Earth Centered Inertial Research Articles

The Kinematic Models of the SINS and Its Errors on the SE(3) Group in the Earth-Centered Inertial Coordinate System.

In this paper, the kinematic models of the Strapdown Inertial Navigation System (SINS) and its errors on the SE(3) group in the Earth-Centered Inertial frame (ECI) are established. On the one hand, with the ECI frame being regarded as the reference, based on the joint representation of attitude and velocity on the SE(3) group, the dynamic of the local geographic coordinate system (n-frame) and the body coordinate system (b-frame) evolve on the differentiable manifold, respectively, and the high-order expansion of the Baker-Campbell-Haussdorff equation compensates for the non-commutative motion errors stimulated by strong maneuverability. On the other hand, the kinematics of the left- and right-invariant errors of the n-frame and the b-frame on the SE(3) group are separately derived, where the errors of the b-frame completely depend on inertial sensor errors, while the errors of the n-frame rely on position errors and velocity errors. In this way, the errors brought by the inconsistency of the reference coordinate system are tackled, and a novel attitude error definition is introduced to separate and decouple the factors affecting the dynamic of the n-frame errors and the b-frame errors for better attitude estimation. Through a turntable experiment and a car-mounted field experiment, the effectiveness of the proposed kinematic models in estimating attitude has been verified, with a remarkable improvement in yaw angle accuracy in the case of large initial misalignment angles, and the models developed have better robustness compared to the traditional SE(3) group-based model.

Formation Flight for Close Satellites With GPS-Based State Estimation Method

This article focuses on the specification of satellite relative state estimations with an Extended Kalman filter (EKF) using the target and tracker satellite positioning derived from the Newton Raphson Method (NRM). Although not true in real applications, it is assumed that the system dynamic model and measurements are correct in state estimations with EKF approximation. In this study, target and tracker satellite initial condition values have been taken from simulation data. Orbital states are Keplerian orbital parameters, and Global Positioning System (GPS) receiver values are the Earth-Centered Inertial (ECI) reference frame. The Pseudo-ranging approach has been used to find the positions of the target and tracker satellites. The random errors have also been considered for the actual distance between the target and tracker satellites. In this study, satellite relative state vectors have been estimated with an analyzed scenario which the target and tracker satellites localization have been estimated individually.

Testing light speed invariance by measuring the one-way light speed on Earth

We show that the one-way speed of light, along an open contour AB on the uniformly rotating Earth, is observable. Our approach exploits the property of the Sagnac effect that can measure the Earth's local angular velocity and, correspondingly, the peripheral speed of the contour AB relative to Earth Centered Inertial (ECI) frame. Light speed variations, measured on the rotating Earth, may be related to the velocity w of the ECI frame relative to a hypothetical preferred frame. Since it is possible to test Einstein's postulate of a universal light speed, standard special relativity is confirmed to be a viable falsifiable theory.

One-Step Initial Alignment Algorithm for SINS in the ECI Frame Based on the Inertial Attitude Measurement of the CNS.

To solve the problem of high-precision and fast initial alignment for the Strapdown Inertial Navigation System (SINS) under both dynamic and static conditions, the high-precision attitude measured by the celestial navigation system (CNS) is used as the reference information for the initial alignment. The alignment algorithm is derived in the Earth-centered inertial (ECI) frame. Compared with the alignment algorithm in the navigation frame, it is independent of position parameters and avoids the influence of the approximate error caused by the dynamic deflection angle. In addition, hull deformation is considered in attitude optimal estimation, which can realize initial the alignment of the SINS installed in various parts of the carrier. On this basis, the velocity measurement information is added to the alignment process, which further improves the accuracy and speed of the initial alignment under static conditions. The experimental results show that the algorithms proposed in this paper have better performance in alignment accuracy, speed, and stability. The attitude and velocity matching algorithm in the ECI frame can achieve alignment accuracy better than 0.6′. The attitude matching algorithm in the ECI frame has better robustness and can be used for both dynamic and static conditions, which can achieve alignment accuracy better than 1.3′.

GNSS Signal Processing with EKF and UKF for Stationary User Position Estimation

Precise and accurate estimation of state vectors is an important process during position determination. In this study, Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) of stationary user, state vectors defined in Earth Centered Inertial (ECI) coordinate system, accompanied by GNSS measurement data. It is aimed to make estimations with methods. EKF and UKF methods were compared with each other. In this study, the effects of nonlinear motion analysis and linearization methods on state vector estimations were investigated. Thanks to this study, estimations of the positioning information required during the specific tasks of many moving platforms have been made.

Earth Rotation-Aware Non-Stationary Satellite Communication Systems: Modeling and Analysis

In this paper, we propose a non-stationary satellite communication system model considering the impacts of Earth rotation by adopting the Earth-centered inertial (ECI), and the Earth-centered Earth-fixed (ECEF) coordinates. The position variations of a satellite (S) and a ground user (U) via coordinate transformations are demonstrated. Considering the variations of the distance between S and U, the instantaneous outage probability (OP) and channel capacity are calculated, as well as the system throughput within finite communication time. A simplified case is considered and analyzed while ignoring the Earth's rotation. Furthermore, the asymptotic expressions for the OP, capacity, and throughput are developed in the high signal-to-noise ratio (SNR) regime to obtain some insights. We also provide new definitions for throughput and OP within a short communication duration. The application and future research directions based on the derived results, including resource allocation, satellite handover, communication scenarios with multiple satellites and mobile users, are also discussed. Finally, some selected numerical results are provided to validate our proposed analysis models.

Detection of the Ether Using the Global Positioning System

In this paper the latest evidence for the existence of a cosmic ether obtained using modern technology is reviewed. The synchronized clocks of the GPS are applied in the search for ether drift by direct measurement of light travel times in the East-West direction. This method reveals that light travels faster West than East and therefore indicates the existence of an Earth-bound ether which we identify as the Earth-centered Inertial (ECI) frame for light transmission. The GPS clocks are then applied in the search for ether drift by direct measurement of light travel times in a modified Michelson-Morley experiment. The East-West light speed difference enables the unambiguous detection of ether drift and the direct confirmation of the existence of a preferred frame. The range equation of the GPS that operates in the Earth-centered inertial (ECI) frame is employed to demonstrate ether drift for rotational motion and Time Transfer technology involving a geo-stationary GPS satellite provides further confirmation of ether drift resulting from the rotating Earth. Finally, using a model applicable in the sun-centered inertial (SCI) frame with Coordinated Universal Time, light speed variation arising from the Earth's orbital motion for light reflected from planets and spacecraft and received at the surface of the Earth is demonstrated. The evidence then is that modern technology has detected ether drift for rotational and orbital motion from the frame of the moving Earth.

Collision Probability Prediction and Orbit Maneuvering Probability Determination of Non-Cooperative Space Object Orbit

Probability of collision between non-cooperative space object (NCSO) and the reference spacecraft (RS) has been increased drastically over the past few decades. The traditional method is difficult to identify the maneuvering of non-cooperative space object. In the present paper, not only positions and velocities, but also accelerations of non-cooperative space object are estimated as parameters by the extended Kalman filtering based on setting up the state linear equation and measurement model of the non-cooperative space object. The algorithm for predicting collision probability is derived from position error ellipsoid, and the algorithm for determining maneuvering probability is derived from maneuvering acceleration and its error ellipsoid, which can be employed to identify whether the upcoming space object is being maneuvered. An epoch Earth-centered inertial (EECI) coordinate system is suggested to replace Earth-centered inertial (ECI) to simplify coordinate transformation. Finally, a set of simulations were conducted to validate the proposed algorithms with the simulated measurement data of the reference spacecraft space-borne millimeter-wave (MMW) radar.

Formation Design for Single-Pass GEO InSAR Considering Earth Rotation Based on Coordinate Rotational Transformation

The single-pass geosynchronous synthetic aperture radar interferometry (GEO InSAR) adopts the formation of a slave satellite accompanying the master satellite, which can reduce the temporal decorrelation caused by atmospheric disturbance and observation time gap between repeated tracks. Current formation design methods for spaceborne SAR are based on the Relative Motion Equation (RME) in the Earth-Centered-Inertial (ECI) coordinate system (referred to as ECI-RME). Since the Earth rotation is not taken into account, the methods will lead to a significant error for the baseline calculation while applied to formation design for GEO InSAR. In this paper, a formation design method for single-pass GEO InSAR based on Coordinate Rotational Transformation (CRT) is proposed. Through CRT, the RME in Earth-Centered-Earth-Fixed (ECEF) coordinate system (referred to as ECEF-RME) is derived. The ECEF-RME can be used to describe the accurate baseline of close-flying satellites for different orbital altitudes, but not limited to geosynchronous orbit. Aiming at the problem that ECEF-RME does not have a regular geometry as ECI-RME does, a numerical formation design method based on the minimum baseline error criterion is proposed. Then, an analytical formation design method is proposed for GEO InSAR, based on the Minimum Along-track Baseline Criterion (MABC) subject to a fixed root mean square of the perpendicular baseline. Simulation results verify the validity of the ECEF-RME and the analytical formation design method. The simulation results also show that the proposed method can help alleviate the atmospheric phase impacts and improve the retrieval accuracy of the digital elevation model (DEM) compared with the ECI-RME-based approach.

Spacecraft formation flying in the port-Hamiltonian framework

The problem of controlling the relative position and velocity in multi-spacecraft formation flying in the planetary orbits is an enabling technology for current and future research. This paper proposes a family of tracking controllers for different dynamics of Spacecraft Formation Flying (SFF) in the framework of port-Hamiltonian (pH) systems through application of timed Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). The leader–multi-follower architecture is used to address this problem. In this regard, first we model the spacecraft motion in the pH framework in the Earth Centered Inertial frame and then transform it to the Hill frame which is a special local coordinate system. By this technique, we may present a unified structure which encompasses linear/nonlinear dynamics, with/without perturbation. Then, using the timed IDA-PBC method and the contraction analysis, a new method for controlling a family of SFF dynamics is developed. The numerical simulations show the efficiency of the approach in two different cases of missions.

Relative hovering analysis about an elliptical perturbed orbit with consideration of dynamic air drag and oblate earth

Accurate model in close and far distance hovering operation is a crucial and important issue due to considerable perturbation effects in long time flights. In this paper a new close/far distance hovering model is proposed with consideration of non-simplified perturbed target in low earth orbit (LEO) that suitable for long-term accurate mission simulation. New equations are precisely extracted to express the dynamics of relative hovering for a perturbed elliptical reference orbit without any simplification, by employing all zonal harmonics perturbation of the Earth's oblateness and air drag force. These perturbation terms have a significant impact on operations in long-term hovering operation in low earth orbits. Thus, applying these terms in the equations can help to investigate the problem more precisely. To validate the presented model, another model has been built as an Earth-Centered-Inertial (ECI) based Relative Motion (ERM) model. After validating the presented model, several useful examples have been provided for analyzing the effect of several parameters such as the amount of fuel consumption, optimal positioning of the follower satellite, minimum and maximum required thrust. Furthermore, the location of the follower satellite will be determined using a hybrid Invasive Weed Optimization/Particle Swarm Optimization (IWO/PSO) algorithm, with the aim of finding minimum and maximum required thrust during the operation. High speed of convergence, not to be located in local minimums, and success in implementing of complex problems, are the advantages of the proposed algorithm.

ANALYSIS ON ATTITUDE POSITION OF EARTH CENTERED INERTIAL (ECI) BASED ON RAZAKSAT DATA

Attitude determination system (ADS) is a process to control the orientation of satellite to make sure that the orientation of satellite is relative to inertial reference frame such as Earth. Earth Centered Inertial (ECI) is one of reference frame for satellite that determines the attitude in three dimensional spacecraft. Since RazakSAT orbits on earth, ECI coordinate system will be used for satellite relative to earth rotation. This paper is about the analysis on attitude position of ECI and velocity at X, Y and Z axis based on RazakSAT data. Satellite Tools Kit (STK) is used to estimate the attitude and velocity based on Two Line Elements (TLE) of RazakSAT. The result is compared with RazakSAT measurement data to observe the accuracy of estimation by using STK.

Stellar Refraction–Based SINS/CNS Integrated Navigation System for Aerospace Vehicles

AbstractStellar refraction-based celestial navigation is an attractive method with high accuracy and low cost. Most existing research works only focus on orbit determination problems for orbital vehicles in outer space while its working domain is limited. In order to expand its applicable area and improve information utilization, a new fault-tolerant stellar refraction-based inertial/celestial integrated navigation system is designed in this work, which is supposed to provide accurate position, velocity, and attitude information for aerospace vehicles that either make a maneuvering flight in near space or move in a predetermined Earth orbit in outer space. First, a new nonlinear navigation system dynamic model is established by error-prorogation equations of the strapdown inertial navigation system (SINS) in the Earth-centered inertial (ECI) frame, in which additive quaternion with less model error is used for attitude computation. Secondly, celestial navigation subsystem (CNS) is used to indirectly sense...

Erroneous UTC Maintained by Timing Labs due to Mix-Up between Relativistic and Absolute Time

In Relativity the Newtonian notions of absolute motion, absolute time, and absolute reference frame have been replaced with relative motion, relativistic time, and inertial reference frames in motion. Relativity is essentially rooted in the assumed isotropy of light speed in earth centered inertial (ECI) reference frame. Under the current procedures of satellite based time transfer, due to the assumed isotropy of light speed in ECI frame, an e-synchronous or relativistic time gets distributed to the master clocks of Timing laboratories located all over the globe. That is, the master clocks in all Timing Labs get e-synchronized instead of achieving absolute synchronization. It is erroneous to compute UTC from the weighted average of relativistic time maintained by these master clocks. The absolute synchronization mismatch between two e-synchronized clocks is given by the relation (D.U)/c2, where D is the separation distance between the two clocks and U is the absolute velocity vector (unknown) of the earth. This absolute synchronization offset between the master clocks at two distant Timing Labs can be physically measured with an appropriate portable clock and such measurements can be used to determine the unknown absolute velocity U of earth. By incorporating the absolute velocity U in the time transfer software, we can account for the anisotropic speed of light in the ECI frame and thereby ensure the distribution of absolute time to different clocks all over the globe. In that case it should even be possible to achieve absolute synchronization in space clocks in deep space flights.

Modified iterated extended Kalman particle filter for single satellite passive tracking

Single satellite-to-satellite passive tracking techniques have great significance in space surveillance systems. A new passive modified iterated extended Kalman particle filter (MIEKPF) using bearings-only measurements in the Earth-Centered Inertial Coordinate System is proposed. The modified iterated extended Kalman filter (MIEKF), with a new maximum likelihood iteration termination criterion, is used to generate the proposal distribution of the MIEKPF. Moreover, a new measurement update equation of the MIEKF is derived by modifying the objective function of the Gauss--Newton iteration. The approximated second-order linearized state propagation equation, Jacobian matrix of state transfer, and measurement equations are derived in satellite 2-body movement. The tracking performances of the MIEKPF, iterated extended Kalman particle filter (IEKPF), extended Kalman particle filter (EKPF), and extended Kalman filter (EKF) are compared via Monte Carlo simulations through simulated data from STK8.1. The simulation results indicate that the proposed MIEKF is capable of passively tracking a low earth circular orbit satellite with a high earth orbit satellite using bearings-only measurements and has higher tracking precision than the traditional algorithms.

Passive tracking from the combined set of bearings and frequency measurements by single satellite

In this paper, a new passive modified iterated extended Kalman filter (MIEKF) using the combined set of bearings and frequency measurements in Earth Centered Inertial (ECI) coordinate is proposed. A new measurement update equation of MIEKF is derived by modifying the objective function of the Gauss-Newton iteration. A new gain equation and iteration termination criteria are acquired by applying the property of the maximum likelihood estimate. The approximated second order linearized state propagation equation, Jacobian matrix of state transfer and measurement equations are derived in satellite two-body movement. The tracking performances of MIEKF, iterated extended Kalman filter (IEKF) and extended Kalman filter (EKF) are compared via Monte Carlo simulations through simulated data from STK8.1. Simulation results indicate that the proposed MIEKF is possible to passively track low earth circular orbit satellite by a high earth orbit satellite, and has higher tracking precision than the IEKF and EKF.

Enhanced GPS Measurements Simulation for Space-oriented Navigation System Design

At the stage of preliminary scheme and algorithm design for spaceborne navigation systems, a precise and high-fidelity software global positioning system (GPS) simulator is a necessary and feasible testing facility in laboratory environments, with consideration of the tradeoffs where possible. This article presents a software GPS measurements simulator on the L1 C/A code and carrier signal for space-oriented navigation system design. The simulator, coded in MATLAB language, generates both C/A code pseudorange and carrier phase measurements. Mathematical models in the Earth centered inertial (ECI) frame are formulated to simulate the GPS constellation and to generate GPS measurements. A series of efficient measures are investigated and utilized to rationalize the enhanced simulator, in terms of ephemeris data selection, space ionospheric model and range rate calculation, etc. Such an enhanced simulator has been facilitating our current work for designing a space integrated GPS/inertial navigation system (INS) navigation system. Consequently, it will promote our future research on space-oriented navigation system.

On the Sagnac Effect in Wave Propagation

According to the classical propagation model, the Sagnac effect is due to the movement of receiver during wave propagation. Based on this, we derive the GPS (global positioning system) Sagnac pseudorange correction, the Sagnac effect in interferometry, and the Doppler frequency shift. Then, we examine these formulas with the propagation experiments associated with the Sagnac effect, particularly the reference frame of the receiver velocity and the influences of earth's rotational and orbital motions. Thereby, it is found that the propagation of electromagnetic wave in these experiments is in accord with the classical model with the propagation frame being stationary in an ECI (earth-centered inertial) frame.

An Elegant State Transition Matrix

The analytic Keplerian 6 × 6 Earth Centered Inertial state transition matrix developed by Goodyear, Battin, and other mathematicians is well known. The primary objectives of this paper are to present a relatively simple formulation of the Keplerian state transition matrix, and to show that the higher-order secular terms associated with the state transition matrices of Goodyear and Battin can be eliminated. Analytic and numerical results indicate that the concern about these unbounded secular terms for numerical integration is not warranted. Comparison of Keplerian, Vinti, and numerical state transition matrices in the Hill-Clohessy-Wiltshire coordinate system (local-vertical coordinate system) further supports this contention since the two secular elements of the Keplerian state transition matrix match closely with those of the Vinti and numerical state transition matrices.

Missing Relativity Terms in GPS

The current GPS relativity corrections were derived by a committee of experts chosen by the Air Force. The original derivation assumed that the GPS reference frame is inertial, with its origin at the center of the earth. It also assumed that the user obtains inertial GPS coordinate time from the satellites. A new approach in relativity is taken to determine the relativity effects for both GPS satellites and GPS receivers. To the first-order approximation, there are no new relativity effects missing in the GPS algorithm if one converts all observations to the earth-centered inertial (ECI) frame and the GPS inertial coordinate time. However, there are some uncompensated effects required to make the complete transformation from the noninertial frame of a GPS receiver to the inertial frame of GPS. These effects and their magnitudes are examined in this paper.