Earth-centered Inertial Frame 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.

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′.

Trajectory Planning for Optical Satellite’s Continuous Surveillance of Geostationary Spacecraft

This paper focuses on the trajectory planning that optical satellites approach the geostationary (GEO) spacecraft and the trajectory planning that the optical satellite flies around the GEO spacecraft for continuous surveillance of the GEO spacecraft The optical satellite approaches the GEO spacecraft with continuous thrust and flies around the GEO spacecraft without thrust. The observation angles of the optical satellite to the GEO spacecraft are less than 30 degrees throughout the entire fly-around mission. Two methods are used in fly-around trajectory planning: trajectory planning based on classical orbital elements and trajectory planning based on CW equations. The advantage of trajectory planning based on classical orbital elements is that the position of the optical satellite relative to the GEO spacecraft is clear in earthcentered-inertial frame and the calculation method of fly-around orbital elements is simple. If the orbital inclination of the GEO spacecraft is less than 0.02 degrees and the semi-major axis of the elliptical relative trajectory of the fly-around formation is greater than 60 km, fly-around trajectory planning based on classical orbital elements is more efficient and easier than that based on CW equations. Optimal control theory is used in the trajectory planning for optical satellites approaching GEO spacecraft and the trajectory optimization is a two-point boundary-value problem. Simulations have proved that this method is feasible, which is highly effective in engineering. This method can be used for space situational awareness and on-orbit servicing missions of GEO spacecraft.

Capturing a spacecraft around a flyby asteroid using Hamiltonian-structure-preserving control

The potentially hazardous asteroid is not only the target of planetary defense missions but also an ideal one for scientific exploration of near-Earth objects due to minimal launching budget during its close encounter with Earth. The flyby segment of asteroid shows to be hyperbolic in an Earth-centred inertial frame. To guide the spacecraft captured by the asteroid under the common Earth's gravity, the Hamiltonian-structure-preserving (HSP) controller, based only on the feedback of relative positions, is constructed to transform the topological structure of the equilibrium point from hyperbolic to elliptic. On account of the small mass of flyby asteroid, a hyperbolic two-body model is used to derive the form of controller. The controller is also proved to work effectively in the hyperbolic three-body model. Critical control gains for both transient and long-term stabilities are investigated, as well as the controlled frequencies caused by different gains. To avoid collision with an asteroid, the artificial potential function (APF) is employed to improve the HSP controller.

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.

Design of Bounded Relative Trajectories in the Earth Zonal Problem

Numerical methods have proven to be reliable tools in generating long-term bounded relative trajectories in nontrivial dynamical environments. Yet, none of the existing procedures have a systematic approach to come up with all of the bounded relative trajectories associated with a variety of satellites in Earth orbit. In this paper, such a systematic procedure is developed based on the assumption that any trajectory, except at critical inclinations, is either periodic or quasi periodic in the four-dimensional Routh reduced system describing the motion of a mass particle about an axisymmetric body. This allows key design parameters to be identified, such as the nodal period and the node drift per nodal period, which are later used to constrain a differential corrector scheme looking for families of quasi-periodic invariant tori that yield bounded relative motion in the Earth-centered inertial frame. Numerical simulations show that the computed solutions are good candidates for cluster flight missions at a large range of altitude and inclination values.

Attitude Consensusability in Multispacecraft Systems Using Magnetic Actuators

The leaderless attitude consensusability of multispacecraft systems by using magnetic actuators is addressed. By proposing a nonlinear consensus protocol for the actuators magnetic dipoles, it is shown that the multispacecraft system is average controllable under the consensus protocol if it is maneuvering at orbits with persistently exciting magnetic fields in the Earth-centered inertial frame. Then, under some conditions, achieving consensus in the multispacecraft system is studied. Simulation results for a team of spacecraft validate the proposed consensus strategy.

Optimal tracking and formation keeping near a general Keplerian orbit under nonlinear perturbations

This study presents a semi-analytic approach for optimal tracking and formation keeping with high precision. For a continuous-thrust propulsion system, optimal formation keeping problems near a general Keplerian orbit are formulated with respect to a reference trajectory which is an explicit function of time. A nonlinear optimal tracking control law is then derived in generic form as a function of the states by employing generating functions in the theory of Hamiltonian systems. The applicability of the overall process is not affected by the complexity of dynamics and the selection of coordinates. As it allows us to design a nonlinear optimal feedback controller in the Earth-centered inertial frame, a variety of nonlinear perturbations can be incorporated easily without complicated coordinate transformations. Numerical experiments demonstrate that the nonlinear tracking control logic achieves superior tracking accuracy and cost reduction by accommodating higher-order nonlinearities.

Geometric Analysis and Design Method for Discontinuous Coverage Satellite Constellations

A new geometric approach for discontinuous coverage constellations in circular orbits is presented. Satellite constellations for zonal coverage with revisit times of more than one to two orbital periods are considered. The method is based on two-dimensional maps of visibility properties. The space dimensions are the right ascension of the ascending node (in the Earth-centered inertial frame) and the time. A new geometric pattern named a coverage belt is introduced. The coverage belt with sawtooth boundaries is formed by visibility intervals for a satellite at consecutive revolutions and the two-dimensional space is divided into two parts, with the revisit time near to and more than one orbital period, respectively. The revisit times for the satellite constellations are analytically computed as the sum of two terms: 1) the revisit time between straight-line envelopes of adjacent sawtooth coverage belts, and 2) the relative shift of the teeth in the belts. The last term is more than one and less than two orbital periods. Various types of interfaces between coverage belts are considered. Qualitative aspects of the solutions for well-known Walker-type constellations are discussed. Results are validated using extensive computer simulation and known solutions. Numerical examples of constellations with three to four satellites for discontinuous coverage of a specified latitude band are presented.

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.

GPS Satellite State Vector Determination in ECI Coordinate System using the Civil Navigation Message

A closed form of an algorithm to determine a Global Positioning System (GPS) satellite's position, velocity and acceleration is proposed, and an Earth Centred Earth Fixed (ECEF) to Earth Centred Inertial (ECI) transformation result using the Civil Navigation (CNAV) message is presented in this paper. To obtain the closed form of the GPS satellite velocity and acceleration determination algorithm using the CNAV, we analytically differentiated the IS-GPS-200F position determination function. The calculated data are transformed from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF) using an equinox-based transform algorithm that is defined in the IAU-2000 resolution system using the Earth Orientation Parameter (EOP) data. To verify the correctness of the proposed velocity and acceleration determination algorithm, the analytical results are compared to the numerical result. The equinox-based transformation result is compared to simple rotation about the z-axis, which does not use the EOP. The results show that by using the proposed algorithm and the equinox-based transformation together, the user can obtain more accurate navigation data in the ECI frame.

One-Way Speed of Light Relative to a Moving Observer

The one-way speed of light relative to a moving observer is determined using the range equation of the Global Positioning System. This equation has been rigorously tested and verified in the Earth-Centred Inertial frame where light signals propagate in straight lines at constant speed c. The result is a simple demonstration of light speed anisotropy that is consistent with light speed anisotropy detected in other experiments and inconsistent with the principle of light speed constancy. This light speed anisotropy was not observed before because there has been no direct one-way measurement of light speed relative to a moving observer.

Another Test of the Light Speed Invariance Postulate

In a paper in 1910 Tolman pointed out that the light speed invariance postulate of special relativity requires that the time for light to traverse a fixed distance between two points is independent of the movement of those points relative to the light source. The range equation of the GPS is used to directly test this proposition. This equation has been rigorously tested and verified in the Earth-Centred Inertial frame where light signals propagate in straight lines at constant speed c. The result is a simple demonstration of light speed anisotropy that is consistent with light speed variation detected in other experiments and inconsistent with the light speed invariance postulate.

One-Way Light Speed Determination Using the Range Measurement Equation of the GPS

The one-way speed of light is determined using the range measurement equation of the Global Positioning System. This equation has been rigorously and extensively tested and verified in the Earth-Centred Inertial frame, a frame that moves with the Earth as it revolves around the Sun but does not share its rotation. The result is a simple demonstration of one-way light speed anisotropy depending on the direction of propagation relative to the rotating Earth.

Relativistic effects in earth-orbiting Doppler lidar return signals

Frequency shifts of side-ranging lidar signals are calculated to high order in the small quantities (v/c), where v is the velocity of a spacecraft carrying a lidar laser or of an aerosol particle that scatters the radiation back into a detector (c is the speed of light). Frequency shift measurements determine horizontal components of ground velocity of the scattering particle, but measured fractional frequency shifts are large because of the large velocities of the spacecraft and of the rotating earth. Subtractions of large terms cause a loss of significant digits and magnify the effect of relativistic corrections in determination of wind velocity. Spacecraft acceleration is also considered. Calculations are performed in an earth-centered inertial frame, and appropriate transformations are applied giving the velocities of scatterers relative to the ground.