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

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

and the control input matrix is expressed as

## 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.