Image Measurement Errors in Visual Servoing: Estimating the Induced Positioning Error - Visual Servoing via Advanced Numerical Methods - page 150

The SMR is useful because it allows one to establish positivity of polynomials by
solving a convex optimization problem with LMIs. Indeed, y ( x )

n

≥ ∈ R

0forall x

,...,

if y ( x ) is a sum of squares of polynomials (SOS), i.e. if there exists y 1 ( x ) y k ( x )

such that

k
i =1 y i ( x ) 2

y ( x )= .

It turns out that y ( x ) is SOS if and only if there exists α such that

Y + L ( α ) ≥ 0

which is an LMI. As explained for example in [2], the feasible set of an LMI is
convex, and to establish whether such a set is nonempty amounts to solving a convex
optimization problem.
See e.g. [13, 7, 8] for details on the SMR and for algorithms for constructing
SMR matrices.

9.3 Computation of the Bounds

Let us denote with p ∗ and p the available estimates of p ∗ and p corrupted by image
noise. According to (9.6), the goal condition in visual servoing is

p ∗ ∞ ≤ ε .

p − (9.10)

The estimates p ∗ and p are related to p ∗ and p by
p = p + n
p ∗ = p ∗ + n ∗

2 N are vectors containing position errors due to image noise. Sup-
pose that n and n ∗ are bounded by

n ∗ ∈ R where n ,

n ∞ ≤ ζ

n ∗ ∞ ≤ ζ

where ζ ∈ R is a bound of the position errors in both current and desired views.

One has:

p ∗ ∞ = p ∗ − n + n ∗ ∞

p − p −

p ∗ ∞ + n ∗ ∞

≤ p − n ∞ +

p ∗ ∞ + 2

ζ .
This implies that the condition (9.10) ensures only

≤ p −

p ∗ ∞ ≤ ε p − + 2 ζ .

p ∗ ∞

Hence, one cannot guarantee that the real image error p − converges to

a value smaller than 2 ζ . This clearly motivates the investigation of the robot