Page 267 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 253

Equation (6.73) becomes

1 T T T T T T T
˙
˙
V ≤ σ Mσ − σ Cσ − σ K σ σ − σ E − σ K s sgn(σ) + σ J λ
2
n
T
˜ ˆ ˆ ˆ
+ {W Mi } • ({S Mi }−{S Mci }−{S Mσi })˙νσ i
i=1
n n
T T
˜
˜
+ {C Mi } •{SW Mci }˙νσ i + { Mi } •{SW Mσi }˙νσ i

i=1 i=1
n
T
ˆ
ˆ
ˆ
˜
+ {W Ci } • ({S Ci }−{S Cci }−{S Cσi })νσ i
i=1
n n
T T
˜
˜
+ {C Ci } •{SW Cci }νσ i + { Ci } •{SW Cσi }νσ i

i=1 i=1
n n n
T
ˆ
ˆ
˜
˜ T ˆ
+ W (S Gi − S Gci − S Gσi )σ i + C SW Gci σ i + SW Gσi σ i
Gi
Gi
˜ T

Gi
i=1 i=1 i=1
n n n
˜ T
˜ T
−1 ˙ −1 ˙ −1 ˙
˜ T
ˆ
ˆ
+ W W Mi + W W Ci + W ˆ
Mi Mi Ci Ci Gi Gi W Gi
i=1 i=1 i=1
n n n
˜ T
−1 ˙ −1 ˙ −1 ˙
˜ T
˜ T
ˆ
ˆ
ˆ
+ C Mi C Mi + C Ci C Ci + C Gi C Gi
Ci
Mi
Gi
i=1 i=1 i=1
n n n
T −1 ˙ T −1 ˙ T −1 ˙
˜
˜
˜
+ ˆ Mi + ˆ Ci + ˆ Gi
Mi Mi Ci Ci Gi Gi
i=1 i=1 i=1
n n n n n

˜ ˜ ˜
− 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
˙ ˙ ˙ T
b c b c + γ
b m b m + γ
+ γ −1 ˜ ˆ −1 ˜ ˆ −1 ˜ ˆ (6.75)
b g b g + ρ 3 µ ˙µ
bm bc bg
Substituting the weight vectors updating laws (6.53)–(6.55), the center vectors
updating laws (6.56)–(6.58), the width vectors updating laws (6.59)–(6.61), and
the constant parameters updating laws (6.62)–(6.64) into (6.75) yields
T
T
T
T
T T
˙
V ≤−σ K σ σ − σ E − σ K s sgn(σ) + σ J λ + ρ 3 µ ˙µ (6.76)
T
T
Noting that k sii ≥|E i | > 0, it is obvious that [−σ E − σ K s sgn(σ)]≤ 0. In
J λ and from (6.28),
addition, from (6.27), we know that ˙µ =−ρ 2 µ − ρ −1 T
3
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 253 — #25