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With the continuous development of Low Earth Orbit (LEO) satellite constellations, their application in the Global Navigation Satellite System (GNSS) has been increasingly emphasized. Minimizing the Geometric Dilution of Precision (GDOP), a key metric for evaluating positioning configurations, along with ensuring a sufficient number of positioning satellites, presents new opportunities and challenges for the selection of GNSS positioning satellites. While additional observation satellites can lower the GDOP value, an excessive number can increase the computational burden on receivers. The recent surge in the number of LEO satellites enhances the potential for GDOP minimization, bringing theoretical optimal spatial distributions within closer reach. In this paper, we address the challenge of GDOP minimization under observation-constrained conditions through a geometric analysis. We first develop a set of nonlinear algebraic equations to determine the conditions for minimized GDOP. These conditions are articulated by segregating satellites into high and low selection regions based on a predefined ratio, ensuring maximal satisfaction of various equation sets within each region. An intelligent optimization algorithm is then crafted to meet this geometric condition, with the concurrent fulfillment of all equation sets as its primary goal. This strategy successfully realizes a GDOP minimization algorithm that is resilient to occlusions. Finally, we assess the positioning performance under a range of occlusion scenarios, as well as with varying numbers and configurations of satellites. The findings offer valuable insights for the enhancement of future satellite positioning systems within expansive LEO satellite constellations.
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