ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören

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1 ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2( yx, ) tn y/ x, x 2 2 y 0 tn 2( y, x), n n t n n ( n) ( ) ( ) n n n n n n m m n n m n t n t n n, n n ; n n n n n n( / 2) n( / 2) n( 2 ) n, n( ) n( ), n n n m n ( )/ n n m n n i i i t n i i m n i j j cos nij sin ; nij i j i j, i j i t t t j j cos n ji sin ; n ji ji j i, i j ˆ ( k, k) k k C, ( m) ( m ), (, b) ( b, ) b/ C C V Jˆ q t i ( k) ( k) rk, k sk k ( m) ( m) n, s 2 ˆ ˆ n n ˆ t J P ˆ ( b) ( b, ) (, b) b/ C C ˆ M. K. Özgörn

2 MANIPULATO OF CONCEN θ P A E ( ), s OA O E θ, A, O θ, E θ POBLEM Consir th mnipltor shown in th figr. Its joint rrngmnt is P 4. It is shown with its complt si viw n its prtil top n front viws on th ppr n lowr sis of th figr. On th figr, lbl ll th nit vctors n th criticl points. srv th lttr for th bs frm origin. Also, inict clrly ll th joint vribls n th nonzro constnt prmtrs. Thn, fill in th following Dnvit-Hrtnbrg tbl. D-H Tbl k k 2 k k 4 k 5 k 6 k 0 k k or k s k or k O k 2 M. K. Özgörn

3 POBLEM 2 Dfin n show tht th orinttion of th hn n th loctions of th wrist n tip points cn b xprss by th following forwr kinmtics qtions with rspct to th bs frm. ˆ /2 C r s 6 2 p r SOLUTION Th link-to-link orinttion mtrics r: ˆ (0,) 0 ( cos sin ) C, ˆ (,2) 0 0 C Iˆ, ˆ (2,) C /2, ˆ (,4) C, C (4,5) 5 /2, C (5,6) /2 ˆ ˆ 0 Hnc, th orinttion mtrics of th links with rspct to th bs frm cn b obtin s follows. Cˆ (0,) ˆ (0,2) ˆ (0,) ˆ (,2) ˆ C C C I ˆ (0,) ˆ (0,2) ˆ (2,) /2 /2 C C C ˆ (0,4) ˆ (0,) ˆ(,4) /2 4 /2 2 4 C C C ˆ (0,5) ˆ (0,4) ˆ (4,5) C C C /2 ˆ ˆ (0,6) ˆ (0,5) ˆ (5,6) / /2 C C C C Th wrist point loction with rspct to th bs frm is xprss s scrib blow. () (2) () (4) r O OA AE E s (/0) (2/0) (/0) (4/0) r r s r s Cˆ Cˆ Cˆ Cˆ (0,) (0,2) (0,) (0,4) / r s r s r s ( cos sin ) As for th tip point loction, it is obtin s follows. (6) 6 6 p r r p p r r Cˆ r C ˆ (6/0) (0,6) / p r r M. K. Özgörn

4 POBLEM Drw th si, top, n front viws of th mnipltor whn ll th nglr joint vribls r zro. Vrify th forwr kinmtics qtions givn in Problm in this vry spcil pos of th mnipltor. SOLUTION Whn ll th nglr joint vribls r zro, th si, top, n front viws of th mnipltor ppr s shown blow. A E,, P P E, s OA O A, O E P Whn ll th nglr joint vribls r zro, th forwr kinmtics qtions tk th following forms. ˆ /2 C r ( ) s p ( ) ( ) s Consiring th joint vribl, th constnt prmtrs (,,, ), n th bsis vctors of th bs frm, which r ll shown on th sktchs, it is sn by inspction tht th bov qtions r vrifi. 4 M. K. Özgörn

5 POBLEM 4 Th mnipltor is rqir to b in pos sch tht, for som spcifi vl of s 2, th links AE n E r lign so tht thy r both prlll to vrticlly own, n th grippr is in horizontl position so tht, th link is irct n n. In this prticlr pos, trmin th vls of th nglr joint vribls, othr thn 6, first by inspction n thn vrify yor inspct vls by sing th forwr kinmtics qtions givn in Problm 2. Aftrwr, trmin 6 too from th givn orinttion qtion. SOLUTION Th givn figr of th mnipltor implis tht, /2, 4 /2, With ths ngls, th forwr kinmtics qtions bcom: Finlly, 2 2 / r s ( ) r ( ) ( s ) (OK. It is s rqir) /2 ( /2) p ( ) ( s ) 2 p ( ) ( s ) (OK. It is s rqir) ˆ /2 ( /2) /2 C ˆ /2 2( /2 6 ) /2 C ˆ /2 ( /2 ) /2 /2 ( /2 ) /2 2 2 C (0/0) (OK. It is s rqir) ˆ /2 ( /2 ) /2 /2 ( /2 ) /2 /2 C n ( /2 ) ( /2) ( /2 ) ( /2) n It is rqir to hv n (0/0) cn b flfill if ( 6 / 2) / 2 or 6 0. Th prcing qtion implis tht this rqirmnt 5 M. K. Özgörn

6 POBLEM 5 In orr to oprt th mnipltor with th longst possibl xtnsion of th wrist point, th links AE n E r kpt lign so tht /2. ) In sch n oprtion, show tht th gnrl forwr kinmtics qtions tk th following spcil forms. ˆ /2 /2 C r s ( cos sin ) ; p r 6 ; 4 5 b) Th loction of th wrist point is spcifi with rspct to th bs frm with th coorints r, r 2, n r. With ths coorints, show tht th wrist point qtion givn in Prt () ls to th following sclr qtions. r cos r sin 2 24 cos r cos r sin s sin r c) From th qtions givn in Prt (b), trmin th joint vribls, 4, n s2. At th sm tim, intify th mltipl soltions n illstrt thm with simpl sktchs. Also, trmin th llowbl rngs of th coorints r, r 2, n r. ) Sinc on of th joint vribls is kpt fix s /2, on of th grs of from of th mnipltor gts block. In othr wors, th mnipltor oprts with fiv grs of from. Thrfor, t th spcifi loction of th wrist point, th pproch vctor cnnot b orint rbitrrily s sir. Consiring th bs-frm xprssion of givn in Prt (), xplin why n how th orinttion of is rstrict. Hint: A nit vctor ns two fr n inpnnt vribls to b orint rbitrrily. 0 ) It is rqir to orint th pproch vctor horizontlly so tht. It is lso rqir tht b forwrly pointing vctor. For this prpos, trmin th ngl 5 in trms of th ngls n 4, which hv lry bn trmin to flfill th wrist point rqirmnt spcifi in Prt (b). Frthrmor, trmin th rslting horizontl orinttion of s xprss by its ngl with rspct to. SOLUTION ) With /2, th forwr kinmtics qtions tk th following forms. ˆ / /2 C ˆ 4 / /2 4 5 /2 26 /2 C 6 M. K. Özgörn

7 b) ˆ /2 /2 C (ED) 2 2 / r s ( cos sin ) r s [ cos sin ] r s ( cos sin ) ; (ED) ˆ /2 /2 6 6 p r C r / p r r p r 4 5 p r 6 ; 4 5 (ED) Th r qtion cn b writtn gin s s ( cos sin ) r Whn th bov qtion is pr-mltipli by, 2, n, th following sclr qtions r obtin. c) 2 24 t t t r cos r sin () cos r cos r sin (2) s sin r () To fin from Eq. (), introc n r 2 sch tht r r2 cos n r2 r2 sin (4) Eq. St (4) implis tht r 2 n r known s r r r n tn 2( r2, r ) (5) With th nw vribls, Eq. () bcoms cos( ) cos / r (6) Eq. (6) implis tht r r2 24 r2 sin( ) sin / ; (7) Eq. (7), in trn, implis th first workspc constrint, which is r r (8) With th prsnc of Inq. (8), Eqtions (6) n (7) l to tn 2(, ) (9) 7 M. K. Özgörn

8 With fon, Eq. (2) cn b writtn s cos ( r cos r sin ) / (0) Eq. (0) implis tht sin ; 4 () Eq. (), in trn, implis th scon workspc constrint, which is ( r cos r sin ) sin( ) / r (2) 4 2 With th prsnc of Inq. (2), Eqtions (0) n () l to tn (, ) () Finlly, with fon 4, Eq. () givs s r sin (4) Not tht min mx s s s. Thrfor, r mst b sch tht min mx 2 4 sin4 2 4 sin4 s r s (5) Th mltipl soltions ssocit with th sign vribls n 2 r illstrt blow. (σ ) A (σ +) (σ ) (σ +) A (σ +) O (σ ) 8 M. K. Özgörn

9 ) As givn in Prt (), 4 5 In th bov xprssion, n 4 hv lry bn trmin s in Prt (c). In othr wors, s nit vctor, hs only on fr vribl, which is 5. Thrfor, cnnot b orint rbitrrily s sir. ) Th stt rqirmnt cn lso b xprss n work ot s shown blow. t 0 t t t t t ( 4)( 5 ) ( cos sin )( cos sin ) 0 sin4sin5 0 Th bov qtion implis two css: (i) If sin4 0, thn sin or 5 (ii) If sin4 0, thn 5 bcoms fr to choos rbitrrily. In cs (i), with 5 0, bcoms: 4 cos 2 sin ngl In this cs, will b forwrly pointing if is ct, i.. if cos 0. In cs (i), with 5, bcoms: ( ) ( ) 4 4 cos( ) sin( ) ngl 2 In this cs, will b forwrly pointing if cos( ) cos 0 or cos 0, i.. if is obts. In cs (ii), with 4 0 bt with n rbitry 5, bcoms: ( ) 5 5 cos( 5) 2 sin( 5) In this cs, will b forwrly pointing if cos( 5) 0, i.. if 5 5 is ct. Hnc, pon choosing th most ct vl for 5, i.. 5 0, 5 is obtin s 5. As notic, in ll th css consir bov, is confin to pln tht is prlll to th horizontl -2 pln of th bs frm. 9 M. K. Özgörn