Earth-centered Earth-fixed Research Articles

Relativistic time transfer in the vicinity of the Earth and in the solar system

The algorithms for relativistic time transfer in the vicinity of the Earth and in the solar system are derived. The concepts of proper time and coordinate time are distinguished. The coordinate time elapsed during the transport of a clock and the propagation of an electromagnetic signal is analysed in three coordinate systems: an Earth-Centred Inertial (ECI) coordinate system, an Earth-Centred Earth-Fixed (ECEF) coordinate system and a barycentric coordinate system. The timescales of Geocentric Coordinate Time (TCG), Terrestrial Time (TT) and Barycentric Coordinate Time (TCB) are defined and their relationships are discussed. Some numerical examples are provided to illustrate the magnitudes of the effects.

Hybrid analytical resolution approach based on ambiguity function for attitude determination

When satellite navigation receivers are equipped with multiple antennas, they can deliver attitude information. In previous researches, carrier phase differencing measurement equations were built in the earth-centered, earth-fixed (ECEF) coordinate, and attitude angles could be obtained through the rotation matrix between the body frame (BF) and the local level frame (LLF). Different from the conventional methods, a hybrid algorithm is presented to resolve attitude parameters utilizing the single differencing (SD) carrier phase equations established in LLF. Assuming that the cycle integer ambiguity is known, the measurement equations have attitude analytical resolutions by using simultaneous single difference equations for two in-view satellites. In addition, the algorithm is capable of reducing the search integer space into countable 2D discrete points and the ambiguity function method (AFM) resolves the ambiguity function within the analytical solutions space. In the case of frequency division multiple access (FDMA) for the Russian Global Orbiting Navigation Satellite System (GLONASS), a receiver clock bias estimation is employed to evaluate its carrier phase. An evaluating variable and a weighted factor are introduced to assess the integer ambiguity initialization. By static and dynamic ground experiments, the results show that the proposed approach is effective, with enough accuracy and low computation. It can satisfy attitude determination in cases of GPS alone and combined with GLONASS.

Autonomous navigation of formation flying spacecrafts in deep space exploration and communication by hybrid navigation utilizing neural network filter

Autonomous navigation of spacecrafts is a difficult task, however, which is a must in future deep space exploration. With multiple spacecrafts flying in space, this aim can be achieved by formation flying spacecraft (FFS) utilizing inverse time difference of arrival (ITDOA) and inverse difference Doppler (IDD) methods, which can locate the position of earth-station from one-way uplink signals in the FFS coordinate, and by way of conversion of coordinates, the position of FFS is achieved in earth-centered earth-fixed (ECEF) coordinate. The ability of neural network (NN) filter in navigation to extract position of spacecrafts from random measuring noise of signal arrival time and Doppler shift is studied with different radius of FFS and surveying parameters. The NN filter used by spacecraft group is new way of unidirectional autonomous navigation and is of high precision of hybrid navigation.

Using high fix rate GPS data to determine the relationships between fix rate, prediction errors and patch selection

The global positioning system (GPS) is increasingly being used to monitor animal movement and to develop more sustainable land management practices. Reliable and accurate GPS data are required to develop robust animal resource selection functions (RSFs). This study used Earth Centred Earth Fixed (ECEF) high fix rate (4 Hz) GPS data collected from Fleck™ 2 GPS collars fitted to six cows over a four-day period. The GPS data were used to derive speed and mean speed histograms. Maximum mean speed values were used to model the maximum distance travelled for a range of sample intervals. Maximum distance travelled was used to estimate the minimum area that could be predicted. To extend the methods further, we explored the use of probability density functions to describe the speed profiles; a Gamma probability density function (PDF) was shown to provide the best description. By modelling animal speeds using both the raw data and the PDF, it was possible to estimate the minimum area occupied for varying GPS sample frequencies and patch areas. Overall, the cows spent the greatest proportion (77%) of their time either stationary or moving very slowly (less than 0.2 m s −1). Despite the slow speeds, the prediction errors were around 90% for sample frequencies of greater than half an hour and a patch area of 100 m 2. If study patch areas are between 10 and 100 m 2 the GPS fix rate would need to be less than 10 s. When GPS fix intervals are recorded at 1 h intervals, the probability of accurately predicting selection within a 10 000 m 2 patch is approximately 30%. Researchers and land managers that use GPS technology to monitor animal behaviour should be aware of the relationship between GPS fix rate and accuracy in predicting landscape use especially when attempting to develop resource selection functions. One approach to using low GPS fix rates would be to gather short duration, high fix rate GPS data to derive speed histograms for establishing probability density functions. These speed histograms could then be used in conjunction with the low fix rate data to establish the probability of an animal visiting a given patch within a landscape.

Computing satellite velocity using the broadcast ephemeris

GPS has been used for applications where satellite velocities are needed as well as satellite positions. Traditionally, the position was determined from noisy code pseudoranges while the velocity came from delta carrier phase measurements. This motivates the need to have a velocity model to accompany the velocity data. With velocity data, the velocity states are directly observed in a Kalman filter. While processing can be carried out using either post-processed or predicted ephemerides from the international GPS service (IGS), it is convenient to use broadcast ephemerides for real-time applications. In this paper, a set of equations is derived which will allow one to compute the velocity vector for a GPS satellite using time derivatives of the Keplerian elements and correction terms broadcast from the GPS satellites, as described in the GPS interface control document (ICD-GPS-200). It will be assumed that the reader is familiar with the equations listed in this document for computing the X, Y, Z earth-centered, earth-fixed (ECEF) coordinates of a satellite (the reader may obtain a PDF version of the ICDGPS-200 from the US Coast Guard website at http:// www.navcen,uscg.gov/pubs/gps/icd200/default.htm). A complete main program written in C is included on the GPS Toolbox website (http://www.ngs.noaa.gov/gps-toolbox) to illustrate how these equations can be easily incorporated into source code to yield accurate velocity components. As a check, the velocity calculations were done numerically as well. Derivation of the velocity equations

Algorithm of earth-centered earth-fixed coordinates to geodetic coordinates

The transformation between Earth-centered Earth-fixed (ECEF) coordinates and geodetic coordinates is required in many applications. This correspondence presents an iterative approach using the Newton-Raphson (NR) method that has good efficiency and accuracy and is free from singularity and nonconvergence except in a small region near the center of the Earth. Comparisons with other well-known approximate methods are made in complexity and performance.

3-D track initiation in clutter using 2-D radar measurements

We present an algorithm for initiating 3-D tracks using range and azimuth (bearing) measurements from a 2-D radar on a moving platform. The work is motivated by the need to track possibly low-flying targets, e.g., cruise missiles, using reports from an aircraft-based surveillance radar. Previous work on this problem considered simple linear motion in a flat Earth coordinate frame. Our research extends this to a more realistic scenario where the Earth's curvature is also considered. The target is assumed to be moving along a great circle at a constant altitude. After the necessary coordinate transformations, the measurements are nonlinear functions of the target state and the observability of target altitude is severely limited. The observability, quantified by the Cramer-Rao lower bound (CRLB), is very sensitive to the sensor-to-target geometry. The paper presents a Maximum Likelihood (ML) estimator for estimating the target motion parameters in the Earth-centered Earth-fixed (ECEF) coordinate frame from 2-D range and angle measurements. In order to handle the possibility of false measurements and missed detections, which was not considered previously, we use the Probabilistic Data Association (PDA) algorithm to weight the detections in a frame. The PDA-based modified global likelihood is optimized using a numerical search. The accuracies obtained by the resulting ML-PDA estimator are quantified using the CRLB for different sensor-target configurations. It is shown that the proposed estimator is efficient, that is, it meets the CRLB. Of particular interest is the achievable accuracy for estimating the target altitude, which is not observed directly by the 2-D radar, but can be only inferred from the range and bearing observations.

Transformation of Cartesian to Geodetic Coordinates without Iterations

A noniterative transformation of earth-centered, earth-fixed (ECEF) Cartesian coordinates to geodetic coordinates of a point is presented. The transformation is based on ellipsoidal coordinates. The ECEF Cartesian coordinates of a point are first transformed to the ellipsoidal coordinates by closed formulas. Second, the reduced latitude referred to the reference ellipsoid takes the approximation to the reduced latitude referred to a confocal ellipsoid. Third, the approximation can be improved by a correction for the purpose of higher accuracy. The accuracy of the transformation is analyzed in the paper and it is shown that the derived geodetic coordinates are sufficiently accurate for the most geodetic purposes.

Sensor alignment with Earth-centered Earth-fixed (ECEF) coordinate system

In this work, we formulate the multiple sensor alignment problem in an Earth-centered Earth-fixed (ECEF) coordinate system. The alignment algorithm maps the sensor measurements to the ECEF coordinates using a geodetic transformation, and attributes the discrepancies in the Earth-referenced system reported by each sensor to the sensor biases. Sensor biases are then estimated using the least squares (LS) technique. Simulated and real-life radar data are used to evaluate the performance of the proposed algorithm. Comparisons are made to those algorithms based on the standard stereographic projection.

Performance of sensor alignment with earth-referenced coordinate systems for multisensor data fusion

We consider the problem of sensor alignment for multisensor data fusion. A least squares (LS) sensor alignment algorithm is formulated using the earth-centered earth-fixed (ECEF) coordinate system. Perturbation analysis is carried for the ECEF-LS algorithm in the presence of sensor elevation measurement errors. A modified generalized least squares (MGLS) alignment approach is proposed that is applicable to 2-D sensor fusion systems and a computationally stable algorithm is developed to obtain the MGLS solution. Computer simulations are used to show the effectiveness of the proposed alignment algorithm.

Converting Earth-centered, Earth-fixed coordinates to geodetic coordinates

An improved algorithm to convert Earth-centered, Earth-fixed (ECEF) coordinates to geodetic coordinates is presented. It is faster than other methods and has accuracy equal to that of one of the most accurate existing algorithms, for heights greater than -3,000,000 m.

Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates

The transformations between Earth-centered Earth-fixed (ECEF) coordinates and geodetic coordinates are required in many applications, for example, in NAVSTAR/GPS navigation and geodesy. The transformation from ECEF coordinates to geodetic coordinates is usually carried out by approximation methods in practice, and the exact transformation methods are not used frequently. In this paper, several exact transformation formulas from ECEF coordinates to geodetic coordinates are reviewed and compared with the approximation methods in complexity and in sensitivity to computer round-off error. The relationship among some exact transformation solutions and the approaches are pointed out.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Equation for optimal power-limited spacecraft trajectories

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