Earth-centered Earth-fixed Coordinate System Research Articles

Dual-UAV Collaborative High-Precision Passive Localization Method Based on Optoelectronic Platform

Utilizing the optical characteristics of the target for detection and localization does not require actively emitting signals and has the advantage of strong concealment. Once the optoelectronic platform mounted on the unmanned aerial vehicle (UAV) detects the target, the vector pointing to the target in the camera coordinate system can estimate the angle of arrival (AOA) of the target relative to the UAV in the Earth-centered Earth-fixed (ECEF) coordinate system through a series of rotation transformations. By employing two UAVs and the corresponding AOA measurements, passive localization of an unknown target is possible. To achieve high-precision target localization, this paper investigates the following three aspects. Firstly, two error transfer models are established to estimate the noise distributions of the AOA and the UAV position in the ECEF coordinate system. Next, to reduce estimation errors, a weighted least squares (WLS) estimator is designed. Theoretical analysis proves that the mean squared error (MSE) of the target position estimation can reach the Cramér–Rao lower bound (CRLB) under the condition of small noise. Finally, we study the optimal placement problem of two coplanar UAVs relative to the target based on the D-optimality criterion and provide explicit conclusions. Simulation experiments validate the effectiveness of the localization method.

A broadcast ephemeris design of LEO navigation augmentation satellites based on the integration-type ephemeris model

Low Earth orbit (LEO) satellites are a promising type of navigation augmentation satellite for current global navigation satellite systems. Aiming at the navigation function, an effective broadcast ephemeris model needs to be designed for LEO satellites. An enhanced integration-type broadcast ephemeris model is proposed in this study. First, the short-term periodical variation characteristics of LEOs’ accelerations in the Earth-centered Earth-fixed coordinate system are analyzed. The Chebyshev polynomials and harmonic functions are then applied to represent the variation perturbation of accelerations. Tests using simulated and real data from LEO satellites at altitudes from 600 to 1400 km are conducted to evaluate the fit accuracy of the proposed models in terms of arc length, integration method, integration step length, orbital altitude, inclination, eccentricity, etc. The fit accuracy is dramatically improved compared to that of the current GLONASS integration-type broadcast model, where fit errors less than 10 cm are achieved with an arc length of 20 min.

An Application Research of Kalman Filter Based Algorithms in ECEF Coordinate System for Motion Models of Sensors

Kalman filter based algorithms such as unscented Kalman filter (UKF), and the unbiased conversion measurement Kalman filter (UCMKF) are the most popular nonlinear filters which used in tracking, navigation, estimation and information fusion. Applying those filters directly in East-North-Up (ENU) coordinates with motion models of sensors causes the degradation or even divergence of filter performance. To address this issue, we first analyzed and discussed the motion model consistency of moving sensor with a constant velocity (CV). Next, we proposed to extend the application of common filter algorithms to Earth Centered Earth Fixed (ECEF) coordinates to filter random errors. We verified the validity of our proposed method by filtering random errors in a constant velocity motion model of radar. The theoretical analysis and simulation results show that the extended algorithms provide better efficiency and compatibility in moving sensors.

A sequential converted measurement Kalman filter in the ECEF coordinate system for airborne Doppler radar

For an airborne Doppler radar, the carrying platform is moving and has time-varying attitudes. The filtering algorithms applicable to local coordinate system are not suitable anymore. In this paper, we propose a sequential converted measurement Kalman filter (SCMKF) with Doppler, which is based on the earth-centered earth-fixed (ECEF) coordinate system. First, to effectively utilize Doppler measurement with probable correlation with slant range, the correlated Doppler and range are decorrelated by the Cholesky factorization. The range component remains unchanged while the Doppler component is changed. Second, by a series of coordinate transformations with unchanged range component and other observations, the converted position measurement is obtained. Meanwhile, the corresponding error covariance is derived by using Taylor series expansion. With the resulting converted measurement and covariance, the target state is filtered by the converted measurement Kalman filter (CMKF), which is extended to the ECEF system. Finally, the filtered state is updated sequentially with the changed Doppler. Compared with the posterior Cramer–Rao lower bound as well as the CMKF, the proposed SCMKF is shown to be an efficient solution, and is superior to the CMKF. Although the gain brought from low-accuracy Doppler is limited, the introduction of high-accuracy Doppler can improve tracking performance significantly.

A Sequential Converted Measurement Kalman Filter with Doppler Measurements in ECEF Coordinate System

When there is the correlation between Doppler and slant range, previous literatures have presented some sequential filter algorithms. However, they are only applied to simplified fixed radar’ s local coordinate system. As a result, a Sequential converted measurement Kalman filter (SCMKF) with Doppler measurements based on the Earth centered earth fixed (ECEF) coordinate system applicable to a moving airborne platform which has time varying attitude is proposed. Firstly, the correlated Doppler and range are decorrelated using the Cholesky factorization, then the converted position measurements are obtained by a series of coordinate transformation with unchanged range component and other observations, such as azimuth and elevation angles; the corresponding error covariances are derived which are used to the Converted measurement Kalman filter (CMKF). Finally the sequential filter is implemented for changed pseudo-Doppler measurements. The proposed method is validated through Monte Carlo test compared with the performance of CMKF with just converted position measurements and traditional SCMKF with Doppler which ignores the correlation between Doppler and range noises, and the conclusion is obtained that utilizing Doppler information correctly can improve tracking performance, nevertheless, the improvement gain of filter accuracy is limited, which can provide some references for engineering application.

GEO objects spatial density and collision probability in the Earth-centered Earth-fixed (ECEF) coordinate system

The geostationary (GEO) ring is a valuable orbital region contaminated with an alarming number of space debris. Due to its particular orbital characters, the GEO objects spatial distribution is very susceptible to local longitude regions. Therefore the local longitude distribution of these objects in the Earth-centered Earth-fixed (ECEF) coordinate system is much more stable and useful in practical applications than it is in the J2000 inertial coordinate system. In previous studies of space debris environment models, the spatial density is calculated in the J2000 coordinate system, which makes it impossible to identify the spatial distribution in different local longitude regions. For GEO objects, this may bring potent inaccuracy. In order to describe the GEO objects spatial distribution in different local longitude regions, this paper introduced a new method which can provide the spatial density distribution in the ECEF coordinate system. Based on 2014/12/10 two line element (TLE) data provided by the US Space Surveillance Network, the spatial density of cataloged GEO objects are given in the ECEF coordinate system. Combined with the previous studies of “Cube” collision probability evaluation, the GEO region collision probability in the ECEF coordinate system is also given here. The examination reveals that GEO space debris distribution is not uniform by longitude; it is relatively centered about the geopotential wells. The method given in this paper is also suitable for smaller debris in the GEO region. Currently the longitudinal-dependent analysis is not represented in GEO debris models such as ORDEM or MASTER. Based our method the further version of space debris environment engineering model (SDEEM) developed by China will present a longitudinal independent GEO space debris environment description in the ECEF coordinate system.