Geometry of Steering Laws in Cooperative Control

Transcription

1 Geomet of Steein Laws in Coopeative Contol Eic W. Jsth, P.S. Kishnapasad, F. Zhan Institte fo Sstems Reseach & ECE Depatment Univesit of Maland Collee Pak, MD 74 Wokshop on Lie Gop Methods and Contol Theo Intenational Cente fo Mathematical Sciences 4, India Steet Edinbh, Scotland, Jne 9, 4 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

2 James Clek Mawell (83-879) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

3 Mawell and Contol Mawell s (857) essa on Satn s ins contained the fist se of the chaacteistic polnomial to assess stabilit. This late led to his se of the same method in desinin a speed oveno a phsical feedback contol device in a poblem of accate electical measements. Mawell stdied the poblem of Satn s ins nde Newtonian attaction. Coheence of a in of atificial eath satellites can be achieved b snthetic inteactions (ealized via feedback contol laws). Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

4 Mawell and Contol It is etaodina that, as we ae atheed hee to discss eometic inteation and contol, the spacecaft Cassini-Hens is takin the closest look eve at Satn s ins (and shephed moons), a timph of action at a distance via pecise contol. New data on bendin waves, densit waves and spokes in the in sstem ma lead to fthe theoetical wok on the poblem of Satn s ins. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

5 Mawell and Goscopic Inteaction Mawell s eqations fo the electomanetic field ae complemented b the eqation of Loentz fo the foce on a chaed paticle in an electomanetic field. In a eion whee the electic field vanishes, the paticle motion is ovened b a pel oscopic Laanian iven in tems of the manetic vecto potential. Goscopic foces leave the kinetic ene invaiant. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

6 O sbect This talk is abot the eploitation of pel oscopic inteactions to achieve coheent pattens in the motion of paticles. We do this in the lanae most natal to the poblem, the lanae of movin fames. The net few slides constitte a bief eview of some aspects of movin fames. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

7 Fenet-Seet Fame Rela Cve and s γ () s C 3 γ B N T Fenet-Seet eqations γ () s Ts () T () s κ() s N() s N () s κ() st() s + τ() s B() s Bs () τ () sns () Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

8 Natal Fenet o Relativel Paallel Adapted Fame T Rela Cve M M s s γ () s C T Eqations fo elativel paallel adapted fame M M γ () s T() s T () s k () s M () s + k () s M () s M () s k () s T() s M () s k () s T() s s R. L. Bishop, Ame. Math. Month. (975), 8(3):46-5 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

9 Contol Sstem on SE(3) γ γ ξ e whee [ TNB] o [ TMM ] and κ ξ κ τ τ o k k k k and e Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

10 Invesion κ() s γ () s τ () s ( ) ( κ() s ) γ () s γ () s γ () s (FS) k () s γ () s M () k ( σγ ) () s γ ( σ) dσ i i i i, s (RPAF) κ () s k () s + k () s θ () s τ (whee θ a ( k ) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

11 Nomal (k-plane) Development of γ Fame inteaction Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

12 Isokinetic Motion F q( Ε + υ Β) υ Ε L # υυ + q υα,, B Α Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

13 Otline Motivation: Flin in spatial pattens (fomations) Model: Inteaction laws fo nit-speed paticles Fomations as shape eqilibia Conveence to specific fomations - Analsis fo two paticles (vehicles) - Simlations fo n vehicles Inteaction with sfaces - obstacle-avoidance and bondafollowin Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

14 Acknowledements Collaboatos: Jeff Hee, La Schette, David Tempe Naval Reseach Laboato Fndin: NRL: Motion Plannin and Contol of Small Aile Fomations AFOSR: Dnamics and Contol of Aile Fomations ARO: Commnicatin Netwoked Contol Sstems Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

15 Red Aows Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

16 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Plana Model (Natal Fenet Fames),,..., n ae cvate (i.e., steein) contol inpts. n n n n n n n n n n n Unit speed assmption Specifin,,..., n as feedback fnctions of (,, ), (,, ),..., ( n, n, n ) defines a contol law.

17 n 3-D Model (Natal Fenet Fames) n z n z n z Unit speed assmption The natal cvates (,v ), (,v ),...,( n,v n ) ae contols. + z v z n n v + z z v + z z n n n n n n n n v n n n Relativel paallel adapted fames in the sense of R. L. Bishop, Ame. Math. Month. (975), 8(3):46-5 v v Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

18 Chaacteization of Eqilibim Shapes Poposition (Jsth, Kishnapasad): Sppose the contols,,, n ae invaiant nde iid motions in the plane. Fo eqilibim shapes (i.e. elative eqilibia of the nedced dnamics),... n, and thee ae onl two possibilities: (a)... n : all vehicles head in the same diection (with abita elative positions), o (b)... n : all vehicles move on the same cicla obit (with abita chodal distances between them). (a) 5 4 (b) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

19 3-D Eqilibim Shapes The contol laws ae assmed to be invaiant nde iid motions in theedimensional space. Shape vaiables capte elative distances and anles between vehicles. Shape eqilibia coespond to stead-state fomations ectilinea fomation 5 ciclin fomation 3 helical fomation Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

20 Pai Eqilibia Rectilinea fomation (motion pependicla to the baseline) Collinea fomation Ciclin fomation (sepaation eqals the diamete of the obit) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

21 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Plana Contol Law fo paticles ) ( + µ η f ) ( + µ η f > > η µ f( )

22 ρ φ Chane of vaiables Dot podcts can be epessed as sines and cosines in the new vaiables: sinφ cosφ sinφ cosφ φ sin( φ φ ) sin( φ φ ) Sstem afte chane of vaiables: ρ sinφ sinφ φ η sinφ cosφ + f ( ρ)cosφ + µ sin( φ φ) + (/ ρ)(cosφ cosφ) φ η sinφ cosφ f ( ρ)cosφ + µ sin( φ φ) + (/ ρ)(cosφ cosφ ) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

23 Lapnov Fnction V ln(cos( ) ( ) φ ) pai φ φ + + h ρ ρ φ A Lapnov fnction is whee f (ρ) dh/dρ. The deivative of V pai with espect to time alon taectoies of the sstem is V pai V φ. pai V φ + ϕ pai This Lapnov fnction is the ke to povin a conveence eslt fo the two-paticle sstem. V pai φ + ρ ρ Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

24 Conveence eslt O Lapnov fnction mst be adiall nbonded, meanin V pai as ρ and as ρ. (Some mino technical assmptions ae also needed.) Poposition (Jsth, Kishnapasad): Fo an initial condition satisfin φ - φ π and ρ >, the sstem convees to the set of eqilibia, which has the fom {( ρ, π, π ), ρ > } {( ρ, π, π ), ρ > } (,,) { ρ e f ( ρ ) } e Poof: Uses LaSalle s Invaiance Pinciple. Eamples of (elative) eqilibia: ρ e ρ e Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

25 Intition Steein contol eqation fo vehicle #: η f ( ) + µ Alin each vehicle pependicla to the baseline between the vehicles. Stee towad o awa fom the othe vehicle to maintain appopiate sepaation. Alin with the othe vehicle s headin. Bioloical analo (swamin, schoolin): - Deceasin esponsiveness at lae sepaation distances. - Switch fom attaction to eplsion based on sepaation distance o densit. - Mechanism fo alinment of headins. D. Günbam, Schoolin as a state fo tais in a nois envionment, in Animal Gops in Thee Dimensions, J.K. Paish and W.M. Hamne, eds., Cambide Univesit Pess, 997. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

26 Ke ideas fo two-vehicle poblem Unit-speed motion with steein contol. - Goscopic foces peseve kinetic ene of each paticle. - In mechanics, oscopic foces ae associated with vecto potentials. Shape vaiables: elative distances and anles. Lapnov fnction conveence eslt fo the shape dnamics. Eqilibia of the shape dnamics elative eqilibia of the vehicle dnamics. Paticle e-labelin smmet. Lie op fomlation: - The dnamics of each paticle can be epessed as a left-invaiant sstem evolvin on SE(), the op of iid motions in the plane. - GSE() is a smmet op fo the dnamics: the contol law is invaiant nde iid motions of the entie fomation. -V pai is also invaiant nde G. Theefoe, we can conside the edced sstem evolvin on shape space (G G)/G G. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

27 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Goscopic Foces and Vecto Potentials.,, R, ) ( T T n K K M M K M L > + Conside the Laanian: Ele-Laane eqations: ( ), ) ( ) ( K Q M K M K M dt d L L dt d T T Kinetic ene Scala potential Vecto potential (linea-in-velocit tem) ). ( ) ( ) ( Q Q Q T T ( ) Note: Laanian with linea-in-velocit tem skew tem in the dnamics, bt the convese onl holds if Q() can be epessed as in ( ) fo some ().

28 Goscopicall inteactin paticles Fo a sinle paticle: position, v, a, m mass, H kinetic ene v, ma F. Note: F is a oscopic foce (Recall Loentz foce law fo a chaed paticle in a manetic field) Note that ma be a complicated fnction of time, and ma involve feedback. H, H H θ. cos θ, sin θ Restict to the level-set of H iven b H/. Fenet-Seet eqations Fo mltiple paticles, the kinetic ene of each paticle is conseved, and the paticles inteact via oscopic foces. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

29 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Shape space fo n paticles....,,, n....,,, n....,,, (),, n se + ξ ξ Dnamics Gop vaiables,,..., n G SE(), the op of iid motions in the plane. copies n G G G S Confiation space.,...,, ~ n Shape space copies n G G G R Fenet-Seet Eqations Assme the contols,,..., n ae fnctions of shape vaiables onl. Shape vaiables capte elative positions and oientations. Shape vaiables

30 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Two-paticle law: Lie op settin Dnamics on confiation space SG G, whee GSE():,, cos sin sin cos cos sin sin cos θ θ θ θ θ θ θ θ ()., se Ad ξ ξ ξ ξ ξ ξ., A A ( ). A A + ξ ( ). A A + ξ (), se ξ ξ Shape vaiable: Dnamics on shape space RG: Contols as fnctions of the shape vaiable : ( ) ]. [, ) ( ) (, ], [, ) ( ) ( i i f f µ η µ η ( ). V V pai pai

31 Lapnov Fnction V ( ) ln + + h( ) Penalize headin-diection misalinment o h(ρ) f (ρ) ρ Penalize inte-vehicle distances which ae too lae o small V depends onl on shape vaiables. Idea: show that fo sitable choice of contol law, dv/dt. Fo two vehicles, lobal conveence eslts ae obtained (Jsth and Kishnapasad, TR, CDC 3, SCL 4, CDC 4) Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

32 Genealization to 3-D Law Natal cvates fo vehicle #: v η f( ) + + µ η f( ) z + z + µ z Natal cvates fo vehicle #: v η f ( ) + µ η f( ) z z + µ z Baseline alinment Collision avoidance Headin alinment Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

33 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Lie Gop Settin....,,, n....,,, n....,,, (),, n se + ξ ξ Dnamics Gop vaiables Fenet-Seet Eqations,,..., n G SE(), the op of iid motions in the plane. copies n G G G S Confiation space.,...,, ~ n Shape vaiables Shape space copies n G G G R Shape vaiables capte elative vehicle positions and oientations. Assme the contols,,..., n ae fnctions of shape vaiables onl.

34 Lie Gop Settin (3-D Taectoies) Repesent each vehicle taecto as a fnction on the Lie op SE(3) of iid motions: z, z SE(3) Define the shape vaiable: z ( ) z ( ) z z z z ( ) z Left-invaiant sstems on SE(3): ξ ( A + A + A v ), ξ se(3) ξ ( A + A + A v ), ξ se(3) ξ ξ ξ ξ ξ ξ se - ( Ad - ) ( ), (3) A, A, A Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

35 Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland Fomation Contol fo n vehicles Genealization of the two-vehicle fomation contol law to n vehicles: k k k k k k k f F n ) ( ),,,, ( n,,, At pesent, it is conected (based on simlation eslts) that sch contol laws stabilize cetain fomations. Howeve, analtical wok is onoin.

36 Rectilinea Contol Law Simlation Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

37 Rectilinea Contol Law Simlations Simlations with vehicles (fo diffeent andom initial conditions). Leade-followin behavio: the ed vehicle follows a pescibed path (dashed line). On-the-fl modification of the sepaation paamete. o Nomalized Sepaation Paamete vs. Time 3 time Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

38 Ciclin Contol Law Simlation Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

39 Ciclin Contol Law k k ± η f( ) n k k k k On-the-fl modification of the sepaation paamete. Simlations with vehicles (fo diffeent andom initial conditions). o Beacon-ciclin behavio: the vehicles espond to a beacon, as well as to each othe. 3 Nomalized Sepaation Paamete vs. Time time Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

40 3-D Simlations Rectilinea Law: Ciclin Law: k k k η f( ) + µ n k k k k k k k k k η f ( ) n k k k k k k k + µ k + k k k η whee k k, µ > >, o f ( k ) α, α > k Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

41 Obstacle Avoidance Idea: contol inpts fo the movin vehicle ae detemined b the taecto of the closest point on the obstacle sface. Goals: bonda followin and non-collision. z z F. Zhan, E.W. Jsth, and P.S. Kishnapasad, CDC 4. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

42 Bonda-Followin Simlation Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

43 Pefomance Citeia Faithfl followin of wapointspecified taectoies Sfficient sepaation between vehicles (to avoid collisions) Intevehicle distances Wapoints time Minimize steein: fo UAVs, tnin eqies consideabl moe ene than staiht, level fliht. Maneveabilit is also limited. ma Steein contols time Steein Ene time - ma Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

44 Towad Pactical Missions Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

45 Conveence Reslt fo n > We conside ectilinea elative eqilibia, and the Lapnov fnction V n [ ln( + cos( θ θk )) + h( k ) ] k< Conveence Reslt (Jsth, Kishnapasad): Thee eists a sblevel set Ω of V and a contol law (dependin onl on shape vaiables) sch that V on Ω. With this Lapnov fnction, we cannot pove lobal conveence fo n >. Althoh we obtain an eplicit fomla fo the contols,,...,n, thee is no aantee that this paticla choice of contols will eslt in conveence to a paticla desied eqilibim shape in Ω. Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

46 Vecto field (in pola coodinates): d / dt cosθ sinθ. / dθ dt Continm Model Continit eqation (Lioville eqation): ( ρ) ρ cosθ + t θ sin θ Consevation of matte: G ρ Ene fnctional: V c ( t,, θ ) ddθ, t. ρ. This continm fomlation onl involves two scala fields: the densit ρ(t,,θ) and the steein contol (t,,θ). Howeve, the ndelin space is 3-dimensional (fo plana fomations). Incopoatin time and/o spatial deivatives in the eqation fo ields a copled sstem of PDEs fo ρ and. ~ ~ ~ [ ln( + cos( θ θ )) + h( ~ ) ] ρ( t,, θ ) ρ( t, ~, θ ) ddθd ~ d. ( t) θ G G Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland

47 Refeences. E.W. Jsth and P.S. Kishnapasad, A simple contol law fo UAV fomation flin, Institte fo Sstems Reseach Technical Repot TR - 38, (see E.W. Jsth and P.S. Kishnapasad, Steein laws and continm models fo plana fomations, Poc. 4 nd IEEE Conf. Decision and Contol, pp , E.W. Jsth and P.S. Kishnapasad, Eqilibia and steein laws fo plana fomations, Sstems and Contol Lettes, Vol. 5, pp. 5-38, E.W. Jsth and P.S. Kishnapasad, Fomation contol in thee dimensions, pepint, F. Zhan, E.W. Jsth, and P.S. Kishnapasad, Bonda followin sin oscopic contol, Sbmitted to IEEE Conf. Decision and Contol, 4. See also Boston Univesit--Havad Univesit--Univesit of Illinois--Univesit of Maland