Page 259 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 245

ˆ
ˆ
In addition, b m , b c , and b g are the estimates of constants b , b , and b ,
ˆ
∗
∗
∗
m c g
respectively, which are defined by

∗
∗
b = max{b } > 0, b ∗ } (6.38)
m m ij m ij = max{w m ij , c m ij , σ m ij
i,j

∗
∗
b = max{b } > 0, ∗ } (6.39)
c c ij b = max{w c ij , c c ij , σ c ij
c ij
i,j

b = max{b } > 0, ∗ } (6.40)
∗
∗
g g i b = max{w g i , c g i , σ g i
g i
i
are known positive functions defined by
and ¯ φ m ij , ¯ φ c ij , and ¯ φ g i
T T
= 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 (6.41)
ˆ
ˆ
¯ φ m ij
c m ij σ m ij c m ij σ m ij
T T
= S ˆ ˆ + S ˆ ˆ + S ˆ + S ˆ (6.42)
¯ φ c ij W c ij W c ij ˆ c c ij ˆ σ c ij + n rc ij
c c ij σ c ij c c ij σ c ij
T T
ˆ + S
ˆ ˆ ˆ ˆ ˆ ˆ
= S W g i + S W g i + S c g i ˆ σ g i + n rg i (6.43)
¯ φ g i
c g i σ g i c g i σ g i
Using the “GL” matrix (denoted by upright and bold symbol with curly
bracket) and operator (denoted by “•”) introduced in Reference 32, the function
emulators (6.35)–(6.37) can be collectively expressed as
∗ T
M(q) =[{W } •{S M }] + E M (6.44)
M
∗ T
C(q, ˙q) =[{W } •{S C }] + E C (6.45)
C
∗ T
G(q) =[{W } •{S G }] + E G (6.46)
G
∗
where [{W }, {S M }], [{W }, {S C }], and [{W }, {S G }] are the desired weights
∗
∗
M C G
and basis function GL matrices pairs of the NF emulation of M(q), C(q, ˙q),
and G(q), respectively; and E M , E C , E G are the collective NF reconstruction
errors, respectively.
ˆ
ˆ
The estimates M(q), C(q, ˙q), G(q), can, accordingly, be expressed as
ˆ
T
ˆ
ˆ
ˆ
M(q) =[{W M } •{S M }] (6.47)
T ˆ
ˆ
ˆ
C(q, ˙q) =[{W C } •{S C }] (6.48)
T
ˆ
ˆ
ˆ
G(q) =[{W G } •{S G }] (6.49)
Note that in real implementation, the actual control torque τ must be
provided rather than Bτ given in (6.34). There are various approaches available
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 245 — #17