Closed form position of airborne targets in new design of geometric constrained problem by bearing measurements

Positioning involves a wide range of military and industrial applications. In a positioning based on directional measurement, the transmitter's position is the intersection of the hypothetical orientation lines that send out the sensors to the transmitter. In the absence of noise, the location of these lines will be unique. But in reality, existing noise causes uncertainty in measurements and positioning. So far, various approaches have been introduced for orientation measurement based positioning. Most of these methods are based on numerical algorithms and the traditional maximum likelihood in order to estimate the optimal target position. These methods are equivalent to minimizing the total measurement errors by assuming Gaussian noise. Although TML is accurate, but due to the use of a recursive numerical algorithm, when the noise measurements are large or the geometry of the problem is undesirable, it causes a divergence of algorithm. Because of the use of numerical algorithms, these methods do not have a mathematical closed form for the answer. In addition to these methods, limited efforts are made to solve the problem and extract the closed form answer based on the geometric properties of the problem, most notably the Stansfield method. This algorithm is based on the assumption that the Gaussian noise is small and that the transmitter's approximate distance from the sensors is known. In this paper, attention is paid to the geometric properties of the problem in order to extract the closed form answer. But the assumption of the small Gaussian noise and the approximate distance between the transmitter and the sensors are not required. In most papers, the response of the TML algorithm as a benchmark has been compared in order to determine the precision of the proposed method. In this paper, it is shown in several scenarios that the proposed algorithm has a lower RMSE error than the TML, in addition to providing a mathematical closed form answer.