Positioning configurations with the lowest GDOP and their classification

Abstract

The positioning configuration optimization is a basic problem in surveying, and the geometric dilution of precision (GDOP) is a key index to handle this problem. Simplex graphs as regular polygons and regular polyhedrons are the well-known configurations with the lowest GDOP. However, it has been proved that there are at most five kinds of regular polyhedrons. We analytically solve the GDOP minimization problem with arbitrary observational freedom to extend the current knowledge. The configuration optimization framework established is composed of the algebraic and geometric operators (including combination, reflection, collinear mapping, projection and three kinds of equivalence relations), basic properties to GDOP minimization (including continuity, combination invariant, reflection invariant, rotation invariant and collinear invariant) and the lowest GDOP configurations (including cones, regular polygons, regular polyhedrons, Descartes configuration, helical configuration and generalized Walker configuration, and their reflections and combinations). GDOP minimization criterion and D-maximization criterion both reduce to the same criterion matrices that the optimization becomes the problem for solving an underdetermined quadratic equation system. Making use of the concepts for solving underdetermined linear equation system, the concepts of base configuration (single classification) and general configuration (combined classification) are applied to the GDOP minimization to analytically solve the quadratic equation system. Firstly, the problems are divided into two subproblems by two kinds of GDOP to reveal the impact of the clock-offset on the configuration optimization, and it shows that the symmetry and uniformity play a key role in identifying the systematic errors. Then, the solution of the GDOP minimization is classified by the number of symmetry axes, that the base configurations with at least one symmetry axis and the general configurations without symmetry axis are categorized to be two large classifications. Complex configurations can be then generated by the combination and the reflection of those base configurations with simplex structure, and this indicates that completely solving the GDOP minimization needs to solve the simplex classifications primarily. Ultimately, constrained or unconstrained configuration optimization examples including GDOP distribution analysis, single-global satellite navigation system (GNSS) or multi-GNSS constellation design, configuration optimization of pseudolites and configuration design of buoys for underwater positioning are performed by employing the properties, lemmas, theorems and corollaries proposed.