Page 264 - Autonomous Mobile Robots

250 Autonomous Mobile Robots

Similarly, we have the following inequalities for other approximation
errors as

T
T
∗ T
σ ([{W C } •{S C }]−[{W } •{S C }])ν
ˆ
ˆ
C
T T T
ˆ
˜
˜
ˆ
ˆ
≤ σ ([{W C } • ({S C }−{S Cc }−{S Cσ })]+[{C C } •{SW Cc }]

n n

T
+[{ C } •{SW Cσ }])ν + b ∗ c ¯ φ c ij |σ i ν j | (6.70)
˜

i=1 j=1
T T ˆ ∗ T
σ ([{W G } •{S G }]−[{W } •{S G }])
ˆ
G
T T T
ˆ
ˆ
˜
˜
ˆ
≤ σ ([{W G } • ({S G }−{S Gc }−{S Gσ })]+[{C G } •{SW Gc }]

n
T
˜
+[{ G } •{SW Gσ }]) + b ∗ |σ i | (6.71)

g ¯ φ g i
i=1
ˆ
ˆ
where definition for GL matrices {S Cc }, {S Cσ }, {SW Cc }, {SW Cσ }, {S Gc },
ˆ

ˆ
{S Gσ }, {SW Gc }, and {SW Gσ }, which is omitted here for conciseness, can be

similarly made.
Consider the Lyapunov function candidate
n n n
1 T 1 −1 1 −1 1
˜ T
˜ T
˜ T
˜
˜
V = σ Mσ + W W Mi + W W Ci + W Gi
Ci Ci
Mi Mi
2 2 2 2
i=1 i=1 i=1
n n
1 1
−1 ˜ T −1 ˜ ˜ T −1 ˜
× Gi W Gi + C Mi C Mi + C Ci C Ci
˜
Mi
Ci
2 2
i=1 i=1
n n n
1 1 T −1 1 T −1
˜ T
˜
˜
˜
˜
+ C −1 ˜ Mi Mi + Ci Ci
C Gi +
Ci
Mi
Gi
Gi
2 2 2
i=1 i=1 i=1
n
1 T −1 1 −1 ˜ 2 1 −1 ˜ 2 1 −1 ˜ 2 1 T
˜
˜
b + γ
+ Gi Gi + γ bm m bc b + γ bg b + ρ 3 µ µ
g
c
Gi
2 2 2 2 2
i=1
(6.72)
∗
with (·) = (·) − (·) .
ˆ
˜
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 250 — #22