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Perception is fundamental to many robot application areas especially inservice robotics. Our aim is to perceive and model an unprepared kitchenscenario with many objects. We start with the perception of a single targetobject. The modeling relies especially on fusing and merging of weakinformation from the sensors of the robot in order to localize objects. Thisrequires the representation of various probability distributions of pose in$S_3 \times \mathbb{R}^3$ as orientation and position have to be localized. Inthis thesis I present a framework for probabilistic modeling of poses in $S_3\times \mathbb{R}^3$ that represents a large class of probability distributionsand provides among others the operations of the fusion and the merge ofestimates. Further it offers the propagation of uncertain information data. Iwork out why we choose to represent the orientation part of a pose by a unitquaternion. The translation part is described either by a 3-dimensional vectoror by a purely imaginary quaternion. This depends on whether we define theprobability density function or whether we want to represent a transformationwhich consists of a rotation and a translation by a dual quaternion. A basicprobability den- sity function over the poses is defined by a tangent point onthe hypersphere and a 6-dimensional Gaussian distribution. The hypersphere isembedded to the R4 which is representing a unit quaternions whereas theGaussian is defined over the product of the tangent space of the sphere and ofthe space of translations. The projection of this Gaussian to the hypersphereinduces a distribution over poses in $S_3 \times \mathbb{R}^3$. The set ofmixtures of projected Gaussians can approximate the probability densityfunctions that arise in our application. Moreover it is closed under theoperations introduced in this framework and allows for an efficientimplementation.

Approximation of probability density functions on the Euclidean group parametrized by dual quaternions