Implementation of position determination function

Question

I'm attempting to implement position determination function of an aircraft to get "latitude/longitude azimuth" I attached 3 images for summerized formula as you may see this is 5-step trigonometric equation (Steps 0/4) which is my aim to program. image1; image2 image3

To find aircraft coordinates there are defined 9 input parameters (image1): Station U and S latitude,longitude,altitude; Aircraft altitude and 2 slant ranges. At the end of problem (image3) we will find 3 outputs: Aircraft latitude/longitude azimuth.

Answer 1

This code implements the solution explained for: How can I triangulate a position using two DMEs? on aviation.se. The code is in Python, which I happen to use instead of C, it's easy to convert into another language as required. I've broken down the calculation in smaller units to make code more legible and to ease your understanding.

The problem involves 3 points in 3D space: U and S are the DMEs, A is the aircraft.

As we just need the coordinates of U and S, to determinate A coordinates, I'm using coordinates of 3 well known DME stations. This will allow to check whether the result is correct. View based on the Low Altitude Enroute Chart:

When the program is run, the output is:

CAN: lat 49.17319, lon -0.4552778, alt 82
EVX: lat 49.03169, lon 1.220861, alt 152
P north: lat 49.386910325692874, lon 0.646650777948733, alt 296
P south: lat 48.78949175956114, lon 0.5265322105880027, alt 296

First are the coordinates of points U (CAN DME) and S (EVX DME) we entered, and then two lines for the two possible location of the aircraft.

I made another test with DME at longer distance (1241 km for ARE and 557.1 km for GLA) which worked pretty good too:

ARE: lat 48.33264, lon -3.602472, alt 50
GLA: lat 46.40861, lon 6.244222, alt 1000
P north: lat 48.082101174246304, lon 13.210754399535269, alt 10
P south: lat 41.958725412109445, lon 9.470999690780628, alt 10

The actual location of the aircraft is supposed to be SZA navaid, in south of France: Lat 41.937°, lon 9.399°.

---

from math import asin, sqrt, cos, sin, atan2, acos, pi, radians, degrees

# Earth radius in meters (https://rechneronline.de/earth-radius/)
E_RADIUS = 6367 * 1000  # at 45°N - Adjust as required


def step_0(r_e, h_u, h_s, h_a, d_ua, d_sa):
    # Return angular distance between each station U/S and aircraft
    # Triangle UCA and SCA: The three sides are known,

    a = (d_ua - h_a + h_u) * (d_ua + h_a - h_u)
    b = (r_e + h_u) * (r_e + h_a)
    theta_ua = 2 * asin(.5 * sqrt(a / b))

    a = (d_sa - h_a + h_s) * (d_sa + h_a - h_s)
    b = (r_e + h_s) * (r_e + h_a)
    theta_sa = 2 * asin(.5 * sqrt(a / b))

    # Return angular distances between stations and aircraft
    return theta_ua, theta_sa


def step_1(lat_u, lon_u, lat_s, lon_s):
    # Determine angular distance between the two stations
    # and the relative azimuth of one to the other.

    a = sin(.5 * (lat_s - lat_u))
    b = sin(.5 * (lon_s - lon_u))
    c = sqrt(a * a + cos(lat_s) * cos(lat_u) * b * b)
    theta_us = 2 * asin(c)

    a = lon_s - lon_u
    b = cos(lat_s) * sin(a)
    c = sin(lat_s) * cos(lat_u)
    d = cos(lat_s) * sin(lat_u) * cos(a)
    psi_su = atan2(b, c - d)

    return theta_us, psi_su


def step_2(theta_us, theta_ua, theta_sa):
    # Determine whether DME spheres intersect

    if (theta_ua + theta_sa) < theta_us:
        # Spheres are too remote to intersect
        return False

    if abs(theta_ua - theta_sa) > theta_us:
        # Spheres are concentric and don't intersect
        return False

    # Spheres intersect
    return True


def step_3(theta_us, theta_ua, theta_sa):
    # Determine one angle of the USA triangle

    a = cos(theta_sa) - cos(theta_us) * cos(theta_ua)
    b = sin(theta_us) * sin(theta_ua)
    beta_u = acos(a / b)

    return beta_u


def step_4(ac_south, lat_u, lon_u, beta_u, psi_su, theta_ua):
    # Determine aircraft coordinates

    psi_au = psi_su
    if ac_south:
        psi_au += beta_u
    else:
        psi_au -= beta_u

    # Determine aircraft latitude
    a = sin(lat_u) * cos(theta_ua)
    b = cos(lat_u) * sin(theta_ua) * cos(psi_au)
    lat_a = asin(a + b)

    # Determine aircraft longitude
    a = sin(psi_au) * sin(theta_ua)
    b = cos(lat_u) * cos(theta_ua)
    c = sin(lat_u) * sin(theta_ua) * cos(psi_au)
    lon_a = atan2(a, b - c) + lon_u

    return lat_a, lon_a


def main():
    # Program entry point
    # -------------------

    # For this test, I'm using three locations in France:
    # VOR Caen, VOR Evreux and VOR L'Aigle.
    # The angles and horizontal distances between them are known
    # by looking at the low-altitude enroute chart which I've posted
    # here: https://i.stack.imgur.com/m8Wmw.png
    # We know there coordinates and altitude by looking at the AIP France too.
    # For DMS <--> Decimal degrees, this tool is handy:
    # https://www.rapidtables.com/convert/number/degrees-minutes-seconds-to-degrees.html

    # Let's pretend the aircraft is at LGL
    # lat = 48.79061, lon = 0.5302778

    # Stations U and S are:
    u = {'label': 'CAN', 'lat': 49.17319, 'lon': -0.4552778, 'alt': 82}
    s = {'label': 'EVX', 'lat': 49.03169, 'lon': 1.220861, 'alt': 152}

    # We know the aircraft altitude
    a_alt = 296  # meters

    # We know the approximate slant ranges to stations U and S
    au_range = 45 * 1852  # 1 NM = 1,852 m
    as_range = 31 * 1852

    # Compute angles station - earth center - aircraft for U and S
    # Expected values UA: 0.0130890288 rad
    #                 SA: 0.0090168045 rad
    theta_ua, theta_sa = step_0(
        r_e=E_RADIUS,  # Earth
        h_u=u['alt'],  # Station U altitude
        h_s=s['alt'],  # Station S altitude
        h_a=a_alt, d_ua=au_range, d_sa=as_range  # aircraft data
    )

    # Compute angle between station, and their relative azimuth
    # We need to convert angles into radians
    theta_us, psi_su = step_1(
        lat_u=radians(u['lat']), lon_u=radians(u['lon']),  # Station U coordinates
        lat_s=radians(s['lat']), lon_s=radians(s['lon']))   # Station S coordinates

    # Check validity of ranges
    if not step_2(
            theta_us=theta_us,
            theta_ua=theta_ua,
            theta_sa=theta_sa):
        # Cannot compute, spheres don't intersect
        print('Cannot compute, ranges are not consistant')
        return 1

    # Solve one angle of the USA triangle
    beta_u = step_3(
        theta_us=theta_us,
        theta_ua=theta_ua,
        theta_sa=theta_sa)

    # Compute aircraft coordinates.
    # The first parameter is whether the aircraft is south of the line
    # between U and S. If you don't know, then you need to compute
    # both, once with ac_south = False, once with ac_south = True.
    # You will get the two possible positions, one must be eliminated.

    # North position
    lat_n, lon_n = step_4(
        ac_south=False,  # See comment above
        lat_u=radians(u['lat']), lon_u=radians(u['lon']),  # Station U
        beta_u=beta_u, psi_su=psi_su, theta_ua=theta_ua  # previously computed
    )
    pn = {'label': 'P north', 'lat': degrees(lat_n), 'lon': degrees(lon_n), 'alt': a_alt}

    # South position
    lat_s, lon_s = step_4(
        ac_south=True,
        lat_u=radians(u['lat']), lon_u=radians(u['lon']),
        beta_u=beta_u, psi_su=psi_su, theta_ua=theta_ua)
    ps = {'label': 'P south', 'lat': degrees(lat_s), 'lon': degrees(lon_s), 'alt': a_alt}

    # Print results
    fmt = '{}: lat {}, lon {}, alt {}'
    for p in u, s, pn, ps:
        print(fmt.format(p['label'], p['lat'], p['lon'], p['alt']))

    # The expected result is about:
    #   CAN: lat 49.17319, lon -0.4552778, alt 82
    #   EVX: lat 49.03169, lon 1.220861, alt 152
    #   P north: lat ??????, lon ??????, alt 296
    #   P south: lat 48.79061, lon 0.5302778, alt 296


if __name__ == '__main__':
    main()