Ground Station Access Software

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Description

The ground location software determines the position of a Spacecraft relative to the Ground Station on some celestial body by evaluating the radius, azimuth and elevation angles of the vector between the two objects.

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Example Use Cases

• Ground Location Determination: Determines the relative angles and spherical coordinates of a spacecraft to a ground location. These angles could be used to point an Antenna towards a spacecraft.

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Module Implementation

The module allows the user to specific a single celestial body ground location $G$ and determines the reference frame and guidance frame for the spacecraft to orient towards such location. The position vector is defined in terms of the spherical coordinate range, azimuth and elevation angles as well as in South-East-Zenith (SEZ) coordinates. The velocity of the craft as seen by the location $G$ is defined in these coordinates. The model assumes that the location is fixed to a spherical body with a constant radius and elevation constraints are computed using a conical view of the normal vector from the body located on the surface.

The position of the spacecraft in the SEZ frame, relative to the ground station on the surface, can be determined using the coordinates:

$${\text{r}}_{GB}=x\hat{S}+y\hat{E}+z\hat{Z}$$

where ${\text{r}}_{BG}$ is the position vector from the body $B$ of the spacecraft relative to the ground location $G$. $x$, $y$ and $z$ are the cartesian coordinates of the referenced SEZ frame. These cartesian coordinates are converted to spherical coordinates, centered on the ground station location, where:

$$\begin{gathered} \rho =\sqrt{x^2+y^2+z^2} \\ A=\tan \left(\frac{y}{x}\right) \\ E=\tan \left(\frac{z}{\sqrt{x^2+y^2}}\right) \end{gathered}$$

Here, $\rho$ is the range between the spacecraft and the ground location, $A$ is the azimuth angle about the negative-$\hat{S}$ direction, in a clockwise direction, and $E$ is the elevation above the horizon of the ground station perpendicular normal vector.

The spherical coordinate rates can then be computed by differentiating each of these values with respect to the rotating SEZ frame. To determine the range derivative:

$$\dot{\rho }=\frac{{}^G\mathrm{d}}{\mathrm{d}t}\rho =\frac{x\dot{x}+y\dot{y}+z\dot{z}}{\sqrt{x^2+y^2+z^2}}$$

Here, the range derivative $\dot{\rho }$ can be calculated by the rate of change with respect to the $G$ frame on the celestial body. The azimuth derivative can be calculated as:

$$\dot{A}=\frac{{}^G\mathrm{d}}{\mathrm{d}t}A=\frac{1}{1+\frac{y^2}{x^2}}\left(\frac{y\dot{x}-x\dot{y}}{x^2}\right)$$

The azimuth is calculated only from the $x$ and $y$ coordinates, but as a derivative relative to time, assuming that $\dot{x}$ and $\dot{y}$ are the derivatives of $x$ and $y$ respectively with time. Finally, the elevation derivative change is calculated as:

$$\dot{E}=\frac{{}^G\mathrm{d}}{\mathrm{d}t}E=\frac{1}{1+\frac{z^2}{x^2+y^2}}(\frac{\dot{z}}{\sqrt{x^2+y^2}}-\frac{z(x\dot{x}+y\dot{y})}{(x^2+y^2{)}^{3/2}})$$

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Assumptions/Limitations

• This assumes the ground location is located on a spherical body. No obliqueness is assumed with the planet.