Real-Time Precise Orbit Determination for LEO between Kinematic and Reduced-Dynamic with Ambiguity Resolution

The real-time integer-ambiguity resolution of the carrier-phase observation is one of the most effective approaches to enhance the accuracy of real-time precise point positioning (PPP), kinematic precise orbit determination (KPOD), and reduced-dynamic precise orbit determination (RPOD) for low earth orbit (LEO) satellites. In this study, the integer phase clock (IPC) and wide-lane satellite bias (WSB) products from CNES (Centre National d’Etudes Spatiales) are used to fix ambiguity in real time. Meanwhile, the three models of real-time PPP, KPOD, and RPOD are applied to validate the contribution of ambiguity resolution. Experimental results show that (1) the average positioning accuracy of IGS stations for ambiguity-fixed solutions is improved from about 7.14 to 5.91 cm, with an improvement of around 17% compared to the real-time float PPP solutions, with enhancement in the east-west direction particularly significant, with an improvement of about 29%; (2) the average accuracy of the estimated LEO orbit with ambiguity-fixed solutions in the real-time KPOD and RPOD mode is improved by about 16% and 10%, respectively, with respect to the corresponding mode with the ambiguity-float solutions; (3) the performance of real-time LEO RPOD is better than that of the corresponding KPOD, regardless of fixed- or float-ambiguity solutions. Moreover, the average ambiguity-fixed ratio can reach more than 90% in real-time PPP, KPOD, and RPOD.

Integrity monitoring for precise orbit determination of LEO satellites

Due to an increasing requirement for high accuracy orbital information for low Earth orbit (LEO) satellites, precise orbit determination (POD) of LEO satellites is a topic of growing interest. To assure the safety and reliability of the applications requiring high accuracy LEO orbits in near-real-time, integrity monitoring (IM) is an essential operation of the POD process. In this contribution, the IM strategy for LEO POD in both the kinematic and reduced-dynamic modes is investigated. The overbounding parameters of the signal-in-space range error are investigated for the GPS products provided by the International GNSS Service’s Real-Time Service and the Multi-GNSS Advanced Demonstration of Orbit and Clock Analysis service. Benefiting from the dynamic models used and the improved model strength, the test results based on the data of the LEO satellite GRACE FO-1 show that the average-case mean protection levels (PLs) can be reduced from about 3–4 m in the kinematic mode to about 1 m in the reduced-dynamic mode in the radial, along-track and cross-track directions. The overbounding mean values of the SISRE play the dominant role in the final PLs. In the reduced-dynamic mode and average-case projection, the IM availabilities reach above 99% in the radial, along-track and cross-track directions with the alert limit (AL) set to 2 m. The values are still above 98% with the AL set to 4 m, when the duty cycle of tracking is reduced to 40%, e.g., in the case of power limits for miniature satellites such as CubeSats.

Reduced-Dynamic Precise Orbit Determination of Haiyang-2B Altimetry Satellite Using a Refined Empirical Acceleration Model

The Haiyang 2B (HY-2B) satellite requires precise orbit determination (POD) products for geodetic remote sensing techniques. An improved set of reduced-dynamic (RD) orbit solutions was generated from the onboard Global Positioning System (GPS) measurements over a 14-month period using refined strategies and processing techniques. The key POD strategies include a refined empirical acceleration model, in-flight calibration of the GPS antenna, and the resolution of single-receiver carrier-phase ambiguities. In this study, the potential periodicity of empirical acceleration in the HY-2B POD was identified by spectral analysis. In the along-track direction, a noticeable signal with four cycles per revolution (CPR) was significant. A mixed spectrum was observed for the cross-track direction. To better understand the real in-flight environment, a refined empirical acceleration model was used to cope with the time variability of empirical accelerations in HY-2B POD. Three POD strategies were used for the reprocessing for superior orbit quality. Validation using over one year of satellite laser ranging (SLR) measurements demonstrated a 5.2% improvement in the orbit solution of the refined model. Reliable correction for the GPS antenna phase center was obtained from an over-420-day dataset, and a trend in radial offset change was observed. After application of the in-flight calibration of the GPS antenna, a 26% reduction in the RMS SLR residuals was achieved for the RD orbit solution, and the carrier phase residuals were clearly reduced. The integer ambiguity resolution of HY-2B led to strong geometric constraints for the estimated parameters, and a 15% improvement in the SLR residuals could be inferred compared with the float solution.

Sentinel-6A precise orbit determination using a combined GPS/Galileo receiver

The Sentinel-6 (or Jason-CS) altimetry mission provides a long-term extension of the Topex and Jason-1/2/3 missions for ocean surface topography monitoring. Analysis of altimeter data relies on highly-accurate knowledge of the orbital position and requires radial RMS orbit errors of less than 1.5 cm. For precise orbit determination (POD), the Sentinel-6A spacecraft is equipped with a dual-constellation GNSS receiver. We present the results of Sentinel-6A POD solutions for the first 6 months since launch and demonstrate a 1-cm consistency of ambiguity-fixed GPS-only and Galileo-only solutions with the dual-constellation product. A similar performance (1.3 cm 3D RMS) is achieved in the comparison of kinematic and reduced-dynamic orbits. While Galileo measurements exhibit 30–50% smaller RMS errors than those of GPS, the POD benefits most from the availability of an increased number of satellites in the combined dual-frequency solution. Considering obvious uncertainties in the pre-mission calibration of the GNSS receiver antenna, an independent inflight calibration of the phase centers for GPS and Galileo signal frequencies is required. As such, Galileo observations cannot provide independent scale information and the estimated orbital height is ultimately driven by the employed forces models and knowledge of the center-of-mass location within the spacecraft. Using satellite laser ranging (SLR) from selected high-performance stations, a better than 1 cm RMS consistency of SLR normal points with the GNSS-based orbits is obtained, which further improves to 6 mm RMS when adjusting site-specific corrections to station positions and ranging biases. For the radial orbit component, a bias of less than 1 mm is found from the SLR analysis relative to the mean height of 13 high-performance SLR stations. Overall, the reduced-dynamic orbit determination based on GPS and Galileo tracking is considered to readily meet the altimetry-related Sentinel-6 mission needs for RMS height errors of less than 1.5 cm.

A Second-Order Time-Difference Position Constrained Reduced-Dynamic Technique for the Precise Orbit Determination of LEOs Using GPS

In this paper, we propose a new reduced-dynamic (RD) method by introducing the second-order time-difference position (STP) as additional pseudo-observations (named the RD_STP method) for the precise orbit determination (POD) of low Earth orbiters (LEOs) from GPS observations. Theoretical and numerical analyses show that the accuracies of integrating the STPs of LEOs at 30 s intervals are better than 0.01 m when the forces (<10−5 ms−2) acting on the LEOs are ignored. Therefore, only using the Earth’s gravity model is good enough for the proposed RD_STP method. All unmodeled dynamic models (e.g., luni-solar gravitation, tide forces) are treated as the error sources of the STP pseudo-observation. In addition, there are no pseudo-stochastic orbit parameters to be estimated in the RD_STP method. Finally, we use the RD_STP method to process 15 days of GPS data from the GOCE mission. The results show that the accuracy of the RD_STP solution is more accurate and smoother than the kinematic solution in nearly polar and equatorial regions, and consistent with the RD solution. The 3D RMS of the differences between the RD_STP and RD solutions is 1.93 cm for 1 s sampling. This indicates that the proposed method has a performance comparable to the RD method, and could be an alternative for the POD of LEOs.

General Methodology for the Identification of Reduced Dynamic Models of Barge-Type Floating Wind Turbines

Floating offshore wind turbines (FOWT) are designed to overcome some of the limitations of offshore bottom-fixed ones. The development of computational models to simulate the behavior of the structure and the turbine is key to understanding the wind energy system and demonstrating its feasibility. In this work, a general methodology for the identification of reduced dynamic models of barge-type FOWTs is presented. The method is described together with an example of the development of a dynamic model of a 5 MW floating offshore wind turbine. The novelty of the proposed identification methodology lies in the iterative loop relationship between the identification and validation processes. Diversified data sets are used to select the best-fitting identified parameters by cross evaluation of every set among all validating conditions. The data set is generated for different initial FOWT operating conditions. Indeed, an optimal initial condition for platform pitch was found to be far enough from the system at rest to allow the dynamics to be well characterized but not so far that the unmodeled system nonlinearities were so large that they affected significantly the accuracy of the model. The model has been successfully applied to structural control research to reduce fatigue on a barge-type FOWT.

Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Finding the relative equilibria and analyzing their stabilities are of great significance to revealing the intrinsic properties of mechanical systems and developing effective controller. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a revolution surface. We note that the symmetry group of the bicycle is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints to the bicycle system. Therefore, we can classify the bicycle dynamics as a general Voronets system whose Lagrangian and constraint distribution are kept invariant under the action of the symmetry group. Applying the Voronets equations to the bicycle dynamics, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. This system takes time-reversal and lateral symmetries, and has two kinds of relative equilibria: the static equilibria and the dynamic equilibria. Further theoretical analysis shows that both kinds of relative equilibria form one-parameter solution families, and their Jacobian matrices take some specific properties. We then show that a static equilibrium cannot be stable unless all the eigenvalues of the Jacobian matrix are located at the imaginary axis of the complex plane. The stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold, which is established based on the conservation of energy of the system. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, we employ symbolic computation to carry out numerical simulations in conjunction with the benchmark parameters of a Whipple bicycle. How the revolution surface affects the relative equilibria and their stabilities is then investigated through our numerical simulations.