A nonlinear optimal (H-infinity) control approach is developed for the model of the ballbot. This robotic system consists of a rolling sphere with a rigid body, in the form of an inverted pendulum, mounted on top of the sphere. Because of the nonlinearities and underactuation that are due to the dynamics of both the rolling sphere and of the rotational motion of the rigid body, control of the ballbot is a non-trivial problem. In the proposed control method, the dynamic model of the ballbot undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm. The linearization process makes use of first-order Taylor series expansion and relies also on the computation of the Jacobian matrices of the state-space model of the robotic system. A stabilizing H-infinity feedback controller is designed for the approximately linearized description of the ballbot. An algebraic Riccati equation is solved at each time-step of the control method so as to compute the controller's feedback gains. Through Lyapunov analysis the stability properties of the control scheme are proven. Besides, through the article's results it is demonstrated that the control method retains the advantages of linear optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.
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