Page 227 - Autonomous Mobile Robots

212 Autonomous Mobile Robots

and the σ process

ξ 1 = x
1 3
ξ 2 = sec θ tan φ
l
tan θ
ξ 3 =
x
y
ξ 4 =
x 2

we obtain a system described by equations of the form

˙ ξ 1 = u 1
˙ ξ 2 = u 2
ξ 2 − ξ 3
˙ ξ 3 = u 1
ξ 1
ξ 3 − 2ξ 4
˙ ξ 4 = u 1
ξ 1

thatis, byequationsoftheform(5.25)withn = 4. Thus, usingProposition(5.2),
we design the state feedback control law

u 1 −kξ 1
=
u 2 p 2 ξ 2 + p 3 ξ 3 + p 4 ξ 4

In the original coordinates the feedback law is described by

x
v 1 =−k
cos θ
3 2 x
v 2 = k sin φ tan θ sec θ
l cos θ

1 tan θ y
2 3 3
+ l cos φ cos θ p 2 sec θ tan φ + p 3 + p 4 2
l x x

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 212 — #26