Relative Motion¶

The astrojax.relative_motion module provides functions for analysing satellite proximity operations: transforming between inertial and rotating local frames, and evaluating linearised relative dynamics.

The RTN Frame¶

When studying two satellites flying in formation, it is convenient to describe the deputy's position and velocity relative to the chief in a co-moving frame attached to the chief. The RTN (Radial, Along-Track, Normal) frame — also called the LVLH (Local Vertical Local Horizontal) frame — is defined as:

Axis	Direction
R (Radial)	From Earth's centre toward the chief's position
T (Along-track)	In the orbital plane, perpendicular to R, in the direction of motion
N (Normal / Cross-track)	Along the orbital angular-momentum vector, completing the right-handed triad

The rotation matrix \(R_{\text{RTN} \to \text{ECI}}\) has columns \([\hat{r} \;\; \hat{t} \;\; \hat{n}]\), where

\[ \hat{r} = \frac{\mathbf{r}}{|\mathbf{r}|}, \quad \hat{n} = \frac{\mathbf{r} \times \mathbf{v}}{|\mathbf{r} \times \mathbf{v}|}, \quad \hat{t} = \hat{n} \times \hat{r}. \]

JAX Compatibility¶

All functions use JAX primitives internally and work with jax.jit:

import jax
from astrojax.relative_motion import state_eci_to_rtn

jit_rtn = jax.jit(state_eci_to_rtn)
rel = jit_rtn(chief, deputy)

hcw_derivative and rotation_rtn_to_eci also support jax.vmap for batched evaluation and jax.grad for gradient computation.

float32 precision

All computations use float32 for GPU/TPU compatibility. Rotation matrix operations on position magnitudes of ~7000 km introduce rounding at the metre level. Roundtrip transformations (ECI -> RTN -> ECI) are accurate to ~0.1 mm.