THE FLIGHT PATH
To track the flight path relative to FE, we need the velocity components in the directions of the axes of Fe. These we get by transforming the velocity vector Уд into V); as shown in Appendix A.4.5
Here Leb is the matrix of direction cosines that corresponds to the reverse of the sequence of rotations given above, which are for a transformation from FE to FB. Thus
Leb = Ьг(-ЧГ)Ц(-@)Ьх(-Ф) (4.4,2)
where Lv, Ly, Lz are respectively L„ L2, L3 of Appendix A.4. Using the rotation matrices given in Appendix A.4, and carrying out the multiplication, we get the final result (4.4,3).
sin ф sin 9 cos ф — cos ф sin ф cos ф sin 9 cos ф + sin ф sin ф
sin ф sin 9 sin ф + cos ф cos ф cos ф sin 9 sin ф — sin ф cos ф
sin Ф cos 9 cos ф cos 9
The differential equations for the coordinates of the flight path are then
The position of the vehicle CG is obtained by integrating the preceding equation.
“See. 5.2 of Etkin (1972).
“Appendix E of ANSI/AIAA (1992).
5Recall that the superscript signifies velocity relative to FE, and that the subscript identifies the reference frame in which the components are given.