  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.
