Qianxin Wang

Representation of the rotation parameter estimation errors in the Helmert transformation model

In the Helmert transformation model, the rotation is more difficult to be treated in terms of representation, estimation, and error analysis. First, two classes of representations of the rotation, i.e. the redundant class including the direction cosine matrix and the unit quaternion, and the minimum class including the rotation vector, the Gibbs vector, the modified Rodrigues parameters, and the Euler angles, are reviewed. It is concluded that in general the redundant class should be preferred as they are transcendental-function-free, singularity-free, and discontinuity-free. Second, two classes of estimation errors, i.e. the additive and the multiplicative errors, are defined and compared in detail. While the multiplicative errors are more convenient, the relationship among different representations and the relationship with their additive counterparts are also explored from first principle. It can be seen as a review paper; however, the content concerning the relationship between the additive and the multiplicative errors is believed new.

M-estimator for the 3D symmetric Helmert coordinate transformation

The M-estimator for the 3D symmetric Helmert coordinate transformation problem is developed. Small-angle rotation assumption is abandoned. The direction cosine matrix or the quaternion is used to represent the rotation. The

Baseline configuration for GNSS attitude determination with an analytical least-squares solution

3 \times 1

An alternative framework for the iterated unscented Kalman filter

3×1 multiplicative error vector is defined to represent the rotation estimation error. An analytical solution can be employed to provide the initial approximate for iteration, if the outliers are not large. The iteration is carried out using the iterative reweighted least-squares scheme. In each iteration after the first one, the measurement equation is linearized using the available parameter estimates, the reweighting matrix is constructed using the residuals obtained in the previous iteration, and then the parameter estimates with their variance-covariance matrix are calculated. The influence functions of a single pseudo-measurement on the least-squares estimator and on the M-estimator are derived to theoretically show the robustness. In the solution process, the parameter is rescaled in order to improve the numerical stability. Monte Carlo experiments are conducted to check the developed method. Different cases to investigate whether the assumed stochastic model is correct are considered. The results with the simulated data slightly deviating from the true model are used to show the developed method’s statistical efficacy at the assumed stochastic model, its robustness against the deviations from the assumed stochastic model, and the validity of the estimated variance-covariance matrix no matter whether the assumed stochastic model is correct or not.

New-Generation BeiDou (BDS-3) Experimental Satellite Precise Orbit Determination with an Improved Cycle-Slip Detection and Repair Algorithm

We derived an error analysis of the Davenports q method solving a general Wahbas problem from first principle. The error analysis is direct because it does not use the null attitude simplification, and is general because it is applicable no matter whether the vector measurements are unit. It is shown that the conventional results presented in the prior literature are a special case of those proposed here.

A generalization of the analytical least-squares solution to the 3D symmetric Helmert coordinate transformation problem with an approximate error analysis

The GNSS attitude determination using carrier phase measurements with 4 antennas is studied on condition that the integer ambiguities have been resolved. The solution to the nonlinear least-squares is often obtained iteratively, however an analytical solution can exist for specific baseline configurations. The main aim of this work is to design this class of configurations. Both single and double difference measurements are treated which refer to the dedicated and non-dedicated receivers respectively. More realistic error models are employed in which the correlations between different measurements are given full consideration. The desired configurations are worked out. The configurations are rotation and scale equivariant and can be applied to both the dedicated and non-dedicated receivers. For these configurations, the analytical and optimal solution for the attitude is also given together with its error variance–covariance matrix.

Error analysis of the 3D similarity coordinate transformation

In order to incorporate the time smoothness of ionospheric delay to aid the cycle slip detection, an adaptive Kalman filter is developed based on variance component estimation. The correlations between measurements at neighboring epochs are fully considered in developing a filtering algorithm for colored measurement noise. Within this filtering framework, epoch-differenced ionospheric delays are predicted. Using this prediction, the potential cycle slips are repaired for triple-frequency signals of global navigation satellite systems. Cycle slips are repaired in a stepwise manner; i.e., for two extra wide lane combinations firstly and then for the third frequency. In the estimation for the third frequency, a stochastic model is followed in which the correlations between the ionospheric delay prediction errors and the errors in the epoch-differenced phase measurements are considered. The implementing details of the proposed method are tabulated. A real BeiDou Navigation Satellite System data set is used to check the performance of the proposed method. Most cycle slips, no matter trivial or nontrivial, can be estimated in float values with satisfactorily high accuracy and their integer values can hence be correctly obtained by simple rounding. To be more specific, all manually introduced nontrivial cycle slips are correctly repaired.

Analytical 3D rotation estimation using vector measurements with full variance-covariance matrix

The iterated version of a family of non-linear Kalman filters, named the unscented transform (UT) based unscented Kalman filters (UKF), are revisited. Two existing frameworks of the iterated UKF are analysed and some shortcomings of them are pointed out. A new framework is proposed based on the statistical linear regression (SLR) perspective of the UT and the framework of the iterated extended Kalman filter (IEKF). The virtue of the proposed framework is twofold: first, the observation equation is linearised strictly following the SLR perspective implying that the regression error is also considered; second, it strictly follows the framework of the IEKF implying that in each iteration, the linearised equation is used to correct the a priori estimate rather than the latest estimate. A simple but illustrative benchmark example is simulated to check the feasibility of the proposed framework, and the results demonstrate the efficacy of the proposed framework.