Intrinsic Euler-Lagrange dynamics and control analysis of the ballbot

Research on bipedal locomotion has shown that a dynamic walking gait is energetically more efficient than a statically stable one. Analogously, even though statically stable multi-wheeled robots are easier to control, they are energetically less efficient and have low accelerations to avoid tipping over. In contrast, the ballbot is an underactuated, nonholonomically constrained mobile robot, upward equilibrium point of whose body has to stabilized by active controls. In this work, we derive coordinate-invariant equations of motion for the ballbot. We present the linearized equations of motion followed by its controllability analysis. Excluding the rotary degree of freedom of the ball in the inertial vertical direction, the linear system turns out to be controllable. It follows that the nonlinear system is locally controllable and we provide a proportional-derivative type controller that locally exponentially stabilizes the upward equilibrium point as well as the translation of the ball. The basin of attraction turns out to be large in the simulation studies.