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

DSTO–TN–0640The vector giving the position of B relative to A can be written r B←A, and the followingare all true in general, as well as being useful for writing down vector relationships withoutneeding to draw pictures:r B←A= −r A←Br B←A= r B←C− r A←Cr B←A= r B←C+ r C←A(4.4)The cities’ positions are:Adelaide: latitude α = −34.9 ◦ , longitude ω = 138.5 ◦ ,Brussels: latitude α = 50.8 ◦ , longitude ω = 4.3 ◦ .Use (2.2) to find the position of each city relative to Earth’s centre (which is exactly whatthe ECEF coordinates specify): set C = Earth’s centre.⎡ ⎤⎡ ⎤−3.924.03r A←C= ⎣ 3.47⎦ × 10 6 metres, r B←C= ⎣0.30⎦ × 10 6 metres. (4.5)−3.634.92The position of Brussels relative to Adelaide is then⎡ ⎤7.95r B←A= r B←C− r A←C= ⎣−3.17⎦ × 10 6 metres. (4.6)8.55This vector is still an ECEF vector, although we can consider it to start at Adelaide andpoint to Brussels. To calculate the bearing of Brussels as seen from Adelaide, we needto get the local north and east components of r B←A. To do this, we’ll first calculateAdelaide’s local north, east, and down axis vectors, labelled N, E, D in Figure 5, all inECEF coordinates, using (4.2). For this we need R N0(ω), constructed from (3.7):⎡ ⎤⎡⎤0−0.74896 −0.66262 0N 0 = ⎣0⎦ , ω = 138.5 ◦ , R N0(ω) ≃ ⎣ 0.66262 −0.74896 0⎦ . (4.7)10 0 1This R N0(ω) then acts in (4.2) to produce N 1 , E 1 , D 1 . We then use −E 1 to construct thenext rotation matrix R −E1(α), which finally produces the N, E, D. The steps are easy toprogramme and are omitted; the resulting axes are⎡ ⎤ ⎡ ⎤ ⎡ ⎤−0.429−0.663N = ⎣ 0.379⎦ , E = ⎣−0.749⎦ , D = ⎣0.820 00.614−0.543⎦ . (4.8)0.572The bearing of Brussels is calculated by projecting r B←Ainto the local north–eastplane. The projection can be visualised as an arrow drawn on the ground, so that itsangle from north is the required bearing. The projection in the local north–east plane issimply given by the components of r B←Aon the local north and east axes. These are dotproducts:North component = r B←A·N = 2.4035 × 10 6 metres,East component = r B←A·E = −2.8958 × 10 6 metres. (4.9)14