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 $\theta $and steering angle $\phi$.

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 \begin{displaymath} \begin{array} {rcl} \dot{x} &=& v \cos (\theta) \\ \dot... ...heta) \\ \dot{\theta} &=& \frac{v}{l} \tan \phi,\\ \end{array}\end{displaymath}

(1)

where x and y are cartesian coordinates of the middle point on the rear axle, $\theta $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

 

 \begin{displaymath} \begin{array} {rcl} \dot{x} &=& \cos (\theta)u_1 \\ \dot{y} &=& \sin (\theta)u_1 \\ \dot{\theta} &=& u_2.\\ \end{array}\end{displaymath}

(2)

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

 

 \begin{displaymath} \begin{array} {rcl} \dot{x} &=& \cos (\theta_0)u_1 \\ \d... ... &=& \sin (\theta_0)u_1 \\ \dot{\theta} &=& u_2,\\ \end{array}\end{displaymath}

(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?).