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Positioning configurations with the lowest GDOP and their classification / Shuqiang Xue in Journal of geodesy, vol 89 n° 1 (January 2015)  

Public [article]  inJournal of geodesy > vol 89 n° 1 (January 2015) . - pp 49 - 71 Titre : Positioning configurations with the lowest GDOP and their classification Type de document : Article/Communication Auteurs : Shuqiang Xue, Auteur ; Yuanxi Yang, Auteur Année de publication : 2015 Article en page(s) : pp 49 - 71 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Applications de géodésie spatiale [Termes IGN] affaiblissement géométrique de la précision [Termes IGN] constellation GNSS [Termes IGN] équation linéaire [Termes IGN] erreur systématique [Termes IGN] optimisation (mathématiques) [Termes IGN] orbitographie [Termes IGN] polyèdre [Termes IGN] pseudolite Résumé : (auteur) 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. Numéro de notice : A2015-330 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1007/s00190-014-0760-6 Date de publication en ligne : 14/10/2014 En ligne : https://doi.org/10.1007/s00190-014-0760-6 Format de la ressource électronique : URL article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=76655 [article]

Dynamic positioning configuration and its first-order optimization / Shuqiang Xue in Journal of geodesy, vol 88 n° 2 (February 2014)  

Public [article]  inJournal of geodesy > vol 88 n° 2 (February 2014) . - pp 127 - 143 Titre : Dynamic positioning configuration and its first-order optimization Type de document : Article/Communication Auteurs : Shuqiang Xue, Auteur ; Yuanxi Yang, Auteur ; et al., Auteur Année de publication : 2014 Article en page(s) : pp 127 - 143 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Applications de géodésie spatiale [Termes IGN] espace de Hilbert [Termes IGN] optimisation (mathématiques) [Termes IGN] positionnement par GPS [Termes IGN] réseau de premier ordre [Termes IGN] réseau géodésique Résumé : (Auteur) Traditional geodetic network optimization deals with static and discrete control points. The modern space geodetic network is, on the other hand, composed of moving control points in space (satellites) and on the Earth (ground stations). The network configuration composed of these facilities is essentially dynamic and continuous. Moreover, besides the position parameter which needs to be estimated, other geophysical information or signals can also be extracted from the continuous observations. The dynamic (continuous) configuration of the space network determines whether a particular frequency of signals can be identified by this system. In this paper, we employ the functional analysis and graph theory to study the dynamic configuration of space geodetic networks, and mainly focus on the optimal estimation of the position and clock-offset parameters. The principle of the D-optimization is introduced in the Hilbert space after the concept of the traditional discrete configuration is generalized from the finite space to the infinite space. It shows that the D-optimization developed in the discrete optimization is still valid in the dynamic configuration optimization, and this is attributed to the natural generalization of least squares from the Euclidean space to the Hilbert space. Then, we introduce the principle of D-optimality invariance under the combination operation and rotation operation, and propose some D-optimal simplex dynamic configurations: (1) (Semi) circular configuration in 2-dimensional space; (2) the D-optimal cone configuration and D-optimal helical configuration which is close to the GPS constellation in 3-dimensional space. The initial design of GPS constellation can be approximately treated as a combination of 24 D-optimal helixes by properly adjusting the ascending node of different satellites to realize a so-called Walker constellation. In the case of estimating the receiver clock-offset parameter, we show that the circular configuration, the symmetrical cone configuration and helical curve configuration are still D-optimal. It shows that the given total observation time determines the optimal frequency (repeatability) of moving known points and vice versa, and one way to improve the repeatability is to increase the rotational speed. Under the Newton’s law of motion, the frequency of satellite motion determines the orbital altitude. Furthermore, we study three kinds of complex dynamic configurations, one of which is the combination of D-optimal cone configurations and a so-called Walker constellation composed of D-optimal helical configuration, the other is the nested cone configuration composed of n cones, and the last is the nested helical configuration composed of n orbital planes. It shows that an effective way to achieve high coverage is to employ the configuration composed of a certain number of moving known points instead of the simplex configuration (such as D-optimal helical configuration), and one can use the D-optimal simplex solutions or D-optimal complex configurations in any combination to achieve powerful configurations with flexile coverage and flexile repeatability. Alternately, how to optimally generate and assess the discrete configurations sampled from the continuous one is discussed. The proposed configuration optimization framework has taken the well-known regular polygons (such as equilateral triangle and quadrangular) in two-dimensional space and regular polyhedrons (regular tetrahedron, cube, regular octahedron, regular icosahedron, or regular dodecahedron) into account. It shows that the conclusions made by the proposed technique are more general and no longer limited by different sampling schemes. By the conditional equation of D-optimal nested helical configuration, the relevance issues of GNSS constellation optimization are solved and some examples are performed by GPS constellation to verify the validation of the newly proposed optimization technique. The proposed technique is potentially helpful in maintenance and quadratic optimization of single GNSS of which the orbital inclination and the orbital altitude change under the precession, as well as in optimally nesting GNSSs to perform global homogeneous coverage of the Earth. Numéro de notice : A2014-139 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1007/s00190-013-0683-7 Date de publication en ligne : 03/12/2013 En ligne : https://doi.org/10.1007/s00190-013-0683-7 Format de la ressource électronique : URL article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=33044 [article]

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