Multicriteria Analysis of Visual Servos through Rational Systems, Biquadratic Lyapunov Functions, and LMIs - Visual Servoing via Advanced Numerical Methods - page 176

˜

∀ s ∈ Ξ S , ∀ k ∈ Ξ K , ∀ ( x , χ ) ∈V ( X×X χ ) ,

(2η s , k − 1) +

Ψ 3 ( x Y s , k ≥ Y s , k Ψ 3 ( x , χ )+ , χ ) 0 , (10.47)

− η s , k Γ 1 a k

P s

˜

∀ s ∈ Ξ , ∀ j ∈ Ξ , ∀ ( x , χ ) ∈V ( X ×X χ ) ,

S J s

Ψ 4 , j ( x F s , j ≥

Q ( x , χ) ( P s ,... )+ F s , j Ψ 4 , j ( x , χ )+ , χ ) 0 , (10.48)

( ˜ ∀ s ∈ Ξ S , ∀ j ∈ Ξ J , ∀ ( x , χ ) ∈V X s ×X χ ) ,

Ψ 4 , j ( x G s , j ≥ R ( x , χ) ( P s ,... )+ G s , j Ψ 4 , j ( x , χ )+ , χ ) 0 , (10.49)

˜

∀ ( s a , s b ) ∈ Π , ∀ ( x , χ ) ∈V ( ∂ X s a s b ×X χ )

s a s b P s a −

M s a s b E

E

Ψ 1 ( x P s b + M Ψ

1 ( x

, χ )+ , χ )

s a s b =

O , (10.50)

s a s b

˜

∀ ( s a , s b ) ∈ Π , ∀ ( x , χ ) ∈V ( ∂ X s a s b ×X χ )

N s a s b E

E

Ψ 2 ( x

M ( x , χ) (

2 ( x

P − P s b )+ N Ψ , χ )+ , χ ) s a s b = O ,

(10.51)

s a s b s a s a s b

with:

Γ 1 A 1 Θ ( x , χ) P s + P s Θ ( x , χ) A 1 Γ 1
A 2 Θ ( x , χ) P s

,

• M ( x , χ) ( P s )

O

⎛
⎝

⎞
⎠ ,

Γ 1 − σ 0( s , j ) K 1j +∑ k v ∈ Ξ n v ( s ) σ v ( s , j , k v ) v ( s , k v ) +∑ k z ∈ Ξ n z ( s ) σ z ( s , j , k z ) z ( s , k z ) P s

(2 σ 0( s , j ) ζ j − 1 − 2

∑ k v ∈ Ξ n v ( s ) σ v ( s , j , k v ) )

Q ( x , χ) ( P s ,... )

O

− σ 0( s , j ) K 2j OO

⎛ ⎞

Γ 1 τ 0( s , j ) K 1j +∑ k v ∈ Ξ n v ( s ) τ v ( s , j , k v ) v ( s , k v ) +∑ k z ∈ Ξ n z ( s ) τ z ( s , j , k z ) z ( s , k z ) P s

( − 2 τ 0( s , j ) ζ j − 1 − 2

∑ k v ∈ Ξ n v ( s ) τ v ( s , j , k v ) )

⎝ ⎠ ;

R ( x , χ) ( P s ,... ) O

τ 0( s , j ) K 2j OO

• E

( n + n ) × m

∈ R E

a full-rank matrix spanning the nullspace of s a s b ,i.e.such

Θ

E

s a s b

E s a s b E s a s b = and rank( E s a s b )= m that O ;

E s a s b O
OI n π

;

E

E s a s b O
OO n π

; E

s a s b E

, e.g. E

• E

s a s b full-rank with

E

s a s b =

O

s a s b s a s b

• Ψ 1 ( .,. ) , Ψ 2 ( .,. ) , Ψ 3 ( .,. ) , Ψ 4 , j ( .,. ) , Ψ 5 ( .,. ) and Λ x ( . ) as defined in (10.40) .

Proof. The proof is not included for space reasons, but follows the lines of the
proof of Theorem 10.1 and of [9]. Note that the conservativeness of (10.47) can be
reduced.

˜

A multicriteria basin of attraction satisfying (10.45)-(10.51) can be expanded

in many ways, e.g. through the heuristic maximization of its extent towards a set
of selected points as suggested in Section 10.3.1.2, or by defining an ellipsoid
(10.31) included in

E

˜

E and maximizing a function of its shape matrix E related to

its “size”.