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(OP)
Hello,
I am trying to create a filter that can integrate inertial measurements with GPS measurements. My inertial sensors are a yaw axis gyro, an odometer, and possibly a 3axis accelerometer.
Could you please help me with some guidance? I have read papers, articles and books on the subject but I am still a little confused.
I consider the acceleration constant, so I have a state vector x=[E N v_E v_N a_E a_N], E and N are the positions in NED coordinate system, v_E and v_N are the speed on each axis, and a is the acceleration on each axis
The measurement vector z is [E_GPS N_GPS E_DR N_DR a_E a_N] where E_GPS and N_GPS are the coordinates obtained from the GPS receiver and E_DR and N_DR are the coordinates obtained from the inertial sensors through dead reckoning, and a is the acceleration obtained from the accelerometer.
F, the transformation matrix is: [1 0 dt 0 (dt^2)/2 0; 0 1 0 dt 0 (dt^2)/2; 0 0 1 0 dt 0; 0 0 0 1 0 dt; 0 0 0 0 0 1];
H, the measurement matrix is: [1 0 0 0 0 0; 0 1 0 0 0 0 ; 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1],
Does it make any sense so far? Should I chose a different state vector?
My problem is that I don't know, for this system how can I determine the process noise covariance and measurement noise covariance.
Can you give me some indications in that direction?
Thank you very much.

### RE: Inertial Navigation Integration FIlter

The noise is not a function of the solution method; it's a function of the process and the sensors.  Your Kalman filter does not predict nor alter the actual process noise; you need to know what it is ahead of time.

### RE: Inertial Navigation Integration FIlter

(OP)
So I should introduce an estimate in the equations the measurement and process noise? Like z=Hx+w (where w is the measurement noise) and x(+)=Fx(-)+v (where v is the process noise)?

### RE: Inertial Navigation Integration FIlter

Is this for school?

### RE: Inertial Navigation Integration FIlter

(OP)
It's not for school. It's a project i'm trying to do im my spare time. So I'm far from the topic? Can you give me some guidance please, so I can clear some of the confusion?
Im trying to integrate the output from an IMU(3 axis gyro, yaw axis accelerometer and odometer if needed) with GPS.
Where do I go wrong in the system described before?

### RE: Inertial Navigation Integration FIlter

The noise estimates are what you put in the covariance matrix.  I don't recall, offhand, how the process noises are added.

### RE: Inertial Navigation Integration FIlter

(OP)
The process noise is added to the state update equation:
x(+)=x(-)F+w (w process noise)
and the measurement noise is added to the measurement equation:
z=Hx+v (v measurement noise)
My question is how to determine the values of v and w?
You determine the values experimentally, analytically or you just assume some value by trial and error?

### RE: Inertial Navigation Integration FIlter

Either of the first two.  You usually need to "tune" the filter after fabrication to get the "right" values.

### RE: Inertial Navigation Integration FIlter

(OP)
Thx for the help. :)
For a Inertial system with a gyro and an odometer, should I introduce more states then [position velocity acceleration]?
Should I introduce in the state vector also the heading, gyro and odometer bias? and i get a state vector like [position velocity acc heading gyro_bias odo_bias] ?
Or in case I don't introduce the bias from the odometer and gyro, they will be part of the measurement noise, and they only downside is that I cannot correct the sensors?

### RE: Inertial Navigation Integration FIlter

Probably, it's a trade between computational throughput and accuracy of the model.

### RE: Inertial Navigation Integration FIlter

(OP)
Should also the heading be part of the state vector? How can I determine what is vital to be in the state vector and what can be skipped?
Thank you

### RE: Inertial Navigation Integration FIlter

Well, do you need to know the heading?  How does one navigate without heading?

### RE: Inertial Navigation Integration FIlter

Is this for school?

### RE: Inertial Navigation Integration FIlter

(OP)
No, it's not for school. I graduated 2 years ago from university.
Why is everybody asking me if it is for school? It shouldn't be?

### RE: Inertial Navigation Integration FIlter

Sorry, I forgot I already asked.  The fact that you don't know what to do with heading raised a red flag.

Student postings are not allowed on this site.  This site is supposed to be for engineering professionals asking work related questions.  Some leeway is allowed for cross-discipline work.

### RE: Inertial Navigation Integration FIlter

(OP)
I'm asking if the heading needs to be part of the state vector as I saw different approaches that did not use it.
And my experience with Kalman filters doesn't go too far. I have read about it but it's the first time I'm trying to implement sth like this.
So then a better state vector would be: [position speed acceleration heading gyro_bias odometer_byas]
or should i leave out the acceleration?

### RE: Inertial Navigation Integration FIlter

If you have no other attitude information, then why do you have heading and gyro_bias?  Seems like you picked a bigger problem that necessary for a first project.

A relatively complete set might be:
3 positions
3 velocities
3 accelerations
3 angle
3 angle rates
3 angle accelerations
same with biases

36 states is not unheard-of

### RE: Inertial Navigation Integration FIlter

#### Quote (IRstuff):

36 states is not unheard-of
And how...!

If you're adding in GPS, you'll want:
position
velocity
clock bias
clock drift
wheel speed scale factor (I'm assuming this is vehicle-based)
compass offset

Add all of that into what IRstuff listed and you have a pretty significant-sized matrix.  Luckily, a large percentage of the coefficients are zero (with the majority of non-zero coefficients along the diagonal), so the calculations are faster than one would expect from a typical 30-coefficient+ matrix.

Dan - Owner
http://www.Hi-TecDesigns.com

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