Page 314 - Autonomous Mobile Robots

302 Autonomous Mobile Robots

Condition (8.12) shows that l must be positive in the look-ahead tracking,
that is, f = 1, and negative in the look-behind tracking, that is, f =−1.
To ensure robust and reliable performance, fl should be chosen well away
from the boundaries, 0 |l| d max , so that d can be effectively kept within
the valid range of the sensor. Equation (8.10) gives an interpretation of the
parameter p. At steady state, φ = pγ , and p is a multiplier relating the steering
angle, γ , of the following vehicle and the relative orientation angle, φ.
To obtain the dynamic relationship between the output function z(t) and the
control input µ, take time derivative of (8.5)

∂z ∂z
˙ z = ˙ q = Gµ = E(θ, γ)µ (8.13)
∂q ∂q

where

T
E(θ, γ) = R (θ)E(γ ) (8.14)
¯
with

 
l
1 − tan γ sin pγ −lp sin pγ
 a 
¯   (8.15)

E(γ ) = 
 1 + f l 

+ cos pγ tan γ lp cos pγ
2 a
To ensure the existence of a feedback control, the matrix E(θ, γ) has to
be nonsingular and the following lemma presents such a set of sufficient
conditions.
Lemma 8.2 Consider a car-like vehicle with restricted steering angle, |γ |≤
γ max <π/2, and a vehicle tracking problem formulated as the forward
tracking or the backward tracking. A control input µ exists for (8.13) if the
design parameters l and p are chosen so that the following two conditions are
satisfied:

1. lp = 0

π
1 + f
2. p − <
2 2γ max

Proof The existence of the input µ is guaranteed iff the matrix E(θ, γ)
or, equivalently, the matrix E(γ ) is nonsingular. This is equivalent to the
¯

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 302 — #8