General Velocity Relations : U.A.V

Velocity Relations:

Linear Velocity:

The true airspeed of unmanned aerial vehicle assumed as Vt, 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 Vg. 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 Vw.


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


Where TIW is:



From Equation 1


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

h = -Vz

So from the equation no 1:

h = – [Vt(-SθCαCβ+SфCθSβ+CфCθSαCβ) + Vwz]

h =    Vt(SθCαCβ-SфCθSβ-CфCθSαCβ) – Vwz

h =    Vt[Cβ(SθCα-CфCθSα) – SфCθSβ] – Vwz

h =   hc Ta/Tasd  =   Vt[Cβ(SθCα-CфCθSα) – SфCθSβ] – Vwz

Where hc refers to measure climb rate from air data system, Ta refers to actual temperature at the test altitude, Tasd refers to standard day ambient temperature corresponding to test altitude.

Now more simplified equation:

h =   hc Ta/Tasd  =   Vt[Cβ(SθCα-CфCθSα) – SфCθSβ] – Vwz

                                    =   VtSinγAir mass – Vwz                            Equation 2

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

                                    =   VgSinγinertial

Where γinertial 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 = Ψω K  +  θω J1 + фω I2

Nothing that:

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

J1 = J2 = cosф j – sinф k

I2  =  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:

 фω =    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.

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