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256 Autonomous Mobile Robots

n
n−m
As λ ∈ L ∞ and µ ∈ L , from Equation (6.27), it is obvious that
∞
n
n
˙ µ ∈ L . Thus, from (6.31), we have ˙ν ∈ L . Since ˙z, ¨z ∈ L m have been
∞ ∞ ∞
n
established before, we can conclude from (6.32) that ˙σ ∈ L . Now, with
∞
n
n
σ, µ ∈ L , ˙σ, ˙µ ∈ L , according to Lemma 6.3, we can conclude that σ and µ
2 ∞
asymptotically converge to zero. Hence, from (6.28), it can be concluded that
s → 0as t →∞. According to Lemma 6.3, we can also obtain e z , ˙e z → 0
as t →∞.
n
Since ˙q, ¨q ∈ L , q and ˙q are uniformly continuous. Therefore, from
∞
Property 6.1, we can conclude that matrices M(q), C(q, ˙q), G(q), S(q), J(q),
D(q), C(q, ˙q), and G(q) are uniformly continuous.
ˆ
ˆ
ˆ
Remark 6.5 If Bτ is directly replaced by (6.34) in the dynamic equation (6.1)
without considering the real implementation issue, a wrong conclusion may be
drawn.
Substituting (6.34) and (6.47) to (6.49) into the dynamic equation (6.33)
yields the closed-loop system error equation as
T
∗ T
ˆ
ˆ
M ˙σ + Cσ = ([{W M } •{S M }]−[{W } •{S M }])˙ν
M
∗ T
T
ˆ
ˆ
+ ([{W C } •{S C }]−[{W } •{S C }])ν
C
T
∗ T
ˆ
ˆ
+ ([{W G } •{S G }]−[{W } •{S G }])
G
T
− K σ σ + (1 + k λ )J e λ − E − K s sgn(σ)
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
(6.86)
which is misleading as it seems there is control effort applied to force error
e λ and the wrong conclusion of asymptotic convergence of e λ may be drawn.
T T
This is due to the ignorance of the inherent property R J = 0. Thus, for the
proposed scheme in this chapter, one can only guarantee the boundedness of
e λ , which will be confirmed in the simulation study.
6.5 SIMULATION STUDIES
Consideramobilerobot moving on ahorizontalplane, driven by two rearwheels
mounted on the same axis, and having one front passive wheel. The dynamic

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