  Certain wheeled mobile robots e.g., electric wheelchairs, can operate through
indirect joystick controls from users. Correct steering angle becomes essential
when the user should determine the vehicle direction and velocity, in
particular for differential wheeled vehicles since the vehicle velocity and
direction are controlled with only two actuating wheels. This problem gets more
challenging when complex curves should be realized by the user. A novel
assistive controller with safety constraints is needed to address these
problems. Also, the classic control methods mostly require the desired states
beforehand which completely contradicts human's spontaneous decisions on the
desired location to go. In this work, we develop a novel assistive control
strategy based on differential geometry relying on only joystick inputs and
vehicle states where the controller does not require any desired states. We
begin with explaining the vehicle kinematics and our designed Darboux frame
kinematics on a contact point of a virtual wheel and plane. Next, the geometric
controller using the Darboux frame kinematics is designed for having smooth
trajectories under certain safety constraints. We experiment our approach with
different participants and evaluate its performance in various routes.
