Absolute Orbit Research Articles

Model predictive orbit control of a Low Earth Orbit satellite using Gauss’s variational equations

In this paper, an autonomous orbit control of a satellite in Low Earth Orbit is investigated using model predictive control. The absolute orbit control problem is transformed to a relative orbit control problem in which the desired states of the reference orbit are the orbital elements of a virtual satellite which is not affected by undesirable perturbations. The relative motion is modeled by Gauss’s variational equations including J2 and drag perturbations which are the dominant perturbations in Low Earth Orbit. The advantage of using Gauss’s variational equations over the Cartesian formulations is that not only the linearization errors are much smaller, but also each orbital element can be controlled independently. Model predictive control finds the finite horizon optimal firing times of the satellite thrusters. The problem of orbit control has been cast as a linear programming which is a subset of convex optimization problems. As a result, model predictive control can maintain and control orbits of Low Earth Orbit satellites in optimal way, and this modern control technique can be an alternative for traditional analytical-based orbit control methods. Also, a comparison between model predictive control and linear quadratic regulator orbit control showed the superiority of MPC in fuel consumption.

Rotordynamic model for electromagnetic excitation caused by an eccentric and angular rotor core in an induction motor

The paper shows a rotordynamic model for electromagnetic excitation caused by an eccentric and angular rotor core in an induction motor. It is shown that an eccentric rotor core leads to an electromagnetic force and an angular rotor core to an electromagnetic moment, which both force the rotor to vibrate. For these two kinds of magnetic unbalance, a rotordynamic model was developed considering the influence of the oil film stiffness and damping of the sleeve bearings, the stiffness of the end-shields and bearing housings, the stiffness of the rotor, the electromagnetic stiffness—radial and angular electromagnetic stiffness—the mass moment of inertia and the gyroscopic effect of the rotor. With this model, the absolute orbits of the shaft centre, the shaft journals and the bearing housings can be calculated, as well as the relative orbits between the shaft journals and the bearing housings. Additionally, the bearing housing velocities can also be computed. In addition to the mathematical derivation of the model, also a numerical example is shown for clarification. The aim of the paper is, on the one hand, to show the mathematical coherences—based on an analytical model—between rotordynamics and the electromagnetics for an induction motor with an eccentric and angular rotor core and, on the other hand, to derive a calculation method for evaluating the vibration sensitivity regarding these two different kinds of magnetic unbalance.

Flight Results of Precise Autonomous Orbit Keeping Experiment on PRISMA Mission

The autonomous orbit keeping experiment on the PRISMA mission was executed successfully from 18 July to 16 August 2011 and has demonstrated the capability of autonomous precise absolute orbit control. Using GPS-based absolute navigation data, the onboard controller commanded thruster activations in the orbital frame to autonomously control the satellite’s longitude of the ascending node within a predefined window. The main performance requirement of the experiment was a control accuracy of 10 m () with a maneuver velocity increment and decrement available budget of . After a four day commissioning phase, the reference orbit was acquired. A 3.5 day controller tuning was then followed by the fine orbit control phase, which commenced on 30 July 2011 and continued until the end of the experiment. The control accuracy requirement was fulfilled. The mean value of the longitude of the ascending node’s deviation was with a standard deviation of 9.5 m during the fine control phase. The total spent during the entire experiment was , corresponding to 27% of the maneuvers budget allocated.

Mathematical Rotordynamic Model Regarding Excitation Due to Elliptical Shaft Journals in Electrical Motors Considering the Gyroscopic Effect

The paper presents a mathematical rotordynamic model regarding excitation due to elliptical shaft journals in sleeve bearings of electrical motors also considering the gyroscopic effect. For this kind of excitation, a mathematical rotordynamic model was developed considering the influence of the oil film stiffness and damping of the sleeve bearings, the stiffness of the end-shields and bearing housings, the stiffness of the rotor, the electromagnetic stiffness in the air gap of the electrical motor and the mass moment of inertia of the rotor and therefore also considering the gyroscopic effect. The solution of the linear differential equation system leads to the mathematical description of the absolute orbits of the shaft centre, the shaft journals and the bearing housings and to the relative orbits between the shaft journals and the bearing housings. Additionally, the bearing housing velocities can also be derived with this mathematical rotordynamic model.

Relative State Estimation and Observability Analysis for Formation Flying Satellites

TOAUTONOMOUSLY control a satellite formation, the relative position and velocity of the satellites needs to be estimated. For tight formations, all the required information needs to be obtained and processed onboard the satellites. In case absolute position measurements, such as provided by the Global Positioning System (GPS), cannot be obtained, estimation of the absolute or relative state requires the satellites to autonomously measure the intersatellite range and/or line-of-sight angles. Markley [1], Herklotz [2], Psiaki [3], Liu and Liu [4], and Yim et al. [5] have analyzed the problem of autonomous absolute orbit determination using intersatellite vector measurements whereas Markley [1], Woffinden and Geller [6–8], Chen and Xu [9], Doolittle et al. [10], Chavez and Lovell [11], Holt and Lightsey [12], Kang et al. [13], and Matko et al. [14] studied the problem of autonomous relative orbit determination using intersatellite measurements. Several authors furthermore studied the effect of intersatellite ranging to augment external measurements like those from GPS and ground stations for either absolute or relative orbit determination. Amongst these are Holt and Lightsey [15], Huxel [16], and Huxel and Bishop [17,18]. Although previous work has been quite extensive, the understanding of the results obtained in some of these works needs to be improved by performing a detailed study of the observability of the system as a function of the relative state and of sensor suite properties, such as sensor accuracy and sensor placement. In addition, as no reference has been found in the literature, a further objective of this research is to provide a derivation of the expected value and variance, in three dimensions, in the estimation of the relative position of two objects. To isolate the basic properties of interest, a simple setting is implemented. The rationale behind this is that more advanced and realistic effects add only minor contributions and effects to the findings while they add considerable complexity to the analysis of the results. Two satellites, a chief and a deputy, areflying in formation in a low-Earth orbit (LEO) and perform intersatellite range measurements using locally generated one-way radio frequency ranging signals. The chief uses either one or three receiver (Rx) antennas to acquire and track the ranging signal transmitted by the deputy and to determine the relative range(s). The range measurements are treated together with a linearized dynamic model of the satellites’ relative motion to estimate their relative orbit using an iterative batch leastsquares (LSQ) algorithm. An observability analysis is performed to gain a deeper insight in the obtained results.

Theoretical vibration analysis of soft mounted electrical machines regarding rotor eccentricity based on a multibody model

The paper presents a simplified multibody model of a soft mounted electrical machine for theoretical vibration analysis, considering different kinds of rotor eccentricity. The most important dynamic eccentricities in electrical machines—eccentricity of rotor mass, magnetic eccentricity, and bent rotor deflection—are analyzed. The aim of the paper is not to replace a detailed three-dimensional finite-element calculation by a simplified plane multibody model, but to show the mathematical correlation between rotor dynamics, electromagnetic influence, oil film characteristics of the sleeve bearings, the influence of a soft foundation, and excitations by different kinds of rotor eccentricity, based on a simplified analytical model. Beside the absolute orbits of the rotor, stator, shaft journals, and bearing housings, also the relative orbits between rotor and stator and between shaft journals and bearing housings are mathematically described. In addition to these theoretical considerations, a numerical example for a soft mounted 2-pole induction motor is also shown. Finally, the paper shows the possibility of optimizing the system—electrical machine and foundation—regarding vibrations, caused by rotor eccentricities.

Lateral vibrations of soft mounted converter-fed induction motors caused by dynamic air gap torques: a mathematical analysis of orbital movements

This paper shows that dynamic air gap torques in converter-fed induction motors may not only cause torsional vibrations in the drive train, but may also cause lateral vibrations of the motor itself—including bending vibrations of the rotor shaft—, if the motor is mounted on a soft foundation. Based on a simplified analytical model, the mathematical correlation between rotor-stator interaction, oil film characteristics of sleeve bearings, the influence of a soft foundation—coupling angular and lateral displacement of the stator—and excitation from dynamic air gap torques is shown. In addition to the absolute orbits of the rotor, stator, shaft journals and bearing housings, also the relative orbits between rotor and stator and between shaft journals and bearing housings are mathematically described. Further, the paper shows the influence of the oil film coefficients of the sleeve bearings on the elliptical shape of the orbits. Therefore, the paper presents a possibility to optimize the whole system—converter, motor and foundation—concerning lateral vibrations, caused by dynamic air gap torques.

High-Precision, Symplectic Numerical, Relative Orbit Propagation

This paper presents a numerical method to propagate relative orbits. It can handle an arbitrary number of zonal and tesseral terms in the geopotential. This method relies on defining a relative Hamiltonian, which describes both the absolute and the relative motion of two satellites. The solution is separated into an analytical Keplerian part and a symplectic numerical integration part. The algorithm is designed to conserve the constants of the motion, resulting in better long-term accuracy. We present results for a broad range of scenarios with large separations and show that submeter accuracy is possible over five days of propagation, with a geopotential model containing 36 terms in tesseral and zonal harmonics. These results are valid for eccentricities reaching 0.5. Furthermore, the relative propagation scheme is significantly faster than differencing two absolute orbit propagations.

Interferometric Baseline Performance Estimations for Multistatic Synthetic Aperture Radar Configurations Derived from GRACE GPS Observations

Recent studies have demonstrated the usefulness of global positioning system (GPS) receivers for relative positioning of formation-flying satellites using dual-frequency carrier-phase observations. The accurate determination of distances or baselines between satellites flying in formation can provide significant benefits to a wide area of geodetic studies. For spaceborne radar interferometry in particular, such measurements will improve the accuracy of interferometric products such as digital elevation models (DEM) or surface deformation maps. The aim of this study is to analyze the impact of relative position errors on the interferometric baseline performance of multistatic synthetic aperture radar (SAR) satellites flying in such a formation. Based on accuracy results obtained from differential GPS (DGPS) observations between the twin gravity recovery and climate experiment (GRACE) satellites, baseline uncertainties are derived for three interferometric scenarios of a dedicated SAR mission. For cross-track interferometry in a bistatic operational mode, a mean 2D baseline error (1σ) of 1.4 mm is derived, whereas baseline estimates necessary for a monostatic acquisition mode with a 50 km along-track separation reveal a 2D uncertainty of approximately 1.7 mm. Absolute orbit solutions based on reduced dynamic orbit determination techniques using GRACE GPS code and carrier-phase data allows a repeat-pass baseline estimation with an accuracy down to 4 cm (2D 1σ). To assess the accuracy with respect to quality requirements of high-resolution DEMs, topographic height errors are derived from the estimated baseline uncertainties. Taking the monostatic pursuit flight configuration as the worst case for baseline performance, the analysis reveals that the induced low-frequency modulation (height bias) fulfills the relative vertical accuracy requirement (σ<1 m linear point-to-point error) according to the digital terrain elevation data level 3 (DTED-3) specifications for most of the baseline constellations. The use of a GPS-based reduced dynamic orbit determination technique improves the baseline performance for repeat-pass interferometry. The problem of fulfilling the DTED-3 horizontal accuracy requirements is still an issue to be investigated. DGPS can be used as an operational navigation tool for high-precision baseline estimation if a geodetic-grade dual-frequency spaceborne GPS receiver is assumed to be the primary instrument onboard the SAR satellites. The possibility of using only single-frequency receivers, however, requires further research effort.

Absolute proper motions and Galactic orbits of M 5, M 12 and M 15 from Schmidt plates

Absolute proper motions and Galactic orbits of M 5, M 12 and M 15 from Schmidt plates

Precision orbit determination for TOPEX/Poseidon using TDRSS doppler tracking data

Precision orbit determination on the TOPEX/Poseidon (T/P) altimeter satellite is now being routinely achieved with sub-5cm radial and sub-15 cm total positioning accuracy using state-of-the-art modeling with precision tracking provided by a combination of: (a) global Satellite Laser Ranging (SLR) and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), or (b) the Global Positioning System (GPS) Constellation which provides pseudo-range and carrier phase observations. The geostationary Tracking and Data Relay Satellite System (TDRSS) satellites are providing the operational tracking and communication support for this mission. The TDRSS Doppler data are of high precision (0.3 mm/s nominal noise levels). Unlike other satellite missions supported operationally by TDRSS, T/P has high quality independent tracking which enables absolute orbit accuracy assessments. In addition, the T/P satellite provides extensive geometry for positioning a satellite at geostationary altitude, and thus the TDRSS-T/P data provides an excellent means for determining the TDRS orbits. Arc lengths of 7 and 10 days with varying degrees of T/P spacecraft attitude complexity are studied. Sub-meter T/P total positioning error is achieved when using the TDRSS range-rate data, with radial orbit errors of 10.6 cm and 15.5 cm RMS for the two arcs studied. Current limitations in the TDRSS precision orbit determination capability include mismodeling of numerous TDRSS satellite-specific dynamic and electronic effects, and in the inadequate treatment of the propagation delay and bending arising from the wet troposphere and ionosphere.

Relativistic effects in many-body systems of finite size, internal structure, and internal motions. II. The determination of the inertial and rest masses of binary stars

We complete the theoretical framework initiated in an earlier paper for the post-Newtonian dynamical description of realistic binary stars. We evaluate the absolute orbits of the members, the corresponding orbital elements and apsidal motions, and generalize the concept of the classical mass function. Moreover we propose star models expressing the inertial mass of a star in terms of its rest mass. So the rest mass, which for stars of equal inertial masses depends on the star's nature, can in principle be determined from the observational data concerning the inertial masses. An application to the binary pulsar PSR 1913+16 shows that the star models proposed are quite satisfactory for distinguishing the rest from the inertial masses with the present accuracy of the observational data.

On the Determination of the Radius Vector in the Absolute Orbit of the Planets

On the Determination of the Radius Vector in the Absolute Orbit of the Planets