Page 325 - Autonomous Mobile Robots

Unified Control Design for Autonomous Vehicle 313

1 1 2 2
1 + f
u s =− ( ¨ θ + 2ξλ( ˙ θ + pω)) + (( ˙ θ + pω) − λ ) tan p − γ
p p 2
cos(((1 + f )/2)γ ) l 2 2
− (( ˙ θ + pω) + λ ) tan γ
p cos[(p − ((1 + f )/2))γ ] a

sin(φ − ((1 + f )/2)γ ) tan γ
+ f − cos(pγ − φ)
l cos(((1 + f )/2)γ ) a
2
2
¨
˙
×[d + 2ξλd + λ d − d( ˙ θ + ˙ φ) ]

cos(φ − ((1 + f )/2)γ ) tan γ
+ f − sin(pγ − φ)
l cos(((1 + f )/2)γ ) a

˙
×[d( ¨ θ + ¨ φ) + 2(d + ξλd)( ˙ θ + ˙ φ)] (8.57)
The above development can be summarized as follows.
Theorem 8.2 Consider the car-like mobile robot performing forward track-
ing, shown in Figure 8.1, and backward tracking, shown in Figure 8.2. The
dynamic motion of these tracking maneuvers is defined collectively as the
dynamics (8.3) of both vehicles and the virtual intervehicular connection (8.4).
Define the tracking error ˜z in (8.5) as the difference between the output of
the follower vehicle (8.5) and the virtual intervehicular connection (8.4). The
tracking target performance in ˜z is defined by the stable second-order system
(8.42) and can be ensured if the nonlinear controls (8.56) for driving and (8.57)
for steering are applied and the following necessary conditions are satisfied.

• Forward tracking: f = 1, λ> 0, ξ> 0, l and p satisfying (8.39)
• Backward tracking: f =−1, λ> 0, ξ> 0, l and p satisfying (8.40)

Proof The target performance (8.42) also guarantees that the tracking error
˙
¨
˜ z(t) and its derivatives ˜z(t) and ˜z(t) are all convergent to zero. Combining
all the conditions from Lemmas 8.1, 8.2, and 8.3, we can obtain the similar
conditions (8.39) for the look-ahead tracking and (8.40) for the look-behind
tracking.

8.3.3 Requirement of Measurements

As stated in the development of the kinematics- and dynamics-based control-
lers, besides the vehicular state feedbacks such as velocity/acceleration and
steering angle, some measurements are required including relative distance
¨
between two vehicles d, velocity d, acceleration d, and relative angle φ as well
˙

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 313 — #19