Position

The time derivative of the position is simply the velocity, rotated into the coordinate system of the position variables. For simulators that store the aircraft's CG position in local coordinates, the kinematic equations for position are:

   

$$\dot x_c$$ = C₁₁u + C₁₂v + C₁₃w (33)

$$\dot y_c$$ = C₂₁u + C₂₂v + C₂₃w (34)

$$\dot z_c$$ = C₃₁u + C₃₂v + C₃₃w (35)

Simulators that store position in a Cartesian coordinate system other than local coordinates substitute components of the appropriate rotation matrix into Equations 33-35.

Simulators that store aircraft position in geocentric spherical coordinates can calculate the spherical coordinate time derivatives using the using the local coordinate time derivatives as intermediate variables $^{\ref{ref:larcsim}}$:

   

$$\dot r -\dot z^L_c$$ = (36)

$$\dot\theta \dot y^L_c/(r \cos\phi')$$ = (37)

$$\dot\phi' \dot x^L_c/r$$ = (38)