Modelling of Rotor drag dependency on quadrotor velocity and thrust

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\dot{p} = v

\dot{R} = R\hat{\omega}

m\bold{\dot{v}} = -mg\bold{z_{W}}+TR\bold{e_3}+R\bold{F_a}

I\dot{\omega} = -\omega \times I\omega +\tau_{g} + \tau + \tau_{a}

F_{a} = \sum_{i=1}^{n}F_{a}^{i} = \sum_{i=1}^{n}-\sqrt{T_i}c_{d1}\pi_{e3}V_{a}^{i}

V_{a}^{i} = R^{T}v + \omega \times d_{i}, \pi_{e3} = I - e_{3}e_{3}^{T}, T_i = k_T \omega_i^{2}

\sqrt{T_i}

Current models

T_i = T_h = \frac{mg}{n}

F_{a} = \sum_{i=1}^{n}F_{a}^{i} = \sum_{i=1}^{4}-\sqrt{\frac{mg}{4}}c_{d1}\pi_{e3}V_{a}^{i}

F_{a} = -4 \sqrt{\frac{mg}{4}}c_{d1}\pi_{e3}V_{a}^{i}  (4 rotors)

Final translation dynamics:

m\bold{\dot{v}} = -mg\bold{z_{W}}+TR\bold{e_3}-RAR^{T}\bold{v}

A = 4\sqrt{\frac{mg}{4}}c_{d1}\pi_{e3}

Linearization

T_i = T_h

T_i > T_h for example T_i = 2T_h

k,q = argmin_{k,q}( \sqrt(T_i) - y), a<T_i<b

y = kT_i +q

Finally translation dynamics:

m\dot{v} = -mgz_{w}+TRe_3 -kTRBR^{T}v +nqRBR^{T}v

B = c_{d1}\pi_{e3}

 

Angular velocity model

t_s = 10s

t_r = 0.3s

Mp = 0.065

Transfer function:

H(s) = \frac{36}{s^{2} + 9s + 36}

\ddot{\omega_x} + 9\dot{\omega_x} + 36\omega_x = 36u

x = [\omega_x, \dot{\omega_x}]^{T}

\dot{x} = Ax + Bu

u = -Kx

 

Desired acceleration vector (for control)

Basic drone model

\bold{\dot{a_{des}}} = \bold{a_{fb}} + \bold{a_{ref}} +g\bold{z_{W}}

\bold{a_{fb}} = Kp(\bold{p} - \bold{p_{ref}}) + Kd(\bold{v} - \bold{v_{ref}})

With velocity dependent rotor drag

\bold{\dot{a_{des}}} = \bold{a_{fb}} + \bold{a_{ref}} -\bold{a_{drag}}+g\bold{z_{W}}

\bold{a_{drag}}= R_{ref}AR_{ref}^{T}\bold{v_{ref}}

 

 

 

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