202210112131

Clohessy-Wiltshire Equations

Also known as Hill’s equations can be written as follows:

${\displaystyle \left[\begin{array}{c}{\mathbf{R}}^{LVLH}(t_0+\mathrm{\Delta }t)\\ {\mathbf{V}}^{LVLH}(t_0+\mathrm{\Delta }t)\end{array}\right]=\left[\begin{array}{cc}{\mathrm{\Phi }}_{rr}(\mathrm{\Delta }t)& {\mathrm{\Phi }}_{rv}(\mathrm{\Delta }t)\\ {\mathrm{\Phi }}_{vr}(\mathrm{\Delta }t)& {\mathrm{\Phi }}_{vv}(\mathrm{\Delta }t)\end{array}\right]\left[\begin{array}{c}{\mathbf{R}}^{LVLH}(t_0)\\ {\mathbf{V}}^{LVLH}(t_0)+\mathrm{\Delta }v\end{array}\right]+{\int }_0^{\mathrm{\Delta }t}\left[\begin{array}{cc}{\mathrm{\Phi }}_{rr}(\tau )& {\mathrm{\Phi }}_{rv}(\tau )\\ {\mathrm{\Phi }}_{vr}(\tau )& {\mathrm{\Phi }}_{vv}(\tau )\end{array}\right]\left[\begin{array}{c}0\\ a_d\end{array}\right]d\tau }$

where we assume $t$ is the time between maneuvers, $v$ is the maneuver delta-v, and $a_{d}$ is a random disturbance representing unmodeled differential accelerations due to solar radiation pressure, atmospheric drag, plume effects, and $J_{2}$ and higher-order gravity terms. If the x, y, and z components of $^{LVLH}$ are altitude, downrange, and cross track, respectively, the above transition matrices are given by:

${\displaystyle {\mathrm{\Phi }}_{rr}(\mathrm{\Delta }t)=\left[\begin{array}{ccc}4-3cos({\omega }_{orb}\mathrm{\Delta }t)& 0& 0\\ 6sin({\omega }_{orb}\mathrm{\Delta }t)-6{\omega }_{orb}\mathrm{\Delta }t& 1& 0\\ 0& 0& cos({\omega }_{orb}\mathrm{\Delta }t)\end{array}\right]}$

${\displaystyle {\mathrm{\Phi }}_{rv}(\mathrm{\Delta }t)=\left[\begin{array}{ccc}sin({\omega }_{orb}\mathrm{\Delta }t\mathrm{/}{\omega }_{orb}& 2\{1-cos({\omega }_{orb}\mathrm{\Delta }t)\}\mathrm{/}{\omega }_{orb})& 0\\ 2\{cos({\omega }_{orb}\mathrm{\Delta }t)-1\}\mathrm{/}{\omega }_{orb}& 4sin({\omega }_{orb}\mathrm{\Delta }t)\mathrm{/}{\omega }_{orb}-3\mathrm{\Delta }t& 0\\ 0& 0& sin({\omega }_{orb}\mathrm{\Delta }t)\mathrm{/}{\omega }_{orb}\end{array}\right]}$

${\displaystyle {\mathrm{\Phi }}_{vr}(\mathrm{\Delta }t)=\left[\begin{array}{ccc}3{\omega }_{orb}sin({\omega }_{orb}\mathrm{\Delta }t)& 0& 0\\ 6{\omega }_{orb}\{cos({\omega }_{orb}\mathrm{\Delta }t)-1\}& 0& 0\\ 0& 0& -{\omega }_{orb}sin({\omega }_{orb}\mathrm{\Delta }t)\end{array}\right]}$

${\displaystyle {\mathrm{\Phi }}_{vv}(\mathrm{\Delta }t)=\left[\begin{array}{ccc}cos({\omega }_{orb}\mathrm{\Delta }t)& 2sin({\omega }_{orb}\mathrm{\Delta }t& 0\\ -2sin({\omega }_{orb}\mathrm{\Delta }t& 4cos({\omega }_{orb}\mathrm{\Delta }t)-3& 0\\ 0& 0& cos({\omega }_{orb}\mathrm{\Delta }t)\end{array}\right]}$

I wrote all of this mostly as $$ practice. It took a long time.

[!warning] WIP: Write the CW equations in the form of the Self Rescue Strategies for EVA paper which is a more ubiquitous form. Separate this note by paper