This paper represents orbit propagation and determination of low Earth orbit (LEO) satellites. Satellite global positioning system (GPS) configured receiver provides position and velocity measures by navigating filter to get the coordinates of the orbit propagation (OP). The main contradictions in real-time orbit which is determined by the problem are orbit positioning accuracy and the amount of calculating two indicators. This paper is dedicated to solving the problem of tradeoffs. To plan to use a nonlinear filtering method for immediate orbit tasks requires more precise satellite orbit state parameters in a short time. Although the traditional extended Kalman filter (EKF) method is widely used, its linear approximation of the drawbacks in dealing with nonlinear problems was especially evident, without compromising Kalman filter (unscented Kalman Filter, UKF). As a new nonlinear estimation method, it is measured at the estimated measurements on more and more applications. This paper will be the first study on UKF microsatellites in LEO orbit in real time, trying to explore the real-time precision orbit determination techniques. Through the preliminary simulation results, they show that, based on orbit mission requirements and conditions using UKF, they can satisfy the positioning accuracy and compute two indicators.
The satellite orbit determination (OD) estimates discrete observation of the position and velocity of an orbiting object. The set of observations includes the measurements from the space based GPS receiver (GPSR) that is located on the object itself. Satellite orbit propagation (OP) estimates the future state of motion of an object, whose orbit has been determined from past observations. The satellite’s motion is described by a set of approximate equations of motion. The degree of approximation depends on the intended use of orbital information. Observations are subject to systematic and random uncertainties; therefore, orbit determination and propagation are probabilistic.
The satellite is influenced by a variety of external forces, including terrestrial gravity, atmospheric drag, multibody gravitation, solar radiation pressure, tides, and spacecraft thrusters. Selection of forces for modeling depends on the accuracy and precision required from the OD process and the amount of available data. The complex modeling of these forces results in a highly nonlinear set of dynamical equations. Many physical and computational uncertainties limit the accuracy and precision of the object state that may be determined. Similarly, the observational data are inherently nonlinear with respect to the state of motion of the object and some influences might not have been included in models of observation of the state of motion.
The remainder of this paper is organized as follows. Section
Three basic strategies are presently used to determine precise LEO orbits with GPS. They are the dynamic, kinematic or nondynamic, and hybrid or reduced-dynamic strategies.
The dynamic orbit determination approach [
The kinematic or geometric approach does not require the description of the dynamics except for possible interpolation between solution points for the user object, and the orbit solution is referenced to the phase center of the on-board GPS antenna instead of the space vehicle's center of mass. Yunck and Wu [
The previous two strategies each have counterbalancing disadvantages: various mismodelling errors in dynamic OD and GPS measurement noise in kinematic OD. A hybrid dynamic and kinematic OD strategy would down weight the errors caused by each strategy but still utilize the strengths of each. One such strategy has been devised and is referred to as reduced dynamic orbit determination. The reduced-dynamic approach [
The orbit propagation algorithm can be divided into two main tasks: orbit determination and orbit prediction (propagation). The general diagram of orbit propagation algorithm is described in Figure
Orbit propagation algorithm diagram.
The orbit determination algorithm is based on Unscented Kalman Filter (UKF) and estimates the object state vector
Orbit prediction algorithm calculates the future state of motion of a vehicle
The OP solution init time and state vector the time moment the set of observations
The following input data are obtained from the previous OP solutions calculation: the last OP solution the time the covariance matrix
The maximal time of continuous propagation
A dynamic model of the object motion essentially adds a priori knowledge from the equations of the orbital motion to the kinematic position knowledge as obtained from the raw GPS measurements. In our case, the dynamic model incorporates the complex Earth gravity field (EGM 96) truncated to order and degree 18. Furthermore, the Sun and Moon gravitation and atmospheric drag are accounted.
The differential dynamic equation of motion is given by
The user coordinates are in the rotating Earth-fixed frame (ECEF). Although this adds some complexity, especially due to the Coriolis and centrifugal acceleration in the dynamic model, no reference system transformations are required in the main program since input (initial position and velocity) and OP output are consistently referring to the Earth-fixed frame. In this way, reference system transformations may completely be encapsulated in the dynamic model. Moreover, some dynamic algorithms, which compute the acceleration due to the Earth’s gravity field and the atmospheric drag, may be formulated simpler in an Earth-fixed than in an inertial frame.
The integration is carried out by using the simple fourth order Runge-Kutta algorithm without any mechanism of step adjustment or error control. The fourth order Runge-Kutta is considered an adequate numerical integrator due to its simplicity, fair accuracy, and low computational burden. Numerical integration is performed in the rotating Earth-fixed frame (ECEF).
According to [
We can see that for altitudes less than 600 km solar radiation pressure is significantly smaller than atmospheric drag. Furthermore, atmospheric drag decreases with altitude and it becomes negligible for altitudes higher than 700 km (Table
SV altitude, km | 400 | 500 | 600 | 700 | 800 |
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Solar radiation to atmospheric drag ratio | 0.018 | 0.08 | 0.27 | 0.8 | 2.1 |
The gravitational potential function for the solid-body mass distribution of the Earth is generally expressed in terms of a spherical harmonic expansion, referred to as the geopotential in the Earth-fixed reference frame (ECEF). The gravitational potential function
In (
The series is theoretically valid for
The acceleration due to geopotential can be defined as
The gravitational perturbations of the Sun, Moon, and other planets can be modeled with sufficient accuracy using point mass approximations. In the geocentric inertial coordinate system, the
The values of the Sun and Moon position vectors
All equations of this item are given according to [
A near-Earth space vehicle of arbitrary shape moving with some velocity
The parameter
Different empirical dynamical atmospheric models can be used for computing the atmospheric density. These include the Jacchia 71 [
According to aforesaid in the current project local atmosphere density model [
Equations for density calculation are
Orbit prediction algorithm calculates the future state of motion of a vehicle whose orbit has been determined from past observations. Moreover, the covariance matrix is propagated.
To construct the future object trajectory, the orbit prediction algorithm uses the dynamic equation of motion given in Section
The object state derivatives vector is defined using dynamic motion model which is described in Section
The covariance matrix propagation is defined below.
The differential dynamic equations of motion are given by
The propagation of
The orbit determination algorithm applies an UKF to estimate the state vector which comprises 8 components: object velocity object position receiver clock bias receiver clock drift
The diagram of orbit determination algorithm is described as follows.
The following process and measurement models can be established:
The variables in the above equation will be described:
The measurement vector is denoted by
The system error covariance matrix
The measuring error covariance matrix
Here,
The parameter is a scaling parameter defined as
The positive constants
Condition predicted average is
Covariance matrix is
Observation measurement predicted average is
The flow chart of the UKF is as shown in Figure
The algorithm flow chart of UKF.
The time update phase of the UKF includes the propagation of state vector
The subsequent measurement update adjusts the state vector
The measurement residual and sensitivity matrix are found by forming the computed observation equation.
The model for a GPS pseudorange measurement is given by
Using the abovementioned nonlinear measurement equation, the sensitivity matrix is given by the Jacobian matrix of partial derivatives of nonlinear measurement vector with respect to the state vector
Using the abovementioned nonlinear measurement equation, the sensitivity matrix is given by the Jacobian matrix of partial derivatives of nonlinear measurement vector with respect to the state vector
If both pseudorange and Doppler measurement are used, the sensitivity matrix will be composed of
The measurement update phase uses the Kalman equations to incorporate the information given by the measurements themselves and obtains improved estimates of the state and of the covariance as follows:
The QR-decomposition algorithm is applied to inverse matrix calculation. The general idea of this algorithm is described in detail.
The basic measurement types that will be employed in the current project are GPS pseudorange and Doppler in
The numerically controlled oscillator (NCO), which controls the carrier tracking loop, provides an indication of the observed frequency shift of the received signal. This observed frequency differs from the nominal
The Doppler can be converted to a pseudorange rate observation given by
Another possible GPS measurement model is a linear combination of GPS
The maximal time of continuous propagation
The maximal integration time step
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Altitude, km | |||
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500 | 1200 | |||
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1 |
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10 | 0.009 |
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20 | 0.16 |
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0.09 |
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30 | 0.9 |
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0.5 |
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50 | 9 |
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5 |
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100 | 150 | 0.15 | 80 | 0.075 |
The maximal half interval of multibody acceleration fixing
Altitude, km | Half interval of fixing | |||||||
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10 | 30 | 60 | 300 | |||||
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500 | 0.169 | 0.000185 | 0.551 | 0.000602 | 1.168 | 0.00128 | 5.806 | 0.00635 |
800 | 0.189 | 0.000193 | 0.615 | 0.000631 | 1.304 | 0.00134 | 6.489 | 0.00666 |
1200 | 0.216 | 0.000204 | 0.706 | 0.000668 | 1.497 | 0.00142 | 7.454 | 0.00705 |
The fixed multibody acceleration components are available on time interval
The minimal count of available Navigation SV
The discrete state-noise covariance matrix
Using the Kalman algorithm to estimate orbit propagation and determination in this section has been proposed to simulate and validate the derivation of the formula. Simulation results are shown in Figure
Microsatellite altitude 500 km, longitude 108° and latitude 35°, sampling time 4 sec, on-modulator magnitude = 2, satellite attitude motion trajectory is shown in Figure
Figure
Simulation results show that, when the microsatellite attitude and orbit maintain balance, satellite orbital position and velocity are in the estimated and the measured value via GPS satellite computer considerable amount. Initial value changes in a relatively small error, the maximum error of 10°. If the number of GPS by the change, the number of GPS consists of four changes into three or less, the error is relatively large when the attitude changes, the maximum error is instantaneous 32°. In the academic theory and engineering practice, a systematic analysis is generally considered to be consistent with the conclusion that it is feasible.
This paper represents original, unpublished material that is not under editorial consideration elsewhere, in whole or in part, other than in abstract form.
The authors declare that there is no conflict of interests regarding the publication of this paper.