Hill-Clohessy-Wiltshire Dynamics¶

For a chief satellite on a circular orbit with mean motion \(n = \sqrt{\mu / a^3}\), the linearised equations of relative motion (HCW model) are:

\[ \ddot{x} = 3n^2 x + 2n\dot{y}, \quad \ddot{y} = -2n\dot{x}, \quad \ddot{z} = -n^2 z. \]

hcw_derivative returns the 6-element state derivative \([\dot{x}, \dot{y}, \dot{z}, \ddot{x}, \ddot{y}, \ddot{z}]\) and is designed to plug directly into a numerical integrator:

import jax.numpy as jnp
from astrojax.constants import R_EARTH, GM_EARTH
from astrojax.relative_motion import hcw_derivative

sma = R_EARTH + 500e3
n = jnp.sqrt(GM_EARTH / sma**3)
state = jnp.array([100.0, 0.0, 0.0, 0.0, 0.0, 0.0])

deriv = hcw_derivative(state, n)

Because it is a pure JAX function, hcw_derivative is compatible with jax.jit, jax.vmap, and jax.grad, making it suitable for batched simulation and differentiable control.

State Transition Matrix¶

Because the HCW equations are linear and time-invariant, the system \(\dot{\mathbf{x}} = A\mathbf{x}\) has a closed-form state transition matrix \(\Phi(t) = e^{At}\) that maps an initial state directly to the state at elapsed time \(t\):

\[ \mathbf{x}(t) = \Phi(t)\,\mathbf{x}_0 \]

Writing \(c = \cos(nt)\) and \(s = \sin(nt)\), the 6 \(\times\) 6 matrix is:

\[ \Phi(t) = \begin{bmatrix} 4-3c & 0 & 0 & s/n & 2(1-c)/n & 0 \\ 6(s-nt) & 1 & 0 & -2(1-c)/n & (4s-3nt)/n & 0 \\ 0 & 0 & c & 0 & 0 & s/n \\ 3ns & 0 & 0 & c & 2s & 0 \\ -6n(1-c) & 0 & 0 & -2s & 4c-3 & 0 \\ 0 & 0 &-ns & 0 & 0 & c \end{bmatrix} \]

hcw_stm returns this matrix and can be used for fast, exact propagation without numerical integration:

import jax.numpy as jnp
from astrojax.constants import R_EARTH, GM_EARTH
from astrojax.relative_motion import hcw_stm

sma = R_EARTH + 500e3
n = jnp.sqrt(GM_EARTH / sma**3)
state0 = jnp.array([100.0, 0.0, 0.0, 0.0, 0.0, 0.0])

state_10min = hcw_stm(600.0, n) @ state0

The STM satisfies the semigroup (composition) property: \(\Phi(t_1 + t_2) = \Phi(t_2)\,\Phi(t_1)\). This means you can chain shorter propagation steps without accumulating integration error. It is also volume-preserving (\(\det\Phi = 1\)), as expected for a Hamiltonian system.