Unified formation and obstacle avoidance control for ball-balancing robots via trajectory modulation

This paper presents a unified nonlinear formation control framework for multiple ball-balancing robots (BBRs) subject to underactuation, input saturation, and obstacle-avoidance constraints. Existing BBR control approaches often separately address formation tracking, posture stabilisation, and obstacle avoidance, such that overall closed-loop stability does not necessarily guarantee simultaneous achievement of all objectives under uncertainties. Moreover, conventional artificial potential field (APF)-based methods inherently suffer from local minima (deadlocks), while obstacle-avoidance maneuvers may destabilise the balancing dynamics. To address these challenges, this paper introduces two main technical contributions. First, a deadlock-mitigating trajectory modulation strategy is developed by incorporating a tangential velocity component into the reference trajectory generation process. Unlike conventional APF-based methods, the proposed strategy reshapes the reference path to enable continuous obstacle avoidance without stagnation. Second, a unified virtual-angle-based control framework is proposed to simultaneously handle formation tracking, posture stabilisation, and obstacle avoidance for BBRs under input saturation and nonlinear uncertainties. Lyapunov analysis proves that all closed-loop signals are uniformly ultimately bounded. Simulation results validate reliable formation tracking and continuous obstacle avoidance under uncertainties and input saturation.