Clohessy-Wiltshire equations

The Clohessy-Wiltshire equations describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is very useful in planning rendezvous of the chaser with the target.[1]

${\displaystyle {\ddot {x}}=3n^{2}x+2n{\dot {y}}}$

${\displaystyle {\ddot {y}}=-2n{\dot {x}}}$

${\displaystyle {\ddot {z}}=-n^{2}z}$

${\displaystyle n={\sqrt {\frac {\mu }{a^{3}}}}}$

which is the orbital rate of the target body. ${\displaystyle a}$ is the radius of the target body's circular orbit, and ${\displaystyle \mu }$ is the standard gravitational parameter.

For illustration, in low earth orbit, ${\displaystyle \mu =3.986E14{\frac {m^{3}}{s^{2}}}}$ and ${\displaystyle a=6378137m+415000m=6793137m}$, so ${\displaystyle n=0.00113s^{-1}}$, corresponding to an orbital period of about 93 minutes.

Solution

We can obtain closed form solutions of these coupled differential equations in matrix form, allowing us to find the position and velocity of the chaser at any time given the initial position and velocity.[2]

${\displaystyle \delta {\vec {r}}(t)=[\Phi _{rr}(t)]\delta {\vec {r_{0}}}+[\Phi _{rv}(t)]\delta {\vec {v_{0}}}}$

${\displaystyle \delta {\vec {v}}(t)=[\Phi _{vr}(t)]\delta {\vec {r_{0}}}+[\Phi _{vv}(t)]\delta {\vec {v_{0}}}}$

where

${\displaystyle \Phi _{rr}(t)={\begin{bmatrix}4-3\cos {nt}&0&0\\6(\sin {nt}-nt)&1&0\\0&0&\cos {nt}\end{bmatrix}}}$

${\displaystyle \Phi _{rv}(t)={\begin{bmatrix}{\frac {1}{n}}\sin {nt}&{\frac {2}{n}}(1-\cos {nt})&0\\{\frac {2}{n}}(\cos {nt}-1)&{\frac {1}{n}}(4\sin {nt}-3nt)&0\\0&0&{\frac {1}{n}}\sin {nt}\end{bmatrix}}}$

${\displaystyle \Phi _{vr}(t)={\begin{bmatrix}3n\sin {nt}&0&0\\6n(\cos {nt}-1)&0&0\\0&0&-n\sin {nt}\end{bmatrix}}}$

${\displaystyle \Phi _{vv}(t)={\begin{bmatrix}\cos {nt}&2\sin {nt}&0\\-2\sin {nt}&4\cos {nt}-3&0\\0&0&\cos {nt}\end{bmatrix}}}$

Note that ${\displaystyle \Phi _{vr}(t)={\frac {d}{dt}}\Phi _{rr}(t)}$ and ${\displaystyle \Phi _{vv}(t)={\frac {d}{dt}}\Phi _{rv}(t)}$

Since these matrices are easily invertible, we can also solve for the initial conditions given only the final conditions and the properties of the target vehicle's orbit.