Stabilization of the wheeled inverted pendulum on a soft surface

We study dynamics of an wheeled inverted pendulum under a proportional-integral-derivative controller on horizontal, inclined and soft surfaces. An oscillatory area and conditions of the stability for the control are shown on the phase portraits of the dynamical systems. Particularly, we study a differential inclusion for moving on the soft surface, and we find semi-stable stationary solutions in our mathematical model. Due to rounding errors of the numerical modelling or external perturbations of robotics equipment the semistability looks as a limit cycle in simulations.