Example:
Car steering
The steering system of a car can be modeled as
Figure 1: The geometry of the carlike robot, with position (x,y), orientation and steering angle . 

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(1) 
where x and y are cartesian coordinates of the middle point on the rear axle, is the orientation angle, v is the longitudinal velocity measured at that point, l is the distance of the two axles, and f is the steering angle. In this case v and f are the two controls.
Let us reduce the complexity by defining u_{1}=v, u_{2}=v/l tan f, then


(2) 
Sometimes, this is called a unicycle model. If we linearize (2) around a point (x_{0}, y_{0},q_{0}), we have


(3) 
which is not controllable. However, using geometric tools we will show in Chapter 8 that the nonlinear system (2) is controllable (This is what you, as a driver, expected, right?).