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Chang G.,China University of Mining and Technology | Chang G.,Xian Research Institute of Surveying and Mapping
Advances in Space Research | Year: 2015

An improved two-position initial alignment model for strapdown inertial navigation system is proposed. In addition to the velocity, angular rates are incorporated as measurements. The measurement equations in full three channels are derived in both navigation and body frames and the latter of which is found to be preferred. The cross-correlation between the process and the measurement noises is analyzed and addressed in the Kalman filter. The incorporation of the angular rates, without introducing additional device or external signal, speeds up the convergence of estimating the attitudes, especially the heading. In the simulation study, different algorithms are tested with different initial errors, and the advantages of the proposed method compared to the conventional one are validated by the simulation results. © 2015 COSPAR.

Chang G.,Xian Research Institute of Surveying and Mapping
Acta Geodaetica et Geophysica | Year: 2016

Least-squares (LS) solutions to the 3D Helmert and symmetric Helmert transformations with rotational invariant covariance structure are studied in a unified framework. This is an extension of the 3D Helmert transformation with naïve identity covariance and the counterpart of the 2D symmetric Helmert transformation with rotational invariant covariance. It is found that the closed form LS solution still exists and that the rotation parameters are still the same between the Helmert and symmetric Helmert transformations. © 2015, Akadémiai Kiadó.

Xue S.,Chinese Academy of Surveying and Mapping | Xue S.,Changan University | Yang Y.,Xian Research Institute of Surveying and Mapping | Dang Y.,Chinese Academy of Surveying and Mapping
Journal of Geodesy | Year: 2014

The Newton method has been widely used for solving nonlinear least-squares problem. In geodetic adjustment, one would prefer to use the Gauss-Newton method because of the parallel with linear least-squares problem. However, it is proved in theory as well as in practice that the Gauss-Newton method has slow convergence rate and low success rate. In this paper, the over-determined pseudo-distance equations are solved by nonlinear methods. At first, the convergence of decent methods is discussed after introducing the conditional equation of nonlinear least squares. Then, a compacted form of the Hessian matrix from the second partial derivates of the pseudo-distance equations is given, and a closed-form of Newton method is presented using the compacted Hessian matrix to save the computation and storage required by Newton method. At last, some numerical examples to investigate the convergence and success rate of the proposed method are designed and performed. The performance of the closed-form of Newton method is compared with the Gauss-Newton method as well as the regularization method. The results show that the closed-form of Newton method has good performances even for dealing with ill-posed problems while a great amount of computation is saved. © 2014 Springer-Verlag Berlin Heidelberg.

Chang G.,Tianjin Institute of Hydrographic Surveying and Charting | Chang G.,Xian Research Institute of Surveying and Mapping
Journal of Geodesy | Year: 2015

In this note, the 3D similarity datum transformation problem with Gauss–Helmert model, also known as the 3D symmetric Helmert transformation, is studied. The closed-form least-squares solution, i.e., without iteration, to this problem is derived. It is found that the rotation parameters in this solution are the same to that for the transformation with Gauss–Markov model, while the scale and translation parameters differ from each other. © 2015, Springer-Verlag Berlin Heidelberg.

Xue S.,Changan University | Xue S.,Chinese Academy of Surveying and Mapping | Yang Y.,Xian Research Institute of Surveying and Mapping | Dang Y.,Chinese Academy of Surveying and Mapping | Chen W.,Hong Kong Polytechnic University
Journal of Geodesy | Year: 2014

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. © 2013 Springer-Verlag Berlin Heidelberg.

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