Convert covariance matrix in ecef to LLA

I would like to take a covariance matrix (sigma from a 2 dimension Kalman Filter estimate) and convert it from ECEF to LLA coordinate systems.

Answer 1

[the cyclist]

According to ChatGPT, this code will do it.

DISCLAIMER: I have no idea if this code is accurate. I don't know what ECEF or LLA coordinates are.

It provided the following commentary to the code:

"Below is an example MATLAB code that demonstrates how to convert a covariance matrix from ECEF to LLA coordinates using a simplified transformation for illustration purposes. Please note that this example assumes a flat Earth for simplicity, and you might need to use more accurate equations depending on your requirements.

Make sure to replace cov_xx, cov_xy, etc., x, y, z, and R with your actual values. Additionally, if your Kalman filter operates in a non-linear setting, you might need to linearize the transformation equations using methods like Taylor series expansion.

Keep in mind that this code is a basic example, and for more accurate results, especially when dealing with the Earth's curvature, you should consider using more sophisticated coordinate transformation equations."

Perhaps it will at least provide a code base for you to modify.

% Sample covariance matrix in ECEF
sigma_ecef = [cov_xx, cov_xy, cov_xz;
              cov_xy, cov_yy, cov_yz;
              cov_xz, cov_yz, cov_zz];

% Sample ECEF coordinates
x_ecef = [x; y; z];

% Transformation equations (simplified for illustration)
lat = asin(z / norm(x_ecef));
lon = atan2(y, x);
alt = norm(x_ecef) - R; % R is the Earth's radius

% Jacobian matrix
partial_lat_partial_x = z / norm(x_ecef);
partial_lat_partial_y = 0;
partial_lat_partial_z = x / norm(x_ecef);

partial_lon_partial_x = -y / (x^2 + y^2);
partial_lon_partial_y = x / (x^2 + y^2);
partial_lon_partial_z = 0;

partial_alt_partial_x = x / norm(x_ecef);
partial_alt_partial_y = y / norm(x_ecef);
partial_alt_partial_z = z / norm(x_ecef);

Jacobian = [partial_lat_partial_x, partial_lat_partial_y, partial_lat_partial_z;
            partial_lon_partial_x, partial_lon_partial_y, partial_lon_partial_z;
            partial_alt_partial_x, partial_alt_partial_y, partial_alt_partial_z];

% Transform covariance matrix
sigma_lla = Jacobian * sigma_ecef * Jacobian';

% Print the result
disp('Covariance Matrix in LLA:');
disp(sigma_lla);

[Paul]

ECEF - Earth Centered Earth Fixed

LLA - Latitude, Longitude, Altitude.

These coordinate systems kind of inherently assume that the earth is some type of spheroid, i.e., not flat as stated in the response.

Having said that, it looks like the equations might be correct assuming a perfectly spherical earth. I say "might" because I didn't really check and I always forget which side of the transformation gets the transposed Jacobian.

[Brittany Decker]

right, would be better if they folded in the model for the earth available within matlab...

[the cyclist]

A google search turns up some related MATLAB functions, in the Mapping Toolbox. For example, take a look at Comparison of 3D Coordinate Systems.