| qdq2av_c |
|
Table of contents
Procedure
qdq2av_c (Quaternion and quaternion derivative to a.v.)
void qdq2av_c ( ConstSpiceDouble q [4],
ConstSpiceDouble dq [4],
SpiceDouble av [3] )
AbstractDerive angular velocity from a unit quaternion and its derivative with respect to time. Required_ReadingROTATION KeywordsMATH POINTING ROTATION Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- q I Unit SPICE quaternion. dq I Derivative of `q' with respect to time. av O Angular velocity defined by `q' and `dq'. Detailed_Input
q is a unit length 4-vector representing a
SPICE-style quaternion. See the discussion of
quaternion styles in -Particulars below.
dq is a 4-vector representing the derivative of
`q' with respect to time.
Detailed_Output
av is 3-vector representing the angular velocity
defined by `q' and `dq', that is, the angular velocity
of the frame defined by the rotation matrix
associated with `q'. This rotation matrix can be
obtained via the CSPICE routine q2m_c; see the
-Particulars section for the explicit matrix
entries.
`av' is the vector (imaginary) part of the
quaternion product
*
-2 * q * dq
This angular velocity is the same vector that could
be obtained (much less efficiently ) by mapping `q'
and `dq' to the corresponding C-matrix `r' and its
derivative `dr', then calling the CSPICE routine
xf2rav_c.
`av' has units of
radians / T
where
1 / T
is the unit associated with `dq'.
ParametersNone. Exceptions
Error free.
1) A unitized version of input quaternion is used in the
computation. No attempt is made to diagnose an invalid
input quaternion.
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
About this routine
==================
Given a time-dependent SPICE quaternion representing the
attitude of an object, we can obtain the object's angular
velocity `av' in terms of the quaternion `q' and its derivative
with respect to time `dq':
*
av = I * ( -2 * q * dq ) (1)
That is, `av' is the vector (imaginary) part of the product
on the right hand side (RHS) of equation (1). The scalar part
of the RHS is zero.
We'll now provide an explanation of formula (1). For any
time-dependent rotation, the associated angular velocity at a
given time is a function of the rotation and its derivative at
that time. This fact enables us to extend a proof for a limited
subset of rotations to *all* rotations: if we find a formula
that, for any rotation in our subset, gives us the angular
velocity as a function of the rotation and its derivative, then
that formula must be true for all rotations.
We start out by considering the set of rotation matrices
r(t) = m(t) * k (2)
where `k' is a constant rotation matrix and m(t) represents a
matrix that "rotates" with constant, unit magnitude angular
velocity and that is equal to the identity matrix at t = 0.
For future reference, we'll consider `k' to represent a coordinate
transformation from frame `f1' to frame `f2'. We'll call `f1' the
"base frame" of `k'. We'll let `avf2' be the angular velocity of
m(t) relative to `f2' and `avf1' be the same angular velocity
relative to `f1'.
Referring to the axis-and-angle decomposition of m(t)
2
m(t) = I + sin(t)*omega + (1-cos(t))*omega (3)
(see the Rotation Required Reading for a derivation) we
have
d(m(t))|
-------| = omega (4)
dt |t=0
Then the derivative of r(t) at t = 0 is given by
d(r(t))|
-------| = omega * k (5)
dt |t=0
The rotation axis `a' associated with `omega' is defined by (6)
a(0) = - omega(1,2)
a(1) = omega(0,2)
a(2) = - omega(0,1)
Since the coordinate system rotation m(t) rotates vectors about `a'
through angle `t' radians at time `t', the angular velocity `avf2' of
m(t) is actually given by
avf2 = - a (7)
This angular velocity is represented relative to the image
frame `f2' associated with the coordinate transformation `k'.
Now, let's proceed to the angular velocity formula for
quaternions.
To avoid some verbiage, we'll freely use 3-vectors to represent
the corresponding pure imaginary quaternions.
Letting qr(t), qm(t), and `qk' be quaternions representing the
time-dependent matrices r(t), m(t) and `k' respectively, where
qm(t) is selected to be a differentiable function of `t' in a
neighborhood of t = 0, the quaternion representing r(t) is
qr(t) = qm(t) * qk (8)
Differentiating with respect to `t', then evaluating derivatives
at t = 0, we have
d(qr(t))| d(qm(t))|
--------| = --------| * qk (9)
dt |t=0 dt |t=0
Since qm(t) represents a rotation having axis `a' and rotation
angle `t', then (according to the relationship between SPICE
quaternions and rotations set out in the Rotation Required
Reading), we see qm(t) must be the quaternion (represented as the
sum of scalar and vector parts):
cos(t/2) + sin(t/2) * a (10)
where `a' is the rotation axis corresponding to the matrix
`omega' introduced in equation (3). By inspection
d(qm(t))|
--------| = 1/2 * a (11)
dt |t=0
which is a quaternion with scalar part zero. This allows us to
rewrite the quaternion derivative
d(qr(t))|
--------| = 1/2 * a * qk (12)
dt |t=0
or for short,
dq = 1/2 * a * qk (13)
Since from (7) we know the angular velocity `avf2' of the frame
associated with qm(t) is the negative of the rotation axis
defined by (3), we have
dq = - 1/2 * avf2 * qk (14)
Since
avf2 = k * avf1 (15)
we can apply the quaternion transformation formula
(from the Rotation Required Reading)
*
avf2 = qk * avf1 * qk (16)
Now we re-write (15) as
*
dq = - 1/2 * ( qk * avf1 * qk ) * qk
= - 1/2 * qk * avf1 (17)
Then the angular velocity vector `avf1' is given by
*
avf` = -2 * qk * dq (18)
The relation (18) has now been demonstrated for quaternions
having constant, unit magnitude angular velocity. But since
all time-dependent quaternions having value `qk' and derivative
`dq' at a given time `t' have the same angular velocity at time `t',
that angular velocity must be `avf1'.
Examples
The numerical results shown for this example 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) The following test program creates a quaternion and quaternion
derivative from a known rotation matrix and angular velocity
vector. The angular velocity is recovered from the quaternion
and quaternion derivative by calling qdq2av_c and by an
alternate method; the results are displayed for comparison.
Example code begins here.
/.
Program qdq2av_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
#include "SpiceZfc.h"
int main()
{
/.
Local constants
./
/.
Local variables
./
SpiceDouble angle [3];
SpiceDouble av [3];
SpiceDouble avx [3];
SpiceDouble dm [3][3];
SpiceDouble dq [4];
SpiceDouble expav [3];
SpiceDouble m [3][3];
SpiceDouble mout [3][3];
SpiceDouble q [4];
SpiceDouble qav [4];
SpiceDouble xtrans [6][6];
SpiceInt i;
/.
Pick some Euler angles and form a rotation matrix.
./
angle[0] = -20.0 * rpd_c();
angle[1] = 50.0 * rpd_c();
angle[2] = -60.0 * rpd_c();
eul2m_c ( angle[2], angle[1], angle[0], 3, 1, 3, m );
m2q_c ( m, q );
/.
Choose an angular velocity vector.
./
expav[0] = 1.0;
expav[1] = 2.0;
expav[2] = 3.0;
/.
Form the quaternion derivative.
./
qav[0] = 0.0;
vequ_c ( expav, qav+1 );
qxq_c ( q, qav, dq );
vsclg_c ( -0.5, dq, 4, dq );
/.
Recover angular velocity from `q' and `dq' using qdq2av_c.
./
qdq2av_c ( q, dq, av );
/.
Now we'll obtain the angular velocity from `q' and
`dq' by an alternate method.
Convert `q' back to a rotation matrix.
./
q2m_c ( q, m );
/.
Convert `q' and `dq' to a rotation derivative matrix. This
somewhat messy procedure is based on differentiating the
formula for deriving a rotation from a quaternion, then
substituting components of `q' and `dq' into the derivative
formula.
./
dm[0][0] = -4.0 * ( q[2]*dq[2] + q[3]*dq[3] );
dm[0][1] = 2.0 * ( q[1]*dq[2] + q[2]*dq[1]
- q[0]*dq[3] - q[3]*dq[0] );
dm[0][2] = 2.0 * ( q[1]*dq[3] + q[3]*dq[1]
+ q[0]*dq[2] + q[2]*dq[0] );
dm[1][0] = 2.0 * ( q[1]*dq[2] + q[2]*dq[1]
+ q[0]*dq[3] + q[3]*dq[0] );
dm[1][1] = -4.0 * ( q[1]*dq[1] + q[3]*dq[3] );
dm[1][2] = 2.0 * ( q[2]*dq[3] + q[3]*dq[2]
- q[0]*dq[1] - q[1]*dq[0] );
dm[2][0] = 2.0 * ( q[3]*dq[1] + q[1]*dq[3]
- q[0]*dq[2] - q[2]*dq[0] );
dm[2][1] = 2.0 * ( q[2]*dq[3] + q[3]*dq[2]
+ q[0]*dq[1] + q[1]*dq[0] );
dm[2][2] = -4.0 * ( q[1]*dq[1] + q[2]*dq[2] );
/.
Form the state transformation matrix corresponding to `m'
and `dm'.
Upper left block:
./
for ( i = 0; i < 3; i++ )
{
vequ_c ( m[i], xtrans[i] );
}
/.
Upper right block:
./
for ( i = 0; i < 3; i++ )
{
vpack_c ( 0.0, 0.0, 0.0, xtrans[i]+3 );
}
/.
Lower left block:
./
for ( i = 0; i < 3; i++ )
{
vequ_c ( dm[i], xtrans[3+i] );
}
/.
Lower right block:
./
for ( i = 0; i < 3; i++ )
{
vequ_c ( m[i], xtrans[3+i]+3 );
}
/.
Now use xf2rav_c to produce the expected angular velocity.
./
xf2rav_c ( xtrans, mout, avx );
/.
The results should match to nearly full double precision.
./
printf ( "Original angular velocity:\n"
" %19.14f %19.14f %19.14f\n"
"qdq2av_c's angular velocity:\n"
" %19.14f %19.14f %19.14f\n"
"xf2rav_c's angular velocity:\n"
" %19.14f %19.14f %19.14f\n",
expav[0], expav[1], expav[2],
av [0], av [1], av [2],
avx [0], avx [1], avx [2] );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Original angular velocity:
1.00000000000000 2.00000000000000 3.00000000000000
qdq2av_c's angular velocity:
1.00000000000000 2.00000000000000 3.00000000000000
xf2rav_c's angular velocity:
1.00000000000000 2.00000000000000 3.00000000000000
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.
-CSPICE Version 1.0.1, 27-FEB-2008 (NJB)
Updated header; added information about SPICE
quaternion conventions.
-CSPICE Version 1.0.0, 31-OCT-2005 (NJB)
Index_Entriesangular velocity from quaternion and derivative |
Fri Dec 31 18:41:11 2021