Three-Dimensional Experimental Kinematics

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1 Notes_5_3 o 8 Three-Dmesoal Epermetal Kematcs Dgte locatos o ladmarks { r } o bod or pots to at gve tme t All pots must be o same bod Use ladmark weghtg actor = pot k s avalable at tme t. Use = pot k ot avalable at gve tme t. r r r at gve tme t. Determe { } { } { } Mea values mea { r } = { r } / = = = = = / / / / k mea { r } { r } mea { r } { r } mea { r } { r } mea { r } { r } = mea [ X ] ({ r } { r } ){ r } { r } [ M] = ([ I3 ] trace( [ X] )) [ X] mea T ( ) / Veloct [ V] = mea T ( ) / { r } { r } { r } 32 { ω } = [ M] V V = ω ω = orm{ ω } V V 3 2 V V ω ω

2 ω [ ω ] = ω ω { û} = { ω} ω / ω ω ω Notes_5_3 2 o 8 ot ISA s o the stataeous screw as or bod at the root o the perpedcular rom the cetrod o the ladmarks. Note that the ISA s ot attached to the bod. A pot o the bod cocdet wth the ISA has traslatoal veloct s alog the ISA. ISA ( ) or a pot attached to bod { r } = s { û} + [ ω ]{ r } { r} ISA mea mea 2 { r} = { r } + [ ω]{ r } / ω s = { û} T { r } mea Accelerato [ A] = mea T ( ) / { r } { r } { r } [ B] = [ A] [ ω ][ ω ][ X] 32 { ω } = [ M] B B = ω ω = orm{ ω } B B 3 2 B B ω ω [ ω ] = ω ω [ β ] = [ ω ] ω ω ω ω ot IA s the stataeous accelerato pole or bod. Note that the IA s ot attached to the bod. ot o the bod cocdet wth IA has ero accelerato. IA ( ) { r } = [ β ]{ r } { r} or a pot attached to bod IA mea mea _ IA { r} = { r } [ β ] { r } or { r } _ at =

3 Notes_5_3 3 o 8 Jerk [ J] = mea T ( ) / { r } { r } { r } ( )[ X] [ H] = [ J] 2[ ω ] 32 { ω } = [ M] H H = ω ω = orm{ ω } H H 3 2 H H ω ω [ ω ] = ω ω [ η ] = [ ω ] + 2[ ω ] ω ω ω ω ot IJ s the stataeous jerk pole or bod. Note that the IJ s ot attached to the bod. ot o the bod cocdet wth IJ has ero jerk. IJ ( ) { r } = [ η ]{ r } { r} or a pot attached to bod IJ mea mea _ IJ { r} = { r } [ η ] { r } or { r} _ at = Secod Order Screw As Ω Ω 2 { Ω } = Ω = [ ω]{ ω }/ ω Ω = orm{ Ω} { tˆ } = { Ω} / Ω d = T IA IHA T ({ tˆ } [ β]{ ( r} { r} )/ { tˆ } [ β]{ û} { c} = { r} IHA + d{ û} ( ) T ({ } [ ω IA 2 tˆ ][ β]{ ( c} { r} )/ ω ) s Ω ω S = /

4 Notes_5_3 4 o 8 t_lm2k3d.m - test 3D kematcs rom ladmark moto HJSIII, 4..4 clear eample data - CRS = [ ]; r = [ ; ; ]; rd = [ ; ; ]; rdd = [ ; ; ]; rddd = [ ; ; ]; vel_test = [ ; ; ]; accel_test = [ ; ; ]; jerk_test = [.5. ; ; ]; aode_test = [ ; ; ]; eample data - RSUR = [ ]; r = [ ; ; ]; rd = [ ; ; ]; rdd = [ ; ; ]; rddd = [ ; ; ]; vel_test = [ ; ; ]; accel_test = [ ; ; ]; jerk_test = [ ; ;

5 Notes_5_3 5 o ]; aode_test = [ ; ; ]; test ucto [ vel, accel, jerk, aode ] = lm2k3d(, r, rd, rdd, rddd ) bottom o t_lm2k3d

6 Notes_5_3 6 o 8 ucto [ vel, accel, jerk, aode ] = lm2k3d(, r, rd, rdd, rddd ) 3D kematcs o a rgd bod rom ladmark moto HJSIII, 4..4 USAGE [ vel, accel, jerk, aode ] = lm2k3d(, r, rd, rdd, rddd ) INUTS - vector o weghts - (j)= meas data vald, (j)= meas data ot avalable r - 3 matr o,, ladmark locato rd - 3 matr o,, ladmark veloct rdd - 3 matr o,, ladmark accelerato rddd - 3 matr o,, ladmark jerk OUTUTS vel = [ w_vec risa sisa ] w_vec = 3 agular veloct vector risa = 3 locato o ISA at root o perpedcular rom cetrod o ladmarks sisa = sldg veloct vector alog ISA accel = [ wd_vec ria rdd_at_ia ] wd_vec = 3 agular accelerato vector ria = 3 locato o accelerato pole rdd_at_ia = 3 accelerato o pot o bod at IA jerk = [ wdd_vec rij rddd_at_ij ] wdd_vec = 3 agular jerk vector rij = 3 locato o jerk pole rddd_at_ij = 3 jerk o pot o bod at IJ aode = [ OMEGA_vec c Sd ] OMEGA_vec = 3 rotato o secod order screw c = 3 cetral pot o geerator Sd = 3 sldg veloct vector alog secod order screw costats eps = e-4; umber o coordates ad ladmarks [ coord, ] = se( r ); mea values mat = dag(); s = trace( mat ); rm = sum( mat*r' )' /s; rdm = sum( mat*rd' )' /s; rddm = sum( mat*rdd' )' /s; rdddm = sum( mat*rddd' )' /s; cetered locato rc = r - rm*oes(,); X = rc * mat * rc' /s; M = trace(x) * ee(coord) - X; Mv = v( M ); veloct V = rd * mat * rc' /s; w_vec = Mv * [ V(3,2)-V(2,3) ; V(,3)-V(3,) ; V(2,)-V(,2) ]; w = orm( w_vec ); w_mat = skew_sm( w_vec ); geeral veloct soluto w > eps, u = w_vec / w; sd = u' * rdm; risa = rm + w_mat * rdm / w^2; sisa = sd * u; specal case - w=, pure traslato risa s at cetrod o ladmarks, sisa s traslato veloct sd = orm( rdm ); u = rdm / sd;

7 Notes_5_3 7 o 8 risa = rm; sisa = rdm; ed accelerato A = rdd * mat * rc' / s; B = A - w_mat*w_mat * X; wd_vec = Mv * [ B(3,2)-B(2,3) ; B(,3)-B(3,) ; B(2,)-B(,2) ]; wd = orm( wd_vec ); wd_mat = skew_sm( wd_vec ); beta_mat = wd_mat + w_mat*w_mat; geeral accelerato soluto abs(det(beta_mat)) > eps; ria = rm - v(beta_mat) * rddm; rdd_at_ia = eros(coord,); specal case - w=, wd=, pure traslato ria s at cetrod o ladmarks, rdd_at_ia s traslato accelerato w < eps, wd < eps, sdd = orm( rddm ); e = rddm / sdd; ria = rm; rdd_at_ia = rddm; specal case 2 - w=, wd>, pure agular accelerato smlar to geeral agular veloct soluto ria s at root o perpedcular to agular accelerato vector rom cetrod o ladmarks rdd_at_ia s traslato accelerato e = wd_vec / wd; sdd = e' * rddm; ria = rm + wd_mat * rddm / wd^2; rdd_at_ia = sdd * e; ed specal case 3 - w>, wd=, pure agular veloct smlar to ero agular veloct soluto wd < eps, e = u; sdd = e' * rddm; rdd_at_ia = sdd * e; ria = rm + (rddm-rdd_at_ia) / w*w; specal case 4 - w>, wd>, w_vec parallel to wd_vec e = wd_vec / wd; sdd = e' * rddm; rdd_at_ia = sd * e; ed ed ed w2a = w*w*ee(3) - wd_mat; ria = rm + v(w2a) * (rddm-rdd_at_ia); jerk J = rddd * mat * rc' / s; eta_mat_mwdd = 2*wd_mat*w_mat + w_mat*wd_mat + w_mat*w_mat*w_mat; H = J - eta_mat_mwdd * X; wdd_vec = Mv * [ H(3,2)-H(2,3) ; H(,3)-H(3,) ; H(2,)-H(,2) ]; wdd = orm( wdd_vec ); h = wdd_vec / wdd; wdd_mat = skew_sm( wdd_vec ); eta_mat = eta_mat_mwdd + wdd_mat; rij = rm - v(eta_mat) * rdddm; rddd_at_ij = eros(coord,); secod order screw

8 Notes_5_3 8 o 8 OMEGA_vec = w_mat * wd_vec /w/w; OMEGA = orm( OMEGA_vec ); t = OMEGA_vec / OMEGA; d = t' * beta_mat * (ria-risa) / (t' * beta_mat * u ); c = risa + d * u; Sd = ( t' * w_mat * beta_mat * (c-ria) /w/w ) - sd * OMEGA /w; Sd_vec = Sd * t; retur argumets vel = [ w_vec risa sisa ]; accel = [ wd_vec ria rdd_at_ia ]; jerk = [ wdd_vec rij rddd_at_ij ]; aode = [ OMEGA_vec c Sd_vec ]; bottom o lm2k3d

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