Cross-backstepping control with prescribed performance for input-coupled underactuated systems under arbitrary initial conditions

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Underactuated robotic systems with input coupling are widely applied in practice. However, designing controllers for such systems poses significant challenges due to the complexities arising from input coupling. To address this issue, this paper proposes a novel cross-backstepping control strategy based on a composite error transformation. The transformation compresses the system’s initial error into a definable range to deal with the issue of initial value dependency of prescribed performance control, while maintaining the transient and steady state of the system. Based on such a technique, a cross-backstepping controller is proposed whose core idea lies in constructing a virtual term that crosses with coupled states to generate a unified form, which serves as a bridge between multiple coupled states and a single actual control input. Furthermore, a nonlinear disturbance observer is introduced to improve the robustness of the system. The stability of the closed-loop system is proved according to the Lyapunov theorem, and two numerical examples of underactuated robotic systems with coupled inputs are presented to demonstrate the effectiveness of the proposed approach.