**1. Introduction**

Ballbot has a body that balances on a single spherical wheel (ball). The robot uses a drive mechanism consisting of three omnidirectional wheels (OWs) for a ball [1] to ensure both stabilizing and transferring. The Ballbot can free travel in any direction on plat plane, even if the robot is crowded with people in a common human-coexisting environment.

An inherent property of the Ballbot system is a naturally nonlinear underactuated multi-input multi-output system, in which the number of the control input signal is less than the number of output signals. This property introduces the challenge related to Ballbot control design. Several control designs were accomplished using a simplified model (linear model) to overcome the complexities of the mathematical equations. The linear model is the first way to deal with the complexity of the mathematical equation of the Ballbot system. Reference [2] designed a control system based on a

state feedback controller to control stabilizing and transferring. The experimental results showed the ability to stabilize and travel with loads in any direction. However, it can be led only to small external disturbances. Studies in [3, 4] employed a doubleloop linear control system to stabilize a Ballbot. Their control system had two controllers: a PI inner-loop controller and a linear quadratic regulator (LQR) outer-loop controller. Kantor et al. [5] proposed a double-loop control system. Both inner-loop and outer-loop controllers were PID controllers. Sukvichai and Parnichkun [6] used a linear controller based on an LQR control scheme for a double-level Ballbot.

The second way to deal with the complexity of the mathematical equation of the Ballbot system is to use nonlinear and intelligent controls such as partial feedback linearization control (FLC), sliding mode control (SMC), or fuzzy control. Lotfiani et al. [7] employed an SMC controller along fuzzy trajectory planning to control a Ballbot to track the desired trajectory. Moreover, a collocated PFL method [8] was introduced for a Ballbot. The open-loop trajectory generation with the collocated PFL controller was simulated which showed a low position error.

Some researchers have studied intelligent control techniques. Reference [9] suggested an intelligent tracking control system combined with a dual Mamdani-type fuzzy control strategy and supervisory control technique for an omnidirectional spherical mobile platform. The experiment results showed the position tracking response of the robot. A fuzzy wavelet cerebellar-model-articulation controller [10] was proposed for a team of multiple Ballbots.

SMC is a well-known and robust nonlinear control scheme. To enhance robustness for both actuated [11] and under-actuated [12–19] systems, several controllers have been employed the SMC control method. The application of SMC for Ballbot control can be found in several previous studies. Ching-Wen et al. [20] introduced a hierarchical SMC based on backstepping to control stabilizing and agile trajectory tracking of a Ballbot with exogenous disturbance. Reference [21] enhanced Ching-Wen's backstepping SMC using interval type 2 fuzzy neural networks for motion control of a Ballbot with a four-motor inverse mouse ball-diving mechanism. However, these studies only presented numerical simulations.

As mentioned, the Ballbot is naturally an under-actuated system in which the number of the control inputs is less than the number of the outputs. The Ballbot system is nonlinear. Moreover, the control design problem also faces uncertainties, un-modeling, and disturbances. Thus, the SMC control scheme is a suitable approach for the Ballbot system.

In this study, we designed a robust controller to control a Ballbot. The proposed control scheme is a basis of the hierarchical SMC technique to execute two main tasks including stabilizing the body on the top of the ball and maneuvering the ball on the floor.
