Index Page q2m_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 q2m_c ( ConstSpiceDouble  q[4],
                    SpiceDouble       r[3][3] )

#### Abstract

       Find the rotation matrix corresponding to a specified unit
       quaternion.

#### Required_Reading

       ROTATION

#### Keywords

       MATH
       MATRIX
       ROTATION

#### Brief_I/O

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

#### Detailed_Input

       q              is a unit-length SPICE-style quaternion representing
                      a rotation. `q' has the property that

                         || q ||  =  1

                      See the discussion of quaternion styles in
                      Particulars below.

#### Detailed_Output

       r              is a 3 by 3 rotation matrix representing the same
                      rotation as does `q'. See the discussion titled
                      "Associating SPICE Quaternions with Rotation
                      Matrices" in Particulars below.

#### Parameters

       None.

#### Exceptions

       Error free.

       1)  If `q' is not a unit quaternion, the output matrix `r' is
           unlikely to be a rotation matrix.

#### Files

       None.

#### Particulars

       If a 4-dimensional vector `q' satisfies the equality

          || q ||   =  1

       or equivalently

              2          2          2          2
          q(0)   +   q(1)   +   q(2)   +   q(3)   =  1,

       then we can always find a unit vector `q' and a scalar `theta' such
       that

          q =

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

       We can interpret `a' and `theta' as the axis and rotation angle of a
       rotation in 3-space. If we restrict `theta' to the range [0, pi],
       then `theta' and `a' are uniquely determined, except if theta = pi.
       In this special case, `a' and -a are both valid rotation axes.

       Every rotation is represented by a unique orthogonal matrix; this
       routine returns that unique rotation matrix corresponding to `q'.

       The CSPICE routine m2q_c is a 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 set of Euler angles that represent a rotation
           specified by a quaternion:

              Suppose our rotation `r' is represented by the quaternion
              `q'.  To find angles `tau', `alpha', `delta' such that

                 r  =  [ tau ]  [ pi/2 - delta ]  [ alpha ]
                              3                 2          3

              we can use the code fragment

                 q2m_c   ( q, r );
                 m2eul_c ( r, 3, 2, 3, tau, delta, alpha );

                 delta = halfpi_c() - delta;

#### Restrictions

       None.

#### Literature_References

       [1]    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 1.3.2, 27-FEB-2008 (NJB)

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

       -CSPICE Version 1.3.1, 06-FEB-2003 (EDW)

           Corrected typo error in Examples section.

       -CSPICE Version 1.3.0, 24-JUL-2001   (NJB)

           Changed prototype:  input q is now type (ConstSpiceDouble [4]).
           Implemented interface macro for casting input q to const.

       -CSPICE Version 1.2.0, 08-FEB-1998 (NJB)

          Removed local variables used for temporary capture of outputs.
          Removed tracing calls, since the underlying Fortran routine
          is error-free.

       -CSPICE Version 1.0.0, 25-OCT-1997 (NJB)

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

#### Index_Entries

       quaternion to matrix

#### Link to routine q2m_c source file q2m_c.c

    Wed Apr  5 17:54:41 2017