motor model scratch work

Motors

We have the following constants:

• the motor resistance $R$
• the motor inductance $L$
• the constant $K_\tau$ relating torque to current
• the back-emf constant $K_m$ relating voltage to angular velocity

Let $e_a$ be the applied voltage, $e_m$ the back-emf voltage, $i$ the current, $\tau$ the torque, and $\theta$ the angle. Let $E_a$, $E_m$ $I$, $T$, and $\Theta$ be their Laplace transforms, with frequency parameter $s$. Then

$$I(s) = \frac{E_a(s) - E_m(s)}{Ls + R}$$ $$E_m(s) = K_m s\Theta(s)$$ $$T(s) = K_\tau I(s) = \frac{K_\tau(E_a(s) - E_m(s))}{Ls + R} = \frac{K_\tau}{Ls+R}E_a(s) - \frac{K_\tau K_ms}{Ls + R}\Theta(s)$$

To "close the loop" from $T(s)$ back to $\Theta(s)$ requires characterizing the load, which we do in the next section.

Parameter estimates

The motors we buy come with the following specifications:

• Nominal voltage $e_{nom}$
• No-load current $i_{free}$
• No-load speed $\omega_{free}$
• Stall current $i_{stall}$
• Stall torque $\tau_{stall}$

We can compute the constants in our model from these (except inductance, which hopefully is small):

$$R = e_{nom} / i_{stall}$$ $$K_\tau = \tau_{stall} / i_{stall}$$ $$K_m = e_{nom} / \omega_{free}$$

Mecanum Wheels

TODO: Read https://doi.org/10.1002/zamm.201900173.