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q2m

Table of contents
Procedure
Abstract
Required_Reading
Keywords
Declarations
Brief_I/O
Detailed_Input
Detailed_Output
Parameters
Exceptions
Files
Particulars
Examples
Restrictions
Literature_References
Author_and_Institution
Version

Procedure

     Q2M ( Quaternion to matrix )

     SUBROUTINE Q2M ( Q, R )

Abstract

     Find the rotation matrix corresponding to a specified unit
     quaternion.

Required_Reading

     ROTATION

Keywords

     MATH
     MATRIX
     ROTATION

Declarations

     IMPLICIT NONE

     DOUBLE PRECISION      Q ( 0 : 3 )
     DOUBLE PRECISION      R ( 3,  3 )

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. 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
         the rotation matrix that is the result of converting
         normalized Q to a rotation matrix.

     2)  If Q is the zero quaternion, the output matrix R is
         the identity 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 A and a scalar r such that

        Q = ( cos(r/2), sin(r/2)A(1), sin(r/2)A(2), sin(r/2)A(3) ).

     We can interpret A and r as the axis and rotation angle of a
     rotation in 3-space. If we restrict r to the range [0, pi],
     then r and A are uniquely determined, except if r = 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 SPICELIB routine M2Q is a one-sided inverse of this routine:
     given any rotation matrix R, the calls

        CALL M2Q ( R, Q )
        CALL Q2M ( Q, R )

     leave R unchanged, except for round-off error. However, the
     calls

        CALL Q2M ( Q, R )
        CALL M2Q ( 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


     SPICELIB subroutine interfaces ALWAYS use SPICE quaternions.
     Quaternions of any other style must be converted to SPICE
     quaternions before they are passed to SPICELIB routines.


     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 SPICELIB routines

        Q2M {quaternion to matrix}
        M2Q {matrix to quaternion}

     M2Q 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 quaternion

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

            to a rotation matrix, we can use the code fragment

               Q(0) =  DSQRT(2)/2.D0
               Q(1) =  0.D0
               Q(2) =  0.D0
               Q(3) = -DSQRT(2)/2.D0

               CALL Q2M ( Q, R )

            The matrix R will be set equal to

               +-              -+
               |  0     1    0  |
               |                |
               | -1     0    0  |.
               |                |
               |  0     0    1  |
               +-              -+

            Why?  Well, Q represents a rotation by some angle r about
            some axis vector A, where r and A satisfy

               Q =

               ( cos(r/2), sin(r/2)A(1), sin(r/2)A(2), sin(r/2)A(3) ).

            In this example,

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

            so

               cos(r/2) = sqrt(2)/2.

            Assuming that r is in the interval [0, pi], we must have

               r = pi/2,

            so

               sin(r/2) = sqrt(2)/2.

            Since the second through fourth components of Q represent

               sin(r/2) * A,

            it follows that

               A = ( 0, 0, -1 ).

            So Q represents a transformation that rotates vectors by
            pi/2 about the negative z-axis. This is equivalent to a
            coordinate system rotation of pi/2 about the positive
            z-axis; and we recognize R as the matrix

               [ pi/2 ] .
                       3


     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


               CALL Q2M    ( Q, R )

               CALL M2EUL  ( R,   3,      2,       3,
              .                   TAU,    DELTA,   ALPHA  )

               DELTA = HALFPI() - DELTA

Restrictions

     None.

Literature_References

     None.

Author_and_Institution

     N.J. Bachman       (JPL)
     J. Diaz del Rio    (ODC Space)
     W.L. Taber         (JPL)
     F.S. Turner        (JPL)

Version

    SPICELIB Version 1.2.0, 12-APR-2021 (JDR)

        Added IMPLICIT NONE statement.

        Edited the header to comply with NAIF standard. Corrected the
        output argument name in $Exceptions section.

    SPICELIB Version 1.1.2, 26-FEB-2008 (NJB)

        Updated header; added information about SPICE
        quaternion conventions.

    SPICELIB Version 1.1.1, 13-JUN-2002 (FST)

        Updated the $Exceptions section to clarify exceptions that
        are the result of changes made in the previous version of
        the routine.

    SPICELIB Version 1.1.0, 04-MAR-1999 (WLT)

        Added code to handle the case in which the input quaternion
        is not of length 1.

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

        Comment section for permuted index source lines was added
        following the header.

    SPICELIB Version 1.0.0, 30-AUG-1990 (NJB)
Fri Dec 31 18:36:40 2021