Page 263 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 249

and D Mu =[d muij ] with

T T
∗
∗
∗
ˆ
ˆ
|d muij |≤ c · S ˆ W m ij + σ · S ˆ W m ij + W · S ˆ ˆ c m ij
m ij c m ij m ij σ m ij m ij c m ij
+ W · S ˆ ˆ σ m ij + W 1 (6.67)
∗
∗
m ij σ m ij m ij
Noting Assumption 6.2, the following can be obtained:
T T
∗
∗
∗
ˆ
c · S ˆ W m ij + σ · S ˆ W m ij + W · S ˆ ˆ c m ij
ˆ
m ij c m ij m ij σ m ij m ij c m ij
∗
∗
+ W · S ˆ ˆ σ m ij + W 1
m ij σ m ij m ij
T T
S ˆ ˆ S ˆ ˆ
≤ c m ij W m ij + σ m ij W m ij + w m ij
c m ij σ m ij

× ( S ˆ ˆ c m ij + S ˆ ˆ σ m ij + n rm ij )
c m ij σ m ij
T T
∗
ˆ
ˆ
≤ b ( S ˆ W m ij + S ˆ W m ij + S ˆ ˆ c m ij + S ˆ ˆ σ m ij + n rm ij )
m ij c m ij σ m ij c m ij σ m ij
∗ ¯
= b φ m ij (6.68)
m ij
Thus, (6.66) becomes
T T ˆ ∗ T
σ ([{W M } •{S M }]−[{W } •{S M }])˙ν
ˆ
M
T T T
˜
ˆ
ˆ
ˆ
˜
= σ ([{W M } • ({S M }−{S Mc }−{S Mσ })]+[{C M } •{SW Mc }]

n n
T
˜
+[{ M } •{SW Mσ }])˙ν + σ i d uij ˙ν j

i=1 j=1
T ˜ T ˆ ˆ ˆ ˜ T
≤ σ ([{W M } • ({S M }−{S Mc }−{S Mσ })]+[{C M } •{SW Mc }]

n n

T
∗
¯
˜
+[{ M } •{SW Mσ }])˙ν + b φ m ij |σ i ˙ν j |

m ij
i=1 j=1
T T T
ˆ
˜
ˆ
˜
ˆ
≤ σ ([{W M } • ({S M }−{S Mc }−{S Mσ })]+[{C M } •{SW Mc }]

n n
T
+[{ M } •{SW Mσ }])˙ν + b ∗ |σ i ˙ν j | (6.69)
˜
m ¯ φ m ij

i=1 j=1
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 249 — #21