Ephemeris LVLH Translation Software

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Description

The Ephemeris LVLH Translation Software is a flight software module that converts spacecraft position and velocity data from inertial ephemeris representation into the relative translational state expressed in the target’s LVLH (Local Vertical Local Horizontal), or Hill, reference frame. This transformation is essential for formation flying, rendezvous and proximity operations, relative navigation, and any application requiring the chaser’s state relative to a target spacecraft. The module computes the relative position and velocity in the target’s LVLH frame and the direction cosine matrix from the inertial frame to that frame, based on the chaser and target ephemeris messages.

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Example Use Cases

• Formation Flying: Provide relative position and velocity in the target’s LVLH frame for formation-keeping control laws and constraint monitoring.
• Rendezvous and Proximity Operations: Supply relative state in LVLH for approach guidance, hold-point targeting, and collision avoidance.
• Relative Navigation: Feed relative state in a natural orbital frame for filters, observers, or sensor processing that assume LVLH coordinates.
• Conjunction Assessment: Express close-approach geometry in radial, along-track, and cross-track components for risk analysis.
• Docking and Berthing: Provide LVLH-relative state for final approach and docking axis alignment.

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Module Implementation

The Ephemeris LVLH Translation Software is a Software component that processes chaser and target ephemeris messages to produce an LVLHTranslationMessage. It computes the relative state in the inertial frame, constructs the target’s LVLH frame from the target’s position and velocity, and projects the relative state into that frame.

Input Messages

The module requires two input messages:

Input Message	Type	Description
In_ChaserEphemerisMsg	EphemerisMessage	Position and velocity of the chaser spacecraft in the inertial frame
In_TargetEphemerisMsg	EphemerisMessage	Position and velocity of the target spacecraft in the inertial frame

The chaser is the object whose state is expressed relative to the target; the target defines the origin and orientation of the LVLH frame.

LVLH (Hill) Frame Definition

The target’s LVLH frame $\mathcal{H}$ is defined at the target’s position:

1. Radial ($x$-axis): Unit vector from the central body (e.g. planet) through the target, pointing away from the central body.
2. Along-track ($y$-axis): The direction of the target’s velocity.
3. Cross-track ($z$-axis): Orbit normal, aligned with the target’s specific angular momentum vector (perpendicular to the orbital plane).

Let ${\mathbf{r}}_{T/N}^{\mathcal{N}}$ and ${\mathbf{v}}_{T/N}^{\mathcal{N}}$ be the target’s position and velocity in the inertial frame $\mathcal{N}$. The unit vectors are:

$${\hat{\mathbf{x}}}_{\mathcal{H}}=\frac{{\mathbf{r}}_{T/N}^{\mathcal{N}}}{r_T},r_T=\Vert {\mathbf{r}}_{T/N}^{\mathcal{N}}\Vert$$ $${\hat{\mathbf{y}}}_{\mathcal{H}}={\hat{\mathbf{z}}}_{\mathcal{H}}\times {\hat{\mathbf{x}}}_{\mathcal{H}}$$ $${\hat{\mathbf{z}}}_{\mathcal{H}}=\frac{{\mathbf{r}}_{T/N}^{\mathcal{N}}\times {\mathbf{v}}_{T/N}^{\mathcal{N}}}{\Vert {\mathbf{r}}_{T/N}^{\mathcal{N}}\times {\mathbf{v}}_{T/N}^{\mathcal{N}}\Vert }$$

Relative State in the Inertial Frame

The relative position and velocity of the chaser with respect to the target, expressed in the inertial frame, are:

$${\mathbf{r}}_{C/T}^{\mathcal{N}}={\mathbf{r}}_{C/N}^{\mathcal{N}}-{\mathbf{r}}_{T/N}^{\mathcal{N}}$$ $${\mathbf{v}}_{C/T}^{\mathcal{N}}={\mathbf{v}}_{C/N}^{\mathcal{N}}-{\mathbf{v}}_{T/N}^{\mathcal{N}}$$

where ${\mathbf{r}}_{C/N}^{\mathcal{N}}$ and ${\mathbf{v}}_{C/N}^{\mathcal{N}}$ are the chaser’s position and velocity in the inertial frame.

Direction Cosine Matrix (Inertial to LVLH)

The direction cosine matrix $[NH]$ (or $DCM\_NH$) rotates vectors from the inertial frame $\mathcal{N}$ to the target’s LVLH frame $\mathcal{H}$. Its rows are the unit vectors of the LVLH frame expressed in inertial coordinates:

$$[NH]=\left[\begin{array}{c}{\hat{\mathbf{x}}}_{\mathcal{H}}^T\\ {\hat{\mathbf{y}}}_{\mathcal{H}}^T\\ {\hat{\mathbf{z}}}_{\mathcal{H}}^T\end{array}\right]$$

A vector ${\mathbf{a}}^{\mathcal{N}}$ in inertial coordinates is transformed to LVLH by:

$${\mathbf{a}}^{\mathcal{H}}=[NH]{\mathbf{a}}^{\mathcal{N}}$$

Relative Position and Velocity in LVLH

The relative position and velocity in the target’s LVLH frame are obtained by rotating the inertial relative vectors:

$${\mathbf{r}}_{C/T}^{\mathcal{H}}=[NH]{\mathbf{r}}_{C/T}^{\mathcal{N}}$$ $${\mathbf{v}}_{C/T}^{\mathcal{H}}=[NH]{\mathbf{v}}_{C/T}^{\mathcal{N}}$$

Components in $\mathcal{H}$ are:

• X: Radial (positive away from central body).
• Y: Along-track (direction of target motion).
• Z: Cross-track (orbit normal).

Output Message

The module produces an LVLHTranslationMessage containing:

Field	Symbol	Units	Description
Position_BT_H	${\mathbf{r}}_{C/T}^{\mathcal{H}}$	m	Relative position of chaser w.r.t. target in target’s LVLH frame
Velocity_BT_H	${\mathbf{v}}_{C/T}^{\mathcal{H}}$	m/s	Relative velocity of chaser w.r.t. target in target’s LVLH frame
DCM_NH	$[NH]$	-	Direction cosine matrix from inertial frame to target’s LVLH frame

Update Sequence

At each simulation time step, the module performs the following operations:

1. Validate Inputs: Check that both chaser and target ephemeris messages are connected; skip update if either is missing.
2. Compute Relative State: Calculate relative position and velocity of chaser with respect to target in the inertial frame.
3. Construct LVLH Frame: From the target’s position and velocity, compute the radial, cross-track, and along-track unit vectors and form $[NH]$ (DCM_NH).
4. Transform to LVLH: Rotate the relative position and velocity into the target’s LVLH frame.
5. Update Output Message: Populate the LVLH translation message with Position_BT_H, Velocity_BT_H, and DCM_NH.

Geometric Interpretation

• The target defines the origin of the LVLH frame; the chaser is the object whose state is reported relative to the target.
• Radial is along the target–central-body line; along-track is in the orbital plane in the direction of motion; cross-track is normal to the orbit.
• This frame is non-inertial (it rotates with the target’s orbit); relative dynamics in this frame are commonly used for formation flying and rendezvous analysis.
• The same DCM can be used to transform other vectors (e.g. thrust or measurements) from the inertial frame to the target’s LVLH frame.

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Assumptions/Limitations

• Both chaser and target ephemeris messages are assumed to be expressed in the same inertial reference frame; no frame conversion between different inertials is performed.
• The LVLH frame is undefined if the target’s position or velocity is zero, or if position and velocity are parallel (radial trajectory); behaviour in such degenerate cases is implementation-dependent.
• The module does not account for the rotating nature of the LVLH frame when interpreting rates; it outputs the relative velocity expressed in LVLH components (inertial relative velocity resolved in LVLH axes), not necessarily the time derivative of Position_BT_H in a rotating-frame sense.
• No interpolation or prediction is performed; outputs reflect the instantaneous states from the connected ephemeris messages at the current time step.
• The central body is implicit in the geometry (e.g. origin for radial direction); the module does not take a separate planet or central body message.
• The chaser and target are assumed to be two distinct objects; no check is performed that the ephemeris messages refer to different spacecraft.
• Numerical singularities may occur for near-radial target trajectories or when the target’s angular momentum magnitude is very small.