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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.

This chapter presents a solution to the problem of autonomous pizza tossing and catching. Under the assumption that robotic fingers grasp the pizza dough with soft contact, the grasp constraints are formulated and used to derive the individual and combined Euler-Lagrange dynamic equations of motion of the robotic manipulator and the dough. In particular, the dynamics of the dough is a modified version of the rigid-body dynamics, taking into account the change of inertia due to its deformation. Through these mathematical models, the two control problems of tossing and catching are formulated. For the tossing phase, an exponentially convergent controller that stabilizes a desired velocity of the dough as it leaves the fingers, is derived. On the other hand, to catch the dough, an optimal trajectory for the end-effector of the robotic manipulator is generated. Finally, the control laws to make the optimal trajectory exponentially attractive are derived. The developed theory is demonstrated with an elaborate simulation of the tossing and catching phases. This chapter is based on the work presented in [1].