Index of Functions: A B C D E F G H I J K L M N O P Q R S T U V W X
Index Page eul2xf_c

Table of contents

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

       eul2xf_c ( Euler angles and derivative to transformation )

       void eul2xf_c ( ConstSpiceDouble    eulang[6],
                       SpiceInt            axisa,
                       SpiceInt            axisb,
                       SpiceInt            axisc,
                       SpiceDouble         xform [6][6] )

#### Abstract

       Compute a state transformation from an Euler angle factorization
       of a rotation and the derivatives of those Euler angles.

#### Required_Reading

       ROTATION

#### Keywords

       ANGLES
       DERIVATIVES
       STATE

#### 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,
       axisb,
       axisc       are the axes desired for the factorization of `r'.

                   All must be in the range from 1 to 3. Moreover
                   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 matrix corresponding to `r'
                   and dr/dt as described above. Pictorially,

                      .-             -.
                      |       |       |
                      |   r   |   0   |
                      |       |       |
                      |-------+-------|
                      |       |       |
                      | dr/dt |   r   |
                      |       |       |
                      `-             -'

                   where `r' is a rotation matrix that varies with respect to
                   time and dr/dt is its time derivative.

#### Parameters

       None.

#### Exceptions

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

              { 1, 2, 3 }

           an error is signaled by a routine in the call tree of this
           routine.

#### Files

       None.

#### Particulars

       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 intended to provide an inverse for xf2eul_c.

       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

       The numerical results shown for this example may differ across
       platforms. The results depend on the SPICE kernels used as
       input, the compiler and supporting libraries, and the machine
       specific arithmetic implementation.

       1) 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 example will perform the desired
          computation.

          Example code begins here.

          /.
             Program eul2xf_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main( )
          {

             /.
             Local variables.
             ./
             SpiceDouble          abgang [6];
             SpiceDouble          rpyang [6];
             SpiceDouble          xform  [6][6];

             SpiceBoolean         unique;

             /.
             Define the initial set of Euler angles.
             ./
             abgang[0] =  0.01;
             abgang[1] =  0.03;
             abgang[2] =  0.09;
             abgang[3] = -0.001;
             abgang[4] = -0.003;
             abgang[5] = -0.009;

             /.
             Compute the equivalent angles and derivatives for a
             1-2-3 rotation.
             ./
             eul2xf_c ( abgang, 3, 1, 3, xform );
             xf2eul_c ( xform, 1, 2, 3, rpyang, &unique );

             if ( unique )
             {
                printf( "1-2-3 equivalent rotation to input (radians):\n" );
                printf( "Roll   %12.9f, droll/dt   %12.9f\n",
                                         rpyang[0], rpyang[3] );
                printf( "Pitch  %12.9f, dpitch/dt  %12.9f\n",
                                         rpyang[1], rpyang[4] );
                printf( "Yaw    %12.9f, dyaw/dt    %12.9f\n",
                                         rpyang[2], rpyang[5] );
             }
             else
             {
                printf( "The values in `rpyang' are not uniquely determined.\n" );
             }

             return ( 0 );
          }

          When this program was executed on a Mac/Intel/cc/64-bit
          platform, the output was:

          1-2-3 equivalent rotation to input (radians):
          Roll    0.029998501, droll/dt   -0.002999550
          Pitch  -0.000299950, dpitch/dt   0.000059980
          Yaw     0.099995501, dyaw/dt    -0.009998650

#### Restrictions

       None.

#### Literature_References

       None.

#### Author_and_Institution

       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)
       W.L. Taber          (JPL)
       E.D. Wright         (JPL)

#### Version

       -CSPICE Version 2.0.2, 10-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard. Added complete
           code example based on existing example.

       -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

    Fri Dec 31 18:41:06 2021