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'[EE]:: Tilt sensing using linear accelerometers'
2008\03\04@040948 by Apptech

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Good paper on the issues involved in using linear
accelerometers as tilt sensors.
Having your feet / legs / wheels on the ground at the time
rather helps.


       http://www.freescale.com/files/sensors/doc/app_note/AN3461.pdf


           Russell McMahon

2008\03\04@105820 by Sean Breheny

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Thanks! I do wish they had discussed more of the physics, though.
There are a lot of misconceptions out there about what accelerometers
can and cannot do (for example, as you said, you need to know
something about the way it is moving or not moving in order to extract
tilt information from the measured accelerations. Actually, what I'd
like to see is a general app note on inertial navigation, covering,
for example:

1) That accelerometers do NOT measure gravity, but rather all the
forces other than gravity which are applied to an object. (In general:
what parts of a system state can and cannot be observed with different
combinations of accelerometers, rate gyros, compasses, and GPS units)
2) Models for the typical noise and offset errors of accelerometers
and rate gyros.
3) Euler angles vs. rotation matrices vs. quaternions
4) Schuler tuning
5) Kalman filters
6) Other types of filters (like sigma point filters)
7) GPS-INS integration, GPS error models

I have yet to find all this information in one location (or even a
significant subset of it, explained in a practical way).

Sean



On Tue, Mar 4, 2008 at 3:03 AM, Apptech <spam_OUTapptechTakeThisOuTspamparadise.net.nz> wrote:
{Quote hidden}

>  --

2008\03\04@120541 by Dave Tweed

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Sean Breheny <.....shb7KILLspamspam@spam@cornell.edu> wrote:
> Thanks! I do wish they had discussed more of the physics, though.
> There are a lot of misconceptions out there about what accelerometers
> can and cannot do (for example, as you said, you need to know
> something about the way it is moving or not moving in order to extract
> tilt information from the measured accelerations. Actually, what I'd
> like to see is a general app note on inertial navigation, covering,
> for example:

That's not an app note, that's a whole shelf full of books!

> 1) That accelerometers do NOT measure gravity, but rather all the
>    forces other than gravity which are applied to an object.

Huh? The accelerometers I have here measure gravity just fine!

Perhaps you meant to say that they measure the net sum all of the
accelerations of an object, including gravity.

> 2) Models for the typical noise and offset errors of accelerometers
>    and rate gyros.

Very dependent on the implementation technology.

> 3) Euler angles vs. rotation matrices vs. quaternions

Mainly a question of which works better for the math you need to do in
your overall applciation.

> 4) Schuler tuning
> 5) Kalman filters
> 6) Other types of filters (like sigma point filters)
> 7) GPS-INS integration, GPS error models

Like I said, each one a topic for an entire book, and the choices made
depend very strongly on the overall application.

For example, I'm working with a client who uses GPS interferometry to
measure vehicle absolute attitude (roll, pitch and yaw) at low sample
rate, then combines this with relative INS measurements in a Kalman
filter to produce an interpolated solution at high sample rate.
Fascinating stuff, but he handles all of the theoretical stuff and the
application software. I just build the underlying hardware that it runs
on.

-- Dave Tweed

2008\03\04@141342 by Sean Breheny

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Hi Dave,

On Tue, Mar 4, 2008 at 12:05 PM, Dave Tweed <picspamKILLspamdtweed.com> wrote:
> Sean Breheny <.....shb7KILLspamspam.....cornell.edu> wrote:
>  > Thanks! I do wish they had discussed more of the physics, though.
>  > There are a lot of misconceptions out there about what accelerometers
>  > can and cannot do (for example, as you said, you need to know
>  > something about the way it is moving or not moving in order to extract
>  > tilt information from the measured accelerations. Actually, what I'd
>  > like to see is a general app note on inertial navigation, covering,
>  > for example:
>
>  That's not an app note, that's a whole shelf full of books!
>

I'd contend that a basic, practical-level intro to it would not be.
You're not going to be good enough to build ICBMs with it, but you can
start playing with INS for robotics-type applications with this level
of knowledge.

{Quote hidden}

No. If they measured the net sum of all the accelerations, then they
would measure zero for a stationary object.

If you have an accelerometer in free-fall in a vacuum, accelerating
under the influence of gravity, it measures zero. If you then set it
on a table so that the sum of the forces on it is zero, but there is 1
newton of gravity and 1 newton worth of normal force acting back from
the table, it will measure ONLY the force from the table (1 newton).
This EQUALS gravity because you know that it is sitting on a table and
not being allowed to accelerate. It is only under this assumption
(that it is not accelerating) that you can relate its measurement to
gravity.

>  > 2) Models for the typical noise and offset errors of accelerometers
>  >    and rate gyros.
>
>  Very dependent on the implementation technology.
>

Yes, but most of the technologies available to the hobbyist are
similar (almost all MEMS with similar specs).

>
>  > 3) Euler angles vs. rotation matrices vs. quaternions
>
>  Mainly a question of which works better for the math you need to do in
>  your overall applciation.
>

Yes, I didn't mean that it would tell you which to pick, only give a
quick overview of how to use each and their advantages and
disadvantages.
>
>  > 4) Schuler tuning
>  > 5) Kalman filters
>  > 6) Other types of filters (like sigma point filters)
>  > 7) GPS-INS integration, GPS error models
>
>  Like I said, each one a topic for an entire book, and the choices made
>  depend very strongly on the overall application.
>

I know of a very good practical reference for Kalman filters which is
only about 10 pages long. Sigma point is even easier to implement.

Schuler tuning can be explained (again, at a beginner level) in a page.

GPS sources of error can be modeled easily at a basic level.

Sean


{Quote hidden}

2008\03\04@154157 by Dave Tweed

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Sean Breheny <EraseMEshb7spam_OUTspamTakeThisOuTcornell.edu> wrote:
> On Tue, Mar 4, 2008 at 12:05 PM, Dave Tweed <picspamspam_OUTdtweed.com> wrote:
> > Sean Breheny <@spam@shb7KILLspamspamcornell.edu> wrote:
> >  > Thanks! I do wish they had discussed more of the physics, though.
> >  > There are a lot of misconceptions out there about what accelerometers
> >  > can and cannot do (for example, as you said, you need to know
> >  > something about the way it is moving or not moving in order to extract
> >  > tilt information from the measured accelerations. Actually, what I'd
> >  > like to see is a general app note on inertial navigation, covering,
> >  > for example:
> >
> >  That's not an app note, that's a whole shelf full of books!
>
> I'd contend that a basic, practical-level intro to it would not be.
> You're not going to be good enough to build ICBMs with it, but you can
> start playing with INS for robotics-type applications with this level
> of knowledge.

OK, I'm confused. Why are we having this conversation?

On one hand, you seem to already have sources for the level of knowledge
you seek. It sounds like you could quickly put together such an app note
yourself.

> If you have an accelerometer in free-fall in a vacuum, accelerating
> under the influence of gravity, it measures zero. If you then set it
> on a table so that the sum of the forces on it is zero, but there is 1
> newton of gravity and 1 newton worth of normal force acting back from
> the table, it will measure ONLY the force from the table (1 newton).
> This EQUALS gravity because you know that it is sitting on a table and
> not being allowed to accelerate. It is only under this assumption
> (that it is not accelerating) that you can relate its measurement to
> gravity.

On the other hand, you seem to be a bit confused about gravity and
acceleration.

First of all, gravity does not have units of Newtons. It is indeed, an
acceleration field, with its magnitude measured in units of length per
time squared.

Yes, any real accelerometer measures its own acceleration *relative* to
the local gravity field. As Einstein showed, there are no *absolute*
references. Gravity in one frame of reference is in no way distinguishable
from a constant linear acceleration in another, so I'm not sure what point
you're trying to make. Perhaps we're in violent agreement.

When I set my accelerometer on a table, it reports that it is accelerating
at 9.8 m/s^2 upward. It is my interpretation, based on the lack of motion
in my larger external reference frame, that attributes this acceleration
to the gravity field and not to actual movement. If I lift it up, the
reading momentarily increases, and if I lower it, the reading decreases.
The accelerometer cannot by itself separate which part of the reading is
due to gravity and which is due to my movements. It's a fundamental system
startup issue -- actual movement is indistinguishable from sensor bias and
scale factor errors. An independent source of information is always needed.

-- Dave Tweed

2008\03\04@191227 by Apptech

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>> 1) That accelerometers do NOT measure gravity, but rather
>> all the
>>    forces other than gravity which are applied to an
>> object.

> Huh? The accelerometers I have here measure gravity just
> fine!

> Perhaps you meant to say that they measure the net sum all
> of the
> accelerations of an object, including gravity.

Both people mean the same thing, probably.

Explanation by thought experiment:

1. Toss your unit in the air so that it is not contacting
the ground. What acceleration forces does it measure.

2.    Place unit on a table in a vibration free environment.
What acceleration forces are reported?

In the case of the table the force contributed to react
against gravity is key to the accelerometer being able to
detect gravity.

If you had the standard hypothetical "totally evacuated
tunnel through the centre of the perfectly uniform and
spherical earth" and dropped your assembly into it, it would
happily oscillate to and fro forever, JUST reaching the
surface on the far side and then returning to you and so on.
At the extremes it would be at zero V and 1g. In the middle
it would be at high V and zero g.

At all stages in the above situation the on board
accelerometer would measure zero acceleration in all
directions.

A zero thickness object doing a close pass slingshot past a
neutron star (assuming no 'atmosphere') such that it is
accelerated to say 10,000g at closest approach, will
experience zero gravity "on board" throughout the pass. The
various stories on neutron star disasters during
slingshotting occur due to gravity differentials across the
objects (ships people etc). Any POINT sees zero net g.



       Russell

2008\03\04@192249 by Sean Breheny

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Hi Dave,


On Tue, Mar 4, 2008 at 3:41 PM, Dave Tweed <KILLspampicKILLspamspamdtweed.com> wrote:
>  OK, I'm confused. Why are we having this conversation?
>
>  On one hand, you seem to already have sources for the level of knowledge
>  you seek. It sounds like you could quickly put together such an app note
>  yourself.
>

With some research, I could put together such an app note. It's not so
much for me to get this info, it is more to see a whole bunch of info
that I've found in various places put together in one place for my
reference and for other's education :)

>  On the other hand, you seem to be a bit confused about gravity and
>  acceleration.
>
>  First of all, gravity does not have units of Newtons. It is indeed, an
>  acceleration field, with its magnitude measured in units of length per
>  time squared.

You are right, I was being sloppy with units.

>
>  Yes, any real accelerometer measures its own acceleration *relative* to
>  the local gravity field. As Einstein showed, there are no *absolute*
>  references. Gravity in one frame of reference is in no way distinguishable
>  from a constant linear acceleration in another, so I'm not sure what point
>  you're trying to make. Perhaps we're in violent agreement.
>

Except that a constant linear acceleration from almost any other
source (e.g. a rocket engine) WOULD produce an output on the
accelerometer but gravity would not.

If you are out in space and being attracted by the gravity of a planet
nearby, accelerating at 1 m/s^2, the accelerometer would read the same
thing as it would if the planet were not there.

If you are in empty space and fire a rocket motor and accelerate at 1
m/s^2, your accelerometer WOULD register 1 m/s^2.

{Quote hidden}

I would put it this way: when it is sitting on the table, it is
reading the normal force. When you pick it up in your hand, it is
reading the force your hand is putting on it (holding it up). When you
accelerate it upward, it will still read the force your hand is
putting on it (which is now slightly higher).

Sean


>
>
>  -- Dave Tweed
>  --

2008\03\05@134438 by Michael Rigby-Jones

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> -----Original Message-----
> From: RemoveMEpiclist-bouncesTakeThisOuTspammit.edu [spamBeGonepiclist-bouncesspamBeGonespammit.edu] On
Behalf
{Quote hidden}

I disagree.  If the planet provided an acceleration of 1 m/s^2 due to
it's gravity, and the accelerometer was actually accelerating at the
same rate toward the planet, then it's output would be zero (1-1=0).  If
the accelerometer was subject to the same acceleration without the
influence of the planets gravity (by e.g. a rocket motor), or was
subject to the planets gravity but prevented from physically moving then
it would most certainly be producing an output.

Regards

Mike

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2008\03\05@144421 by Denny Esterline

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>
>
> I know of a very good practical reference for Kalman filters which is
> only about 10 pages long. Sigma point is even easier to implement.
>
>
At the risk of dragging this thread further afield....
I'd be very interested in a practical referance for Kalman filters.

-Denny

2008\03\05@191626 by Apptech

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>> If you are out in space and being attracted by the
>> gravity of a planet
>> nearby, accelerating at 1 m/s^2, the accelerometer would
>> read the same
>> thing as it would if the planet were not there.

> I disagree.

What you say is essentially correct, but you are disagreeing
with something that he didn't say or mean to imply.

> If the planet provided an acceleration of 1 m/s^2 due to
> it's gravity, and the accelerometer was actually
> accelerating at the
> same rate toward the planet,

Necessary condition as, if it were not accelerating at this
rate then some other forces would be involved

> then it's output would be zero (1-1=0).

Yes.
But

> If the accelerometer was subject to the same acceleration
> without the
> influence of the planets gravity (by e.g. a rocket motor),
> or was
> subject to the planets gravity but prevented from
> physically moving then
> it would most certainly be producing an output.

Yes, but that is not what he said.

First let's deal with what he said and then with why the
cases are different.

He said:

1.    Hang in "empty space". No masses within sensitivity of
detection of g meter. No meter reading.

2.    Hang in "empty space". Magically add planet close
enough to apply a 1m/s/s acceleration field. No meter
reading.

The above reflect what you would see.

______________________

Now:

1.    Hang in "empty space". No masses within sensitivity of
detection of g meter. No meter reading.

2.    Start rocket motor so ship accelerates at 1m/s/s.
Meter reads 1m/s/s.

In the latter case the rocket motor reacts against the ships
structure and the g meter reacts against the ship. The
forces are not applied to the whole structure. If you could
find a way of applying the forces that the acceleration
"field" applied to all objects within the ship
simultaneously then a g meter will not see the acceleration
relative to itself.



       Russell McMahon



2008\03\05@204810 by Sean Breheny

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Hi Denny,

Here's the one I mentioned:

http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf

It is 16 pages long and covers both a regular Kalman filter (used for
linear systems or those which can be modeled as such) and the extended
Kalman filter, used for significantly nonlinear systems. Be forewarned
that the EKF is difficult to work with (very cumbersome math required
and it is often very difficult to get it to converge). If one needs to
do anything more than simple cases, one should learn about other types
of estimators (sigma point filters, for example) so he can choose the
best one for a given situation.

Sean


On Wed, Mar 5, 2008 at 2:43 PM, Denny Esterline <TakeThisOuTdesterlineEraseMEspamspam_OUTgmail.com> wrote:
{Quote hidden}

2008\03\05@205245 by Sean Breheny

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Hi Russell,

Thanks for helping to settle the dispute.

I think it sounded like I was saying something I wasn't. I meant that
accelerometers do not measure gravity directly. You can certainly use
one to measure gravity, but only when certain other assumptions are
met. The only way it has of measuring acceleration is by compression
of some kind of spring, so the force must be transmitted THRU that
spring (which gravity itself is not but the force which acts against
gravity to hold something still IS).

For example, if you managed to charge both the proof mass and the body
of an accelerometer electrically, and then attracted it to an
oppositely charged object, it would not read the correct acceleration
either (since it is in a "force field" rather than under the influence
of an externally applied force). In general relativity there is
probably a neat, elegant, and profound (but not simple to understand)
way to state this but I do not know it.

Sean


On Wed, Mar 5, 2008 at 7:13 PM, Apptech <RemoveMEapptechspamTakeThisOuTparadise.net.nz> wrote:
{Quote hidden}

>  --

2008\03\05@230727 by Apptech

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> Thanks for helping to settle the dispute.


I see/saw it more as a misunderstanding of what the other
was saying. Only when people understand each other can they
disagree ;-).


       Russell

2008\03\06@011950 by Christopher Head

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Hi all,
I've been following this thread for a bit but haven't really taken the
time to think about the situation until now, and here's what I came up with:

I would argue that the best way to describe what the accelerometer
measures is that it measures the difference in net forces between the
ball and the outside casing (assuming the basic model of a ball
connected to a spring connected to the casing).

When you're in open space with nothing around, net force on both is zero.

When a rocket engine pushes the accelerometer in open space at 1g, the
rocket body applies force to the casing but not the ball. Hence, the
accelerometer measures the 1g difference.

When the accelerometer is in freefall towards a planet pulling at 1g,
the 1g gravitational force is applied to both the casing and the ball,
and the accelerometer measures a difference of zero.

When the accelerometer is sitting on a table in a g-field of 1g, the 1g
gravitational force is applied to both the casing and the ball, but the
table's normal force of 1g is applied only to the casing and not the
ball. The accelerometer measures the difference of 1g.

I hope this is sufficiently elegant! It definitely does eliminate the
apparent distinction between gravity and "everything else" - the
distinction is that gravity pulls on the ball and the casing, while
other forces push or pull only on the casing.

Chris

Sean Breheny wrote:
| Hi Russell,
|
| Thanks for helping to settle the dispute.
|
| I think it sounded like I was saying something I wasn't. I meant that
| accelerometers do not measure gravity directly. You can certainly use
| one to measure gravity, but only when certain other assumptions are
| met. The only way it has of measuring acceleration is by compression
| of some kind of spring, so the force must be transmitted THRU that
| spring (which gravity itself is not but the force which acts against
| gravity to hold something still IS).
|
| For example, if you managed to charge both the proof mass and the body
| of an accelerometer electrically, and then attracted it to an
| oppositely charged object, it would not read the correct acceleration
| either (since it is in a "force field" rather than under the influence
| of an externally applied force). In general relativity there is
| probably a neat, elegant, and profound (but not simple to understand)
| way to state this but I do not know it.
|
| Sean
|
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2008\03\06@015942 by Apptech

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> I've been following this thread for a bit but haven't
> really taken the
> time to think about the situation until now, and here's
> what I came up with:
...

{Quote hidden}

Yes. All that was essentially correct (IMHO of course :-) ).

There is one twist which may interest people.

> When the accelerometer is sitting on a table in a g-field
> of 1g, the 1g
> gravitational force is applied to both the casing and the
> ball, but the
> table's normal force of 1g is applied only to the casing
> and not the
> ball. The accelerometer measures the difference of 1g.

Now stand in a sealed room with your ultra high resolution
ultra high accuracy g meter.
Place the g meter on the floor.
Note that it reads 1.000000000000000000 g
Now raise the g meter to the ceiling and measure again.
Note that it still reads 1.000000000000000000 g.

1.    Given 2 options, are you able to conclude whether you
are in a rocket accelerating at 1g or stationary near a
planetary surface

2.    Does the answer seem to conflict with what Einstein
says about the equivalence of gravitational fields.

3.    If the second reading above had been about 1 part in
10^12 smaller than the first reading would it have changed
your answer to question 1?



       Russell McMahon

2008\03\06@030534 by Peter Todd

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On Thu, Mar 06, 2008 at 07:56:53PM +1300, Apptech wrote:
{Quote hidden}

Assuming one heck of an accurate sensor, perhaps immersed in liquid
helium, mounted to a Canadian Space Agency designed vibration damper,
and being flown accross the barren tundra in a small prop plane, yes.

> 2.    Does the answer seem to conflict with what Einstein
> says about the equivalence of gravitational fields.

Nope.

> 3.    If the second reading above had been about 1 part in
> 10^12 smaller than the first reading would it have changed
> your answer to question 1?

Yes. Einsteins equivalence is assuming a planer gravitational field, not
one eminating from a spherical object. Now, admittedly I cheated a bit,
and happened to have been reading about the sensor technologies designed
by one company for mineral prospecting, in this case, diamonds up north
where my parents live, and noticed that their stated unit of measure was
a oddly named thing called an Eotvos. A bit of wikipedia finds that
an Eotvos == 1e-9 galileo per centimetre, and a galieo is 1e-2 m/s²,
or in SI units 1e-9 m/s² per m

Getting to the point... F = G(m1m2)/r² Now with m1 being the earth,
6e+24kg, and m2 being are standard test weight, 1kg, and the earths
radius being 6e+6m we get:

F = G*6e+24kg*1kg/6e+6m²

F = 6e-11Nm²kg-2 *  6e+24kg² / 6e+12m²

(notice every unit but the newton cancels out, and it's all 6e
something)

F = 6e+1N (adding up exponents -11 + 24 - 12)

Now, lets try adding 1m to the radius. Since the 1m is 1e-12 in
porportion to the radius, it will change that result by 2e-12... yup,
roughly what Russel said.

Now trying that stunt on the surface of the sun, with it's 6e+8m radius,
will rquire one to measure a difference of a part in 10^16... Might want
to find something other than prospecting gear... That, and a good
aspestos suit. Anyway, the skin cancer would kill ya long before lung
cancer.

- --
http://petertodd.org
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2008\03\06@033504 by Peter Todd

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On Thu, Mar 06, 2008 at 03:00:18AM -0500, Peter Todd wrote:

Gotta stop doing physics problems at 3am... noticed I missed a few
steps...

{Quote hidden}

Which gives us an acelleration of.. 60m/s²... er... right, think I
rounded a wee bit much there... well anyway, it's in the 10^1 range.
Also that's accelleration relative to the earth, and therefore should be
porportional to the ratio of the masses, minor error there, 10^-12
minor.

More importantly though, the stated survey gear had a resolution of
something like 1e-9m/s², which is 3 orders of magnitude too small for
what I got below. But we can ask the pilot to gain some altitude and
hope that the resulting mass loss in the gas tanks doesn't muck up our
results...

{Quote hidden}

> -

2008\03\06@042126 by Alan B. Pearce

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>this case, diamonds up north where my parents live

Hah, we have the "Ice Road Truckers" TV series on here in the UK at the
moment. Now that is a job they can keep ...

2008\03\06@131741 by Brooke Clarke

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face
Hi:

The thing I find very interesting is that the newer 3-axis accelerometers are
just starting to be used for free fall detection.  The first generation units
from Freescale had that function in the eval board.  Within a few inches of
free fall from a static drop a LED turns on indicating free fall.  The key
applications may be HDD head parking in laptop computers, human fall detection,
cell phone menu navigation.   App Note on Free Fall Sensing:
http://www.freescale.com/files/sensors/doc/app_note/AN3459.pdf?tid=tslpGSEL

But the eval module can be fooled by dynamics such as a spin ball toss which
requires sensing other things.  Free scale has
http://www.prc68.com/I/Sensors.shtml

For the 8-Ball video (8 MB) see:
www.freescale.com/files/multimedia/TSP-12837_8Ball_071008.wmv
If you know of documentation on the 8-Ball or where to get one let me know.

Another interesting thing about gyros is that there long term drift rate is a
function of the volume of the sensor independent of the type of sensor.  I
posted at table at:
http://www.prc68.com/I/Sensors.shtml#Gyroscopic

--
Have Fun,

Brooke Clarke
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2008\03\06@181415 by Michael Rigby-Jones

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> -----Original Message-----
> From: piclist-bouncesEraseMEspam.....mit.edu [EraseMEpiclist-bouncesspammit.edu] On
Behalf
{Quote hidden}

If the "hanging" is preventing the accelerometer from physically moving
with respect to the "magicaly appearing" planet, why would you see no
reading?

Regards

Mike

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2008\03\06@200732 by Apptech

face
flavicon
face
>> 2.    Hang in "empty space". Magically add planet close
>> enough to apply a 1m/s/s acceleration field. No meter
>> reading.

>> The above reflect what you would see.

> If the "hanging" is preventing the accelerometer from
> physically moving
> with respect to the "magicaly appearing" planet, why would
> you see no
> reading?

If it did it wouldn't :-).
The "hang" was intended in the identical sense to the
identical usage in the prior paragraph.
For better sense replace "Hang" with eg "Exist" or "Be
located" or "Starting".


       Russell

2008\03\07@144218 by Sean Breheny

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Hi Brooke,

There is no proposed theoretical basis for the statement below, is
there? It sounds like it was arrived at by just surveying existing
technology.

I've used QRS rate gyros which are around 1 deg/sec/hr drift, which is
almost 10 times better than what is given in your table. (They cost
$$$$)

MEMS rate gyros have a trade-off between bandwidth and noise. Air (or
any gas) inside the sensor causes noise by random motion of the
molecules. However, it also provides damping (this is basically
Johnson noise which all dampers have), thus broadening the bandwidth,
which is desirable. Thus, the noise floor (and partly the drift) is
somewhat determined by the necessary bandwidth. One could probably
take two rate gyros (a low noise, low bandwidth one and a high noise,
high-bandwidth one) and combine their outputs with a Kalman filter or
somesuch and get the benefits of both.

I also don't think that laser ring gyros are as small as they could
possibly get (although I know little about them).

Sean


On Thu, Mar 6, 2008 at 1:17 PM, Brooke Clarke <RemoveMEbrookeEraseMEspamEraseMEpacific.net> wrote:
{Quote hidden}

>  --

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