Nadir 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 module constructs an output reference frame $\mathcal{R}$ that aligns with the Local-Vertical-Local-Horizontal (LVLH) reference frame $\mathcal{L}:[{\hat{i}}_r,{\hat{i}}_h,{\hat{i}}_{\theta }]$ where:

${\hat{i}}_r$: points radially outward in the direction that connects the center of the planet with the parent object, as per SPICE data.

${\hat{i}}_h$: is defined normal to the orbital plane in the direction of the angular momentum

${\hat{i}}_{\theta }$: completes the right-handed triad.

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Example Use Cases

• Compute the LVLH reference frame for pointing applications: Construct a reference frame commonly applied for nadir pointing operations e.g. Earth Observation.
• Supports navigation translation errors modelling: Accepts general navigation messages, which may provide body state estimations from sensors, flight computers, etc. or the actual body state directly from the parent Spacecraft.

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Module Implementation

Reference Frame Definition

This module assumes a standard reference frame, where ${\mathbf{R}}_s$ is the position vector of the spacecraft in the inertial frame $\mathcal{N}:[{\hat{n}}_1,{\hat{n}}_2,{\hat{n}}_3]$ and ${\mathbf{R}}_p$ is the position vector of the celestial body in the inertial frame. The relative position and velocity of the spacecraft in the planet-centered inertial frame are then:

$$\begin{gathered} \mathbf{r}={\mathbf{R}}_s-{\mathbf{R}}_p \\ \mathbf{v}={\mathbf{v}}_s-{\mathbf{v}}_p \end{gathered}$$

The Hill frame orientation is then defined as:

$$\begin{gathered} {\hat{\mathbf{i}}}_r=\frac{\mathbf{r}}{r} \\ {\hat{\mathbf{i}}}_h=\frac{\mathbf{r}\times \mathbf{v}}{rv} \\ {\hat{\mathbf{i}}}_{\theta }={\hat{\mathbf{i}}}_h\times {\hat{\mathbf{i}}}_r \end{gathered}$$

The direction cosine matrix that maps $\mathcal{R}$ to $\mathcal{N}$ is then $[RN]=[{\hat{i}}_r,{\hat{i}}_{\theta },{\hat{i}}_h]$.

Reference Frame Angular Velocity Vector

The angular velocity of the original reference frame ${\mathcal{R}}_0$ is ${\omega }_{R_0/N}$. The angular rate and acceleration vectors are then defined by the true anomaly $f$ of the orbit as:

$$\begin{gathered} {\omega }_{R_0/N}=\dot{f}{\hat{\mathbf{i}}}_h=\frac{h}{r^2}{\hat{\mathbf{i}}}_h \\ {\dot{\omega }}_{R_0/N}=\ddot{f}{\hat{\mathbf{i}}}_h=-2\frac{\mathbf{v}\cdot {\hat{\mathbf{i}}}_r}{r}\dot{f}{\hat{\mathbf{i}}}_h \end{gathered}$$

The angular rate and acceleration in the output reference frame are then computed as:

$$\begin{gathered} {\omega }_{R/N}={\omega }_{R/R_0}+{\omega }_{R_0/N}=\dot{f}{\hat{\mathbf{i}}}_h \\ {\dot{\omega }}_{R/N}=\ddot{f}{\hat{\mathbf{i}}}_h \end{gathered}$$

Expressed in the inertial frame:

$${}^{\mathcal{N}}{\omega }_{R/N}=[NR{]}^{\mathcal{R}}{\omega }_{R/N}$$ $${}^{\mathcal{N}}{\dot{\omega }}_{R/N}=[NR{]}^{\mathcal{R}}{\dot{\omega }}_{R/N}$$

where $[NR]=[RN{]}^T$.

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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.