I need to \"model\" a Low Earth Orbit satellite\'s position at for a given time

Question

I need to "model" a Low Earth Orbit satellite's position at for a given time in the future. However I do not have a TLE for the desired satellite, I simply have a state-vector (Epoch and XYZ Position and Velocity). I am aware that this may not be a accurate projection for any significant amount of time, but this is all I have to work with.

What would the recommended steps be for working this out? Could I convert the state-vector into orbital elements somehow and then use those elements to calculate the position in respect to time? I am rather new to orbital mechanics so I apologize if this is a simple question. I have done some research and it appears the the orbital elements route seems like the route that most others have taken. However once I get those orbital elements, how do I calculate the satellites position?

Once I have an algorithm worked out I have other satellites with TLE's I plan to decompose into state-vectors and then compare the calculated state vector position with the calculated TLE position to ensure I am in the right ballpark.

Explanation / Answer

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. The satellite is influenced by a variety of external forces, including terrestrial gravity, atmospheric drag, multi-body 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.

Three basic strategies are presently used to determine precise LEO orbits with GPS. They are the dynamic, kinematic or non-dynamic, and hybrid or reduced-dynamic strategies.

The dynamic orbit determination approach requires precise models of the forces acting on user object. This technique has been applied to many successful space vehicle missions and has become the mainstream of precision OD (POD) approach. Dynamic model errors are the limiting factor for this technique, such as the geo-potential model errors and atmospheric drag model errors, depending on the dynamic environment of the user space vehicle. With the continuous, global, and high precision GPS tracking data, dynamic model parameters, such as geo-potential parameters, can be tuned effectively to reduce the effects of dynamic model error in the context of dynamic approach. The dense tracking data also allows for the frequent estimation of empirical parameters to absorb the effects of un-modeled or mis-modeled dynamic errors.

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. A geometric method that uses the continuous record of object position changes obtained from the GPS carrier phase to smooth the position measurements made with pseudo range. This approach assumes the accessibility of -codes at both the end frequencies. A developed a kinematic orbit determination algorithm using double- and triple-differenced GPS carrier phase measurements. Kinematic solutions are more sensitive to geometrical factors, such as the direction of the GPS satellites and the GPS orbit accuracy, and they require the resolution of phase ambiguities.

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 utilize the strengths of each. One such strategy has been devised and is referred to as reduced dynamic orbit determination. The reduced-dynamic approach uses both geometric and dynamic information and weighs their relative strength by solving local geometric position corrections using a process noise model to absorb dynamic model errors.

The orbit determination algorithm estimates the object state vector and covariance matrix from discrete observations. The set of observations includes the measurements from the space based GPS receiver that is located on the space vehicle itself. The orbit determination algorithm includes the orbit prediction task as time update stage of UFK.

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. A numerical integration of the dynamic model is applied for orbit prediction.

The OP solution is output data of orbit propagation algorithm. The OP solution and covariance matrix can be obtained as from prediction task as from determination task. The following external data are required for OP solution calculation:

(i)            init time and state vector tinit , xinit for algorithm initialization/re-initialization;

(ii)           the time moment tk+1 to new OP solution xk+1 calculation;

(iii)          the set of observations NSVk+1 for new estimation xk+1 calculation.

The following input data are obtained from the previous OP solutions calculation:

(i)            the last OP solution tk, xk;

(ii)           the time test of last calculation of estimation xk+1 ;

(iii)          the covariance matrix Pk .

The maximal time of continuous propagation Tprpmax , maximal integration time step hmax , minimal count of available Navigation SV NSVmin , and default covariance matrix Pdef are used for algorithm control.

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 this model 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:

f (x,t) =

vx                            axGEO + axNbod + Fx / m + omega2ex + 2omega ey

vy                                           ayGEO + ayNbod + Fy / m + omega2ey – 2 omegaex

vz                                           azGEO + azNbod + Fz / m

x                     =                                    vx

y                                                        vy

z                                                         vz

b                                                        d

d                                                        0

where vx,, vy and vz are the ECEF velocity components of object x,y and z are the ECEF radius vector components of object, b is the receiver clock bias, d is the receiver clock drift, aN-body is the Sun and Moon gravitation forces, aGEO is the acceleration due to geopotential, Fdrag = { Fx, Fy,Fz} is a perturbing force due to aerodynamic drag, and omegae is the angular velocity of Earth rotation. 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).

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