Table of contents
CSPICE_M2Q calculates a unit quaternion corresponding to a
specified 3x3 double precision, rotation matrix.
Given:
r a rotation matrix.
help, r
DOUBLE = Array[3,3]
the call:
cspice_m2q, r, q
returns:
q a unit-length SPICE-style quaternion representing `r'.
help, q
DOUBLE = Array[4]
See the discussion of quaternion styles in -Particulars below.
`q' is a 4-dimensional vector. If `r' rotates vectors in the
counterclockwise sense by an angle of theta radians about a unit
vector `a', where
0 < theta < pi
- -
then letting h = theta/2,
q = ( cos(h), sin(h)a , sin(h)a , sin(h)a ).
1 2 3
The restriction that theta must be in the range [0, pi]
determines the output quaternion `q' uniquely except when
theta = pi; in this special case, both of the quaternions
q = ( 0, a , a , a )
1 2 3
and
q = ( 0, -a , -a , -a )
1 2 3
are possible outputs.
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
and the machine specific arithmetic implementation.
1) Create a 3-dimensional rotation matrix of 90 degrees about the
Z axis and convert it to a unit quaternion. Verify that the
norm of the quaternion is equal to 1.
Example code begins here.
PRO m2q_ex1
;;
;; Create a rotation matrix of 90 degrees about the Z axis.
;;
cspice_rotate, cspice_halfpi(), 3, r
print, 'Rotation matrix: '
print, r
;;
;; Convert the matrix to a quaternion.
;;
cspice_m2q, r, q
print, 'Unit quaternion: '
print, q
print, ''
;;
;; Confirm || q || = 1.
;;
print, '|| q || = ', q ## transpose(q)
END
When this program was executed on a Mac/Intel/IDL8.x/64-bit
platform, the output was:
Rotation matrix:
6.1232340e-17 1.0000000 0.0000000
-1.0000000 6.1232340e-17 0.0000000
0.0000000 0.0000000 1.0000000
Unit quaternion:
0.70710678 0.0000000 0.0000000 -0.70710678
|| q || = 1.0000000
Note, that the call sequence:
cspice_m2q, r, q
cspice_q2m, q,r
preserves 'r' except for round-off error. Yet, the call
sequence:
cspice_q2m, q,r
cspice_m2q, r, q
may preserve 'q' or return '-q'.
A unit quaternion is a 4-dimensional vector for which the sum of
the squares of the components is 1. Unit quaternions can be used
to represent rotations in the following way: given a rotation
angle theta, where
0 < theta < pi
- -
and a unit vector `a', we can represent the transformation that
rotates vectors in the counterclockwise sense by theta radians
about `a' using the quaternion `q', where
q = ( cos(theta/2), sin(theta/2)a , sin(theta/2)a , sin(theta/2)a )
1 2 3
As mentioned in Detailed Output, our restriction on the range of
theta determines `q' uniquely, except when theta = pi.
The Icy routine cspice_q2m is an one-sided inverse of this routine:
given any rotation matrix `r', the calls
cspice_m2q, r, q
cspice_q2m, q, r
leave `r' unchanged, except for round-off error. However, the
calls
cspice_q2m, q, r
cspice_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
Icy routine interfaces ALWAYS use SPICE quaternions. Quaternions of any
other style must be converted to SPICE quaternions before they are passed to
Icy 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[0],
sin(theta/2) a[1],
sin(theta/2) a[2] )
while the engineering quaternions representing `m' are
(+/-) ( -sin(theta/2) a[0],
-sin(theta/2) a[1],
-sin(theta/2) a[2],
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*( q0 + q1 ) 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 -n2 n1 |
| |
omega = | n2 0 -n0 |
| |
| -n1 n0 0 |
`- -'
The vector N of matrix entries (n0, n1, n2) 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*n0, s*n1, s*n2 )
The mappings between quaternions and the corresponding rotations
are carried out by the Icy routines
cspice_q2m {quaternion to matrix}
cspice_m2q {matrix to quaternion}
cspice_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
1) If `r' is not a rotation matrix, the error SPICE(NOTAROTATION)
is signaled by a routine in the call tree of this routine.
2) If the input argument `r' is undefined, an error is signaled
by the IDL error handling system.
3) If the input argument `r' is not of the expected type, or it
does not have the expected dimensions and size, an error is
signaled by the Icy interface.
4) If the output argument `q' is not a named variable, an error
is signaled by the Icy interface.
None.
None.
ICY.REQ
ROTATION.REQ
None.
J. Diaz del Rio (ODC Space)
E.D. Wright (JPL)
-Icy Version 1.0.2, 10-AUG-2021 (JDR)
Edited the header to comply with NAIF standard. Added example's
problem statement and reformatted example's output.
Added -Parameters, -Exceptions, -Files, -Restrictions,
-Literature_References and -Author_and_Institution sections.
Removed reference to the routine's corresponding CSPICE header from
-Abstract section.
Added arguments' type and size information in the -I/O section.
-Icy Version 1.0.1, 06-NOV-2005 (EDW)
Updated -Particulars section to include the
"About SPICE Quaternions" description. Recast
the -I/O section to meet Icy format standards.
-Icy Version 1.0.0, 16-JUN-2003 (EDW)
matrix to quaternion
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