| Re: Averaging quaternions [message #38624] |
Fri, 19 March 2004 08:24  |
Arnold Neumaier
Messages: 5
Registered: March 2004
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Junior Member |
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John Lansberry wrote:
> "Craig Markwardt" <craigmnet@REMOVEcow.physics.wisc.edu> wrote in message
> news:on65d167y8.fsf@cow.physics.wisc.edu...
>
>> GrahamWilsonCA@yahoo.ca (Graham) writes:
>>> Does anyone know if it is possible to take an average of regularly
>>> sampled quaternions to get a mean orientation (i.e. a mean rotation
>>> matrix)? I seem to recall there being a trick involved but beyond
>>> re-normalizing the resuling (averaged) quaternion, I cannot remember
>>> what it is.
>
>> quaternions are degenerate. For each unique rotation, there are two
>> possible quaternions whose components have opposite signs. This is
>> because a positive rotation about axis V is identical to a negative
>> rotation about axis -V.
>>
>> If your system is capable of both signs indiscriminately, then you
>> must make the sign conventions uniform. For example, by always making
>> one component positive.
This will not work if the component is zero.
Averaging (0,q) and (0,-q) which are the same rotation gives 0,
which is meaningless.
Thus the average and scale procedure makes only sense if all quaternions
are oriented the same way. One way to achieve this for any set of
unit quaternions that do not stray too much is the following:
1. apply to all quaternions a rotation that moves one of them to 1
(for example one that is closest to the trivial average),
2. orient all results to positive real part,
3. average the results,
4. rotate back the result,
5. normalize.
More costly but completely reliable.
Arnold Neumaier
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| Re: Averaging quaternions [message #38627 is a reply to message #38624] |
Fri, 19 March 2004 07:22  |
John Lansberry
Messages: 3
Registered: March 2004
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Junior Member |
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Sorry, I didn't finish before sending. I should have mentioned, however,
that Craig's suggestion to just "average the components and normalize" is,
in fact, a common approach (see, for example, CLAUS GRAMKOW, "On Averaging
Rotations", International Journal of Computer Vision 42(1/2), 7-16, 2001).
So Craig's suggestion is certainly one method (just not one I happen to like
very much).
No scolding intended.
John
"John Lansberry" <john.lansberry@jhuapl.edu> wrote in message
news:c3f1hp$fto$1@aplcore.jhuapl.edu...
>
> "Craig Markwardt" <craigmnet@REMOVEcow.physics.wisc.edu> wrote in message
> news:on65d167y8.fsf@cow.physics.wisc.edu...
>>
>> GrahamWilsonCA@yahoo.ca (Graham) writes:
>>
>>> Does anyone know if it is possible to take an average of regularly
>>> sampled quaternions to get a mean orientation (i.e. a mean rotation
>>> matrix)? I seem to recall there being a trick involved but beyond
>>> re-normalizing the resuling (averaged) quaternion, I cannot remember
>>> what it is.
>>
>> I am sure I will be scolded by somebody, but I believe that you can
>> average the quaternion components, and then normalize as you say.
>> This is assumes that you are noise dominated.
>>
> Averaging components is a bad idea no matter what, since the result is
never
> a "quaternion." The OP doesn't imply anything about "noise."
>
>> Also, there is one trick that I can think of, which is that
>> quaternions are degenerate. For each unique rotation, there are two
>> possible quaternions whose components have opposite signs. This is
>> because a positive rotation about axis V is identical to a negative
>> rotation about axis -V.
>>
>> If your system is capable of both signs indiscriminately, then you
>> must make the sign conventions uniform. For example, by always making
>> one component positive.
>
> You are correct that q and -q represent the same rotation - that's not
> "degenerate", it's just not "unique." Typically, the "scalar" part of the
> quaternion, cos(theta/2), is chosen to be the component that's always
> positive.
>
> John
>
>
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| Re: Averaging quaternions [message #38628 is a reply to message #38627] |
Fri, 19 March 2004 07:14 |
John Lansberry
Messages: 3
Registered: March 2004
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Junior Member |
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"Graham" <GrahamWilsonCA@yahoo.ca> wrote in message
news:eda30d78.0403181434.229b3b53@posting.google.com...
> Does anyone know if it is possible to take an average of regularly
> sampled quaternions to get a mean orientation (i.e. a mean rotation
> matrix)? I seem to recall there being a trick involved but beyond
> re-normalizing the resuling (averaged) quaternion, I cannot remember
> what it is.
>
> Cheers,
> Graham
Hi Graham,
It might help a lot to explain a little more about what you mean by
"regularly sampled" and "averaging."
Do you mean that you have time samples of quaternions over time? In other
words, you have q(t1), q(t2), q(t3), etc., where t1, t2, t3, etc. are evenly
spaced time points (not that even spacing is all that important).
Is each sample a valid quaternion?
By "averaging," what do you really want?
SLERP (as suggested by OUP) is generally used to interpolate between two
(valid) quaternions. The result is always a valid quaternion. SLERP is in
fact valid for any two unit vectors of equal dimension (not just
quaternions). So, you can use SLERP to find the quaternion at the midpoint
of your time interval and call that an "average".
My guess is that you need something more complex - try Googling on
"averaging quaternions".
John
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| Re: Averaging quaternions [message #38629 is a reply to message #38628] |
Fri, 19 March 2004 06:53 |
John Lansberry
Messages: 3
Registered: March 2004
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Junior Member |
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"Craig Markwardt" <craigmnet@REMOVEcow.physics.wisc.edu> wrote in message
news:on65d167y8.fsf@cow.physics.wisc.edu...
>
> GrahamWilsonCA@yahoo.ca (Graham) writes:
>
>> Does anyone know if it is possible to take an average of regularly
>> sampled quaternions to get a mean orientation (i.e. a mean rotation
>> matrix)? I seem to recall there being a trick involved but beyond
>> re-normalizing the resuling (averaged) quaternion, I cannot remember
>> what it is.
>
> I am sure I will be scolded by somebody, but I believe that you can
> average the quaternion components, and then normalize as you say.
> This is assumes that you are noise dominated.
>
Averaging components is a bad idea no matter what, since the result is never
a "quaternion." The OP doesn't imply anything about "noise."
> Also, there is one trick that I can think of, which is that
> quaternions are degenerate. For each unique rotation, there are two
> possible quaternions whose components have opposite signs. This is
> because a positive rotation about axis V is identical to a negative
> rotation about axis -V.
>
> If your system is capable of both signs indiscriminately, then you
> must make the sign conventions uniform. For example, by always making
> one component positive.
You are correct that q and -q represent the same rotation - that's not
"degenerate", it's just not "unique." Typically, the "scalar" part of the
quaternion, cos(theta/2), is chosen to be the component that's always
positive.
John
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| Re: Averaging quaternions [message #38640 is a reply to message #38635] |
Thu, 18 March 2004 15:39 |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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GrahamWilsonCA@yahoo.ca (Graham) writes:
> Does anyone know if it is possible to take an average of regularly
> sampled quaternions to get a mean orientation (i.e. a mean rotation
> matrix)? I seem to recall there being a trick involved but beyond
> re-normalizing the resuling (averaged) quaternion, I cannot remember
> what it is.
I am sure I will be scolded by somebody, but I believe that you can
average the quaternion components, and then normalize as you say.
This is assumes that you are noise dominated.
Also, there is one trick that I can think of, which is that
quaternions are degenerate. For each unique rotation, there are two
possible quaternions whose components have opposite signs. This is
because a positive rotation about axis V is identical to a negative
rotation about axis -V.
If your system is capable of both signs indiscriminately, then you
must make the sign conventions uniform. For example, by always making
one component positive.
Some advertising since you crossposted on the IDL newsgroup: I do have
a fairly comprehensive quaternion IDL library on my web page.
Craig
P.S. http://cow.physics.wisc.edu/~craigm/idl/idl.html (under Math)
--
------------------------------------------------------------ --------------
Craig B. Markwardt, Ph.D. EMAIL: craigmnet@REMOVEcow.physics.wisc.edu
Astrophysics, IDL, Finance, Derivatives | Remove "net" for better response
------------------------------------------------------------ --------------
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| Re: Averaging quaternions [message #38791 is a reply to message #38624] |
Mon, 29 March 2004 12:45 |
GrahamWilsonCA
Messages: 12
Registered: January 2003
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Junior Member |
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I have now made a few attempts at averaging my quaternions. For what
it is worth, the quaternion data I have is from a star-tracker on
board a satellite and each quaternion represents the inferred
spacecraft attitude from an algebraic computation using the positions
of 3-6 stars on a CCD.
The paper by Curtis et al. in the 1993 IEEE Annual virtual reality
international symposium was probably the easiest for me to actually
understand so I started with it. However, their approach only works
for averaging 2 quaternions and I have 15-1000 to average in time.
Moreover, I need to weight each quaternion by its relative precision
(i.e. the number of stars on the CCD and the amount of stray light) at
any given time. Both of these points are eluted to in the future work
section of this paper but I cannot find any further papers by these
people.
From the suggested approach:
> 1. apply to all quaternions a rotation that moves one of them to 1
> (for example one that is closest to the trivial average),
> 2. orient all results to positive real part,
> 3. average the results
> 4. rotate back the result,
> 5. normalize.
It isn't clear why steps 1 and 4 are required as they can be combined
with 2 using a dot product? I'm not entirely sure of that but the
following seems to work ok:
Quaternion_t AverageQuaternion(int numRotations,
const Quaternion_t *sourceQ)
{
Quaternion_t averageQ;
int i;
/* initialise result with first rotation */
averageQ = (sourceQ[0]);
/* loop over remaining rotations */
for(i = 1; i<numRotations; ++i)
{
/* Check for identical rotations and add to the average quaternion
*/
if(DotQuaternion(averageQ, sourceQ[i])>0.0f)
averageQ = AddQuaternion (averageQ, sourceQ[i]);
else
averageQ = AddQuaternion (averageQ,
ReverseQuaternion(sourceQ[i]) );
}
/* Normalize */
return NormalizeQuaternion (averageQ);
}
Does anyone have a suggestions on how do I can weight the different
quaternions to get a weighted average rotation?
A previous post on averaging rotation matrices suggested:
> I'd suggest transformation of the rotation matrices into
> quaternions. The quaternion coefficients can be regarded as forming a
> unit vector in 4-space. Your observations should give a cluster of
> such vectors. The centroid of this cluster should give the mean
> rotation.
I quite like this description, but I have no idea how to find the
centroid of a cluster of vectors in 4-space. Can anyone point me to a
book/paper/letter that a mere physicist might be able to understand.
Cheers,
Graham
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