Aircraft Kinematic Relations
This section derives the kinematic equations of aircraft motion used in flight dynamics and control. See Elkin  and Nelson  for further details.
9.6.1 Aircraft position rate
The rate of the position RO in Earth axes equals its velocity Ue in Earth axes.
^Ro = Ue (9.10)
In practice this is expressed as
since the velocity is defined as Ub in the body axes. Integration of (9.11) to obtain the aircraft trajectory RO(t) therefore requires not only the aircraft velocity Ub(t), but also the concurrent Euler angles ф, в, фр)
which are needed to compute the transformation matrix Tb(t) at each step in time.
9.6.2 Aircraft orientation rate