Page 294 - Autonomous Mobile Robots

Adaptive Control of Mobile Robots 281

Denote L = L − L, R = R − R, K a = K a − K a , and K N = K N − K N .
ˆ ˜
ˆ
˜
ˆ
ˆ ˜
˜
Substituting (7.38) into (7.27), we have the error dynamics for the actuator
dynamic subsystem
˜
L˙e I =−K I e I + LI d + RI + K a Q + H (7.46)
˜
˜ ˙
The closed-loop stability is summarized in Theorem 7.2.
Theorem 7.2 For a nonholonomic system described by (7.25)–(7.27), using
the control law (7.41) with the virtual control (7.33) and the parameter adapta-
tion laws (7.35), (7.40), (7.42)–(7.44), Z is globally asymptotically stabilizable
at the origin Z = 0.

Proof For the convenience of proof, define the following three functions:

r
1 T 1 T −1 1 −1 2
V 1 = ˜u M 4 (Z)˜u + θ ˜ θ + γ N K Ni K NInvi (7.47)
˜
˜
2 2 2
i=1
m−1
1 1 1
2 2 2
V 2 = z j,2 + z j,3 + ··· + z (7.48)
2 ρ j,1 n j −2 j,n j
j=1 i=1 ρ j,i
r r
1 T 1 −1 2 1 −1 2
˜
˜
V 3 = e Le I + γ L L + γ R R i
I
i
2 2 2
i=1 i=1
r r
1 −1 2 1 −1 2
˜
˜
+ γ a K + γ N K Ni (7.49)
ai
2 2
i=1 i=1
˙ ˙
where ˜ θ = ˆ θ − θ. Since θ is a constant vector, we have ˆ θ = ˜ θ
The derivative of V 1 along Equation (7.45) is given as
1 T T −1 ˙
T
˙
˙
V 1 =˜u M 4 (Z)˜u + u M 4 (Z)˜u + ˜ θ ˜ θ
˙
˜
2
T
T
T −1 ˙
T
T
=˜u (S 1 θ − S K e S 3 ˜u − + S B 3 K N K NInv τ md ) + ˜ θ ˜ θ
˜
˜
3 3 3
r
−1 ˙ T T
˜
˜
+ γ K Ni K NInvi K NInvi +˜u S B 3 K N e I (7.50)
N 3
i=1
T
where the property that M 4 − 2S C 3 is skew-symmetric has been used.
˙
3
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 281 — #15