====== Motor ======

===== DC motor model =====

$$ u = Ri + L \frac{di}{dt} + E $$

$$ i = \frac{1}{K_t} \tau_m $$
$$ E = K_t \omega_m $$

Where:

  * $u$ is the voltage applied to the armature of the motor. $[V]$
  * $R$ is the armature resistance. $[\Omega]$
  * $E$ is the back EMF. $[V]$
  * $K_t$ is the torque constant. $[Nm/A]$ or $[Vs]$ 
  * $\omega_m$ is the angular velocity of the motor. $[\mathrm{rad/s}]$
  * $\tau_m$ is the motor torque. $[Nm]$

This can be rewritten as:

$$ \frac{K_t}{R} u = \tau_m + T_e \frac{d\tau_m}{dt} + \frac{K_t^2}{R} \omega_m $$