13.2.4.3 Smooth differential drive

A second-order differential drive model can be made by defining actions $u_l$ and $u_r$ that accelerate the motors, instead of directly setting their velocities. Let $\omega_l$ and $\omega_r$ denote the left and right motor angular velocities, respectively. The resulting state transition equation is

$${\dot x} = \frac{r}{2} (\omega_l + \omega_r) \cos \theta \qquad {\dot \omega}_l = u_l$$

$${\dot y} = \frac{r}{2} (\omega_l + \omega_r) \sin \theta \qquad {\dot \omega}_r = u_r$$ (13.49)

$${\dot \theta} = \frac{r}{L} (\omega_r - \omega_l).$$

In summary, an important technique for making existing models somewhat more realistic is to insert one or more integrators in front of any action variables. The dimension of the phase space increases with the introduction of each integrator. A single integrator forces an original action to become continuous over time. If the new action is bounded, then the rate of change of the original action is bounded in places where it is differentiable (it is Lipschitz in general, as expressed in (8.16)). Using a double integrator, the original action is forced to be $C^1$ smooth. Chaining more integrators on an action variable further constrains its values. In general, $k$ integrators can be chained in front of an original action to force it to be $C^{k-1}$ smooth and respect Lipschitz bounds.

One important limitation, however, is that to make realistic models, other variables may depend on the new phase variables. For example, if the simple car is traveling fast, then we should not be able to turn as sharply as in the case of a slow-moving car (think about how sharply you can turn the wheel while parallel parking in comparison to driving on the highway). The development of better differential models ultimately requires careful consideration of mechanics. This provides motivation for Sections 13.3 and 13.4.