# ECI to LVLH conversion

I want to transform the position and velocity of two satellites from ECI mean of date mean of equinox to LVLH(Local Vertical Local Horizontal). The problem is that there is little documentation that I could find, allowing me to understand that it is a system that has its origin at the center of one of the satellites (target) and one could find the position of the second satellite(chaser) with respect to the first satellite. Is there a site or a book where someone explains what is the algorithm needed to be implemented for this procedure and also more information for a proper understanding of the system?

## 1 Answer

An LVLH frame is easy to construct. One construction:

• Let $$\boldsymbol r$$ and $$\boldsymbol v$$ denote the spacecraft's position and velocity with respect to the center of the planet as expressed in an inertial frame.
• Construct $$\hat {\boldsymbol x}$$ as the unit vector directed along the spacecraft position vector:
$$\hat {\boldsymbol x} = \boldsymbol r/||\boldsymbol r||$$.
• Construct $$\hat {\boldsymbol z}$$ as the unit vector directed along the spacecraft's orbital angular momentum vector:
$$\hat {\boldsymbol z} = \boldsymbol r \times \boldsymbol v / ||\boldsymbol r \times \boldsymbol v||$$.
• Construct $$\hat{\boldsymbol y}$$ as the unit vector that completes a right handed coordinate system:
$$\hat {\boldsymbol y} = \hat {\boldsymbol z} \times \hat{\boldsymbol x}$$.

Note very well: This construction is not unique. You will see some references having what I denoted as $$\hat{\boldsymbol x}$$ and $$\hat{\boldsymbol z}$$ negated. The ordering of and the names of the axes also vary from one reference to another. So take care when reading references on this subject.

A spacecraft-centered LVLH frame is both an accelerating and rotating frame of reference. This would appear to make the equations of motion rather complicated. The trick is to linearize the equations of motion for an object (typically called the chaser or chase vehicle) very close to the spacecraft (typically called the target vehicle.) Ignoring non-spherical gravity, drag, perturbations from other planets, and assuming the target vehicle is in a circular orbit, the linearized equations of motion for a chaser located at $$\boldsymbol r_{\text{rel}} = x\hat{\boldsymbol x} + y\hat{\boldsymbol y} + z\hat{\boldsymbol z}$$ become the Clohessy-Wiltshire equations (also here, derivation here), also known as Hill's equations,

\begin{aligned} \ddot x(t) &= \frac{F_x(t)}{m} + 3\omega^2 x(t) + 2\omega\,\dot y(t) \\ \ddot y(t) &= \frac{F_y(t)}{m} \phantom{3\omega^2 x(t)\ \quad} -2\omega\,\dot x(t) \\ \ddot z(t) &= \frac{F_z(t)}{m} -\phantom{3}\omega^2 z(t) \end{aligned} where $$\boldsymbol F(t) = F_x(t)\,\hat{\boldsymbol x} + F_y(t)\,\hat{\boldsymbol y} + F_z(t)\,\hat{\boldsymbol z}$$ is the time-varying thrust generated by the chaser.

• +1 I added the links; I'd never seen this before and found them helpful. – uhoh Jan 24 at 1:29