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q2m_c

 Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input Detailed_Output Parameters Exceptions Files Particulars Examples Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

q2m_c ( Quaternion to matrix )

void q2m_c ( ConstSpiceDouble  q[4],
SpiceDouble       r[3][3] )

#### Abstract

Find the rotation matrix corresponding to a specified unit
quaternion.

ROTATION

MATH
MATRIX
ROTATION

#### 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 representing
a rotation. `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.

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.

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 `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

#### Examples

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

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.

None.

#### Author_and_Institution

N.J. Bachman        (JPL)
J. Diaz del Rio     (ODC Space)
W.L. Taber          (JPL)
E.D. Wright         (JPL)

#### Version

-CSPICE Version 1.3.3, 10-AUG-2021 (JDR)

Edited the header to comply with NAIF standard. Moved ROTATIONS required

Updated entry #1 and added entry #2 to -Exceptions section.

-CSPICE Version 1.3.2, 27-FEB-2008 (NJB)

-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) (WLT)

Based on SPICELIB Version 1.0.1, 10-MAR-1992 (WLT)

#### Index_Entries

quaternion to matrix
Fri Dec 31 18:41:11 2021