Page 268 - Autonomous Mobile Robots

254 Autonomous Mobile Robots

T
T
T T
σ = s R + µ . Thus, we have
T T T T T T T
σ J λ + ρ 3 µ (1 + k λ ) ˙µ =−ρ 2 ρ 3 µ µ + s R J λ (6.77)
T T
Noting R J = 0 from (6.4), we then have
T
T
V ≤−σ K σ σ − ρ 2 ρ 3 µ µ ≤ 0 (6.78)
˙
As V ≥ 0 and V ≤ 0, V ∈ L ∞ . From the definition of V, it follows that σ,
˙
n
n i
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
µ ∈ L , W Mi , W Ci , W Gi , C Mi , C Ci , C Gi , Mi , Ci , Gi ∈ L ∞ , i = 1, ... , n
∞
ˆ ˆ
ˆ
with n i denoting the compatible size of the vectors, and b m , b c , b g ∈ L ∞ .
Integrating both sides of (6.78), we have
t
T
σ K σ σ ≤ V(0) − V(t) ≤ V(0) (6.79)
0
n
Hence σ ∈ L .
2
T
R (σ − µ), hence s ∈ L
From (6.28), we have s = (R R) −1 T m since R is
∞
m
bounded. From Lemma 6.2, it can be concluded that e z , ˙e z ∈ L .
∞
From (6.27), (6.29), and (6.31), we have
ˆ ˙
ˆ
ˆ
ˆ
ˆ
M ˙ν + Cν + G = M(R˙z r + R¨z r −˙µ) + C(R˙z r − µ) + G
ˆ
T
ˆ
ˆ
ˆ
ˆ
ˆ ˙
= M(R˙z r + R¨z r ) + CR˙z r + G − Cµ + ρ −1 MJ λ (6.80)
3
From (6.26), it is known that
˙ q = R˙z r + Rs (6.81)
˙
˙
¨ q = R˙z r + R¨z r + Rs + R˙s (6.82)
T
Replacing τ by (6.51) in dynamic equation (6.1) by noting f = J (q)λ,
Equations (6.80)–(6.82), the closed-loop system becomes
MRs + MR˙s + CRs − (M − M)(R˙z r + R¨z r ) − (C − C)R˙z r − (G − G)
ˆ
˙
˙
ˆ
ˆ
n n

ˆ ˆ
+ Cµ + K σ σ + K s sgn(σ) + b m ¯ φ m ij |σ i ˙ν j |
i=1 j=1
n n n

ˆ ˆ
+ b c ¯ φ c ij |σ i ν j |+ b g ¯ φ g i |σ i |− τ d
i=1 j=1 i=1
T
ˆ
= (ρ −1 M + I n )J λ (6.83)
3
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 254 — #26