Page 319 - Autonomous Mobile Robots

Unified Control Design for Autonomous Vehicle 307

If both eigenvalues λ 1 and λ 2 are real numbers, the conditions for them to
be negative real numbers are

λ 1 + λ 2
≥ 0 and λ 1 λ 2 > 0 and < 0
2
Since Lemma 8.1 implies fl > 0, which lead to (((1 + f )/2)a + fl)> 0, the
above conditions are equivalent to

(((1 + f )/2)a + l) 2
0 < lp ≤ and fv d > 0
4a

Likewise, if both eigenvalues are a pair of complex conjugates, then the
conditions are

λ 1 + λ 2
< 0 and < 0
2

which lead to

(((1 + f )/2)a + l) 2
lp > > 0 and fv d > 0
4a

In summary, matrix A(v d ) has both eigenvalues with negative real part
iif lp > 0 and fv d > 0. Under these conditions, the system (8.22) is uni-
formly asymptotically stable. In other words, the tracking error ˜η converges
to zero.

The condition fv d > 0 in Lemma 8.3 implies that vehicle-following man-
euver is feasible and successful only if the leader vehicle moves forward
(v d > 0) in the look-ahead tracking mode ( f = 1) and moves backward
(v d < 0) in the look-behind tracking mode ( f =−1). This condition is sat-
isfied automatically based on the formulations of the forward tracking and
backward tracking defined earlier in Section 8.2.2. Condition lp > 0 implies
that the vehicle tracking must be in the formations defined in Figure 8.1
and Figure 8.2 for forward tracking and backward tracking, respectively.
In other words, look-ahead control can only be used for forward tracking
formation and look-behind control can only be used for backward tracking
control.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 307 — #13