iterative least squares two dimensional geolocation in python

Iterative Least Squares (ILS) is an optimization algorithm that is commonly used in geolocation to estimate the position of a target. In a two dimensional geolocation scenario, ILS can be used to estimate the latitude and longitude of a target based on the time of arrival (TOA) or time difference of arrival (TDOA) measurements from several sensors.

Here's an example implementation of ILS in Python:

main.py
import numpy as np

def ils_2d(sensor_locations, measurements, initial_guess):
    """
    :param sensor_locations: np.array shape=(n_sensors, 2)
    :param measurements: np.array shape=(n_sensors,)
    :param initial_guess: np.array shape=(2,)
    :returns estimated_position: np.array shape=(2,)
    """
    n_sensors = sensor_locations.shape[0]
    estimated_position = initial_guess

    while True:
        # construct the linear system to solve
        A = np.zeros((n_sensors, 2))
        B = np.zeros((n_sensors, 1))
        for i in range(n_sensors):
            delta_x = sensor_locations[i, 0] - estimated_position[0]
            delta_y = sensor_locations[i, 1] - estimated_position[1]
            r = np.sqrt(delta_x**2 + delta_y**2)
            A[i, 0] = delta_x / r
            A[i, 1] = delta_y / r
            B[i, 0] = r - measurements[i]

        # solve using least squares
        delta = np.linalg.lstsq(A, B, rcond=None)[0]
        estimated_position -= delta.flatten()

        # check for convergence
        if np.linalg.norm(delta) < 1e-6:
            break

    return estimated_position
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In this implementation, sensor_locations is a numpy array with shape (n_sensors, 2), where each row corresponds to the x-y coordinates of a sensor. measurements is a numpy array with shape (n_sensors,), where each element corresponds to a measurement taken at the corresponding sensor. initial_guess is a numpy array with shape (2,) representing an initial estimate of the target's position.

The function iteratively solves a linear system that approximates the distance between each sensor and the target using the current estimate of the target's position. The solution to the linear system is then used to update the estimate of the target's position.

The iteration terminates when the change in the estimated position falls below a predefined threshold.

Note that this implementation assumes that the speed of light is constant, and that the sensors provide accurate measurements. In practice, these assumptions may not hold and additional calibration may be required.