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Table of contents
Procedure
Q2M ( Quaternion to matrix )
SUBROUTINE Q2M ( Q, R )
Abstract
Find the rotation matrix corresponding to a specified unit
quaternion.
Required_Reading
ROTATION
Keywords
MATH
MATRIX
ROTATION
Declarations
IMPLICIT NONE
DOUBLE PRECISION Q ( 0 : 3 )
DOUBLE PRECISION R ( 3, 3 )
Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
Q I A unit quaternion.
R O A rotation matrix corresponding to Q.
Detailed_Input
Q is a unit-length SPICE-style quaternion. Q has the
property that
|| Q || = 1
See the discussion of quaternion styles in
$Particulars below.
Detailed_Output
R is a 3 by 3 rotation matrix representing the same
rotation as does Q. See the discussion titled
"Associating SPICE Quaternions with Rotation
Matrices" in $Particulars below.
Parameters
None.
Exceptions
Error free.
1) If Q is not a unit quaternion, the output matrix R is
the rotation matrix that is the result of converting
normalized Q to a rotation matrix.
2) If Q is the zero quaternion, the output matrix R is
the identity matrix.
Files
None.
Particulars
If a 4-dimensional vector Q satisfies the equality
|| Q || = 1
or equivalently
2 2 2 2
Q(0) + Q(1) + Q(2) + Q(3) = 1,
then we can always find a unit vector A and a scalar r such that
Q = ( cos(r/2), sin(r/2)A(1), sin(r/2)A(2), sin(r/2)A(3) ).
We can interpret A and r as the axis and rotation angle of a
rotation in 3-space. If we restrict r to the range [0, pi],
then r and A are uniquely determined, except if r = pi. In this
special case, A and -A are both valid rotation axes.
Every rotation is represented by a unique orthogonal matrix; this
routine returns that unique rotation matrix corresponding to Q.
The SPICELIB routine M2Q is a 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 quaternion
Q = ( sqrt(2)/2, 0, 0, -sqrt(2)/2 )
to a rotation matrix, we can use the code fragment
Q(0) = DSQRT(2)/2.D0
Q(1) = 0.D0
Q(2) = 0.D0
Q(3) = -DSQRT(2)/2.D0
CALL Q2M ( Q, R )
The matrix R will be set equal to
+- -+
| 0 1 0 |
| |
| -1 0 0 |.
| |
| 0 0 1 |
+- -+
Why? Well, Q represents a rotation by some angle r about
some axis vector A, where r and A satisfy
Q =
( cos(r/2), sin(r/2)A(1), sin(r/2)A(2), sin(r/2)A(3) ).
In this example,
Q = ( sqrt(2)/2, 0, 0, -sqrt(2)/2 ),
so
cos(r/2) = sqrt(2)/2.
Assuming that r is in the interval [0, pi], we must have
r = pi/2,
so
sin(r/2) = sqrt(2)/2.
Since the second through fourth components of Q represent
sin(r/2) * A,
it follows that
A = ( 0, 0, -1 ).
So Q represents a transformation that rotates vectors by
pi/2 about the negative z-axis. This is equivalent to a
coordinate system rotation of pi/2 about the positive
z-axis; and we recognize R as the matrix
[ pi/2 ] .
3
2) Finding a set of Euler angles that represent a rotation
specified by a quaternion:
Suppose our rotation R is represented by the quaternion
Q. To find angles TAU, ALPHA, DELTA such that
R = [ TAU ] [ pi/2 - DELTA ] [ ALPHA ] ,
3 2 3
we can use the code fragment
CALL Q2M ( Q, R )
CALL M2EUL ( R, 3, 2, 3,
. TAU, DELTA, ALPHA )
DELTA = HALFPI() - DELTA
Restrictions
None.
Literature_References
None.
Author_and_Institution
N.J. Bachman (JPL)
J. Diaz del Rio (ODC Space)
W.L. Taber (JPL)
F.S. Turner (JPL)
Version
SPICELIB Version 1.2.0, 12-APR-2021 (JDR)
Added IMPLICIT NONE statement.
Edited the header to comply with NAIF standard. Corrected the
output argument name in $Exceptions section.
SPICELIB Version 1.1.2, 26-FEB-2008 (NJB)
Updated header; added information about SPICE
quaternion conventions.
SPICELIB Version 1.1.1, 13-JUN-2002 (FST)
Updated the $Exceptions section to clarify exceptions that
are the result of changes made in the previous version of
the routine.
SPICELIB Version 1.1.0, 04-MAR-1999 (WLT)
Added code to handle the case in which the input quaternion
is not of length 1.
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 18:36:40 2021