Single Camera Structure and Motion Estimation - Visual Servoing via Advanced Numerical Methods - page 218

Choosing gains ρ 3 ( t ), ρ 4 ( t ), ρ 5 ( t ) as given by (12.56), (12.57) and (12.58) yields

V

≤− ρ 1 e 1 − ρ 2 e 2 − ρ 7 e 3 − ρ 8 e 4 − ρ 9 e 5 − ρ 6 e 6 +

Π (12.77)

where ρ i ∈ R , ∀ i = { 7 , 8 , 9 } are strictly positive constants, and Π ( t ) ∈ R is

Π ( t ) ( e 1 − ρ 4 e 4 ) Ψ 1 +( e 2 − ρ 5 e 5 ) Ψ 2 +( y 2 y 3 e 3 + y 2 u 1 e 4 + y 2 u 2 e 5 ) ˜ ω 1

− ( y 1 y 3 e 3 + y 1 u 1 e 4 + y 1 u 2 e 5 ) ˜ ω 2 . (12.78)

Thus, (12.77) can be upper bounded as

V

≤− λ V + Π . (12.79)

0. Taking the time derivative of V ( e ) and utilizing
(12.64), (12.65), (12.69)-(12.71), (12.74) yields

Case 2: y 3 ( t ) > y 3 and φ ( t ) >

V = e 1 ( e 4

− y 1 b 3 e 3 − ρ

1 e 1 +

Ψ

1 )+ e 2 ( e 5

− y 2 b 3 e 3 − ρ

2 e 2 +

Ψ

2 )+

α e 3 (12.80)

e 3 + e 4 [ b 3 u 1 e 3 + y 3 b 3 e 4 +( y 2 ω 1 −

y 3 − y 3

− φ y 1 ω 2 ) e 4 + y 2 u 1 ˜ ω 1

δ

− y 1 u 1 ˜ ω 2 + u 1 J 4 e 4 + q ( u 1 ) e 4 − ρ 4 ( e 4 + Ψ 1 ) − e 1 ]

+ e 5 [ b 3 u 2 e 3 + y 3 b 3 e 5 +( y 2 ω 1 − y 1 ω 2 ) e 5 + y 2 u 2 ˜ ω 1 − y 1 u 2 ˜ ω 2

+ u 2 J 5 e 5 + q ( u 2 ) e 5 − ρ 5 ( e 5 + Ψ 2 ) − e 2 ]

+ e 6 ( c 1 b 3 e 3 − ρ 6 e 6 ) .

0and y 3 − y 3
δ

Since 0, the bracketed term in (12.80) is strictly

positive, and the expression for V ( t ) can be upper bounded as in (12.77).
Case 3: y 3 ( t )

φ ( t ) > 0, e 3 ( t ) < <

0. Taking the time derivative of V ( e ) and utilizing
(12.64), (12.65), (12.69)-(12.71), (12.75) yields

< y 3 and φ ( t ) <

V = e 1 ( e 4 −

y 1 b 3 e 3 − ρ 1 e 1 + Ψ 1 )+ e 2 ( e 5 − y 2 b 3 e 3 − ρ 2 e 2 + Ψ 2 )+ α e 3

e 3 + e 4 [ b 3 u 1 e 3 + y 3 b 3 e 4 +( y 2 ω 1 −

y 3 − y 3

− φ y 1 ω 2 ) e 4 + y 2 u 1 ˜ ω 1

δ

− y 1 u 1 ˜ ω 2 + u 1 J 4 e 4 + q ( u 1 ) e 4 − ρ 4 ( e 4 + Ψ 1 ) − e 1 ]

+ e 5 [ b 3 u 2 e 3 + y 3 b 3 e 5 +( y 2 ω 1 − y 1 ω 2 ) e 5 + y 2 u 2 ˜ ω 1 − y 1 u 2 ˜ ω 2

+ u 2 J 5 e 5 + q ( u 2 ) e 5 − ρ 5 ( e 5 + Ψ 2 ) − e 2 ]

+ e 6 ( c 1 b 3 e 3 − ρ 6 e 6 ) .

0and y 3 − y 3
δ

Since φ ( t ) < 0, e 3 ( t ) > < 0, the bracketed term in (12.80) is strictly

positive, and the expression for ·

V ( t ) can be upper bounded as in (12.77).
The results in Section 12.4.2.1 indicate that

˜

ω ( t ) → 0as t → ∞ , and hence,

Ψ 1 ( t ) , Ψ 2 ( t ) , Π ( t ) → 0as t → ∞ . Thus, the expression (12.79) is a linear system