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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 22 "Singularity Loci of 3-" 
}{TEXT 259 1 "R" }{TEXT 260 2 "RR" }{TEXT 261 90 " Planar Parallel Man
ipulators\nGeneral Case: Distal and Proximal Links of Different Length
s" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 55 "Ilian A. Bonev and Clement \+
M. Gosselin\nOctober 24, 2000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "restart: with(linalg):" }}}{EXCHG {PARA 3 "" 0 "" {TEXT 257 26 "
Part I: Deriving the Expre" }{TEXT 31 0 "" }{TEXT 258 29 "ssion for th
e Jacobian Matrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 149 "In this cas
e we cannot proceed symbolically but instead assign random integer num
erical values to the\nparameters describing the parallel manipulator.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "RandNum := evalf(rand(
-13..12)):\nfor i from 1 to 3 do\n  a||i := RandNum(): b||i := RandNum
():\n  c||i := RandNum(): d||i := RandNum():\nod:\nl1 := 1: l2 := 1+ab
s(RandNum()):\na1 := 0: b1 := 0: c1 := 0: d1 := 0:\nd2 := b2:" }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 34 "1. Unit Vectors Along Distal Links
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 52 "In all the derivations, we co
nsider only one branch." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 
"for i from 1 to 3 do\n  p||i := (rho||i + l1^2 - l2^2)/(2*l1):\n  Gam
ma||i := rho||i - p||i^2:\n  sinT||i := (p||i*(Y+b||i) + (X+a||i)*Delt
a||i)/rho||i:\n  cosT||i := (p||i*(X+a||i) - (Y+b||i)*Delta||i)/rho||i
:\n  rho||i := (X+a||i)^2 + (Y+b||i)^2:\n  n||i := [X + a||i - l1*cosT
||i, Y + b||i - l1*sinT||i]:  \nod:" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
-1 16 "2 . Vectors E*si" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "
for i from 1 to 3 do\n  es||i := [-d||i, c||i];\nod:" }}}{EXCHG {PARA 
4 "" 0 "" {TEXT -1 19 "3 . Jacobian Matrix" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 175 "J := matrix(3,3,[n1[1],n1[2],n1[1]*es1[1] + n1[2]*
es1[2],\n                 n2[1],n2[2],n2[1]*es2[1] + n2[2]*es2[2],\n  \+
               n3[1],n3[2],n3[1]*es3[1] + n3[2]*es3[2]]):" }}}{EXCHG 
{PARA 3 "" 0 "" {TEXT 262 40 "Part II: Obtaining the Singularity Locus
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Eq := simplify(numer(de
t(J))):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 40 "1 . Getting Rid of The
 First Square Root" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 169 "After rear
ranging Eq and raising to power two, we cancel the common factors rho1
, rho2, and rho3. Note\nthat these comprise in fact the numerator of t
he determinant of J." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Eq
1 := expand(coeff(Eq,Delta1,1)^2*Gamma1 - coeff(Eq,Delta1,0)^2):\nEq1 \+
:= factor(Eq1/rho1):\nEq1 := expand(subs(Delta3^2=Gamma3, subs(Delta2^
2=Gamma2, Eq1) )):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 41 "2 . Getting
 Rid of The Second Square Root" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 147 "Eq2 := expand(coeff(Eq1,Delta2,1)^2*Gamma2) - expand(coeff(Eq
1,Delta2,0)^2):\nEq2 := factor(Eq2/rho2^2):\nEq2 := simplify(subs(Delt
a3^2=Gamma3,Eq2)):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 40 "3 . Getting
 Rid of The Third Square Root" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 103 "Eq3 := expand(coeff(Eq2,Delta3,1)^2*Gamma3) - expand(coeff(Eq2,
Delta3,0)^2):\nEq3 := factor(Eq3/rho3^4):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "degree(Eq3,X), degree(Eq3,Y), degree(Eq3,[X,Y]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#UF#F#" }}}}{MARK "16 0 0" 96 }
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