Page 262 - Autonomous Mobile Robots

248 Autonomous Mobile Robots

1
T
Proof The time derivative of σ Mσ along (6.52) is
2
T T T T T T T T
σ M ˙σ =−σ K σ σ − σ E − σ K s sgn(σ) + σ J λ − σ Cσ
T
∗ T
T
ˆ
+ σ ([{W M } •{S M }]−[{W } •{S M }])˙ν
ˆ
M
T
T
∗ T
+ σ ([{W C } •{S C }]−[{W } •{S C }])ν
ˆ
ˆ
C
T T ∗ T
ˆ
ˆ
+ σ ([{W G } •{S G }]−[{W } •{S G }])
G
n n n n n

ˆ ˆ ˆ
− 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
(6.65)
Using the properties (6.17) and (6.18) given in Lemma 6.1, we have the
following property for the NF approximation error:
T T ∗ T
ˆ
ˆ
σ ([{W M } •{S M }]−[{W } •{S M }])˙ν
M
T T ˆ ˆ ˆ
˜
= σ ([{W M } • ({S M }−{S Mc }−{S Mσ })]
T
T
˜
+[{C M } •{SW Mc }]+[{ M } •{SW Mσ }] + D Mu )˙ν (6.66)
˜

ˆ
where GL matrices {S Mc }, {S Mσ }, {SW Mc }, {SW Mσ }, and matrix D Mu are
ˆ

defined respectively as:
ˆ ˆ
   
{S Mc1 } {S Mσ1 }
. .
   
ˆ
ˆ
{S Mc }= . . , {S Mσ }= . .
   
 ˆ   ˆ 
{S Mcn } {S Mσn }
   

{SW Mc1 } {SW Mσ1 }
. .
   
{SW Mc }= . . , {SW Mσ }= . .

   
   
{SW Mcn } {SW Mσn }

with

ˆ
ˆ
ˆ
{S Mci }={ ˆ S Mci1 ··· S Mcin }, S Mcij = S ˆ ˆ c m ij
c m ij

ˆ
ˆ
ˆ
{S Mσi }={ ˆ ··· S Mσin }, S Mσij = S ˆ ˆ σ m ij
S Mσi1
σ m ij
T
ˆ
{SW Mci }={SW Mci1 ··· SW Mcin }, SW Mcij = S ˆ W m ij

c m ij
T
ˆ
ˆ
{SW Mσi }={SW Mσi1 ··· SW Mσin }, SW Mσij = S W m ij

σ m ij
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 248 — #20