Introduction to Robotics (Fag 3480)

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1 Intouton to Robot (Fg 8) Vå Robet Woo (Hw Engneeeng n pple Sene-B) Ole Jkob Elle PhD (Mofe fo IFI/UIO) Føtemnuen II Inttutt fo Infomtkk Unvetetet Olo Sekjonlee Teknolog Intevenjonenteet Olo Unvetetkehu (Rkhoptlet)

2 Ch. : Fow n Invee Knemt e Fg 8 - Intouton to Robot

3 Fg 8 - Intouton to Robot e Rep: The Denvt-Htenbeg (DH) Conventon Repeentng eh nvul homogeneou tnfomton the pout of fou b tnfomton: z z α α α α α α α α α α α Rot Tn Tn Rot

4 Rep: the phl b fo DH pmete : lnk length tne between the o n o (pojete long ) α : lnk twt ngle between z n z (meue oun ) : lnk offet tne between o n o (pojete long z ) : jont ngle ngle between n (meue oun z ) e Fg 8 - Intouton to Robot

5 Genel poeue fo etemnng fow knemt Lbel jont e z z n- ( z jont fo jont ) Chooe be fme: et o on z n hooe n ung ght-hne onventon Fo :n- Ple o whee the noml to z n z - nteet z. If z nteet z - put o t nteeton. If z n z - e pllel ple o long z uh tht the ommon noml though o o noml to the plne fome b z - n z f the two nteet Detemne ung ght-hne onventon Ple the tool fme: et z n pllel to z n- Fo :n fll n the tble of DH pmete Fom homogeneou tnfomton mte Cete T n tht gve the poton n oentton of the en-effeto n the netl fme e Fg 8 - Intouton to Robot

6 Emple : thee-lnk lnl obot DOF: nee to gn fou oonte fme Chooe z ( of otton fo jont be fme) Chooe z ( of tnlton fo jont ) Chooe z ( of tnlton fo jont ) Chooe z (tool fme) Th gn bt fo th e ne we hve ebe no wt/gppe Inte efne z pllel to z e Fg 8 - Intouton to Robot

7 Fg 8 - Intouton to Robot e 7 Emple : thee-lnk lnl obot Now efne DH pmete Ft efne the ontnt pmete α Seon efne the vble pmete lnk α -9 T

8 Emple : phel wt DOF: nee to gn fou oonte fme w pth oll ( ) ll nteetng t one pont o (wt ente) Chooe z ( of otton fo jont ) Chooe z ( of otton fo jont ) Chooe z ( of otton fo jont ) Chooe tool fme: z () ollne wth z () n the eton the gppe loe (n) hoen wth ght-hne onventon e 8 Fg 8 - Intouton to Robot

9 Fg 8 - Intouton to Robot e 9 Emple : phel wt lnk α -9 9 Now efne DH pmete Ft efne the ontnt pmete α Seon efne the vble pmete T

10 Emple : lnl obot wth phel wt DOF: nee to gn even oonte fme But we le th fo the pevou two emple o we n fll n the tble of DH pmete: lnk α -9 o o o e ll t the me pont o -9 9 e Fg 8 - Intouton to Robot

11 Fg 8 - Intouton to Robot e Emple : lnl obot wth phel wt Note tht z ( fo jont ) ollne wth z ( fo jont ) thu we n mke the followng ombnton: z T T T z

12 Emple : the Stnfo mnpulto DOF: nee to gn even oonte fme: Chooe z ( of otton fo jont be fme) Chooe z -z e (e of otton/tnlton fo jont -) Chooe e Chooe tool fme Fll n tble of DH pmete: lnk α e Fg 8 - Intouton to Robot

13 Fg 8 - Intouton to Robot e Emple : the Stnfo mnpulto Now etemne the nvul homogeneou tnfomton:

14 Fg 8 - Intouton to Robot e Emple : the Stnfo mnpulto Fnll ombne to gve the omplete epton of the fow knemt: z T [ ] [ ] [ ] [ ] z

15 Emple : the SCR mnpulto DOF: nee to gn fve oonte fme: Chooe z ( of otton fo jont be fme) Chooe z -z e (e of otton/tnlton fo jont -) Chooe e Chooe tool fme Fll n tble of DH pmete: lnk α 8 e Fg 8 - Intouton to Robot

16 Fg 8 - Intouton to Robot e Emple : the SCR mnpulto Now etemne the nvul homogeneou tnfomton: T

17 Inutl obot Hgh peon n epettve tk Pk n ple pntng et Hzou envonment e 7 Fg 8 - Intouton to Robot

18 Common onfguton: elbow mnpulto nthopomoph m: BB IRB o KUK Ve ml to the lb m NCHI (RRR) Fg 8 - Intouton to Robot e 8

19 Smple emple: ontol of DOF pln mnpulto Move fom home poton n follow the pth B wth ontnt ontt foe F ll ung vul feebk e 9 Fg 8 - Intouton to Robot

20 Fg 8 - Intouton to Robot e Coonte fme & fow knemt Thee oonte fme: Poton: Oentton of the tool fme: n o t n n o o ) o( ) n( ) n( ) o( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ R ˆ ˆ ) o( ) n( ˆ ) n( ) o( ˆ

21 Invee Knemt Fn the vlue of jont pmete tht wll put the tool fme t ee poton n oentton (wthn the wokpe) Gven H: H Fn ll oluton to: R o SE Notng tht: T T n ( q q ) H... n n ( q... q ) ( q ) ( q ) n n n Th gve (nontvl) equton wth n unknown e Fg 8 - Intouton to Robot

22 Fg 8 - Intouton to Robot e Fo gven H: Fn : One oluton: π/ π/. π/ π/.7. H Emple: the Stnfo mnpulto [ ] [ ] [ ] [ ].7.

23 Invee Knemt The pevou emple how how ffult t woul be to obtn loe-fom oluton to the equton Inte we evelop temt metho be upon the mnpulto onfguton Fo the fow knemt thee lw unque oluton Potentll omple nonlne funton The nvee knemt m o m not hve oluton Soluton m o m not be unque Soluton m volte jont lmt Cloe-fom oluton e el! Fg 8 - Intouton to Robot e

24 Ovevew: knemt eouplng ppopte fo tem tht hve n m wt Suh tht the wt jont e e lgne t pont Fo uh tem we n plt the nvee knemt poblem nto two pt: Invee poton knemt: poton of the wt ente Invee oentton knemt: oentton of the wt Ft ume DOF the lt thee nteetng t o R o ( q... q ) ( q... q ) o Ue the poton of the wt ente to etemne the ft thee jont ngle R e Fg 8 - Intouton to Robot

25 Fg 8 - Intouton to Robot e Ovevew: knemt eouplng Now ogn of tool fme o tne tnlte long z (ne z n z e ollne) Thu the th olumn of R the eton of z (w/ epet to the be fme) n we n wte: Rengng: Cllng o [o o o z ] T o [ z ] T R o o o o R o oo o o o z z

26 Ovevew: knemt eouplng Sne [ z ] T e etemne fom the ft thee jont ngle ou fow knemt epeon now llow u to olve fo the ft thee jont ngle eouple fom the fnl thee. Thu we now hve R Note tht: R R R To olve fo the fnl thee jont ngle: R T ( R ) R ( R ) R Sne the lt thee jont fo phel wt we n ue et of Eule ngle to olve fo them e Fg 8 - Intouton to Robot

27 Invee poton Now tht we hve [ z ] T we nee to fn q q q Solve fo q b pojetng onto the - - plne olve tg poblem Two emple: elbow (RRR) n phel (RRP) mnpulto Fo emple fo n elbow mnpulto to olve fo pojet the m onto the plne e 7 Fg 8 - Intouton to Robot

28 Bkgoun: two gument tn We ue tn( ) nte of tn( ) to ount fo the full nge of ngul oluton Clle fou-qunt tn tn ( ) tn π tn tn π ( ) < < unefne > Fg 8 - Intouton to Robot e 8

29 Emple: RRR mnpulto To olve fo pojet the m onto the plne ( ) tn tn π ( ) e 9 Fg 8 - Intouton to Robot

30 Cvet: ngul onfguton offet If unefne.e. n vlue of wll wok If thee n offet then we wll hve two oluton fo : left m n ght m Howeve wt ente nnot nteet z e Fg 8 - Intouton to Robot

31 Fg 8 - Intouton to Robot e Left m: Rght m: Left m n ght m oluton tn tn - α φ α φ tn tn tn π β α β α

32 Fg 8 - Intouton to Robot e Theefoe thee e n genel two oluton fo Fnng n entl to the pln two-lnk mnpulto we hve een pevoul Theefoe we n fn two oluton fo : Left m n ght m oluton D z z o o D D ± tn

33 Fg 8 - Intouton to Robot e The two oluton fo oepon to the elbow-own n elbow-up poton epetvel Now olve fo : Thu thee e two oluton fo the p ( ) Left m n ght m oluton z tn tn tn tn

34 RRR: Fou totl oluton In genel thee wll be mmum of fou oluton to the nvee poton knemt of n elbow mnpulto E: PUM e Fg 8 - Intouton to Robot

35 Emple: RRP mnpulto Sphel onfguton Solve fo ung me metho wth RRR tn ( ) gn f thee n offet thee wll be left-m n ght-m oluton Solve fo : ( ) tn z Solve fo : ( z ) e Fg 8 - Intouton to Robot

36 Net l Complete the uon of nvee knemt Invee oentton Intouton to othe metho Intouton to velot knemt n the Jobn Fg 8 - Intouton to Robot e

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