| qxq_c |
|
Table of contents
Procedure
qxq_c ( Quaternion times quaternion )
void qxq_c ( ConstSpiceDouble q1 [4],
ConstSpiceDouble q2 [4],
SpiceDouble qout [4] )
AbstractMultiply two quaternions. Required_ReadingROTATION KeywordsMATH POINTING ROTATION Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- q1 I First SPICE quaternion factor. q2 I Second SPICE quaternion factor. qout O Product of `q1' and `q2'. Detailed_Input
q1 is a 4-vector representing a SPICE-style quaternion.
See the discussion of "Quaternion Styles" in the
-Particulars section 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.
q2 is a second SPICE-style quaternion.
Detailed_Output
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.
ParametersNone. ExceptionsError free. FilesNone. Particulars
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
CSPICE function interfaces ALWAYS use SPICE quaternions.
Quaternions of any other style must be converted to SPICE
quaternions before they are passed to CSPICE functions.
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*( 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 -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
cth = cos(theta/2)
sth = sin(theta/2)
Then the unit quaternions `q' corresponding to `m' are
q = +/- ( cth, sth*n0, sth*n1, sth*n2 )
The mappings between quaternions and the corresponding rotations
are carried out by the CSPICE routines
q2m_c {quaternion to matrix}
m2q_c {matrix to quaternion}
m2q_c 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 `quat1' and `quat2' be SPICE quaternions with respective scalar
and vector parts `s1', `s2' and `v1', `v2':
quat1 = s1 + v1
quat2 = 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
quat1*quat2 = s1*s2 - <v1,v2> + s1*v2 + s2*v1 + (v1 x v2)
If `quat1' and `quat2' represent the rotation matrices `m1' and `m2'
respectively, then the quaternion product
quat1*quat1
represents the matrix product
m1*m2
Examples
The numerical results shown for these examples may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.
1) Given the "basis" quaternions:
qid: ( 1.0, 0.0, 0.0, 0.0 )
qi : ( 0.0, 1.0, 0.0, 0.0 )
qj : ( 0.0, 0.0, 1.0, 0.0 )
qk : ( 0.0, 0.0, 0.0, 1.0 )
the following quaternion products give these results:
Product Expected result
----------- ----------------------
qi * qj ( 0.0, 0.0, 0.0, 1.0 )
qj * qk ( 0.0, 1.0, 0.0, 0.0 )
qk * qi ( 0.0, 0.0, 1.0, 0.0 )
qi * qi (-1.0, 0.0, 0.0, 0.0 )
qj * qj (-1.0, 0.0, 0.0, 0.0 )
qk * qk (-1.0, 0.0, 0.0, 0.0 )
qid * qi ( 0.0, 1.0, 0.0, 0.0 )
qi * qid ( 0.0, 1.0, 0.0, 0.0 )
qid * qj ( 0.0, 0.0, 1.0, 0.0 )
The following code example uses QXQ to produce these results.
Example code begins here.
/.
Program qxq_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
/.
Local variables
./
SpiceDouble qout [4];
/.
Let `qid', `qi', `qj', `qk' be the "basis"
quaternions.
./
SpiceDouble qid [4] = { 1.0, 0.0, 0.0, 0.0 };
SpiceDouble qi [4] = { 0.0, 1.0, 0.0, 0.0 };
SpiceDouble qj [4] = { 0.0, 0.0, 1.0, 0.0 };
SpiceDouble qk [4] = { 0.0, 0.0, 0.0, 1.0 };
/.
Compute:
qi x qj = qk
qj x qk = qi
qk x qi = qj
./
qxq_c ( qi, qj, qout );
printf( "qi x qj = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qk = %7.1f %7.1f %7.1f %7.1f\n",
qk[0], qk[1], qk[2], qk[3] );
printf( " \n" );
qxq_c ( qj, qk, qout );
printf( "qj x qk = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qi = %7.1f %7.1f %7.1f %7.1f\n",
qi[0], qi[1], qi[2], qi[3] );
printf( " \n" );
qxq_c ( qk, qi, qout );
printf( "qk x qi = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qj = %7.1f %7.1f %7.1f %7.1f\n",
qj[0], qj[1], qj[2], qj[3] );
printf( " \n" );
/.
Compute:
qi x qi == -qid
qj x qj == -qid
qk x qk == -qid
./
qxq_c ( qi, qi, qout );
printf( "qi x qi = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qid = %7.1f %7.1f %7.1f %7.1f\n",
qid[0], qid[1], qid[2], qid[3] );
printf( " \n" );
qxq_c ( qj, qj, qout );
printf( "qj x qj = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qid = %7.1f %7.1f %7.1f %7.1f\n",
qid[0], qid[1], qid[2], qid[3] );
printf( " \n" );
qxq_c ( qk, qk, qout );
printf( "qk x qk = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qid = %7.1f %7.1f %7.1f %7.1f\n",
qid[0], qid[1], qid[2], qid[3] );
printf( " \n" );
/.
Compute:
qid x qi = qi
qi x qid = qi
qid x qj = qj
./
qxq_c ( qid, qi, qout );
printf( "qid x qi = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qi = %7.1f %7.1f %7.1f %7.1f\n",
qi[0], qi[1], qi[2], qi[3] );
printf( " \n" );
qxq_c ( qi, qid, qout );
printf( "qi x qid = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qi = %7.1f %7.1f %7.1f %7.1f\n",
qi[0], qi[1], qi[2], qi[3] );
printf( " \n" );
qxq_c ( qid, qj, qout );
printf( "qid x qj = %7.1f %7.1f %7.1f %7.1f\n",
qout[0], qout[1], qout[2], qout[3] );
printf( " qj = %7.1f %7.1f %7.1f %7.1f\n",
qj[0], qj[1], qj[2], qj[3] );
printf( " \n" );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
qi x qj = 0.0 0.0 0.0 1.0
qk = 0.0 0.0 0.0 1.0
qj x qk = 0.0 1.0 0.0 0.0
qi = 0.0 1.0 0.0 0.0
qk x qi = 0.0 0.0 1.0 0.0
qj = 0.0 0.0 1.0 0.0
qi x qi = -1.0 0.0 0.0 0.0
qid = 1.0 0.0 0.0 0.0
qj x qj = -1.0 0.0 0.0 0.0
qid = 1.0 0.0 0.0 0.0
qk x qk = -1.0 0.0 0.0 0.0
qid = 1.0 0.0 0.0 0.0
qid x qi = 0.0 1.0 0.0 0.0
qi = 0.0 1.0 0.0 0.0
qi x qid = 0.0 1.0 0.0 0.0
qi = 0.0 1.0 0.0 0.0
qid x qj = 0.0 0.0 1.0 0.0
qj = 0.0 0.0 1.0 0.0
2) Compute the composition of two rotation matrices by
converting them to quaternions and computing their
product, and by directly multiplying the matrices.
Example code begins here.
/.
Program qxq_ex2
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
/.
Local variables
./
SpiceDouble cmout [3][3];
SpiceDouble q1 [4];
SpiceDouble q2 [4];
SpiceDouble qout [4];
SpiceDouble cmat1 [3][3] = { {1.0, 0.0, 0.0},
{0.0, -1.0, 0.0},
{0.0, 0.0, -1.0} };
SpiceDouble cmat2 [3][3] = { {0.0, 1.0, 0.0},
{1.0, 0.0, 0.0},
{0.0, 0.0, -1.0} };
/.
Convert the C-matrices to quaternions.
./
m2q_c ( cmat1, q1 );
m2q_c ( cmat2, q2 );
/.
Find the product.
./
qxq_c ( q1, q2, qout );
/.
Convert the result to a C-matrix.
./
q2m_c ( qout, cmout );
printf( "Using quaternion product:\n" );
printf( "%9.4f %9.4f %9.4f\n",
cmout[0][0], cmout[0][1], cmout[0][2] );
printf( "%9.4f %9.4f %9.4f\n",
cmout[1][0], cmout[1][1], cmout[1][2] );
printf( "%9.4f %9.4f %9.4f\n",
cmout[2][0], cmout[2][1], cmout[2][2] );
/.
Multiply `cmat1' and `cmat2' directly.
./
mxm_c ( cmat1, cmat2, cmout );
printf( "Using matrix product:\n" );
printf( "%9.4f %9.4f %9.4f\n",
cmout[0][0], cmout[0][1], cmout[0][2] );
printf( "%9.4f %9.4f %9.4f\n",
cmout[1][0], cmout[1][1], cmout[1][2] );
printf( "%9.4f %9.4f %9.4f\n",
cmout[2][0], cmout[2][1], cmout[2][2] );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Using quaternion product:
0.0000 1.0000 0.0000
-1.0000 0.0000 0.0000
0.0000 0.0000 1.0000
Using matrix product:
0.0000 1.0000 0.0000
-1.0000 0.0000 0.0000
0.0000 0.0000 1.0000
RestrictionsNone. Literature_ReferencesNone. Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) Version
-CSPICE Version 1.0.2, 10-AUG-2021 (JDR)
Edited the header to comply with NAIF standard.
Created complete code examples from existing example and code
fragments.
-CSPICE Version 1.0.1, 27-FEB-2008 (NJB)
Updated header; added information about SPICE
quaternion conventions.
-CSPICE Version 1.0.0, 27-OCT-2005 (NJB)
Index_Entriesquaternion times quaternion multiply quaternion by quaternion |
Fri Dec 31 18:41:11 2021