Page 269 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 255

Invoking (6.44)–(6.46) and (6.47)–(6.49), Equation (6.83) then becomes

T
∗ T
ˆ
MRs + MR˙s + CRs − ([{W M } •{S M }]−[{W } •{S M }])(R˙z r + R¨z r )
˙
˙
ˆ
M
T ∗ T
ˆ ˆ
− ([{W C } •{S C }]−[{W } •{S C }])R˙z r
C
∗ T
T
ˆ
− ([{W G } •{S G }]−[{W } •{S G }]) + Cµ
ˆ
ˆ
G
n n n n

ˆ ˆ
+ K σ σ + K s sgn(σ) + b m ¯ φ m ij |σ i ˙ν j |+ b c ¯ φ c ij |σ i ν j |
i=1 j=1 i=1 j=1
n

ˆ ˙
+ b g ¯ φ g i |σ i |+ E M (R˙z r + R¨z r ) + E C R˙z r + E G − τ d
i=1
−1 T
ˆ
= (ρ M + I n )J λ (6.84)
3
Since M(q) is nonsingular, multiplying J(q)M −1 (q) on both sides of (6.84)
yields

T
∗ T
ˆ
ˆ
JRs + JM −1 CRs − ([{W M } •{S M }]−[{W } •{S M }])(R˙z r + R¨z r )
˙
˙
M
T ∗ T
ˆ ˆ
− ([{W C } •{S C }]−[{W } •{S C }])R˙z r
C
T
∗ T
ˆ
ˆ
− ([{W G } •{S G }]−[{W } •{S G }]) + Cµ
ˆ
G
n n n n

ˆ ˆ
+ K σ σ + K s sgn(σ) + b m ¯ φ m ij |σ i ˙ν j |+ b c ¯ φ c ij |σ i ν j |
i=1 j=1 i=1 j=1
n !

ˆ ˙
+ b g ¯ φ g i |σ i |+ E M (R˙z r + R¨z r ) + E C R˙z r + E G − τ d
i=1
−1 −1 T
= JM (ρ M + I n )J λ (6.85)
ˆ
3
m
Since we have established that e z , ˙e z ∈ L , from Assumption 6.4 and
∞
m
(6.24), it can be concluded that ˙z r (t), ¨z r (t) ∈ L .As r is shown to be
∞
n
bounded, so is ˙z from (6.26). Hence, ˙q(t) = R˙z(t) ∈ L . It follows that
∞
n
ˆ
M(q), M(q), C(q, ˙q), C(q, ˙q) ∈ L n×n , and G(q), G(q) ∈ L . Thus, the left
ˆ
ˆ
∞ ∞
hand side of (6.85) is bounded. In fact, ρ 3 can be properly chosen to keep
−1
ˆ
(ρ M + I n ) on the right hand side of (6.85) from being singular. Hence, we
3
have λ ∈ L n−m .As λ d is bounded, so are e λ and Bτ.
∞
n
From (6.1), we can conclude that ¨q ∈ L .
∞
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 255 — #27