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304 Autonomous Mobile Robots

˙
and its derivative ˜z(t) as well as the steering rate ω to zero, that is,
˙
lim ˜z(t) = lim ˜z(t) = lim ω = 0
t→∞ t→∞ t→∞
In addition to the conditions in Lemmas 8.1 and 8.2, the parameters l, p, and f
are necessary to satisfy lp > 0 and fv d > 0.
Proof When the leader vehicle moves on a straight path (γ d = 0) at a speed
v d = 0, its heading angle θ d will stay as constant ( ˙ θ d = 0),wehave

v d cos θ d T v d
˙ z d = = R (θ d ) (8.20)
v d sin θ d 0
We also define the following tracking errors:

˜ θ θ − θ d
˜ η = η − η d = = (8.21)
˜ γ γ − γ d
We now prove Lemma 8.3 in two steps. First, we prove that the convergence
of the tracking error ˜z(t) to zero implies that ˜η converges to zero, that is,

lim ˜η = lim ˜ θ = lim ˜γ = 0
t→∞ t→∞ t→∞

Second, we derive the necessary conditions of parameter l, p, and f for the
tracking stability of ˜η defined in (8.21).

1. The convergence of ˜z implies the convergence of ˜η.

γ
Suppose γ =˜ converges to γ d = 0, that is, the follower vehicle eventually
moves on a straight path. This implies θ converges to a constant and

1 0 v v
T T T
¯
lim ˙z = lim R (θ)E(γ )µ = lim R (θ) = R (θ)
t→∞ t→∞ t→∞ 0 lp ω 0
where we have used the assumption lim t→∞ ω = 0. Furthermore

v T v d
T
˙
lim ˜z = lim (˙z −˙z d ) = R (θ) − R (θ d )
t→∞ t→∞ 0 0

v − v d cos ˜ θ
T
= R (θ) = 0
v d sin ˜ θ
© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 304 — #10