Page 260 - Autonomous Mobile Robots

246 Autonomous Mobile Robots

in the literature to solve τ from (6.34), either analytically or numerically. In this
chapter, the following scheme is applied to compute the control torque τ with
rigor and rationality.
Define

u = Bτ (6.50)

T
Premultiplying both sides of (6.50) by R , we obtain

T
T
R u = R Bτ
T
FromAssumption6.3, itisknown that R B isnonsingular. Thus, τ isobtained as

T
R u
τ = (R B ) −1 T (6.51)
Substituting (6.51) and (6.47)–(6.49) into the dynamic equation (6.33) yields
the closed-loop system error equation as

T
∗ T
ˆ
ˆ
M ˙σ + Cσ = ([{W M } •{S M }]−[{W } •{S M }])˙ν
M
T
∗ T
ˆ
ˆ
+ ([{W C } •{S C }]−[{W } •{S C }])ν
C
T ˆ ∗ T
+ ([{W G } •{S G }]−[{W } •{S G }])
ˆ
G
T
− K σ σ + J λ − E − K s sgn(σ)
n n n n n

ˆ ˆ ˆ
− b m ¯ φ m ij |σ i ˙ν j |− b c ¯ φ c ij |σ i ν j |− b g ¯ φ g i |σ i |
i=1 j=1 i=1 j=1 i=1
(6.52)
where E = E M ˙ν + E C ν + E G − τ d .
The stability of the closed-loop system will be illustrated in the following
theorem.
Theorem 6.1 Consider the nonholonomic mobile robot system described by
dynamic equation (6.1) and the (n−m) independent nonholonomic constraints
(6.2). If the control law is chosen by (6.34), and the parameter adaptation laws

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FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 246 — #18