Index Page eul2xf_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 eul2xf_c ( ConstSpiceDouble    eulang[6],
                       SpiceInt            axisa,
                       SpiceInt            axisb,
                       SpiceInt            axisc,
                       SpiceDouble         xform [6][6] )
#### Abstract
       This routine computes a state transformation from an Euler angle
       factorization of a rotation and the derivatives of those Euler
       angles.
#### Required_Reading
       ROTATION
#### Keywords
       ANGLES
       STATE
       DERIVATIVES
#### Brief_I/O
       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       eulang     I   An array of Euler angles and their derivatives.
       axisa      I   Axis A of the Euler angle factorization.
       axisb      I   Axis B of the Euler angle factorization.
       axisc      I   Axis C of the Euler angle factorization.
       xform      O   A state transformation matrix.
#### Detailed_Input
       eulang      is the set of Euler angles corresponding to the
                   specified factorization.

                   If we represent r as shown here:

                       r =  [ alpha ]     [ beta ]     [ gamma ]
                                     axisa        axisb         axisc

                   then

                      eulang[0] = alpha
                      eulang[1] = beta
                      eulang[2] = gamma
                      eulang[3] = dalpha/dt
                      eulang[4] = dbeta/dt
                      eulang[5] = dgamma/dt

       axisa       are the axes desired for the factorization of r.
       axisb       All must be in the range from 1 to 3.  Moreover
       axisc       it must be the case that axisa and axisb are distinct
                   and that axisb and axisc are distinct.

                   Every rotation matrix can be represented as a product
                   of three rotation matrices about the principal axes
                   of a reference frame.

                       r =  [ alpha ]     [ beta ]     [ gamma ]
                                     axisa        axisb         axisc

                   The value 1 corresponds to the X axis.
                   The value 2 corresponds to the Y axis.
                   The value 3 corresponds to the Z axis.
#### Detailed_Output
       xform       is the state transformation corresponding r and dr/dt
                   as described above.  Pictorially,

                        [       |        ]
                        |  r    |    0   |
                        |       |        |
                        |-------+--------|
                        |       |        |
                        | dr/dt |    r   |
                        [       |        ]

                   where r is a rotation that varies with respect to time
                   and dr/dt is its time derivative.
#### Parameters
       None.
#### Exceptions
       All erroneous inputs are diagnosed by routines in the call
       tree to this routine.  These include

       1)   If any of axisa, axisb, or axisc do not have values in

               { 1, 2, 3 },

            then the error SPICE(INPUTOUTOFRANGE) is signaled.
#### Files
       None.
#### Particulars
       This function is intended to provide an inverse for the function
       xf2eul_c.

       A word about notation:  the symbol

          [ x ]
               i

       indicates a coordinate system rotation of x radians about the
       ith coordinate axis.  To be specific, the symbol

          [ x ]
               1

       indicates a coordinate system rotation of x radians about the
       first, or x-, axis; the corresponding matrix is

          +-                    -+
          |  1      0       0    |
          |                      |
          |  0    cos(x)  sin(x) |.
          |                      |
          |  0   -sin(x)  cos(x) |
          +-                    -+

       Remember, this is a COORDINATE SYSTEM rotation by x radians; this
       matrix, when applied to a vector, rotates the vector by -x
       radians, not x radians.  Applying the matrix to a vector yields
       the vector's representation relative to the rotated coordinate
       system.

       The analogous rotation about the second, or y-, axis is
       represented by

          [ x ]
               2

       which symbolizes the matrix

          +-                    -+
          | cos(x)   0   -sin(x) |
          |                      |
          |  0       1      0    |,
          |                      |
          | sin(x)   0    cos(x) |
          +-                    -+

       and the analogous rotation about the third, or z-, axis is
       represented by

          [ x ]
               3

       which symbolizes the matrix

          +-                    -+
          |  cos(x)  sin(x)   0  |
          |                      |
          | -sin(x)  cos(x)   0  |.
          |                      |
          |  0        0       1  |
          +-                    -+

       The input matrix is assumed to be the product of three
       rotation matrices, each one of the form

          +-                    -+
          |  1      0       0    |
          |                      |
          |  0    cos(r)  sin(r) |     (rotation of r radians about the
          |                      |      x-axis),
          |  0   -sin(r)  cos(r) |
          +-                    -+

          +-                    -+
          | cos(s)   0   -sin(s) |
          |                      |
          |  0       1      0    |     (rotation of s radians about the
          |                      |      y-axis),
          | sin(s)   0    cos(s) |
          +-                    -+

       or

          +-                    -+
          |  cos(t)  sin(t)   0  |
          |                      |
          | -sin(t)  cos(t)   0  |     (rotation of t radians about the
          |                      |      z-axis),
          |  0        0       1  |
          +-                    -+

       where the second rotation axis is not equal to the first or
       third.  Any rotation matrix can be factored as a sequence of
       three such rotations, provided that this last criterion is met.

       This routine is related to the routine eul2xf_c which produces
       a state transformation from an input set of axes, Euler angles
       and derivatives.

       The two function calls shown here will not change xform except for
       round off errors.

          xf2eul_c ( xform,  axisa, axisb, axisc, eulang, &unique );
          eul2xf_c ( eulang, axisa, axisb, axisc, xform           );

       On the other hand the two calls

          eul2xf_c ( eulang, axisa, axisb, axisc, xform           );
          xf2eul_c ( xform,  axisa, axisb, axisc, eulang, &unique );

       will leave eulang unchanged only if the components of eulang
       are in the range produced by xf2eul_c and the Euler representation
       of the rotation component of xform is unique within that range.
#### Examples
       Suppose you have a set of Euler angles and their derivatives
       for a 3 1 3 rotation, and that you would like to determine
       the equivalent angles and derivatives for a 1 2 3 rotation.

           r = [alpha]  [beta]  [gamma]
                      3       1        3

           r = [roll]  [pitch]  [yaw]
                     1        2      3

       The following code fragment will perform the desired computation.

          abgang[0] = alpha;
          abgang[1] = beta;
          abgang[2] = gamma;
          abgang[3] = dalpha;
          abgang[4] = dbeta;
          abgang[5] = dgamma;

          eul2xf_c ( abgang, 3, 1, 3, xform  );
          xf2eul_c ( xform,  1, 2, 3, rpyang, &unique );

          roll     = rpyang[0];
          pitch    = rpyang[1];
          yaw      = rpyang[2];
          droll    = rpyang[3];
          dpitch   = rpyang[4];
          dyaw     = rpyang[5];
#### Restrictions
       None.
#### Literature_References
       None.
#### Author_and_Institution
       N.J. Bachman    (JPL)
       W.L. Taber      (JPL)
#### Version
       -CSPICE Version 2.0.1, 25-APR-2007 (EDW)

          Corrected code in Examples section, example showed
          a xf2eul_c call:

                xf2eul_c( xform,  1, 2, 3, rpyang);

          The proper form of the call:

                xf2eul_c( xform,  1, 2, 3, rpyang, &unique );

       -CSPICE Version 2.0.0, 31-OCT-2005 (NJB)

          Restriction that second axis must differ from the first
          and third was removed.

       -CSPICE Version 1.0.1, 03-JUN-2003 (EDW)

          Correct typo in Procedure line.

       -CSPICE Version 1.0.0, 18-MAY-1999 (WLT) (NJB)
#### Index_Entries
       State transformation from Euler angles and derivatives
#### Link to routine eul2xf_c source file eul2xf_c.c

    Wed Jun  9 13:05:23 2010