  Perception is fundamental to many robot application areas especially in
service robotics. Our aim is to perceive and model an unprepared kitchen
scenario with many objects. We start with the perception of a single target
object. The modeling relies especially on fusing and merging of weak
information from the sensors of the robot in order to localize objects. This
requires the representation of various probability distributions of pose in
$S_3 \times \mathbb{R}^3$ as orientation and position have to be localized. In
this thesis I present a framework for probabilistic modeling of poses in $S_3
\times \mathbb{R}^3$ that represents a large class of probability distributions
and provides among others the operations of the fusion and the merge of
estimates. Further it offers the propagation of uncertain information data. I
work out why we choose to represent the orientation part of a pose by a unit
quaternion. The translation part is described either by a 3-dimensional vector
or by a purely imaginary quaternion. This depends on whether we define the
probability density function or whether we want to represent a transformation
which consists of a rotation and a translation by a dual quaternion. A basic
probability den- sity function over the poses is defined by a tangent point on
the hypersphere and a 6-dimensional Gaussian distribution. The hypersphere is
embedded to the R4 which is representing a unit quaternions whereas the
Gaussian is defined over the product of the tangent space of the sphere and of
the space of translations. The projection of this Gaussian to the hypersphere
induces a distribution over poses in $S_3 \times \mathbb{R}^3$. The set of
mixtures of projected Gaussians can approximate the probability density
functions that arise in our application. Moreover it is closed under the
operations introduced in this framework and allows for an efficient
implementation.
