Table of Contents

Software: Ground Pointing

Description

The ground location pointing software determines a reference and guidance frame between the spacecraft and a particular location on the ground. A common example would be to point the spacecraft towards a ground station, reading in the ground states a message from the ground station and being able to point towards that location on a celestial body.

Example Use Cases

  • Ground Location Imaging: Imaging a particular location on the surface of the Earth by pointing a camera towards a target.
  • Ground Station Communications: Pointing an antenna towards a ground station network for communicating data and uplinking commands.

Module Implementation

The module takes in some location on a spherical body \(G\) and the spacecraft’s inertial position \(B\) and is able to evaluate the pointing angle between the two locations. The vector pointing from the satellite location \(\textbf{r}_{BG}\) can be determined as:

\[ \textbf{r}_{BG} = \textbf{r}_{BN} - \textbf{r}_{GN} \]

where \(\textbf{r}_{BN}\) is the spacecraft’s coordinates in the inertial frame and \(\textbf{r}_{GN}\) is the ground location’s coordinates in the inertial frame. Then, from this vector, \(\hat{\textbf{r}}_{BG}\) is the normalized heading vector relative to the location. The unit vector \(\hat{\textbf{p}}\) is the body-fixed vector and defines the body axis to point towards the desired location \(G\). The module performs a 2-degree-of-freedom attitude guidance solution. The \(\hat{\textbf{p}}\) axis is rotated towards \(\hat{\textbf{r}}_{BG}\) given by the eigen rotation \(\hat{\textbf{e}}\).

\[ \hat{\textbf{e}} = {\hat{\textbf{p}} \times \hat{\textbf{r}}_{BG} \over |\hat{\textbf{p}} \times \hat{\textbf{r}}_{BG}|} \]

The principle rotation angle \(\phi\) can then be defined by:

\[ \phi = \arccos(\hat{\textbf{p}} \cdot\hat{\textbf{r}}_{BG}) \]

Using these values, the attitude tracking error, which is required for the feedback controller software, \(\sigma_{RB}\) from the reference frame \(R\) to the body frame \(B\) can be determined via the MRP principle:

\[ \sigma_{RB} = -\tan\bigg({\phi \over 4}\bigg)\hat{\textbf{e}} \]

Finally, the attitude tracking error rates \(\omega_{RB}\) are calculated via the numerical differentiation of the feedback loop using the PID values. In the first loop of the software, as the data has not yet propagated to the feedback loop, these values remain as \(0\). Generally, since this is a 2D approximation, the heading angle about the target direction is not damped. As such, the spacecraft would tumble about the heading angle when facing the ground location. To adjust this, the Boresight Rate Damping parameter can be set to a non-zero value, allowing the body rates about the desired heading angle to be added to the tracking error, yielding an approximation to the control solution.

Assumptions/Limitations

  • The module assumes that the celestial body is located on a spherical planet.
  • The model uses a 2D solution to the pointing angle. Without using boresight damping, the spacecraft can spin about the target frame axis.