###### Abstract  
In this paper, a neural-network-based control method to achieve trajectory tracking and balancing of a ball-balancing robot with uncertainty is presented. Because the ball-balancing robot is an underactuated system and has nonlinear couplings in the dynamic model, it is challenging to design a controller for trajectory tracking and balancing. Thus, various approaches have been proposed to solve these problems. However, there are still problems such as the complex control system and instability. Therefore, the objective of this paper was to propose a solution to these problems. To this end, we developed a virtual angle-based control scheme. Because the virtual angle was used as the reference angle to achieve trajectory tracking while keeping the balance of the ball-balancing robot, we could solve the underactuation problem  
technique [6] was proposed to relax this restriction, the control system using the multi-loop method becomes complex due to additional feed-forward controllers for robustness against uncertainties [7]. Sliding mode control (SMC) is a representative nonlinear control method that is robust to uncertainty. Thus, the hierarchical sliding mode control (HSMC) method [8], which is composed of combinations of independent sliding surfaces, was proposed to address underactuated systems. The HSMC method based on the two-dimensional model of the ball-balancing robot was presented in [9] and then applied to the aggregated model to consider the combined effects [10]. An integral backstepping technique using the HSMC structure was proposed in [11], and an artificial neural network-based method [12] was presented to deal with uncertainty and input saturation. The hierarchical structure used in the above papers can solve the underactuation problem and has robustness for uncertainties. However, the hierarchical structure has the local minimum problem [13], which may not guarantee the convergence of all errors. Therefore, a different approach was needed to achieve simultaneous trajectory tracking and balancing for the ball-balancing robot. In this regard, we proposed a virtual angle-based approach [14]. The virtual angle composed of the tracking errors was used as a reference angle to achieve simultaneous trajectory tracking and balancing using the single-loop control system. However, the upper bounds of uncertainties for robustness should be known, which may be impossible in the real world. Artificial intelligence can be a solution to address this problem. In [15], deterministic artificial intelligence (DAI) was used for underactuated systems with nonlinear couplings. In [16], a artificial neural network was employed to deal with model uncertainty and the external disturbance of underactuated surface vessels. The field of non-stochastic or DAI involves the assertion of self-awareness statements and uses optimal learning to compensate the deleterious effects of error sources [15]. On the other hand, an artificial neural network (ANN), which is in the field of stochastic artificial intelligence, does not need to devise an algorithm to perform a particular task because the learning process of a neural network takes place by itself [17]. Motivated by these observations, we proposed a neural network-based control for trajectory tracking and balancing of the ball-balancing robot with uncertainty. A virtual angle-based control method was used to design a single-loop control system to achieve trajectory tracking and balancing of the robot. Unlike in [14], the radial basis function networks (RBFNs) [18], a kind of ANN, were used to compensate the uncertainties without knowing their upper bounds, and the dynamic surface control (DSC) method [19] was employed to design the controller. From the Lyapunov stability theory, it was proven that all errors of the closed-loop control system were uniformly ultimately bounded. Finally, the performance of the proposed method was demonstrated through the simulation results. The main contributions of this paper are summarized as follows:  
1.