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Procedure
Abstract
Required_Reading
Keywords
Brief_I/O
Detailed_Input
Detailed_Output
Parameters
Exceptions
Files
Particulars
Examples
Restrictions
Literature_References
Author_and_Institution
Version
Index_Entries

Procedure

   void qxq_c ( ConstSpiceDouble    q1   [4],
                ConstSpiceDouble    q2   [4],
                SpiceDouble         qout [4]  ) 

Abstract

 
   Multiply two quaternions. 
    

Required_Reading

 
   ROTATION 
  

Keywords

 
   MATH 
   POINTING 
   ROTATION 
 

Brief_I/O

 
   VARIABLE  I/O  DESCRIPTION 
   --------  ---  -------------------------------------------------- 
   q1         I   First SPICE quaternion factor. 
   q2         I   Second SPICE quaternion factor. 
   qout       O   Product of `q1' and `q2'. 
 

Detailed_Input

 
   q1             is a 4-vector representing a SPICE-style quaternion.
                  See the discussion of "Quaternion Styles" in the
                  Particulars section below.

                  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.
 
   q2             is a second SPICE-style quaternion. 
 

Detailed_Output

 
   qout           is 4-vector representing the quaternion product  
 
                     q1 * q2 
 
                  Representing q(i) as the sums of scalar (real) 
                  part s(i) and vector (imaginary) part v(i) 
                  respectively, 
 
                     q1 = s1 + v1 
                     q2 = s2 + v2 
 
                  qout has scalar part s3 defined by 
 
                     s3 = s1 * s2 - <v1, v2> 
 
                  and vector part v3 defined by 
 
                     v3 = s1 * v2  +  s2 * v1  +  v1 x v2 
 
                  where the notation < , > denotes the inner 
                  product operator and x indicates the cross 
                  product operator. 
 

Parameters

 
   None. 
 

Exceptions

 
   Error free. 
 

Files

 
   None. 
 

Particulars

     
   Quaternion Styles
   -----------------

   There are different "styles" of quaternions used in
   science and engineering applications. Quaternion styles
   are characterized by

      - The order of quaternion elements

      - The quaternion multiplication formula

      - The convention for associating quaternions
        with rotation matrices

   Two of the commonly used styles are

      - "SPICE"

         > Invented by Sir William Rowan Hamilton
         > Frequently used in mathematics and physics textbooks

      - "Engineering"

         > Widely used in aerospace engineering applications


   CSPICE function interfaces ALWAYS use SPICE quaternions.
   Quaternions of any other style must be converted to SPICE
   quaternions before they are passed to CSPICE functions.


   Relationship between SPICE and Engineering Quaternions
   ------------------------------------------------------

   Let M be a rotation matrix such that for any vector V,

      M*V

   is the result of rotating V by theta radians in the
   counterclockwise direction about unit rotation axis vector A.
   Then the SPICE quaternions representing M are

      (+/-) (  cos(theta/2),
               sin(theta/2) A(1),
               sin(theta/2) A(2),
               sin(theta/2) A(3)  )

   while the engineering quaternions representing M are

      (+/-) ( -sin(theta/2) A(1),
              -sin(theta/2) A(2),
              -sin(theta/2) A(3),
               cos(theta/2)       )

   For both styles of quaternions, if a quaternion q represents
   a rotation matrix M, then -q represents M as well.

   Given an engineering quaternion

      QENG   = ( q0,  q1,  q2,  q3 )

   the equivalent SPICE quaternion is

      QSPICE = ( q3, -q0, -q1, -q2 )


   Associating SPICE Quaternions with Rotation Matrices
   ----------------------------------------------------

   Let FROM and TO be two right-handed reference frames, for
   example, an inertial frame and a spacecraft-fixed frame. Let the
   symbols

      V    ,   V
       FROM     TO

   denote, respectively, an arbitrary vector expressed relative to
   the FROM and TO frames. Let M denote the transformation matrix
   that transforms vectors from frame FROM to frame TO; then

      V   =  M * V
       TO         FROM

   where the expression on the right hand side represents left
   multiplication of the vector by the matrix.

   Then if the unit-length SPICE quaternion q represents M, where

      q = (q0, q1, q2, q3)

   the elements of M are derived from the elements of q as follows:

        +-                                                         -+
        |           2    2                                          |
        | 1 - 2*( q2 + q3 )   2*(q1*q2 - q0*q3)   2*(q1*q3 + q0*q2) |
        |                                                           |
        |                                                           |
        |                               2    2                      |
    M = | 2*(q1*q2 + q0*q3)   1 - 2*( q1 + q3 )   2*(q2*q3 - q0*q1) |
        |                                                           |
        |                                                           |
        |                                                   2    2  |
        | 2*(q1*q3 - q0*q2)   2*(q2*q3 + q0*q1)   1 - 2*( q1 + q2 ) |
        |                                                           |
        +-                                                         -+

   Note that substituting the elements of -q for those of q in the
   right hand side leaves each element of M unchanged; this shows
   that if a quaternion q represents a matrix M, then so does the
   quaternion -q.

   To map the rotation matrix M to a unit quaternion, we start by
   decomposing the rotation matrix as a sum of symmetric
   and skew-symmetric parts:

                                      2
      M = [ I  +  (1-cos(theta)) OMEGA  ] + [ sin(theta) OMEGA ]

                   symmetric                   skew-symmetric


   OMEGA is a skew-symmetric matrix of the form

                 +-             -+
                 |  0   -n3   n2 |
                 |               |
       OMEGA  =  |  n3   0   -n1 |
                 |               |
                 | -n2   n1   0  |
                 +-             -+

   The vector N of matrix entries (n1, n2, n3) is the rotation axis
   of M and theta is M's rotation angle.  Note that N and theta
   are not unique.

   Let

      C = cos(theta/2)
      S = sin(theta/2)

   Then the unit quaternions Q corresponding to M are

      Q = +/- ( C, S*n1, S*n2, S*n3 )

   The mappings between quaternions and the corresponding rotations
   are carried out by the CSPICE routines

      q2m_c {quaternion to matrix}
      m2q_c {matrix to quaternion}

   m2q_c always returns a quaternion with scalar part greater than
   or equal to zero.


   SPICE Quaternion Multiplication Formula
   ---------------------------------------

   Given a SPICE quaternion

      Q = ( q0, q1, q2, q3 )

   corresponding to rotation axis A and angle theta as above, we can
   represent Q using "scalar + vector" notation as follows:

      s =   q0           = cos(theta/2)

      v = ( q1, q2, q3 ) = sin(theta/2) * A

      Q = s + v

   Let Q1 and Q2 be SPICE quaternions with respective scalar
   and vector parts s1, s2 and v1, v2:

      Q1 = s1 + v1
      Q2 = s2 + v2

   We represent the dot product of v1 and v2 by

      <v1, v2>

   and the cross product of v1 and v2 by

      v1 x v2

   Then the SPICE quaternion product is

      Q1*Q2 = s1*s2 - <v1,v2>  + s1*v2 + s2*v1 + (v1 x v2)

   If Q1 and Q2 represent the rotation matrices M1 and M2
   respectively, then the quaternion product

      Q1*Q2

   represents the matrix product

      M1*M2

 

Examples

 
   1)  Let qid, qi, qj, qk be the "basis" quaternions 
 
          qid  =  ( 1, 0, 0, 0 ) 
          qi   =  ( 0, 1, 0, 0 ) 
          qj   =  ( 0, 0, 1, 0 ) 
          qk   =  ( 0, 0, 0, 1 ) 
 
       respectively.  Then the calls 
 
          qxq_c ( qi, qj, ixj );
          qxq_c ( qj, qk, jxk );
          qxq_c ( qk, qi, kxi );
 
       produce the results 
 
          ixj == qk 
          jxk == qi 
          kxi == qj 
 
       All of the calls 
 
          qxq_c ( qi, qi, qout );
          qxq_c ( qj, qj, qout );
          qxq_c ( qk, qk, qout );
 
       produce the result 
 
          qout  ==  -qid
 
       For any quaternion Q, the calls 
 
          qxq_c ( qid, q,   qout );
          qxq_c ( q,   qid, qout );
 
       produce the result 
 
          qout  ==  q 
 
 
 
   2)  Composition of rotations:  let `cmat1' and `cmat2' be two 
       C-matrices (which are rotation matrices).  Then the 
       following code fragment computes the product cmat1 * cmat2: 
 
 
          /. 
          Convert the C-matrices to quaternions. 
          ./
          m2q_c ( cmat1, q1 );
          m2q_c ( cmat2, q2 );
 
          /.
          Find the product. 
          ./ 
          qxq_c ( q1, q2, qout ); 
 
          /.
          Convert the result to a C-matrix. 
          ./ 
          q2m_c ( qout, cmat3 );
 
          /.
          Multiply `cmat1' and `cmat2' directly. 
          ./ 
          mxm_c ( cmat1, cmat2, cmat4 );
 
          /.
          Compare the results.  The difference `diff' of 
          `cmat3' and `cmat4' should be close to the zero 
          matrix. 
          ./ 
          vsubg_c ( 9, cmat3, cmat4, diff );

 

Restrictions

 
   None. 
 

Literature_References

 
   None. 
 

Author_and_Institution

 
   N.J. Bachman    (JPL) 
 

Version

 
   -CSPICE Version 1.0.1, 27-FEB-2008 (NJB)

      Updated header; added information about SPICE 
      quaternion conventions.

   -CSPICE Version 1.0.0, 27-OCT-2005 (NJB)

Index_Entries

 
   quaternion times quaternion 
   multiply quaternion by quaternion 
Wed Apr  5 17:54:41 2017