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244 Autonomous Mobile Robots

From (6.26), (6.28), and (6.29), we have

σ + ν = R˙z (6.30)

The time derivatives of ν and σ are given by

˙ ν = R˙z r + R¨z r −˙µ (6.31)
˙
˙
˙ σ = R˙z + R¨z −˙ν (6.32)

From the dynamic equation (6.8) together with (6.30) and (6.32), we have

T
M(q)˙σ + C(q, ˙q)σ + M(q)˙ν + C(q, ˙q)ν + G(q) = B(q)τ + J (q)λ + τ d
(6.33)

Consider the control law as

T
T
ˆ
ˆ
Bτ = M(q)˙ν + C(q, ˙q)ν + G(q) − K σ σ − J λ d + k λ J e λ − K s sgn(σ)
ˆ
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
(6.34)
where matrix K σ > 0, constant k λ > 0, matrix K s = diag{k sii } with k sii ≥|E i |
ˆ
ˆ
and E i is the element of vector E (defined later), M(q), C(q, ˙q), and G(q) are
ˆ
the estimates of M(q), C(q, ˙q), and G(q), respectively, the elements of which,
that is, m ij (q), c ij (q, ˙q), and g i (q) can be expressed by NF networks as
T
∗ ∗ ∗
m ij (q) = W S(q, c , σ ) + m ij (q) (6.35)
m ij m ij m ij
T
∗ ∗ ∗
c ij (q, ˙q) = W S(q, ˙q, c , σ ) + c ij (q, ˙q) (6.36)
c ij c ij c ij
T
∗ ∗ ∗
g i (q) = W S(q, c , σ ) + g i (q) (6.37)
g i g i g i
∗
∗
∗
where W , W , W are ideal constant weight vectors, c , c , c are the
∗
∗
∗
m ij c ij g i m ij c ij g i
∗
∗
∗
ideal constant center vectors, σ , σ , σ are the ideal constant width vectors,
m ij c ij g i
(q) are the approximation errors.
and m ij (q), c ij (q, ˙q), g i
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 244 — #16