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274 Autonomous Mobile Robots

where

M 2 (X) = M(q)| −1
q=T 1 (X)
S 2 (X) = S(q)T 2 (q)| −1
q=T (X)
1
˙
˙
C 2 (X, X) = C 1 (q, ˙q)T 2 + M(q)S(q)T 2 (q)| −1
q=T (X)
1
G 2 (X) = G(q)| −1
q=T (X)
1
B 2 (X) = B(q)| −1
q=T (X)
1
J 2 (X) = J(q)| −1
q=T (X)
1
The actuator dynamics is transformed to
dI
L + RI + K a Q(u, µ, X) = ν (7.17)
dt
where
Q = µT 2 (q)u| −1
q=T (X)
1
Next, letusfurthertransformthechainedformintoskew-symmetricchained
form for the convenience of controller design. This transformation is the simple
extension of the transformation of the one-generation, two-inputs, single-
chained system given by Samson [18]. As shown in References 18, 23, and
24 by introducing the skew-symmetric chained form, via Lyapunov-like ana-
T
lysis, it is easier to design U 2 =[u 2 , ... , u m ] and a time-varying control u 1
T
to globally stabilize [x 1 , X 2 , ... , X m ] of the kinematic subsystem, as will be
detailed later.
The kinematic model of chained form (7.13) can be equivalently written as

m

˙
X = h 1 (X)u 1 + h 2,j u j = h 1 (X)u 1 + h 2 U 2 (7.18)
j=2
where

T
,0]
h 1 (X) =[1, x 1,3 , ... , x 1,n1 ,0, ... , x m−1,3 , ... , x m−1,n m−1
T
h 2 =[h 2,2 , ... , h 2,m ]
j

and h 2,j , j = 2, ... , m is an n-dimensional vector with the 1 + (n i − 1)th
i=1
element being 1 and other elements being zero.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 274 — #8