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Unified Control Design for Autonomous Vehicle 305

implies lim t→∞ θ = lim t→∞ (θ −θ d ) = 0, given the constraint | ˜ θ| <π/2, and
˜
lim t→∞ (v − v d ) = 0. This shows lim t→∞ ˜η = 0.
On the other hand, suppose that γ diverges from γ d = 0. The assumption
lim t→∞ ω = 0 implies lim t→∞ γ = c, with c being a nonzero angle. The
instantaneous turning radius of the follower vehicle

a
r =
tan γ

converges to a finite constant. The follower vehicle eventually moves on a
circular path while the leader vehicle moves on a straight path. This contra-
dicts the assumption of lim t→∞ ˜z = 0, or, equivalently, lim t→∞ d = fl,by
Lemma 8.1.
These show that lim t→∞ ˜z = 0 implies lim t→∞ ˜η = 0. The reverse
may not be true because two vehicles may be moving on two separate and
parallel straight paths (˜η = 0), whereas the tracking error ˜z is not zero.

2. Necessary conditions on l, p, and f for ˜η to converge to zero.

Since ˙ θ d = 0 and γ = γ d +˜γ =˜γ , the error ˜η is computed as follows:

v  1 
tan γ tan γ 0
˙ ˜ η =˙η −˙η d =˙η = a =  a  µ
ω 0 1
1
 
γ
tan ˜ 0
=  a  µ = Q( ˜γ)µ (8.22)
0 1
˙
When the convergence of ˜z(t) and ˜z(t) to zero are achieved, we have

˙
0 ≡ ˜z =˙z −˙z d = E(θ, γ)µ −˙z d
Since the matrix E is nonsingular (Lemma 8.2), we obtain

µ = E −1 (θ, γ)˙z d = E ¯ −1 (γ )R(θ)˙z d (8.23)

Noting ˙z d in (8.20) and γ =˜γ , (8.23) becomes

T
µ = E ¯ −1 ( ˜γ)R(θ)R (θ d ) v d = E ¯ −1 ( ˜γ)R( ˜ θ) v d (8.24)
0 0

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 305 — #11