Time‐optimal trajectory generation and observer‐based hierarchical sliding mode control for ballbots with system constraints

Abstract This paper introduces a comprehensive motion planning–tracking–safety constraint scheme for a 3D ballbot system. A nonlinear control for the 3D ballbot system is designed based on three separate planes by utilizing extended state observer (ESO) to estimate coupling mechanisms. Three virtual control signals are generated from these distinct planes and can be used for formulating actual control signals. To overcome the complexity of nonlinear motion equations, flatness theory is used to construct the time‐optimal trajectory through an optimization problem, facilitating smooth movement of the ballbot, and obstacle avoidance based on RRT* waypoints. Furthermore, our work manipulates the hierarchical sliding mode controller (HSMC) as the nominal controller to ensure that the ballbot tracks to the optimal trajectory, unifying with the exponential control barrier function (ECBF) to address safety constraints in the body's deflection angle. Through extensive simulations and comparative analysis, the system demonstrates its effectiveness and safe operation in various working conditions.