qxq_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

```   void qxq_c ( ConstSpiceDouble    q1   ,
ConstSpiceDouble    q2   ,
SpiceDouble         qout   )

```

#### Abstract

```
Multiply two quaternions.
```

```
ROTATION
```

```
MATH
POINTING
ROTATION

```

#### Brief_I/O

```
VARIABLE  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.
```

```
None.
```

```
Error free.
```

```
None.
```

#### 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(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)  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

qxq_c ( qi, qj, ixj );
qxq_c ( qj, qk, jxk );
qxq_c ( qk, qi, kxi );

produce the results

ixj == qk
jxk == qi
kxi == qj

All of the calls

qxq_c ( qi, qi, qout );
qxq_c ( qj, qj, qout );
qxq_c ( qk, qk, qout );

produce the result

qout  ==  -qid

For any quaternion Q, the calls

qxq_c ( qid, q,   qout );
qxq_c ( 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:

/.
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, cmat3 );

/.
Multiply `cmat1' and `cmat2' directly.
./
mxm_c ( cmat1, cmat2, cmat4 );

/.
Compare the results.  The difference `diff' of
`cmat3' and `cmat4' should be close to the zero
matrix.
./
vsubg_c ( 9, cmat3, cmat4, diff );

```

```
None.
```

```
None.
```

#### Author_and_Institution

```
N.J. Bachman    (JPL)
```

#### Version

```
-CSPICE Version 1.0.1, 27-FEB-2008 (NJB)

```
`Wed Apr  5 17:54:41 2017`