Empirical Characterization of Convergence Properties for Kernel-based Visual Servoing - Visual Servoing via Advanced Numerical Methods - page 23

signal I ( w , x ( t )) ∈ R . The spatial indexing parameter of the 2D image is w ∈I =

2 . The signal is represented as a function of the spatial indexing and the position
which is a function of time. Although one might expect to see the signal written only
as a function of time, I ( w

R

,

t ), we will be more explicit and write it as a function of
the robot configuration which is in turn a function of time, I ( w

x ( t )).Forthesakeof
notational simplicity and without loss of generality we assume that the image plane
is a unit distance away from the scene.
Using the modified notation, the signal at any translated position is related to the
signal at the goal

,

I ( w , x ( t )) = I ( w − x ( t ) , 0 )= I 0 ( w − x ( t )) . (2.2)

n

In the two dimensional case, the kernel-projected value is a function ξ : I → R

where

= ξ K ( w ) I ( w , t ) d w

I

=

(2.3)

K ( w ) I 0 ( w − x ( t )) d w ,

I

and the kernel-projected value at the goal is at x 0 . The dimension n refers to

the number of kernels being used such that the kernels represent a function K :

ξ 0 = ξ

2

R →

n . Note that using the change of coordinates w = w

R − x ( t ), (2.3) can be written as

= ξ K ( w + x ( t )) I 0 ( w ) d w . (2.4)

I

(2.3) and (2.4) show that the the kernel-projected value is the same for the fixed
kernel with an image taken after the camera has been moved and a kernel shifted in
the opposite direction with the goal image. This can be seen pictorially in Figure 2.2
for a one-dimensional signal and kernel. The change of coordinates in (2.4) will be
used later to allow us to take time derivatives of the kernel-projected value: we can
easily design kernels with known and analytically determined derivatives, whereas
the time derivatives of the images are unknown.
The goal of the KBVS method is to find a control input u , a function of

ξ ,to

move the robot such that

lim
t → ∞

x ( t )= x 0 .

To find such a control input, we consider

V = 1 ξ − ξ 0 ) T P (

2 (

ξ − ξ 0 ) (2.5)

as a Lyapunov function candidate, where P is any positive n n matrix. To choose

the control input, u , we analyze the time derivative of the Lyapunov function using
the shifted kernel representation of the kernel-projected value from (2.4):

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