Kinematics of Spherical Robots Rolling Over 3D Terrains

Although the kinematics and dynamics of spherical robots (SRs) on flat horizontal and inclined 2D surfaces are thoroughly investigated, their rolling behavior on generic 3D terrains has remained unexplored. This paper derives the kinematics equations of the most common SRs configurations rolling over 3D surfaces. First, the kinematics equations for a geometrical sphere rolling over a 3D surface are derived along with the characterization of the modeling method. Next, a brief review of current SR mechanical configurations is presented as well as a novel classification for spherical robots based on their kinematics. Then, considering the mechanical constraints of each category, the kinematics equations for each group of spherical robots are presented. Afterwards, a path tracking method is utilized for a desired 3D trajectory. Finally, simulations are carried out to validate the developed models and the effectiveness of the proposed control scheme.