While mobile ground robots have now the physical capacity of travelling in unstructured challenging environments such as extraterrestrial surfaces or devastated terrains, their safe and efficient autonomous navigation has yet to be improved before entrusting them with complex unsupervised missions in such conditions. Recent advances in machine learning applied to semantic scene understanding and environment representations, coupled with modern embedded computational means and sensors hold promising potential in this matter. This paper therefore introduces the combination of semantic understanding, continuous implicit environment representation and smooth informed path-planning in a new method named COSMAu-Nav. It is specifically dedicated to autonomous ground robot navigation in unstructured environments and adaptable for embedded, real-time usage without requiring any form of telecommunication. Data clustering and Gaussian processes are employed to perform online regression of the environment topography, occupancy and terrain traversability from 3D semantic point clouds while providing an uncertainty modeling. The continuous and differentiable properties of Gaussian processes allow gradient based optimisation to be used for smooth local path-planning with respect to the terrain properties. The proposed pipeline has been evaluated and compared with two reference 3D semantic mapping methods in terms of quality of representation under localisation and semantic segmentation uncertainty using a Gazebo simulation, derived from the 3DRMS dataset. Its computational requirements have been evaluated using the Rellis-3D real world dataset. It has been implemented on a real ground robot and successfully employed for its autonomous navigation in a previously unknown outdoor environment.