Kalman Filter and INS

> Two questions:
>
> One: Why is acceleration a state? �I thought you were measuring it.

Acceleration should definitely be one of the states.  Simply because
you are measuring acceleration does not somehow disqualify it from
also being a state.

>
> Two: If you _are_ measuring acceleration, how do you know what direction
> it's coming from? �IOW with an accelerometer you can measure
> acceleration with respect to your vehicle frame of motion, but doesn't
> that rotate with respect to the local inertial frame?
>

As posed by the OP, the problem is one of estimating distance
travelled along the frame of reference defined by the heading of the
vehicle/robot/whatever.  Currently, the OP has not asked for an
estimation of current position in some inertial frame.

If it were, then your point is completely valid, and changes the
design of the state model etc., such that roll pitch and heading etc
would probably form part of the state vector.

In answer to the original question, which is:

"Here is what i have, a measurement of a distance from an external
sensor at
a lower sampling rate and acceleration measurements from accelerometer
(
IMU at a higher rate). How can i use a kalman filter to fuse these to
produce a good estimate of the position travelled."

Ans: Change your H matrix each time interval so that it matches to the
current measurement(s) being taken.  For example, if your current
measurement is a high-rate measurement of acceleration only, then your
H matrix is [0 0 1].  If your current measurement is both position and
acceleration, then your H matrix is [1 0 1] (correct dimensionality
omitted for purposes of emphasis).

Malachy Moses wrote:
>> Two questions:
>>
>> One: Why is acceleration a state?  I thought you were measuring it.
> 
> Acceleration should definitely be one of the states.  Simply because
> you are measuring acceleration does not somehow disqualify it from
> also being a state.

That would depend on how you're modeling the system, and how you're 
measuring acceleration.  If the higher-order inputs and dynamics are 
unknown then you probably want to treat acceleration as a measurement, 
not a state.

>> Two: If you _are_ measuring acceleration, how do you know what direction
>> it's coming from?  IOW with an accelerometer you can measure
>> acceleration with respect to your vehicle frame of motion, but doesn't
>> that rotate with respect to the local inertial frame?
>>
> 
> As posed by the OP, the problem is one of estimating distance
> travelled along the frame of reference defined by the heading of the
> vehicle/robot/whatever.  Currently, the OP has not asked for an
> estimation of current position in some inertial frame.

As posed by the OP, frames of reference are not mentioned.  In other 
posts on similar topics this same OP has shown confusion about the need 
for tracking heading along with position and velocity.

> If it were, then your point is completely valid, and changes the
> design of the state model etc., such that roll pitch and heading etc
> would probably form part of the state vector.
> 
> In answer to the original question, which is:
> 
> "Here is what i have, a measurement of a distance from an external
> sensor at
> a lower sampling rate and acceleration measurements from accelerometer
> (
> IMU at a higher rate). How can i use a kalman filter to fuse these to
> produce a good estimate of the position travelled."
> 
> Ans: Change your H matrix each time interval so that it matches to the
> current measurement(s) being taken.  For example, if your current
> measurement is a high-rate measurement of acceleration only, then your
> H matrix is [0 0 1].  If your current measurement is both position and
> acceleration, then your H matrix is [1 0 1] (correct dimensionality
> omitted for purposes of emphasis).
> 
Or find a good treatment of a mixed continuous/discrete Kalman filter, 
extend it to a multi-rate filter, and find how you can avoid a lot of 
computations when you're not getting position fixes.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

>As posed by the OP, frames of reference are not mentioned.  In other 
>posts on similar topics this same OP has shown confusion about the need 
>for tracking heading along with position and velocity.
>

Thanks for all the replies guys. As Tim has mentioned, i have had other
posts in the forum talking about using just accelerometer ( 3 axis) to
measure distance without any external sensor and that idea for valid
reasons was quashed to death ( such as the bias grows at 1/2*t^2 when
integrated etc). I am relatively new at the INS ( less than 6 weeks) and i
am involved in a project of developing a low cost INS to measure the
distance travelled, hence trying to get as much info as possible. 

IMU i am using spits out 3 axis acc, gyro readings. This coupled with
another external sensor i have and 3d compass, i am trying to develop a low
cost robust INS.

Ideally i would have liked to just use the MEMS unit without any external
sensors for ease of installation but as i mentioned Tim and others who have
more experience than i do in this have strongly suggested thats not
possible with low cost IMU.

Can you guys think of any non-contact speed sensors which i might be able
to replace my current speed sensor which has some tedious installation? 

On Wed, 15 Apr 2009 11:29:00 -0500, arvkr wrote:

>>As posed by the OP, frames of reference are not mentioned.  In other
>>posts on similar topics this same OP has shown confusion about the need
>>for tracking heading along with position and velocity.
>>
>>
> Thanks for all the replies guys. As Tim has mentioned, i have had other
> posts in the forum talking about using just accelerometer ( 3 axis) to
> measure distance without any external sensor and that idea for valid
> reasons was quashed to death ( such as the bias grows at 1/2*t^2 when
> integrated etc). I am relatively new at the INS ( less than 6 weeks) and
> i am involved in a project of developing a low cost INS to measure the
> distance travelled, hence trying to get as much info as possible.
> 
> IMU i am using spits out 3 axis acc, gyro readings. This coupled with
> another external sensor i have and 3d compass, i am trying to develop a
> low cost robust INS.
> 
> Ideally i would have liked to just use the MEMS unit without any
> external sensors for ease of installation but as i mentioned Tim and
> others who have more experience than i do in this have strongly
> suggested thats not possible with low cost IMU.
> 
> Can you guys think of any non-contact speed sensors which i might be
> able to replace my current speed sensor which has some tedious
> installation?

If you have a 6-axis IMU, a distance measurement from a reference point, 
and a vehicle that's moving enough, you should be able to settle to an 
trajectory that is accurate but for a cylindrically symmetric uncertainty 
centered around the reference point (i.e., you'll have the right shape 
with respect to the reference point but you won't know which way north 
is).

Add in a compass or distances to two references, and you should be there.

-- 
http://www.wescottdesign.com

 
>If you have a 6-axis IMU, a distance measurement from a reference point,

>and a vehicle that's moving enough, you should be able to settle to an 
>trajectory that is accurate but for a cylindrically symmetric uncertainty

>centered around the reference point (i.e., you'll have the right shape 
>with respect to the reference point but you won't know which way north 
>is).
>
>Add in a compass or distances to two references, and you should be
there.
>
>-- 
>http://www.wescottdesign.com
>

Thanks Tim. I have a slightly off topic question. Do you know of any good
doppler radar based speed sensor which consumes the lowest power and lowest
in size etc?

Tim Wescott wrote:


> If you have a 6-axis IMU, a distance measurement from a reference point, 
> and a vehicle that's moving enough, you should be able to settle to an 
> trajectory that is accurate but for a cylindrically symmetric uncertainty 
> centered around the reference point (i.e., you'll have the right shape 
> with respect to the reference point but you won't know which way north 
> is).
> 
> Add in a compass or distances to two references, and you should be there.

But would the INS be any useful if you have compass (or a steering wheel 
position sensor) and odometer? The navigation by the trivial dead 
reckoning is many orders more accurate then MEMS INS.

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com