CS 4758 Robot Kinematics. Ashutosh Saxena

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1 CS 4758 Rt Kemt Ahuth Se

2 Kemt tude the mt f de

3 e re tereted tw emt tp Frwrd Kemt (ge t pt ht u re gve: he egth f eh he ge f eh t ht u fd: he pt f pt (.e. t (,, rdte Ivere Kemt (pt t ge ht u re gve: he egth f eh he pt f me pt the rt ht u fd: he ge f eh t eeded t t tht pt

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5 (, rt g fwed rtt θ θ θ θ θ (te :, re retve t the rg rdte frme. rt fwed rtt dfferet th rtt fwed trt. I ther wrd, wg the rdte f pt (, me rdte frme ( u fd the pt f tht pt retve t ur rg rdte frme (.

6 θ θ θ θ HMGEEUS RERESEAI uttg t t Mtr θ θ θ θ θ θ θ θ ht we fud dg trt d rtt ddg wth d Smpfg t mtr frm θ θ θ θ H Hmgeu Mtr fr rt pe, fwed Rtt rud the -

7 Rtt Mtre 3D R θ θ θ θ Rtt rud the Z-A R θ θ θ θ Rtt rud the -A R θ θ θ θ Rtt rud the -A

8 H Hmgeeu Mtre 3D H 44 mtr tht dere trt, rtt, r th e mtr rt wthut rtt H Z Z A A Rtt wthut trt Rtt prt: Cud e rtt rud -, -, - r mt f the three.

9 F r w r d K e m t

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11 Empe rem: u re hve three rm tht trt ut ged the -. Eh h egth,, 3, repetve. u te the frt e t mve, d the dgrm ugget. Fd the Hmgeeu mtr t get the pt f the ew dt the frme H R ( * ( * R ( * ( * R ( 3.e. Rttg w put u the frme. rte the g the. Rttg w put u the frme. d ut u re the 3 3 frme. he pt f the ew dt retve t the 3 3 frme (,. Mutpg H tht pt vetr w gve u the rdte f the ew pt retve the the frme.

12 Sght vrt the t ut: Me the ew dt the rg f ew rdte 4 4 frme H R ( * ( * R ( * ( * R ( 3 * 3 ( 3 Z H h te u frm the frme t the 4 4 frme. he pt f the ew dt retve t the 4 4 frme (,. te tht mutpg the (,,, vetr w equ the t um f the H mtr.

13 I v e r e K e m t Frm t t Age

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15 Ivere Kemt f w L Mputr (, Gve:,,, Fd:, Redud: A uque ut t th prem de t et. te, tht ug the gve tw ut re pe. Smetme ut pe. (,

16 he Gemetr Sut (pt α (, Ug the Lw f Ce: r θ (θ (θ θ (8 θ (8 ( C Ug the Lw f Ce: rt α α θ θ (θ θ (8 θ C B rt (θ r θ Redudt e θ ud e the frt r furth qudrt. Redud ued e θ h tw pe vue

17 ( ( ( r θ ( ( ( ( ( ( ( ( he Ager Sut (pt (, θ θ (3θ ( ( θ (θ θ Uw ( ( ( ( ( ( ( ( ( ( : te

18 ( ( ( ( ( ( ( ( ( ( : te ( ( ( ( e w wht θ frm the prevu de. e eed t ve fr θ. w we hve tw equt d tw uw ( θ d θ ( ( ( ( ( ( ( ( ( ( Suttutg fr d mpfg m tme te th the w f e d e reped ( r θ

19 Hgher DF rt? ed frm ut fr hgh DF rt. Atertve pprhe?

20 Ed f eture

21 Qu Mth Revew Dt rdut: Gemetr Repreett: A B θ B θ A B A Ut etr etr the dret f he vetr ut whe mgtude. B B u B Mtr Repreett: B A B u B

22 Mtr Mutpt: Qu Mtr Revew A (m mtr A d ( p mtr B, e mutped e the umer f um f A equ t the umer f rw f B. -Cmmuttve Mutpt AB equ t BA d e g f h ( e g ( f h ( e dg ( f dh Mtr Addt: d e g f h ( e ( f ( g ( d h

23 B rfrmt Mvg Betwee Crdte Frme rt Ag the -A (, dte etwee the d rdte pe tt:

24 rtg term f

25 rt g the -A d -A

26 Ug B etr B vetr re ut vetr tht pt g rdte Ut vetr g the -A Ut vetr g the -A Mgtude f the vetr θ θ θ (9 θ

27 Rtt (rud the Z-A Z Age f rtt etwee the d rdte

28 Ut vetr g -A C e dered wth repet t the rdte r rdte α α ( (Suttutg fr ug the d mpet f the vetr ( ( (θ (θ ((θ (θ 9

29 Smr. α (9 α ( ( ( (θ (θ (θ θ ((9 S. (θ (θ (θ (θ rtte Mtr Frm θ θ θ θ Rtt Mtr ut the -

30 Fdg the Hmgeeu Mtr E. Z J I K A A A K J I Z t retve t the --A frme t retve t the --Z frme t retve t the I-J-K frme A K J I A K J I

31 A H A Hmgeeu Ctued. he (,, pt f pt retve t the urret rdte frme u re. he rtt d trt prt e med t ge hmgeeu mtr IF d L IF th re retve t the me rdte frme. A

32 Z J I K A A J I Z K J I Z Suttutg fr K J I A Z

33 H A Z H rdut f the tw mtre te tht H e wrtte : H H (rt retve t the Z frme * (Rtt retve t the Z frme * (rt retve t the IJK frme * (Rtt retve t the IJK frme

34 he Hmgeeu Mtr tet f umeru trt d rtt Z e mre vrt fdg H: A A H (Rtte tht the - ged wth * ( rte g the ew t- (mgtude f * ( Rtte tht the t- ged wth * ( rte g the p- * ( Rtte tht the p- ged wth the - h methd mght eem t fug, ut t tu eer w t ve ur prem gve the frmt we hve. Here empe

35 Mre Frwrd Kemt Devt - Hrteerg rmeter

36 Devt-Hrteerg tt Z ( - ( - Z ( - ( - d ( - IDEA: Eh t ged rdte frme. Ug the Devt- Hrteerg tt, u eed 4 prmeter t dere hw frme ( rete t prevu frme ( -. HE ARAMEERS/ARIABLES:,, d,

37 he rmeter Z ( - ( - ( - ( - d Z u g the tw ut ug the 4 prmeter ( - (- eh Deft: (- the egth f the perpedur etwee the t e. he t e the e rud whh revut te pe whh re the Z (- d Z ( e. hee tw e e vewed e pe. he mm perpedur the hrtet e etwee the tw -e d perpedur t th -e.

38 (- t... u Apprh - A w t vue the prmeter (- t mge epdg der whe the Z (- - whe the der ut tuhe the t the rdu f the der equ t (-. (Mputr Kemt It Uu the Dgrm Apprh - If the dgrm red pefe the vru rdte frme, the the mm perpedur uu the (-. S (- ut the dpemet g the (- t mve frm the (- frme t the frme. If the prmt, the (- vre, t prmeter. Z ( - ( - Z ( - ( - ( - d

39 (- eh Deft: Amut f rtt rud the mm perpedur tht the t e re pre..e. Hw muh u hve t rtte rud the (- tht the Z (- ptg the me dret the Z. tve rtt fw the rght hd rue. 3 d (- eh Deft: he dpemet g the Z eeded t g the (- mm perpedur t the mm perpedur. I ther wrd, dpemet g the Z t g the (- d e. Z ( - ( - ( - ( - ( - d Z 4 Amut f rtt rud the Z eeded t g the (- wth the.

40 he Devt-Hrteerg Mtr θ θα θα ( ( θ θ α θ α ( ( α α ( ( ( α α ( ( d d Jut e the Hmgeeu Mtr, the Devt-Hrteerg Mtr trfrmt mtr frm e rdte frme t the et. Ug ere f D-H Mtr mutpt d the D-H rmeter te, the f reut trfrmt mtr frm me frme t ur t frme. ut the trfrmt here Z ( - ( - ( - ( - ( - d Z

41 3 Revute Jt Z Z d Devt-Hrteerg L rmeter e te tht the te h tw ue: dere the rt wth t vre d prmeter. dere me tte f the rt hvg umer vue fr the vre. α (- (- d θ θ θ -9 d θ

42 Z Z α (- (- d θ d θ θ -9 d θ Z ( ( ( Z te: the D-H mtr wth (- d.

43 θ θ θ θ α (- (- d θ θ θ -9 d θ h ut rtt rud the Z θ θ θ θ θ θ d θ θ h trt fwed rtt rud the Z h trt d the d fwed rtt rud the d Z ( ( (

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