Hill-Clohessy-Wiltshire Equations

The linearized relative dynamics between a chaser satellite and a target object in orbit can be described by the following set of equations:

$$\begin{align*} \ddot{x}(t) &= 2n \cdot \dot{y}(t) + 3n^2 \cdot x(t) \; [\;+\; \frac{F_x}{m} \; ] \\[0.3em] \ddot{y}(t) &= -2n \cdot \dot{x}(t) \; [\;+\; \frac{F_y}{m} \; ] \\[0.3em] \ddot{z}(t) &= -n^2 \cdot z(t) \; [\;+\; \frac{F_z}{m} \; ] \end{align*}$$

with the terms appearing inside the brackets being optional (controlled case) [1].

Assumptions:

• Distance from target to chaser is much much smaller than the orbital radius, i.e., $\vert \boldsymbol{\rho}(t) \vert < < \vert \mathbf{r}_{tgt} \vert \Rightarrow \boldsymbol{r} _{ch} \approx \mathbf{r} _{tgt} $
• A first order approximation on the term $\left( 1 + \frac{2x}{r_{tgt}} \right)^{-3/2} \approx 1 - \frac{3x}{r_{tgt}}$
• Reduce to time invariant by assuming target orbit’s eccentricity equal to zero.

Discrete System Equations

The discrete, analytical solution to the system can be expressed as a traditional LTI system of the form $ \mathbf{x}_{k+1} = \mathbf{A}_d \mathbf{x}_k + \mathbf{B}_d \mathbf{u}_k$ [2], where the system matrix is of the form

$$\begin{align*} \mathbf{A}_d = \begin{bmatrix} 4-3\cos(nT) & 0 & 0 & \frac{1}{n}\sin(nT) & \frac{2}{n}(1-\cos(nT)) & 0 \\\\[0.3em] 6(\sin(nT)-nT) & 1 & 0 & -\frac{2}{n}(1-\cos(nT)) & \frac{1}{n}(4\sin(nT)-3nT) & 0 \\\\[0.3em] 0 & 0 & \cos(nT) & 0 & 0 & \frac{1}{n}\sin(nT) \\\\[0.3em] 3n\sin(nT) & 0 & 0 & \cos(nT) & 2\sin(nT) & 0 \\\\[0.3em] -6n(1-\cos(nT)) & 0 & 0 & -2\sin(nT) & 4\cos(nT)-3 & 0 \\\\[0.3em] 0 & 0 & -n\sin(nT) & 0 & 0 & \cos(nT) \\\\[0.3em] \end{bmatrix}, \end{align*}$$

and the control input matrix is expressed as

$$\begin{align*} \mathbf{B}_d = \begin{bmatrix} \frac{1}{n}\sin(nT) & \frac{2}{n}(1-\cos(nT)) & 0 \\\\[0.3em] -\frac{2}{n^2}(nT - \sin(nT)) & \frac{4}{n^2}(1-\cos(nT))-\frac{3}{2} T^2 & 0 \\\\[0.3em] 0 & 0 & \frac{1}{n^2}(1-\cos(nT)) \\\\[0.3em] \frac{1}{n}\sin(nT) & \frac{2}{n}(1-\cos(nT)) & 0 \\\\[0.3em] -\frac{2}{n}(1-cos(nT)) & \frac{4}{n}\sin(nT) - 3T & 0 \\\\[0.3em] 0 & 0 & \frac{1}{n}\sin(nT) \\\\[0.3em] \end{bmatrix}. \end{align*}$$

Bibliography

1. [1]T. A. Lovell and D. A. Spencer, “Relative orbital elements formulation based upon the Clohessy-Wiltshire equations,” The Journal of the Astronautical Sciences, vol. 61, no. 4, pp. 341–366, 2014.
2. [2]C. Jewison and D. W. Miller, “Probabilistic Trajectory Optimization Under Uncertain Path Constraints for Close Proximity Operations,” Journal of Guidance, Control, and Dynamics, pp. 1–16, 2018.