A Variational Approach to Trajectory Planning in Visual Servoing - Visual Servoing via Advanced Numerical Methods - page 115

Π

n f

X

f

d

x

x

f

F

F

Rb

Fig. 7.1 Geometry of two views

The Euclidean homography can be decomposed into a rotation matrix and a rank 1
matrix [11]:

b
d f n fT

H = R + , (7.8)

where R and b represent the rotation matrix and the translation vector between the
current and the desired camera frames (denoted

f

respectively), n f

F and F is the

f ,and d f

unitary normal to the virtual plane expressed in F is the distance from Π to

f (see Figure 7.1). From G and K , it is thus possible to determine the
camera motion parameters ( i.e. , the rotation R and the scaled translation b d f =

the origin of F

b

d f )
and the normal vector n f , by using for example one of the algorithms proposed in
[11] or in [30].
In the sequel we assume that given an initial image and a desired image of the
scene, some image features can be extracted and matched. This framework is the
classical one in visual servoing. From the extracted image features, the collineation
matrix at time t 0 , G 0 , can be computed [10, 4]. Note also that, when the desired
configuration is reached (at time t f ), the collineation matrix is proportional to the
identity matrix:

G f I 3 × 3 .

In the next sections, we consider the problem of finding a path of the collineation
matrix between G 0 and G f corresponding to an optimal camera path with respect to
criteria which will be specified in the sequel. The image trajectories are then derived
from the collineation path.

∝

7.2.3 The Unconstrained Problem

The constrained variational problem we will solve in the sequel requires an initializ-
ing trajectory. We will use the trajectory provided by the method proposed in [23] by