In a previous Flight Vehicle Dynamics tutorial (Traditional Aircraft Axes and Orbital Mechanics - Orbit Parameters), we had talked about traditional aircraft (AC) axes such as body, stability, and wind coordinate systems, and we had briefly talked about Earth-centric Earth-fixed (ECEF) as well as Earth centric inertial (ECI) coordinate systems. Now we will continue the discussion of coordinate systems even further and do some recap.
The first coordinate system we will talk about is North-West-Up, or NWU. This system has the x-axis pointing to the North, the y-axis pointing to the West, and the z-axis pointing away from the surface of the Earth.
Fig. 1 - NWU Coordinate System Graphic
Now in NWU, there are two important degrees needed to find the velocity vector of a craft: $ \psi $ which is the heading from North and $ \gamma $ which is the flight path angle. This gives
[ 1 ]
$$ v_{NED} = \begin{Bmatrix} v_x \\ v_y \\ v_z \end{Bmatrix} $$ $$ = \begin{Bmatrix} v \cos \gamma \cos \psi \\ -v \cos \gamma \sin \psi \\ v \sin \gamma \end{Bmatrix} $$
We can use Figure 2 below to help us visualize this.
Fig. 2 - NWU coordinate system visualization.
North-East-Down (NED) is similar to NWU in that the x direction points North, but it differs because the y-axis points to the East and the z-axis point down towards Earth.
Fig. 3 - NED Coordinate System Graphic
NED uses a very similar vector compared to NWU, except since the y and z-axes are flipped by 180°, the vectors will only different by a negative sign for the y and z components.
[ 2 ]
$$ v_{NED} = \begin{Bmatrix} v_x \\ v_y \\ v_z \end{Bmatrix} $$ $$ = \begin{Bmatrix} v \cos \gamma \cos \psi \\ v \cos \gamma \sin \psi \\ -v \sin \gamma \end{Bmatrix} $$
We can visualize the vector for NED using Figure 4.
Fig. 4 - NED coordinate system visualization.
Earth-centric Earth-fixed (ECEF) is a non-inertial coordinate system because it rotates with the Earth. The z-axis points along the rotating axis of the Earth while the x-axis insects with Greenwich and the equatorial plane. Greenwich marks the Greenwich Meridian at 0° longitude.
Earth centric inertial (ECI) on the otherhand is an inertial coordinate system. The z-axis points along the rotating axis of the Earth and the x-axis points to the Vernal equinox. This coordinate system stays fixed because it is aligned with distant stars that do not move significantly from Earth's perspective view.
You can think of these coordinate systems as centered at the same origin, Earth, but ECEF rotates with the Greenwich Meridian and ECI remains fixed in the same spot. Both of these coordinate system as well as NED, NWU, and the traditional AC axes are very important for future analysis and orbital maneuvers!