Index Page m2q_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 m2q_c (  ConstSpiceDouble  r[3][3],
                     SpiceDouble       q[4]     )

#### Abstract

       Find a unit quaternion corresponding to a specified rotation
       matrix.

#### Required_Reading

       ROTATION

#### Keywords

       MATH
       MATRIX
       ROTATION

#### Brief_I/O

       Variable  I/O  Description
       --------  ---  --------------------------------------------------
       r          I   A rotation matrix.
       q          O   A unit quaternion representing `r'.

#### Detailed_Input

       r              is a rotation matrix.

#### Detailed_Output

       q              is a unit-length SPICE-style quaternion representing
                      `r'. See the discussion of quaternion styles in
                      Particulars below.

                      `q' is a 4-dimensional vector. If `r' rotates vectors in
                      the counterclockwise sense by an angle of `theta' radians
                      about a unit vector `a', where

                         0 < theta < pi
                           -       -

                      then letting h = theta/2,

                         q = ( cos(h), sin(h)a ,  sin(h)a ,  sin(h)a ).
                                              1          2          3

                      The restriction that `theta' must be in the range [0, pi]
                      determines the output quaternion `q' uniquely
                      except when theta = pi; in this special case, both of
                      the quaternions

                         q = ( 0,  a ,  a ,  a  )
                                    1    2    3
                      and

                         q = ( 0, -a , -a , -a  )
                                    1    2    3

                     are possible outputs.

#### Parameters

       None.

#### Exceptions

       1)   If `r' is not a rotation matrix, the error SPICE(NOTAROTATION)
            is signaled.

#### Files

       None.

#### Particulars

       A unit quaternion is a 4-dimensional vector for which the sum of
       the squares of the components is 1. Unit quaternions can be used
       to represent rotations in the following way: given a rotation
       angle `theta', where

          0 < theta < pi
            -       -

       and a unit vector `a', we can represent the transformation that
       rotates vectors in the counterclockwise sense by theta radians about
       `a' using the quaternion `q', where

          q = ( cos(theta/2), sin(theta/2)a , sin(theta/2)a , sin(theta/2)a )
                                           1               2               3

       As mentioned in Detailed Output, our restriction on the range of
       `theta' determines `q' uniquely, except when theta = pi.

       The CSPICE routine q2m_c is an one-sided inverse of this routine:
       given any rotation matrix `r', the calls

          m2q_c ( r, q );
          q2m_c ( q, r );

       leave `r' unchanged, except for round-off error.  However, the
       calls

          q2m_c ( q, r );
          m2q_c ( r, q );

       might preserve `q' or convert `q' to -q.

       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)  A case amenable to checking by hand calculation:

              To convert the rotation matrix

                       +-              -+
                       |  0     1    0  |
                       |                |
                 r  =  | -1     0    0  |
                       |                |
                       |  0     0    1  |
                       +-              -+

              also represented as

                 [ pi/2 ]
                         3

              to a quaternion, we can use the code fragment

                 rotate_c (  halfpi_c(),  3,  r );
                 m2q_c    (  r,               q );

              m2q_c will return `q' as

                 ( sqrt(2)/2, 0, 0, -sqrt(2)/2 )

              Why?  Well, `r' is a reference frame transformation that
              rotates vectors by -pi/2 radians about the axis vector

                  a = ( 0, 0, 1 )

              Equivalently, `r' rotates vectors by pi/2 radians in
              the counterclockwise sense about the axis vector

                 -a = ( 0, 0, -1 )

              so our definition of `q',

                 h = theta/2

                 q = ( cos(h), sin(h)a , sin(h)a , sin(h)a  )
                                      1         2         3

              implies that in this case,

                 q =  ( cos(pi/4),  0,  0,  -sin(pi/4) )

                   =  ( sqrt(2)/2,  0,  0,  -sqrt(2)/2 )

       2)  Finding a quaternion that represents a rotation specified by
           a set of Euler angles:

              Suppose our original rotation `r' is the product

                 [ tau ]  [ pi/2 - delta ]  [ alpha ] .
                        3                 2          3

              The code fragment

                 eul2m_c  ( tau,   halfpi_c() - delta,   alpha,
                            3,     2,                    3,      r );

                 m2q_c    ( r, q );

              yields a quaternion `q' that represents `r'.

#### Restrictions

       None.

#### Literature_References

       NAIF document 179.0, "Rotations and their Habits", by
       W. L. Taber.

#### Author_and_Institution

       N.J. Bachman   (JPL)
       E.D. Wright    (JPL)

#### Version

       -CSPICE Version 1.1.1, 27-FEB-2008 (NJB)

          Updated header; added information about SPICE
          quaternion conventions. Made minor edits throughout
          header.

       -CSPICE Version 1.1.0, 21-OCT-1998 (NJB)

           Made input matrix const.

       -CSPICE Version 1.0.1, 13-FEB-1998 (EDW)

           Minor corrections to header.

       -CSPICE Version 1.0.0, 08-FEB-1998 (NJB)

           Based on SPICELIB Version 1.0.1, 10-MAR-1992 (WLT)

#### Index_Entries

       matrix to quaternion

#### Link to routine m2q_c source file m2q_c.c

    Wed Apr  5 17:54:38 2017