m2q

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

#### Procedure

```      M2Q ( Matrix to quaternion )

SUBROUTINE M2Q ( R, Q )
```

#### Abstract

```     Find a unit quaternion corresponding to a specified rotation
matrix.
```

```     ROTATION
```

#### Keywords

```     MATH
MATRIX
ROTATION
```

#### Declarations

```
DOUBLE PRECISION      R ( 3,  3 )
DOUBLE PRECISION      Q ( 0 : 3 )

```

#### Brief_I/O

```     Variable  I/O  Description
--------  ---  --------------------------------------------------
R          I   A rotation matrix.
Q          O   A unit quaternion representing R.
```

#### Detailed_Input

```     R              is a rotation matrix.
```

#### Detailed_Output

```     Q              is a unit-length SPICE-style quaternion
representing R. See the discussion of quaternion
styles in Particulars below.

Q is a 4-dimensional vector. If R rotates vectors
in the counterclockwise sense by an angle of theta

0 < theta < pi
-       -

then letting h = theta/2,

Q = ( cos(h), sin(h)A ,  sin(h)A ,  sin(h)A ).
1          2          3

The restriction that theta must be in the range
[0, pi] determines the output quaternion Q
uniquely except when theta = pi; in this special
case, both of the quaternions

Q = ( 0,  A ,  A ,  A  )
1    2    3
and

Q = ( 0, -A , -A , -A  )
1    2    3

are possible outputs.
```

#### Parameters

```     None.
```

#### Exceptions

```     1)   If R is not a rotation matrix, the error SPICE(NOTAROTATION)
is signaled.
```

#### Files

```     None.
```

#### Particulars

```     A unit quaternion is a 4-dimensional vector for which the sum of
the squares of the components is 1. Unit quaternions can be used
to represent rotations in the following way: given a rotation
angle theta, where

0 < theta < pi
-       -

and a unit vector A, we can represent the transformation that
rotates vectors in the counterclockwise sense by theta radians
about A using the quaternion Q, where

Q =

( cos(theta/2), sin(theta/2)a , sin(theta/2)a , sin(theta/2)a )
1               2               3

As mentioned in Detailed Output, our restriction on the range of
theta determines Q uniquely, except when theta = pi.

The SPICELIB routine Q2M is an one-sided inverse of this routine:
given any rotation matrix R, the calls

CALL M2Q ( R, Q )
CALL Q2M ( Q, R )

leave R unchanged, except for round-off error.  However, the
calls

CALL Q2M ( Q, R )
CALL M2Q ( 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

SPICELIB subroutine interfaces ALWAYS use SPICE quaternions.
Quaternions of any other style must be converted to SPICE
quaternions before they are passed to SPICELIB routines.

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 SPICELIB routines

Q2M {quaternion to matrix}
M2Q {matrix to quaternion}

M2Q 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

CALL ROTATE (  HALFPI(),  3,  R  )
CALL M2Q    (  R,             Q  )

M2Q 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 quaternion that represents a rotation specified by
a set of Euler angles:

Suppose our original rotation R is the product

[ TAU ]  [ pi/2 - DELTA ]  [ ALPHA ]
3                 2          3

The code fragment

CALL EUL2M  ( TAU,   HALFPI() - DELTA,   ALPHA,
.              3,     2,                  3,      R )

CALL M2Q    ( R, Q )

yields a quaternion Q that represents R.
```

#### Restrictions

```     None.
```

#### Literature_References

```     None.
```

#### Author_and_Institution

```     N.J. Bachman   (JPL)
W.L. Taber     (JPL)

```

#### Version

```    SPICELIB Version 2.0.1, 27-FEB-2008 (NJB)

quaternion conventions. Made various minor edits

SPICELIB Version 2.0.0, 17-SEP-1999 (WLT)

The routine was re-implemented to sharpen the numerical
stability of the routine and eliminate calls to SIN
and COS functions.

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

Comment section for permuted index source lines was added
`Wed Apr  5 17:46:54 2017`