This thesis covers the derivation of a non-linear model and sliding mode controller for a ball-balancing robot with three omniwheels. A quaternion-based model is derived using Lagrangian mechanics, with the quaternion and ball position as the generalized coordinates. The quaternion unit norm constraint is enforced with a Lagrange multiplier. The quaternion model is used for sliding mode controller design of an orientation stabilizing controller that considers a quaternion error function based on the desired quaternion and angular velocity reference. Two sliding surfaces are proposed and compared in simulation. The derived controller is verified in both simulation and in practice, using a 16 kg ballbot prototype, Kugle V1. The controller and the necessary estimators, two extended Kalman filters for quaternion and velocity estimation, are implemented in an embedded firmware.The controller is confirmed to work as designed and the simulation model is verified against the practical results, using a Vicon motion capture system. Continuously changing references can be tracked with a tracking error of less than 1 degree up to 1 Hz. The controller is furthermore tested in a cascaded configuration with first a velocity LQR controller and subsequently a shape-accelerated path-following MPC, which generate quaternion and angular velocity references and enable station-keeping, velocity tracking and path-following. Velocity references up to 1 m/s and 1 rad/s are tested and confirmed trackable in practice. The thesis concludes that it is indeed possible to derive and use a quaternion model for ballbot control even though it complicates the derivation and is deemed unnecessary due to the operating envelope. All material, including the MATLAB code, simulations, ROS drivers and the embedded firmware, is open source and available on GitHub.