Page 315 - Autonomous Mobile Robots

Unified Control Design for Autonomous Vehicle 303

¯
determinant of matrix E(γ ) being nonzero, that is,
1 + f
¯
det(E) = lp cos pγ + lp tan γ sin pγ = 0 (8.16)
2
Since f only takes two values −1 or 1, we have the following equality:

1 + f
1 + f
tan γ = tan γ (8.17)
2 2

and (8.16) becomes
1 + f

det(E) = lp cos pγ + lp tan γ sin pγ
¯
2
cos[(p − ((1 + f )/2))γ ]
= lp = 0 (8.18)
cos((1 + f /2)γ )
Condition (8.18) is satisfied if the following two conditions are satisfied:


1. lp = 0



 π

2. p − |γ |≤ p − max <
 1 + f 1 + f
γ

2 2 2
(8.19)


 π

 1 + f
⇒ p − <



2 2γ max

For practical car-like wheeled mobile robots, the steering γ is restricted
by |γ |≤ γ max <π/2. Condition 1 in Lemma 8.2 requires (1) (l = 0),
that is, the focus point P r cannot be fixed at the front center point P f of the
follower vehicle in forward tracking or at the back point P b in backward track-
ing; and (2) (p = 0), that is, P r cannot be fixed on the longitudinal center
axis. Condition 2 in Lemma 8.2 indicates that the selectable range of parameter
p is bounded.
Lemmas 8.1 and 8.2 provide some sufficient conditions in choosing the
design parameters l and p. It can be expected that vehicle tracking stability
require more conditions on l and p. By examining the basic maneuvers, we can
gain some insights and necessary conditions on l and p for tracking stability.
Vehicle tracking along a straight path is a basic maneuver and its requirement
on stability will offer some insight and a set of necessary conditions. In the
following Lemma 8.3, a set of such conditions are derived for this purpose.
Lemma 8.3 Consider a basic maneuver of vehicle following along a straight
path (γ d = 0) at a speed (v d = 0). Suppose there exists a feedback vehicle-
following controller that guarantees the convergence of the tracking error ˜z(t)
© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 303 — #9