Fix LuGre passivity in DC motor doc

[giusenso]

Correct the LuGre passivity description in the DC motor technical documentation.
Since tau is defined as the torque applied to the mechanism (eq. 9b of the pdf), the passive resisting effort is -tau. The documented map is therefore corrected from omega -> tau to omega -> -tau.

[giusenso]

For a more formal proof

Proof

The document defines the torque applied to the mechanism as

$$ \tau=-(\sigma_0z+\sigma_1\dot z+\sigma_2\omega). $$

Let $F=-\tau$ be the resisting friction effort and

$$ W(z)=\frac{\sigma_0}{2}z^2 $$

the stored bristle energy. Using

$$ \dot z=\omega-\sigma_0\frac{|\omega|}{g(\omega)}z, $$

we obtain

$$ F\omega-\dot W = \sigma_2\omega^2 \quad+\sigma_1\omega^2 \left(1-\frac{\sigma_0z,\mathrm{sgn}(\omega)}{g(\omega)}\right)\quad+\frac{|\omega|}{g(\omega)}(\sigma_0z)^2. $$

For the standard LuGre domain, $\sigma_1\ge0$, $\tau_s\ge\tau_c>0$, $|\sigma_0z|\le\tau_s$, and $\tau_c\le g(\omega)\le\tau_s$. Therefore,

$$ F\omega-\dot W \ge \left( \sigma_2-\sigma_1\frac{\tau_s-\tau_c}{\tau_c} \right)\omega^2. $$

If

$$ \rho= \sigma_2-\sigma_1\frac{\tau_s-\tau_c}{\tau_c}>0, $$

then

$$ \dot W\le(-\tau)\omega-\rho\omega^2, $$

which is the input-strict passivity inequality for the map
$\omega\mapsto-\tau$.