Reply by Michael Plante●September 2, 20092009-09-02
>On 2 Sep, 17:29, "Michael Plante" <michael.pla...@gmail.com> wrote:
>
>> Based on what you're suggesting, that's the correct thread. =A0It
wasn't =
>a
>> link, just a title: =A0"Unscented Filtering in a Unit Quaternion Space
fo=
>r
>> Spacecraft Attitude Estimation". =A0My intent in posting it was to
attack
>> the root problem (a UKF on attitude), not to get too general about
>> directional statistics.
>
>Thanks. You wouldn't by any chance have more tips on
>UKFs for attitude estimation?
I have two more references on that exact topic, but no real tips, as I
don't consider myself sufficiently experienced. I guess one thing I will
pass along is that almost every author, whether using the EKF or UKF, goes
to great lengths to deal with the nearly-singular covariance matrix you
eventually get if you put the whole quaternion in the state vector, and
that there are multiple different approaches out there to working around
that, usually involving some trick to take an NxN covariance matrix down to
(N-1)x(N-1).
This thesis has a lot of information, and some general utility. Note that
the UKF stuff is split into multiple sections throughout the paper, and
that it discusses other filters:
http://www.control.auc.dk/~danji/research/publications/bhanderi_phd_thesis.pdf
This paper is a little more to the point than that thesis (as papers
always are):
http://hep1.physik.uni-bonn.de/fileadmin/Publications/Mqube/Kraft_FusionPaperWeb.pdf
Reply by Rune Allnor●September 2, 20092009-09-02
On 2 Sep, 17:29, "Michael Plante" <michael.pla...@gmail.com> wrote:
> Based on what you're suggesting, that's the correct thread. �It wasn't a
> link, just a title: �"Unscented Filtering in a Unit Quaternion Space for
> Spacecraft Attitude Estimation". �My intent in posting it was to attack
> the root problem (a UKF on attitude), not to get too general about
> directional statistics.
Thanks. You wouldn't by any chance have more tips on
UKFs for attitude estimation?
Rune
Reply by Michael Plante●September 2, 20092009-09-02
>On 2 Sep, 12:51, illywhacker <illywac...@gmail.com> wrote:
>> On 2 Sep, 12:35, Rune Allnor <all...@tele.ntnu.no> wrote:
>>
>>
>>
>>
>>
>> > On 2 Sep, 12:08, illywhacker <illywac...@gmail.com> wrote:
>>
>> > > On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
>>
>> > > > On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>>
>> > > > Illywhacker,
>>
>> > > > A few monts ago you posted a link to a paper on directional
>> > > > statistics, in the context of unscented Kalman filters. The
>> > > > search engine on google groups is broken so I can't find the
>> > > > original thread, and I can't find anything that looks relevant
>> > > > in IEEExplore.
>>
>> > > > Could you please re-post the link to the paper?
>>
>> > > > Rune
>>
>> > > Hi Rune,
>>
>> > > Do you recall whether it was a paper or a Wikipedia link?
>>
>> > > illywhacker;
>>
>> > I'm pretty sure it was a paper; I might be wrong, though.
>>
>> > Rune- Hide quoted text -
>>
>> > - Show quoted text -
>>
>> Is this the thread?
>>
>>
>>
>> I have a couple of posts in this thread. There is a response to a
>> question you posed, but there are no references.
>
>The time is about right, but I remember somebody posting
>a reference to a journal paper. One reason I remember
>it was that Tim commented that the paper seemed to be
>worth actually purchasing. For some reason, I found that
>particular comment noteworthy.
>
>In the thread above you were the first to mention the
>term 'directional statistics' - maybe that's why I remember
>a signifcant contribution from you. It could have been
>somebody else who posted the reference.
>
>Rune
Based on what you're suggesting, that's the correct thread. It wasn't a
link, just a title: "Unscented Filtering in a Unit Quaternion Space for
Spacecraft Attitude Estimation". My intent in posting it was to attack
the root problem (a UKF on attitude), not to get too general about
directional statistics.
Reply by illywhacker●September 2, 20092009-09-02
On 2 Sep, 13:19, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 2 Sep, 12:51, illywhacker <illywac...@gmail.com> wrote:
>
>
>
>
>
> > On 2 Sep, 12:35, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > On 2 Sep, 12:08, illywhacker <illywac...@gmail.com> wrote:
>
> > > > On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > > > On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>
> > > > > Illywhacker,
>
> > > > > A few monts ago you posted a link to a paper on directional
> > > > > statistics, in the context of unscented Kalman filters. The
> > > > > search engine on google groups is broken so I can't find the
> > > > > original thread, and I can't find anything that looks relevant
> > > > > in IEEExplore.
>
> > > > > Could you please re-post the link to the paper?
>
> > > > > Rune
>
> > > > Hi Rune,
>
> > > > Do you recall whether it was a paper or a Wikipedia link?
>
> > > > illywhacker;
>
> > > I'm pretty sure it was a paper; I might be wrong, though.
>
> > > Rune- Hide quoted text -
>
> > > - Show quoted text -
>
> > Is this the thread?
>
> >http://groups.google.co.uk/group/comp.dsp/browse_frm/thread/30ab091e8...
>
> > I have a couple of posts in this thread. There is a response to a
> > question you posed, but there are no references.
>
> The time is about right, but I remember somebody posting
> a reference to a journal paper. One reason I remember
> it was that Tim commented that the paper seemed to be
> worth actually purchasing. For some reason, I found that
> particular comment noteworthy.
>
> In the thread above you were the first to mention the
> term 'directional statistics' - maybe that's why I remember
> a signifcant contribution from you. It could have been
> somebody else who posted the reference.
I think it must have been. Sorry not to be more help.
illywhacker;
Reply by Rune Allnor●September 2, 20092009-09-02
On 2 Sep, 12:51, illywhacker <illywac...@gmail.com> wrote:
> On 2 Sep, 12:35, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > On 2 Sep, 12:08, illywhacker <illywac...@gmail.com> wrote:
>
> > > On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > > On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>
> > > > Illywhacker,
>
> > > > A few monts ago you posted a link to a paper on directional
> > > > statistics, in the context of unscented Kalman filters. The
> > > > search engine on google groups is broken so I can't find the
> > > > original thread, and I can't find anything that looks relevant
> > > > in IEEExplore.
>
> > > > Could you please re-post the link to the paper?
>
> > > > Rune
>
> > > Hi Rune,
>
> > > Do you recall whether it was a paper or a Wikipedia link?
>
> > > illywhacker;
>
> > I'm pretty sure it was a paper; I might be wrong, though.
>
> > Rune- Hide quoted text -
>
> > - Show quoted text -
>
> Is this the thread?
>
> http://groups.google.co.uk/group/comp.dsp/browse_frm/thread/30ab091e8...
>
> I have a couple of posts in this thread. There is a response to a
> question you posed, but there are no references.
The time is about right, but I remember somebody posting
a reference to a journal paper. One reason I remember
it was that Tim commented that the paper seemed to be
worth actually purchasing. For some reason, I found that
particular comment noteworthy.
In the thread above you were the first to mention the
term 'directional statistics' - maybe that's why I remember
a signifcant contribution from you. It could have been
somebody else who posted the reference.
Rune
Reply by illywhacker●September 2, 20092009-09-02
On 2 Sep, 12:35, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 2 Sep, 12:08, illywhacker <illywac...@gmail.com> wrote:
>
>
>
>
>
> > On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>
> > > Illywhacker,
>
> > > A few monts ago you posted a link to a paper on directional
> > > statistics, in the context of unscented Kalman filters. The
> > > search engine on google groups is broken so I can't find the
> > > original thread, and I can't find anything that looks relevant
> > > in IEEExplore.
>
> > > Could you please re-post the link to the paper?
>
> > > Rune
>
> > Hi Rune,
>
> > Do you recall whether it was a paper or a Wikipedia link?
>
> > illywhacker;
>
> I'm pretty sure it was a paper; I might be wrong, though.
>
> Rune- Hide quoted text -
>
> - Show quoted text -
On 2 Sep, 12:08, illywhacker <illywac...@gmail.com> wrote:
> On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>
> > Illywhacker,
>
> > A few monts ago you posted a link to a paper on directional
> > statistics, in the context of unscented Kalman filters. The
> > search engine on google groups is broken so I can't find the
> > original thread, and I can't find anything that looks relevant
> > in IEEExplore.
>
> > Could you please re-post the link to the paper?
>
> > Rune
>
> Hi Rune,
>
> Do you recall whether it was a paper or a Wikipedia link?
>
> illywhacker;
I'm pretty sure it was a paper; I might be wrong, though.
Rune
Reply by illywhacker●September 2, 20092009-09-02
On 2 Sep, 11:33, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
>
> Illywhacker,
>
> A few monts ago you posted a link to a paper on directional
> statistics, in the context of unscented Kalman filters. The
> search engine on google groups is broken so I can't find the
> original thread, and I can't find anything that looks relevant
> in IEEExplore.
>
> Could you please re-post the link to the paper?
>
> Rune
Hi Rune,
Do you recall whether it was a paper or a Wikipedia link?
illywhacker;
Reply by Rune Allnor●September 2, 20092009-09-02
On 2 Sep, 10:50, illywhacker <illywac...@gmail.com> wrote:
Illywhacker,
A few monts ago you posted a link to a paper on directional
statistics, in the context of unscented Kalman filters. The
search engine on google groups is broken so I can't find the
original thread, and I can't find anything that looks relevant
in IEEExplore.
Could you please re-post the link to the paper?
Rune
Reply by illywhacker●September 2, 20092009-09-02
On 1 Sep, 15:16, "dvsarw...@yahoo.com"
<dvsarw...@gmail.com> wrote:
> On Sep 1, 4:35 am, illywhacker <illywac...@gmail.com> wrote:
>
> > You seem to be ruling out Brownian motion as a useful
> > model, which is a bit drastic!
>
> I did not rule out Brownian motion, which is a different
> issue entirely. As you well know, Brownian motion is
> not a wide-sense-stationary process, and the usual
> notion of power spectral density as it is used in the
> second-order theory of random processes in linear
> systems is not applicable.
Brownian motion strictly defined indeed has a fixed
starting point. However, without a fixed starting point, it
can also be viewed as a measure on functions on the real
line. Without the fixed point at zero, the variance is of
course infinite. (This is no more mysterious than the fact
that a definite integral as a function of its upper limit
has an unknown value if the lower limit is not fixed.) One
solution to this is to fix a point. Another is to regard it
as a measure on functions modulo constants. In the
particular case at hand, however, all this is moot, because
there will be a fixed starting point for the integration.
> The question posed by the
> OP was about the power spectral density of the output
> of a linear time-invariant system driven by white noise.
> The linear system is stated to be an integrator, that is,
> the impulse response is the unit step function, and so
> the output of the linear system at time t is the integral
> of the input from -infinity to t (not from 0 to t as is the
> case for Brownian motion). The usual second-order
> theory is not applicable because the linear system is
> not BIBO-stable, but if one does go through the motions
> anyway (as engineers are apt to do), the resulting
> process turns out to have infinite variance. Even this
> would not faze some people, and so I reminded people
> that if the input were Gaussian white noise, then the
> output process would be a Gaussian process, and
> would consist of Gaussian random variables all of
> which have infinite variance, thus making it a bit hard
> to write down the probability density functions.
A minor point: you cannot write down probability density
functions (in a mathematically strict sense, which is what
you seem to favour) for white noise either: you can only
write down correlation functions.
A major point: you seem to be wrong anyway. The
Wiener-Khinchin theorem guarantees exactly what the OP
wants, as a glance at the 'Applications' section of the
relevant Wikipedia page would tell you.
This result can be found in a far easier way, however, by
heeding the advice of Gauss, as follows. While it is good
for our sense of our own cleverness and power to be able to
manipulate infinite mathematical objects directly, one must
remember that infinity is not a number, and does not exist
except as a limit. Thus it is a cause of errors to take a
limit too early: limits should always and only be taken at
the end of a calculation. The existence of a limit then
just shows that the results at finite values of the
parameter that is going to the limit are not too sensitive
to the precise value provided it is large enough. Of
course, once we are familiar enough with certain types of
limit, we can take them earlier (e.g. we manipulate
derivatives directly), but we must never forget that there
is always the possibility of errors creeping in; if this
happens we must render everything finite and only take the
limits at the end. This is what engineers and physicist
effectively do when hand-waving.
In the present case, by imposing low- and high-frequency
cut-offs, everything is rendered finite. The above result
can then be derived and the cut-offs removed. This is a
better way to do it than using more complex theory:
simpler, more transparent, and more illustrative of the
meaning of the limit.
> >What does not make sense is systems of infinite power, of
> >infinite extent, or that extend down to infinitely small
> >distances (e.g. the value at a point), except as limits
> >that are useful in some circumstances, obstructive in
> >others.
>
> True, and one might say that a linear time-invariant
> system whose impulse response is a step function
> does not make sense because it assumes that the
> it has been busily integrating its input from -infinity
> up till now, which is not true in any practical sense.
> But many commonly used mathematical models
> do give us systems whose response extends to infinity.
> The mathematical model for a capacitor discharging
> through a resistor says that there is residual charge at
> all finite times; it is only the limiting value of the charge
> (as t approaches infinity) that is zero . Most engineers
> would say that after some time has elapsed (say 5 time
> constants), the capacitor is for all practical purposes
> fully discharged.
The advice of Gauss applies. There is nothing wrong with
infinities, provided they are always regarded as limits,
and if necessary, regularized at intermediate stages of a
calculation.
illywhacker;