Relative Pointing Software

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Description

This module is designed to work as the pointing program in an ADCS software chain as shown in the figure below.

This model provides a general method to establish a pointing frame between two known positions. Specifically, this module constructs a reference frame $\mathcal{R}$ that aligns a vector (${}^{\mathcal{B}}\mathbf{a}$) defined by the user in the body frame $\mathcal{B}$ of the parent object with a vector that points to a target object (${}^{\mathcal{B}}\mathbf{p}$) in the body frame. This is for one Spacecraft to point towards another.

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Example Use Cases

• Space-based Observations: Establish a pointing reference frame between a nearby space object and an on-board sensor.
• Terrestrial Observations: Establish a pointing reference frame between a plane based on received ADS-B (flight tracking) position information.

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Module Implementation

The relative pointing frame is constructed about a principle pointing axis, which is derived from the inertial position of the parent spacecraft ${}^{\mathcal{N}}{\mathbf{r}}_p$ and target ${}^{\mathcal{N}}{\mathbf{r}}_t$:

$${}^{\mathcal{N}}\hat{\mathbf{p}}=\frac{{}^{\mathcal{N}}{\mathbf{r}}_p{-}^{\mathcal{N}}{\mathbf{r}}_t}{|{|}^{\mathcal{N}}{\mathbf{r}}_p{-}^{\mathcal{N}}{\mathbf{r}}_t||}$$

A coordinate system about this principle axis is then built such that:

$$\begin{gathered} \hat{\mathbf{x}}=\hat{\mathbf{p}} \\ \hat{\mathbf{y}}=\hat{\mathbf{z}}\times \hat{\mathbf{p}} \\ \hat{\mathbf{z}}=\hat{\mathbf{x}}\times \hat{\mathbf{y}} \end{gathered}$$

Where $\hat{\mathbf{z}}=[0,0,1{]}^T$ and the DCM is $[RN]=[{\hat{\mathbf{x}}}^T,{\hat{\mathbf{y}}}^T,{\hat{\mathbf{z}}}^T]$. The angular velocity of the frame may then simply be calculated from the time-step $t$ as:

$${\dot{\sigma }}_{RN}=\frac{{\sigma }_{RN}(t)-{\sigma }_{RN}(t-1)}{\Delta t}$$

Using this result, and applying equation 3.164 in [1], the angular velocity of the $\mathcal{R}$ frame with respect to $\mathcal{N}$ in the $\mathcal{R}$ frame components may be computed as:

$${}^{\mathcal{R}}{\omega }_{RN}=\frac{4}{(1+{\sigma }^2{)}^2}\left[(1-{\sigma }^2)[I_{3\times 3}]-2[\sigma ]+2\sigma {\sigma }^T\right]{\dot{\sigma }}_{RN}$$ $${}^{\mathcal{N}}{\dot{\omega }}_{RN}=\frac{{}^{\mathcal{N}}{\omega }_{RN}(t){-}^{\mathcal{N}}{\omega }_{RN}(t-1)}{\Delta t}$$

The above definitions will align the reference frame’s x-axis with the target. To align with a user-specific vector, we create a second coordinate frame $R_1$. In the case that the $\mathcal{B}$ frame aligns with the $R_1$ frame, ${}^{\mathcal{B}}\mathbf{a}$ has to align with ${}^{\mathcal{N}}\mathbf{p}$. For this reason, an $\mathcal{A}$ frame is built around the ${}^{\mathcal{B}}\mathbf{a}$ vector, such that:

$$\begin{gathered} \hat{\mathbf{x}}=\hat{\mathbf{a}} \\ \hat{\mathbf{y}}=\hat{\mathbf{z}}\times \hat{\mathbf{a}} \\ \hat{\mathbf{z}}=\hat{\mathbf{x}}\times \hat{\mathbf{y}} \end{gathered}$$

where all the above are in $\mathcal{B}$. The resultant DCM $[AB]=[{\hat{\mathbf{x}}}^T,{\hat{\mathbf{y}}}^T,{\hat{\mathbf{z}}}^T]$. Converting to MRPs, the final rotation between ${\mathcal{R}}_1$ and $\mathcal{N}$ is simply:

$${\sigma }_{R_{1}N}={\sigma }_{RN}+{\sigma }_{BA}$$

The angular velocity and angular acceleration remain the same.

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References

[1] Hanspeter Schaub and John L. Junkins. Analytical Mechanics of Space Systems. AIAA Education Series, Reston, VA, 3rd edition, 2014.