Index Page qxq_c

A B C D E F G H I J K L M N O P Q R S T U V W X

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

#### Link to routine qxq_c source file qxq_c.c

    Wed Apr  5 17:54:41 2017