> σ, σ j, j σ j, σ j j σ σ j σ j (t) = σ (t ) σ j (t) = σ () j(t ) n j σ, σ j R lm σ = σ j, j V (8) t σ R σ d R lm σ = σ d V (9) t Fg.. Communcaton ln

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1 IIC-- Dstrbuted Cooperatve Atttude Control for Multple Rgd Bodes wth Communcaton Delay Yoshhro achbana, oru Namerkawa (Keo Unversty) Abstract hs paper descrbes dstrbuted cooperatve atttude consensus and synchronzaton for multple rgd bodes wth communcaton delay. Mult-agent system s composed of multple autonomous agents and they exchanges nformaton for each other. If there are tme delays n the communcaton lnes, the system may become unstable and may not acheve the control objectve. he atttudes are represented by modfed Rodrguez parameters. Frst, we show the proposed control law guarantees consensus among agents. Second, t can be shown that atttude synchronzaton to reference atttude can be guaranteed va the proposed consensus control law. Fnally, smulaton results show effectveness of the two proposed control laws. (mult-agent system, communcaton delay, rgd body, atttude synchronzaton ). (MAS) MAS () () () () () () MAS MAS (7) (8) (9) () () () () () (9) (MRP) (). v = v v v R v v v = v v () v v w = w w w R v w = v w a, b R a b a =, a a = (MRP) MRP σ R λ R θ R σ = λ tan θ () () () σ = F(σ )ω J ω = ω J ω + τ F(σ ) = σ σ I + σ + σ σ () ω R J R τ R /

2 > σ, σ j, j σ j, σ j j σ σ j σ j (t) = σ (t ) σ j (t) = σ () j(t ) n j σ, σ j R lm σ = σ j, j V (8) t σ R σ d R lm σ = σ d V (9) t Fg.. Communcaton lne n v V = {v, v,, v n } E V V (v j,v ) v j v N = {v j V : (v j, v ) E} v v j v v j v v j A D L A = a j (v j N ) a j = () (v j N ) D v d = N D = dag(d, d,, d n ) () MAS L L = D A (7) () L n = n = R n () L. () () τ (t) = F k j (σ (t)) a j (σ (t) σ j (t )) + κ σ (t) d () j= κ R > R > A = a j k j d a j = a j, k j = k j κ > k j () n () () MAS Proof. σ (t) σ (t ) σ (t ) = σ (t) σ j (t) σ (t + µ)dµ () σ j (t) = σ (t) σ j (t) () () τ (t) = F (σ (t)) k j a j (σ j (t) + d j= σ j (t + µ)dµ) + κ σ (t) () (t) V = κ + d ω J ω + κ = = j= k ja j = j= µ k j a j σ jσ j σ (t + s) σ (t + s)dsdµ() /

3 V ω, σ j, σ c > {(ω, σ j, σ ) V c } ω, σ j, σ V t V = κ d ω J ω + κ k j a j σ jσ j = + = j= = κ d ω + = + = = = j= k ja j ( σ σ σ (t + µ) σ (t + µ) ) dµ ( F (σ ) j= k j d a j (σ j σ j (t + µ)dµ) + κ σ ) + κ k ja j j= k ja j j= = κ d σ σ + κ = = = = κ κ = = k j a j σ j= k ja j j= σ σ dµ = k j a j σ σ j j= σ (t + µ) σ (t + µ)dµ = j= a j σ σ + j= k j a j σ j= k ja j σ σ σ j (t + µ)dµ σ (t + µ) σ (t + µ)dµ = k ja j σ σ j= σ j (t + µ)dµ k ja j σ (t + µ) σ (t + µ)dµ = j= = a j (κ k j ) σ σ = j= ( κ a j = j= σ σ + κk j σ σ j (t + µ) ) +k j σ j (t + µ) σ j (t + µ) dµ = a j (κ k j ) σ σ = j= ( ) κ a j σ + k j σ j (t + µ) = j= ( ) κ σ + k j σ j (t + µ) dµ () κ > k j > V V σ σ = F(σ )ω ω () τ σ, τ σ, τ () τ = F (σ) ( L I )σ + ( A I ) σ(t + µ)dµ + κ σ () F (σ) = dagf (σ ),, F n (σ n ) A = k j d a j L A σ σ τ τ () F (σ) ( L I )σ + ( A I ) σ(t + µ)dµ (7) ( A I ) t σ(µ)dµ σ(t) t t = µ σ(µ) ( A I ) t σ(µ)dµ (7) t ( L I )σ (8) σ = σ = = σ n t σ (t) σ j (t), ω (t) () τ (t) = F k j (σ (t)) a j (σ (t) σ j (t )) d j= +k (n+) a (n+) (σ (t) σ d ) + κ σ (t) (9) σ d a (n+) k (n+) κ > k j σ d (9) n () () MAS Proof. (), () (9) τ (t) = F k j (σ (t)) a j (σ j (t) d j= + σ j (t + µ)dµ) + k (n+) a (n+) (σ (t) σ d ) + κ σ (t) () (t) V = κ d ω J ω + κ k j a j σ jσ j = = j= + κ d k (n+) a (n+) (σ σ d ) (σ σ d ) + = = j= k ja j µ σ (t + s) σ (t + s)dsdµ() V ω, σ j, σ σ d, σ c > {(ω, σ j, σ σ d, σ ) V c } ω, σ j, σ σ d, σ V t V = κ d ω J ω + κ k j a j σ jσ j = = j= +κ d k (n+) a (n+) σ (σ σ d ) = /

4 + k ja j ( σ σ σ (t + µ) σ (t + µ) ) dµ = j= = a j (κ k j ) σ σ = j= ( ) κ a j σ + k j σ j (t + µ) = j= ( ) κ σ + k j σ j (t + µ) dµ () κ > k j > V V σ σ = F(σ )ω ω () τ σ, τ σ, τ () τ = F (σ) (M I )(σ n σ d ) +( A I ) σ(t + µ)dµ + κ σ () κ > k j =. (), κ =.,. able. Rgd body specfcatons J kgm.9..;...;...9 J kgm...;...;... J kgm..7.;.7..;... J kgm.9..;...7;..7. J kgm...;...;... J kgm...;..9.;...7 M = L + dagk (n+) a (n+),, k n(n+) a n(n+) σ σ τ τ () F (σ) (M I )(σ n σ d ) +( A I ) σ(t + µ)dµ () ( A I ) t σ(µ)dµ σ(t) t t = µ σ(µ) ( A I ) t σ(µ)dµ () t Fg.. Communcaton topology (M I )(σ n σ d ) () M z R z = z,, z n z ( L I )z = z = z j σ d k (n+) a (n+) > z (M I )z = z ( L I )z + n = k (n+) a (n+) z z z (M I )z = z = M M I σ n σ d t σ (t) σ d, ω (t) Fg.. Intal atttudes of rgd bodes. n = a 7 = a 7 = ( ) σ d k j = k 7 = σ d =... =.s κ 8, κ =., 7 κ =. 8 σ ( j) ω ( j) ( j) σ ω j,, 7 κ /

5 8 9 # σ d κ, 9, κ σ d =... σ (). σ ().. σ ()... σ () σ () σ () (κ =.) Fg.. Atttude (κ =.) ω () (κ =.) Fg.. Atttude (κ =.).. ω (). ω ()... 7 (κ =.) Fg. 7. Angular velocty (κ =.) ω () ω () ω ().... (κ =.) Fg.. Angular velocty (κ =.) Fg Fnal atttudes of rgd bodes. /

6 σ ().. σ ().. σ () ω (). 9 (κ =.) Fg. 9. Atttude (κ =.). ω () ω ().... (κ =.) Fg.. Angular velocty (κ =.) Fg.. Fnal atttudes of rgd bodes. Namerkawa: Consensus Problem for Mult-agent Systems and Cooperatve Catpurng Behavor, Systems, Control and Infomaton, Vol., No., pp. -8 (9) :, //, Vol., No., pp. -8 (9) C. Yoshoka and. Namerkawa: Consensus Problem for Mult-agent System and Its Applcaton to Formaton Control, ransactons of the Socety of Instrument and Control Engneers, Vol., No.8, pp. -9 (8) :,, Vol., No.8, pp. -9 (8) J.. Y. Wen and K. Kreutz-Delgado: he atttude control problem, IEEE ransactons on Automatc Control, vol., no., pp. 8- (99) W. Ren: Dstrbuted atttude algnment n spacecraft formaton flyng, Internatonal Journal of Adaptve Control and Sgnal Processng, vol., no. -, pp. 9- (7) J.-J. E. Slotne and M. D. D. Benedetto: Hamltonan adaptve control of spacecraft, IEEE ransactons on Automatc Control, vol., no. 7, pp (99) W. Ren: Dstrbuted Cooperatve Atttude Synchronzaton and rackng for Multple Rgd Bodes, IEEE ransactons on Control Systems echnology, vol. 8, no., pp. 8-9 () 7 K. Peng and Y. Yang: Leader-followng consensus problem wth a varyng-velocty leader and tme-varyng delays, Physca A: Statstcal Mechancs and ts Applcatons, Vol. 88, pp. 9-8 (9) 8 J. Hu and Y. Hong: Leader-followng coordnaton of multagent systems wth couplng delays, Physca A: Statstcal Mechancs and ts Applcatons, Vol. 7, pp. 8-8 (7) 9 U. Munz, A. Papachrstodoulou and F. Allgöwer: Delay- Dependent Rendezvous and Flockng of Large Scale Mult- Agent Systems wth Communcaton Delays, Proceedngs of the 7th IEEE Conference on Decson and Control, pp. 8- (8) H. Wang and Y. Xe: On Atttude Synchronzaton of Multple Rgd Bodes wth me Delays Preprnts of the 8th IFAC World Congress, pp () Z. Meng, Z. You, G. L, and C. Fan: Cooperatve Atttude Control of Multple Rgd Bodes wth Multple me-varyng Delays and Dynamcally Changng opologes, Hndaw Publshng Corporaton, Mathematcal Problems n Engneerng, vol., do:.//9 () Y. Igarash,. Hatanaka, M. Fujta and M. W. Spong: Passvty-Based Atttude Synchronzaton n S E(), IEEE ransactons on Control Systems echnology, vol. 7, no., pp.9- (9) /

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10_4.dvi Vol.44, No.1, 1/7 28 Synchronzed Control for Blateral Teleoperaton wth Dfferent Confguratons and Communcaton Delays Hsanosuke Kawada,KoueYoshda and Toru Namerkawa Ths paper addresses the problem of the