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# Quaternion Toolbox

18 Dec 2001 (Updated 26 Jul 2009)

Vectorized quaternion functions

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Version 1.3 (JASP) 26-Jul-2009

For purposes of these tools, a quaternion, q, is just a four element vector where q(1:3) is the "imaginary" or "vector" portion of the hypercomplex number, and q(4) is the "real" or "scalar" portion. Consequently, if q represents a rotation, then:

q(1) = v1*sin(phi/2)
q(2) = v2*sin(phi/2)
q(3) = v3*sin(phi/2)
q(4) = cos(phi/2)

where phi is the amount of rotation about the unit vector [v1 v2 v3].

All tools are vectorized, so "vectors" of quaternions (4xN or Nx4 matrices) can be handled as well. Since it is most common to work with normalized quaternions (also referred to as "unit quaternions" and "versors"), if a set of 4 quaternions, i.e., a 4x4 matrix, is input, the tools will attempt to determine the shape of the component quaternions (4x1 or 1x4) based on whether the rows or columns are normalized.

Of course, some of the tools, like QDECOMP, only make sense for normalized quaternions, and thus those tools enforce normality via QNORM.

isq - determines whether or not input is a quaternion
isnormq - determines whether or not input is a normalized quaternion

qconj - quaternion conjugate
qnorm - normalize quaternion
qmult - multiply quaternions

qdecomp - decompose quaternion into unit vector and rotation angle

qcvq - operation on vector: qconj(q) v q
qvqc - operation on vector: q v qconj(q)

Because the author uses the convention described in "Spacecraft Attitude Determination and Control" (Wertz, 1978), the following aliases exist:

qvxform - quaternion/vector transform (alias for qcvq)
qvrot - quaternion/vector rotation (alias for qvqc)

Likewise, the following operations assume the relationship between the DCM and the quaternion is: R*v = qvxform(q, v) = qcvq(q, v). That is, the q that performs the equivalent operation on v is the "right hand quaterion".

q2dcm - quaternion to direction cosine matrix
dcm2q - direction cosine matrix to quaternion

Note that many more recent applications, particularly computer graphics libraries, choose the opposite convention. That is, the equivalent "q" is the "left hand quaternion" and consequently the qvxform/qvrot aliases and q2dcm and dcm2q functions would be "backward".

MATLAB release MATLAB 7.5 (R2007b)
Other requirements If you want to run the unit tests, you will need the test_tools package found elsewhere in this archive.
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Comments and Ratings (8)
23 Oct 2009

Works well but not very quickly. I think I'll try to find a mex implementation.

02 Aug 2009

Lars Barring: It looks like you have a different "qnorm" in your search path prior to the quaternions toolbox. Do a "which qnorm" at the MATLAB command line.

-Jay

28 Jul 2009

QNORM fails the unit test with the following output (R2008b):
============================================
Unit Test for: qnorm
============================================
----------------------------------------
Insufficient Arguments
----------------------------------------
Function Call: qnorm

Expected Error:
qnorm() requires one input argument

error ~= expected_error ***FAILED***

ans =
Input argument "p" is undefined.

err_string =
qnorm() requires one input argument

----------------------------------------
Invalid Input
----------------------------------------
Function Call: qnorm(1)

Expected Error:
Invalid input: must be a quaternion or a vector of quaternions

ans =
Inf

error ~= expected_error ***FAILED***

ans =
Input argument "p" is undefined.

err_string =
Invalid input: must be a quaternion or a vector of quaternions

----------------------------------------
Column of two quaternions
----------------------------------------

q1 =
1 2 3 4
q2 =
4 3 2 1

??? Error using ==> qnorm at 13
A probability should be 0<=p<=1, please!

Error in ==> check_value at 39
l_test_value = evalin('base', test_value);

Error in ==> test_qnorm at 36
failures=failures+check_value(truth_value, test_value);

Error in ==> unit_test at 25
test_qnorm; Failures=Failures+failures;

20 Aug 2007

NOTE:
A much better approach than testing for absolute equality is to calculate (q_calc - q_exp)/(q_calc + q_exp) and compare that value with eps('double')

09 Dec 2006

dcm2q([sqrt(.5) 0 -sqrt(.5) 0 ]) returns
[0.7071,0, 0.7071, 0] instead of
[-sqrt(.5) 0 sqrt(.5) 0 ]

25 Jan 2006

wrong answer for rotation. qvrot function is wrong.

04 Nov 2005

nice one!

16 Jul 2003

nicely structured! makes life a lot easier!!
thank you.