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) .

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 $\mathbf{x}_{k+1} = \mathbf{A}_d \mathbf{x}_k + \mathbf{B}_d \mathbf{u}_k$ , where the system matrix is of the form

and the control input matrix is expressed as

## Bibliography

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. 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.