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Table of contents
Procedure
M2Q ( Matrix to quaternion )
SUBROUTINE M2Q ( R, Q )
Abstract
Find a unit quaternion corresponding to a specified rotation
matrix.
Required_Reading
ROTATION
Keywords
MATH
MATRIX
ROTATION
Declarations
IMPLICIT NONE
DOUBLE PRECISION R ( 3, 3 )
DOUBLE PRECISION Q ( 0 : 3 )
Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
R I A rotation matrix.
Q O A unit quaternion representing R.
Detailed_Input
R is a rotation matrix.
Detailed_Output
Q is a unit-length SPICE-style quaternion
representing R. 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.
Parameters
None.
Exceptions
1) If R is not a rotation matrix, the error SPICE(NOTAROTATION)
is signaled.
Files
None.
Particulars
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 SPICELIB routine Q2M is an 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 rotation matrix
+- -+
| 0 1 0 |
| |
R = | -1 0 0 |
| |
| 0 0 1 |
+- -+
also represented as
[ pi/2 ]
3
to a quaternion, we can use the code fragment
CALL ROTATE ( HALFPI(), 3, R )
CALL M2Q ( R, Q )
M2Q will return Q as
( sqrt(2)/2, 0, 0, -sqrt(2)/2 )
Why? Well, R is a reference frame transformation that
rotates vectors by -pi/2 radians about the axis vector
A = ( 0, 0, 1 )
Equivalently, R rotates vectors by pi/2 radians in
the counterclockwise sense about the axis vector
-A = ( 0, 0, -1 )
so our definition of Q,
h = theta/2
Q = ( cos(h), sin(h)A , sin(h)A , sin(h)A )
1 2 3
implies that in this case,
Q = ( cos(pi/4), 0, 0, -sin(pi/4) )
= ( sqrt(2)/2, 0, 0, -sqrt(2)/2 )
2) Finding a quaternion that represents a rotation specified by
a set of Euler angles:
Suppose our original rotation R is the product
[ TAU ] [ pi/2 - DELTA ] [ ALPHA ]
3 2 3
The code fragment
CALL EUL2M ( TAU, HALFPI() - DELTA, ALPHA,
. 3, 2, 3, R )
CALL M2Q ( R, Q )
yields a quaternion Q that represents R.
Restrictions
None.
Literature_References
None.
Author_and_Institution
N.J. Bachman (JPL)
J. Diaz del Rio (ODC Space)
W.L. Taber (JPL)
Version
SPICELIB Version 2.1.0, 24-AUG-2021 (JDR)
Added IMPLICIT NONE statement.
Edited the header to comply with NAIF standard.
SPICELIB Version 2.0.1, 27-FEB-2008 (NJB)
Updated header; added information about SPICE
quaternion conventions. Made various minor edits
throughout header.
SPICELIB Version 2.0.0, 17-SEP-1999 (WLT)
The routine was re-implemented to sharpen the numerical
stability of the routine and eliminate calls to SIN
and COS functions.
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)
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Fri Dec 31 18:36:33 2021