## SGPS ORBIT DETERMINATION

#### SGPS orbit determination (OD) refers to determining a spacecraft's state, e.g., three dimensional position and velocity. This can be done directly with just position and velocity solutions from the receiver or for high accuracy results using the measurements from the spaceborne receiver, terrestrial receivers and dynamic models in a procedure that has come to be known as precise orbit determination (POD). This page will concentrate on the latter.

### Benefits of GPS:

#### GPS provides relatively inexpensive, accurate, continuous, three dimensional spatial information to the orbit determination process.

### Methodology:

#### Three basic strategies are presently in use to determine precise LEO orbits with GPS. They are the dynamic, the kinematic or non-dynamic, and the hybrid or reduced-dynamic strategies. Each is briefly described below.

#### DYNAMIC STRATEGY

#### In the dynamic strategy, mathematical models of the forces acting on the LEO and mathematical models of the LEO's physical properties (altogether usually referred to as dynamic models) are used to compute a model of the LEO's acceleration over time via the constraints of Newton's second law of motion. Double integration of this model using a nominal spacecraft state vector produces a nominal trajectory ( thus developing the equations of motion of the LEO. A model trajectory is then estimated by selecting the LEO state that best fits (e.g., in a least squares sense) the pre-processed (undifferenced or differenced) GPS tracking measurements.

#### An example of the most accurate SGPS dynamic orbit solution is for the ~1300 km altitude TOPEX/Poseidon satellite. Sub-decimetre radial, along-track and cross-track precision position components have been estimated.

#### This OD strategy also allows for the simultaneous estimation of other parameters to improve the fit between the nominal trajectory and the tracking data, while still preserving available measurement strength by means of the dynamic models. These parameters can be classified as perturbing force and geometric parameters (e.g., gravity coefficients and terrestrial observing station coordinates), and empirical parameters (e.g., once- or twice-per-orbit revolution accelerations). Over a long data arc, the effect of noisy instantaneous tracking measurements on the solution are reduced, given that the dynamic models are adequate. However, errors in these models will result in steadily growing systematic errors in the LEO state for longer data arc lengths. For example, empirical parameter estimation indicates weakness in the dynamic models, which generally increases with decreasing LEO altitude, and increasing LEO dynamics.

#### KINEMATIC STRATEGY

#### In the kinematic or non-dynamic strategy, the trajectory smoothing caused by the dynamic constraints in the estimation process is removed. The rationale for this is that, particularly at lower altitudes, the actual path of the LEO may be closer to the precise GPS position estimates than the trajectory determined via the dynamics. This strategy can be applied by estimating in a Kalman filter formulation, along with the spacecraft state, a process noise vector representing three force corrections at each measurement epoch. Increasing the process noise can reduce almost completely the effects of the dynamic models.

#### Sub-decimetre position component precision can also be attained with this strategy.

#### The kinematic OD strategy is therefore actually based on an underlying dynamic formulation, however dynamic modelling errors are circumvented. The strategy relies almost entirely on the precision of the GPS observations and the strength of the observing geometry ( that is, the relative location of the LEO and terrestrial receivers with respect to the GPS constellation, and the continuous GPS satellite tracking from the SGPS receiver and the terrestrial GPS receiver array.

#### HYBRID STRATEGY

#### The previous two strategies each have counterbalancing disadvantages: various mis-modelling 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 utilise the strengths of each. One such strategy has been devised and is referred to as reduced dynamic orbit determination.

#### Its basis is again the kinematic correction of the dynamic solution with continuous GPS data. However, by not completely removing the LEO dynamic and spacecraft models, a more accurate solution is possible because sensitivity to mis-modelling and GPS measurement error are both reduced. The weighting of the kinematic and dynamic data is performed again via the Kalman filter process noise. The process noise model contains two primary parameters: a time constant that defines the correlation in the dynamic model error over one update interval, and the dynamic model steady state variance. When the time constant approaches infinite and the steady state variance approaches 0, the technique reduces to the dynamic strategy, and when time constant approaches 0 and the steady state variance approached infinite, it approximates the kinematic strategy.

#### Orbit determination results using the reduced dynamic technique for the TOPEX/Poseidon satellite have been consistent with results obtained with conventional dynamic techniques using GPS, and SLR and DORIS tracking data. Moreover, using refined dynamic models produces solutions that are even more similar.

#### In the hybrid strategy, the proper weights of the process noise parameters must be chosen to give the most accurate orbit solution. These values can be derived from computer simulations, covariance analysis, or can be determined from real data. Once the correction parameter values are used, this strategy provides equal or better accuracy compared to the other two strategies.

### Mission examples:

#### - The TOPEX/Poseidon mission has undergone the most rigorous GPS POD and POD research. Orbits that are accurate to approximately one decimetre in each position component exist for this and most other POD missions.

### Related Internet Sites:

### References:

#### Yunck, T. P. (1996). "Orbit determination." In Global Positioning System: Theory and Applications Volume 2, Eds. B.W. Parkinson, J.J. Spilker Jr., Progress in Astronautics and Aeronautics Volume 164, American Institute of Aeronautics and Astronautics, Inc., Washington, D.C., U.S.A., pp. 559-592.

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