{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Titl e" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 1 2 258 1 }{PSTYLE "AC - Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 259 1 }{PSTYLE "AC - Note" -1 258 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "AC - Normal Text" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Se ction Heading" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 260 1 }{PSTYLE "AC - Disclaimer " -1 261 1 {CSTYLE "" -1 -1 "Times" 1 9 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 12 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Author" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 259 1 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 261 62 "Finding the Shortest but Smoot h Path for the Path of a Robot. " }{TEXT 257 0 "" }}{PARA 263 "" 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 144 "Nathan Collier, Autar Kaw, Jai Paul , Michael Keteltas, University of South Florida , kaw@eng.usf.ed u , http://numericalmethods.eng.usf.edu/mws " }}{PARA 258 "" 0 "" {TEXT -1 221 "NOTE: This worksheet demonstrates the use of Maple for f inding the shortest but smooth path for the path of a robot in the are a of manufacturing. It illustrates how spline interpolation can be use d to determine this path." }}{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "I ntroduction" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 259 445 "The fol lowing example illustrates the real world use of interpolation to find the shortest but smooth path of a robot. A robot arm equipped with a \+ laser is doing a quick quality check of the radius on six holes on a r ectangular plate 15\" x 10\". The center locations of the six holes ar e given as (2, 7.2), (4.25, 7.1), (5.25, 6), (7.81, 5), (9.2, 3.5), (1 0.6, 5). Find the shortest but smooth path for the robot from the firs t to the last point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 17 "Section I : Data." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The following is the data (x-y) coordinate data of the center of the \+ six holes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xy:=[[2,7.2], [4.25,7.1],[5.25,6],[7.81,5],[9.2,3.5],[10.6,5]]:" }}}}{PARA 3 "" 0 " " {TEXT 263 18 "Plotting the data:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot(xy,x=0..12,y=0..8,style=POINT,symbol=CIRCLE,symb olsize=20,title=\"Plot of the data points.\");" }}{PARA 13 "" 1 "" {GLPLOT2D 412 346 346 {PLOTDATA 2 "6(-%'CURVESG6$7(7$$\"\"#\"\"!$\"3;+ ++++++s!#<7$$\"3+++++++]UF-$\"3k*************4(F-7$$\"3+++++++]_F-$\" \"'F*7$$\"3h************4yF-$\"\"&F*7$$\"3G*************>*F-$\"3++++++ ++NF-7$$\"3)*************f5!#;F;-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*FM-%' SYMBOLG6$%'CIRCLEG\"#?-%&TITLEG6#Q9Plot~of~the~data~points.6\"-%&STYLE G6#%&POINTG-%+AXESLABELSG6$Q\"xFWQ\"yFW-%%VIEWG6$;FM$\"#7F*;FM$\"\")F* " 1 5 4 1 20 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 27 "Section II: Linea r Splines." }}{PARA 0 "" 0 "" {TEXT -1 275 "Connecting the consecutive data points using linear splines will give the shortest path. However , this path is not smooth as the derivatives will be discontinuous at \+ the interior data points. However, this will form a baseline for other calculations in the next two sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "linear_spline:=spline([ 2,4.25,5.25,7.81,9.2,10.6],[7.2,7.1,6,5,3.5,5],x,linear);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.linear_splineG-%*PIECEWISEG6'7$,&$\"+*)))))) G(!\"*\"\"\"*&$\"+WWWWW!#6F-%\"xGF-!\"\"2F2$\"$D%!\"#7$,&$\"+++]x6!\") F-*&$\"+++++6F,F-F2F-F32F2$\"$D&F77$,&$\"+]7y]!)F,F-*&$\"+++D1R!#5F-F2 F-F32F2$\"$\"yF77$,&$\"+bd!GM\"F:B>N:F:nyyy yy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::fyyyyya:nYf::G:I:wAyA::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::g jhuavwIq]Vr;V:>j@Z:j:vCSmlJ::::::::::OJ;@jyy yyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<::::::=J:nF>: V:Y:G:;:wA?:Jyyyyg:n:nY>;F;N;;j ?J@j@>:W:YJ:><:a:c:e:gJ:v<Z:B:::LJ@J:::::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV @cYH_WV>Z:::::::V:C:;:::::::::JHJ;Nw:>q:iA]AVXF:ZrK:dJ;@RJN\\=`F>Ihj:Dk;A@bRf[:FjGJMQ:WD;F:M:Ic::EZ:F[^:NZ :vYxI:;J:<::::::CJ:D_mlVH[KR<:;B:::::::JFNZ;V:;J:[F;B:F:Y<>Z:>:::::::::J?B:yay=J:B::::::nYyA<::::::::::::jysy:> :<::::::::j:E:=b:yyyyq\\:B:;xyyqTyyyyY:lZ:ft>j?<:G;Sj`@Pt\\Pd`QrP@@J>>:oi:;JBB:JrHJ\\Ij< jC>o>V[>^:f?;:^rwM:<:[V:>Z<>ZaTX UeRYEUXQZB:aE=OREgrJKxWyrJKxW;CjOxW>@C:UK;>;;JSdJOIJOEj:>:_YXs:qQ:uI;B :>Lb:DZJVDvGZ:>iCN[`o\\=?Sy:;;:f?AJb=VvBF:vxs?;N@oE;AI<=:k YXK:_K_Djk]j:JuaQ>JS>BQd;=J:vvs?;N@om_Tj:JnaQ>JSNCQd;=:GYXK:_KJHjXPj:j O`Q>JSJv@j:>:uSXs:qQJxI;B:>L=:Cb:;N@KU:ug:=J:Nfs?;N@;H;yg:=:=UXs:qQ:[:JB;Cb:^D:::y:;;Z:jPV:C:[Q ;>;N@;k?eJ;Jl\\Q>>:_K`LJQXJ;>:C=N@Wc:Qs;?:CUXK:_KJ>B=B:Vds_=VYZ:JB?:[:JSjRdj:JJ`Q>JSjiLj:jR`QHjw?>B:>LB:Cb::::jIJ J:B:f?AJaj:ZsO;n>N;yayI:>:s:qQBv::[Hw_jNDN^YQ<>:UK;^:>XAj;B:;:::Ja@Na`>;B::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::2:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:= 10.6:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The lengt h of the linear spline, 'Slinear' is :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Slinear:= int((1+diff(linear_spline,x)^2)^0.5,x=a..b);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(SlinearG$\"+8cSe5! \")" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 38 "Section III: Polynomial Interpolation." } }{PARA 0 "" 0 "" {TEXT -1 255 "Since the robot has to pass through six data points, we can interpolate the data to a fifth order polynomial. Since a fifth order polynomial has continuous first derivatives, the \+ path given by the polynomial would be smooth. The fifth order polynomi al is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "polynomial_function:=interp([2,4.25,5.25,7.81,9.2,10. 6],[7.2,7.1,6,5,3.5,5],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4polyn omial_functionG,.*&$\"+rvVMT!\")\"\"\"%\"xGF*F*$\"+%*)>)*3$F)!\"\"*&$ \"+i$yae\"F)F*)F+\"\"#F*F.*&$\"+.6B'y#!\"*F*)F+\"\"$F*F**&$\"+&fQ\"4B! #5F*)F+\"\"%F*F.*&$\"+@7M#H(!#7F*)F+\"\"&F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The length of the polynomial function, 'Spoly' is :" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Spoly:= int((1+diff(polynomial_function,x)^2)^0.5,x=a..b);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&SpolyG$\"+BaL78!\")" }}}}{PARA 4 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 266 33 "Section IV: Spline Interpolation." }}{PARA 0 "" 0 "" {TEXT -1 282 "Clearly the length of the curve from the polynomial interpolation is larger than the length obtained from linear spline interpolation. \+ Let us use cubic spline interpolation. Since a cubic spline interpolan t has continuous first derivative, the path given by it would also be \+ smooth." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "cubic_spline:=spline([2 ,4.25,5.25,7.81,9.2,10.6],[7.2,7.1,6,5,3.5,5],x,cubic);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%-cubic_splineG-%*PIECEWISEG6'7$,*$\"+qw07k!\"* \"\"\"*&$\"3.+++\\;rRR!#=F-%\"xGF-F-*&$\"3[n;6V'fu%R!#LF-),&F2F-\"\"#! \"\"F9F-F:*&$\"3m+++<91g')!#>F-)F8\"\"$F-F:2F2$\"$D%!\"#7$,*$\"+d@a,6! \")F-*&$\"3n******Hmv7#*F1F-F2F-F:*&$\"3_))*))pXTb%eF1F-),&$FCFDF:F2F- F9F-F:*&$\"3;+++(3)HeSF1F-)FQF@F-F-2F2$\"$D&FD7$,*$\"+F'p#e5FIF-*&$\"3 G+++$GX*G()F1F-F2F-F:*&$\"3iJ%G/!GNHjF1F-),&$FYFDF:F2F-F9F-F-*&$\"3*)* ****f`=lt\"F1F-)F_oF@F-F:2F2$\"$\"yFD7$,*$\"+.'QsJ\"FIF-*&$\"3)******* f-SY5!#+++SK@1 " 0 "" {MPLTEXT 1 0 52 "Scubic:= int((1+diff(cubic _spline,x)^2)^0.5,x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Scubi cG$\"+7[)G4\"!\")" }}}}{PARA 4 "" 0 "" {TEXT -1 0 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pline~Interpolation~PathFS-%'SYMBOLG6$%'CIRCLEG\"#?-%*THICKNESSG6#\"\" %-%&TITLEG6#Q]oComparison~of~Linear~Spline,~Polynomial~Function~and~Cu bic~Spline.FS-%+AXESLABELSG6$Q!FSFc]n-%%VIEWG6$;F(Fbbl;FJ$\"#5F*" 1 2 4 1 20 4 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Center of Holes" "Linear Spline Interpolation Path" "Polynomial Interpolation Path" "Cu bic Spline Interpolation Path" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }{TEXT 270 23 "Section VI: Conclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "Maple \+ helped us to apply our knowledge of numerical methods of interpolation to find the shortest but smooth path of the robot. Using Maple functi ons and plotting routines made it easy to find a solution efficiently. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Ca n you check the length of the path by using Quadratic Splines? Is the path shorter than what we obtained using cubic splines and fifth orde r polynomial interpolation?" }}}{PARA 4 "" 0 "" {TEXT 271 10 "Referenc es" }}{PARA 0 "" 0 "" {TEXT -1 4 "[1] " }{TEXT 272 156 "Autar Kaw and \+ Michael Keteltas, Holistic Numerical Methods Institute, See http://num ericalmethods.eng.usf.edu/mws/ind/05inp/mws_ind_inp_spe_shortestpath.p df" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }{TEXT 260 10 "Disclaimer" }{TEXT 273 1 ":" }{TEXT -1 248 " While ever y effort has been made to validate the solutions in this worksheet, Un iversity of South Florida and the contributors are not responsible for any errors contained and are not liable for any damages resulting fro m the use of this material." }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }