Mathematical Modeling of the Spatial Kinematics of an Omni-platform Balancing on a Spherical Wheel

This article presents a mathematical modeling the spatial kinematics of an omni-platform balancing on a spherical wheel. The design of a platform with three omni-wheels, symmetrically located on a circle, is being considered. The purpose of the work is to construct formulas for the angular velocities of omni-wheels during the spatial motion of a platform balancing on a spherical wheel are derived from the equations of non-slip conditions of the interacting bodies contact points. In the particular case of motion of a robot spherical wheel on a horizontal surface with the platform positioned in the upper equilibrium position, forward and inverse kinematics equations are obtained. Based on the results of numerical modeling, estimates of the maximum speeds of the omni-platform are obtained for given constraints on the angular speeds of the omni-wheels. The results of the work can be used in the design and manufacture of new Ballbot.