This paper introduces a delayed-feedback curve tracking control strategy utilizing radial basis function neural networks (RBFNN) for a two-wheel self-balancing robot (TWSBR). Initially, a dynamic model of the TWSBR navigating an inclined surface is developed via the Lagrange equation. The resulting six-dimensional dynamic system is decomposed into two distinct subsystems: a four-dimensional subsystem governing forward movement and a two-dimensional subsystem controlling steering dynamics. Subsequently, an integral transformation technique is applied to effectively convert the input-delayed system into an equivalent delay-free system. Delay-feedback controllers are designed for each subsystem by integrating linear quadratic regulator (LQR) control with integral sliding mode control techniques. To enhance robustness and substantially reduce the undesirable ‘chattering’ typically associated with sliding mode control, RBFNN is employed to adaptively approximate the upper bound of uncertainties within the system. Numerical simulations validate the efficacy and practicality of the proposed control strategy by demonstrating successful circular trajectory tracking, with simulation outcomes aligning closely with theoretical predictions.