The Functional Gradient Description Method of Space Coordinate Transformation

The 3-dimension coordinate datum transformation is very important in geomatics.Generally,there are small angle coordinate transformation model (i.e.,Bursa model,et al) and large angle coordinate transformation model (i.e.,Rodrigo matrix model,et al).However,these papers only focus on the fixed and static situation regardless of the small or large angle.This paper proposes the conception of coordinate transformation gradient field,which can realize the space coordinate transformation from small angle to arbitrary angle,and from static to dynamical.Based on the equivalent characteristics of the unit quaternion rotation matrix and the Rodrigo matrix,through studying the mathematical relationship between the spatial coordinate transformation and the functional gradient,we derive a functional gradient expression of the arbitrary transformation formula in space.The study shows that the essence of spatial coordinate transformation is a kind of potential field mathematically.This potential field conception can unify all the space coordinate transformations.It is the theoretical foundation for the further study of time continuous space coordinate transformation,and this study gives a new solution for the attitude determination of motion carriers.

A new inter‑system double‑difference RTK model applicable to both overlapping and non‑overlapping signal frequencies

Aiming at the problem that the traditional inter-system double-difference model is not suitable for non-overlapping signal frequencies,we propose a new inter-system double-difference model with single difference ambiguity estimation,which can be applied for both overlapping and non-overlapping signal frequencies.The single difference ambiguities of all satellites and Differential Inter-System Biases(DISB)are first estimated,and the intra-system double difference ambiguities,which have integer characteristics,are then fixed.After the ambiguities are successfully fixed,high-precision coordinates and DISB can be obtained with a constructed transformation matrix.The model effectively avoids the DISB parameter filtering discontinuity caused by the reference satellite transformation and the low precision of the reference satellite single difference ambiguity calculated with the code.A zero-baseline using multiple types of receivers is selected to verify the stability of the estimated DISB.Three baselines with different lengths are selected to assess the positioning performance of the model.The ionospheric-fixed and ionospheric-float models are used for short and medium-long baselines,respectively.The results show that the Differential Inter-System Code Biases(DISCB)and Differential Inter-System Phase Biases(DISPB)have good stability regardless of the receivers type and the signal frequency used and can be calibrated to enhance the strength of the positioning model.The positioning results with three baselines of different lengths show that the proposed inter-system double-difference model can improve the positioning accuracy by 6–22%compared with the intra-system double-difference model which selects the reference satellite independently for each system.The Time to First Fix(TTFF)of the two medium-long baselines is reduced by 30%and 29%,respectively.