Kinematics of Wheeled Robots

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1 1 Kinemaics of Wheeled Robos

2 hps:// 2

3 Wheeled Mobile Robos robo can hae one or more wheels ha can proide seering direcional conrol power eer a force agains he ground an ideal wheel is perfecl round perimeer 2πr moes in he direcion perpendicular o is ais 3

4 Wheel 4

5 Deiaions from Ideal 5

6 Insananeous Cener of Curaure for smooh rolling moion, all wheels in ground conac mus follow a circular pah abou a common ais of reoluion each wheel mus be poining in is correc direcion reole wih an angular eloci consisen wih he moion of he robo each wheel mus reole a is correc speed 6

7 Insananeous Cener of Curaure 7 a 3 wheels wih roll aes inersecing a a common poin he insananeous cener of curaure,. b No eiss. A robo haing wheels shown in a can ehibi smooh rolling moion, whereas a robo wih wheel arrangemen b canno.

8 Casor Wheels proide suppor bu no seering nor propulsion 8

9 Differenial Drie wo independenl drien wheels mouned on a common ais 9

10 Differenial Drie angular eloci abou he defines he wheel ground elociies and l disance beween and righ wheel r R R 2 2 disance beween and lef wheel hps://opencurriculum.org/5481/circular-moion-linear-and-angular-speed/ 10

11 Differenial Drie 11 gien he wheel ground elociies i is eas o sole for he radius, R, and angular eloci ω ineresing cases: l l R r r r 2

12 Tracked Vehicles similar o differenial drie bu relies on ground slip or skid o change direcion kinemaics poorl deermined b moion of reads 12 hp://en.wikipedia.org/wiki/file:tucker-kien-varians.jpg

13 Seered Wheels: Biccle f d 90 h r 13

14 Seered Wheels: Biccle imporan o remember he assumpions in he kinemaic model smooh rolling moion in he plane does no capure all possible moions hp:// 14

Mecanum Wheel a normal wheel wih rollers mouned on he circumference hp://blog.makezine.com/archie/2010/04/3d-prinable-mecanum-wheel.hml hps://www.

15 Mecanum Wheel a normal wheel wih rollers mouned on he circumference hp://blog.makezine.com/archie/2010/04/3d-prinable-mecanum-wheel.hml hps:// hps:// hp://fp.mi.fu-berlin.de/pub/rojas/omniwheel/diegel-bade-brigh-pogieer-tlale.pdf 15

16 Mecanum Wheel AndMark Mecanum wheel specificaion shee hp://d1prrjwm20z9.cloudfron.ne/mecanumwheelspecshee.pdf 16

17 Forward Kinemaics serial manipulaors gien he join ariables, find he pose of he end-effecor mobile robo gien he conrol ariables as a funcion of ime, find he pose of he robo for he differenial drie he conrol ariables are ofen aken o be he ground elociies of he lef and righ wheels i is imporan o noe ha he wheel elociies are needed as funcions of ime; a differenial drie ha moes forward and hen urns righ ends up in a er differen posiion han one ha urns righ hen moes forward! 17

18 Forward Kinemaics robo wih pose [ θ] T moing wih eloci V in a direcion θ measured relaie he ais of {W}: V θ {W} 18

19 Forward Kinemaics for a robo saring wih pose [ 0 0 θ 0 ] T moing wih eloci V in a direcion θ : V θ V cos θ V sin θ V cos V sin d d d 19

20 for differenial drie: Forward Kinemaics 20 r r r d d d sin 2 1 cos 2 1

21 Sensiii o Wheel Veloci r 1 N 1 N , 2 0, σ = 0.05 σ =

22 Sensiii o Wheel Veloci L = 0.2; sigma = 0.05; figure hold on for i = 1:1000 R = 1 + normrnd0, sigma; L = 1 + normrnd0, sigma; hea = 0; = 0; = 0; d = 0.1; for = 0.1:d:10 = * R + L * coshea * d; = * R + L * sinhea * d; hea = hea + 1 / L * R L * d; R = 1 + normrnd0, sigma; L = 1 + normrnd0, sigma; end plo,, 'b.'; end 22

23 23 Mobile Robo Forward Kinemaics

24 Forward Kinemaics : Differenial Drie wha is he posiion of he in {W}? R θ V {W} 24

25 Forward Kinemaics : Differenial Drie Rsin R cos R θ V {W} 25

26 Forward Kinemaics : Differenial Drie assuming smooh rolling moion a each poin in ime he differenial drie is moing in a circular pah cenered on he hus, for a small ineral of ime δ he change in pose can be compued as a roaion abou he R P+ P 26

27 Forward Kinemaics : Differenial Drie compuing he roaion abou he 1. ranslae so ha he moes o he origin of {W} 2. roae abou he origin of {W} 3. ranslae back o he original R P+ P 27

28 Forward Kinemaics : Differenial Drie compuing he roaion abou he 1. ranslae so ha he moes o he origin of {W} 2. roae abou he origin of {W} 3. ranslae back o he original Rsin R cos 28

29 Forward Kinemaics : Differenial Drie compuing he roaion abou he 1. ranslae so ha he moes o he origin of {W} 2. roae abou he origin of {W} 3. ranslae back o he original how much roaion oer he ime ineral? angular eloci * elapsed ime = cos sin sin cos 29

30 Forward Kinemaics : Differenial Drie compuing he roaion abou he 1. ranslae so ha he moes o he origin of {W} 2. roae abou he origin of {W} 3. ranslae back o he original cos sin sin cos 30

31 Forward Kinemaics : Differenial Drie 31 wha abou he orienaion? jus add he roaion for he ime ineral new pose which can be wrien as cos sin sin cos cos sin 0 sin cos

32 Forward Kinemaics: Differenial Drie 32 he preious equaion is alid if i.e., if he differenial drie is no raelling in a sraigh line if hen L R R L sin cos

33 Sensiii o Wheel Veloci r 1 N 1 N , 2 0, σ = 0.05 σ =

34 Sensiii o Wheel Veloci gien he forward kinemaics of he differenial drie i is eas o wrie a simulaion of he moion we need a wa o draw random numbers from a normal disribuion in Malab randnn reurns an n-b-n mari conaining pseudorandom alues drawn from he sandard normal disribuion see mnrnd for random alues from a muliariae normal disribuion 34

35 Sensiii o Wheel Veloci POSE = []; % final pose of robo afer each rial sigma = 0.01; % noise sandard deiaion L = 0.2; % disance beween wheels d = 0.1; % ime sep TRIALS = 1000; % number of rials for rial = 1:TRIALS end -run each rialsee ne slide 35

36 Sensiii o Wheel Veloci r = 1; % iniial righ-wheel eloci l = 1; % iniial lef-wheel eloci pose = [0; 0; 0]; % iniial pose of robo for = 0:d:10 -moe he robo one ime sep - see ne slide end POSE = [POSE pose]; % record final pose afer rial 36

37 Sensiii o Wheel Veloci hea = pose3; if r == l pose = pose + [r * coshea * d; r * sinhea * d; 0]; else omega = r l / L; R = L / 2 * r + l / r l; = pose + [-R * sinhea; R * coshea; 0]; pose = rzomega * d * pose + + [0; 0; omega * d]; end r = 1 + sigma * randn1; l = 1 + sigma * randn1; 37

Brock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension

Brock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension Brock Uniersiy Physics 1P21/1P91 Fall 2013 Dr. D Agosino Soluions for Tuorial 3: Chaper 2, Moion in One Dimension The goals of his uorial are: undersand posiion-ime graphs, elociy-ime graphs, and heir