Using Rotations to Build Aerospace Coordinate Systems - Defence ...

DSTO–TN–0640ZNDEN 0E 0D 0 N 1E 1ωαYXD 1Figure 5: Constructing local north–east–down axes N, E, D at a given pointof known latitude and longitude. Start with the three vectors N 0 , E 0 , D 0 .Construct the intermediate set N 1 , E 1 , D 1 by rotating the original setthrough the longitude ω about N 0 . Then rotate N 1 , E 1 , D 1 about −E 1 by thelatitude α, according to the right hand rule, to give the final set N, E, D. Orequivalently, rotate the intermediate set about E 1 through minus the latitude.The resulting vectors are the local north–east–down axesthe act of rotating each of the vectors that represent the axes of some frame.There are two basic tasks to consider in this type of aerospace calculation. First, givena place on Earth (usually in lat–long–height coordinates), we need to construct the localgeographic frame’s axes, such as NED. Second, an aircraft can be introduced into thisNED frame, and its orientation found relative to that frame.4.1 Constructing NED Axes for the Local Geographic FrameGiven the lat–long–height of a place on or near Earth, the first thing we wish to do is findthe local NED axes. The technique used is the primary one of this report. We build aninitial set of NED axes in a place where it’s simple to determine them, and then we rotatethese around to the required place.The simplest place to construct an initial set of NED axes is on the junction of theEquator and the prime meridian, since this has a latitude and longitude of α = ω = 0 ◦ .Represent each axis by a unit vector in the ECEF frame, which all calculations are beingdone within:⎡ ⎤0⎡ ⎤0⎡ ⎤−1N 0 = ⎣0⎦ , E 0 = ⎣1⎦ , D 0 = ⎣ 0⎦ . (4.1)1 0 0The scenario is shown in Figure 5. Now rotate each of the N 0 , E 0 , D 0 vectors in two steps.12