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Adaptive Control of Mobile Robots 273

where s i (q) are smooth, linearly independent input vector fields. There exist
state transformation X = T 1 (q) and feedback v = T 2 (q)u on some open set
n
U ⊂ R to transform the system into an (m−1)-chain, single-generator chained
form, if and only if there exists a basis f 1 , ... , f m for
0 := span{s 1 , ... , s m }
which has the form

n

i
f 1 = (∂/∂q 1 ) + f (q)∂/∂q i
1
i=2
n
i
f j = f (q)∂/∂q i , 2 ≤ j ≤ m
j
i=2
such that the distributions

i i
G j = span{ad f 2 , ... ,ad f m :0 ≤ i ≤ j},
f 1 f 1
0 ≤ j ≤ n − 1

have constant dimension on U, are all involutive, and G n−1 has dimension n−1
on U [9,34].

Using the constructive method given in Reference 14, a two input
controllable system, that is,

˙ q = s 1 (q)v 1 + s 2 (q)v 2 (7.14)
n
T
where s 1 (q), s 2 (q) are linearly independent and smooth, q ∈ R , v =[v 1 , v 2 ] ,
can be transformed into chained form (7.13) as

˙ x 1 = u 1
˙ x 2 = u 2
˙ x 3 = x 2 u 1 (7.15)
.
.
.
˙ x n = x n−1 u 1

Under Assumption 7.1, that is, the existence of transformations X =
T 1 (q), v = T 2 (q)u, dynamic subsystem (7.9) is correspondingly converted into

T
M 2 (X)S 2 (X)˙u + C 2 (X, X)u + G 2 (X) = B 2 (X)K N I + J (X)λ (7.16)
˙
2

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 273 — #7