NONLINEAR CONTROL OF WHEELED MOBILE ROBOTS


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1 NONLINEAR 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 AUTOLOCALIZAÇÃO (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 prdtrmind prdtrmind 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 prdtrmind; obstacls not considrd; may lad to larg nxpctd paths) trajctory tracking (rqirs prplannd path) irtal hicl tracking (rqirs prplannd trajctory) PotntialFild lik mthods potntial filds (holonomic hicls) gnralizd potntial filds (nonholonomic hicls) modifid potntial filds (nonholonomic hicls) Vctor Fild Histogram (VHF) lik mthods narnss diagram naigation (holonomic hicls) frzon (nonholonomic hicls) NONLINEAR 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! nonlinar 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 nonlinar? 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 timarying 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 noncontrollabl 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 closdloop 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 timfr 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 ) SrrtFrnt 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 xaxis 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 xaxis Th rror dynamics can b drid writing th hicl kinmatic modl in th SrrtFrnt fram:! cos θ s c(s) " "! sinθ! cos θc(s) θ w c(s) " PROBLEM FORMULATION Gin a path in th xy plan and th mobil robot translational locity, (t), (assmd to b bondd) togthr with its timdriati 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 >! Closdloop 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 closdloop 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 nonlinar 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 zz 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) timarying nonlinar fdback k(z,t) Picwis continos control laws k(z) Timarying 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
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