| eul2m |
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Table of contents
Procedure
EUL2M ( Euler angles to matrix )
SUBROUTINE EUL2M ( ANGLE3, ANGLE2, ANGLE1,
. AXIS3, AXIS2, AXIS1, R )
Abstract
Construct a rotation matrix from a set of Euler angles.
Required_Reading
ROTATION
Keywords
MATRIX
ROTATION
TRANSFORMATION
Declarations
IMPLICIT NONE
DOUBLE PRECISION ANGLE3
DOUBLE PRECISION ANGLE2
DOUBLE PRECISION ANGLE1
INTEGER AXIS3
INTEGER AXIS2
INTEGER AXIS1
DOUBLE PRECISION R ( 3, 3 )
Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
ANGLE3,
ANGLE2,
ANGLE1 I Rotation angles about third, second, and first
rotation axes (radians).
AXIS3,
AXIS2,
AXIS1 I Axis numbers of third, second, and first rotation
axes.
R O Product of the 3 rotations.
Detailed_Input
ANGLE3,
ANGLE2,
ANGLE1,
AXIS3,
AXIS2,
AXIS1 are, respectively, a set of three angles and three
coordinate axis numbers; each pair ANGLEx and AXISx
specifies a coordinate transformation consisting of a
rotation by ANGLEx radians about the coordinate axis
indexed by AXISx. These coordinate transformations are
typically symbolized by
[ ANGLEx ] .
AXISx
See the $Particulars section below for details concerning
this notation.
Note that these coordinate transformations rotate vectors
by
-ANGLEx
radians about the axis indexed by AXISx.
The values of AXISx may be 1, 2, or 3, indicating the X,
Y, and Z axes respectively.
Detailed_Output
R is a rotation matrix representing the composition of the
rotations defined by the input angle-axis pairs.
Together, the three pairs specify a composite
transformation that is the result of performing the
rotations about the axes indexed by AXIS1, AXIS2, and
AXIS3, in that order. So,
R = [ ANGLE3 ] [ ANGLE2 ] [ ANGLE1 ]
AXIS3 AXIS2 AXIS1
See the $Particulars section below for details concerning
this notation.
The resulting matrix R may be thought of as a coordinate
transformation; applying it to a vector yields the
vector's coordinates in the rotated system.
Viewing R as a coordinate transformation matrix, the
basis that R transforms vectors to is created by rotating
the original coordinate axes first by ANGLE1 radians
about the coordinate axis indexed by AXIS1, next by
ANGLE2 radians about the coordinate axis indexed by
AXIS2, and finally by ANGLE3 radians about coordinate
axis indexed by AXIS3. At the second and third steps of
this process, the coordinate axes about which rotations
are performed belong to the bases resulting from the
previous rotations.
Parameters
None.
Exceptions
1) If any of AXIS3, AXIS2, or AXIS1 do not have values in
{ 1, 2, 3 }
the error SPICE(BADAXISNUMBERS) is signaled.
Files
None.
Particulars
A word about notation: the symbol
[ x ]
i
indicates a 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 |
`- -'
From time to time, (depending on one's line of work, perhaps) one
may happen upon a pair of coordinate systems related by a
sequence of rotations. For example, the coordinate system
defined by an instrument such as a camera is sometime specified
by RA, DEC, and twist; if alpha, delta, and phi are the rotation
angles, then the series of rotations
[ phi ] [ pi/2 - delta ] [ alpha ]
3 2 3
produces a transformation from inertial to camera coordinates.
This routine is related to the SPICELIB routine M2EUL, which
produces a sequence of Euler angles, given a rotation matrix.
This routine is a "left inverse" of M2EUL: the sequence of
calls
CALL M2EUL ( R, AXIS3, AXIS2, AXIS1,
. ANGLE3, ANGLE2, ANGLE1 )
CALL EUL2M ( ANGLE3, ANGLE2, ANGLE1,
. AXIS3, AXIS2, AXIS1, R )
preserves R, except for round-off error.
On the other hand, the sequence of calls
CALL EUL2M ( ANGLE3, ANGLE2, ANGLE1,
. AXIS3, AXIS2, AXIS1, R )
CALL M2EUL ( R, AXIS3, AXIS2, AXIS1,
. ANGLE3, ANGLE2, ANGLE1 )
preserve ANGLE3, ANGLE2, and ANGLE1 only if these angles start
out in the ranges that M2EUL's outputs are restricted to.
Examples
1) Create a coordinate transformation matrix by rotating
the original coordinate axes first by 30 degrees about
the z axis, next by 60 degrees about the y axis resulting
from the first rotation, and finally by -50 degrees about
the z axis resulting from the first two rotations.
C
C Create the coordinate transformation matrix
C
C o o o
C R = [ -50 ] [ 60 ] [ 30 ]
C 3 2 3
C
C All angles in radians, please. The SPICELIB
C function RPD (radians per degree) gives the
C conversion factor.
C
C The z axis is `axis 3'; the y axis is `axis 2'.
C
ANGLE1 = RPD() * 30.D0
ANGLE2 = RPD() * 60.D0
ANGLE3 = RPD() * -50.D0
AXIS1 = 3
AXIS2 = 2
AXIS3 = 3
CALL EUL2M ( ANGLE3, ANGLE2, ANGLE1,
. AXIS3, AXIS2, AXIS1, R )
2) A trivial example using actual numbers.
The code fragment
CALL EUL2M ( 0.D0, 0.D0, HALFPI(),
. 1, 1, 3, R )
sets R equal to the matrix
+- -+
| 0 1 0 |
| |
| -1 0 0 |.
| |
| 0 0 1 |
+- -+
3) Finding the rotation matrix specified by a set of `clock,
cone, and twist' angles, as defined on the Voyager 2 project:
Voyager 2 narrow angle camera pointing, relative to the
Sun-Canopus coordinate system, was frequently specified
by a set of Euler angles called `clock, cone, and twist'.
These defined a 3-2-3 coordinate transformation matrix
TSCTV as the product
[ twist ] [ cone ] [ clock ] .
3 2 3
Given the angles CLOCK, CONE, and TWIST (in units of
radians), we can compute TSCTV with the code fragment
CALL EUL2M ( TWIST, CONE, CLOCK,
. 3, 2, 3, TSCTV )
4) Finding the rotation matrix specified by a set of `right
ascension, declination, and twist' angles, as defined on the
Galileo project:
Galileo scan platform pointing, relative to an inertial
reference frame, (EME50 variety) is frequently specified
by a set of Euler angles called `right ascension (RA),
declination (Dec), and twist'. These define a 3-2-3
coordinate transformation matrix TISP as the product
[ Twist ] [ pi/2 - Dec ] [ RA ] .
3 2 3
Given the angles RA, DEC, and TWIST (in units of radians),
we can compute TISP with the code fragment
CALL EUL2M ( TWIST, HALFPI()-DEC, RA,
. 3, 2, 3, TISP )
Restrictions
1) Beware: more than one definition of "RA, DEC and twist"
exists.
Literature_References
[1] W. Owen, "Galileo Attitude and Camera Models," JPL
Interoffice Memorandum 314-323, Nov. 11, 1983. NAIF document
number 204.0.
Author_and_Institution
N.J. Bachman (JPL)
J. Diaz del Rio (ODC Space)
L.S. Elson (JPL)
W.L. Taber (JPL)
Version
SPICELIB Version 1.3.0, 06-JUL-2021 (JDR)
Added IMPLICIT NONE statement.
Edited the header to comply with NAIF standard. Removed
unnecessary entries from $Revisions section.
SPICELIB Version 1.2.1, 26-DEC-2006 (NJB)
Corrected header typo.
SPICELIB Version 1.2.0, 25-AUG-2005 (NJB)
Updated to remove non-standard use of duplicate arguments
in ROTMAT calls.
SPICELIB Version 1.1.2, 14-OCT-2004 (LSE)
Corrected a typo in the header.
SPICELIB Version 1.1.1, 10-MAR-1992 (WLT)
Comment section for permuted index source lines was added
following the header.
SPICELIB Version 1.1.0, 02-NOV-1990 (NJB)
Names of input arguments changed to reflect the order in
which the rotations are applied when their product is
computed. The header was upgraded to describe notation in
more detail. Examples were added.
SPICELIB Version 1.0.0, 30-AUG-1990 (NJB)
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Fri Dec 31 18:36:21 2021