Convert ENU velocity to ECEF

[microgorilla]

Homework Statement

This is not actually a homework problem, but it should be a fairly simple problem (hopefully). My apologies if this is better suited for the engineering forum.
I have been given an aircraft heading and a set of velocities in the ENU coordinate system. I need to convert the velocities from ENU to ECEF and calculate the aircraft track. As far as I can tell I am doing the velocity conversions correctly, but I cannot get the aircraft heading and the aircraft track to agree within +/- 15 degrees.

Homework Equations

I am using the following direction cosine matrix, which I arrived at by starting with the DCM for converting from ECEF to NED, swapping the first and second rows, negating the third row and taking the transpose:

CN2E = | -sin(lon) -sin(lat)*cos(lon) cos(lat)*cos(lon) |
| cos(lon) -sin(lat)*sin(lon) cos(lat)*sin(lon) |
| 0 cos(lat) sin(lat) |

lat = 34.2547 %degrees
lon = -96.6522 %degrees
height = 3690 %meters
velocity_ENU = [76.8213; -42.7113; 1.5302] %meters
heading = 123.2568 %degrees

The Attempt at a Solution

Multiplying CN2E * vel_ENU yields:
vel_x = 73.3726
vel_y = -34.0346
vel_z = -34.4414

This gives an aircraft speed of approx 170 knots; i don't have our exact aircraft speed at the time the measurement was made, but that's in the ballpark.
Finally, taking atan2(vel_y/vel_x) and converting to degrees yields an aircraft track of -24.8847 degrees, which is not even close to the heading of 123.2568 degrees.
Can anyone see where I might have gone wrong?

 

[KataKoniK]

It looks like you have correctly converted the velocities from ENU to ECEF using the given direction cosine matrix. However, it seems that your issue lies in calculating the aircraft track. The formula you have used, atan2(vel_y/vel_x), will only give you the direction of the velocity vector, not the heading of the aircraft. To calculate the aircraft track, you will need to take into account the heading and the direction of the velocity vector.

One way to do this is to use the dot product between the velocity vector and the heading vector. This will give you the component of the velocity in the direction of the heading. Then, you can use the magnitude of the velocity vector and the component in the heading direction to calculate the aircraft track using the formula: track = acos(component/magnitude). This will give you the angle between the aircraft heading and the direction of the velocity vector.

I hope this helps and I wish you luck in solving your problem. If you continue to have issues, I would suggest seeking help from an engineer or a fellow scientist who may have experience with this type of conversion. Good luck!