Band 17, Heft 3

The integration of Inertial Navigation Systems and Global Navigation Satellite Systems (GNSS) represents the core navigation unit for mobile platforms in open sky environments. A realistic assessment of the accuracy of the navigation solution depends on the accurate modelling of inertial sensor errors. Sensor noise and biases contribute most to short-term navigation errors. For the latter, different models can be used, varying in complexity. This paper investigates how the use of two different models for the accelerometer bias affects the accuracy of the state estimate in an extended Kalman filter. For this purpose, the Allan variance technique is applied to a data sequence from a specific inertial sensor to identify and quantify the underlying noise processes. The estimated noise parameters are used to characterise a bias model for the accelerometers that in addition to the static bias model takes non-white noise processes of the inertial sensor under investigation into account. This detailed accelerometer bias model is compared to a classical modelling approach that only considers static biases. Both approaches are evaluated based on simulation studies for continuous and intermittent GNSS coverages. The results show no significant difference between the two modelling approaches in terms of horizontal position and attitude precision. Furthermore, the correctness of the accelerometer bias estimates is not significantly affected by the modelling approach. All in all, it can be concluded that a detailed bias model of the accelerometers does not outperform the classical modelling approach.

The errors-in-variables (EIV) model is used for data processing in the field of geodesy. However, the EIV model may be ill-posed. By analyzing the decreasing regularization (D-regularization) characteristic of solutions for EIV models, algorithms using traditional methods such as singular value decomposition or the Tikhonov function can directly determine the irrationality of a model. When an EIV model is ill-posed, solutions in which the observation errors in the coefficient matrix are simulated by variables make the ill-posed nature of the model more serious. This is because the simulated observation errors are subtracted from the coefficient matrix in subsequent computations, which reduces the singular value of the normal matrix. This point is verified using an example. To account for the D-regularization of solutions in EIV models, a modified algorithm is derived by classifying the models into two categories, and the regularization parameters are iteratively revised based on the mean squared error. Finally, some conclusions are drawn from two separate examples.

Many applications today require the precise determination of the position and orientation of a moving platform over time. However, especially in safety-critical areas, it is also important to derive quality characteristics of the trajectory estimation. This allows verification that sensors are operating within the precision and accuracy required for the application. In this paper, we propose a methodology for trajectory evaluation and address the challenges involved. Our approach is based on repeated measurements obtained using a closed loop rail track and allows the evaluation of the trajectory estimation in terms of precision and accuracy. Starting with the chronologically ordered raw data, the methodology first spatially sorts the measurements and then approximates them to a mean trajectory. The deviations between the single pose observations and the mean trajectory indicate the precision of the observed poses. With the addition of a higher-order reference, our methodology also determines the accuracy of the system under test. The applicability of our method is demonstrated by an exemplary evaluation of a low-cost inertial navigation system.

The availability of high-degree and recent global geopotential models is a crucial resource for different geodetic and geophysical applications such as modelling of geoid and quasi-geoid and establishing global reference and height systems, estimating Moho depth, gravity anomaly and tectonics, other geopotential functional, and oceanography, which emphasizes the importance for model evaluation. We have therefore estimated complete Bouguer anomalies and Moho whose results are evaluated with the terrestrial values. We validate the recently released satellite-only and high-degree combined GGMs over Ethiopia using terrestrial gravity data based on a statistical comparison of the Bouguer anomaly, its residual and Moho depth. The terrestrial-derived gravity anomaly is more accurately estimated by EIGEN6C4. The assessment against a recently conducted high resolution (∼3 km) terrestrial and airborne gravimetric survey over Ethiopia shows that EIGEN6C4/SGG_UGM_1 and 2 have the highest accuracy (∼3.28/3.27 mGal). However, the comparison with such data hardly discriminates the qualities of other GGMs that have or are truncated to the same degree and order. Whereas, the validation results of GGMs against terrestrial and airborne data are identical. EIGEN6C4, SGG_UGM_2, XGM2016, XGM2019e_2159/SGG_UGM_1 have the best quality, and the accuracy of associated Moho is 4.89/4.90 km, and this value changes to 4.98/4.91/5.51 km when the EGM08/ITSG_Grace2018s/GOCO06S are assessed.

The precision index is the primary basis for judging the GNSS positioning result, and the positioning mean error is usually used as the precision index in practical applications. In order to solve the problem of mismatch between positioning deviation and mean error in the priori situation, this paper proposed a positioning precision calculation method based on the posteriori unit weight variance and deduced the formulas combining sequential adjustment or Kalman filter, respectively. This method uses the characteristic that there are system errors in error corrections to calculate and screen the posterior unit weight variance. This method introduces the system error’s influence into the mean error, which can improve positioning precision. The application of static difference and RTK proved that this method has remarkable effects, which can significantly alleviate the problem of false high precision and improve the reliability of positioning mean errors.

The subject of this paper is the analysis of the stability of BeiDou system clocks; currently only signals from two blocks, BSD-2 and BDS-3, are available. For elaboration, 30 s clock corrections from the 2014 to 2020 period for 37 satellites were used (9 IGSO, 28 MEO). Four different Allan variances were used to determine stability, and additionally, the type of noise characteristic for each satellite was also determined. Based on the calculations, it was shown that the BDS-2 segment has a significantly lower stability than BDS-3. Moreover, it was possible to notice a difference in the course of the graphs of the same satellites using different variances. BDS-2 satellites were mostly characterised by the presence of WFM noise, while BDS-3 satellites were characterised by WFM noise for the shortest averaging times and RWFM for the other intervals. Accuracy varies between 10 −10  s to 10 −6  s for a rubidium clocks in general, in case of the hydrogen masers in is between 10 −14  s to 10 −10  s.

In this study, positioning quality is tested with the use of low-cost in-house developed receivers. The analyzes consider the practical use of low-cost devices in surveying works. In the network solution, the accuracy of the GNSS positioning based on low-cost receivers can be characterized by the repeatability of the baseline length of 1 and 6 mm in 24 h and 10 min observation sessions, respectively. The field experiment of 4 GNSS receivers and 3 GNSS low-cost receivers allowed for establishing a precise geodetic control network. The accuracy of the control point coordinates determined with low-cost GNSS receivers equals a maximum of 17 and 40 mm for the horizontal and height components, respectively. Therefore, low-cost GNSS receivers can provide positioning accuracy at the some centimeter level and can support land surveying and geodetic monitoring activities.

Displacements in typical monitoring applications occur in 3D but having sensors capable of measuring such 3D deformations with areal coverage is rare. One way could be to combine three or more line-of-sight measurements carried out from different locations at the same time and derive 3D displacement vectors. Automotive Multiple-Input-Multiple-Output Synthetic Aperture Radar (MIMO-SAR) systems are of interest for such monitoring applications as they can acquire line-of-sight displacement measurements with areal coverage and are associated with low cost and high flexibility. In this paper, we present a set of algorithms deriving 3D displacement vectors from line-of-sight displacement measurements while applying spatial and temporal least squares adjustments. We evaluated the algorithms on simulated data and tested them on experimentally acquired MIMO-SAR acquisitions. The results showed that especially spatial parametric and non-parametric least squares adjustments worked very well for typical displacements occurring in geomonitoring and structural monitoring (e.g. tilting, bending, oscillating, etc.). The simulations were confirmed by an experiment, where a corner cube was moved step-wise. The results show that acquisitions of off-the-shelf automotive-grade MIMO-SAR systems can be combined to derive 3D displacement vectors with high accuracy.

For the development of various geodetic tasks within a state, determining the Height Reference Surface by the geoid model is extremely important. Considering this, one of the main task of geodesy is to determine the geoid, which is defined as an equipotential surface of the Earth’s gravity field, as a result, it corresponds on average to the sea level. The aim of this study is to analyze the best-fitting geoid model for the territory of the Republic of Albania. In this study, DFHRS (Digital Finite Element Height Reference Surface) method was used (www.dfhbf.de), developed by Reiner Jäger [Jäger R. State of the art and present developments of a general approach for GPS-based height determination. East Africa: University of Applied Sciences, Faculty Geoinformationswesen, Department of Surveying and Geomatics. Paper Presented at the First Workshop on GPS and Mathematical Geodesy in Tanzania (Kilimanjaro Expedition 1999); 1999] to determine the most suitable geoid model for the territory of Albania. This approach allows the conversion of ellipsoidal heights determined by GNSS into the standard heights, which refer to the height reference surface (HRS) of an orthometric. The DFHRS is defined as continuous HRS in arbitrarily large areas by bivariate polynomials over an irregular grid [Jäger R, Schneid S. Online and postprocessed GPS heighting based on the concept of a digital height reference surface (DFHRS), in vertical reference systems. In: IAG Symposium. Cartagena, Colombia, Heidelberg: Springer; 2001, vol 124:203–8 pp]. The DFHRS approach uses a wide range of input data (Geometric and Physical) and in our case, there were 151 GPS/levelling height data as well as physical derivatives from different global geopotential models. The main focus of this study is placed on the calculation of the most suitable geoid model for the territory of Albania using global geopotential models (EGM96, EGM2008, EIGEN04, EIGEN6C4 and European Gravimetric Geoid Model 1997 (EGG97)). After analyzing the results and comparing the models among themselves, the Albanian DFHRS-EIGEN6C4 model was selected as the most suitable model for the territory of Albania.

Integrated GPS/INS using Kalman filter is the best technique for improving navigation accuracy. Assuming that the covariance matrices are known and constant, a conventional Kalman filter (CKF) is usually used, however, when they are unknown and time-varying, several adaptive estimation approaches have to be developed to estimate the statistical information of the measurement ( R ), process ( Q ), and state ( P ) covariance matrices. In many situations, blunders/faults in the measurement model and/or sudden changes in the dynamic model may occur during the navigation period. Therefore, the CKF, as well as the adaptive Kalman filter (AKF) will exhibit abnormal behavior and may lead the filter to be suboptimal or even diverge. In this study, the Sage-Husa adaptive Kalman filter (SHAKF) and innovation-based adaptive Kalman filter (IAKF) approaches are employed for adapting the measurement covariance matrix( R ). In the case of abrupt changes in the dynamic model, the state covariance matrix ( P ) is adapted using the strong tracking filter (STF). The performance of these adaptive approaches is evaluated before and after simulating a fault of different sizes in the measurement and dynamic models. The results show that with a large window width, the SHAKF outperforms the CKF and IAKF. However, when the system encounters any fault either in the measurement or dynamic model, the SHAKF loses its optimality and diverges. The sensitivity of the SHAKF to the fault is because the R matrix accumulates with the propagation of the recursive noise estimator. On the other hand, the IAKF and STF provide better performance than both the CKF and SHAKF because the gain matrix is adaptively adjusted to mitigate the influence of the fault, and therefore, they behave normally when a fault of any size occurs in the measurement and/or dynamic model.