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236 Autonomous Mobile Robots

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=1
such that the distributions

i
i
G j = span{ad f 2 , ... ,ad f m :0 ≤ i ≤ j}, 0 ≤ j ≤ n − 1
f 1 f 1
have constant dimension on U and are all involutive, and G n−1 has dimension
n − 1 on U [38,40].

For a two-input controllable system, a constructive method was reproduced
in Reference 10 and it is given here for completeness. Consider

˙ q = r 1 (q)˙z 1 + r 2 (q)˙z 2 (6.12)

n
where r 1 (q), r 2 (q) are linearly independent and smooth, q ∈ R , and ˙z =
T
[˙z 1 , ˙z 2 ] .
Define

0 := span{r 1 , r 2 ,ad r 1 2 , ... ,ad n−2 r 2 }
r
r 1
n−2
r
1 := span{r 2 ,ad r 1 2 , ... ,ad r 2 }
r 1
r
2 := span{r 2 ,ad r 1 2 , ... ,ad n−3 r 2 }
r 1
n n
If 0 (q) = R , ∀q ∈ U (where U is some open set of R ), 1 and 2
are involutive on U, and r 1 (q) satisfies [r 1 , 1 ]⊂ 1 , then there exist two
independent functions h 1 : U → R and h 2 : U → R which satisfy the following
relationships:

dh 1 · 1 = 0, dh 1 · r 1 = 1
n−2
dh 2 · 2 = 0, dh 2 · ad r 2 = 0
r 1

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 236 — #8