;+ ; NAME: ; QTERP ; ; 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: ; Smoothly interpolate from a grid of quaternions (spline or slerp) ; ; MAJOR TOPICS: ; Geometry ; ; CALLING SEQUENCE: ; QNEW = QTERP(TGRID, QGRID, TNEW, [/SLERP], QDIFF=, [/RESET]) ; ; DESCRIPTION: ; ; The function QTERP is used to interplate from a set of known unit ; quaternions specified on a grid of independent values, to a new set ; of independent values. For example, given a set of quaternions at ; specified key times, QTERP can interpolate at any points between ; those times. This has applications for computer animation and ; spacecraft attitude control. ; ; The "grid" of quaternions can be regularly or irregularly sampled. ; The new values can also be regularly or irregularly sampled. ; ; The simplest case comes when one wants to interpolate between two ; quaternions Q1 and Q2. In that case the user should specify the ; gridded quaterion as QGRID = [[Q1], [Q2]], with grid points at ; TGRID = [0d, 1d]. Then the user can sample any intermediate ; orientation by specifying TNEW anywhere between 0 and 1. ; ; The user has the option of performing pure spline interpolation of ; the quaternion components (the default technique). The resulting ; interpolants are normalized to be unit quaternions. This option is ; useful for fast interpolation of quaternions, but suffers if the ; grid is not well sampled enough. Spline interpolation will not ; strictly find the shortest path between two orientations. ; ; The second option is to use Spherical Linear IntERPolation, or ; SLERPing, to interpolate between quaternions (by specifying the ; SLERP keyword). This technique is guaranteed to find the shortest ; path between two orientations, but is somewhat slower than spline ; interpolation. This approach involves computing a finite ; difference of the data. To avoid repeated computation of the ; difference on every call, users can pass a named variable in the ; QDIFF keyword. This value can be reset with the RESET keyword. ; ; 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. ; ; Users must have the VALUE_LOCATE() function, available either in ; IDL 5.3 or later, or from the Markwardt web page. ; ; INPUTS: ; ; TGRID - a vector of N independent variable values. In the ; simplest case, this can be [0, 1, ...] up to the number of ; quaternions in the grid. The grid sampling does not have ; to be uniform. ; ; QGRID - an 4xN array of unit quaternions specified on the grid. ; ; TNEW - a vector of M desired independent variable values which ; sample the grid specified by TGRID. The desired values do ; not have to be uniformly sampled. ; ; RETURNS: ; ; A 4xM array of unit quaternions, where is M is the number of ; desired samples. ; ; ; KEYWORD PARAMETERS: ; ; SLERP - if set, then spherical linear interpolation is performed. ; The default is to perform spline interpolation on the ; quaternion coefficients. ; ; QDIFF - upon return, QDIFF is filled with finite difference values ; which can be used to speed computations in subsequent ; calls. Users should be aware that QDIFF may be ; inadvertently reused from one call to the next. When the ; difference data should no longer be reused, the named ; variable passed to the QDIFF keyword should be set to a ; scalar, or the /RESET keyword should be used. ; ; RESET - if set, then the QDIFF finite difference will be forced to ; be recalculated, even if there is already data present and ; passed to the QDIFF keyword. ; ; ; EXAMPLE: ; ; This example starts with two quaternions representing rotations of ; 0 degrees and 45 degrees, and forms 1001 quaternions which are ; smooth interpolations between 0 and 45 degrees. ; ; ;; Create a grid of two quaternions at times 0 and 1 ; Q0 = qtcompose([1,0,0], 0D) & T0 = 0D ; Q1 = qtcompose([1,0,0], !dpi/4) & T1 = 1D ; ; ;; Put the grid elements into an array ; TGRID = [T0, T1] ; QGRID = [[Q0], [Q1]] ; ; ;; Make an array of 11 values smoothly varying from 0 to 1 ; TNEW = dindgen(11)/10d ; ; ;; Perform spherical linear interpolation ; QNEW = QTERP(TGRID, QGRID, TNEW, /SLERP) ; ; ---> (interpolated results in QNEW) ; ; 0.0000000 0.0000000 0.0000000 1.0000000 ; 0.039259816 0.0000000 0.0000000 0.99922904 ; 0.078459096 0.0000000 0.0000000 0.99691733 ; 0.11753740 0.0000000 0.0000000 0.99306846 ; 0.15643447 0.0000000 0.0000000 0.98768834 ; 0.19509032 0.0000000 0.0000000 0.98078528 ; 0.23344536 0.0000000 0.0000000 0.97236992 ; 0.27144045 0.0000000 0.0000000 0.96245524 ; 0.30901699 0.0000000 0.0000000 0.95105652 ; 0.34611706 0.0000000 0.0000000 0.93819134 ; 0.38268343 0.0000000 0.0000000 0.92387953 ; ; ; SEE ALSO ; QTANG, QTAXIS, QTCOMPOSE, QTERP, QTEXP, QTFIND, QTINV, QTLOG, ; QTMAT, QTMULT, QTPOW, QTVROT ; ; MODIFICATION HISTORY: ; Written, July 2001, CM ; Documented, Dec 2001, CM ; Usage message; check for 0- and 1-length quaternions; handle case ; when quaternions are GE 180 degrees apart; handle case of ; interpolating beyond end of known grid, 15 Mar 2002, CM ; Use simplified QTMULT with /INV, 21 Sep 2007, CM ; Added sample output, 29 Sep 2008, CM ; Handle pathalogical case when some input quaternions were NAN, ; 2012-10-10, CM ; ; \$Id: qterp.pro,v 1.9 2012/10/10 23:27:05 cmarkwar Exp \$ ; ;- ; Copyright (C) 2001, 2002, 2007, 2008, 2012, 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 qterp, t0, q0, t1, qdiff=qdiff, reset=reset, slerp=slerp if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'QNEW = QTERP(TGRID, QGRIDJ, TNEW, [/SLERP, QDIFF=, /RESET])',\$ info=info return, 0 endif nq = n_elements(q0)/4 if nq EQ 0 then goto, USAGE ;; If there is only one quaternion, replicate it, since there is ;; nothing to interpolate if nq EQ 1 then \$ return, rebin(reform(q0,4,1),4,n_elements(t1)) if keyword_set(slerp) then begin if n_elements(qdiff)/4 NE nq-1 OR keyword_set(reset) then begin qdiff = qtmult(q0(*,0:nq-2), /inv1, q0(*,1:*)) ;; Normalize the quaternions to get the smallest path wh = where(qdiff(3,*) LT 0, ct) if ct GT 0 then qdiff(*,wh) = -qdiff(*,wh) endif ii = value_locate(t0, t1) < (nq-2) > 0 hh = (t1-t0(ii))/(t0(ii+1)-t0(ii)) return, qtmult(q0(*,ii), qtpow(qdiff(*,ii),hh)) endif q1 = (q0(*,0) # t1) & q1(*) = 0 for i = 0, 3 do \$ q1(i,*) = spl_interp(t0, q0(i,*), spl_init(t0, q0(i,*)), t1) tot = sqrt(total(q1^2,1)) for i = 0, 3 do \$ q1(i,*) = q1(i,*) / tot return, q1 end