Image Measurement Errors in Visual Servoing: Estimating the Induced Positioning Error - Visual Servoing via Advanced Numerical Methods - page 151

positioning error introduced by image measurement errors. For this reason, from
now on we will consider the computation of the errors s r ( δ ) and s t ( δ ) where δ ,

defined as

ζ , δ = ε + 2 (9.11)

represents the total image error.
Before proceeding, let us parameterize the rotation matrix through the Cayley
parameter. Let us define the function

3 3 × 3

Γ : R → R

( a )= I 3

− 1 I 3 +[ a ]

Γ − [ a ]

×

(9.12)

×

3 is called Cayley parameter. It turns out that this
parametrization satisfies the following properties:
1.

a 3 ) T where a =( a 1 , a 2 , ∈ R

3 ;

∈ ∈ R

Γ ( a ) SO (3) for all a

3

∈

θ < π

∈ R

2. for any R SO (3) such that θ in (9.8) satisfies , there exists a such

( a ).
Moreover, (9.12) can be rewritten as

that R = Γ

( a )= Ω

( a )

Γ (9.13)

2

1 + a

3 × 3 is the matrix quadratic polynomial given by

3

where Ω ( a ) : R → R

⎛ ⎞

a 1 − a 2 − a 3 + 12( a 1 a 2 − a 3 ) 2 ( a 1 a 3 + a 2 )

⎝ ⎠ .

a 1 + a 2 − a 3 + 12( a 2 a 3

− −

Ω ( a )= 2 ( a 1 a 2 + a 3 ) a 1 ) (9.14)

a 1 − a 2 + a 3 + 1 2 ( a 1 a 3 − a 2 ) 2 ( a 2 a 3 + a 1 ) −

9.3.1 Upper Bounds

Let us consider first the computation of upper bounds of s r ( ) in (9.7)-

(9.9). We will show that this step can be solved by exploiting convex optimization.
Indeed, consider the constraint

δ ) and s t ( δ

p ∗ ∞ < δ

p − in the computation of s r ( δ ) and s t ( δ ).

From (9.1), (9.3) and (9.5) it follows that for the i -th point we can write

( a ) T q i ( a ) T t

Ω − Ω q i

e 3 q i

p ∗ i = A p i −

( a ) T t ) −

A (9.15)

e 3 ( Ω ( a ) T q i − Ω

where it has been taken into account that q i is expressed with respect to the desired
camera frame F ∗ , which coincides with the absolute frame F abs . Hence, we have that

p ∗ i ∞ ≤ δ p i −

if and only if

⎧
⎨

| f i , 3 g i , 1 ( a , t ) − f i , 1 g i , 3 ( a , t ) |≤ δ f i , 3 g i , 3 ( a , t )

| f i , 3 g i , 2 ( a , t ) − f i , 2 g i , 3 ( a , t ) |≤ δ f i , 3 g i , 3 ( a , t ) (9.16)

⎩

g i , 3 ( a , t ) > 0

where f i , j ∈ R is a constant and g i , j ( a , t ) is a polynomial given by