This paper investigated the dynamics and control of an underactuated rimless wheel (RW) with semicircular feet driven by two pendulum-like arms. We focused on the entrainment property of the RW walking to the periodic oscillations of the arms and analyzed how different arm coordination patterns, i.e., in-phase and anti-phase swingings, affect the walking stability and performance of the system. A mathematical model was developed that captured the continuous dynamics, collision events, and control method for this hybrid mechanical system. Numerical simulations demonstrated that the anti-phase arm swinging, similar to human walking, provided superior performance compared to the in-phase swinging. Specifically, the anti-phase coordination yielded a wider range of entrainment, as well as a wider range of stable walking, across various control parameters including the forcing frequency and the arm weight. Through rigorous Poincaré map stability analysis, we further quantified the system’s robustness to perturbations, revealing that anti-phase arm swinging maintains stability even with increased arm mass, while in-phase swinging becomes unstable. The stable walking of the anti-phase coordination was supported by the opposing movements of the left and right arms, which effectively counterbalance the forward and backward momentums to suppress the ground reaction forces. These results may provide design principles for controlling underactuated walking robots with arms.

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