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Problem with Euler angles |
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| Aug12-04, 08:29 AM | #1 |
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Problem with Euler angles
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nI encountered the following problem:\nA transformation T is given in the form\nT = Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0)\nwhere Ry(teta) is rotation around (current) y axis by angle teta and\nRz(phi) is rotation around (current) z axis by angle phi. Angles teta0\n.... tetaN, phi1 ... phiN are given. Task: express T in the form\nT = Rz(gama)Ry(beta)Rz(alfa)\nwhere alfa, beta, and gama are the corresponding Euler angles.\nAny ideas?\nComment: The transformation T can be more general (arbitrary series of\nrotations). I do not know if the special case I present here makes the\nsolution simpler. Anyway, I am interested only in this special case.\n\nSincerely,\nItsik\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I encountered the following problem:
A transformation T is given in the form T = Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0) where Ry(teta) is rotation around (current) y axis by angle teta and [itex]Rz(\phi)[/itex] is rotation around (current) z axis by angle [itex]\phi[/itex]. Angles teta0 .... tetaN, phi1 ... phiN are given. Task: express T in the form [itex]T = Rz(gama)Ry(\beta)Rz(alfa)[/itex] where alfa, [itex]\beta,[/itex] and gama are the corresponding Euler angles. Any ideas? Comment: The transformation T can be more general (arbitrary series of rotations). I do not know if the special case I present here makes the solution simpler. Anyway, I am interested only in this special case. Sincerely, Itsik |
| Aug13-04, 05:41 AM | #2 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Itsik Weissman" <itsikw@advisol.co.il> wrote in message\nnews:996cc19e.0408110627.148206c7@posting.google.com...\n>\ n>\n>\n>\n> I encountered the following problem:\n> A transformation T is given in the form\n> T = Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0)\n> where Ry(teta) is rotation around (current) y axis by angle teta and\n> Rz(phi) is rotation around (current) z axis by angle phi. Angles teta0\n> ... tetaN, phi1 ... phiN are given. Task: express T in the form\n> T = Rz(gama)Ry(beta)Rz(alfa)\n> where alfa, beta, and gama are the corresponding Euler angles.\n> Any ideas?\n> Comment: The transformation T can be more general (arbitrary series of\n> rotations). I do not know if the special case I present here makes the\n> solution simpler. Anyway, I am interested only in this special case.\n>\n> Sincerely,\n> Itsik\n\nItsik,\n\nIf you have Mathematica, I have at my web site a Mathematica Application on\nRotations and Euler Angles. It has a Mathematica package and four notebooks\nwith many graphics and animations illustrating rotations. One animation, for\nexample, allows the user to rotate two books in side by side panels using\ndifferent rotation sequences.\n\nThe third notebook on Euler Angles demonstrates routines in the package that\nallow the user to calculate all 24 possible Euler angle decompositions.\nThere are 12 different rotation sequences and 2 decompositions for each\nsequence. If your input angles are all numeric, you can multiply all your\ninitial matrices and then decompose into whichever sequence you desire.\n\nIf you send me a set of input angles, I will do a calculation for you.\n\nDavid Park\ndjmp@earthlink.net\nhttp://home.earthlink.net/~djmp/\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Itsik Weissman" <itsikw@advisol.co.il> wrote in message
news:996cc19e.0408110627.148206c7@posting.google.com... > > > > > I encountered the following problem: > A transformation T is given in the form > T = Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0) > where Ry(teta) is rotation around (current) y axis by angle teta and > [itex]Rz(\phi)[/itex] is rotation around (current) z axis by angle [itex]\phi.[/itex] Angles teta0 > ... tetaN, phi1 ... phiN are given. Task: express T in the form > [itex]T = Rz(gama)Ry(\beta)Rz(alfa)[/itex] > where alfa, [itex]\beta,[/itex] and gama are the corresponding Euler angles. > Any ideas? > Comment: The transformation T can be more general (arbitrary series of > rotations). I do not know if the special case I present here makes the > solution simpler. Anyway, I am interested only in this special case. > > Sincerely, > Itsik Itsik, If you have Mathematica, I have at my web site a Mathematica Application on Rotations and Euler Angles. It has a Mathematica package and four notebooks with many graphics and animations illustrating rotations. One animation, for example, allows the user to rotate two books in side by side panels using different rotation sequences. The third notebook on Euler Angles demonstrates routines in the package that allow the user to calculate all 24 possible Euler angle decompositions. There are 12 different rotation sequences and 2 decompositions for each sequence. If your input angles are all numeric, you can multiply all your initial matrices and then decompose into whichever sequence you desire. If you send me a set of input angles, I will do a calculation for you. David Park djmp@earthlink.net http://home.earthlink.net/~djmp/ |
| Aug14-04, 06:58 AM | #3 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOn 12 Aug 2004 09:29:04 -0400, itsikw@advisol.co.il (Itsik Weissman)\nwrote:\n\n>\n>\n>\n>\n>I encountered the following problem:\n>A transformation T is given in the form\n>T = Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0)\n>where Ry(teta) is rotation around (current) y axis by angle teta and\n>Rz(phi) is rotation around (current) z axis by angle phi. Angles teta0\n>... tetaN, phi1 ... phiN are given. Task: express T in the form\n>T = Rz(gama)Ry(beta)Rz(alfa)\n>where alfa, beta, and gama are the corresponding Euler angles.\n>Any ideas?\n>Comment: The transformation T can be more general (arbitrary series of\n>rotations). I do not know if the special case I present here makes the\n>solution simpler. Anyway, I am interested only in this special case.\n>\n>Sincerely,\n>Itsik\nThe Euler angles are characterized by a "gimbal order" 323 or zyz,\nbeing rotation matrices about z then y then z. Quite obviously there\nis no general method of reducing these to functions of the angles you\ncite.\n\nBUT, take a cascade of 3 successive sets of Euler angles and you can\ncondense them somewhat by linear combination of the X\'s:\nXYX *XYX * XYX\n= XY(XX)Y(XX)YX\n= XYX\'YX"YX\nYou can make linear combinations of the abutting XX\'s = X+X.\n\nSo that does satisfy your "Comment: The transformation T can be more\ngeneral (arbitrary series of >rotations)". I have made use of the only\nlinear combinations available.\n\nAny other investigation might be to sent a unit vector through the\nnumerical matrix and from the resultant, find the eigenvector, etc.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On 12 Aug 2004 09:29:04 [itex]-0400,[/itex] itsikw@advisol.co.il (Itsik Weissman)
wrote: > > > > >I encountered the following problem: >A transformation T is given in the form [itex]>T =[/itex] Ry(tetaN)Rz(phiN)Ry(tetaN-1)Rz(phiN-1)...Ry(teta1)Rz(phi1)Ry(teta0) >where Ry(teta) is rotation around (current) y axis by angle teta and [itex]>Rz(\phi)[/itex] is rotation around (current) z axis by angle [itex]\phi[/itex]. Angles teta0 >... tetaN, phi1 ... phiN are given. Task: express T in the form [itex]>T = Rz(gama)Ry(\beta)Rz(alfa)[/itex] >where alfa, [itex]\beta,[/itex] and gama are the corresponding Euler angles. >Any ideas? >Comment: The transformation T can be more general (arbitrary series of >rotations). I do not know if the special case I present here makes the >solution simpler. Anyway, I am interested only in this special case. > >Sincerely, >Itsik The Euler angles are characterized by a "gimbal order" 323 or zyz, being rotation matrices about z then y then z. Quite obviously there is no general method of reducing these to functions of the angles you cite. BUT, take a cascade of 3 successive sets of Euler angles and you can condense them somewhat by linear combination of the X's: XYX [itex]*XYX *[/itex] XYX [itex]= XY(XX)Y(XX)YX[/itex] = XYX'YX"YX You can make linear combinations of the abutting XX's [itex]= X+X[/itex]. So that does satisfy your "Comment: The transformation T can be more general (arbitrary series of >rotations)". I have made use of the only linear combinations available. Any other investigation might be to sent a unit vector through the numerical matrix and from the resultant, find the eigenvector, etc. |
| Aug31-04, 03:55 AM | #4 |
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Problem with Euler angles
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <mqkph0tvikohojqkc7fsboqhsmd27e339e@4ax.com>,\nJohn C. Polasek <jpolasek@cfl.rr.com> wrote:\n>\n\n\nThere\'s a neat newsgroup: comp.graphics.algorithms\n\nThere are some true geniuses there, it appears,\nthree of the most prolific marked by the following\ntrn regexp:\n\n/eberly\\/faqs\\|broeker/tf\n\nThe guy (who calls him/herself ...d\'Faqs) has recently (last three\nweeks?) written the clearest set of (long) articles about\nEuler angles and, in detail, why in many cases they\'re\nnot the things to use, where as quaternions work a lot\nbetter.\n\nNo, not the usual treatment you see in books, seems to me,\nand *very worth reading*.\n\nAgain, in the last three or four weeks there\'s been\ncome incredible threads related to one or another\naspect of Euler angles.\n\nEnjoy!\n\nDavid\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <mqkph0tvikohojqkc7fsboqhsmd27e339e@4ax.com>,
John C. Polasek <jpolasek@cfl.rr.com> wrote: > There's a neat newsgroup: comp.graphics.algorithms There are some true geniuses there, it appears, three of the most prolific marked by the following trn regexp: [tex]/eberly\/faqs\|broeker/tf[/tex] The guy (who calls him/herself ...d'Faqs) has recently (last three weeks?) written the clearest set of (long) articles about Euler angles and, in detail, why in many cases they're not the things to use, where as quaternions work a lot better. No, not the usual treatment you see in books, seems to me, and [itex]*very[/itex] worth reading*. Again, in the last three or four weeks there's been come incredible threads related to one or another aspect of Euler angles. Enjoy! David |
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