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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 orbit determination process consists of obtaining values of the parameters which completely specify the motion of an orbiting body, like artificial satellites, based on a set of observations of the body. It involves nonlinear dynamical and nonlinear measurement systems, which depends on the tracking system, and estimation technique (e.g., the Kalman filtering or least squares [

The function of the transition matrix is to relate the rectangular coordinate variations for the times

In Chiaradia [

In the present work, the spherical harmonic coefficients of degree and order up to 50 and the drag effect are included in the reference orbit provided by the RK78 numerical integrator (Runge-Kutta with Fehlberg coefficients of order 7-8). For this study, the atmospheric density is considered constant. For the simulations made here, circular and elliptical low (up to 300 km) orbits were used. To analyze the results, the reference orbit is compared with the two methods implemented in this work.

The differential equation for the Keplerian motion is expressed by

Goodyear [

According to Goodyear [

Given ^{−12} is achieved. The variation of the eccentric anomaly is calculated and reduced to the interval 0 to 2

First of all, it is necessary to calculate the secular component _{4}_{5}

Sometimes the inverse matrix is required, such as in backward filters. It is also easily accomplished as follows. The inverse matrix

Markley's method uses two states, one at the _{0}

Performing successive derivatives of (

First of all, the reference orbit is integrated using the numerical integrator Runge-Kutta of eighth order with automatic step size control (RK78). This integrator is implemented to integrate simultaneously the stated vectors considering the spherical harmonic coefficients up to 50th order and degree and the transition matrix considering the Earth's flattening and the atmospheric drag effects.

The transition matrix generated by the numerical integrator is used as the reference to compare the transition matrix generated by the two methods. To compare the accuracy of them, let us define the global relative error as [

The first method considers the pure Keplerian motion and the second one considers the Keplerian motion and the

Processing time for analytical calculation of the transition matrix.

Steps | CPU TIME(s) | ||

First method | Second method | ||

1 | 86,400 | 19.8 | 26.2 |

10 | 8,640 | 2.1 | 2.7 |

30 | 2,880 | 0.7 | 0.9 |

60 | 1,440 | 0.4 | 0.4 |

To analyze the accuracy of the methods, six comparisons are done, using two kinds of orbits: one circular and one elliptical. The transition matrix generated by each of the methods is compared with the reference generated by the RK78. The circular orbit is from the Topex/Poseidon satellite [

Keplerian parameters of the test orbits.

Parameter | Topex/Poseidon | Molniya |
---|---|---|

Semiaxis major | 7714423.46 m | 26563000.0 m |

Eccentricity | 0.75 | |

Inclination | 66°.039 | 63°.435 |

Longitude of node | 236°.72 | 0°.0 |

Argument of perigee | 102°.83 | 270°.0 |

Mean anomaly | 153°.54 | 0°.0 |

Besides, the RK4 (Runge-Kutta of fixed 4th order) numerical method for computing the transition matrix is also implemented and the transition matrix generated by the RK4 is also included in the comparison. The RK4 used the same dynamical model of the reference RK78. The results of those three comparisons for the circular orbit are shown in Table

Comparison for a circular orbit (Topex).

Global error | RK4 | ||

First method | Second method | ||

1 | |||

10 | |||

30 | |||

60 | |||

300 | |||

600 |

Comparison for an elliptical orbit (Molniya).

Global error | RK4 | ||

First method | Second method | ||

1 | |||

10 | |||

30 | |||

60 | |||

300 | |||

600 |

The goal of this research was to compare and choose the most suited method to calculate the transition matrix used to propagate the covariance matrix of the position and velocity of the state estimator, (e.g., Kalman filtering or least squares), within the procedure of the artificial satellite orbit determination. The methods were evaluated according to accuracy, processing time, and handling complexity of the equations for two kinds of orbits: circular and elliptical. The processing time of the second method is around 30% larger than the first one; however, this difference is not considered enough to harm the computer load in the orbit determination tasks. The second method is more accurate for short intervals, as

The authors wish to express their appreciation for the support provided by UNESP (Universidade Estadual Paulista “Júlio de Mesquita Filho”) of Brazil and INPE (Brazilian Institute for Space Research).