Optimal reduction and conversion of range-difference measurements for positioning

For positioning an object with m references, there are m−1 linearly independent range differences and measuring them is essential. However, none of m(m−1) possible range differences should be considered redundant unless their measurements are free of noise and locations of the references are exactly known. From all available range-difference measurements, m range measurements are obtained for positioning based on the least squares principle. The problem formulation treats missing and weighted range-difference measurements simultaneously. The exact relationships among several formulations of least squares positioning are established. A numerical example illustrates the results.