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300 Autonomous Mobile Robots

P b P r = l in backward tracking). The focus point P r has a directional angle
defined by the longitudinal axis of the vehicle (P b P f ) and the focus line P f P r
(in the forward tracking as shown in Figure 8.1) or P b d (in the backwardP
tracking as shown in Figure 8.2). It is p times as much as the steering angle γ .
This focus point P r , in both cases, can be expressed with respect to P b as follows

1+f
 
x + a cos θ + l cos(θ + pγ)
2
P r = z =   (8.5)
1+f
y + a sin θ + l sin(θ + pγ)
2
where l and p are two system parameters that will affect the performance of the
vehicle tracking system. The focus point as defined in (8.5) can be viewed as a
nonlinear output function of the posture of the follower vehicle.
An output tracking error can be defined as the difference between the output
of the follower vehicle (8.5) and the virtual intervehicular connection (8.4) as
follows

l cos(θ + pγ) − fd cos(θ + φ)
˜ z = z − z d =
l sin(θ + pγ) − fd sin(θ + φ)
l cos pγ − fd cos φ
T
= R (θ) (8.6)
l sin pγ − fd sin φ
where R(θ) is a standard rotation matrix of θ as follows

cos θ sin θ
R(θ) = (8.7)
− sin θ cos θ
Lemma 8.1 Consider a car-like vehicle with restricted steering angle, |γ |≤
γ max <π/2, and a vehicle tracking problem formulated as forward tracking
or backward tracking. If the parameter p is chosen such that |p| <π/(2γ max )
and l is chosen as a finite constant, then the following two statements are
equivalent:

1. The vehicle tracking error converges to zero, that is,

lim ˜z(t) = 0
t→∞

2. The relative orientation angle φ converges to pγ , that is,

lim (φ − pγ) = 0
t→∞

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 300 — #6