scholarly journals Comparison between Two Methods to Calculate the Transition Matrix of Orbit Motion

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Ana Paula Marins Chiaradia ◽  
Hélio Koiti Kuga ◽  
Antonio Fernando Bertachini de Almeida Prado
Keyword(s):  
Orbit Determination ◽  
Transition Matrix ◽  
Numerical Approximation ◽  
State Transition ◽  
Artificial Satellite ◽  
Computational Cost ◽  
Satellite Orbit ◽  
Computational Costs ◽  
Numerical Integrator ◽  
Determination Procedure

Two methods to evaluate the state transition matrix are implemented and analyzed to verify the computational cost and the accuracy of both methods. This evaluation represents one of the highest computational costs on the artificial satellite orbit determination task. The first method is an approximation of the Keplerian motion, providing an analytical solution which is then calculated numerically by solving Kepler's equation. The second one is a local numerical approximation that includes the effect of . The analysis is performed comparing these two methods with a reference generated by a numerical integrator. For small intervals of time (1 to 10 s) and when one needs more accuracy, it is recommended to use the second method, since the CPU time does not excessively overload the computer during the orbit determination procedure. For larger intervals of time and when one expects more stability on the calculation, it is recommended to use the first method.

Satellite Orbit Determination Using Short Arcs of GPS Data

2014 ◽  
Vol 706 ◽  
pp. 206-221
Author(s):  
Ana Paula Marins Chiaradia ◽  
Hélio Koiti Kuga ◽  
Bruna Yukiko Pinheiro Lopes Masago
Keyword(s):  
Orbit Determination ◽  
Computational Cost ◽  
Satellite Orbit ◽  
Single Frequency ◽  
Gps Data ◽  
Force Model ◽  
Gps Receiver ◽  
Tracking Station ◽  
Low Computational Cost ◽  
Special Case

This work is concerned with short arcs orbit determination using GPS signals. A special case of truncated arcs assuming that GPS data is only available when the satellite carrying the GPS receiver passes over a ground tracking station is presented. The behaviour of an Extended Kalman filter (EKF) in real time satellite orbit determination using short arcs of data is analysed. The algorithm is a simplified and compact model with low computational cost, and uses the EKF to estimate the state vector, composed of position and velocity components, and GPS receiver clock parameters. The algorithm may use different step-sizes between the GPS signal measurements. Its force model in the motion equations considered the perturbations as being due to the geopotential up to the 10thorder and degree of the spherical harmonics. The algorithm has been formerly qualified using raw single frequency pseudorange GPS measurements of the Topex/Poseidon (T/P) satellite, and used as reference in this work. However, the GPS data are truncated as if they had been collected by a single ground tracking station. In other words, the data are obtained only when the satellite T/P is within the viewing area of the station. The research results are presented showing the degradation of performance when compared to a full arc orbit determination.

A New Algorithm for Onboard Autonomous Orbit Determination of Navigation Satellites

2011 ◽  
Vol 64 (S1) ◽  
pp. S162-S179 ◽  
Author(s):  
Haihong Wang ◽  
Zhonggui Chen ◽  
Jinjun Zheng ◽  
Haibin Chu
Keyword(s):  
Data Processing ◽  
Orbit Determination ◽  
Transition Matrix ◽  
State Transition ◽  
State Transition Matrix ◽  
Processing Unit ◽  
Data Processing Unit ◽  
Autonomous Orbit Determination ◽  
Computing Method

Autonomous orbit determination of a navigation constellation is the process by which the orbit parameters of navigation satellites are autonomously calibrated onboard the satellites without the need for external aids. It commonly uses a satellite onboard data processing unit and a filtering method to process the measurements of inter-satellite ranges. The onboard data processing unit is the main module of autonomous navigation systems. In this paper, the two main factors that affect the accuracy of autonomous orbit determination for a navigation constellation are discussed first, and then a distributed onboard algorithm for autonomous orbit determination of navigation satellites is proposed. This method is based on a long-term ephemeris prediction and is suitable for the satellite hardware capability. The main feature of this method is that both the distributed computing method and an onboard analytical state transition matrix are used to process inter-satellite range measurements. One of the main advantages of this approach is high-speed computing since the amount of calculations needed is significantly less than that of the centralised computing method and those distributed methods that need to use an onboard numerical integrator. Another advantage of this approach is that the use of the onboard analytical state transition matrix algorithm can save a great amount of resources for both ground-to-satellite data transmissions and data storage units in satellites’ hardware. This could result in substantial cost reduction for space missions. Finally, a simulation method used for testing the proposed algorithm is presented. Results of tests over a period of 90 days show that the user range error of autonomous orbit determination derived from the proposed method is less than three metres.

scholarly journals Approximate State Transition Matrix and Secular Orbit Model

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. P. Ramachandran
Keyword(s):  
Orbit Determination ◽  
Transition Matrix ◽  
Numerical Experiments ◽  
State Transition ◽  
Orbital Motion ◽  
The State ◽  
State Transition Matrix ◽  
Reference Orbit ◽  
Determination System ◽  
Orbit Model

The state transition matrix (STM) is a part of the onboard orbit determination system. It is used to control the satellite’s orbital motion to a predefined reference orbit. Firstly in this paper a simple orbit model that captures the secular behavior of the orbital motion in the presence of all perturbation forces is derived. Next, an approximate STM to match the secular effects in the orbit due to oblate earth effect and later in the presence of all perturbation forces is derived. Numerical experiments are provided for illustration.

scholarly journals Onboard and Real-Time Artificial Satellite Orbit Determination Using GPS

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ana Paula Marins Chiaradia ◽  
Hélio Koiti Kuga ◽  
Antonio Fernando Bertachini de Almeida Prado
Keyword(s):  
Real Time ◽  
Orbit Determination ◽  
Artificial Satellite ◽  
Time Estimation ◽  
Satellite Orbit ◽  
Estimation Algorithm ◽  
Measurement Model ◽  
Integration Scheme ◽  
Force Model ◽  
Step Size

An algorithm for real-time and onboard orbit determination applying the Extended Kalman Filter (EKF) method is developed. Aiming at a very simple and still fairly accurate orbit determination, an analysis is performed to ascertain an adequacy of modeling complexity versus accuracy. The minimum set of to-be-estimated states to reach the level of accuracy of tens of meters is found to have at least the position, velocity, and user clock offset components. The dynamical model is assessed through several tests, covering force model, numerical integration scheme and step size, and simplified variational equations. The measurement model includes only relevant effects to the order of meters. The EKF method is chosen to be the simplest real-time estimation algorithm with adequate tuning of its parameters. In the developed procedure, the obtained position and velocity errors along a day vary from 15 to 20 m and from 0.014 to 0.018 m/s, respectively, with standard deviation from 6 to 10 m and from 0.006 to 0.008 m/s, respectively, with the SA either on or off. The results, as well as analysis of the final adopted models used, are presented in this work.

Artificial satellite orbit computations

1966 ◽  
Vol 25 ◽  
pp. 363-371
Author(s):  
P. Sconzo
Keyword(s):  
Artificial Satellite ◽  
Satellite Orbit ◽  
Artificial Satellites ◽  
Orbital Elements ◽  
Differential Correction ◽  
Gravitational Effects ◽  
Short Intervals ◽  
The Subject ◽  
Introductory Discussion ◽  
Orbit Computation

In this paper an orbit computation program for artificial satellites is presented. This program is operational and it has already been used to compute the orbits of several satellites.After an introductory discussion on the subject of artificial satellite orbit computations, the features of this program are thoroughly explained. In order to achieve the representation of the orbital elements over short intervals of time a drag-free perturbation theory coupled with a differential correction procedure is used, while the long range behavior is obtained empirically. The empirical treatment of the non-gravitational effects upon the satellite motion seems to be very satisfactory. Numerical analysis procedures supporting this treatment and experience gained in using our program are also objects of discussion.

scholarly journals Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Atmosphere ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 444 ◽  
Author(s):  
Jinxi Li ◽  
Jie Zheng ◽  
Jiang Zhu ◽  
Fangxin Fang ◽  
Christopher. Pain ◽  
...  
Keyword(s):  
Unstructured Mesh ◽  
Computational Cost ◽  
Adaptive Mesh ◽  
Galerkin Finite Element Method ◽  
Adaptive Meshes ◽  
Galerkin Finite Element ◽  
Computational Costs ◽  
Cut Cell ◽  
Discontinuous Galerkin Finite Element ◽  
Terrain Following

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

scholarly journals Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Khalid ◽  
M. N. Naeem ◽  
P. Agarwal ◽  
A. Ghaffar ◽  
Z. Ullah ◽  
...  
Keyword(s):  
Numerical Approximation ◽  
Analytic Solution ◽  
Linear Equations ◽  
Computational Cost ◽  
Spline Method ◽  
System Of Linear Equations ◽  
Approximation Solution ◽  
B Spline ◽  
Novel Technique ◽  
Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

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