Page 226 - Autonomous Mobile Robots

Stabilization of Nonholonomic Systems 211

The kinematic model of a car with rear tires aligned with the car and front
tires allowed to spin about the vertical axis [7] is

˙ x = cos θv 1
˙ y = sin θv 1
(5.30)
1
˙ θ = tan θv 1
l
˙ φ = v 2

where (x, y) denotes the location of the center of the axle between the two rear
wheels, θ the angle of the car body with respect to the x-axis, φ the steering
angle with respect to the car body, and v 1 and v 2 the forward velocity of the rear
wheels and the velocity of the steering wheels, respectively (see Figure 5.2).
Applying the control transformation

 
u 1
v 1
cos θ
 
v 2 3 2 2 3
− sin φ tan θ sec θu 1 + l cos φ cos θu 2
l

Y f

l

u
y

O x X

FIGURE 5.2 Model of an automobile.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 211 — #25