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Abstract
I/O
Examples
Particulars
Required Reading
Version
Index_Entries

Abstract


   CSPICE_QDQ2AV derives angular velocity from a unit quaternion and 
   its derivative with respect to time.

   For important details concerning this module's function, please refer to
   the CSPICE routine qdq2av_c.

I/O

   
   Given:

      q    is a unit length double precision 4-vector representing
           a SPICE-style quaternion.

           Note that multiple styles of quaternions are in use.
           This routine will not work properly if the input
           quaternions do not conform to the SPICE convention.
           See the Particulars section for details.
 
      dq   is a double precision 4-vector representing the derivative
           of 'q' with respect to time.

   the call:

      cspice_qdq2av, q, dq, av
   
   returns:

      av   is a double precision 3-vector representing the angular 
           velocity defined by 'q' and 'dq', that is, the angular 
           velocity of the frame defined by the rotation matrix 
           associated with 'q'. 

Examples


   Any numerical results shown for this example may differ between
   platforms as the results depend on the SPICE kernels used as input
   and the machine specific arithmetic implementation.

      ;;
      ;; Pick some Euler angles and form a rotation matrix.
      ;;
      angle =  [ -20.d, 50.d, -60.d  ] * cspice_rpd()

      cspice_eul2m, angle[2], angle[1], angle[0], 3, 1, 3, m
         
      cspice_m2q, m, q

      ;;
      ;; Choose an angular velocity vector.  
      ;;
      expav = [ 1.d, 2., 3. ]

      ;; 
      ;; Form the quaternion derivative.
      ;;
      qav =  [ 0.0, expav ]

      cspice_qxq, q, qav, dq

      dq = -0.5d * dq

      ;;
      ;; Recover angular velocity from 'q' and 'dq' using cspice_qdq2av.
      ;;
      cspice_qdq2av, q, dq, av

      ;;
      ;; Now we'll obtain the angular velocity from 'q' and
      ;; 'dq' by an alternate method.
      ;;
      ;; Convert 'q' back to a rotation matrix.
      ;;
      cspice_q2m, q, m

      ;;
      ;; Convert 'q' and 'dq' to a rotation derivative matrix.  This
      ;; somewhat messy procedure is based on differentiating the
      ;; formula for deriving a rotation from a quaternion, then
      ;; substituting components of 'q' and 'dq' into the derivative
      ;; formula.
      ;;

      dm = dblarr(3,3)

      dm[0,0]  =  -4.d0  * (   q[2]*dq[2]  +  q[3]*dq[3]  )

      dm[1,0]  =   2.d0  * (   q[1]*dq[2]  +  q[2]*dq[1]  $
                             - q[0]*dq[3]  -  q[3]*dq[0]  )

      dm[2,0]  =   2.d0  * (   q[1]*dq[3]  +  q[3]*dq[1]  $
                             + q[0]*dq[2]  +  q[2]*dq[0]  )

      dm[0,1]  =   2.d0  * (   q[1]*dq[2]  +  q[2]*dq[1]  $
                             + q[0]*dq[3]  +  q[3]*dq[0]  )

      dm[1,1]  =  -4.d0  * (   q[1]*dq[1]  +  q[3]*dq[3]  )

      dm[2,1]  =   2.d0  * (   q[2]*dq[3]  +  q[3]*dq[2]  $
                             - q[0]*dq[1]  -  q[1]*dq[0]  )

      dm[0,2]  =   2.d0  * (   q[3]*dq[1]  +  q[1]*dq[3]  $
                             - q[0]*dq[2]  -  q[2]*dq[0]  )

      dm[1,2]  =   2.d0  * (   q[2]*dq[3]  +  q[3]*dq[2]  $
                             + q[0]*dq[1]  +  q[1]*dq[0] )

      dm[2,2]  =  -4.d0  * (   q[1]*dq[1]  +  q[2]*dq[2]  )

      xtrans = dblarr(6,6)

      ;;
      ;; Upper left block: 
      ;;
      xtrans[0,0] = m[*,0]
      xtrans[0,1] = m[*,1]
      xtrans[0,2] = m[*,2]

      ;;
      ;; Lower left block: 
      ;;
      xtrans[0,3] = dm[*,0]
      xtrans[0,4] = dm[*,1]
      xtrans[0,5] = dm[*,2]

      ;;
      ;; Lower right block: 
      ;;
      xtrans[3,3] = m[*,0]
      xtrans[3,4] = m[*,1]
      xtrans[3,5] = m[*,2]

      ;;
      ;; Now use cspice_xf2rav to produce the expected angular velocity.
      ;; 
      cspice_xf2rav, xtrans, mout, avx
         
      ;;
      ;; The results should match to nearly full double precision.
      ;;
      print, "Original angular velocity     : ", expav
      print, "cspice_qdq2av angular velocity: ", av
      print, "cspice_xf2rav angular velocity: ", avx

   IDL outputs:
   
      Original angular velocity     :   1.0000000   2.0000000   3.0000000
      cspice_qdq2av angular velocity:   1.0000000   2.0000000   3.0000000
      cspice_xf2rav angular velocity:   1.0000000   2.0000000   3.0000000

Particulars


   About SPICE quaternions 
   ======================= 
 
   There are (at least) two popular "styles" of quaternions; these 
   differ in the layout of the quaternion elements, the definition 
   of the multiplication operation, and the mapping between the set 
   of unit quaternions and corresponding rotation matrices. 
 
   SPICE-style quaternions have the scalar part in the first 
   component and the vector part in the subsequent components. The 
   SPICE convention, along with the multiplication rules for SPICE 
   quaternions, are those used by William Rowan Hamilton, the 
   inventor of quaternions. 
 
   Another common quaternion style places the scalar component 
   last.  This style is often used in engineering applications. 

Required Reading


   ICY.REQ
   ROTATIONS.REQ

Version


   -Icy Version 1.0.0, 06-NOV-2005, EDW (JPL)


Index_Entries

 
   angular velocity from  quaternion and derivative 



Wed Apr  5 17:58:03 2017