rotation matrix in 3-D, analyze the convergence properties, and quantitatively characterize the resulting collective motions in 3-D, namely, convergence to a point, circular patterns with concentric orbits, and logarithmic spiral curves lying on a plane perpendicular to the Euler axis, over a general network topology. The resulting collective motions are expected to have applications in rendezvous, persistent surveillance, and coverage control with teams of heterogeneous vehicles. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. Our analysis relies on algebraic graph theory, matrix theory, and properties of the Kronecker product. In particular, we will show that the convergence result in [13] is a special case of the results in this note and the convergence result in [13] can be recovered by exploiting the properties of circulant matrices and the Kronecker product. A preliminary version of the work has appeared in [16].
II. BACKGROUND AND PRELIMINARIES A. Graph Theory Notions It is natural to model interaction among vehicles by directed graphs. Suppose that a team consists of n vehicles. A weighted directed graph G consists of a node set V = f1; . . . ; ng, an edge set E V 2V , and a weighted adjacency matrix A = [aij ] 2 n2n . An edge (i; j ) denotes that vehicle j can obtain information from vehicle i, but not necessarily vice versa. Weighted adjacency matrix A associated with G is defined such that aij is a positive weight if (j; i) 2 E , while aij = 0 if (j; i) 62 E . A directed path is a sequence of edges in a directed graph of the form (i1 ; i2 ); (i2 ; i3 ); . . ., where ij 2 V . A directed graph has a directed spanning tree if there exists at least one node having a directed path to all other nodes. Let nonsymmetric Laplacian matrix L = [`ij ] 2 n2n associated with A be defined as `ii = n a and `ij = 0aij , j =1;j 6=i ij i 6= j [17]. B. Existing Consensus Algorithm for Double-Integrator Dynamics Consider vehicles with double-integrator dynamics given by
r_i = vi ;
v_ i = ui ; i = 1; . . . ; n
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(1)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009
where ri 2 m and vi 2 m are, respectively, the position and velocity of the ith vehicle, and ui 2 m is the control input. A consensus algorithm for (1) is studied in [9], [18] as
ui
n
=0 j
=1
(
aij ri 0 rj
) 0 v ; i
i
= 1; . . . ; n
(2)
where aij is the (i; j )th entry of weighted adjacency matrix A associated with weighted directed graph G , and is a positive damping gain. Consensus is reached for (1) using (2) if for all ri (0) and vi (0), ri (t) ! rj (t) and vi (t) ! 0 as t ! 1. III. CONSENSUS WITH CARTESIAN COORDINATE COUPLING In this section, we consider a consensus algorithm for double-integrator dynamics (1) with Cartesian coordinate coupling as ui
n
=0 j
=1
(
aij C ri 0 rj
) 0 v ; i
i
= 1; . . . ; n
(3)
where aij and are defined as in (2), and C 2 m2m denotes a Cartesian coordinate coupling matrix. We assume that all vehicles know C and a priori and the vehicles’ positions and velocities are represented in a common reference frame. Note that (2) corresponds to the case where C = Im , where Im denotes the m 2 m identity matrix. That is, using (2), the components of ri (i.e., the Cartesian coordinates of vehicle i) can be decoupled while using (3) the components of ri are coupled. A. Convergence Result In this subsection, we analyze the convergence properties of (3). We focus on the case where C is a rotation matrix while a similar analysis can be extended to the case where C is a general matrix. Before moving on, we need the following lemmas and definition: Lemma 3.1: Let U 2 p2p , V 2 q2q , X 2 p2p , and Y 2 q 2q . Then (U V )(X Y ) = U X V Y , where denotes the Kronecker product. Let A 2 p2p have eigenvalues i with associated eigenvectors fi 2 p , i = 1; . . . ; p, and let B 2 q2q have eigenvalues j with associated eigenvectors gj 2 q , j = 1; . . . ; q . Then the pq eigenvalues of A B are i j with associated eigenvectors fi gj , i = 1; . . . ; p, j = 1; . . . ; q . Lemma 3.2: [3] Let L be the nonsymmetric Laplacian matrix associated with weighted directed graph G . Then L has at least one zero eigenvalue and all its nonzero eigenvalues have positive real parts. Furthermore, L has a simple zero eigenvalue and all other eigenvalues have positive real parts if and only if G has a directed spanning tree. In addition, there exist 1n , where 1n is the n 2 1 column vector of all ones, satisfying L1n = 0 and p 2 n satisfying p 0, pT L = 0, and pT 1n = 1. 1 Definition 3.1: Let i , i = 1; . . . ; n, be the ith eigenvalue of 0L with associated right eigenvector wi and left eigenvector i . Also let arg(i ) = 0 for i = 0 and arg(i ) 2 (=2; 3=2) for all i 6= 0, where arg(1) denotes the phase of a number. Without loss of generality, suppose that i is labeled such that arg(1 ) arg(2 ) 1 1 1 arg(n). 2 Lemma 3.3: (see e.g., [19]) Given a rotation matrix R 2 323 , let a = [a1 ; a2 ; a3 ]T and denote, respectively, the Euler axis (i.e., the unit vector in the direction of rotation) and Euler angle (i.e., the rotation angle). The eigenvalues of R are 1, e , and e0 , where denotes the imaginary unit, with the associated right eigenvectors given by, respectively, &1 = a, &2 = [(a22 + a23 ) sin2 (=2); 0a1 a2 sin2 (=2)+ a3 sin(=2)jsin(=2)j; 0a1 a3 sin2 (=2)0 a2 sin(=2)jsin(=2)j]T, and &3 = & 2 , where 1 denotes the complex conjugate of a number. 1That is, 1 and p are, respectively, the right and left eigenvectors of L associated with the zero eigenvalue. 2It
follows from Lemma 3.2 that
= 0, w = 1
, and
= p.
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The associated left eigenvectors are, respectively, $1 = &1 , $2 = &2 , and $3 = &3 . Lemma 3.4: Let A 2 n2n with eigenvalues i and associated right and left eigenvectors qi and si , respectively. Also let
02
In n A 0 In , where n2n denotes the n 2 n zero matrix and is a positive scalar. Then the eigenvalues of B are 0 2
i = with associated right given by 2i01
=
B
0
n
= (
and left eigenvectors
2i
= (0 0
2
q
+
and
q
+( 4 ) +2 )
s
s
, respectively, and
++ 4) )=2, with associated right and left eigenveci
and ( s s , respectively. Proof: Suppose that is an eigenvalue of B with an asson . It follows that ciated right eigenvector fg , where f; g 2
tors
q
q
02
f , which implies g f and g 2 Af 0 g g . It thus follows that Af f . Noting
i qi , we let f that Aqi qi and 2
i . That is, each eigenvalue of A, i , corresponds to two eigenvalues of B , denoted by 0 6 2 i = . Because g f , it follows that 2i01;2i the right eigenvectors associated with 2i01 and 2i are, respectively, n
f g
In
n
0 In
A
= = =(
=
= =( + ) + = =
= +4 ) 2
q and q q . A similar analysis can be used to find the left q eigenvectors of B associated with 2i01 and 2i . Theorem 3.2: Suppose that weighted directed graph G has a directed spanning tree. Let the control algorithm for (1) be given by (3), where ri = [xi ; yi ; zi ]T and vi = [vxi ; vyi ; vzi ]T . Let i , wi , i , and arg(i ) be defined in Definition 3.1, p be defined in Lemma 3.2, and a = [a1 ; a2 ; a3 ]T , &k , and $k be defined in Lemma 3.3. 1) Suppose that C = I3 . Then all vehicles will eventu1 ally rendezvous if and only if > c , where c = 2 maxi ji j sin (arg(i ))= 0 cos(arg(i )). The rendezvous position is given by
pT
(0) + v (0)
x
x
;p
T
y
(0) + v (0)
;
pT
z
y
(0) + v (0) z
(4)
where x, y , z , vx , vy , and vz are, respectively, column stack vectors of xi , yi , zi , vxi , vyi , and vzi . 2) Suppose that C = R, where R is the 3 2 3 rotation matrix defined in Lemma 3.3, and > c . Given ji j, i = 2; . . . ; n, let il 2 (=2; ) (respectively, iu 2 (; 3=2)) be the solution to ji j sin2 ( i ) + 2 cos( i ) = 0 if arg(i ) 2 (=2; ] 1 (respectively, arg(i ) 2 [; 3=2)). If jj < dc , where dc = u minarg( )2[;3=2) ( i 0 arg(i )), then all vehicles will eventually rendezvous at the position given by (4). 3) Under the assumption of 2), if jj = dc and there 2 [; 3=2) such that exists a unique arg( ) u = dc , then all vehicles will eventually 0 arg( ) move on circular orbits with center given by (4) and period =j sin( u )j. The radius of the orbit for vehicle i is given 2jw(i) pcT [r(0)T ; v(0)T ]T j a22 + a23 sin2 (=2), by where w(i) is the ith component of w and pc = 1=(2c +u )T w $2T &2 ( + )( $ $ ) , where c = (2j j sin( )= ). The relative radii of the orbits are equal to the relative magnitudes of w(i) . The relative phases of the vehicles on their orbits are equal to the relative phases of w(i) . The circular orbits are on a plane perpendicular to Euler axis a. 4) Under the assumption of 2), if there exists a unique arg( ) 2 [; 3=2) such that u 0u arg( ) = dc and dc < jj < minarg( )2[;3=2);i6= ( i 0 arg(i )), then the vehicles will eventually move along logarithmic spiral curves with center given ), where Re(1) denotes the real part of a by (4), growing rate Re(sp number and s = (0 + 2 + 4s )=2 with s = ejj , and period 2=jIm(s )j, where Im(1) represents the imaginary part
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of a number. The radius of the logarithmic spiral curve for vehicle T i is 2jw(i) pTs [r (0)T ; v (0)T ] eRe( )t a22 + a23 sin2 (=2), T where ps = 1=(2s + ) w $2T &2 ( + )( $ $ ) . The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of w(i) . The relative phases of the vehicles on their curves are equal to the relative phases of w(i) . The curves are on a plane perpendicular to Euler axis a. Proof: 1) For the first statement, if C = I3 , then (1) using (3) can be written in matrix form as 0n2n In
I3 r_ r 0L 0 In = (5) v_ v 0
T T where r = [r1T ; . . . ; rnT ] , v = [v1T ; . . . ; vnT ] , and L is the nonsymmetric Laplacian matrix associated with G . It follows from the proof of Theorem 5.1 in [18] that the vehicles will eventually rendezvous if and only if 0 defined in (5) has a simple zero eigenvalue and all other eigenvalues have negative real parts. Note from Lemma 3.4 that each eigenvalue i of 0L corresponds to two eigenvalues of 0 given by 2i01 = (0 + 2 + 4i )=2 with associated right and left eigenvectors w w and ( + ) , respectively, and
= (0 0 2 + 4i )=2, with associated right and left eigen ) , respectively, where i = 1; . . . ; n. vectors ww and ( + Because weighted directed graph G has a directed spanning tree, it follows from Lemma 3.2 that 0L has a simple zero eigenvalue and all other eigenvalues have negative real parts. According to Definition 3.1, we let 1 = 0 and Re(i ) < 0, i = 2; . . . ; n. Note from Lemma 3.2 that w1 = 1n and 1 = p. It thus follows that 1 = 0 with associated right and left eigenvectors given by 10 and pp , respectively, and 2 + 4i 2 = 0 . Note that 2 < 0 if > 0. Also noting that all have nonnegative real parts, it follows that all 2i , i = 2; . . . ; n, have negative real parts if > 0. It is left to show conditions under which 3 2i01 , i = 2; . . . ; n, have negative real parts. Suppose that i is the critical value for such that 2i01 , i = 2; . . . ; n, is on the imaginary axis. Let 2i01 = i , where i 2 , i = 2; . . . ; n. After some manip2 ji j sin (arg(i ))= 0 cos(arg(i )) ulation, it follows that 3i = and i = 2ji j sin(arg(i ))= , i = 2; . . . ; n. It is straightforward to verify that if > 3i (respectively, < 3i ), then 2i01 , i = 2; . . . ; n, has a negative (respectively, positive) real part. Therefore, all 2i01 , i = 2; . . . ; n, have negative real parts if and only if > maxi=2;...;n 3i . Combining the above arguments shows that 0 has a simple zero eigenvalue and all other eigenvalues have negative real parts if and only if c > . Matrix 0 can be written in Jordan canonical form as SJ S 01 , where the columns of S , denoted by sk , k = 1; . . . ; 2n, can be chosen to be the right eigenvectors or generalized right eigenvectors of 0 associated with eigenvalue k , k = 1; . . . ; 2n, the rows of S 01 , denoted by hkT , k = 1; . . . ; 2n, can be chosen to be the left eigenvectors or generalized left eigenvectors of 0 associated with eigenvalue k such that hkT sk = 1 and hkT s` = 0, k 6= `, and J is the Jordan block diagonal matrix with k T ]T and being the diagonal entries. We can choose s1 = [1Tn ; 0n T T T T h1 = [p ; (1= )p ] . It can be verified that h1 s1 = 1. It = thus follows that limt!1 vr((tt)) = limt!1 (e0t I3 ) vr(0) (0) 1 r (0) T T [( 0 [p (1= )p ]) I3 ] v(0) , which implies that xi (t) ! pT x(0) + (1= )pT vx (0), yi (t) ! pT y(0) + (1= )pT vy (0), T T zi (t) ! p z (0) + (1= )p vz (0), vxi (t) ! 0, vyi (t) ! 0, and vzi (t) ! 0 as t ! 1. Equivalently, it follows that all vehicles will eventually rendezvous at the position given by (4). 2) For the second statement, using (3), (1) can be written in matrix form as 2i
r_
v_
=
03n23n ( R)
0 L
I3n
r
n
0 I3
v
:
(6)
It follows from Lemmas 3.1 and 3.3 and Definition 3.1 that the eigenvalues of 0(L R) are i , i e , and i e0 with associated right eigenvectors wi &1 , wi &2 , and wi &3 , respectively, and associated left eigenvectors i $1 , i $2 , and i $3 , respectively. That is, the eigenvalues of 0(L R) correspond to the eigenvalues of 0L rotated by angles 0, , and 0 , respectively. Let ` , ` = 1; . . . ; 3n, denote the `th eigenvalue of 0(L R). Without loss of generality, let 0 , i = 1; . . . ; n, be the 3i02 = i , 3i01 = i e , and 3i = i e eigenvalues of 0(L R). Note from Lemma 3.4 that each k correspondspto two eigenvalues of 6, defined in (6), given by 2k01;2k = (0 6 2 + 4k )=2, k = 1; . . . ; 3n. Because 1 = 0, it follows that 1 = 2 = 3 = 0, which in turn implies that 1 = 3 = 5 = 0 and 2 = 4 = 6 = 0 . Similar to the proof of the first statement, all 2k , k = 1; . . . ; 3n, have negative real parts if > 0. Given arg( ) , i = 2; . . . ; n, l and u are the critical > 0 and i = ji je i i values for arg( i ) 2 [0; 2 ) such that (0 + 2 + 4 i )=2 is on the imaginary axis. In particular, if arg( i ) = il (respectively, iu ), then (0 + 2 + 4 i )=2 = (2ji j sin(arg( il )= ) (respectively, u l u (2ji j sin(arg( i )= )), i = 2; . . . ; n. If arg( i ) 2 ( i ; i ) (reu l 2 spectively, arg( i ) 2 [0; i ) [ ( i ; 2 )), then (0 + + 4 i )=2 have negative (respectively, positive) real parts. Because > c , the first statement implies that all (0 + 2 + 4i )=2, i = 2; . . . ; n, have negative real parts, which in turn implies that arg(i ) 2 ( il ; iu ), c i = 2; . . . ; n. If j j < d , then arg(3i02 ), arg(3i01 ), and arg(3i ) l u are all within ( i ; i ), which implies that 6i05 , 6i03 , and 6i01 , c i = 2; . . . ; n, all have negative real parts. Therefore, if j j < d , then 6 has exactly three zero eigenvalues and all other eigenvalues have negative real parts. Similar to the proof of the first statement, we write 6 in Jordan canonical form as M J M 01 , where the columns of M , denoted by mk , k = 1; . . . ; 6n, can be chosen to be the right eigenvectors or generalized right eigenvectors of 6 associated with eigenvalue k , the rows of M 01 , denoted by pkT , k = 1; . . . ; 6n, can be chosen to be the left eigenvectors or generalized left eigenvectors of 6 associated with eigenvalue k such that pkT mk = 1 and pkT m` = 0, k 6= `, and J is the Jordan block diagonal matrix with k being the diagonal entries. Recall that the right and left eigenvectors of 0(L R) associated with eigenvalue ` = 0 are, respectively, 1n &` and p $` , where ` = 1, 2, 3. It in turn follows from Lemma 3.4 that the right and left eigenvectors of 6 associated with 2`01 = 0 are, respectively, 10 & and p $ , where ` = 1, 2, 3. We can choose m 1 & and 2`01 = p $ 0 p ($ =$ & ) p2`01 = , where ` = 1, 2, 3. It can be verified that p ($ = $ & ) T T p2`01 m2`01 = 1 and p2`01 m2k01 = 0, where k; ` = 1, 2, 3 and k 6= r(t) `. Noting that 2`01 = 0, ` = 1, 2, 3, it follows that limt!1 v (t) = 3 r (0) Jt 0 1 r (0) T limt!1 M e M v(0) ! ( `=1 m2`01 p2`01 ) v(0) , which implies that xi (t) ! pT x(0) + (1= )pT vx (0), yi (t) ! pT y (0) + (1= )pT vy (0), zi (t) ! pT z (0) + (1= )pT vz (0), vxi (t) ! 0, vyi (t) ! 0, and vzi (t) ! 0 as t ! 1. Equivalently, it follows that all vehicles will eventually rendezvous at the position given by (4). 3) For the third statement, if = dc (respectively, = 0dc ) and there exists a unique arg( ) 2 [; 3=2) such that u c = j je (re 0 arg( ) = d , then 301 = e 0 spectively, 3 = e = j je ), which implies that u 603 = (0 + 2 + 4301 )=2 = (2j j sin( )= ) (respecp 2 tively, 601 = (0 + + 43 )=2 = (2j j sin( u )= )). Noting that the complex eigenvalues of 6 are in pairs, it follows that 6 has an eigenvalue equal to 603 = 0(2j j sin( u )= ) (respectively, 601 = 0(2j j sin( u )= )), denoted by 3 for simplicity. In this case, 6 has exactly three zero eigenvalues, two nonzero eigenvalues on the imaginary axis, and all other eigenvalues have negative real parts. In the following, we focus on = dc since the analysis for = 0dc is similar except that all vehicles will move in reverse directions. Note from Lemma 3.4 that the right and left eigen& vectors associated with 603 are, respectively, w (
w & ) and
6
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m p = w $ & p m m p p p m m e ! ct m p m p e t ct mp k c t e t ct k ; ;n c w & p r ;v e i ; ;n ` & ` & c t j& w p r ;v j = t j j w p r ;v & i ; ;n ` x t ! x = v c t = v c t y t ! y z t ! z c t = v t k c t ;c t ;c t k jw p r ;v j a a =
& . We can choose 603 = w (
w & ) and ( + )( $ ) T T . It can 603 = (1 (2 603 + ) 2 2) $ T be verified that 603 603 = 1. Similarly, it follows that 3 and 3 corresponds to 3 are given by 3 3= 603 and 3 = 603 . It T ) r(0) + ( ) ( follows that vr((tt)) = 6t vr(0) 2 ` 0 1 2 ` 0 1 `=1 (0) v(0) 1 for large , where ( ) = ( (2j j sin( )= )t 603 6T03 + 0(2j j sin( )= ) 3 3T ) r(0) . Let k ( ) be the th compov(0) = 1 . . . 6 . It follows that 3(i0T1)+` ( ) = nent of ( ), 2Re( (2j j sin( )= )t (i) 2(`) T603 [ (0)T (0)T ] ), where = 1 . . . , = 1, 2, 3, and 2(`) denotes the th component of 2 . After some manipulation, it follows that 3(i01)+` ( ) = cos((2 sin( u ) ) + 2 2(`) (i) T603 [ (0)T (0)TT]T T T T arg( (i) 603 [ (0) (0) ] ) + arg( T2(`) )), = 1 . . . T , = 1, p (0) + (1 )p x (0) + 2, 3. Therefore, it follows that i ( ) pT (0) + (1 )pT y (0) + 3i01 ( ), and i( ) 3i02 ( ), pT (0) + (1 )pT z (0) + 3i ( ) for large . After some i( ) manipulation, it can be verified that [ 3i02 ( ) 3i01 ( ) 3i ( )]T = 2 (i) 6T03 [ (0)T (0)T ]T 22 + 23 sin2 ( 2), which is a constant. Therefore, it follows that all vehicles will eventually move on circular orbits with center give by (4) and period u sin( ) . The radius of the orbit for vehicle is given by T 2 (i) 603 [ (0)T (0)T ]T 22 + 23 sin2 ( 2). The relative radii of the orbits are equal to the relative magnitudes of (i) . In addition, the relative phases of the vehicles are equal to the relative phases of (i) . Note from Lemma 3.3 that Euler axis a is orthogonal to both Re( 2 ) and Im( 2 ) are applied componentwise. It can thus be verified that a is orthogonal to [ 3i02 ( ) 3i01 ( ) 3i ( )]T , which implies that the circular orbits are on a plane perpendicular to a. 4) For the fourth statement, if there exists a unique [ 3 2) such that u arg( ) = dc and arg( ) c minarg( )2[;3u=2);i6= ( iu arg( i )) (respectively, d c arg( i )) minarg( )2[;3=2);i6= ( i d ), (arg( )+) (respectively, then 301 = = u 0 = (arg( )0) ), where arg( ) + 3 = u (respectively, arg( ) ), which implies that 2 +4 (respectively, = ( + 301 ) 2 603 = ( + 2 + 4 3 ) 2) has a positive real part. 601 A similar argument as above shows that 6 has exactly three zero eigenvalues and two eigenvalues with positive real parts and all other eigenvalues have negative real parts. By following a similar procedure to the proof of the third statement, we can show that all vehicles will eventually move along logarithmic spiral curves with center given by (4), growing rate Re( 603 ), and period 2 Im( 603 ) . The radius of the logarithmic spiral curve for vehicle is given by 2 (i) 6T03 [ (0)T (0)T ]T Re( )t 22 + 23 sin2 ( 2). The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of (i) . In addition, the relative phases of the vehicles on their curves are equal to the relative phases of (i) . A similar argument to that for the third statement shows that the curves are on a plane perpendicular to Euler axis a. Example 3.3: To illustrate, consider four vehicles with network topology shown by Fig. 1. Let associated with be given by (
+ )(
$
=j jw p
$
r
j
)
j a a
;v
w &
&
c
i
=
t ;c
w
t ;c t
0 2 ; = e j je 0 > = 0 0 p =
jw p
=j i a a =
je
r ;v w
j
w
G
L
G
1:5 0 01:1 00:4 01:2 1:2 0 0 : 00:1 00:5 0:6 0 01 0 0 1
:
R
It can be computed that dc = 0 3557 rad. Let be the rotation matrix corresponding to Euler axis a = (1 14)[1 2 3]T and Euler angle = c c = 0 3626. We let = c + 0 5. d . It can also be computed that and ( ). We can see that Fig. 2 shows the eigenvalues of ) correspond to the eigenvalues of the eigenvalues of ( rotated by angles 0, , and . Fig. 3 shows the eigenvalues of 6.
0L R 0
=
0L
:
;;
Fig. 1. Network topology for four vehicles. An arrow from j to i denotes that vehicle i can receive information from vehicle j .
0 L R
:
0L
0L
0 L
Fig. 2. Eigenvalues of and ( R) with = . Circles denote the eigenvalues of while x-marks denote the eigenvalues of ( R). The eigenvalues of ( R) correspond to the eigenvalues of rotated by angles 0, , and , respectively. In particular, the eigenvalues obtained by rotating by angles 0, , and are shown by, respectively, the solid line, the dashed line, and the dashdot line.
0L 0 L
0 0
0L
0 L
0 L R p 0 6
We can see that each eigenvalue of ( ), k , corresponds to two 2 +4 eigenvalues of 6, 2k01;2k , where 2k01;2k = ( k ) 2, c , two nonzero eigenvalues = 1 . . . 12. Because = dc and of 6 are located on the imaginary axis as shown in Fig. 2.
k
; ;
>
=
B. Discussion and Extension In this subsection, we discuss the results in Section III-A and show an extension to single-integrator kinematics. In existing consensus algorithms for double-integrator dynamics (e.g., [9], [18]), the Cartesian coordinates of a vehicle are decoupled. We have shown in Theorem 3.2 that different collective motions can result from the Cartesian coordinate coupling. In addition, the first statement of Theorem 3.2 generalizes Theorem 5.1 in [18], which gives only a sufficient condition for , by giving a necessary and sufficient condition. The results in Section III-A extend [13] threefold, namely, extension from 2-D to 3-D, extension from single-integrator kinematics to double-integrator dynamics without the knowledge of relative velocities, and extension from a unidirectional ring topology to a general network topology. While [13] proposes an extension from single-integrator kinematics to double-integrator dynamics to deal with a formation control problem, the extension relies not only on relative positions but also on relative velocities between neighbors. In contrast, algorithm (3) does not rely on relative velocities between neighbors. For vehicles with nonholonomic constraints, algorithm (3) can still be applied if the vehicle dynamics can be feedback linearized as double-integrator dynamics. For single-integrator kinematics given by
r_i = ui ; i = 1; . . . ;n
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p 6. Squares denote the eigenvalues computed by = (0 + + p4 )=2 while diamonds denote the eigenvalues computed by = (0 0 + 4 )=2, k = 1; . . . ; 12. In particular, the eigenvalues of 6 correspond to = , = e , and = e are shown by, respectively, the solid line, the dashed line, and the dashdot line. Because = , two nonzero eigenvalues of 6 are on the imaginary axis. Fig. 3. Eigenvalues of
where ri = [xi ; yi ; zi ]T is the position and ui 2 3 is the control input associated with the ith vehicle, a consensus algorithm with Cartesian coordinate coupling takes in the form ui
=0
n j =1
(
)
aij R ri 0 rj ;
i
= 1; . . . ; n
0
Fig. 4. Trajectories of the four vehicles using (3) with = 0:2. Circles denote the starting positions of the vehicles while the squares denote the snapshots of the vehicles at t = 30.
= [x ; y ] and R is the 2 2 2 rotation matrix cos( ) sin() given by R() = 0 sin() cos() . 1) If jj < (=n), the vehicles will eventually rendezvous at position (p x; p y), where x = [x1 ; . . . ; x ] and y = [y1 ; . . . ; y ] . 2) If jj = =n, all vehicles will eventually move on the same circular orbit with center (p x; p y ), period =(sin(=n)), and radius 2jw ( ) ( = w [1=2; 0(1=2)])r(0)j. 3 In addition, the vehicles will eventually be evenly distributed on the orbit. 3) If (=n) < jj < (2=n), all vehicles will eventually move along logarithmic spiral curves with center (p x; p y ), growing rate 2 sin(=n)sin(jj 0 (=n)), period = sin(=n)cos(jj 0 =n), and radius 2jw ( ) (( = w ) [1=2; 0(1=2)])r(0)j given by (8), where ri
T
i
i
T
T
T
(8)
where R is the 3 2 3 rotation matrix. We can adopt a similar approach to that used in Theorem 3.2 to analyze (8). Due to space limitation, we present the following theorem with its proof omitted. Theorem 3.4: Suppose that weighted directed graph G has a directed spanning tree. Let the control algorithm for (7) be given by (8). Let i , wi , i , and arg(i ) be defined in Definition 3.1, p be defined in Lemma 3.2, and a = [a1 ; a2 ; a3 ]T , &k , and $k be defined in Lemma 3.3. 1 1) If jj < sc , where sc = (3=2) 0 arg(n ), the vehicles will eventually rendezvous at position (pT x; pT y; pT z ), where x, y , and z are, respectively, column stack vectors of xi , yi , and zi . 2) If jj = sc and arg(n ) is the unique maximum phase of i , all vehicles will eventually move on circular orbits with center (pT x; pT y; pT z ) and period 2=jn j. The radius of the orbit for vehicle i is given by 2jwn(i) (nT =nT wn
$2T =$2T &2 )r(0)j a22 + a23 sin2 (=2), where wn(i) is the ith component of wn . The relative radii of the orbits are equal to the relative magnitudes of wn(i) . The relative phases of the vehicles on their orbits are equal to the relative phases of wn(i) . The circular orbits are on a plane perpendicular to Euler axis a. 3) If arg(n ) is the unique maximum phase of i and sc < jj < move along (3=2) 0 arg(n01 ), all vehicles willT eventually logarithmic spiral curves with center (p x; pT y; pT z ), growing rate jn j cos(arg(n )+ jj), and period 2=(jn sin(arg(n )+ jj)j). The radius of the logarithmic spiral curve for vehicle i is given by 2jwn(i) (nT =nT wn $2T =$2T &2 )r(0)j e[j j cos(arg( )+jj)]t a22 + a23 sin2 (=2). The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of wn(i) . The relative phases of the vehicles on their curves are equal to the relative phases of wn(i) . The logarithmic spiral curves are on a plane perpendicular to Euler axis a. Corollary 3.5: Suppose that weighted directed graph G is a unidirectional ring (i.e., a cyclic pursuit topology). Also suppose that aij = 1 if (j; i) 2 E and aij = 0 otherwise. Let the control algorithm for (7) be
T
n
n i
T n
T n
n
T
n
T
2 sin(=n) sin(jj0(=n))t
n i
e
T
T n
T n
T
n
. In addition, the phases of all vehicles will eventually be evenly distributed. Proof: Note that if weighted directed graph G is a unidirectional ring and aij = 1 if (j; i) 2 E and aij = 0 otherwise, then L is a circulant matrix. Also note that a circulant matrix can be diagonalized by a Fourier matrix. The proof then follows Theorem 3.4 directly by use of the properties of the eigenvalues of a circulant matrix and the properties of the Fourier matrix. Corollary 3.5 was proved in [13] by use of parametric spectral analysis of some special types of circulant matrices. Here we have shown that the convergence result in [13] is a special case of Theorem 3.4 and can be recovered by exploiting the properties of the circulant matrices and the Kronecker product. When G is a unidirectional ring but different positive weights are chosen for aij , where (j; i) 2 E , all vehicles will move on orbits with different radii and their phases will not be evenly distributed. IV. SIMULATION In this section, we study collective motions of four vehicles using (3). Suppose that the network topology is given by Fig. 1 and L is defined in Example 3.3. Let dc , a, and be given in Example 3.3. It can be verified that there exists a unique arg(4 ) 2 [; 3=2) such that 4u 0 arg(4) = dc (i.e., = 4 in Theorem 3.2). It can also be computed that the right eigenvector of 0L associated with eigenvalue 4 is w4 =
[00:2847 0 0:2820; 0:7213; 00:2501+0:1355; 0:4809+0:0837] and p = [0:2502; 0:1911; 0:4587; 0:1001] . Figs. 4, 5, and 6 show, respectively, the trajectories of the four vehicles using (3) with = 0 0:2, = , and = + 0:2. It can T
T
c d
3In
this case, all w
,i
c d
c d
= 1; . . . ; n, have the same magnitude.
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also demonstrated collective motions of four vehicles using the introduced algorithm in simulation. In future work, we will apply the algorithm in experiments in motion coordination of robotic networks. The robustness of the case of circular orbits can be improved by letting the Euler angle vary slightly below or above its critical value rather than remain constant at its critical value in case that the resulting trajectories spiral out or spiral in. We will also study a synthesis problem, namely, how to design the network topology such that some trajectories with desired relative radii and relative phases can be achieved.
REFERENCES
=
Fig. 5. Trajectories of the four vehicles using (3) with . Circles denote the starting positions of the vehicles while the squares denote the snapshots of . the vehicles at t
= 30
= +02
Fig. 6. Trajectories of the four vehicles using (3) with : . Circles denote the starting positions of the vehicles while the squares denote the snap. shots of the vehicles at t
= 10
be seen that all vehicles eventually rendezvous at the position given by (4) when = dc 0 0:2, move on circular orbits when = dc , and move along logarithmic spiral curves when = dc +0:2. Also observe that when = dc , the relative radii of the circular orbits (respectively, the relative phases of the vehicles) are equal to the relative magnitudes (respectively, phases) of the components of w4 . In addition, the trajectories of all vehicles are perpendicular to Euler axis a in all cases. A similar pattern for the relative radii and phases can also be observed for the logarithmic spiral curves.
[1] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Automat. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [2] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. [3] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Syst. Mag., vol. 27, no. 2, pp. 71–82, 2007. [4] G. Lafferriere, A. Williams, J. Caughman, and J. J. P. Veerman, “Decentralized control of vehicle formations,” Syst. Control Lett., vol. 54, no. 9, pp. 899–910, 2005. [5] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” IEEE Trans. Automati. Control, vol. 51, no. 3, pp. 401–420, Mar. 2006. [6] W. Ren and E. M. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” Int. J. Robust Nonlin. Control, vol. 17, no. 10–11, pp. 1002–1033, 2007. [7] D. Lee and M. W. Spong, “Stable flocking of multiple inertial agents on balanced graphs,” IEEE Trans. Automat. Control, vol. 52, no. 8, pp. 1469–1475, Aug. 2007. [8] H. G. Tanner, A. Jadbabaie, and G. J. Pappas, “Flocking in fixed and switching networks,” IEEE Trans. Automat. Control, vol. 52, no. 5, pp. 863–868, May 2007. [9] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Int. J. Robust Nonlin. Control, vol. 17, no. 10–11, pp. 941–959, 2007. [10] Z. Lin, M. Broucke, and B. Francis, “Local control strategies for groups of mobile autonomous agents,” IEEE Trans. Automat. Control, vol. 49, no. 4, pp. 622–629, Apr. 2004. [11] A. Sinha and D. Ghose, “Generalization of linear cyclic pursuit with application to rendezvous of multiple autonomous agents,” IEEE Trans. Automat. Control, vol. 51, no. 11, pp. 1819–1824, Nov. 2006. [12] J. A. Marshall, M. E. Broucke, and B. A. Francis, “Pursuit formations of unicycles,” Automatica, vol. 42, no. 1, pp. 3–12, 2006. [13] M. Pavone and E. Frazzoli, “Decentralized policies for geometric pattern formation and path coverage,” ASME J. Dyn. Syst., Meas., Control, vol. 129, no. 5, pp. 633–643, 2007. [14] E. W. Justh and P. S. Krishnaprasad, “Equilibria and steering laws for planar formations,” Syst. Control Lett., vol. 52, pp. 25–38, 2004. [15] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grünbaum, and J. K. Parrish, “Oscillator models and collective motion,” IEEE Control Syst. Mag., vol. 27, no. 4, pp. 89–105, 2007. [16] W. Ren, “Collective motion from consensus with Cartesian coordinate coupling—Part i: Single-integrator kinematics & part ii: Double-integrator dynamics,” in Proc. IEEE Conf. Decision Control, Cancun, Mexico, Dec. 2008, pp. 1006–1017. [17] R. Agaev and P. Chebotarev, “On the spectra of nonsymmetric Laplacian matrices,” Linear Algebra Appl., vol. 399, pp. 157–178, 2005. [18] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Trans. Automat. Control, vol. 53, no. 6, pp. 1503–1509, Jul. 2008. [19] [Online]. Available: http://www.mathpages.com/home/kmath593/ kmath593.htm
V. CONCLUSION We have introduced Cartesian coordinate coupling to a consensus algorithm for double-integrator dynamics by a rotation matrix in 3-D. We have shown conditions under which rendezvous, circular patterns, and logarithmic spiral patterns can be achieved using the algorithm with Cartesian coordinate coupling under a general network topology and quantitatively characterize the resulting collective motions. We have
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Collective Motion From Consensus With Cartesian Coordinate Coupling Wei Ren, Member, IEEE
Abstract—Collective motions including rendezvous, circular patterns, and logarithmic spiral patterns can be achieved by introducing Cartesian coordinate coupling to existing consensus algorithms. We study the collective motions of a team of vehicles in 3-D by introducing a rotation matrix to an existing consensus algorithm for double-integrator dynamics. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound, and the Euler angle is below, equal, or above a critical value, the vehicles will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane perpendicular to the Euler axis. In particular, when the vehicles eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the vehicles on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric Laplacian matrix. Simulation results are presented to demonstrate the theoretical results. Index Terms—Collective motion, consensus, cooperative control, distributed algorithms, multi-vehicle systems.
I. INTRODUCTION Coordination of robotic networks has received significant attention in recent years due to its potential impact in numerous civilian, homeland security, and military applications. Consensus plays an important role in achieving distributed coordination. The basic idea of consensus is that a team of vehicles reaches an agreement on a common value by negotiating with their neighbors. Consensus algorithms are studied for both single-integrator kinematics [1]–[3] and double-integrator dynamics [4]–[9], to name a few. Related to consensus is the cyclic pursuit strategy, where each vehicle pursues only one other vehicle with the network topology forming a unidirectional ring. Cyclic pursuit is studied for single-integrator kinematics in [10], [11] while for wheeled vehicles subject to nonholonomic constraints in [12]. Ref. [13] generalizes the cyclic pursuit strategy by letting each vehicle pursue one other vehicle along the line of sight rotated by a common offset angle. It is shown that depending on the common offset angle, the vehicles can achieve different symmetric formations, namely, convergence to a single point, a circle, or a logarithmic spiral pattern. Other researchers also study symmetric formations by adopting models based on the Frenet-Serret equations of motion [14] or by exploring the connections between phase models of coupled oscillators and kinematic models of self-propelled particle groups [15]. While the strategy proposed in [13] generates interesting symmetric formation patterns, there are limitations. First, the results in [13] are
Manuscript received March 27, 2008; revised October 04, 2008 and January 14, 2009. First published May 27, 2009; current version published June 10, 2009. This work was supported by a National Science Foundation CAREER Award (ECCS-0748287). Recommended by Associate Editor M. Egerstedt. The author is with the Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322 USA (e-mail: [email protected]. edu). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TAC.2009.2015544
limited to 2-D. However, for applications involving unmanned aerial vehicles, it will be more natural to study motions in 3-D. Second, the results in [13] primarily focus on single-integrator kinematics. While an extension to double-integrator dynamics is proposed to deal with a formation control problem, the extension relies not only on relative positions but also on relative velocities between neighbors. However, the requirement for the knowledge of relative velocities might be too restrictive for some applications. Third, the results in [13] rely on a unidirectional ring topology and the resulting circular and logarithmic spiral patterns are evenly spaced. However, for a team consisting of heterogenous vehicles with different sensing/communciation capabilities, it might not always be desirable that the vehicles are evenly spaced and move along the same orbit with an identical radius. It will also be interesting to study the motions resulting from a general (not necessarily unidirectional ring) network topology. To address these limitations, the strategy proposed in [13] needs to be extended. In this note, we extend the results in [13] threefold, namely, i) extension from 2-D to 3-D, ii) extension from single-integrator kinematics to double-integrator dynamics without the knowledge of relative velocities, and iii) extension from a unidirectional ring topology to a general network topology to generate possibly non-evenly-spaced circular and logarithmic spiral patterns on concentric orbits with possibly nonidentical radii. In particular, we introduce Cartesian coordinate coupling to an existing consensus algorithm for double-integrator dynamics through a rotation matrix in 3-D, analyze the convergence properties, and quantitatively characterize the resulting collective motions in 3-D, namely, convergence to a point, circular patterns with concentric orbits, and logarithmic spiral curves lying on a plane perpendicular to the Euler axis, over a general network topology. The resulting collective motions are expected to have applications in rendezvous, persistent surveillance, and coverage control with teams of heterogeneous vehicles. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. Our analysis relies on algebraic graph theory, matrix theory, and properties of the Kronecker product. In particular, we will show that the convergence result in [13] is a special case of the results in this note and the convergence result in [13] can be recovered by exploiting the properties of circulant matrices and the Kronecker product. A preliminary version of the work has appeared in [16].
II. BACKGROUND AND PRELIMINARIES A. Graph Theory Notions It is natural to model interaction among vehicles by directed graphs. Suppose that a team consists of n vehicles. A weighted directed graph G consists of a node set V = f1; . . . ; ng, an edge set E V 2V , and a weighted adjacency matrix A = [aij ] 2 n2n . An edge (i; j ) denotes that vehicle j can obtain information from vehicle i, but not necessarily vice versa. Weighted adjacency matrix A associated with G is defined such that aij is a positive weight if (j; i) 2 E , while aij = 0 if (j; i) 62 E . A directed path is a sequence of edges in a directed graph of the form (i1 ; i2 ); (i2 ; i3 ); . . ., where ij 2 V . A directed graph has a directed spanning tree if there exists at least one node having a directed path to all other nodes. Let nonsymmetric Laplacian matrix L = [`ij ] 2 n2n associated with A be defined as `ii = n a and `ij = 0aij , j =1;j 6=i ij i 6= j [17]. B. Existing Consensus Algorithm for Double-Integrator Dynamics Consider vehicles with double-integrator dynamics given by
r_i = vi ;
v_ i = ui ; i = 1; . . . ; n
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(1)
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where ri 2 m and vi 2 m are, respectively, the position and velocity of the ith vehicle, and ui 2 m is the control input. A consensus algorithm for (1) is studied in [9], [18] as
ui
n
=0 j
=1
(
aij ri 0 rj
) 0 v ; i
i
= 1; . . . ; n
(2)
where aij is the (i; j )th entry of weighted adjacency matrix A associated with weighted directed graph G , and is a positive damping gain. Consensus is reached for (1) using (2) if for all ri (0) and vi (0), ri (t) ! rj (t) and vi (t) ! 0 as t ! 1. III. CONSENSUS WITH CARTESIAN COORDINATE COUPLING In this section, we consider a consensus algorithm for double-integrator dynamics (1) with Cartesian coordinate coupling as ui
n
=0 j
=1
(
aij C ri 0 rj
) 0 v ; i
i
= 1; . . . ; n
(3)
where aij and are defined as in (2), and C 2 m2m denotes a Cartesian coordinate coupling matrix. We assume that all vehicles know C and a priori and the vehicles’ positions and velocities are represented in a common reference frame. Note that (2) corresponds to the case where C = Im , where Im denotes the m 2 m identity matrix. That is, using (2), the components of ri (i.e., the Cartesian coordinates of vehicle i) can be decoupled while using (3) the components of ri are coupled. A. Convergence Result In this subsection, we analyze the convergence properties of (3). We focus on the case where C is a rotation matrix while a similar analysis can be extended to the case where C is a general matrix. Before moving on, we need the following lemmas and definition: Lemma 3.1: Let U 2 p2p , V 2 q2q , X 2 p2p , and Y 2 q 2q . Then (U V )(X Y ) = U X V Y , where denotes the Kronecker product. Let A 2 p2p have eigenvalues i with associated eigenvectors fi 2 p , i = 1; . . . ; p, and let B 2 q2q have eigenvalues j with associated eigenvectors gj 2 q , j = 1; . . . ; q . Then the pq eigenvalues of A B are i j with associated eigenvectors fi gj , i = 1; . . . ; p, j = 1; . . . ; q . Lemma 3.2: [3] Let L be the nonsymmetric Laplacian matrix associated with weighted directed graph G . Then L has at least one zero eigenvalue and all its nonzero eigenvalues have positive real parts. Furthermore, L has a simple zero eigenvalue and all other eigenvalues have positive real parts if and only if G has a directed spanning tree. In addition, there exist 1n , where 1n is the n 2 1 column vector of all ones, satisfying L1n = 0 and p 2 n satisfying p 0, pT L = 0, and pT 1n = 1. 1 Definition 3.1: Let i , i = 1; . . . ; n, be the ith eigenvalue of 0L with associated right eigenvector wi and left eigenvector i . Also let arg(i ) = 0 for i = 0 and arg(i ) 2 (=2; 3=2) for all i 6= 0, where arg(1) denotes the phase of a number. Without loss of generality, suppose that i is labeled such that arg(1 ) arg(2 ) 1 1 1 arg(n). 2 Lemma 3.3: (see e.g., [19]) Given a rotation matrix R 2 323 , let a = [a1 ; a2 ; a3 ]T and denote, respectively, the Euler axis (i.e., the unit vector in the direction of rotation) and Euler angle (i.e., the rotation angle). The eigenvalues of R are 1, e , and e0 , where denotes the imaginary unit, with the associated right eigenvectors given by, respectively, &1 = a, &2 = [(a22 + a23 ) sin2 (=2); 0a1 a2 sin2 (=2)+ a3 sin(=2)jsin(=2)j; 0a1 a3 sin2 (=2)0 a2 sin(=2)jsin(=2)j]T, and &3 = & 2 , where 1 denotes the complex conjugate of a number. 1That is, 1 and p are, respectively, the right and left eigenvectors of L associated with the zero eigenvalue. 2It
follows from Lemma 3.2 that
= 0, w = 1
, and
= p.
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The associated left eigenvectors are, respectively, $1 = &1 , $2 = &2 , and $3 = &3 . Lemma 3.4: Let A 2 n2n with eigenvalues i and associated right and left eigenvectors qi and si , respectively. Also let
02
In n A 0 In , where n2n denotes the n 2 n zero matrix and is a positive scalar. Then the eigenvalues of B are 0 2
i = with associated right given by 2i01
=
B
0
n
= (
and left eigenvectors
2i
= (0 0
2
q
+
and
q
+( 4 ) +2 )
s
s
, respectively, and
++ 4) )=2, with associated right and left eigenveci
and ( s s , respectively. Proof: Suppose that is an eigenvalue of B with an asson . It follows that ciated right eigenvector fg , where f; g 2
tors
q
q
02
f , which implies g f and g 2 Af 0 g g . It thus follows that Af f . Noting
i qi , we let f that Aqi qi and 2
i . That is, each eigenvalue of A, i , corresponds to two eigenvalues of B , denoted by 0 6 2 i = . Because g f , it follows that 2i01;2i the right eigenvectors associated with 2i01 and 2i are, respectively, n
f g
In
n
0 In
A
= = =(
=
= =( + ) + = =
= +4 ) 2
q and q q . A similar analysis can be used to find the left q eigenvectors of B associated with 2i01 and 2i . Theorem 3.2: Suppose that weighted directed graph G has a directed spanning tree. Let the control algorithm for (1) be given by (3), where ri = [xi ; yi ; zi ]T and vi = [vxi ; vyi ; vzi ]T . Let i , wi , i , and arg(i ) be defined in Definition 3.1, p be defined in Lemma 3.2, and a = [a1 ; a2 ; a3 ]T , &k , and $k be defined in Lemma 3.3. 1) Suppose that C = I3 . Then all vehicles will eventu1 ally rendezvous if and only if > c , where c = 2 maxi ji j sin (arg(i ))= 0 cos(arg(i )). The rendezvous position is given by
pT
(0) + v (0)
x
x
;p
T
y
(0) + v (0)
;
pT
z
y
(0) + v (0) z
(4)
where x, y , z , vx , vy , and vz are, respectively, column stack vectors of xi , yi , zi , vxi , vyi , and vzi . 2) Suppose that C = R, where R is the 3 2 3 rotation matrix defined in Lemma 3.3, and > c . Given ji j, i = 2; . . . ; n, let il 2 (=2; ) (respectively, iu 2 (; 3=2)) be the solution to ji j sin2 ( i ) + 2 cos( i ) = 0 if arg(i ) 2 (=2; ] 1 (respectively, arg(i ) 2 [; 3=2)). If jj < dc , where dc = u minarg( )2[;3=2) ( i 0 arg(i )), then all vehicles will eventually rendezvous at the position given by (4). 3) Under the assumption of 2), if jj = dc and there 2 [; 3=2) such that exists a unique arg( ) u = dc , then all vehicles will eventually 0 arg( ) move on circular orbits with center given by (4) and period =j sin( u )j. The radius of the orbit for vehicle i is given 2jw(i) pcT [r(0)T ; v(0)T ]T j a22 + a23 sin2 (=2), by where w(i) is the ith component of w and pc = 1=(2c +u )T w $2T &2 ( + )( $ $ ) , where c = (2j j sin( )= ). The relative radii of the orbits are equal to the relative magnitudes of w(i) . The relative phases of the vehicles on their orbits are equal to the relative phases of w(i) . The circular orbits are on a plane perpendicular to Euler axis a. 4) Under the assumption of 2), if there exists a unique arg( ) 2 [; 3=2) such that u 0u arg( ) = dc and dc < jj < minarg( )2[;3=2);i6= ( i 0 arg(i )), then the vehicles will eventually move along logarithmic spiral curves with center given ), where Re(1) denotes the real part of a by (4), growing rate Re(sp number and s = (0 + 2 + 4s )=2 with s = ejj , and period 2=jIm(s )j, where Im(1) represents the imaginary part
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of a number. The radius of the logarithmic spiral curve for vehicle T i is 2jw(i) pTs [r (0)T ; v (0)T ] eRe( )t a22 + a23 sin2 (=2), T where ps = 1=(2s + ) w $2T &2 ( + )( $ $ ) . The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of w(i) . The relative phases of the vehicles on their curves are equal to the relative phases of w(i) . The curves are on a plane perpendicular to Euler axis a. Proof: 1) For the first statement, if C = I3 , then (1) using (3) can be written in matrix form as 0n2n In
I3 r_ r 0L 0 In = (5) v_ v 0
T T where r = [r1T ; . . . ; rnT ] , v = [v1T ; . . . ; vnT ] , and L is the nonsymmetric Laplacian matrix associated with G . It follows from the proof of Theorem 5.1 in [18] that the vehicles will eventually rendezvous if and only if 0 defined in (5) has a simple zero eigenvalue and all other eigenvalues have negative real parts. Note from Lemma 3.4 that each eigenvalue i of 0L corresponds to two eigenvalues of 0 given by 2i01 = (0 + 2 + 4i )=2 with associated right and left eigenvectors w w and ( + ) , respectively, and
= (0 0 2 + 4i )=2, with associated right and left eigen ) , respectively, where i = 1; . . . ; n. vectors ww and ( + Because weighted directed graph G has a directed spanning tree, it follows from Lemma 3.2 that 0L has a simple zero eigenvalue and all other eigenvalues have negative real parts. According to Definition 3.1, we let 1 = 0 and Re(i ) < 0, i = 2; . . . ; n. Note from Lemma 3.2 that w1 = 1n and 1 = p. It thus follows that 1 = 0 with associated right and left eigenvectors given by 10 and pp , respectively, and 2 + 4i 2 = 0 . Note that 2 < 0 if > 0. Also noting that all have nonnegative real parts, it follows that all 2i , i = 2; . . . ; n, have negative real parts if > 0. It is left to show conditions under which 3 2i01 , i = 2; . . . ; n, have negative real parts. Suppose that i is the critical value for such that 2i01 , i = 2; . . . ; n, is on the imaginary axis. Let 2i01 = i , where i 2 , i = 2; . . . ; n. After some manip2 ji j sin (arg(i ))= 0 cos(arg(i )) ulation, it follows that 3i = and i = 2ji j sin(arg(i ))= , i = 2; . . . ; n. It is straightforward to verify that if > 3i (respectively, < 3i ), then 2i01 , i = 2; . . . ; n, has a negative (respectively, positive) real part. Therefore, all 2i01 , i = 2; . . . ; n, have negative real parts if and only if > maxi=2;...;n 3i . Combining the above arguments shows that 0 has a simple zero eigenvalue and all other eigenvalues have negative real parts if and only if c > . Matrix 0 can be written in Jordan canonical form as SJ S 01 , where the columns of S , denoted by sk , k = 1; . . . ; 2n, can be chosen to be the right eigenvectors or generalized right eigenvectors of 0 associated with eigenvalue k , k = 1; . . . ; 2n, the rows of S 01 , denoted by hkT , k = 1; . . . ; 2n, can be chosen to be the left eigenvectors or generalized left eigenvectors of 0 associated with eigenvalue k such that hkT sk = 1 and hkT s` = 0, k 6= `, and J is the Jordan block diagonal matrix with k T ]T and being the diagonal entries. We can choose s1 = [1Tn ; 0n T T T T h1 = [p ; (1= )p ] . It can be verified that h1 s1 = 1. It = thus follows that limt!1 vr((tt)) = limt!1 (e0t I3 ) vr(0) (0) 1 r (0) T T [( 0 [p (1= )p ]) I3 ] v(0) , which implies that xi (t) ! pT x(0) + (1= )pT vx (0), yi (t) ! pT y(0) + (1= )pT vy (0), T T zi (t) ! p z (0) + (1= )p vz (0), vxi (t) ! 0, vyi (t) ! 0, and vzi (t) ! 0 as t ! 1. Equivalently, it follows that all vehicles will eventually rendezvous at the position given by (4). 2) For the second statement, using (3), (1) can be written in matrix form as 2i
r_
v_
=
03n23n ( R)
0 L
I3n
r
n
0 I3
v
:
(6)
It follows from Lemmas 3.1 and 3.3 and Definition 3.1 that the eigenvalues of 0(L R) are i , i e , and i e0 with associated right eigenvectors wi &1 , wi &2 , and wi &3 , respectively, and associated left eigenvectors i $1 , i $2 , and i $3 , respectively. That is, the eigenvalues of 0(L R) correspond to the eigenvalues of 0L rotated by angles 0, , and 0 , respectively. Let ` , ` = 1; . . . ; 3n, denote the `th eigenvalue of 0(L R). Without loss of generality, let 0 , i = 1; . . . ; n, be the 3i02 = i , 3i01 = i e , and 3i = i e eigenvalues of 0(L R). Note from Lemma 3.4 that each k correspondspto two eigenvalues of 6, defined in (6), given by 2k01;2k = (0 6 2 + 4k )=2, k = 1; . . . ; 3n. Because 1 = 0, it follows that 1 = 2 = 3 = 0, which in turn implies that 1 = 3 = 5 = 0 and 2 = 4 = 6 = 0 . Similar to the proof of the first statement, all 2k , k = 1; . . . ; 3n, have negative real parts if > 0. Given arg( ) , i = 2; . . . ; n, l and u are the critical > 0 and i = ji je i i values for arg( i ) 2 [0; 2 ) such that (0 + 2 + 4 i )=2 is on the imaginary axis. In particular, if arg( i ) = il (respectively, iu ), then (0 + 2 + 4 i )=2 = (2ji j sin(arg( il )= ) (respectively, u l u (2ji j sin(arg( i )= )), i = 2; . . . ; n. If arg( i ) 2 ( i ; i ) (reu l 2 spectively, arg( i ) 2 [0; i ) [ ( i ; 2 )), then (0 + + 4 i )=2 have negative (respectively, positive) real parts. Because > c , the first statement implies that all (0 + 2 + 4i )=2, i = 2; . . . ; n, have negative real parts, which in turn implies that arg(i ) 2 ( il ; iu ), c i = 2; . . . ; n. If j j < d , then arg(3i02 ), arg(3i01 ), and arg(3i ) l u are all within ( i ; i ), which implies that 6i05 , 6i03 , and 6i01 , c i = 2; . . . ; n, all have negative real parts. Therefore, if j j < d , then 6 has exactly three zero eigenvalues and all other eigenvalues have negative real parts. Similar to the proof of the first statement, we write 6 in Jordan canonical form as M J M 01 , where the columns of M , denoted by mk , k = 1; . . . ; 6n, can be chosen to be the right eigenvectors or generalized right eigenvectors of 6 associated with eigenvalue k , the rows of M 01 , denoted by pkT , k = 1; . . . ; 6n, can be chosen to be the left eigenvectors or generalized left eigenvectors of 6 associated with eigenvalue k such that pkT mk = 1 and pkT m` = 0, k 6= `, and J is the Jordan block diagonal matrix with k being the diagonal entries. Recall that the right and left eigenvectors of 0(L R) associated with eigenvalue ` = 0 are, respectively, 1n &` and p $` , where ` = 1, 2, 3. It in turn follows from Lemma 3.4 that the right and left eigenvectors of 6 associated with 2`01 = 0 are, respectively, 10 & and p $ , where ` = 1, 2, 3. We can choose m 1 & and 2`01 = p $ 0 p ($ =$ & ) p2`01 = , where ` = 1, 2, 3. It can be verified that p ($ = $ & ) T T p2`01 m2`01 = 1 and p2`01 m2k01 = 0, where k; ` = 1, 2, 3 and k 6= r(t) `. Noting that 2`01 = 0, ` = 1, 2, 3, it follows that limt!1 v (t) = 3 r (0) Jt 0 1 r (0) T limt!1 M e M v(0) ! ( `=1 m2`01 p2`01 ) v(0) , which implies that xi (t) ! pT x(0) + (1= )pT vx (0), yi (t) ! pT y (0) + (1= )pT vy (0), zi (t) ! pT z (0) + (1= )pT vz (0), vxi (t) ! 0, vyi (t) ! 0, and vzi (t) ! 0 as t ! 1. Equivalently, it follows that all vehicles will eventually rendezvous at the position given by (4). 3) For the third statement, if = dc (respectively, = 0dc ) and there exists a unique arg( ) 2 [; 3=2) such that u c = j je (re 0 arg( ) = d , then 301 = e 0 spectively, 3 = e = j je ), which implies that u 603 = (0 + 2 + 4301 )=2 = (2j j sin( )= ) (respecp 2 tively, 601 = (0 + + 43 )=2 = (2j j sin( u )= )). Noting that the complex eigenvalues of 6 are in pairs, it follows that 6 has an eigenvalue equal to 603 = 0(2j j sin( u )= ) (respectively, 601 = 0(2j j sin( u )= )), denoted by 3 for simplicity. In this case, 6 has exactly three zero eigenvalues, two nonzero eigenvalues on the imaginary axis, and all other eigenvalues have negative real parts. In the following, we focus on = dc since the analysis for = 0dc is similar except that all vehicles will move in reverse directions. Note from Lemma 3.4 that the right and left eigen& vectors associated with 603 are, respectively, w (
w & ) and
6
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m p = w $ & p m m p p p m m e ! ct m p m p e t ct mp k c t e t ct k ; ;n c w & p r ;v e i ; ;n ` & ` & c t j& w p r ;v j = t j j w p r ;v & i ; ;n ` x t ! x = v c t = v c t y t ! y z t ! z c t = v t k c t ;c t ;c t k jw p r ;v j a a =
& . We can choose 603 = w (
w & ) and ( + )( $ ) T T . It can 603 = (1 (2 603 + ) 2 2) $ T be verified that 603 603 = 1. Similarly, it follows that 3 and 3 corresponds to 3 are given by 3 3= 603 and 3 = 603 . It T ) r(0) + ( ) ( follows that vr((tt)) = 6t vr(0) 2 ` 0 1 2 ` 0 1 `=1 (0) v(0) 1 for large , where ( ) = ( (2j j sin( )= )t 603 6T03 + 0(2j j sin( )= ) 3 3T ) r(0) . Let k ( ) be the th compov(0) = 1 . . . 6 . It follows that 3(i0T1)+` ( ) = nent of ( ), 2Re( (2j j sin( )= )t (i) 2(`) T603 [ (0)T (0)T ] ), where = 1 . . . , = 1, 2, 3, and 2(`) denotes the th component of 2 . After some manipulation, it follows that 3(i01)+` ( ) = cos((2 sin( u ) ) + 2 2(`) (i) T603 [ (0)T (0)TT]T T T T arg( (i) 603 [ (0) (0) ] ) + arg( T2(`) )), = 1 . . . T , = 1, p (0) + (1 )p x (0) + 2, 3. Therefore, it follows that i ( ) pT (0) + (1 )pT y (0) + 3i01 ( ), and i( ) 3i02 ( ), pT (0) + (1 )pT z (0) + 3i ( ) for large . After some i( ) manipulation, it can be verified that [ 3i02 ( ) 3i01 ( ) 3i ( )]T = 2 (i) 6T03 [ (0)T (0)T ]T 22 + 23 sin2 ( 2), which is a constant. Therefore, it follows that all vehicles will eventually move on circular orbits with center give by (4) and period u sin( ) . The radius of the orbit for vehicle is given by T 2 (i) 603 [ (0)T (0)T ]T 22 + 23 sin2 ( 2). The relative radii of the orbits are equal to the relative magnitudes of (i) . In addition, the relative phases of the vehicles are equal to the relative phases of (i) . Note from Lemma 3.3 that Euler axis a is orthogonal to both Re( 2 ) and Im( 2 ) are applied componentwise. It can thus be verified that a is orthogonal to [ 3i02 ( ) 3i01 ( ) 3i ( )]T , which implies that the circular orbits are on a plane perpendicular to a. 4) For the fourth statement, if there exists a unique [ 3 2) such that u arg( ) = dc and arg( ) c minarg( )2[;3u=2);i6= ( iu arg( i )) (respectively, d c arg( i )) minarg( )2[;3=2);i6= ( i d ), (arg( )+) (respectively, then 301 = = u 0 = (arg( )0) ), where arg( ) + 3 = u (respectively, arg( ) ), which implies that 2 +4 (respectively, = ( + 301 ) 2 603 = ( + 2 + 4 3 ) 2) has a positive real part. 601 A similar argument as above shows that 6 has exactly three zero eigenvalues and two eigenvalues with positive real parts and all other eigenvalues have negative real parts. By following a similar procedure to the proof of the third statement, we can show that all vehicles will eventually move along logarithmic spiral curves with center given by (4), growing rate Re( 603 ), and period 2 Im( 603 ) . The radius of the logarithmic spiral curve for vehicle is given by 2 (i) 6T03 [ (0)T (0)T ]T Re( )t 22 + 23 sin2 ( 2). The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of (i) . In addition, the relative phases of the vehicles on their curves are equal to the relative phases of (i) . A similar argument to that for the third statement shows that the curves are on a plane perpendicular to Euler axis a. Example 3.3: To illustrate, consider four vehicles with network topology shown by Fig. 1. Let associated with be given by (
+ )(
$
=j jw p
$
r
j
)
j a a
;v
w &
&
c
i
=
t ;c
w
t ;c t
0 2 ; = e j je 0 > = 0 0 p =
jw p
=j i a a =
je
r ;v w
j
w
G
L
G
1:5 0 01:1 00:4 01:2 1:2 0 0 : 00:1 00:5 0:6 0 01 0 0 1
:
R
It can be computed that dc = 0 3557 rad. Let be the rotation matrix corresponding to Euler axis a = (1 14)[1 2 3]T and Euler angle = c c = 0 3626. We let = c + 0 5. d . It can also be computed that and ( ). We can see that Fig. 2 shows the eigenvalues of ) correspond to the eigenvalues of the eigenvalues of ( rotated by angles 0, , and . Fig. 3 shows the eigenvalues of 6.
0L R 0
=
0L
:
;;
Fig. 1. Network topology for four vehicles. An arrow from j to i denotes that vehicle i can receive information from vehicle j .
0 L R
:
0L
0L
0 L
Fig. 2. Eigenvalues of and ( R) with = . Circles denote the eigenvalues of while x-marks denote the eigenvalues of ( R). The eigenvalues of ( R) correspond to the eigenvalues of rotated by angles 0, , and , respectively. In particular, the eigenvalues obtained by rotating by angles 0, , and are shown by, respectively, the solid line, the dashed line, and the dashdot line.
0L 0 L
0 0
0L
0 L
0 L R p 0 6
We can see that each eigenvalue of ( ), k , corresponds to two 2 +4 eigenvalues of 6, 2k01;2k , where 2k01;2k = ( k ) 2, c , two nonzero eigenvalues = 1 . . . 12. Because = dc and of 6 are located on the imaginary axis as shown in Fig. 2.
k
; ;
>
=
B. Discussion and Extension In this subsection, we discuss the results in Section III-A and show an extension to single-integrator kinematics. In existing consensus algorithms for double-integrator dynamics (e.g., [9], [18]), the Cartesian coordinates of a vehicle are decoupled. We have shown in Theorem 3.2 that different collective motions can result from the Cartesian coordinate coupling. In addition, the first statement of Theorem 3.2 generalizes Theorem 5.1 in [18], which gives only a sufficient condition for , by giving a necessary and sufficient condition. The results in Section III-A extend [13] threefold, namely, extension from 2-D to 3-D, extension from single-integrator kinematics to double-integrator dynamics without the knowledge of relative velocities, and extension from a unidirectional ring topology to a general network topology. While [13] proposes an extension from single-integrator kinematics to double-integrator dynamics to deal with a formation control problem, the extension relies not only on relative positions but also on relative velocities between neighbors. In contrast, algorithm (3) does not rely on relative velocities between neighbors. For vehicles with nonholonomic constraints, algorithm (3) can still be applied if the vehicle dynamics can be feedback linearized as double-integrator dynamics. For single-integrator kinematics given by
r_i = ui ; i = 1; . . . ;n
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(7)
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p 6. Squares denote the eigenvalues computed by = (0 + + p4 )=2 while diamonds denote the eigenvalues computed by = (0 0 + 4 )=2, k = 1; . . . ; 12. In particular, the eigenvalues of 6 correspond to = , = e , and = e are shown by, respectively, the solid line, the dashed line, and the dashdot line. Because = , two nonzero eigenvalues of 6 are on the imaginary axis. Fig. 3. Eigenvalues of
where ri = [xi ; yi ; zi ]T is the position and ui 2 3 is the control input associated with the ith vehicle, a consensus algorithm with Cartesian coordinate coupling takes in the form ui
=0
n j =1
(
)
aij R ri 0 rj ;
i
= 1; . . . ; n
0
Fig. 4. Trajectories of the four vehicles using (3) with = 0:2. Circles denote the starting positions of the vehicles while the squares denote the snapshots of the vehicles at t = 30.
= [x ; y ] and R is the 2 2 2 rotation matrix cos( ) sin() given by R() = 0 sin() cos() . 1) If jj < (=n), the vehicles will eventually rendezvous at position (p x; p y), where x = [x1 ; . . . ; x ] and y = [y1 ; . . . ; y ] . 2) If jj = =n, all vehicles will eventually move on the same circular orbit with center (p x; p y ), period =(sin(=n)), and radius 2jw ( ) ( = w [1=2; 0(1=2)])r(0)j. 3 In addition, the vehicles will eventually be evenly distributed on the orbit. 3) If (=n) < jj < (2=n), all vehicles will eventually move along logarithmic spiral curves with center (p x; p y ), growing rate 2 sin(=n)sin(jj 0 (=n)), period = sin(=n)cos(jj 0 =n), and radius 2jw ( ) (( = w ) [1=2; 0(1=2)])r(0)j given by (8), where ri
T
i
i
T
T
T
(8)
where R is the 3 2 3 rotation matrix. We can adopt a similar approach to that used in Theorem 3.2 to analyze (8). Due to space limitation, we present the following theorem with its proof omitted. Theorem 3.4: Suppose that weighted directed graph G has a directed spanning tree. Let the control algorithm for (7) be given by (8). Let i , wi , i , and arg(i ) be defined in Definition 3.1, p be defined in Lemma 3.2, and a = [a1 ; a2 ; a3 ]T , &k , and $k be defined in Lemma 3.3. 1 1) If jj < sc , where sc = (3=2) 0 arg(n ), the vehicles will eventually rendezvous at position (pT x; pT y; pT z ), where x, y , and z are, respectively, column stack vectors of xi , yi , and zi . 2) If jj = sc and arg(n ) is the unique maximum phase of i , all vehicles will eventually move on circular orbits with center (pT x; pT y; pT z ) and period 2=jn j. The radius of the orbit for vehicle i is given by 2jwn(i) (nT =nT wn
$2T =$2T &2 )r(0)j a22 + a23 sin2 (=2), where wn(i) is the ith component of wn . The relative radii of the orbits are equal to the relative magnitudes of wn(i) . The relative phases of the vehicles on their orbits are equal to the relative phases of wn(i) . The circular orbits are on a plane perpendicular to Euler axis a. 3) If arg(n ) is the unique maximum phase of i and sc < jj < move along (3=2) 0 arg(n01 ), all vehicles willT eventually logarithmic spiral curves with center (p x; pT y; pT z ), growing rate jn j cos(arg(n )+ jj), and period 2=(jn sin(arg(n )+ jj)j). The radius of the logarithmic spiral curve for vehicle i is given by 2jwn(i) (nT =nT wn $2T =$2T &2 )r(0)j e[j j cos(arg( )+jj)]t a22 + a23 sin2 (=2). The relative radii of the logarithmic spiral curves are equal to the relative magnitudes of wn(i) . The relative phases of the vehicles on their curves are equal to the relative phases of wn(i) . The logarithmic spiral curves are on a plane perpendicular to Euler axis a. Corollary 3.5: Suppose that weighted directed graph G is a unidirectional ring (i.e., a cyclic pursuit topology). Also suppose that aij = 1 if (j; i) 2 E and aij = 0 otherwise. Let the control algorithm for (7) be
T
n
n i
T n
T n
n
T
n
T
2 sin(=n) sin(jj0(=n))t
n i
e
T
T n
T n
T
n
. In addition, the phases of all vehicles will eventually be evenly distributed. Proof: Note that if weighted directed graph G is a unidirectional ring and aij = 1 if (j; i) 2 E and aij = 0 otherwise, then L is a circulant matrix. Also note that a circulant matrix can be diagonalized by a Fourier matrix. The proof then follows Theorem 3.4 directly by use of the properties of the eigenvalues of a circulant matrix and the properties of the Fourier matrix. Corollary 3.5 was proved in [13] by use of parametric spectral analysis of some special types of circulant matrices. Here we have shown that the convergence result in [13] is a special case of Theorem 3.4 and can be recovered by exploiting the properties of the circulant matrices and the Kronecker product. When G is a unidirectional ring but different positive weights are chosen for aij , where (j; i) 2 E , all vehicles will move on orbits with different radii and their phases will not be evenly distributed. IV. SIMULATION In this section, we study collective motions of four vehicles using (3). Suppose that the network topology is given by Fig. 1 and L is defined in Example 3.3. Let dc , a, and be given in Example 3.3. It can be verified that there exists a unique arg(4 ) 2 [; 3=2) such that 4u 0 arg(4) = dc (i.e., = 4 in Theorem 3.2). It can also be computed that the right eigenvector of 0L associated with eigenvalue 4 is w4 =
[00:2847 0 0:2820; 0:7213; 00:2501+0:1355; 0:4809+0:0837] and p = [0:2502; 0:1911; 0:4587; 0:1001] . Figs. 4, 5, and 6 show, respectively, the trajectories of the four vehicles using (3) with = 0 0:2, = , and = + 0:2. It can T
T
c d
3In
this case, all w
,i
c d
c d
= 1; . . . ; n, have the same magnitude.
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also demonstrated collective motions of four vehicles using the introduced algorithm in simulation. In future work, we will apply the algorithm in experiments in motion coordination of robotic networks. The robustness of the case of circular orbits can be improved by letting the Euler angle vary slightly below or above its critical value rather than remain constant at its critical value in case that the resulting trajectories spiral out or spiral in. We will also study a synthesis problem, namely, how to design the network topology such that some trajectories with desired relative radii and relative phases can be achieved.
REFERENCES
=
Fig. 5. Trajectories of the four vehicles using (3) with . Circles denote the starting positions of the vehicles while the squares denote the snapshots of . the vehicles at t
= 30
= +02
Fig. 6. Trajectories of the four vehicles using (3) with : . Circles denote the starting positions of the vehicles while the squares denote the snap. shots of the vehicles at t
= 10
be seen that all vehicles eventually rendezvous at the position given by (4) when = dc 0 0:2, move on circular orbits when = dc , and move along logarithmic spiral curves when = dc +0:2. Also observe that when = dc , the relative radii of the circular orbits (respectively, the relative phases of the vehicles) are equal to the relative magnitudes (respectively, phases) of the components of w4 . In addition, the trajectories of all vehicles are perpendicular to Euler axis a in all cases. A similar pattern for the relative radii and phases can also be observed for the logarithmic spiral curves.
[1] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Automat. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [2] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. [3] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Syst. Mag., vol. 27, no. 2, pp. 71–82, 2007. [4] G. Lafferriere, A. Williams, J. Caughman, and J. J. P. Veerman, “Decentralized control of vehicle formations,” Syst. Control Lett., vol. 54, no. 9, pp. 899–910, 2005. [5] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” IEEE Trans. Automati. Control, vol. 51, no. 3, pp. 401–420, Mar. 2006. [6] W. Ren and E. M. Atkins, “Distributed multi-vehicle coordinated control via local information exchange,” Int. J. Robust Nonlin. Control, vol. 17, no. 10–11, pp. 1002–1033, 2007. [7] D. Lee and M. W. Spong, “Stable flocking of multiple inertial agents on balanced graphs,” IEEE Trans. Automat. Control, vol. 52, no. 8, pp. 1469–1475, Aug. 2007. [8] H. G. Tanner, A. Jadbabaie, and G. J. Pappas, “Flocking in fixed and switching networks,” IEEE Trans. Automat. Control, vol. 52, no. 5, pp. 863–868, May 2007. [9] G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” Int. J. Robust Nonlin. Control, vol. 17, no. 10–11, pp. 941–959, 2007. [10] Z. Lin, M. Broucke, and B. Francis, “Local control strategies for groups of mobile autonomous agents,” IEEE Trans. Automat. Control, vol. 49, no. 4, pp. 622–629, Apr. 2004. [11] A. Sinha and D. Ghose, “Generalization of linear cyclic pursuit with application to rendezvous of multiple autonomous agents,” IEEE Trans. Automat. Control, vol. 51, no. 11, pp. 1819–1824, Nov. 2006. [12] J. A. Marshall, M. E. Broucke, and B. A. Francis, “Pursuit formations of unicycles,” Automatica, vol. 42, no. 1, pp. 3–12, 2006. [13] M. Pavone and E. Frazzoli, “Decentralized policies for geometric pattern formation and path coverage,” ASME J. Dyn. Syst., Meas., Control, vol. 129, no. 5, pp. 633–643, 2007. [14] E. W. Justh and P. S. Krishnaprasad, “Equilibria and steering laws for planar formations,” Syst. Control Lett., vol. 52, pp. 25–38, 2004. [15] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grünbaum, and J. K. Parrish, “Oscillator models and collective motion,” IEEE Control Syst. Mag., vol. 27, no. 4, pp. 89–105, 2007. [16] W. Ren, “Collective motion from consensus with Cartesian coordinate coupling—Part i: Single-integrator kinematics & part ii: Double-integrator dynamics,” in Proc. IEEE Conf. Decision Control, Cancun, Mexico, Dec. 2008, pp. 1006–1017. [17] R. Agaev and P. Chebotarev, “On the spectra of nonsymmetric Laplacian matrices,” Linear Algebra Appl., vol. 399, pp. 157–178, 2005. [18] W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Trans. Automat. Control, vol. 53, no. 6, pp. 1503–1509, Jul. 2008. [19] [Online]. Available: http://www.mathpages.com/home/kmath593/ kmath593.htm
V. CONCLUSION We have introduced Cartesian coordinate coupling to a consensus algorithm for double-integrator dynamics by a rotation matrix in 3-D. We have shown conditions under which rendezvous, circular patterns, and logarithmic spiral patterns can be achieved using the algorithm with Cartesian coordinate coupling under a general network topology and quantitatively characterize the resulting collective motions. We have
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