  This paper presents a new kinematic model based on the Darboux-frame for
motion control and planning. In this work, we show that an underactuated system
as a spin-rolling sphere on a plane with three inputs and five states can be
transformed into a fully-actuated model by the given Darboux-frame
transformation. This nonlinear state transformation is a geometric model that
is different from conventional state-space models. First, a kinematic model of
the Darboux frame at the contact point of a rotating object i.e., the sphere,
is established. Next, we propose a virtual surface that is trapped between
sphere and plane surfaces. This virtual surface generates arc-length-based
inputs for controlling the trajectories on the sphere and plane. Finally, we
discuss the controllability of this new system after our introduced
transformation. In the future, we will design a proper geometric path planning
method for the obtained kinematic model.
