Example: Car steering

The steering system of a car can be modeled as

 Figure 1: The geometry of the car-like robot, with position (x,y), orientation and steering angle . <>

 (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 u1=v, u2=v/l tan f, then

 (2)

Sometimes, this is called a unicycle model. If we linearize (2) around a point (x0, y0,q0), 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?).