Page 225 - Autonomous Mobile Robots

210 Autonomous Mobile Robots

and

b 2 =[0 1 0 ··· 0] (5.28)

is stabilizable with the second control input u 2 . Therefore, recalling the results
established in Section 5.3, we give the following statement.

Proposition 5.2 [20] The discontinuous control law

 

−kx 1
u 1
u = =  x n−1  (5.29)
x n
p 2 x 2 + p 3 x 3 + ··· + p n−1 n−3 + p n n−2
u 2
x 1 x 1 x 1
with k > 0 and p =[0, p 2 , p 3 , ... , p n−1 , p n ] such that the eigenvalues of the
matrix A + b 2 p have all negative real part, almost exponentially stabilizes the
n
system (5.22) in the open and dense set 1 ={x ∈ R |x 1 = 0}.
Remark 5.9 If we rewrite the control law (5.29) as

 

−kx 1
u 1
u = =  p 3 p n−1 p n 
p 2 x 2 + x 3 + ··· + x x
u 2 n−3 n−1 + n−2 n
x 1 x 1 x 1
we can regard it as a linear control law with state dependent gains.

5.5.1 An Example: A Car-Like Vehicle
In this section we consider the problem of designing a discontinuous controller
for a prototypical nonholonomic system: a car-like vehicle. For simplicity we
consider an ideal system, that is, the wheels roll without slipping and all pairs
of wheels are perfectly aligned and with the same radius. A thorough analysis
of the phenomena caused by nonideal wheels can be found in Reference 51.
The problem of stabilizing a car-like vehicle has been addressed with different
techniques by several authors, see References 5 and 7 for open loop strategies
and References 9, 12, and 52, for state feedback control laws. In what follows,
exploiting the results in Section 5.3, we design a discontinuous state feedback
controller. This control law, because of its singularity, is not directly implement-
able. However, as discussed in Section 5.3, and in Reference 33 and 53, and
in Section 5.7, it is possible to build modifications yielding uniform ultimate
boundedness or (robust) exponential stability.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 210 — #24