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282 Autonomous Mobile Robots

The time derivative of V 2 is given by

m−1
1 1

˙
˙ z
V 2 (Z 2 ) = z j,2 ˙z j,2 + z j,3 ˙z j,3 + ··· + z j,n j j,n j (7.51)
n j −2
ρ j,1
j=1 i=1 ρ j,i
Substituting (7.25) into (7.51), we have
m−1
1 1
˙
V 2 = z j,2 u 1 z j,3 − z j,3 ρ j,1 u 1 z j,2 + z j,3 u 1 z j,4 + ···
ρ j,1 ρ j,1
j=1
1 1
− n j −3 z j,n j −1 ρ j,n j −3 u 1 z j,n j −2 + nj−3 z j,n j −1 u 1 z j,n j
i=1 ρ j,i i=1 ρ j,i
1
u
+ z j,n j (L h1 z j,n j 1 + u j+1 )
n j −2
i=1 ρ j,i
m−1
1

= z j,n j ((ρ j,n j −2 z j,n j −1 + L h1 z j,n j )u 1 + u j+1 ) (7.52)
n j −2
j=1 i=1 ρ j,i
The time derivative of V 3 is given by

T
T
T
T
˜
˜
˙
V 3 =− e K I e I + e LI d + e RI + e K a Q
˜ ˙
I I I I
r r
T −1 ˙ −1 ˙
˜ ˜
˜ ˜
+ e H + γ L L i L i + γ R R i R i
I
i=1 i=1
r r
−1 ˙ −1 ˙
˜ ˜
˜
˜
+ γ K ai K ai + γ K Ni K Ni (7.53)
a N
i=1 i=1
For stability analysis, let us consider the following Lyapunov function
candidate:
V = V 1 + V 2 + V 3 (7.54)
Combining Equation (7.50), Equation (7.52), and Equation (7.53), and
using the adaptation laws (7.35), (7.40), (7.42) to (7.44), the derivative of V
can be obtained as
m−1
1
˙ 2 T T T
V =− z −˜u S K e S 3 ˜u − e K I e I (7.55)
n j −2 k u j+1 j,n j 3 I
j=1 i=1 ρ j,i
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 282 — #16