{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 255 0 0 1 0 2 2 0 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "Helvetica" 1 10 0 128 0 1 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "Helvetica" 1 10 0 0 255 1 1 2 2 0 0 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "" 0 1 0 128 128 1 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 1 1 2 2 0 0 2 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "helvetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "h elvetica" 1 12 0 0 0 0 2 1 2 0 0 2 0 0 0 1 }1 0 0 -1 8 2 0 0 0 0 0 0 -1 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "helvetica" 1 10 0 0 0 0 1 2 0 0 0 1 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 22 "Singularity Loci of 3-" }{TEXT 259 1 "R" }{TEXT 260 2 "RR" }{TEXT 261 85 " Planar Parallel Man ipulators\nSpecial Case: Distal and Proximal Links of Equal Length" }} }{EXCHG {PARA 19 "" 0 "" {TEXT -1 55 "Ilian A. Bonev and Clement M. Go sselin\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 the Jac obian Matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "l1 := 1: l2 := 1:\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 "" {MPLTEXT 1 0 268 "for i from 1 to 3 do\n p|| i := (rho||i + l1^2 - l2^2)/(2*l1):\n Gamma||i := rho||i - p||i^2:\n \+ sinT||i := (p||i*(Y+b||i) + (X+a||i)*Delta||i)/rho||i:\n cosT||i := \+ (p||i*(X+a||i) - (Y+b||i)*Delta||i)/rho||i:\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 39 "Part II: Obtaining the Singularit y Loci" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Eq := simplify(nu mer(det(J))):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 53 "T he denominator of det(J) is simply 8*rho1*rho2*rho3." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 36 "1 . Getting Rid of The First Radical" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 18 168 "After rearranging Eq and rasing t o power two, we cancel the common factors rho1, rho2, and rho3. Note\n that these comprise in fact the numerator of the determinant of J." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Eq1 := expand( coeff(Eq,De lta1,1)^2*factor(Gamma1/rho1) -\n expand( factor(coeff(Eq,Delta1,0)/r ho1)^2 * rho1 ) ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 234 "If we try \+ to substitute Delta2^2 with Gamma2 and Delta3^2 with Gamma3 directly i n Eq1, the expression\nfor the latter becomes too big to handle. That \+ is why, we partition Eq1 in 9 different parts and perform the\nsubstit utions manually." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 655 "E11 := factor(coeff(coeff(Eq1,Delta2,0),Delta3,0)/(rho2*rho3)):\nE12 := fact or(coeff(coeff(Eq1,Delta2,0),Delta3,1)/(rho2*rho3)):\nE13 := factor(co eff(coeff(Eq1,Delta2,1),Delta3,0)/(rho2*rho3)):\nE14 := factor(coeff(c oeff(Eq1,Delta2,1),Delta3,1)/(rho2*rho3)):\nE15 := factor(coeff(coeff( Eq1,Delta2,0),Delta3,2)/rho2)*factor(Gamma3/rho3):\nE16 := factor(coef f(coeff(Eq1,Delta2,2),Delta3,0)/rho3)*factor(Gamma2/rho2):\nE17 := fac tor(coeff(coeff(Eq1,Delta2,1),Delta3,2)/rho2)*factor(Gamma3/rho3):\nE1 8 := factor(coeff(coeff(Eq1,Delta2,2),Delta3,1)/rho3)*factor(Gamma2/rh o2):\nE19 := coeff(coeff(Eq1,Delta2,2),Delta3,2)*\n factor(Gamma 2/rho2)*factor(Gamma3/rho3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Eq1 := E11 + E12*Delta3 + E13*Delta2 + E14*Delta2*Delta3 + E15 \+ + E16 +\n E17*Delta2 + E18*Delta3 + E19:" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 37 "2 . Getting Rid of The Second Radical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "E21 := E13 + E17:\nE22 := E14:\nE23 := E11 + E15 + E16 + E19:\nE24 := E12 + E18:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Eq2 := E21^2*Gamma2 + E22^2*Gamma2*Gamma3 + 2*E 21*E22*Gamma2*Delta3 -\n (E23^2 + Eq24^2*Gamma3 + 2*E23*Eq24*Del ta3): " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 36 "3 . Getting Rid of The \+ Third Radical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Eq3 := (2 *E21*E22*Gamma2 - 2*E23*E24)^2*Gamma3 -\n (E21^2*Gamma2 + E22^2* Gamma2*Gamma3 - E23^2 - E24^2*Gamma3)^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 97 "Finally, we substitute the expressions fot rho1, rho2, an d rho3 to obtain the desired polynomial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Eq3f := subs(\{rho1=X^2+Y^2, rho2=(X+a2)^2+(Y+b2)^2, \+ rho3=(X+a3)^2+(Y+b3)^2\}, Eq3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "degree(Eq3f,X), degree(Eq3f,Y), degree(Eq3f,[X,Y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#[F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 199 "The degree of the polynomial in Eq3f seems to be 48 since the \+ latter is not expanded. However, if we\nsimplify the coefficients of a ll terms of degree more than 42, we can see that they are all zeros." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "for i from 0 to 48 do\n \+ for j from max(0,43-i) to (48-i) do\n print(simplify(coeff(coeff(E q3f,X,i),Y,j)));\n od;\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 18 81 "S urprisingly, the coefficients of all terms of degree less than 8 are a lso zeros." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i from 0 to 7 do\n for j from 0 to (7-i) do\n print(simplify(coeff(coeff(E q3f,X,i),Y,j)));\n od;\nod;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 18 73 "Therefore, the degree of the polynomial, in fact fewnom ial, in Eq3 is 42." }}}}{MARK "5 0 0" 12 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }