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Although global navigation satellite systems (GNSS) have been routinely applied to determine attitudes, there exists no literature on determining angular velocity and/or angular acceleration from GNSS. Motivated by the invention of computerized accelerometers of the correspondence author and following the success of accurately recovering translational velocity and acceleration waveforms from very high-rate GNSS precise positioning by Xu and his collaborators in 2021, we propose the concept of GNSS gyroscopes and reconstruct angular velocity and acceleration from very high-rate GNSS attitudes by applying regularization under the criterion of minimum mean squared errors. The major results from the experiments can be summarized in the following: (i) angular velocity and acceleration waveforms computed by applying the difference methods to high-rate GNSS attitudes are too noisy and can be physically not meaningful and numerically incorrect. The same can be said about inertial measurement unit (IMU) attitudes, if IMU gyros are not of very high accuracy; (ii) regularization is successfully applied to reconstruct the high-rate angular velocity and acceleration waveforms from 50 Hz GNSS attitudes and significantly outperforms the difference methods, validating the proposed concept of GNSS gyroscopes. By comparing the angular velocity and acceleration results by using the difference methods and regularization, we find that the peak values of angular velocity and acceleration by regularization are much smaller by a maximum factor of 1.57 in the angular velocity to a maximum factor of 8662.53 times in the angular acceleration in the case of high-rate GNSS, and by a maximum factor of 1.26 in the angular velocity to a maximum factor of 2819.85 times in the angular acceleration in the case of IMU, respectively; and (iii) the IMU attitudes apparently lead to better regularized angular velocity and acceleration waveforms than the high-rate GNSS attitudes, which can well be explained by the fact that the former is of better accuracy than the latter. As a result, to suppress the significant amplification of noise in GNSS attitudes, larger regularization parameters have to be chosen for the high-rate GNSS attitudes, resulting in smaller peak angular accelerations by a maximum factor of 37.55 percent in the angular velocity to a maximum factor of 6.20 times in the angular acceleration in comparison of the corresponding IMU results. Nevertheless, the regularized angular acceleration waveforms for both GNSS and IMU look more or less similar in pattern or waveform shape. [ABSTRACT FROM AUTHOR]

Location service demands precise positioning, which can be obtained by applying maximum likelihood or least squares (LS) directly to measurements such as ranges and/or pseudoranges or to the computed quantities of measurements such as squared ranges and their differences. Nonlinearity and the stochastic models of computed quantities of measurements are not yet taken into account in precise location estimation. We propose a bias-corrected weighted LS method for precise location service, which consists of two basic elements. One is to automatically correct both the bias due to model nonlinearity and the induced biases in squared ranges/pseudoranges, and the other is to sequentially estimate unknown location parameters by treating equality constraints as a condition adjustment. As a result, the location estimation is theoretically almost unbiased. The method is applied to ranges, squared ranges and the differences of squared ranges. We also work out bias-corrected precise location estimation from pseudoranges, squared pseudoranges and the differences of pseudoranges. A large scale of simulations is carried out with range- and pseudorange-based models, respectively. The simulation results have clearly indicated: (i) the proposed bias-corrected weighted LS method can indeed result in almost unbiased estimates of location and further improve precise location estimation, depending on model nonlinearity and the ratio of signal to noise; (ii) the bias-corrected weighted LS method with the Gauss—Newton algorithm is shown to perform best in terms of the smallest bias and mean squared errors (MSE), if directly applied to ranges with a well-conditioned configuration, since it avoids the extra nonlinearity of squared ranges/pseudoranges and the induced bias of these computed quantities; (iii) the bias-corrected weighted LS and the weighted LS methods are of almost the same best performance in terms of MSE roots for each type of measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges. Nevertheless, the measurement models with the differences of squared ranges perform less accurate in the vertical component than those with ranges and squared ranges in the example, due to a weaker geometrical constraint; and (iv) the ordinary LS method clearly results in the most biased estimates of location and is of the worst accuracy for all the six measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges, indicating the importance of using correct stochastic models of measurements for location estimation. [ABSTRACT FROM AUTHOR]