NON-LINEAR CONTROL OF WHEELED MOBILE ROBOTS

Transcription

1 NON-LINEAR CONTROL OF WHEELED MOBILE ROBOTS Maria Isabl Ribiro Pdro Lima Institto Sprior Técnico (IST) Institto d Sistmas Robótica (ISR) A.Roisco Pais, 49- Lisboa PORTUGAL May. all rights rsrd - Pdro Lima, M. Isabl Ribiro

2 Cors Otlin taas PLANEAMENTO DE TAREFAS PLANEAMENTO DE MOVIMENTO mapa global actalização do mapa MAPA DO MUNDO FUSÃO SENSORIAL Trajctória dsjada (pos+l+ac) mapa local Caractrísticas do mio nolnt Dtcção d obstáclos AUTO-LOCALIZAÇÃO (long baslin, short baslin, fatr matching) Trajctória stimada (pos+l+ac) CONDUÇÃO (sgir trajctória /itar obstáclos) CONTROLO Posição /o locidads dsjadas informação procssada a partir dos dados dos snsors Informação niada aos actadors SENSORES Sonar, lasr, isão, ncodrs, giroscópios VEÍCULO ACTUADORES Motors, rodas, hélics, sprf ciis d dflcção - Pdro Lima, M. Isabl Ribiro

3 Gidanc Path Plannr targt path or trajctory obstacls Gidanc joint st points (.g., whl locitis) Joint Controllr oprati on point joint torqs (.g., motor inpts) joint stat fdback Vhicl postr stimat Localization snsor masrmnts GUIDANCE tak tak th th robot robot from from th th crrnt crrnt postr postr to to th th dsird dsird postr, postr, possibly possibly following following a pr-dtrmind pr-dtrmind path path or or trajctory, trajctory, whil whil aoiding aoiding obstacls obstacls - Pdro Lima, M. Isabl Ribiro

4 Gidanc Mthodologis Som Gidanc mthodologis Stat(postr)-fdback mthods: postr stabilization (initial and final postrs gin; no path or trajctory pr-dtrmind; obstacls not considrd; may lad to larg nxpctd paths) trajctory tracking (rqirs pr-plannd path) irtal hicl tracking (rqirs pr-plannd trajctory) Potntial-Fild lik mthods potntial filds (holonomic hicls) gnralizd potntial filds (non-holonomic hicls) modifid potntial filds (non-holonomic hicls) Vctor Fild Histogram (VHF) lik mthods narnss diagram naigation (holonomic hicls) frzon (non-holonomic hicls) NON-LINEAR CONTROL DESIGN FOR MOBILE ROBOTS - Pdro Lima, M. Isabl Ribiro 4

5 Control of Mobil Robots Thr distintic problms: Trajctory Tracking or Postr Tracking Path Following Point Stabilization - Pdro Lima, M. Isabl Ribiro 5

6 Trajctory Tracking Vhicl of nicycl typ x! cosθ y! sinθ θ! ω kinmatic modl z (x,y, θ) ω position and orintation with rspct to a fixd fram linar locity anglar locity Is a simplifid modl, bt Captrs th nonholonomy proprty which charactrizs most WMR and is th cor of th difficltis inold in th control of ths hicls Th trajctory tracking problm for a WMR of th nicycl typ is sally formlatd with th introdction of a irtal rnc hicl to b trackd. rnc hicl y θ y {} B θ x x - Pdro Lima, M. Isabl Ribiro 6

7 Trajctory Tracking Kinmatic modl of th Rfrnc Vhicl x! y! θ! ω cosθ sinθ z (x,y, θ ) ω (t) (t) Bondd Bondd driatis Do not tnd to zro as t tnds to infinity Control Objcti Dri th rrors x x θ, y y, - θ to zro Exprss th rrors in th {B} fram cosθ sinθ sinθ cosθ x x y y θ θ Diffrntiating Introdcing th chang of inpts + ω ω cos - Pdro Lima, M. Isabl Ribiro 7

8 8 - Pdro Lima, M. Isabl Ribiro Trajctory Tracking + + ω ω sin! non-linar dynamic systm control ariabls Qstion? Is it possibl to dsign a fdback law f() sch that th rror conrgs to zro? Is this law linar or non-linar? Two diffrnt soltions: Linar fdback control Nonlinar fdback control

9 9 - Pdro Lima, M. Isabl Ribiro Trajctory Tracking LINEAR FEEDBACK CONTROL + + ω ω sin! linariz abot th qilibrim point + ω ω (t) (t)! linar tim-arying dynamic systm (t) (t) ω ω Assming linar tim inariant dynamic systm Is it possibl to dsign a linar fdback law f() sch that th rror conrgs to zro? Is th dynamic systm controllabl?

10 Trajctory Tracking Γ c ω ω ω If ω th SLIT is non-controllabl th rnc robot at rst k th rror cannot b takn to zro in finit tim othrwis K K K K K K K Kij Chosn by pol placmnt ndtrmind systm closd loop pols (s + ξa)(s + ξas + a ) k k sgn( ) k - Pdro Lima, M. Isabl Ribiro ξa If r tnds to zro, k incrass withoth bond k k k ξa a ωr

11 Trajctory Tracking To aoid th prios problm: Th closd-loop pols dpnd on th als of r and w r closd loop pols (s + ξa)(s + ξas + a ) a + w b k k sgn( r ) k k k k ξ b ξ w w + b + b - Pdro Lima, M. Isabl Ribiro

12 Trajctory Tracking NONLINEAR FEEDBACK CONTROL! ω ω + sin + k k ( 4,w sin ) k (,w ) k 4 positi constant k continos fnction strictly positi in RxR-(,) k continos fnction strictly positi in RxR-(,) S th analogy with th linar contol Proprty: This control globally asymptotically stabilizis th origin dmonstration sing Lyapno stability thory k k 4 ( b,w ) k (,w ) ξ w + b - Pdro Lima, M. Isabl Ribiro

13 Path Following Objcti: Str th hicl at a constant forward spd along a prdfind gomtric path that is gin in a tim-fr paramtrization. θ Path y {} B " θ d x Approach: Th controllr shold compt Th distanc of th hicl to th path Th orintation rror btwn th hicl s main axis and th tangnt to th path Th controllr shold act on th anglar locity to dri both to zro - Pdro Lima, M. Isabl Ribiro

14 Path Following y {} B θ Path " η T η N M θ d x M is th orthogonal projction of th robot s position P on th path M xists and is niqly dfind if th path satisfis som conditions ( P, η T, ηn ) Srrt-Frnt fram moing along th path. η T Th ctor is th tangnt ctor to th path in th closst point to th hicl Th ctor η N is th normal " is th distanc btwn P and M s is th signd crilinar distanc along th path, from som initial path to th point M θ d (s) is th angl btwn th hicl s x-axis and th tang to th path at th point M. c(s) is th path s cratr at th point M, assmd niformly bondd and diffrntiabl θ θ θ is th orintation rror d - Pdro Lima, M. Isabl Ribiro 4

15 Path Following A nw st of stat coordinats for th mobil robot ( s, ", θ) Thy coincid with x,y,θ in th particlar cas whr th path coincids with th x-axis Th rror dynamics can b drid writing th hicl kinmatic modl in th Srrt-Frnt fram:! cos θ s c(s) " "! sinθ! cos θc(s) θ w c(s) " PROBLEM FORMULATION Gin a path in th x-y plan and th mobil robot translational locity, (t), (assmd to b bondd) togthr with its tim-driati d(t)/dt, th path following problm consists of finding a (smooth) fdback control law ω k(s, ", θ,(t)) sch that lim" (t) t lim θ(t) t - Pdro Lima, M. Isabl Ribiro 5

16 Path Following! cos θ s c(s) " "! sin θ! cos θc(s) θ w c(s) " cos θc(s) w c(s)" cos s! θ c(s) " "! sin θ! θ Two diffrnt soltions: Linar fdback control Nonlinar fdback control LINEAR FEEDBACK CONTROL Linariz th dynamics arond th qilibrim point (", θ ) "! (t) (t) θ(t)! θ(t) (t) Linarization of th last two qations - Pdro Lima, M. Isabl Ribiro 6

17 Path Following If (t)ct Th linar systm is CONTROLLABLE ASYMPTOTICALY STABILIZABLE BY LINEAR STATE FEEDBACK Linar stabilizing fdback k k " θ k >, k >! Closd-loop diffrntial qation "! + k "! + k " transformation " "' γ γ t dτ Distanc gon by point M along th path " '' ' + k" + k " s + ξas + a - Pdro Lima, M. Isabl Ribiro Dsird closd-loop characristic qation 7

18 Path Following NONLINEAR FEEDBACK CONTROL cos s! θ c(s) " "! sin θ! θ Nonlinar control law k " sin θ θ k() θ with k > k(. ) continos fnction strictly positi In ordr to ha th two (linar and nonlinar) controllrs bha similarly nar ", θ chos k() k with k k ξa a - Pdro Lima, M. Isabl Ribiro 8

19 Path Following Proprty Undr th assmption lim (t) t th non-linar control sinθ k " θ k() θ asymptotically stabilizs (", θ ) proidd that th hicl s initial configration is sch that " ( ) + θ() < k lim sp(c(s) Condition to garant that -c(s)l rmains positi Th hicl s location along th path is charactrizd by th al of s (th distanc gon along th path) Dpnds on (t) This dgr of frdom can b sd to stabiliz s abot a prscribd al s d - Pdro Lima, M. Isabl Ribiro 9

20 Point Stabilization Gin An arbitrary postr z d (x,y, θ) Find A control law ω k(z z d,t) which stabilizs asymptotically z-z d abot zro, whatr th initial robot s postr z() COROLLARY Thr is no smooth control law k(z) that can sol th point stabilization problm for th considrd class of systms. ALTERNATIVES Smooth (diffrntiabl) tim-arying nonlinar fdback k(z,t) Picwis continos control laws k(z) Tim-arying picwis continos control laws k(z,t) - Pdro Lima, M. Isabl Ribiro

21 Rfrncs C. Candas d Wit, H. Khnnof, C. Samson, O. Sordaln, Nonlinar Control Dsign for Mobil Robots in Rcnt Dlopmnts in Mobil Robots, World Scintific, 99. Rading assignmnt Carlos Candas d Wit, Brno Siciliano and Gorgs Bastin (Eds), "Thory of Robot Control", Pdro Lima, M. Isabl Ribiro