1.1 Background

The final project for AE 326 is a high altitude balloon project. GPS technology will be used during that project. GPS technology is also used in everyday life: for example, GPS is used in cellphone’s and car GPS units. In order to locate an object using GPS, one must know the location of the GPS satellites in their respective orbits. One way to determine the location of a GPS satellite is the almanac data.

1.2 Purpose

Use the almanac data to predict where a satellite will be in its orbit. Calculate the margin of error between the calculations from the almanac data and the precise ephemeris data.

2.0 METHODOLOGY

The commented MATLAB code discussed in the following paragraphs is located in 7.0 Appendix I: MATLAB

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Equation (3) also uses the mean anomaly from the almanac data (M_0), the semi-major axis of the orbit (a), and the gravitation parameter (μ). Mean anomaly has units of radians.

M=M¬_0+√(μ/a^3 )*t_k (3)

The mean anomaly (M) calculated in equation (3) is used in equation (4) along with the eccentricity (e) to calculate the eccentric anomaly (E). Eccentric anomaly has units of radians.

M=E-e*sin(E) (4)

The eccentric anomaly (E) calculated in equation (4) is used in equation (5) along with the eccentricity (e) to calculate the true anomaly (θ_k^*). The true anomaly has units of radians.

tan((θ_k^*)/2)=√((1+e)/(1-e)) tan( E/2 ) (5)

The true anomaly (θ_k^*) calculated in equation (5) is used in equation (6) along with the argument of perigee (ω) to calculate the argument of latitude (θ_k). The argument of latitude has units of radians.

θ_k=θ_k^*+ω (6)

Equation (7) is used to calculate the magnitude of the position of the satellite (r) given the semi-major axis (a), the eccentricity (e), and the eccentric anomaly (E). The magnitude of the position of the satellite has units of

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The ECEF position vector was for the almanac data was calculated in equation (11), and the ECEF position vector for the ephemeris data is given in the ephemeris data text file. Equations (16) and (17) show the calculations performed to calculate the latitude and longitude.

Equation (16) calculates the latitude (ϕ) given the z-coordinate of the position vector of the satellite in the ECEF coordinate frame (z_ECEF) and magnitude of the position vector of the satellite (r). Latitude has units of radians, but it is converted to degrees for use in the geoshow function in MATLAB by multiplying by π/〖180〗^o .

ϕ=Sin^(-1) (z_ECEF/r) (16)

Equation (17) calculates the longitude (λ) given the y-coordinate and x-coordinate of the position vector of the satellite in the ECEF coordinate frame (x_ECEF and y_ECEF). Longitude has units of radians, but it is converted to degrees for use in the geoshow function in MATLAB by multiplying by π/〖180〗^o .

λ=Tan^(-1) (y_ECEF/x_ECEF )