The approaches dedicated to the geometric model of the object, in the object-oriented modeling framework, specific to robots, are focused on the computation of the position of the elements, relative to a global reference frame. One important problem, in this regard, is the direct geometry, which starts with the initial data represented by the generalized coordinates and parameters of each element (in its own reference frame) and calculates the configuration of the structure in the global reference frame. This paper extends the Authors’ geometrical modeling approach, referred to as the translated reference frame formalism, by changing the mathematical apparatus of homogeneous transformations with that specific to quaternions. This new approach has three main advantages over the state-of-the art, namely, it simplifies substantially the configuration modeling, the transformation parameters are intuitive, and the computation is substituted with the operation of choosing a suitable transformation that belongs to a set of six homogeneous transformations. The suggested approach is validated by the computation of the geometric models of two illustrative robotic examples. The source codes are available in a public repository.