Frame of reference

[Qacer]

Hi all,

I'm trying to get a good grasp of reference frames. I have a data file with Euler angles (yaw, pitch, and roll). These angles are referenced to true North. I also have another data file with unique coordinates from different sites. These sites have a heading / bearing data that is also reference to true North. I'm trying to find a way to uniquely describe the perspective of each site as it sees the airplane at that instant of time.

For example, one site might see the front of the plane while another site might be seeing the back at the same time. I still have not found a way to do this. I guess I haven't quite found a way to represent that perspective into a "mathematical representation."

I figured I can convert the Euler angles to quaternions, and use the quaternions to relate a unique perspective seen from each site. I figured since the airplane and the sites are referenced to true North that I can somehow do this (?). I don't know how, yet.

My goal is to somehow sort this unique perspective and find out how many times it was encountered by each site. After all, if the plane is doing multiple route loops, then each site will eventually see a similar perspective from another site, but at a different instance of time.

Any tips?

Thanks!

 

[herm0016]

if both sets of data are referenced to true north and are moveing with the same veolicity then they are both in the same refrence frame. if one set referenced the other set that referenced true north then the first set would be in the frame of the second set which is ref. to true north.

so.. you are referencing airplane geometery from ground stations? the ground stations should be able to reference eachothers actual position to find the correct path / geometery for the aircraft.


let me know if im way off base here..

 

[Qacer]

Hi there,

Yes, the plane's heading and the ground station's heading / bearing are both reference to true North. The ground stations are not moving, so that takes away some of the complexities.

My problem is that I don't know how to describe what the ground stations are seeing "mathematically." I'll have to think even more.

Maybe I can reverse my thinking and look at it from the airplane's perspective. If assume that the back, sides, top, bottom, and front of the plane can "see" things, then maybe I can reference it to a flat ground and relate which immediate ground station the back, sides, top, bottom, and front "see" at certain orientations.

I'm so confused. :/

 

[CycloWizard]

Naively, I would approach this problem in the following manner:

1. Convert the Euler angles of the plane to the same coordinate system of the ground stations (or vice versa).
2. Decide how far the ground stations (or the plane) can 'see'.
3. Compare the coordinates of the plane with those of the ground stations for each point in time.
4. If the plane is within sight distance at a given time n, then check its position at points n-1 and n+1 to decide the plane's velocity using a central difference to approximate the derivative of displacement.
5. From the result of 4, you can have an idea of the plane's orientation at time step n and decide what each station could see based on the angle between the ground station and the plane's velocity vector.

There may be practical reasons why this wouldn't work, but off the top of my head it sounds good. 😛

 

[Armitage]

Ok, the Euler angles should represent the orientation of a cartesian reference frame fixed to the aircraft (centered at the CG) to a cartesian reference frame fixed to the earth. You say it's oriented to true north - but that's not enough to define the earth fixed system. Anyway, let's call the earth fixed frame ECR and call the aircraft reference frame A The transformation of a vector from A to ECR is given by the formula:

{Vecr} = [Rz(gamma)][Ry(beta)][Rx(alpha)]{Va}

Where gamma, beta and alpha are your euler angles - change the order of the rotations of course. It isn't standard, though zyx is common.

You say you have a file showing heading and bearing from each site also, right? From that you can define a vector from the aircraft to the sensor in the earth fixed cordinate system. Call it S

What I think you want to do is transform the S vector from the earth fixed frame to the aircraft fixed frame. This simply involves solving the above equation for Va instead of Vecr

Once S in the A cordinate frame you can tel where your sensor sees the plane from. For example, if the x component of Sa is positive, the plane is oriented toward the sensor. If it's negative, it's oriented nose away from the sensor. If z is positive, you see the top of the plane, etc., etc.

 

[Qacer]

I'm still trying to grasp this. What else do I need in order to define the earth fixed system? Do you have a link to an online resource or maybe a name of a book that I can check out? Thanks!

 

[Armitage]

The earth fixed reference frame is typically defined with Z through the rotational pole of the earth - ie. "true north", X through 0 longitude at the equator, and Y completing the right-hand system. One of the best books I've seen on coordinate systems and transformations is "Methods of Orbit Determination" by Escobal. Another good one is "Fundamentals of Astrodynamics and Applications" by Vallado. Not quire the same field and focus you're dealing with though.

But regardless - if all of your measurements are in the same coordinate frame, you probably don't need to get to particular about it. Try the method I outlined above with some test cases and see if you get the expected results.