Ephemeris Geodetic Translation Software

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Description

The Ephemeris Geodetic Translation Software is a flight software module that converts spacecraft position data from an inertial ephemeris representation to a geodetic coordinate system. This transformation is essential for ground track analysis, coverage calculations, communication link geometry, and any application requiring the spacecraft’s position relative to a planetary surface. The module computes latitude, longitude, altitude, and the position vector in the Planet-Centered Planet-Fixed (PCPF) reference frame based on the input ephemeris and planetary state information.

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Example Use Cases

• Ground Track Visualization: Convert orbital position to latitude and longitude for plotting spacecraft ground tracks on a map.
• Communication Link Analysis: Determine spacecraft position relative to ground stations for link budget calculations.
• Coverage Analysis: Compute sub-satellite points for sensor footprint and coverage studies.
• Mission Planning: Provide geodetic coordinates for payload targeting and observation scheduling.
• Re-entry Analysis: Track altitude and geographic position during atmospheric entry phases.

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Module Implementation

The Ephemeris Geodetic Translation Software is a Software component that processes ephemeris and planetary state messages to produce geodetic coordinates. It performs coordinate frame transformations from the inertial frame to the planet-fixed frame and extracts geodetic elements.

Input Messages

The module requires two input messages:

Input Message	Type	Description
In_EphemerisMsg	EphemerisMessage	Spacecraft position and velocity in the inertial frame
In_PlanetStateMsg	PlanetStateMessage	Planetary state including position and rotation angle

By default, the planet state message is initialized to Earth from the Solar System.

Coordinate Frame Definitions

The transformation involves three coordinate frames:

1. Inertial Frame ($\mathcal{N}$): The J2000 Earth-centered inertial reference frame.
2. Planet-Centered Inertial Frame ($\mathcal{I}$): Origin at the planet’s center of mass, axes aligned with the inertial frame.
3. Planet-Centered Planet-Fixed Frame ($\mathcal{P}$): Origin at the planet’s center of mass, rotating with the planet.

Position Relative to Planet

The first step computes the spacecraft position vector relative to the planet’s center in the inertial frame. Given the spacecraft position ${\mathbf{r}}_{B/N}^{\mathcal{N}}$ and planet position ${\mathbf{r}}_{P/N}^{\mathcal{N}}$:

$${\mathbf{r}}_{B/P}^{\mathcal{N}}={\mathbf{r}}_{B/N}^{\mathcal{N}}-{\mathbf{r}}_{P/N}^{\mathcal{N}}$$

where:

• ${\mathbf{r}}_{B/N}^{\mathcal{N}}$ is the spacecraft position relative to the inertial origin
• ${\mathbf{r}}_{P/N}^{\mathcal{N}}$ is the planet position relative to the inertial origin
• ${\mathbf{r}}_{B/P}^{\mathcal{N}}$ is the spacecraft position relative to the planet center, expressed in inertial coordinates

PCI to PCPF Transformation

The position vector is then transformed from the Planet-Centered Inertial (PCI) frame to the Planet-Centered Planet-Fixed (PCPF) frame using the planet’s rotation angle. The rotation matrix $[PN]$ transforms vectors from the inertial frame to the planet-fixed frame:

$$[PN]=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$$

where $\theta$ is the planet’s rotation angle relative to the J2000 epoch (Greenwich Sidereal Angle for Earth).

The position in the PCPF frame is:

$${\mathbf{r}}_{B/P}^{\mathcal{P}}=[PN]{\mathbf{r}}_{B/P}^{\mathcal{N}}$$

PCPF to Geodetic Conversion

The PCPF position vector is converted to geodetic coordinates (latitude, longitude, altitude) using the planet’s reference ellipsoid. For a spherical planet approximation with radius $R_P$:

Longitude

The geodetic longitude $\lambda$ is computed from the PCPF position components:

$$\lambda =\arctan 2(r_y,r_x)$$

where $r_x$ and $r_y$ are the $x$ and $y$ components of ${\mathbf{r}}_{B/P}^{\mathcal{P}}$.

Latitude

The geodetic latitude $\varphi$ is computed as:

$$\varphi =\arctan 2\left(r_z,\sqrt{r_x^2+r_y^2}\right)$$

where $r_z$ is the $z$ component of ${\mathbf{r}}_{B/P}^{\mathcal{P}}$.

Altitude

The altitude $h$ above the reference surface is:

$$h=\Vert {\mathbf{r}}_{B/P}^{\mathcal{P}}\Vert -R_P$$

where $R_P$ is the planet’s reference radius.

Output Message

The module produces a GeodeticMessage containing:

Field	Symbol	Units	Description
Position_BP_P	${\mathbf{r}}_{B/P}^{\mathcal{P}}$	m	Position vector in PCPF frame
Latitude	$\varphi$	rad	Geodetic latitude
Longitude	$\lambda$	rad	Geodetic longitude
Altitude	$h$	m	Altitude above reference surface
Planet	-	-	Name of the reference planet

Update Sequence

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

1. Validate Inputs: Check that both ephemeris and planet state messages are connected; skip update if either is missing.
2. Compute Relative Position: Calculate the spacecraft position relative to the planet center in the inertial frame.
3. Transform to PCPF: Apply the planet rotation to obtain the position in the planet-fixed frame.
4. Extract Geodetic Elements: Convert the PCPF position to latitude, longitude, and altitude.
5. Update Output Message: Populate the geodetic message with computed values and planet name.

Geometric Interpretation

The relationship between the coordinate frames can be visualized as follows:

• The inertial frame provides a non-rotating reference, typically aligned with the vernal equinox and celestial pole at J2000.
• The PCPF frame rotates with the planet, with the $x$-axis passing through the prime meridian and the $z$-axis along the rotation axis.
• Latitude measures the angle north or south of the equatorial plane.
• Longitude measures the angle east or west of the prime meridian.
• Altitude measures the radial distance above the reference surface.

The sub-satellite point is the location on the planet’s surface directly beneath the spacecraft, defined by the latitude and longitude when altitude is zero.

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Assumptions/Limitations

• The planet is modelled as a sphere with uniform radius; oblate spheroid effects (flattening) are not included in the altitude calculation.
• The rotation model assumes simple rotation about the polar axis; precession, nutation, and polar motion are not modelled.
• The J2000 rotation angle from the planet state message is assumed to be accurate and up-to-date.
• No velocity transformation is performed; only position coordinates are converted.
• The module defaults to Earth if no planet state message is explicitly connected.
• Geodetic latitude is equivalent to geocentric latitude for the spherical approximation used.
• The altitude calculation does not account for local terrain elevation or geoid undulations.
• The transformation assumes the planet’s rotation axis is aligned with the $z$-axis of the PCPF frame.
• No interpolation is performed between update steps; geodetic coordinates reflect the instantaneous state.
• The module does not validate that the ephemeris message corresponds to an object orbiting the specified planet.