void q2m_c ( ConstSpiceDouble q[4],
SpiceDouble r[3][3] )
Find the rotation matrix corresponding to a specified unit
quaternion.
ROTATION
MATH
MATRIX
ROTATION
Variable I/O Description
-------- --- --------------------------------------------------
q I A unit quaternion.
r O A rotation matrix corresponding to `q'.
q is a unit-length SPICE-style quaternion representing
a rotation. `q' has the property that
|| q || = 1
See the discussion of quaternion styles in
Particulars below.
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.
None.
Error free.
1) If `q' is not a unit quaternion, the output matrix `r' is
unlikely to be a rotation matrix.
None.
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 `q' and a scalar `theta' such
that
q =
( cos(theta/2), sin(theta/2)a(1), sin(theta/2)a(2), sin(theta/2)a(3) )
We can interpret `a' and `theta' as the axis and rotation angle of a
rotation in 3-space. If we restrict `theta' to the range [0, pi],
then `theta' and `a' are uniquely determined, except if theta = 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 CSPICE routine m2q_c is a one-sided inverse of this routine:
given any rotation matrix `r', the calls
m2q_c ( r, q )
q2m_c ( q, r )
leave `r' unchanged, except for round-off error. However, the
calls
q2m_c ( q, r )
m2q_c ( 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
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(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 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 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) 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
rotate_c ( halfpi_c(), 3, r );
m2q_c ( r, q );
m2q_c 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 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
q2m_c ( q, r );
m2eul_c ( r, 3, 2, 3, tau, delta, alpha );
delta = halfpi_c() - delta;
None.
[1] NAIF document 179.0, "Rotations and their Habits", by
W. L. Taber.
N.J. Bachman (JPL)
E.D. Wright (JPL)
-CSPICE Version 1.3.2, 27-FEB-2008 (NJB)
Updated header; added information about SPICE quaternion
conventions. Made miscellaneous edits throughout header.
-CSPICE Version 1.3.1, 06-FEB-2003 (EDW)
Corrected typo error in Examples section.
-CSPICE Version 1.3.0, 24-JUL-2001 (NJB)
Changed prototype: input q is now type (ConstSpiceDouble [4]).
Implemented interface macro for casting input q to const.
-CSPICE Version 1.2.0, 08-FEB-1998 (NJB)
Removed local variables used for temporary capture of outputs.
Removed tracing calls, since the underlying Fortran routine
is error-free.
-CSPICE Version 1.0.0, 25-OCT-1997 (NJB)
Based on SPICELIB Version 1.0.1, 10-MAR-1992 (WLT)
quaternion to matrix
Link to routine q2m_c source file q2m_c.c
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