Dipartimento di Elettronica Informazione e Bioingegneria Robotics

Transcription

1 Diparimeno di Eleronica Informazione e Bioingegneria Roboics Mobiliy: wheels and 2015

2 Wheeled Mobile robos (WMR) A robo capable of locomoion on a surface solely hrough he acuaion of wheel assemblies mouned on he robo and in conac wih he surface. A wheel assembly is a device which provides or allows moion beween is moun and surface on which i is inended o have a single poin of rolling conac. (Muir and Newman, 1986) AGV Auomaed Guided Vehicle ALV Auonomous Land Vehicle

3 Basic assumpions 1. only rigid pars. 2. each wheel may have a 1 link for seering. 3. seering axes are orhogonal o soil. 4. pure rolling of he wheel abou is axis (x axis) 5. no ranslaion of he wheel This model will be used in he kinemaic model roll y x y Wheel parameers: r radius v linear velociy w angular velociy z moion

4 wheel fixed Canno move in direcions perpendicular o he wheel plane orienable cenered x y caser wheel omnidirecional composed or Svedish

5 WMR Easy model Sensible o errain irregulariies 2 wheels For ouside errains Imprecisions in movemens, in paricular roaions racks Free movemen Complex srucure Omnidirecional

6 ICC (or CIR ICR) ICC Canno move Insananeous Cener of Curvaure or Roaion ICC, ICR, CIR = poin where he wheel axes inersec

7 Represening a pose ICC xm, ym is in he mobile sysem xb, yb is in he base sysem yb ym xm θ xb The roaion θ is abou he z axis

8 WMR Kinemaics DIRECT KINEMATICS Given he conrol parameers (wheels velociies, ec) and a ime of movemen, find he pose (x, y, θ) reached by he robo INVERSE KINEMATICS Given he final pose (x, y, θ) find he conrol parameers o move he robo here in a given ime

9 Direc and inverse kinemaics wheels parameers conrol variables: R angular velociy ω() linear velociy ν() Direc kinemaics: inegraing... x(), y(), θ() robo pose: x(), y(), θ() wheels parameers conrol variables Inverse kinemaics

10 2 wheels differenial drive A robo for RoboCup

11 Differenial drive - 1 pose Conrol inpu v : linear velociy of he robo ω : angular velociy of he robo 2 variables independenly conrolled Vr velociy righ wheel Consrucion: 2 wheels on he same axis 2 moors -one for wheel 3rd passive supporing wheel Vl - velociy lef wheel

12 differenial drive - 2 The wheels L and R follow a circular pah wih ω angular velociy and differen curvaure radius ω(r + L/2) = v R ω(r - L/2) = v L find ω given Vl and Vr Solve for R and equae ω = v R -v L /L PARTICULAR CASES Roaion in place R = 0 VR = -VL Linear movemen R = infinie VR = VL find R solve for ω and equae

13 ICC in differenial drive ICC R The wheels move around ICC on a circumference isananeous radius = R angular velociy = ω Find ICC: ICC = (x +Rcos(θ+π/2), y + Rsin(θ+π/2) = (x - Rsin(θ), y + Rcos(θ)

14 ICC compuaion ICC x' y' θ ' = cos( ωδ) sin( ωδ) 0 sin( ωδ) cos( ωδ) x ICC y ICC θ x y + ICCx ICCy ωδ R P() Ro(z,wd) Trasl(x_ICCx, y _ICCy,θ)+)+Trasl(ICCx, ICCy,ωd) P(+δ) Given pose(), pose(+d) is: roaion a disance R from ICC wih velociy ω. ranslaee ICC in origin roae arounda ICC wih ω velociy Translae back

15 Differenial Drive - equaions R ICC ω (x,y) y l/2 θ x v l v r l v v v v v v v v v l R l R v l R v l r l r l r r l l r = + = + = = + = ω ω ω 2 ) ( ) ( 2 2) / ( 2) / ( ] cos, sin [ θ θ R y R x ICC + =

16 Posiion from velociy ω() y Inegrae o obain posiion V x = V() cos(θ()) V y = V() sin(θ()) Thus, V L 2d V R x x() = V() cos(θ()) d y() = V() sin(θ()) d θ() = ω() d R() robo s urning radius ICC wih ω = ( V R - V L ) / 2d R = 2d ( V R + V L ) / ( V R - V L ) V = ωr = ( V R + V L ) / 2

17 Direc kinemaics - direc drive CIR R P() P(+δ) ' ')] ( ') ( [ 1 ) ( ' ')] ( sin[ ')] ( ') ( [ 2 1 ) ( ' ')] ( cos[ ')] ( ') ( [ 2 1 ) ( = + = + = l r l r l r d v v l d v v y d v v x θ θ θ Using he conrol inpu ' ') ( ) ( ' ')] ( sin[ ') ( ) ( ' ')] ( cos[ ') ( ) ( = = = d d v y d v x ω θ θ θ

18 Inverse kinemaics Key quesion: y Given a desired posiion or velociy, wha can we do o achieve i? Finding some soluion is no hard, bu finding he bes soluion is very difficul... V L () V R () saring posiion x final posiion quickes ime mos energy efficien smoohes velociy profiles V L () define bes... V L ()

19 Inverse kinemaics The equaions of he direc kinemaics describe a consrain on he velociy of he robo ha canno be inegraed ino a posiional consrain (non holonomic consrain) How o find Vr and Vl o ge o he waned pose? underconsrained: infinie combinaions of Vr e Vl he robo moves on a circle passing for (0,0) a ime 0 and (x,y) a ime. Infinie soluions No independen conrol of θ Bu he soluion is srainghforward for a limied class of conrol funcions vr and vl 0,0,0 x,y,

20 Usual approach: decompose he problem and conrol only a few DOF a a ime Differenial Drive (1) urn so ha he wheels are parallel o he line beween he original and final posiion of he robo origin. y -V L () = V R () = V max V L () x V R () saring posiion final posiion

21 Usual approach: decompose he problem and conrol only a few DOF a a ime Differenial Drive y (1) urn so ha he wheels are parallel o he line beween he original and final posiion of he robo origin. V L () x -V L () = V R () = V max (2) drive sraigh unil he robo s origin coincides wih he desinaion V L () = V R () = V max V R () saring posiion final posiion

22 Usual approach: decompose he problem and conrol only a few DOF a a ime V L () V R () y Differenial Drive (1) urn so ha he wheels are parallel o he line beween he original and final posiion of he robo origin. -V L () = V R () = V max V L () x (2) drive sraigh unil he robo s origin coincides wih he desinaion V L () = V R () = V max V R () saring posiion final posiion (3) roae again in order o achieve he desired final orienaion -V L () = V R () = V max

23 equaions 1. If VR differen VL: x() = [v L + v R ]/ [v R -v L ]* L/2*sin [(v R -v L )/L] y() = [v L + v R ]/ [v R -v L ]* L/2*cos [(v R -v L )/L] + [v L + v R ]/ [v R - v L ]* L/2 θ() = /l * (v R -v L ) 2. If VR=VL, R is infinie, θ() consan linear movemen of he robo

24 Synchronous drive 3 wheels acuaed and seered, conrolled ogeher a moor o roll all he wheels, a second moor o roae hem The 3 wheels poin in he same direcion I is possible o conrol direcly θ. The robo can roae on sie abou he cener of he robo Holonomic behavior y θ v() CONTROL VARIABLES: v(), ω() x ICC is always a infiniy

25 synchronous drive kinemaics - 1 DK: roaion around he robo cener a angular velociy ω(), ranslaion a linear velociy v(). ω and v are direcly conrolled ICC is a infinie disance Seering changes he direcion of ICC Paricular cases: v() = 0, w() = w for Δ, he robo roaes in place v() = v, w() = 0 for Δ, he robo moves linearly

26 Synchonous Drive kinemaics - 2 y x( ) = v( ') cos[ θ ( ')] d' 0 y( ) = v( ') sin[ θ ( ')] d' θ v() x θ ( ) = 0 ω( ') d' ω( ) 0 Inverse kinemaics Consider ha 1. v() = 0 and ω() = ω for δ, hen he robo roaes in place 2. v() = v and ω() = 0 for δ, hen he robo moves in he direcion i is poining

27 Inverse kinemaics synchro drive Usual approach: decompose he problem and conrol only a few DOF a a ime Synchro Drive final posiion (1) urn so ha he wheels are parallel o he line beween he original and final posiion of he robo origin. y V() ω() = ω max (2) drive sraigh unil he robo s origin coincides wih he desinaion V() = V max ω() saring posiion x (3) roae again in order o achieve he desired final orienaion ω() = ω max Similar o he special cases of differenial drive

28 Oher seered wheels Tricicle (2 passive wheels, one acuaed and seerable) as in AGV. Robos ha do no use eiher differenial or synchronous drive have boh wheels ha can be seered and fixed wheels Bicycle (1 fixed and 1 searable) ICC

29 Ackermann Seering Car-like (2 fixed and 2 seerable)

30 Oher robo vehicles For oudoor environmens, more degrees of freedom and more complex wheels Omnidirecional: easier o conrol Sojourner Truh 4 seerable wheels + 2 fixed

31 Soil mechanism When a wheel is driven, he driving force urns he wheel and is opposed by he fricional force beween wheel and ground. fricion If driving force > fricional force he wheel may slip The fricional force limis he rae of acceleraion of he vehicle. On sof ground, wheels and legs sink. If a wheel moves forward, i compacs he soil in fron of i using energy. A driven leg has o compac he soil oo, bu has he advanage ha i happens only in he poin of conac, no on all he pah If fricional force is low, a wheel may no be able o generae enough force o compac he soil and will spin. In conras a leg can move over a surface wih low fricion.

32 Tracked vehicles Kinemaics as in differenial drive v of inside and ouside racks r diameer of he wheel v o = rω 0 v i = rω i I is necessary a model of he slipping Tracks eliminae mos of he sinking problems irobo, 4 racks

33 racks

34 Degrees of Freedom Maneuverabiliy is equivalen o he vehicle s degree of freedom (DOF) a body in 2D characerize he posiions ha ca be reached in space Examples: Bicycle: DOF=3 Omni Drive: DOF=3 The robo s independenly achievable velociies = differeniable degrees of freedom (DDOF) Examples: Differenial drive: DDOF = 2, DOF 3 Omni Drive: DDOF=3; DOF=3

35 Degrees of Freedom -> Holonomy DOF degrees of freedom: Robos abiliy o achieve various poses DDOF differeniable degrees of freedom: Robos abiliy o achieve various pah DDOF DOF Holonomic A holonomic kinemaic consrain can be expressed as an explici funcion of posiion variables only A non-holonomic consrain requires a differen relaionship, such as he derivaive of a posiion variable Fixed and seered sandard wheels impose nonholonomic consrains

36 kinemaic holonomic/nonholonomic consrains Kinemaic consrain - a siuaion ha reduces posiions/velociies of he mechanical sysem, so reducing he relaive mobiliy Consrains can be expressed as a se of equaions/disequaions of posiion and velociy of he poins in he sysem: A posiion consrain is called holonomic (posiion) if in he equaion here is no dependence on he velociy. is non holonomic ( mobiliy) if i consrains also velociy This equaion canno be inegraed (Holos in Greek = ineger)

37 Example holonomic consrains rolling cylinder 6 coordinaes, x, y, z, r, θ, φ 5 consrains: z=0, since i rolls on he plane r = consan φ = 0, he plane faces are orhogonal o he plane space s = (x 2 + y 2 ) 1/2 i replaces 2 coordinaes wih 1 s = aθ if he cylinder rolls wihou slipping- in holonomic form i is ds= adθ so s-s0= a(θ θ0), a relaion beween coordinaes so holonomic only 1 degree of mobiliy (6 5), s or θ.

38 Example non holonomic consrain A hin disk rolling on an horizonal plane φ Six coordinaes as before Holonomic consrains: Z=0 R=a=consan Bu i can spin abou boh θ (roll) and, φ (urn) so i is non holonomic 5 degrees of freedom Since he disk can urn (change φ) i can roll from one place o anoher by differen roues and so he disance in he plane space s = (x 2 + y 2 ) 1/2 is no relaed o he angle θ, as i was in he cylinder, and so his consrain does no lead o a definie condiion on he coordinaes.

39 Resric DOF or velociy Each configuraion of a sysem is univocally deermined by n independen parameers q1,..., qn; n degrees of freedom The holonomic consrains subrac a degree of freedom for each consrain equaion The non holonomic consrains resric only velociy: - hey allow o reach any posiion, so hey do no reduce he degrees of freedom. - some pahs are no allowed while any posiion can be reached. imagine a car; whils i is possible for i o be in any posiion on he road, i is no possible for i o move sideways.

40 WMR holonomic and non holonomic Holonomic robo = all is consrains are holonomic A mobile robo on he plane has 3 dof An holonomic WMR wih 3 dof is only he omnidirecional one Many WMR are non holonomic Wheels canno move orhogonally o he direcion of ravel Non holonomic consrains reduce he conrol space (ex: no possible o move laerally) Non-holonomic sysems differenial equaions are no inegrable o he final posiion. he measure of he raveled disance of each wheel is no sufficien o calculae he final posiion of he robo. One has also o know how his movemen was execued as a funcion of ime.

41 Free movemen (holonomic robo) RIV, Ispra Service robo for nuclear wase sore

42 Conrol of WMR The objecive of a kinemaic conroller is o follow a rajecory described by is posiion and/or velociy profiles as funcion of ime. Given 2 poses (q i and q f ) find a rajecory ha follows he kinemaic consrains here are possible rajecories also for non holonomic robos usually he kinemaic model is used usually mobile robos are underacuaed for insance he differenial drive has 2 inpu Vl and Vr and 3 generalized coordinaes x, y, θ Ofen mobile robos use open loop conrol

43 Open Loop Conrol rajecory (pah) divided in moion segmens of clearly defined shape: sraigh lines and segmens of a circle y I conrol problem: pre-compue a smooh rajecory based on line and circle segmens Disadvanages: I is no an easy ask o pre-compue a feasible rajecory limiaions and consrains of he robos velociies and acceleraions does no adap or correc he rajecory if dynamical changes of he environmen occur The resuling rajecories are usually no smooh goal x I

44 Whegs Wheels + legs Simple conrol RHEX, MiniWhegs

45 Scou (umn - Darpa) Lengh: 110 mm Diameer: 40 mm Weigh: 200 g Top Speed: 0.31 m/s Locomoion: Primary: 2 Foam wheels Secondary: Jumping" mechanism Maximum Jump Heigh: 30 cm Sensors: Camera (Color or B/W) Acceleromeers Wheel Encoders Magneomeers Operaional Lifeime: Quiescen Mode: 120 minues Full Speed: 70 minues Jumping Only: 100 jumps

46 LionHell All errain robo Whegs + passive join + acive neck Behavior based

47 Robos o move in air Flying robos Fixed wings vehicles Mobile wings vehicles quadroors

48 Robos o move in waer Acquaic vehicles: wo basic designs are orpedo-like srucures, wih a single propeller Bodys wih a collecion of hrusers New design is bioinspired and uses sof maerials. In general hey operae in a ehered manner, conneced o a suppor vessel ha provides power and communicaion

49 sphero A. Charging Dock The ball ress in a dock ha uses an inducion sysem o ransfer elecriciy o he baeries. B. Wheels Two independenly conrolled rubber-rimmed wheels inside he Sphero seer and drive he ball a up o 1.2 m/sec C. Top Slip Bearing braces he mechanism in order o keep he wheels in firm conac wih he shell. D. Prined Circui Board A processor combines daa from a hree-axis acceleromeer and a gyroscope o produce he precise measuremens of he Sphero s roll, pich, and yaw. These measuremens are required o respond correcly o commands radioed by a smar phone over a Blueooh connecion E. Blueooh Radio and Anenna This sysem, wih a maximum range of 50 meers, is used o communicae wih mobile devices. F. Mulicolor LED The ligh from a single LED package wih red, green, and blue elemens is diffused o make he Sphero glow. Differen colors signal informaion.

50 Passive robo Planes or ocean exploraion Adap (no or limied acuaion) Example: humbleweed (NASA)

51 conclusion Who is invesing in new mobile robos? Media Google (Boson Dynamics) ElecroMechanical companies Sony Honda Tradiional roboics companies irobo Adep Technology Kuka