Collective Motion from Consensus with Cartesian Coordinate Coupling - Part II: Double-integrator Dynamics

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1 Proceedngs of the 47th IEEE Conference on Decson Control Cancun Mexco Dec. 9-8 TuB. Collectve Moton from Consensus wth Cartesan Coordnate Couplng - Part II: Double-ntegrator Dynamcs We Ren Abstract Ths s the second part of a two-part paper on collectve moton from consensus wth Cartesan coordnate couplng. In ths part we study the collectve motons of a team of vehcles n 3D by ntroducng a rotaton matrx to an exstng consensus algorthm for double-ntegrator dynamcs. It s shown that the network topology the dampng gan the value of the Euler angle all affect the resultng collectve motons. In partcular we show a necessary suffcent condton on the dampng gan for rendezvous when there s no Cartesan coordnate couplng. We also explctly show the crtcal value for the Euler angle when there s Cartesan coordnate couplng quanttatvely characterze the resultng collectve motons namely rendezvous crcular patterns logarthmc spral patterns. Smulaton results are presented to demonstrate the theoretcal results. I. INTRODUCTION Takng nto account the fact that equatons of moton of a broad class of vehcles requre a double-ntegrator dynamc model consensus algorthms for double-ntegrator dynamcs are studed n [] [6]. In partcular [] [] derve condtons on the network topology the control gans under whch convergence s guaranteed. Refs. [3] study formaton keepng problems whle [4] [6] study flockng of multple vehcle systems. Motvated by [8] we have ntroduced n the frst part [9] of the two-part paper Cartesan coordnate couplng to an exstng consensus algorthm for sngle-ntegrator knematcs. In ths second part we consder the case of double-ntegrator dynamcs. In contrast to the sngle-ntegrator case the analyss for double-ntegrator dynamcs poses more challenges. The contrbutons of ths second part of the paper are as follows. We study the convergence propertes of a consensus algorthm wth a rotaton matrx ntroduced n 3D for doublentegrator dynamcs over a general network topology. In contrast to the sngle-ntegrator case we show that the network topology the dampng gan the value of the Euler angle all play an mportant role n the resultng collectve motons. In partcular we show a necessary suffcent condton on the dampng gan for rendezvous when there s no Cartesan coordnate couplng. We also explctly show the crtcal value for the Euler angle when there s Cartesan coordnate couplng quanttatvely characterze the resultng collectve motons namely rendezvous crcular patterns logarthmc spral patterns. The results generalze the Cartesan coordnate couplng case for sngle-ntegrator W. Ren s wth the Department of Electrcal Computer Engneerng Utah State Unversty Logan UT 843 USA wren@engneerng.usu.edu Ths work was supported by a Natonal Scence Foundaton CAREER Award (ECCS-74887). knematcs presented n [9] to account for dynamc models also generalze exstng consensus algorthms for doublentegrator dynamcs to acheve dfferent collecton motons. II. BACKGROUND AND PRELIMINARIES A. Graph Theory Notons It s natural to model nteracton among vehcles by drected or undrected graphs. Suppose that a team conssts of n vehcles. A weghted graph G conssts of a node set V = {...n} an edge set E V V a weghted adjacency matrx A = [a j ] R n n. An edge (j) n a weghted drected graph denotes that vehcle j can obtan nformaton from vehcle but not necessarly vce versa. In contrast the pars of nodes n a weghted undrected graph are unordered where an edge (j) denotes that vehcles j can obtan nformaton from each another. Weghted adjacency matrx A of a weghted drected graph s defned such that a j s a postve weght f (j ) E whle a j = f (j ) E. Weghted adjacency matrx A of a weghted undrected graph s defned analogously except that a j = a j j snce (j) E mples (j) E. A drected path s a sequence of edges n a drected graph of the form ( ) ( 3 )... where j V. An undrected path n an undrected graph s defned analogously. A drected graph has a drected spannng tree f there exsts at least one node havng a drected path to all other nodes. An undrected graph s connected f there s an undrected path between every par of dstnct nodes. Let nonsymmetrc Laplacan matrx L = [l j ] R n n assocated wth A be defned as l = n j=j a j l j = a j j. For a weghted undrected graph L s symmetrc postve sem-defnte. However L for a weghted drected graph does not have ths property. B. Exstng Consensus Algorthm Consder vehcles wth double-ntegrator dynamcs gven by ṙ = v v = u =...n () where r R m v R m are respectvely the poston velocty of the th vehcle u R m s the control nput. A consensus algorthm for () s studed n [] [] as n u = a j (r r j ) v =...n () j= where a j s the (j)th entry of weghted adjacency matrx A assocated wth weghted drected graph G s a /8/$5. 8 IEEE

2 47th IEEE CDC Cancun Mexco Dec. 9-8 TuB. postve gan. Consensus s reached usng () f for all r () v () r (t) r j (t) v (t) as t. III. CONSENSUS FOR DOUBLE-INTEGRATOR DYNAMICS WITH CARTESIAN COORDINATE COUPLING In ths secton we consder a consensus algorthm for double-ntegrator dynamcs () wth Cartesan coordnate couplng as n u = a j C(r r j ) v =...n (3) j= where C R m m denotes a Cartesan coordnate couplng matrx. Note that () corresponds to the case where C = I m. That s usng () the components of r (.e. the Cartesan coordnates of vehcle ) are decoupled whle usng (3) the components of r are coupled. In ths secton we focuses on the case where C s a rotaton matrx whle a smlar analyss can be extended to the case where C s a general matrx. Before movng on we need the followng lemmas defnton: Lemma 3.: [] Let U R p p V R q q X R p p Y R q q. Then (U V )(X Y ) = UX V Y. Let A R p p have egenvalues β wth assocated egenvectors f C p =...p let B R q q have egenvalues ρ j wth assocated egenvectors g j C q j =...q. Then the pq egenvalues of A B are β ρ j wth assocated egenvectors f g j =...p j =...q. Lemma 3.: [] Let L be the nonsymmetrc Laplacan matrx assocated wth weghted drected graph G. Then L has at least one zero egenvalue all nonzero egenvalues have postve real parts. Furthermore L has a smple zero egenvalue all other egenvalues have postve real parts f only f G has a drected spannng tree. In addton there exst n where n s the n column vector of all ones satsfyng L n = p R n satsfyng p p T L = p T =. Defnton 3.: Let µ =...n be the th egenvalue of L wth assocated rght egenvector w left egenvector ν. Also let arg(µ ) = for µ = arg(µ ) ( π 3π ) for all µ where arg( ) denotes the phase of a number. Wthout loss of generalty suppose that µ s labeled such that arg(µ ) arg(µ ) arg(µ n ). Lemma 3.3: (see e.g. [3]) Gven a rotaton matrx R R 3 3 let a = [a a a 3 ] T θ denote respectvely the Euler axs (.e. the unt vector n the drecton of rotaton) Euler angle (.e. the rotaton angle). The egenvalues of R are e ιθ e ιθ where ι denotes the magnary unt wth the assocated rght egenvectors gven by respectvely ς = a ς = [(a + a 3) sn ( θ ) a a sn ( θ ) + ιa 3 sn( θ ) sn( θ ) a a 3 sn ( θ ) ιa sn( θ ) sn( θ ) ]T ς 3 = ς where denotes the complex conjugate of a number. The assocated left egenvectors are respectvely = ς = ς 3 = ς 3. That s n p are respectvely the rght left egenvectors of L assocated wth the zero egenvalue. It follows from Lemma 3. that µ = w = n ν = p. Lemma 3.4: Let A R n n wth egenvalues γ assocated rght [ left egenvectors ] q s respectvely. n n I Also let B = n where A I n n denotes the n n n zero matrx s a postve scalar. Then the egenvalues of B are gven by ζ = + +4γ [ wth assocated ] q rght left egenvectors (ζ + )s ζ q respectvely ζ = +4γ [ ] wth assocated [ rght ] q (ζ + )s left egenvectors gven by ζ q s respectvely Proof: Suppose that ζ s [ an egenvalue of B wth an f assocated rght egenvector where f g C g] n. It follows [ n n I that n f f = ζ whch mples g = ζf A I n g g] Af g = ζg. It thus follows that Af = (ζ + ζ)f. Notng that Aq = γ q we let f = q ζ +ζ = γ. That s each egenvalue of A γ corresponds to two egenvalues of B denoted by ζ = ± +4γ. Because g = ζf t follows that the rght egenvectors assocated wth ζ q ζ are respectvely q. A smlar ζ q ζ q analyss can be used to fnd the left egenvectors of B assocated wth ζ ζ. Theorem 3.: Suppose that weghted drected graph G has a drected spannng tree. Let the control algorthm for () be gven by (3) where r = [x y z ] T v = [v x v y v z ] T. Let µ w ν arg(µ ) be defned n Defnton 3. p be defned n Lemma 3. a = [a a a 3 ] T ς k k be defned n Lemma 3.3. ) Suppose that C = I 3. Then all vehcles wll eventually rendezvous f only f > c where c = max µ sn (arg(µ )) cos(arg(µ )). The rendezvous poston s gven by [p T (x() + v x() )pt (y() + v y() )pt (z() + v z() )] (4) where x y z v x v y v z are respectvely stack vectors of x y z v x v y v z. ) Suppose that C = R where R s the 3 3 rotaton matrx defned n Lemma 3.3 > c. Gven µ =...n let ψ l ( π π) (respectvely ψu (π 3π )) be the soluton to µ sn (ψ ) + cos(ψ ) = f arg(µ ) ( π π] (respectvely arg(µ ) [π 3π )). If θ < θc d where θd c = mn arg(µ ) [π 3π ) (ψ u arg(µ )) then all vehcles wll eventually rendezvous at the poston gven by (4). 3) Under the assumpton of ) f θ = θd c there exsts a unque arg(µ κ ) [π 3π ) such that ψκ u arg(µ κ ) = θd c then all vehcles wll eventually move on crcular orbts wth center gven by (4) π perod µ κ sn(ψκ u ). The radus of the orbt for vehcle s gven by w κ() p T c [r() T v() T ] T a + a 3 sn ( θ ) where w κ() s the th component of w κ s 3

3 47th IEEE CDC Cancun Mexco Dec. 9-8 (σc + )(ν p c = κ ) (σ c+)ν where κ T wκ T ς ν κ σ c = ι µκ sn(ψu κ ). The relatve radus of the orbts s equal to the relatve magntude of w κ(). The relatve phase of the vehcles on ther orbts s equal to the relatve phase of w κ(). The crcular orbts are on a plane perpendcular to Euler axs a. 4) If there exsts a unque arg(µ κ ) [π 3π ) such that ψκ u arg(µ κ ) = θd c θc d < θ < mn arg(µ) [π 3π ) κ (ψ u arg(µ )) then the vehcles wll eventually move along logarthmc spral curves wth center gven by (4) growng rate Re(σ s ) where σ s = + +4λ s wth λ s = µ κ e ι θ π perod Im(σ. s) The radus of the logarthmc spral curve for vehcle s w κ() p T s [r() T v() T ] T e Re(σ s)t [ a + a 3 sn ] ( θ ) where (σs + )(ν p s = κ ) (σ s +)νκ T w κ T ς. The relatve ν κ radus of the logarthmc spral curves s equal to the relatve magntude of w κ(). The relatve phase of the vehcles on ther curves s equal to the relatve phase of w κ(). The curves are on a plane perpendcular to Euler axs a. Proof: ) For the frst statement f C = I 3 then () usng (3) can be wrtten n matrx form as [ṙ ] n n I = n r I v L I 3 (5) n v }{{} Γ where r = [r T...r T n] T v = [v T...v T n ] T. It follows from the proof of Theorem 5. n [] that the vehcles wll eventually rendezvous f only f Γ defned n (5) has a smple zero egenvalue all other egenvalues have negatve real parts. Note from Lemma 3.4 that each egenvalue µ of L corresponds to two egenvalues of Γ gven by ζ = + +4µ [ wth ] assocated [ rght ] w left egenvectors gven by (ζ + )ν ζ w ν respectvely ζ = +4µ [ ] wth assocated [ rght ] w (ζ + )ν left egenvectors gven by ζ w ν respectvely where =...n. Because weghted drected graph G has a drected spannng tree t follows from Lemma 3. that L has a smple zero egenvalue all other egenvalues have negatve real parts. Accordng to Defnton 3. we let µ = Re(µ ) < =...n. Note from Lemma 3. that w = n ν = p. It thus follows that ζ [ = ] wth n assocated rght left egenvectors gven by n p respectvely ζ p =. Note that ζ < f >. Also notng that all + 4µ have nonnegatve real parts t follows that all ζ =...n have negatve real parts f >. It s left to show condtons under whch ζ =...n have negatve real parts. Suppose that s the crtcal value for such that ζ =...n s on the magnary axs. Let ζ = η ι where η R =...n. After some manpulaton t follows that = µ sn (arg(µ )) µ sn(arg(µ)) cos(arg(µ )) η = =...n. It s straghtforward to verfy that f > (respectvely < ) then ζ =...n has a negatve (respectvely postve) real part. Therefore all ζ =...n have negatve real parts f only f > max =...n. Combnng the above arguments shows that Γ has a smple zero egenvalue all other egenvalues have negatve real parts f only f > c. Note that Γ can be wrtten n Jordan canoncal form as SJS where the columns of S denoted by s k k =...n can be chosen to be the rght egenvectors or generalzed rght egenvectors of Γ assocated wth egenvalue ζ k k =...n the rows of S denoted by h T k k =...n can be chosen to be the left egenvectors or generalzed left egenvectors of Γ assocated wth egenvalue ζ k such that h T k s k = h T k s l = k l J s the Jordan block dagonal matrx wth ζ k beng the dagonal entres. We can choose s = [ T n T n] T h = [p T pt [] T. ] Note that h T s = r(t). It thus follows that lm t = lm t (e Γt v(t) [( ] r() n [p T r() I 3 ) = v() pt]) I 3 whch n v() mples that x (t) p T x()+ pt v x () y (t) p T y()+ pt v y () z (t) p T z() + pt v z () v x (t) v y (t) v z (t) as t. Equvalently t follows that all vehcles wll eventually rendezvous at the poston gven by (4). ) For the second statement usng (3) () can be wrtten n matrx form as [ṙ ] = v [ 3n 3n I 3n r. (6) (L R) I 3n v] }{{} Σ It follows from Lemmas Defnton 3. that the egenvalues of (L R) are µ µ e ιθ µ e ιθ wth assocated rght egenvectors w ς w ς w ς 3 respectvely assocated left egenvectors ν ν ν 3 respectvely. That s the egenvalues of (L R) correspond to the egenvalues of L rotated by angles θ θ respectvely. Let λ l l =...3n denote the lth egenvalue of (L R). Wthout loss of generalty let λ 3 = µ λ 3 = µ e ιθ λ 3 = µ e ιθ =...n be the egenvalues of (L R). Note from Lemma 3.4 that each λ k corresponds to two egenvalues of Σ defned n (6) gven by σ k k = ± +4λ k k =...3n. Because µ = t follows that λ = λ = λ 3 = whch n turn mples that σ = σ 3 = σ 5 = σ = σ 4 = σ 6 =. Smlar to the proof of the frst statement all σ k k =...3n have negatve real parts f >. Gven > χ = µ e ιarg(χ) =...n ψ l ψu are the crtcal values for arg(χ ) [ π) such that + +4χ s on the magnary axs. In partcular f arg(χ ) = ψ l (respectvely ψ u) then + +4χ = ι µ sn(arg(ψ l ) TuB. (respectvely 4

4 47th IEEE CDC Cancun Mexco Dec. 9-8 TuB. ι µ sn(arg(ψu ) ) =...n. If arg(χ ) (ψ lψu ) (respectvely arg(χ ) [ψ l) (ψu π)) then + +4χ have negatve (respectvely postve) real parts. Because > c + the frst statement mples that all +4µ =...n have negatve real parts whch n turn mples that arg(µ ) (ψ lψu ) =...n. If θ < θc d then arg(λ 3 ) arg(λ 3 ) arg(λ 3 ) are all wthn (ψ lψu ) whch mples that σ 6 5 σ 6 3 σ 6 =...n all have negatve real parts. Therefore f θ < θd c then Σ has exactly three zero egenvalues all other egenvalues have negatve real parts. Smlar to the proof of the frst statement we wrte Σ n Jordan canoncal form as MJM where the columns of M denoted by m k k =...6n can be chosen to be the rght egenvectors or generalzed rght egenvectors of Σ assocated wth egenvalue σ k the rows of M denoted by p T k k =...6n can be chosen to be the left egenvectors or generalzed left egenvectors of Σ assocated wth egenvalue σ k such that p T k m k = p T k m l = k l J s the Jordan block dagonal matrx wth σ k beng the dagonal entres. Note that the rght left egenvectors of (L R) assocated wth egenvalue λ l = are respectvely n ς l p l where l = 3. It n turn follows from Lemma 3.4 that the rght left egenvectors of Σ assocated [ wth ] σ l = n ς are respectvely l p l where l = 3n [ p l ] n ς 3. We can choose m l = l p l = 3n p l Tl ς l where l = 3. Note that p p T l l m l = Tl ς l p T l m k = where k l = 3 k [ l. Notng ] r(t) that σ l = l = 3 t follows that lm t = v(t) r() lm t Me Jt M ( 3 r() v() l= m l p T l ) v() whch mples that x (t) p T x() + pt v x () y (t) p T y()+ pt v y () z (t) p T z()+ pt v z () v x (t) v y (t) v z (t) as t. Equvalently t follows that all vehcles wll eventually rendezvous at the poston gven by (4). 3) For the thrd statement f θ = θd c (respectvely θ = θd c) there exsts a unque arg(µ κ) [π 3π ) such that ψκ u arg(µ κ ) = θd c then λ 3κ = µ κ e ιθ = µ κ e ιψu κ (respectvely λ 3κ = µ κ e ιθ = µ κ e ιψu κ ) whch mples that σ 6κ 3 = + +4λ 3κ = ι µ κ sn(ψ u κ ) (respectvely σ 6κ = + +4λ 3κ = ι µ κ sn(ψ u κ ) ). Notng that the complex egenvalues of Σ are n pars t follows that Σ has an egenvalue equal to σ 6κ 3 = ι µκ sn(ψu κ ) (respectvely σ 6κ = ι µ κ sn(ψ u κ ) ) denoted by σ for smplcty. In ths case Σ has exactly three zero egenvalues two nonzero egenvalues on the magnary axs all other egenvalues have negatve real parts. In the followng we focus on θ = θd c snce the analyss for θ = θc d s smlar except that all vehcles wll move n reverse drectons. Note from Lemma 3.4 that the rght left egenvectors [ assocated ] wth σ[ 6κ 3 are respectvely ] w κ ς (σ6κ 3 + )(ν κ ). σ 6κ 3 (w κ ς ) [ ν κ ] w We can choose m 6κ 3 = κ ς [ σ 6κ 3 (w κ ς ) ] (σ6κ 3 + )(ν p 6κ 3 = κ ) (σ 6κ 3 +)νκ T w κ T ς. ν κ Note that p T 6κ 3m 6κ 3 =. Smlarly t follows that m p correspondng to σ are gven by m ] = m 6κ 3 ] p = p 6κ 3. It follows ] that [ r(t) v(t) = e Σt [ r() v() ( 3 l= m l p T l ) [ r() v() + c(t) for large t where c(t) = (e ι µκ sn(ψu κ ) t m 6κ 3 p T 6κ 3 + e ι µ κ sn(ψκ u ) r() m p T ). Let c v() k (t) be the kth component of c(t) k =...6n. It follows that c 3( )+l (t) = Re(e ι µκ sn(ψu κ ) t w κ() ς (l) p T 6κ 3[r() T v() T ] T ) where =...n l = 3 ς (l) denotes the lth component of ς. After some manpulaton t follows that c 3( )+l (t) = ς (l) w κ() p T 6κ 3[r() T v() T ] T cos( µκ sn(ψu κ ) t + arg(w κ() p T 6κ 3[r() T v() T ] T ) + arg(ς (l) )) =...n l = 3. Therefore t follows that x (t) p T x() + pt v x () + c 3 (t) y (t) p T y() + pt v y () + c 3 (t) z (t) p T z() + pt v z () + c 3 (t) for large t. After some manpulaton t can be verfed that [c 3 (t)c 3 (t)c 3 (t)] T = w κ() p T 6κ 3[r() T v() T ] T a + a 3 sn ( θ ) whch s a constant. Therefore t follows that all vehcles wll eventually move on crcular orbts wth center gve by (4) π perod µ κ sn(ψκ u ). The radus of the orbt for vehcle s gven by w κ() p T 6κ 3[r() T v() T ] T a + a 3 sn ( θ ). The relatve radus of the orbts s equal to the relatve magntude of w κ(). In addton the relatve phase of the vehcles s equal to the relatve phase of w κ(). Note from Lemma 3.3 that Euler axs a s orthogonal to both Re(ς ) Im(ς ) where Re( ) Im( ) representng respectvely the real magnary part of a number are appled componentwse. It can thus be verfed that a s orthogonal to [c 3 (t) c 3 (t)c 3 (t)] T whch mples that the crcular orbts are on a plane perpendcular to a. 4) For the fourth statement f there exsts a unque arg(µ κ ) [π 3π ) such that ψu κ arg(µ κ ) = θ c d θ c d < θ < mn arg(µ ) [π 3π ) κ (ψ u arg(µ )) (respectvely mn arg(µ) [π 3π ) κ (ψ u arg(µ )) < θ < θ c d ) then λ 3κ = µ κ e ιθ = µ κ e ι(arg(µ κ)+θ) (respectvely λ 3κ = µ κ e ιθ = µ κ e ι(arg(µκ) θ) ) where arg(µ κ ) + θ > ψ u κ (respectvely arg(µ κ ) θ > ψ u κ) whch mples that σ 6κ 3 = + +4λ 3κ (respectvely σ 6κ = + +4λ 3κ ) has a postve real part. A smlar argument as above shows that Σ has exactly three zero egenvalues two egenvalues wth postve real parts all other egenvalues have negatve real parts. By followng a smlar procedure to the proof of the thrd statement we can show that all vehcles wll eventually move along 5

5 47th IEEE CDC Cancun Mexco Dec. 9-8 TuB. logarthmc spral curves wth center gven by (4) growng π rate Re(σ 6κ 3 ) perod Im(σ 6κ 3 ). The radus of the logarthmc spral curve for vehcle s gven by w κ() p T 6κ 3[r() T v() T ] T e Re(σ 6κ 3)t a + a 3 sn ( θ ). The relatve radus of the logarthmc spral curves s equal to the relatve magntude of w κ(). In addton the relatve phase of the vehcles on ther curves s equal to the relatve phase of w κ(). A smlar argument to that for the thrd statement shows that the curves are on a plane perpendcular to Euler axs a. Remark 3.3: Note that the frst statement of Theorem 3. generalzes Theorem 5. n [] whch gves only a suffcent condton for by gvng a necessary suffcent condton. Unlke the sngle-ntegrator case the crtcal value for the Euler angle for double-ntegrator dynamcs depend on both L. The crtcal value for the Euler angle n the double-ntegrator case s smaller than that for the sngle-ntegrator case. When ncreases to nfnty the crtcal value for the Euler angle n the double-ntegrator case approaches that for the sngle-ntegrator case. Note that besdes the network topology the Euler angle plays an mportant role n (3). Example 3.4: To llustrate consder four vehcles wth network topology G shown by Fg.. Let L assocated wth G be gven by (7) It can be computed that θd c =.3557 rad. Let R be the rotaton matrx correspondng to Euler axs a = 4 [ 3]T Euler angle θ = θd c. Fg. shows the egenvalues of L (L R). Note that the egenvalues of (L R) correspond to the egenvalues of L rotated by angles θ θ. Fg. 3 shows the egenvalues of Σ. Note that each egenvalue of (L R) λ k correspond to two egenvalues of Σ σ k k where σ k k = ± +4λ k k =... Because θ = θd c two nonzero egenvalues of Σ are located on the magnary axs as shown n Fg Fg.. Network topology for four vehcles. An arrow from j to denotes that vehcle can receve nformaton from vehcle j. IV. SIMULATION In ths secton we study collectve motons of four vehcles usng (3). Suppose that the network topology s gven by Fg. L s gven by (7). Let θs c θd c a be gven n Example 3.4. Usng (3) t can be computed that c =.366. We let = c +.5. Also note that there exsts Imagnary part θ θ Real part Fg.. Egenvalues of L (L R) wth θ = θd c. Crcles denote the egenvalues of L whle x-marks denote the egenvalues of (L R). The egenvalues of (L R) correspond to the egenvalues of L rotated by angles θ θ respectvely. In partcular the egenvalues obtaned by rotatng µ 4 by angles θ θ are shown by respectvely the sold lne the dashed lne the dashdot lne. a unque arg(µ 4 ) [π 3π ) such that ψu 4 arg(µ 4 ) = θd c (.e. κ = 4 n Theorem 3.). Note that the rght egenvector of L assocated wth egenvalue µ 4 s w 4 = [.847.8ι ι ι] T. Also note that p = [ ] T. Fgs show respectvely the trajectores of the four vehcles usng (3) wth θ = θd c. θ = θc d θ = θd c +.. Note that all vehcles eventually rendezvous at the poston gven by (4) when θ = θd c. move on crcular orbts when θ = θd c move along logarthmc spral curves when θ = θd c +.. Also note that when θ = θd c the relatve radus of the crcular orbts (respectvely the relatve phase of the vehcles) s equal to the relatve magntude (respectvely phase) of the components of w 4. In addton the trajectores of all vehcles are perpendcular to Euler axs a n all cases. By comparng the smulaton results wth those n [9] we note that the crtcal values for the Euler angle the perod of the crcular moton the perod growng rate of the logarthmc spral moton are qute dfferent n both cases even f the network topology L are chosen to the same n both cases. V. CONCLUSION We have ntroduced Cartesan coordnate couplng to a consensus algorthm by a rotaton matrx n 3D for doublentegrator dynamcs. The results generalze the results presented n the frst part [9] of the two-part paper exstng results on consensus algorthms for double-ntegrator dynamcs. We have shown that the network topology the dampng gan the value of the Euler angle all affect the resultng collectve motons quanttatvely characterze the resultng collectve motons. Smulaton results have shown the effectveness of theoretcal results. 6

6 47th IEEE CDC Cancun Mexco Dec. 9-8 TuB..5 Vehcle Vehcle Vehcle 3 Vehcle Imagnary part Real part Fg. 3. Egenvalues of Σ wth θ = θd c. Squares denote the egenvalues computed by σ k = + +4λ k whle damonds denote the egenvalues computed by σ k = +4λ k k =.... In partcular the egenvalues of Σ correspond to λ = µ 4 λ = µ 4 e ιθ λ = µ 4 e ιθ are shown by respectvely the sold lne the dashed lne the dashdot lne. Because θ = θd c two nonzero egenvalues of Σ are on the magnary axs. 3 Fg. 5. Trajectores of the four vehcles usng (3) wth θ = θd c. Crcles denote the startng postons of the vehcles whle the squares denote the snapshots of the vehcles at 3 sec. 4 3 Vehcle Vehcle Vehcle 3 Vehcle Vehcle Vehcle Vehcle 3 Vehcle Fg. 4. Trajectores of the four vehcles usng (3) wth θ = θd c.. Crcles denote the startng postons of the vehcles whle the squares denote the snapshots of the vehcles at 3 sec. REFERENCES [] W. Ren E. M. Atkns Dstrbuted mult-vehcle coordnated control va local nformaton exchange Internatonal Journal of Robust Nonlnear Control vol. 7 no. pp. 33 July 7. [] G. Xe L. Wang Consensus control for a class of networks of dynamc agents Internatonal Journal of Robust Nonlnear Control vol. 7 no. - pp July 7. [3] G. Lafferrere A. Wllams J. Caughman J. J. P. Veerman Decentralzed control of vehcle formatons Systems Control Letters vol. 54 no. 9 pp [4] R. Olfat-Saber Flockng for mult-agent dynamc systems: Algorthms theory IEEE Transactons on Automatc Control vol. 5 no. 3 pp. 4 4 March 6. [5] H. G. Tanner A. Jadbabae G. J. Pappas Flockng n fxed Fg. 6. Trajectores of the four vehcles usng (3) wth θ = θd c +.. Crcles denote the startng postons of the vehcles whle the squares denote the snapshots of the vehcles at sec. swtchng networks IEEE Transactons on Automatc Control vol. 5 no. 5 pp May 7. [6] D. Lee M. W. Spong Stable flockng of multple nertal agents on balanced graphs IEEE Transactons on Automatc Control vol. 5 no. 8 pp August 7. [7] M. Pavone E. Frazzol Decentralzed polces for geometrc pattern formaton path coverage ASME Journal of Dynamc Systems Measurement Control vol. 9 no. 5 pp September 7. [8] W. Ren Collectve moton from consensus wth Cartesan coordnate couplng - part : Sngle-ntegrator knematcs n Proceedngs of the IEEE Conference on Decson Control 8. [9] On consensus algorthms for double-ntegrator dynamcs n Proceedngs of the IEEE Conference on Decson Control New Orleans LA December 7 pp [] A. J. Laub Matrx Analyss for Scentsts Engneers. Phladelpha PA: SIAM 5. [] W. Ren R. W. Beard Consensus seekng n multagent systems under dynamcally changng nteracton topologes IEEE Transactons on Automatc Control vol. 5 no. 5 pp May 5. []