Visual Servoing and Pose Estimation with Cameras Obeying the Unified Model - Visual Servoing via Advanced Numerical Methods - page 225

• in Section 13.4, the feature choice to control the 6 DOF of the camera or to

estimate its pose is explained; and

•

in Section 13.5, validations results for visual servoing and pose estimation are
presented. In this way, the pose estimation method using VVS is compared to
linear pose estimation method [1] and an iterative method [2].

13.2 Camera Model

Central imaging systems can be modeled using two consecutive projections: spher-
ical then perspective. This geometric formulation called the unified model was pro-
posed by Geyer and Daniilidis in [12]. Consider a virtual unitary sphere centered
on C m and the perspective camera centered on C p (refer to Figure 13.1). The frames
attached to the sphere and the perspective camera are related by a simple translation
of

Z ) in

− ξ along the Z-axis. Let X be a 3D point with coordinates X =( X , Y ,

F m . The world point is projected onto the image plane at a point with homoge-

neous coordinates p = Km ,where K is a 3

X

3 upper triangular matrix containing
the conventional camera intrinsic parameters coupled with mirror intrinsic parame-
ters and

×

1

1 = m = x

X Y

ξ X , ξ X ,

, y , . (13.1)

Z + Z +

can be obtained after calibration using, for
example, the methods proposed in [22]. In the sequel, the imaging system is as-
sumed to be calibrated. In this case, the inverse projection onto the unit sphere can
be obtained by

The matrix K and the parameter ξ

x

− λ

X ,

s = λ , y , 1 (13.2)

= ξ+ √ 1+(1 − ξ

2 )( x 2 + y 2 )
1+ x 2 + y 2 .
Note that the conventional perspective camera is nothing but a particular case of
this model (when

where λ

= 0). The projection onto the unit sphere from the image plane
is possible for all sensors obeying the unified model.

ξ

13.3 Mathematical Background

This section first introduces pose estimation via VVS, and then moments from
points projected onto the unit sphere.

13.3.1 Visual Servoing and Pose Estimation

In few words, we recall that the time variation s of the visual features s can
be expressed linearly with respect to the relative camera-object kinematics screw
V =( v , ω ):

s = L s V , (13.3)