;+ ; NAME: ; QTFIND ; ; AUTHOR: ; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770 ; craigm@lheamail.gsfc.nasa.gov ; UPDATED VERSIONs can be found on my WEB PAGE: ; http://cow.physics.wisc.edu/~craigm/idl/idl.html ; ; PURPOSE: ; Find quaternion(s) from direction cosine matrix ; ; MAJOR TOPICS: ; Geometry ; ; CALLING SEQUENCE: ; Q = QTFIND(MATRIX) ; ; DESCRIPTION: ; ; The function QTFIND determines one or more unit quaternions from ; direction cosine matrices. ; ; This routine is optimized to avoid singularities which occur when ; any one of the quaternion components is nearly zero. Up to four ; different transformations are attempted to maximize the precision ; of all four quaternion components. ; ; QTFIND and QTMAT are functional inverses: use QTFIND to convert a ; known direction cosine matrix to a new quaternion; use QTMAT to ; convert a known quaternion to matrix representation. ; ; Conventions for storing quaternions vary in the literature and from ; library to library. This library uses the convention that the ; first three components of each quaternion are the 3-vector axis of ; rotation, and the 4th component is the rotation angle. Expressed ; in formulae, a single quaternion is given by: ; ; Q(0:2) = [VX, VY, VZ]*SIN(PHI/2) ; Q(3) = COS(PHI/2) ; ; where PHI is the rotation angle, and VAXIS = [VX, VY, VZ] is the ; rotation eigen axis expressed as a unit vector. This library ; accepts quaternions of both signs, but by preference returns ; quaternions with a positive 4th component. ; ; ; INPUTS: ; ; MATRIX - array of one or more direction cosine matrices. For a ; single matrix, MATRIX should be a 3x3 array. For N ; matrices, MATRIX should be a 3x3xN array. The arrays are ; assumed to be valid rotation matrices. ; ; ; RETURNS: ; ; The resulting unit quaternions. For a single matrix, returns a ; single quaternion as a 4-vector. For N matrices, returns N ; quaternions as a 4xN array. ; ; ; KEYWORD PARAMETERS: ; ; NONE ; ; EXAMPLE: ; ; ;; Form a rotation matrix about the Z axis by 32 degrees ; th1 = 32d*!dpi/180 ; mat1 = [[cos(th1),-sin(th1),0],[sin(th1),cos(th1),0],[0,0,1]] ; ; ;; Form a rotation matrix about the X axis by 116 degrees ; th2 = 116d*!dpi/180 ; mat2 = [[1,0,0],[0,cos(th2),-sin(th2)],[0,sin(th2),cos(th2)]] ; ; ;; Find the quaternion that represents MAT1, MAT2 and the ; composition of the two, MAT2 ## MAT1. ; ; print, qtfind(mat1), qtfind(mat2), qtfind(mat2 ## mat1) ; 0.0000000 0.0000000 0.27563736 0.96126170 ; 0.84804810 0.0000000 0.0000000 0.52991926 ; 0.81519615 -0.23375373 0.14606554 0.50939109 ; ; ; SEE ALSO ; QTANG, QTAXIS, QTCOMPOSE, QTERP, QTEXP, QTFIND, QTINV, QTLOG, ; QTMAT, QTMULT, QTPOW, QTVROT ; ; MODIFICATION HISTORY: ; Written, July 2001, CM ; Documented, Dec 2001, CM ; Re-added check to enforce q(3) GE 0, 15 Mar 2002, CM ; Usage message, error checking, 15 Mar 2002, CM ; ; \$Id: qtfind.pro,v 1.8 2008/12/14 20:00:31 craigm Exp \$ ; ;- ; Copyright (C) 2001, 2002, Craig Markwardt ; This software is provided as is without any warranty whatsoever. ; Permission to use, copy, modify, and distribute modified or ; unmodified copies is granted, provided this copyright and disclaimer ; are included unchanged. ;- function qtfind, amat ; THIS ROUTINE CONVERTS ROTATION MATRIX AMAT INTO QUATERNION AQT ; IT ASSUMES AMAT IS A VALID ROTATION MATRIX ; THIS IS ADAPTED FROM CHAPTER 12 BY F.L.MARKLEY ; ; MODIFIED 11/22/95 TO AVOID SINGULARITIES (E.G., Q4=0.) ; THE SQUARE OF ONE OF THE QUATERNION ELEMENTS MUST BE >= 0.25 ; SINCE THE 4 SUM TO 1. ; MOD 11/24/95 TO MAKE SURE Q4 >= 0 ; MOD 14-DEC-95 TO FIX BUG OF WRONG SIGN OF Q4 IF Q1,Q3,&Q4 < .5 if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'Q = QTFIND(MATRIX)', /info message, ' MATRIX must be a 3x3xN array of direction cosines', \$ info=info return, 0 endif sz = size(amat) if sz(0) LT 2 then begin DIM_ERROR: message, 'ERROR: MATRIX must be a 3x3xN array', /info return, 0 endif if sz(1) NE 3 OR sz(2) NE 3 then goto, DIM_ERROR nq = n_elements(amat)/9 ad0 = amat(0,0,*) & ad1 = amat(1,1,*) & ad2 = amat(2,2,*) a12 = amat(1,2,*) & a21 = amat(2,1,*) a20 = amat(2,0,*) & a02 = amat(0,2,*) a01 = amat(0,1,*) & a10 = amat(1,0,*) n1 = nq q0 = replicate(amat(0)*0+0., nq) & q1 = q0 & q2 = q0 & q3 = q0 qd = 1. + ad0 + ad1 + ad2 wh = where(qd GE 0.99, ct) if ct GT 0 then begin qx = 0.5*sqrt(qd(wh)) q3(wh) = qx qx = qx * 4 q0(wh) = (a12-a21)(wh)/qx q1(wh) = (a20-a02)(wh)/qx q2(wh) = (a01-a10)(wh)/qx n1 = n1 - ct endif if n1 GT 0 then begin qd = 1. + ad0 - ad1 - ad2 wh = where(qd GE 0.99, ct) if ct GT 0 then begin qx = 0.5*sqrt(qd(wh)) q0(wh) = qx qx = qx * 4 q3(wh) = (a12-a21)(wh)/qx q2(wh) = (a20+a02)(wh)/qx q1(wh) = (a01+a10)(wh)/qx n1 = n1 - ct endif endif if n1 GT 0 then begin qd = 1. - ad0 - ad1 + ad2 wh = where(qd GE 0.99, ct) if ct GT 0 then begin qx = 0.5*sqrt(qd(wh)) q2(wh) = qx qx = qx * 4 q1(wh) = (a12+a21)(wh)/qx q0(wh) = (a20+a02)(wh)/qx q3(wh) = (a01-a10)(wh)/qx n1 = n1 - ct endif endif if n1 GT 0 then begin qd = 1. - ad0 + ad1 - ad2 wh = where(qd GE 0.99, ct) if ct GT 0 then begin qx = 0.5*sqrt(qd(wh)) q1(wh) = qx qx = qx * 4 q2(wh) = (a12+a21)(wh)/qx q3(wh) = (a20-a02)(wh)/qx q0(wh) = (a01+a10)(wh)/qx n1 = n1 - ct endif endif wh = where(q3 LT 0, ct) if ct GT 0 then begin q0(wh) = -q0(wh) q1(wh) = -q1(wh) q2(wh) = -q2(wh) q3(wh) = -q3(wh) endif return, transpose([[q0],[q1],[q2],[q3]]) end