Geodetic Object

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Description

A Geodetic Object is a Physical Object fixed to a celestial body. It is defined by latitude, longitude, and altitude, and it holds that location in the rotating planet-fixed frame. As the planet turns, the object’s inertial position and attitude move with the planet. Components such as antennas or sensors can be attached to it.

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Example Use Cases

• Ground Stations: Represents communication facilities on Earth or other planetary surfaces.
• Surface Assets: Models fixed infrastructure or observation points on celestial body surfaces.

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Module Implementation

On creation the object is placed on Earth at latitude $0^{\circ }$, longitude $0^{\circ }$, and altitude $0\mathrm{m}$. A blank planet name is treated as Earth. Latitude is clamped to $[-90^{\circ },90^{\circ }]$ and longitude is clamped to $[-180^{\circ },180^{\circ }]$.

The object subscribes to the planet rotation. Whenever that rotation changes, the inertial position and attitude are rebuilt from the stored planet-fixed state. Latitude, longitude, and the planet-fixed position stay constant. The subscription is released when the object is removed. While it exists, the object is registered with the Planet Coverage System.

Spherical Position

For a spherical planet of radius $R$, a site at geocentric latitude $\varphi$, longitude $\lambda$, and altitude $h$ has planet-centered, planet-fixed position

$$r=R+h$$ $${}^P{\mathbf{r}}_{LP}=r\left[\begin{array}{c}\cos \varphi \cos \lambda \\ \cos \varphi \sin \lambda \\ \sin \varphi \end{array}\right]$$

A planet-fixed position can be supplied directly. It is converted with the attached planet radius:

$$\varphi =\arcsin \left(\frac{z}{r}\right),\lambda =atan2(y,x),h=r-R$$

where $r=|{}^P{\mathbf{r}}_{LP}|$ and $(x,y,z)$ are the planet-fixed components.

Inertial State

Let $[PN]$ be the planet rotation from the inertial frame $N$ to the planet-fixed frame $P$, and let ${}^N{\mathbf{r}}_{PN}$ be the planet center in $N$. The site position relative to the planet, and the inertial position of the site, are

$${}^N{\mathbf{r}}_{LP}=[PN{]}^{\mathrm{T}}{}^P{\mathbf{r}}_{LP}$$ $${}^N{\mathbf{r}}_{LN}={}^N{\mathbf{r}}_{PN}+{}^N{\mathbf{r}}_{LP}$$

Site Orientation

The local frame $L$ is south-east-zenith. Its rotation from $P$ to $L$ depends only on latitude and longitude:

$$[LP]=R_3(\lambda )R_2\left(\frac{\pi }{2}-\varphi \right)$$

The object’s inertial attitude uses a separate planet-fixed basis at the site. Up is the geocentric radial direction. South is taken from a point $0.00001^{\circ }$ due north, at the same longitude and altitude:

$${\hat{\mathbf{e}}}_U=\frac{{}^P{\mathbf{r}}_{LP}}{\left|{}^P{\mathbf{r}}_{LP}\right|}$$ $${\hat{\mathbf{e}}}_S=-\frac{{}^P{\mathbf{r}}_{+}-{}^P{\mathbf{r}}_{LP}}{\left|{}^P{\mathbf{r}}_{+}-{}^P{\mathbf{r}}_{LP}\right|}$$ $${\hat{\mathbf{e}}}_R={\hat{\mathbf{e}}}_S\times {\hat{\mathbf{e}}}_U$$

where ${}^P{\mathbf{r}}_{+}$ is the planet-fixed position of the offset point. ${\hat{\mathbf{e}}}_R$, ${\hat{\mathbf{e}}}_S$, and ${\hat{\mathbf{e}}}_U$ are the right, south, and up axes. The inertial attitude is this basis mapped through the current $[PN]$. Planet rotation leaves the planet-fixed axes unchanged.

Non-Spherical Earth

When the planet is marked non-spherical, the supplied altitude is shifted before the spherical conversion. The shift uses the WGS-84 geocentric radius

$$R_{\mathrm{WGS}}(\varphi )=\sqrt{\frac{a^4{\cos }^2\varphi +b^4{\sin }^2\varphi }{a^2{\cos }^2\varphi +b^2{\sin }^2\varphi }}$$

where $a$ and $b$ are the WGS-84 equatorial and polar radii. The stored altitude becomes

$$h\leftarrow h+R_{\mathrm{WGS}}(\varphi )-R$$

so the geocentric distance is $R_{\mathrm{WGS}}(\varphi )$ plus the altitude that was supplied. A spherical planet keeps the supplied altitude.

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Assumptions/Limitations

• The surface is a sphere of the planet’s published radius. Oblateness enters only as the non-spherical altitude shift.
• That shift uses the Earth WGS-84 radii. It does not convert geodetic latitude to geocentric latitude, and it leaves longitude unchanged.
• Latitude and longitude outside range are clamped. Longitude beyond $\pm 180^{\circ }$ is not wrapped into another meridian.
• The south-east-zenith matrix and the inertial attitude are separate. The attitude south axis uses a fixed latitude offset of $0.00001^{\circ }$.
• The site is fixed to the planet. Travel over the surface is modeled by a Vehicle.
• A planet-fixed position on a non-spherical planet is converted on the sphere, and the altitude shift is then applied.