General Velocity Relations : U.A.V

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_tSinγ_{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_gSinγ_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 J₁ + ф_ω 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:

 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.