**Velocity Relations:**

**Linear Velocity:**

The true airspeed of unmanned aerial vehicle assumed as V_{t}, which is calculated by air data system represents the magnitude of flight path vector relative to surrounding air mass. Flight Path vector aligned with wind coordinate system X axis. In the absence of surrounding air mass movement, the true airspeed can be directly to ground speed, which is assumed as V_{g. }Opposite it, if air mass is moving, the calculation of ground speed will include the velocity of air mass with respect to ground, which is assumed as V_{w}.

**Relationship between air vehicle velocity in inertial frame and air vehicle velocity relative to wind:**

Where T^{IW }is:

Hence:

From Equation 1

Actual climb rate of air vehicle, h is simply the negative of the vertical component of velocity:

h = -V_{z}

So from the equation no 1:

h = – [V_{t}(-SθCαCβ+SфCθSβ+CфCθSαCβ) + V_{wz}]

h = V_{t}(SθCαCβ-SфCθSβ-CфCθSαCβ) – V_{wz}

h = V_{t}[Cβ(SθCα-CфCθSα) – SфCθSβ] – V_{wz}

h = h_{c }T_{a/}T_{asd = } V_{t}[Cβ(SθCα-CфCθSα) – SфCθSβ] – V_{wz}

Where h_{c }refers to measure climb rate from air data system, T_{a }refers to actual temperature at the test altitude, T_{asd }refers to standard day ambient temperature corresponding to test altitude.

Now more simplified equation:

h = h_{c }T_{a/}T_{asd = } V_{t}[Cβ(SθCα-CфCθSα) – SфCθSβ] – V_{wz}

_{ = } V_{t}Sinγ_{Air mass} – V_{wz } Equation 2

Where γ_{Air mass }refers to flight path vector relative to air mass. If no movement in surrounding air mass then Equation 2 can be:

_{ = } V_{g}Sinγ_{inertial}

Where γ_{inertial }refers to air vehicle flight path vector relative to inertial frame.

**Angular Velocity:**

Now, this is time to formulate angular velocity. Let’s consider two frames, first one inertial frame and second one body frame. Three rotations occur through the Euler angles in a complete process. Angular rate related to each rotation can be added in vector form.

Hence, Angular velocity of the body axis with respect to inertial frame is:

Ω = Ψ_{ω} K + θ_{ω} J_{1 }+ ф_{ω} I_{2}

Nothing that:

K = -sinθ i + cosθ sinф j + cosθ cosф k

J_{1 }= J_{2 }= cosф j – sinф k

I_{2 } = i

Where p = dф /dt , q = dθ /dt, r = dΨ /dt

Inverting Equation 1, Yielding the equations of motion relates the rate of change air vehicle orientations to body rotational rates:

** ф _{ω} = p + (q sinф + r cosф ) tanθ **

_{ θω }**= q cosф – r sinф**

_{ Ψω }**= (q sinф _{ }+ r cosф ) secθ**

**Note: **Please refer coordinate system transformation article for vectors formation.