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Unified Control Design for Autonomous Vehicle 301

and the intervehicular spacing d converges to fl, that is,

lim (d − fl) = 0
t→∞

Proof From the definition of the tracking error (8.6), we have

2 2 2
˜z = l + d − 2lfd cos( pγ − φ)
2 2
=[d − fl cos( pγ − φ)] +[ fl sin( pγ − φ)] (8.8)

Note that l is a finite constant, Statement 1 has an equivalent statement as
follows

lim t→∞ sin(φ − pγ) = 0 (i)
lim ˜z(t) = 0 ⇔ (8.9)
t→∞ lim t→∞ [d − fl cos(φ − pγ)]= 0 (ii)
(a) If Statement 2 is true, it is easy to check that Statement 2 ensures
both (8.9-i) and (8.9-ii) satisfied. Hence, lim t→∞ ˜z(t) = 0, that is,
Statement 1 is true.
(b) If Statement 1 is true, we now prove that Statement 2 is true. Since
|γ |≤ γ max and |p| <π/(2γ max ),we have |pγ | <π/2. Further-
more, the relative orientation angle φ is also bounded, |φ|≤ π/2.
Thus, we have

|pγ − φ|≤|pγ |+|φ| <π

As a result, (8.9-i) leads to

lim (φ − pγ) = 0 (8.10)
t→∞
Combining (8.10) with (8.9-ii) produces

lim (d − fl) = 0 (8.11)
t→∞
Lemma 8.1 implies that a control law that ensures the convergence of the
tracking error ˜z(t) can guarantee that the intervehicular spacing ultimately
converges to the desired distance |l|. In practice, sensing range is limited,
0 < d < d max and parameter l must be chosen such that fl lies in the valid
range of the sensor

0 < fl < d max (8.12)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 301 — #7