QXQ (Quaternion times quaternion)
SUBROUTINE QXQ ( Q1, Q2, QOUT )
Multiply two quaternions.
ROTATION
MATH
POINTING
ROTATION
DOUBLE PRECISION Q1 ( 0 : 3 )
DOUBLE PRECISION Q2 ( 0 : 3 )
DOUBLE PRECISION QOUT ( 0 : 3 )
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
Q1 I First SPICE quaternion factor.
Q2 I Second SPICE quaternion factor.
QOUT O Product of Q1 and Q2.
Q1 is a 4-vector representing a SPICE-style
quaternion. See the discussion of quaternion
styles in Particulars below.
Note that multiple styles of quaternions
are in use. This routine will not work properly
if the input quaternions do not conform to
the SPICE convention. See the Particulars
section for details.
Q2 is a second SPICE-style quaternion.
QOUT is 4-vector representing the quaternion product
Q1 * Q2
Representing Q(i) as the sums of scalar (real)
part s(i) and vector (imaginary) part v(i)
respectively,
Q1 = s1 + v1
Q2 = s2 + v2
QOUT has scalar part s3 defined by
s3 = s1 * s2 - <v1, v2>
and vector part v3 defined by
v3 = s1 * v2 + s2 * v1 + v1 x v2
where the notation < , > denotes the inner
product operator and x indicates the cross
product operator.
None.
Error free.
None.
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
1) Let QID, QI, QJ, QK be the "basis" quaternions
QID = ( 1, 0, 0, 0 )
QI = ( 0, 1, 0, 0 )
QJ = ( 0, 0, 1, 0 )
QK = ( 0, 0, 0, 1 )
respectively. Then the calls
CALL QXQ ( QI, QJ, IXJ )
CALL QXQ ( QJ, QK, JXK )
CALL QXQ ( QK, QI, KXI )
produce the results
IXJ = QK
JXK = QI
KXI = QJ
All of the calls
CALL QXQ ( QI, QI, QOUT )
CALL QXQ ( QJ, QJ, QOUT )
CALL QXQ ( QK, QK, QOUT )
produce the result
QOUT = -QID
For any quaternion Q, the calls
CALL QXQ ( QID, Q, QOUT )
CALL QXQ ( Q, QID, QOUT )
produce the result
QOUT = Q
2) Composition of rotations: let CMAT1 and CMAT2 be two
C-matrices (which are rotation matrices). Then the
following code fragment computes the product CMAT1 * CMAT2:
C
C Convert the C-matrices to quaternions.
C
CALL M2Q ( CMAT1, Q1 )
CALL M2Q ( CMAT2, Q2 )
C
C Find the product.
C
CALL QXQ ( Q1, Q2, QOUT )
C
C Convert the result to a C-matrix.
C
CALL Q2M ( QOUT, CMAT3 )
C
C Multiply CMAT1 and CMAT2 directly.
C
CALL MXM ( CMAT1, CMAT2, CMAT4 )
C
C Compare the results. The difference DIFF of
C CMAT3 and CMAT4 should be close to the zero
C matrix.
C
CALL VSUBG ( 9, CMAT3, CMAT4, DIFF )
None.
None.
N.J. Bachman (JPL)
SPICELIB Version 1.0.1, 26-FEB-2008 (NJB)
Updated header; added information about SPICE
quaternion conventions.
SPICELIB Version 1.0.0, 18-AUG-2002 (NJB)
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