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252 Autonomous Mobile Robots

˙
T ˙
As matrix M − 2C is skew-symmetric, σ (M − 2C)σ = 0, ∀x = 0.
Noting that
 ˜ T ˆ 
{W M1 } •{S M1 }˙ν
T ˜ T ˆ  . 
σ [{W M } •{S M }]˙ν = σ 1 σ 2 ··· σ n  . . 
T
{W Mn } •{S Mn }˙ν
ˆ
˜
n
T
˜
ˆ
= {W Mi } •{S Mi }˙νσ i (6.74)
i=1
and similarly
T T
ˆ
ˆ
˜
ˆ
σ [{W M } • ({S M }−{S Mc }−{S Mσ })]˙ν
n
T
ˆ
ˆ
˜
ˆ
= {W Mi } • ({S Mi }−{S Mci }−{S Mσi })˙νσ i
i=1
n
T ˜ T ˜ T
σ [{C M } •{SW Mc }]˙ν = {C Mi } •{SW Mci }˙νσ i

i=1
n
T T T
˜
˜
σ [{ M } •{SW Mσ }]˙ν = { Mi } •{SW Mσi }˙νσ i

i=1
T T
ˆ
ˆ
˜
ˆ
σ [{W C } • ({S C }−{S Cc }−{S Cσ })]ν
n
T
ˆ
ˆ
ˆ
˜
= {W Ci } • ({S Ci }−{S Cci }−{S Cσi })νσ i
i=1
n
T ˜ T ˜ T
σ [{C C } •{SW Cc }]ν = {C Ci } •{SW Cci }νσ i

i=1
n
T T T
˜
˜
σ [{ C } •{SW Cσ }]ν = { Ci } •{SW Cσi }νσ i

i=1
n
T T
ˆ
ˆ
˜ T ˆ
ˆ
ˆ
ˆ
˜
σ [{W G } • ({S G }−{S Gc }−{S Gσ })]= W (S Gi − S Gci − S Gσi )σ i
Gi
i=1
n
T ˜ T
σ [{C G } •{SW Gc }] = ˜ T
C SW Gci σ i
Gi

i=1
n
T T T
˜
˜
σ [{ G } •{SW Gσ }] = SW Gσi σ i

Gi
i=1
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 252 — #24